A higher categorical approach to graded bivariant $K$ -theories for Real $C^{*}$ -algebras

A nupam Datta ^†^†thanks: Universität Göttingen, anupam.datta@mathematik.uni-goettingen.de

Abstract

We present an $\infty$ -categorical refinement of classical $KK$ -theory and $E$ -theory for graded, $G$ -equivariant for a locally compact second countable group $G$ , Real $C^{*}$ -algebras. As a consequence, we obtain a couple of fiber sequences connecting the different $K$ -theory spectra, and explicitly analyse the induced endomorphisms on $K_{0}$ -groups. We also answer precisely some folklore questions in the area. Among other things, we show that graded $KK$ - and $E$ -categories are not a Dwyer-Kan localization of graded $C^{*}$ -algebras.

1 Introduction

This article is devoted to studying bivariant $K$ -theories for $C^{*}$ -algebras found in an index theorist’s toolkit using methods from higher category theory. Recent works of Bunke and coauthors ([13], [17], [11]) have refined the classical $KK$ - and $E$ -categories of $C^{*}$ -algebras to stable $\infty$ -categories, which then enjoy an $\infty$ -categorical version of the classical ( $1$ -categorical) universal properties as mentioned in [33] and [20]. This $\infty$ -categorical enhancement is not just for the satisfaction of the modern homotopy theorist, but also helps segregate the homotopical and analytical aspects of the subject.

In [17], Bunke constructs small stable $\infty$ -categories $KK^{\operatorname{sep}}_{!,\infty}$ (where $!\in\{\operatorname{semi-split(se),exact(ex)}\}$ denotes the kind of excision we want to have) by universally enforcing certain properties Kasparov’s $KK$ - and Connes-Higson’s $E$ -theory are known to satisfy, namely, homotopy invariance, compact stability, $!$ -excision, and Bott periodicity. The same construction can also be done equivariantly in the presence of the action of locally compact second countable group, as mentioned in [16]. That the homotopy category of $KK^{\operatorname{sep}}_{\operatorname{se},\infty}$ (resp. $KK^{\operatorname{sep}}_{\operatorname{ex},\infty}$ ) is equivalent to Kasparov’s $KK$ - (resp. Connes-Higson’s $E$ -)categories can be seen by comparing with the classical universal properties, as is worked out in [17, Section 13]. The advantage of defining bivariant $K$ -categories as a Dwyer-Kan localization is that one can write functors out of the $KK_{!,\infty}^{\operatorname{sep}}$ -categories by verifying that a functor defined on separable $C^{*}$ -algebras satisfies certain properties.

However, [13], [17], [11] have been written exclusively with ungraded $C^{*}$ -algebras. In contrast, most of classical bivariant $K$ -theories have been applied to problems in geometry and topology in their graded avatar. For example, consider the proof of the Baum-Connes conjecture for a-T-menable groups in Higson-Guentner’s article [32], where a myriad of the exotic functorial properties of the graded $E$ -category were used. Nevertheless, it is the ungraded $KK_{!}$ -theories¹¹ 1 we adopt Bunke’s notation and denote by $KK_{!}$ either $KK$ - or $E$ -theory, even in the classical contexts. whose universal properties are classically known. Given the importance of the graded theory and the effectiveness of realizing $KK_{!}$ as Dwyer-Kan localizations as mentioned earlier, one can ask the same for the graded $KK_{!}$ -category. It turns out that the answer is in the negative, as shown in Section 4.1. The author does not know if this fact was known earlier. However, not all is lost, and higher categorical refinements of graded $KK_{!}$ -categories can still be provided. Following ideas in an unpublished note of Bunke-Duenzinger (see also [15, Section 2.4]), we define an $\infty$ -categorical refinement of graded, $G$ -equivariant $KK_{!}$ -category as a certain category of comodules over a distinguished cocommutative coalgebra in $KK^{G_{2}}_{!,\infty}$ , the $G\times C_{2}$ -equivariant, ungraded bivariant $K$ -category. It turns out that this picture captures the formal properties of the classical graded $KK_{!}$ -categories. The details of this higher categorical refinement of graded bivariant $K$ -categories is one of the cornerstones of this article, and is developed in Section 4.

The other aspect missing from the writings of Bunke on higher categorical approaches to $KK_{!}$ -theories is a discussion of Real $C^{*}$ -algebras. Apart from being of independent interest, Real $K(K)$ -theory encodes finer information than its complex counterpart, as for example is clear from the seminal works of Hitchin ([36]) and Rosenberg ([51]). Luckily, the construction of Real $KK_{!,\infty}$ -categories for separable $C^{*}$ -algebras follow the same path as in [17] (see also [11, Section 3.4]), with only one difference: we would have to enforce the well known $(1,1)$ -periodicity at the final step of the construction. The construction of stable $\infty$ -categories refining classical Real $KK_{!}$ -categories is the content of the first few subsections of Section 3.²² 2 In our treatment of graded $KK_{!}$ -categories, we have worked with Real $C^{*}$ -algebras equipped with the action of an external locally compact group. We did not mention this in the last paragraph because the passage from ungraded to graded is much more drastic than the one from complex to Real. And equivariant matters were already considered in works of Bunke. Unless otherwise mentioned, in this article we work with Real $C^{*}$ -algebras by default.

We now mention some applications of the abstract theory developed in Sections 3 and 4 of this article. In order to streamline the presentation, we package all of them in the next theorem. To simplify the exposition, we also ignore the external (compact) group action here. The (more or less straightforward) generalization to the equivariant case is present in Section 5.

Theorem 1.1.

Denote by $\mathbb{K}(-),\mathbb{K}^{C_{2}}(-),\hat{\mathbb{K}}(-)$ the ungraded (classical), $C_{2}$ -equivariant, and graded spectrum valued $K$ -theory functors on the corresponding categories of (Real) $C^{*}$ -algebras. Let $A$ be a $C^{*}$ -algebra with a $C_{2}$ -action $\alpha_{A}$ . Then we have the following³³ 3 the labels $\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}$ are just notation and should not be confused with the multiplication with some constant maps!.

1.

There is a natural fiber sequence of spectra

$\mathbb{K}(A_{u})\xrightarrow{\mathbf{1}}\mathbb{K}^{C_{2}}(A)\xrightarrow{\mathbf{2}}\hat{\mathbb{K}}(A)$

Here $(-)_{u}$ denotes the underlying $C^{*}$ -algebra functor. Furthermore, the fiber sequence splits if $A$ is inner graded.

There is a natural fiber sequence of spectra

\hat{\mathbb{K}}(A)\xrightarrow{\mathbf{3}}\mathbb{K}^{C_{2}}(A)\xrightarrow{\mathbf{4}}\mathbb{K}((A\hat{\otimes}Cl^{0,2})_{u})

Here $A\hat{\otimes}Cl^{0,2}$ denotes the graded tensor product of $A$ and $Cl^{0,2}$ , the Clifford algebra on two generators which square to $+1$ . If $A$ is inner graded, the fiber sequence splits.

3.
There are endomorphisms of the different $K$ -theory spectra of $A$ arising from the above two fiber sequences. At the level of $\pi_{0}$ , their effect can be concretely described when the algebra $A$ is unital:
1. (a)
  
  $\mathbf{3}\circ\mathbf{2}\in\operatorname{End}(K_{0}^{C_{2}})(A)$ is given by mapping the class of a $C_{2}$ -equivariant projective $A$ -module $[M,\alpha_{M}]\in K_{0}^{C_{2}}(A)$ to $[M,\alpha_{M}]-[M,-\alpha_{M}]$ .
2. (b)
  
  $\mathbf{2}\circ\mathbf{3}\in\operatorname{End}(\hat{K}_{0}(A))$ is given by mapping the class of a Fredholm cycle⁴⁴ 4 see Section 2.2. $(\mathcal{E},F)\in K_{0}^{\operatorname{Fr}}(A)$ to $(\mathcal{E}\oplus\mathcal{E},F\oplus-F)\in K_{0}^{\operatorname{Fr}}(A)$ .
3. (c)
  
  When $A$ is inner graded or $A$ is complex, then $\mathbb{K}((A\hat{\otimes}Cl^{0,2})_{u})\simeq\mathbb{K}(A_{u})$ , and the map $\mathbf{4}\circ\mathbf{1}\in\operatorname{End}(K_{0}(A_{u}))$ is given by mapping the class of a finitely generated projective $A$ -module $[M]\in K_{0}(A)$ to $[M\oplus M_{\alpha}]\in K_{0}(A)$ , where $M_{\alpha}$ has the same underlying abelian group $\underline{M}$ as $M$ , and the $A$ -action is given by $\underline{m}.a:=\underline{m\alpha_{A}(a)}.$
4. (d)
  
  Under the same assumption on $A$ as in the last point, the map $\mathbf{1}\circ\mathbf{4}\in\operatorname{End}(K_{0}^{C_{2}}(A))$ is given by mapping the class of a $C_{2}$ -equivariant finitely generated projective $A$ -module $[M,\alpha_{M}]\in K_{0}^{C_{2}}(A)$ to $[M\oplus M,(m,n)\mapsto(\alpha_{M}(n),\alpha_{M}(m))]\in K_{0}^{C_{2}}(A)$ .

Before describing the organization of this article, we mention the relation between our fiber sequences and the Clifford module quotient⁵⁵ 5 the author learnt this sleek term from a talk of Luuk Stehouwer at the Max Planck Institute for Mathematics in 2025. picture of $K$ -theory of a point as described by Atiyah, Bott and Shapiro in [4]. Roughly speaking, one of the main results of loc. cit. (see also [41, Section I.9]) is the following (recall that $Cl^{p,q}$ denotes the Clifford algebra on $p+q$ generators, where $p$ generators square to $-1$ , and $q$ generators square to $+1$ .)

Theorem 1.2.

(Atiyah-Bott-Shapiro) For every non-negative integer $n$ , there is a canonical isomorphism of abelian groups

K^{-n}(*)\cong\frac{\{\operatorname{finitely\penalty\ generated\penalty\ graded\penalty\ Cl^{n,0}-modules}\}^{\operatorname{gp}}}{\{\operatorname{finitely\penalty\ generated\penalty\ graded\penalty\ Cl^{n+1,0}-modules}\}^{\operatorname{gp}}}

It can be seen (Section 5.1) that Theorem 1.2 is asserting that the long exact sequence of homotopy groups of the fiber sequence labeled “ABS” splits off into short exact sequences when $A$ is a Clifford algebra. Our results therefore clarifies the situation for a general graded, Real ( $G$ -equivariant for a compact group $G$ ) $C^{*}$ -algebra: replace the process of taking cokernels in abelian groups by taking cofiber in spectra. This also answers a folklore question pertaining to whether higher $K$ -groups of a $C^{*}$ -algebra can be given a Clifford module quotient description analogous to Theorem 1.2; see Section 5.1 for details.

As the reader may guess, the other fiber sequence labeled “van Daele” has been tagged so because it recovers the classical definition of van Daele $K$ -theory ([60],[61]). After a bit of massaging⁶⁶ 6 by working with the (graded) opposite $C^{*}$ -algebra, this sequence can be seen to recover Karoubi’s definition of graded $K$ -theory ([38, Chapitre II.2.1]) as well. We do not delve into these comparisons (and the inter-comparisons) in this article, and point the curious reader to the work of David Aretz and Luuk Stehouwer ([3]), where such considerations have been discussed in detail. Note that loc. cit. is written in the generality of Banach algebras, and our fiber sequence “ABS” is more or less their definition of the graded $K$ -theory spectrum (see [3, Definitions 3.42 & 3.55]).

Finally, we mention that Ulrich Haag has works ([30],[31]) in this area. Therein, Haag obtains a couple of long exact sequences involving the different flavors of $K$ -groups, but only for complex $C^{*}$ -algebras. It can be seen that exactly one of Haag’s sequences survives the Real case, and the other one does not. The sequence that survives can be readily identified with the long exact sequence of homotopy groups of our sequence “ABS”. In the author’s view, some of the proofs presented in his papers would benefit from a more detailed and transparent exposition. Hence the methods of higher categories used in this paper both generalize and clarify Haag’s works.

1.1 Organization of the article

We now discuss the organization of this article.

Section 2 contains well-known results from equivariant and graded $K$ -theory. The key highlights of this section are Theorem 2.3 and Section 2.2. The non-equivariant case of Theorem 2.3 is the content of [26], and the equivariant generalization is more or less straightforward. Section 2.2 contains some miscellaneous results on the graded $K$ -theory of a distinguished graded $C^{*}$ -algebra $\mathcal{S}$ , and the results therein are going to be really crucial in the later sections.

In Section 3, we emulate the analogous chapters of [17], and construct a stable $\infty$ -categorical refinement of the $G$ -equivariant, Real $KK$ - and $E$ -categories. We actually combine the approaches of [17] and [11]; Section 3.3 is taken from the relevant section of [11]. In Section 3.4, we (have to) deviate from the writings of Bunke and work with a Real variant of the Toeplitz extension so as to enforce $(1,1)$ -periodicity. The category so obtained is denoted by $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ , where $!\in\{\operatorname{se,ex}\}$ denotes whether we enforced excision for semi-split exact or all exact sequences. Section 3.5 discusses the well-known change of group functors: restriction, induction and coinduction. These come in handy while studying the graded theory, although in the simple case with the groups involved being $G$ and $G_{2}:=G\times C_{2}$ . Section 3.6 makes connections with the models of equivariant and graded $K$ -theory as studied in Section 2 with certain corepresentable functors out of $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ (see Theorem 3.14 for a precise statement). Finally in Section 3.7, the extension of $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ to the large category of all $G$ - $C^{*}$ -algebras is considered.

Section 4 contains the higher categorical approach to defining the graded bivariant $K$ -categories, which we denote by $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ . Without apriori knowing that the homotopy category of $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ recovers classical graded $KK$ - and $E$ -theory according as $!=\operatorname{se}$ or $!=\operatorname{ex}$ , we show that the formal consequences of our definition, together with a calculation in classical graded $K$ -theory yields the familiar equivalence between $C_{0}(\mathbb{R}^{0,1})$ and $Cl^{1,0}$ in $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ . We also connect the spectral model of $K$ -theory as discussed in Section 2 with the functor corepresented by $\mathbb{C}$ in $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ .

Section 5 is devoted to deducing Theorem 1.1 as a consequence of the machinery developed in Sections 3 and 4. As advertised before, in this section we also discuss the folklore question of representing graded $K$ -groups of a (graded), (Real) unital $C^{*}$ -algebra in terms of Clifford module quotients.

Finally in Section 6, we compare the homotopy categories of the ungraded and graded $KK^{G}_{!,\infty}$ -categories we developed with the classical ungraded and graded $KK^{G}$ - and $E^{G}$ -categories (upon restricting to separable $C^{*}$ -algebras; the general comparison is discussed in Section 6.1). We give more details in the context of $E$ -theory: the well-known universal property of ungraded $E$ -theory lacks a proof in the foundational writings [20] and [29], and a detailed proof (following the work of Thomsen in [59]) is provided in Section 6.2. In Section 6.3, we discuss an equivariant generalization of a technical result of Dadarlat and Loring, which is an intermediary in making the comparison of the homotopy category of $\widehat{\mathrm{KK}}_{\mathrm{ex},\infty}^{G,\mathrm{sep}}$ (denoted $\widehat{\mathrm{KK}}_{\mathrm{ex},0}^{G,\mathrm{sep}}$ ) and $\widehat{\mathrm{E}}^{G}_{\mathrm{class}}$ . We mention that the comparison of the homotopy category of $\widehat{\mathrm{KK}}_{\mathrm{se},\infty}^{G,\mathrm{sep}}$ (denoted $\widehat{\mathrm{KK}}_{\mathrm{se},0}^{G,\mathrm{sep}}$ ) and $\widehat{\mathrm{KK}}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ is not directly possible using the techniques we used for their $E$ -theoretic counterparts, but instead can be deduced from results of Meyer ([46]).

Acknowledgments: The author is deeply grateful to Johannes Ebert for suggesting this project, his generous guidance throughout, and his thoughtful comments on earlier drafts. Many thanks to David Aretz and Luuk Stehouwer for having several pleasant discussions on topics related to this article, and also for freely sharing some ideas from their recent (and related) project. The author would also like to thank Ulrich Bunke, Benjamin Duenzinger, Michael Joachim, Matthias Lesch, Yuezhao Li and Maxime Ramzi for discussions at several points which clarified some mathematical issues.

This paper forms part of the PhD thesis of the author at the University of Bonn. Throughout the duration of his PhD, the author was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – GZ 2047/1 - 390685813, Hausdorff Center for Mathematics.

2 Background on $K$ -theory spaces, groups, and some calculations

In this section, we discuss the necessary background from classical / equivariant / graded $K$ -theory that will be needed in the later sections. We emphasize that (morally) $K$ -theory is a significantly simpler concept than $KK$ -theory, and we use our knowledge of $K$ -theory as discussed in this section later while studying the bivariant $K$ -categories.

Before proceeding further, we have to introduce the main objects of the entire project, namely, graded, Real, equivariant $C^{*}$ -algebras.

Definition 2.0.

1.

A graded complex $C^{*}$ -algebra $(A,\varepsilon)$ is a complex $C^{*}$ -algebra $A$ equipped with a $*$ -automorphism $\varepsilon$ such that $\varepsilon^{2}=1$ . $\varepsilon$ is called the grading on $A$ . The elements of the $+1$ and $-1$ eigenspaces of $A$ are called even and odd, respectively.
2.

For $G$ a locally compact second countable group, a complex $G$ - $C^{*}$ -algebra $(A,\alpha)$ is a $C^{*}$ -algebra $A$ together with a continuous homomorphism $\alpha:G\rightarrow\operatorname{Aut}(A)$ , where $\operatorname{Aut(A)}$ is equipped with the point-norm topology.
3.

A Real $C^{*}$ -algebra $(A,\tau)$ is a complex $C^{*}$ -algebra $A$ equipped with a $\mathbb{C}$ -anti-linear $*$ -automorphism $\tau$ such that $\tau^{2}=1$ . $\tau$ is called the Real structure on $A$ . ⁷⁷ 7 Sometimes, the Real structure will simply be denoted by $a\mapsto\bar{a}$ . The real $C^{*}$ -algebra associated to $(A,\tau)$ is $A_{\tau}:=\{a\in A\mid a=\tau(a)\}$ , and it can be shown that this induces an equivalence of categories between real and Real $C^{*}$ -algebras.
4.

For any two (or all three) of the adjectives defined before, we combine them in the obvious way by requiring the corresponding structure maps to commute.

Notation 2.0.

We follow the following conventions throughout this article.

•

Unless otherwise mentioned, $G$ will denote a locally compact second countable group.
•

As already mentioned in the introduction, we will by default be working be working with Real $C^{*}$ -algebras. Results in the complex case can be deduced by simply ignoring the Real structure. We do not mention the word “Real” unless there is an explicit need to do so.
•

We denote the category of (ungraded) Real $G$ - $C^{*}$ -algebras (with the obvious morphisms) by $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ . This category is enriched over topological spaces using the point-norm topology on the hom sets (see [26, Section 2.2] for details), and the mapping space is denoted by $\underline{\operatorname{Hom}}_{G}(A,B)$ . The set of path componentns of $\underline{\operatorname{Hom}}_{G}(A,B)$ is denoted by $[A,B]_{G}$ . If it is clear from the context, we suppress the $G$ from the notation.
•

The symmetric monoidal structures on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ will be denoted by $-\otimes_{?}-$ , where $?\in\{\min,\max\}$ corresponds to the minimal and maximal tensor products. If an algebra under consideration is nuclear, we often omit the decoration from the tensor product.
•

We denote the category of graded, Real $G$ - $C^{*}$ -algebras (with the obvious morphisms) by $\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ . Its symmetric monoidal structures will be denoted by $-\hat{\otimes}_{?}-$ , where $?\in\{\min,\max\}$ corresponds to the graded minimal and maximal tensor products. If an algebra under consideration is nuclear, we often omit the decoration from the tensor product (see also [63, Corollary E.2.19]). $\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ is also topologically enriched in a way analogous to the ungraded case mentioned above; the only difference being that the desired equivariance now is with respect to $G\times C_{2}$ .
•

Finally, we remark that in exceptional cases where we do have to work with the category of complex $C^{*}$ -algebras (for example, in Section 5.2), we keep using the same notation as above. This should not create any confusion.

We next mention some examples of graded $C^{*}$ -algebras which we will often encounter. Further examples and discussions can be found in [26]

Example 2.0.

1.

The base field $\mathbb{C}$ is trivially graded, and is always equipped with the trivial $G$ -action. In the Real situation, the Real structure on $\mathbb{C}$ is given by pointwise complex conjugation, and is omitted from the notation.
2.

The Clifford algebras come with a natural grading, and carry the trivial action of the external group $G$ .

$\mathcal{S}:=C_{0}(\mathbb{R})$ with the even-odd grading, trivial $G$ -action, and the Real structure given by pointwise complex conjugation is going to be a very important algebra for us. By choosing an odd, increasing homeomorphism between $(-1,1)$ and $\mathbb{R}$ (the choice of this homeomorphism does not matter as the space of all such homeomorphisms is contractible), we obtain an isomorphism between $\mathcal{S}$ and $\tilde{\mathcal{S}}$ , the bounded transform of $\mathcal{S}$ . We mention another perspective on $\tilde{\mathcal{S}}$ below, which comes very handy in concrete situations. Consider the map

	$\displaystyle\Phi:\tilde{\mathcal{S}}$	$\displaystyle\rightarrow\{f:[0,1)\rightarrow Cl^{0,1}\mid f(0)\in\mathbb{C}\subseteq Cl^{0,1}\}\subseteq C_{0}([0,1),Cl^{0,1})$
	$\displaystyle f$	$\displaystyle\mapsto f_{\operatorname{ev}}\mid_{[0,1)}+\varepsilon f_{\operatorname{odd}}\mid_{[0,1)},$

where $f_{\operatorname{ev}},f_{\operatorname{odd}}$ denote the even and odd parts of $f$ respectively, and $\varepsilon$ is the non-trivial generator of $Cl^{0,1}$ . $\Phi$ can directly be checked to be an isomorphism in $\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ .

4.

A grading on a $C^{*}$ -algebra is said to be inner if it is given by conjugation with a self-adjoint unitary in the multiplier algebra of $A$ . If $A$ is furthermore equivariant and/or Real, the unitary needs to satisfy the required invariance conditions. The algebras of $G$ -continuous bounded and compact operators on a graded $G$ -Hilbert space are naturally inner graded. The Clifford algebra $Cl^{p,q}$ is inner graded if $p-q\equiv 0\penalty\ (\operatorname{mod}4)$ .

Before proceeding, we set up some shorthand notation that we will be using throughout this article.

Notation 2.0.

We denote by $G_{2}$ the group $G\times C_{2}$ (equipped with the product topology).

Remark 2.0.

Although the categories $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ and $\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ are equivalent (by definition!), the symmetric monoidal structures on them are different. See [3, Section 3.1] for a categorical perspective on the symmetric monoidal structures on $\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ .

We now begin our study of $K$ -theory. As is customary in the development of equivariant $K$ -theory, we obtain interesting results when the action is by a compact group. So throughout the remainder of this section, $G$ will denote a compact group.

2.1 Miscellaneous results on equivariant (ungraded) $K$ -theory

In this subsection, we record some well known results in the realm of equivariant $K$ -theory following [8, Chapter 11]. Recall that (by definition) the equivariant $K_{0}$ -group of a unital $G$ - $C^{*}$ -algebra is the Grothendieck group of the abelian monoid (under direct sum) of finitely generated, $G$ -equivariant projective $A$ -modules. We summarize some well-known properties of the equivariant $K_{n}$ -functors in the next remark for the ease of reference later.

Remark 2.0.

The following facts about $G$ -equivariant $K$ -theory are well-known.

1.

$K_{0}^{G}(\mathbb{C})$ is a ring with multiplication obtained from the external tensor product of $G$ -vector spaces, and is isomorphic to $R(G)$ , the representation ring of $G$ . In particular, the ring $K_{0}^{C_{2}}(\mathbb{C})$ is isomorphic to $\frac{\mathbb{Z}[x]}{x^{2}=1}$ , the group ring of $C_{2}$ . The preferred isomorphism maps the trivial representation of $C_{2}$ on $\mathbb{C}$ to $1\in\frac{\mathbb{Z}[x]}{x^{2}=1}$ and the sign representation of $C_{2}$ on $\mathbb{C}$ to $x\in\frac{\mathbb{Z}[x]}{x^{2}=1}$ .
2.

More generally, the external product with an element of $K^{G}_{0}(\mathbb{C})$ refines $K^{G}_{n}(-)$ to functors ⁸⁸ 8 this should not be confused with the “ $G$ -action on $K^{G}$ -groups induced by the action on the algebra”, because in general there is no such action! When $G$ is $C_{2}$ , then this action exists, but can be shown to be always trivial.

$K_{n}^{G}(-):\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}\rightarrow R(G)\text{-}\operatorname{Mod}\forall\penalty\ n\geq 0$
3.

Of importance to us is the Green-Julg theorem, which asserts that there is a natural isomorphism

$K^{G}_{n}(A)\cong K_{n}(A\rtimes G)\penalty\ \forall\penalty\ n\geq 0$

See [8, Theorem 11.7.1] for a proof. For the case of $G=C_{2}$ , note that we have a natural isomorphism $A\rtimes C_{2}\cong(A\hat{\otimes}Cl^{0,1})_{u}$ (see [8, Proposition 14.5.1]) and will sometimes use this identification without mentioning.

For our applications later, we need to compute the $C_{2}$ -equivariant $K$ -groups of the graded $C^{*}$ -algebras $Cl^{0,1}$ and $\mathcal{S}$ . Though the calculations of these groups are surely well known, we include in the proofs some details which will come handy in Section 5.

Proposition 2.0.

$K_{n}^{C_{2}}(Cl^{0,1})\cong K_{n}(\mathbb{C})\penalty\ \forall\penalty\ n\in\mathbb{N}_{\geq 0}$ . The $C_{2}$ -action on $K_{n}^{C_{2}}(Cl^{0,1})$ induced by $K_{0}^{C_{2}}(\mathbb{C})$ (as dictated by Section 2.1.2) is given by the identity.

Proof.

We begin by analyzing the maps

K_{n}^{C_{2}}(\mathbb{C})\xrightarrow{\iota_{*}}K_{n}^{C_{2}}(Cl^{0,1}).

Consider the commutative diagram (with the unlabeled horizontal arrow us so defined to render the diagram commutative)

where the top vertical isomorphisms are induced by the Green-Julg theorem, and the lower vertical isomorphisms are induced by the canonical maps. We claim that the bottom horizontal map is given by addition. To that end, one checks that the inclusion $Cl^{0,1}\rightarrow Cl^{0,2}$ corresponds to the homomorphism

	$\displaystyle\mathbb{C}\oplus\mathbb{C}$	$\displaystyle\rightarrow M_{2}(\mathbb{C})$
	$\displaystyle(x,y)$	$\displaystyle\mapsto\frac{1}{2}\begin{pmatrix}x+y&x-y\\ x-y&x+y\end{pmatrix}.$

under the standard identifications. Precomposing with the two inclusions of $\mathbb{C}$ into $(\mathbb{C}\oplus\mathbb{C})_{u}$ therefore induce the homomorphisms $\mathbb{C}\rightarrow M_{2}(\mathbb{C})_{u}$ given by

\lambda\mapsto\frac{1}{2}\begin{pmatrix}\lambda&\lambda\\ \lambda&\lambda\end{pmatrix}\penalty\ \penalty\ \text{and}\penalty\ \penalty\ \mu\mapsto\frac{1}{2}\begin{pmatrix}\mu&-\mu\\ -\mu&\mu\end{pmatrix}.

The two homomorphisms can be homotoped to the left-upper-corner inclusion and/or the right-lower-corner inclusion. The claim now follows from [63, Lemma 2.7.6].

To understand the $C_{2}$ -action on $K_{n}^{C_{2}}(Cl^{0,1})$ induced by $K_{0}^{C_{2}}(\mathbb{C})$ , one notes that the flip action on $K_{0}^{C_{2}}(\mathbb{C})$ is invariant under the addition map, and that $\iota_{*}$ is an $\mathbb{Z}[C_{2}]$ -module homomorphism. ∎

Corollary 2.0.

$K_{0}^{C_{2}}(\mathcal{S})\cong\mathbb{Z}$ . The $C_{2}$ -action on $K_{0}^{C_{2}}(\mathcal{S})$ induced by $K_{0}^{C_{2}}(\mathbb{C})$ is given by $n\mapsto-n$ .

Proof.

⁹⁹ 9 We probably are not giving the cleanest of this result! Instead, we mention in the proof some ideas which we will use later. We are also implicitly identifying

\mathcal{S}

with the bounded transform

\tilde{\mathcal{S}}

\mathcal{S}

in this proof.

Consider the diagram in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$

(2.1)

where $\Phi$ is as in Section 2.3. Note that the right square is a pullback square (it is the defining square of the mapping cylinder construction; see also Section 3.1.2), and the left square as well as the two triangles involving the dashed edge commute up to homotopy. The rest follows by applying the $K_{0}^{C_{2}}(-)$ functor to Equation 2.1, using the homotopy invariance and exicision properties of equivariant $K$ -theory and the proof of Section 2.1. Observe that the kernel of the addition map $\mathbb{Z}\oplus\mathbb{Z}\rightarrow\mathbb{Z}$ is $\{(n,-n)\mid n\in\mathbb{Z}\},$ which is evidently isomorphic to $\mathbb{Z}$ with the action $n\mapsto-n$ . ∎

We now want to present a generator of $K_{0}^{C_{2}}(\mathcal{S})\cong\mathbb{Z}$ . To that end, we need a concrete model for the $C_{2}$ -equivariant $K_{0}$ -group of a non-unital $C^{*}$ -algebra. This picture is going to be convenient for later use, especially in Section 3.6.

We begin with the relevant definitions.

Definition 2.1.

Let $A$ be a separable $G$ - $C^{*}$ -algebra. Set $QA:=A*A$ , the coproduct in the category of separable $G$ - $C^{*}$ -algebras. Denote by $qA$ the kernel of the fold map $QA\rightarrow A$ . $qA$ gets a natural $G$ -action from $A$ .

Remark 2.1.

The following remarks about $qA$ are in order.

1.

$qA$ fits in a short exact sequence

$0\rightarrow qA\rightarrow QA\rightarrow A\rightarrow 0.$

The sequence splits in two canonical ways, depending on embedding $A$ into one of the components of $QA$ .
2.

The $q$ construction is functorial, and there are two natural maps $qA\rightarrow A$ given by restricting the homomorphisms

$A*A\xrightarrow{(1_{A},0)}A,\penalty\ \penalty\ \penalty\ \penalty\ A*A\xrightarrow{(0,1_{A})}A$

to $qA$ .

Given $A,B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ , a prequasi-homomorphism from $A$ to $B$ consists of a $G$ - $C^{*}$ -algebra $E$ , two $G$ -equivariant $*$ -homomorphisms¹⁰¹⁰ 10 $\overline{\alpha}$ should not be confused with any sort of complex conjugation! $\alpha,\overline{\alpha}:A\rightarrow E$ , an ideal $J\triangleleft E$ , and a $G$ -equivariant $*$ -homomorphism $\mu:J\rightarrow B$ such that $\alpha(x)-\overline{\alpha}(x)\in J\penalty\ \forall\penalty\ x\in A$ . We write it as

By universal properties, a prequasi-homomorphism $A\rightarrow B$ yields a map $\mu\circ q(\alpha,\overline{\alpha}):qA\rightarrow B$ , which we simply denote by $\alpha-\overline{\alpha}$ .

4.

When $A=\mathbb{C}$ , there is a concrete description of $q\mathbb{C}$ :

$q\mathbb{C}\cong\{f\in C_{0}((0,1],M_{2}(\mathbb{C}))\mid f(1)\penalty\ \text{is diagonal}\}.$

This is stated in [8, Exercise 10.11.13.g], and a proof with a lot of details and other background is available in [54]. Technically, loc. cit. gives a concrete function algebra description of the universal unital $C^{*}$ -algebra on two projections, from which our result can be directly deduced in light of the defining short exact sequence of $q\mathbb{C}$ as mentioned in the first remark above. See also [21, Section 6].
5.

The above isomorphism directly shows that, in light of Section 2.3,

$\mathcal{S}\rtimes C_{2}\cong(\mathcal{S}\hat{\otimes}Cl^{0,1})_{u}\cong q\mathbb{C}.$

This bridges the gap between the spectral model (studied in the next subsection) and the Cuntz picture. In particular, the proof of the next proposition can also be deduced from Theorem 2.3 in light of a restriction-coinduction adjunction at the level of the compact-stabilized homotopy categories. Similar ideas are used in the proof of Section 3.6.
6.

An important technical feature of $q\mathbb{C}$ is semi-projectivity, which implies that for all $A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ , the canonical map

$[q\mathbb{C},A]_{G}\xrightarrow{\cong}[[q\mathbb{C},A]]_{G}$

from the $G$ -equivariant homotopy classes of maps to the $G$ -equivariant asymptotic homotopy classes of maps from $q\mathbb{C}$ to $A$ is a bijection of sets. See [8, Exercise 4.7.1, Proposition 25.1.7] and [Blackadar1985Shape] for general results on semi-projective $C^{*}$ -algebras. The proof of $q\mathbb{C}$ being semi-projective is the content of [42, Theorem 7].

Proposition 2.1.

Let $G$ be a compact group and $A$ a $G$ - $C^{*}$ -algebra. Then, there is a natural isomorphism of abelian groups

[q\mathbb{C},A\otimes\mathcal{K}(L^{2}G_{\mathbb{N}})]_{G}\cong K_{0}^{G}(A).

Proof.

The result is well known, see for example [8, Section 17.6], [22, Remark 1]. The compact group action (and the Real structure) do not cause any extra difficulty as $G$ -equivariant¹¹¹¹ 11 $G$ -equivariant adjointable operators are precisely the $G$ -fixed points of the $G$ - $C^{*}$ -algebra of adjointable operators on a $G$ -Hilbert module., $G$ -continuous Fredholm operators on $G$ -Hilbert modules classify $G$ -equivariant $K_{0}$ when $G$ is compact; see Theorem 2.3.5 for example. ∎

Remark 2.1.

We mention that the isomorphism

\displaystyle K_{0}^{G}(A)

\displaystyle\xrightarrow{\cong}[q\mathbb{C},A\otimes\mathcal{K}(L^{2}G_{\mathbb{N}})]_{G}

(2.2)

is given by precomposition with the first map $q\mathbb{C}\rightarrow\mathbb{C}$ as described in Section 2.1.2, see [8, Exercise 10.11.13.f]. By taking $n$ -fold loops on both sides of Equation 2.2, we obtain

K_{n}^{G}(A)\cong\pi_{n}(\underline{\operatorname{Hom}}_{G}(q\mathbb{C},A\otimes\mathcal{K}(L^{2}G_{\mathbb{N}})))

for all $n\in\mathbb{N}$ .

We are now in a position to supply a concrete generator of $K_{0}^{C_{2}}(\mathcal{S})$ . To that end, we consider two distinguised $C_{2}$ -invariant projections in $C([-1,1],M_{2})$ . Note that $C[-1,1]$ with the even-odd grading is the multiplier algebra of $\mathcal{S}$ ¹²¹² 12 we implicitly identified $\mathcal{S}$ with its bounded version and $M_{2}$ has the diagonal-off diagonal grading, thereby inducing the tensor product action on $C([-1,1],M_{2})$ . Set

	$\displaystyle p_{1}:[-1,$	$\displaystyle 1]\rightarrow M_{2}(\mathbb{C})$
		$\displaystyle t\mapsto\begin{pmatrix}\cos^{2}(\frac{\pi}{2}t)&\sin(\frac{\pi}{2}t)\cos(\frac{\pi}{2}t)\\ \sin(\frac{\pi}{2}t)\cos(\frac{\pi}{2}t)&\sin^{2}(\frac{\pi}{2}t)\end{pmatrix}$

and

	$\displaystyle p_{2}:[-1,$	$\displaystyle 1]\rightarrow M_{2}(\mathbb{C})$
		$\displaystyle t\mapsto\begin{pmatrix}0&0\\ 0&1\end{pmatrix}$

It is clear that $p_{1},p_{2}$ are $C_{2}$ -invariant projections in $C([-1,1],M_{2})$ , and that they induce a homomorphism

p_{1}-p_{2}:q\mathbb{C}\rightarrow M_{2}(\mathcal{S})

in the notation of Section 2.1.3.

Proposition 2.2.

The homomorphism $\mathbb{Z}\rightarrow K_{0}^{C_{2}}(\mathcal{S})$ given by $1\mapsto p_{1}-p_{2}$ renders the diagram

commutative, where the right isomorphism is the isomorphism described in Section 2.1.1.

Proof.

From the descriptions of $p_{1}$ and $p_{2}$ and Section 2.1, it is clear that $\operatorname{ev}_{0}(p_{1}-p_{2})$ corresponds to the map $q\mathbb{C}\rightarrow M_{2}$ given by the difference of the $C_{2}$ -equivariant projections $\begin{pmatrix}1&0\\ 0&0\end{pmatrix}$ and $\begin{pmatrix}0&0\\ 0&1\end{pmatrix}$ in $M_{2}$ . Now, $M_{2}$ with the diagonal-off diagonal grading is nothing but the algebra of operators on the $C_{2}$ -vector space $\mathbb{C}^{\operatorname{triv}}\oplus\mathbb{C}^{\operatorname{sign}}$ . The rest follows directly. ∎

Corollary 2.2.

The left vertical arrow above is in fact an isomorphism. We declare $p:=p_{1}-p_{2}$ to be the preferred generator of $K_{0}^{C_{2}}(\mathcal{S})\cong\mathbb{Z}$ .

Proof.

Since we know that $K_{0}^{C_{2}}(\mathcal{S})$ is abstractly isomorphic to $\mathbb{Z}$ , this follows from the elementary observation that the map

\mathbb{Z}\xrightarrow{n\mapsto(n,-n)}\mathbb{Z}\oplus\mathbb{Z}

can only factor through $\mathbb{Z}$ (in the category of abelian groups) through an isomorphism $\mathbb{Z}\rightarrow\mathbb{Z}$ . ∎

2.2 The spectral model of graded, $G$ -equivariant $K$ -theory

Recall that throughtout this subsection, $G$ is a compact group. We want to develop a $G$ -equivariant version of the spectral model of $K$ -theory, originally due to Higson and Guentner in the complex case ([32, Section 1.5]), and discussed with details in the Real case in [26]. The spectral model predates any $\infty$ -categorical refinements of $K(K)$ -theory, and in the present work, we would need to have formal properties of it setup a priori in order to get mileage out of the stable $\infty$ -categories for graded bivariant $K$ -theories constructed in Section 4 (see especially the proof of Theorem 4.4).

We begin with the relevant definitions.

Definition 2.2.

Let $A\in\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ . Define the $G$ -equivariant spectral $K$ -theory space of $A$ , denoted $\mathbb{K}^{G,\operatorname{sp}}(A)$ , to be

\mathbb{K}^{G,\operatorname{sp}}(A):=\underline{\operatorname{Hom}}_{G_{2}}(\mathcal{S},A\hat{\otimes}\hat{\mathcal{K}}_{G}),

the $G\times C_{2}$ -equivariant $*$ -homomorphisms from $\mathcal{S}$ to $A\hat{\otimes}\hat{\mathcal{K}}_{G}$ equipped with the point-norm topology. We define the spectral $K$ -theory groups of $A$ to be the homotopy groups of $\mathbb{K}^{G,\operatorname{sp}}(A)$ ,

K_{n}^{G,\operatorname{sp}}(A):=\pi_{n}(\mathbb{K}^{G,\operatorname{sp}}(A),\underline{0})

Before studying the spectral model, we introduce the so-called Fredholm model, which formalizes the notion of $G$ -equivariant Fredholm operators classifying $G$ -equivariant $K$ -theory. It is also a crucial bridge between showing that the spectral model recovers the classical projective module picture of $K$ -theory when the algebra is inner graded.

Definition 2.2.

Let $A\in\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ .

•

A $G$ -Fredholm cycle over $A$ is a tuple $(\mathcal{E},F)$ , where $\mathcal{E}$ is a graded, countably generated Hilbert $A,G$ -module ¹³¹³ 13 see [46, Section 2.1] for the definition, and $F$ is a $G$ -continuous, $G$ -invariant, odd, self-adjoint Fredholm operator on it.
•

Two $G$ -Fredholm cycles $(\mathcal{E}_{0},F_{0})$ and $(\mathcal{E}_{1},F_{1})$ over $A$ are said to be concordant if there exists a $G$ -Fredholm cycle $(\mathcal{E},F)$ over $A\hat{\otimes}C[0,1]\cong C([0,1],A)$ such that $(\mathcal{E}\hat{\otimes}_{\operatorname{ev}_{i}}A,\operatorname{ev}_{i}(F))\cong(\mathcal{E}_{i},F_{i})$ for $i=0,1$ , where $\operatorname{ev}_{t}:C([0,1],A)\rightarrow A$ is the homomorphism given by $\operatorname{ev}_{t}(f)=f(t)$ .
•

Denote by $K_{0}^{G,\operatorname{Fr}}(A)$ the abelian monoid of concordance classes of $G$ -Fredholm cycles over $A$ .

Theorem 2.3.

Some key features of the spectral model and the Fredholm model of graded, $G$ -equivariant $K$ -theory are mentioned below. In what follows, $A,B,...$ denote arbitrary graded $G$ - $C^{*}$ -algebras, and all the morphisms are in the category $\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ .

1.

$\mathbb{K}^{G,\operatorname{sp}}(A)$ is an $H$ -group. Consequently, all the equivariant spectral $K$ -groups are in fact abelian groups.
2.

A short exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ in $\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ yields a Serre fibration on the associated spectral $K$ -theory spaces.
3.

The functors $K_{n}^{G,\operatorname{sp}}(-):\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}\rightarrow\operatorname{Ab}$ preserve filtered colimits for all $n\in\mathbb{N}_{\geq 0}$ .
4.

The functors

$K_{0}^{G,\operatorname{sp}}(-):(\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg},\hat{\otimes}_{!})\rightarrow(\operatorname{Ab},\otimes)$

are lax monoidal.
5.

We have a natural isomorphism of abelian groups

$\pi_{0}(\mathbb{K}^{G,\operatorname{sp}}(A))\xrightarrow{\cong}K_{0}^{G,\operatorname{Fr}}(A).$
6.

When $A$ is inner graded, we have a natural isomorphism of abelian groups

$\pi_{n}(\mathbb{K}^{G,\operatorname{sp}}(A))\cong K_{n}^{G}(A)\penalty\ \forall\penalty\ n\geq 0.$

Proof sketch.

Assertion 1 follows from the more general statement that given a graded morphism $\varphi:\mathcal{S}\rightarrow A$ , the induced morphism

\mathcal{S}\rightarrow A\hat{\otimes}M_{2}(\mathbb{C})=M_{2}(A),\penalty\ \penalty\ \penalty\ f\mapsto\begin{pmatrix}\varphi(f)&0\\ 0&\alpha(\varphi(f))\end{pmatrix}

is nullhomotopic in $\underline{\operatorname{Hom}}_{G}(\mathcal{S},A\hat{\otimes}M_{2}(\mathbb{C}))$ (here $\alpha$ is the grading on $A$ ), where $M_{2}(\mathbb{C})$ has the grading given by $\operatorname{Ad}(\tau)$ , where $\tau=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$ . See [32, Remark 1.7].

Assertion 2 follows from the non-equivariant assertion proven in [26, Section 3.2] by taking $G$ -fixed points. Note that taking $G$ -fixed points is an exact functor because $G$ is compact.

By similar reductions as done before, to show Assertion 3, it suffices to prove that for a filtered diagram $(A_{i})_{i\in I}$ of non-equivariant graded $C^{*}$ -algebras with colimit $A$ , one has a weak homotopy equivalence of spaces

\operatorname{colim}_{i\in I}\underline{\operatorname{Hom}}(\mathcal{S},A_{i})\rightarrow\underline{\operatorname{Hom}}(\mathcal{S},A).

Upon passing to the Cayley transform picture (see [26, Lemma 3.1]), the result follows from [48, Corollary to Theorem 12].

The key to demonstrating Assertion 4 is the comultiplication map on $\mathcal{S}$ as described in [26, Section 4.2]. The details are in [26, Section 6], and the same proofs go through equivariantly. See also Section 4.

We now discuss Assertion 5. A natural map $\pi_{0}(\mathbb{K}^{G,\operatorname{sp}}(A))\xrightarrow{\cong}K_{0}^{G,\operatorname{Fr}}(A)$ can be written by directly emulating the non-equivariant construction as in [26, Section 8.5]. We mention some aspects of it for ease of reference later.

Firstly, we implicitly replace $\mathcal{S}$ with $\tilde{\mathcal{S}}$ , and the Fredholm cycles with essentially unitary ones. Recall that

A\hat{\otimes}\hat{\mathcal{K}}_{G}\cong\mathcal{K}_{A}(A\otimes\widehat{L^{2}G_{\mathbb{N}}}),

the $G$ -equivariant graded $C^{*}$ -algebra of $A$ -compact operators on the standard graded $G$ -equivariant Hilbert $A$ -module. Given $\varphi:\mathcal{S}\rightarrow A\hat{\otimes}\hat{\mathcal{K}}_{G}\cong\mathcal{K}_{A}(A\otimes\widehat{L^{2}G_{\mathbb{N}}})$ , set

\mathcal{H}_{\varphi}:=\overline{\varphi(\mathcal{S})(A\otimes\widehat{L^{2}G_{\mathbb{N}}})},

and note that $\varphi$ restricts to a non-degenerate homomorphism

\varphi^{\operatorname{nd}}:\mathcal{S}\rightarrow\mathcal{K}_{A}(\mathcal{H}_{\varphi}).

The crucial point here is that $\varphi^{\operatorname{nd}}$ indeed maps to the $A$ -compact operators on the (not necessarily complementable) submodule $\mathcal{H}_{\varphi}\subseteq A\otimes\widehat{L^{2}G_{\mathbb{N}}}$ , as is guaranteed by [46, Lemma 4.1]. Non-degeneracy of $\varphi^{\operatorname{nd}}$ implies that there exists a unique $G$ -equivariant, graded homomorphism $\tilde{\varphi}:C[-1,1]\rightarrow\mathcal{B}(\mathcal{H}_{\varphi})$ extending $\varphi^{\operatorname{nd}}$ . The natural map is now given by

	$\displaystyle\pi_{0}(\mathbb{K}^{G,\operatorname{sp}}(A))\xrightarrow{}$	$\displaystyle K_{0}^{G,\operatorname{Fr}}(A)$
	$\displaystyle\varphi\longmapsto$	$\displaystyle(\mathcal{H}_{\varphi},\tilde{\varphi}(x))$

The proof that this map is bijective now follows with the obvious modifications from [26, Section 8.5].

Finally, we come to Assertion 6. We just discuss the case of $\pi_{0}$ when $A$ is unital and ungraded, the general case follows from the techniques described in [63, Section 2.9]. One proof strategy in the unital, ungraded case would be to show that the natural map $K_{0}^{G}(A)\rightarrow K_{0}^{G,\operatorname{Fr}}(A)$ has an inverse given by taking the $G$ -equivariant index of $G$ -invariant Fredholm operators (as done in the non-equivariant case in [26, Section 8.4]), and then conclude by Assertion 5. However, granted the results of loc. cit. and the Green-Julg theorem, we can quickly obtain the conclusion by leveraging the fact that $A\rtimes G\cong(A\otimes\mathcal{K}(L^{2}G))^{G}$ , as shown in [29, Proposition 11.2]. See also the proof of [27, Proposition 3.1]. ∎

Notation 2.3.

Having established natural isomorphisms between different models of graded $K$ -theory, we henceforth denote the $G$ -equivariant graded $K$ -groups as $\hat{K}^{G}_{*}(-)$ .

Next, we analyse the graded $K_{0}$ -group of $\mathcal{S}$ (ignoring the external group $G$ ). We summarize our results in the following proposition. Note that all the algebras $C_{0}(\mathbb{R}^{n})$ in the below proof (and in general) carry the trivial grading. The author learnt most of these results from Johannes Ebert.

Proposition 2.3.

There is a short exact sequence

0\xrightarrow{}C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\xrightarrow{\psi}\mathcal{S}\xrightarrow{\operatorname{ev}_{0}}\mathbb{C}\rightarrow 0

in $\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ .

For every $A\in\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ , the map

\underline{\operatorname{Hom}}(\mathcal{S},\mathcal{S}\hat{\otimes}A)\xrightarrow{(\operatorname{ev}_{0}\hat{\otimes}1_{A})_{*}}\underline{\operatorname{Hom}}(\mathcal{S},A)

have natural continuous sections $\sigma_{A}$ . Furthermore, $\sigma_{\mathbb{C}}(\operatorname{ev}_{0})=\operatorname{1}_{\mathcal{S}}$ .

For every $A\in\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ , we have naturally split exact sequences

0\rightarrow\hat{K}_{*}(C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\hat{\otimes}A)\xrightarrow{\psi_{*}}\hat{K}_{*}(\mathcal{S}\hat{\otimes}A)\xrightarrow{\operatorname{{ev}_{0}}_{*}}\hat{K}_{*}(A)\rightarrow 0.

4.

For each $n\in\mathbb{N}$ , multiplication by the Bott class $[\beta_{\mathbb{C}}^{n}]\in\hat{K}(C_{0}(\mathbb{R}^{n})\hat{\otimes}Cl^{0,n})$ yields natural isomorphisms

$\hat{K}_{*}(A)\rightarrow\hat{K}_{*}(C_{0}(\mathbb{R}^{n})\hat{\otimes}Cl^{0,n}\hat{\otimes}A).$

In particular, $\hat{K}_{0}(C_{0}(\mathbb{R}^{n})\hat{\otimes}Cl^{0,n})\cong\mathbb{Z}$ , and the isomorphism identifies $[\beta_{\mathbb{C}}^{n}]$ and $1$ . The grading $\alpha$ on $C_{0}(\mathbb{R}^{n})\hat{\otimes}Cl^{0,n}$ induces the map $(-1)^{n}:\mathbb{Z}\rightarrow\mathbb{Z}$ .
5.

$\hat{K}_{0}(\mathcal{S})\cong\mathbb{Z}\oplus\mathbb{Z}$ as abelian groups, with the preferred isomorphism mapping $\psi_{*}[\beta_{\mathbb{C}}]=:u$ to $(0,1)$ and $[\operatorname{1_{\mathcal{S}}}]=:e$ to $(1,0)$ .

The composition

C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\xrightarrow{\psi}\mathcal{S}\xrightarrow{\beta_{\mathbb{C}}}C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}

induces multiplication by $2$ on $\hat{K}_{0}(C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1})\cong\mathbb{Z}$ .

7.

In the context of Assertion 5, we have $\alpha_{*}(u)=-u$ and $\alpha_{*}(e)=e-u$ , where $\alpha$ denotes the grading on $\mathcal{S}$ .
8.

There exists an isomorphism of $C_{2}$ -abelian groups between $\hat{K_{0}}(\mathcal{S})$ and $\mathbb{Z}\oplus\mathbb{Z}$ with the flip action, with the preferred isomorphism sending $[1_{\mathcal{S}}]$ to $(1,0)$ and $[\alpha_{\mathcal{S}}]$ to $(0,1)$ .

Proof.

We begin with Assertion 1. Consider

0\xrightarrow{}J\xrightarrow{}\mathcal{S}\xrightarrow{\operatorname{ev}_{0}}\mathbb{C}\rightarrow 0

where $J:=\ker(\operatorname{ev}_{0})$ . Then, we have

\displaystyle J\xrightarrow{\cong}C_{0}((0,\infty))\hat{\otimes}Cl^{0,1}\penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ f\mapsto f_{\operatorname{ev}}\mid_{(0,\infty)}\hat{\otimes}1+f_{\operatorname{odd}}\mid_{(0,\infty)}\hat{\otimes}\varepsilon,

(2.4)

where $f_{\operatorname{ev}}(x)=\frac{f(x)+f(-x)}{2},f_{{\operatorname{odd}}}(x)=\frac{f(x)-f(-x)}{2}$ , and $\varepsilon$ is the canonical generator of $Cl^{0,1}$ . Take a homeomorphism $h:(0,\infty)\rightarrow\mathbb{R}$ such that

\lim_{t\to 0}h(t)=-\infty,\lim_{t\to\infty}h(t)=\infty.

For concreteness, take $h(x)=\ln(x)$ . We can define our sought after map $\psi:C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\rightarrow\mathcal{S}$ using $h$ and the isomorphism above.

For Assertion 2, note that a direct calculation using the generators $e^{-x^{2}}$ and $xe^{-x^{2}}$ of $\mathcal{S}$ shows that the composite

\operatorname{Hom}(\mathcal{S},A)\xrightarrow{1_{\mathcal{S}}\hat{\otimes}-}\operatorname{Hom}(\mathcal{S}\hat{\otimes}\mathcal{S},\mathcal{S}\hat{\otimes}A)\xrightarrow{\Delta^{*}}\operatorname{Hom}(\mathcal{S},\mathcal{S}\hat{\otimes}A)\xrightarrow{(\operatorname{ev}_{0}\hat{\otimes}1_{A})_{*}}\operatorname{Hom}(\mathcal{S},A)

is the identity, and has the required effect on $\operatorname{ev}_{0}\in\operatorname{Hom}(\mathcal{S},\mathbb{C})$ too.

Assertion 3 follows immediately from Assertion 2.

We just prove the last part of Assertion 4, since the first two parts are the content of [26, Section 9]. Recall that the Bott class is the class in graded $K$ -theory represented by the Bott map, which is the functional calculus of the embedding of $\mathbb{R}^{n}$ in ${Cl^{0,n}}$ as a homomorphism $\mathcal{S}\rightarrow C_{0}(\mathbb{R}^{n},Cl^{0,n})$ .¹⁵¹⁵ 15 strictly speaking, we mean $\mathbb{R}^{0,n}$ here

We begin with the case $n=1$ . To simplify the notation, we denote $\beta_{\mathbb{C}}^{1}$ by $\beta_{\mathbb{C}}$ . We have to show that $\alpha_{*}([\beta_{\mathbb{C}}])=-[\beta_{\mathbb{C}}]$ . By definition of $\beta_{\mathbb{C}}$ , we have

\displaystyle\beta_{\mathbb{C}}:\mathcal{S}

\displaystyle\rightarrow C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\cong C_{0}(\mathbb{R},Cl^{0,1})\penalty\ \penalty\ \penalty\ \penalty\ \penalty\ f\mapsto t\mapsto f(t\varepsilon).

This is same as the map $f\mapsto f^{\operatorname{ev}}+f^{\operatorname{odd}}\varepsilon$ , as can be readily checked on the generators $u$ and $v$ of $\mathcal{S}$ . Therefore, we have

\displaystyle\alpha\circ\beta_{\mathbb{C}}:\mathcal{S}

\displaystyle\rightarrow C_{0}(\mathbb{R},Cl^{0,1}),\penalty\ \penalty\ \penalty\ \penalty\ \penalty\ f\mapsto f^{\operatorname{ev}}-f^{\operatorname{odd}}\varepsilon.

We now identify

\operatorname{Hom}(\mathcal{S},C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1})\cong\operatorname{Map}_{*}(\mathbb{R}^{+},\operatorname{Hom}(\mathcal{S},Cl^{0,1})).

Furthermore, under the Cayley transform isomorphism, we have a homeomorphism¹⁶¹⁶ 16 in what follows, we denote by $\mathbb{R}^{+}$ the one-point compactification of $\mathbb{R}$ . $\infty\in\mathbb{R}^{+}$ is the point at infinity.

$\displaystyle\mathbb{R}^{+}$	$\displaystyle\rightarrow\operatorname{Hom}(\mathcal{S},Cl^{0,1})$	(2.5)
$\displaystyle\infty$	$\displaystyle\mapsto 0$	(2.6)
$\displaystyle t$	$\displaystyle\longmapsto f\mapsto f^{\operatorname{ev}}(t)+f^{\operatorname{odd}}(t)\varepsilon$	(2.7)

Whence, we have a homeomorphism of spaces

\operatorname{Hom}(\mathcal{S},C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1})\cong\operatorname{Map}_{*}(\mathbb{R}^{+},\mathbb{R}^{+}),

under which $\beta_{\mathbb{C}}$ corresponds to the identity, and $\alpha\circ\beta$ corresponds to the negative of identity. Since degree of the negative identity as a map from $\mathbb{R}^{+}$ to itself is $-1$ , the conclusion follows.

For the general case, we consider the commutative diagram of abelian groups (see [26, Section 7.1] for the details)

\begin{array}[]{ccccccc}\hat{K}_{0}\!\left(C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\right)&\otimes&\hat{K}_{0}\!\left(C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\right)&\otimes\cdots\otimes&\hat{K}_{0}\!\left(C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\right)&\longrightarrow&\hat{K}_{0}\!\left(C_{0}(\mathbb{R}^{n})\hat{\otimes}Cl^{0,n}\right)\\[4.30554pt] \downarrow\!{\scriptstyle\alpha_{*}}&&\downarrow\!{\scriptstyle\alpha_{*}}&&\downarrow\!{\scriptstyle\alpha_{*}}&&\downarrow\!{\scriptstyle\alpha_{*}}\\[4.30554pt] \hat{K}_{0}\!\left(C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\right)&\otimes&\hat{K}_{0}\!\left(C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\right)&\otimes\cdots\otimes&\hat{K}_{0}\!\left(C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\right)&\longrightarrow&\hat{K}_{0}\!\left(C_{0}(\mathbb{R}^{n})\hat{\otimes}Cl^{0,n}\right).\end{array}

From [63, Lemma 9.3.4] on the one hand we get

[\beta_{\mathbb{C}}]\times[\beta_{\mathbb{C}}]\times...\times[\beta_{\mathbb{C}}]\mapsto[\beta_{\mathbb{C}}^{n}]\in\hat{K}_{0}(C_{0}(\mathbb{R}^{n})\hat{\otimes}Cl^{0,n}).

Since $\alpha_{*}([\beta_{\mathbb{C}}])=-[\beta_{\mathbb{C}}]$ for the case dealt with above, from the commutativity of the above diagram, we get on the other hand

(-[\beta_{\mathbb{C}}])\times(-[\beta_{\mathbb{C}}])\times...\times(-[\beta_{\mathbb{C}}])\mapsto\alpha_{*}[\beta_{\mathbb{C}}^{n}]\in\hat{K}_{0}(C_{0}(\mathbb{R}^{n})\hat{\otimes}Cl^{0,n}).

This proves Assertion 4.

Assertion 5 follows immediately from Assertions 3 and 4.

We now show Assertion 6. Choose a homeomorphism $h:(0,\infty)\rightarrow\mathbb{R}$ as in the proof of Assertion 1. By definition, the composition $\beta_{\mathbb{C}}\circ\psi$ is

C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1}\xrightarrow{h^{*}}C_{0}((0,\infty))\hat{\otimes}Cl^{0,1}\xrightarrow{}\mathcal{S}\xrightarrow{\beta_{\mathbb{C}}}C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1},

where the unlabelled arrow sends

f\hat{\otimes}1\longmapsto x\mapsto f(|x|),\penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ f\hat{\otimes}\varepsilon\longmapsto x\mapsto s(x)f(|x|),

where $s(x):\mathbb{R}\setminus\{0\}\rightarrow\{-1,1\}$ is the sign function¹⁷¹⁷ 17 strictly speaking these are the formulae when $x\neq 0$ . At $x=0$ we by default assign the value $0$ . We also do this at certain places later without explicitly mentioning. (this function is nothing but the inverse to the isomorphism in Equation 2.4 postcomposed with the inclusion of $J$ into $\mathcal{S}$ ). The whole composite therefore sends

f\hat{\otimes}1\longmapsto f\circ h\circ|\cdot|\hat{\otimes}1,\penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ f\hat{\otimes}\varepsilon\longmapsto(s\cdot f\circ h\circ|\cdot|)\hat{\otimes}\varepsilon.

From the description of the Bott map as in the proof of Assertion 4, the induced map

\displaystyle\operatorname{Hom}(\mathcal{S},C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1})

\displaystyle\xrightarrow{(\beta_{\mathbb{C}}\circ\psi)_{*}}\operatorname{Hom}(\mathcal{S},C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1})

sends the Bott map to the homomorphism

(f\longmapsto f^{\operatorname{ev}}\circ h\circ|\cdot|)\hat{\otimes}1+(s\cdot f^{\operatorname{odd}}\circ h\circ|\cdot|)\hat{\otimes}\varepsilon.

Upon identifying

\operatorname{Hom}(\mathcal{S},C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1})\cong\operatorname{Map}_{*}(\mathbb{R}^{+},\operatorname{Hom}(\mathcal{S},Cl^{0,1})),

the Bott map therefore gets mapped to

x\longmapsto f\mapsto(f^{\operatorname{ev}}(h(|x|))\cdot 1+s(x)f^{\operatorname{odd}}(h(|x|))\cdot\varepsilon).

But this is same as the function

	$\displaystyle\mathbb{R}^{+}$	$\displaystyle\rightarrow\operatorname{Hom}(\mathcal{S},Cl^{0,1})$
	$\displaystyle x$	$\displaystyle\longmapsto f\mapsto\begin{cases}f^{\operatorname{ev}}(h(x))\cdot 1+f^{\operatorname{odd}}(h(x))\cdot\varepsilon\penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \text{if}\penalty\ x>0\\ f^{\operatorname{ev}}(h(-x))\cdot 1-f^{\operatorname{odd}}(h(-x))\cdot\varepsilon\penalty\ \penalty\ \penalty\ \penalty\ \text{if}\penalty\ x<0\\ 0\penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \text{if}\penalty\ x=0,\infty\end{cases}$

Under the identification

\mathbb{R}^{+}\cong\operatorname{Hom}(\mathcal{S},Cl^{0,1})

from Equation 2.5, this corresponds to the function

	$\displaystyle h^{\prime}:\mathbb{R}^{+}$	$\displaystyle\rightarrow\mathbb{R}^{+}$
	$\displaystyle x$	$\displaystyle\mapsto\begin{cases}h(x)\penalty\ \text{if}\penalty\ x>0\\ -h(-x)\penalty\ \text{if}\penalty\ x<0\\ \infty\penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \text{if}\penalty\ x=0,\infty\end{cases}$

The local degree formula yields that the degree of $h^{\prime}$ is $2$ . The proof is now completed by the following argument: consider the diagram

where the bottom horizontal arrow is defined so as to make the diagram commute. Therefore the bottom horizontal arrow sends

\operatorname{id}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}

h^{\prime}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+},

since we showed above that the right-down itinerary maps $\beta_{\mathbb{C}}$ to $h^{\prime}$ , and the left vertical composite identifies $\beta_{\mathbb{C}}$ and $\operatorname{id}$ , as was observed in the proof of Assertion 4. Thus, the map induced by $\beta_{\mathbb{C}}\circ\psi$ on $\hat{K}_{0}(C_{0}(\mathbb{R},Cl^{0,1}))$ is multiplication by the degree of $h^{\prime}$ , which is equal to $2$ .

For Assertion 7, we consider the commutative diagram (we denote by $\alpha$ the grading on the respective algebras)

where the the top horizontal map is negation from Assertion 4, and the bottom horizontal map is the identity since $\mathbb{C}$ has the trivial grading. The commutativity of the top rectangle yields that

\alpha_{*}(u)=-u.

Understanding $\alpha_{*}(e)$ takes a bit more work. From Assertion 5 and the commutativity of the lower rectangle in the above diagram, we can say that

\alpha_{*}(e)=e+nu

for some $n\in\mathbb{Z}$ . To compute the value of $n$ , we consider the commutative diagram

From Assertion 4 and Assertion 6, we can say that $\beta_{*}(e)=[\beta_{\mathbb{C}}]$ and $\beta_{*}(u)=2[\beta_{\mathbb{C}}]$ . Tracing the fate of $e$ under the two itineraries from $\hat{K}_{0}(\mathcal{S})$ to $\hat{K}_{0}(C_{0}(\mathbb{R})\hat{\otimes}Cl^{0,1})$ , we obtain the equation

[\beta_{\mathbb{C}}]+2n\cdot[\beta_{\mathbb{C}}]=-[\beta_{\mathbb{C}}],

which shows

n=-1.

This finishes the proof of Assertion 7.

To prove Assertion 8, we note that Assertion 7 shows (in light of Assertion 5) that $(e,\alpha_{*}e)$ is also a $\mathbb{Z}$ -basis of $\hat{K}_{0}(\mathcal{S})$ , and the grading $\alpha_{\mathcal{S}}$ evidently swaps the basis elements. ∎

3 Equivariant (ungraded) Real $KK_{!}$ -theory

Throughout this section, we work with Real $C^{*}$ -algebras as always. Recall that $!\in\{\operatorname{se,ex}\}$ denotes the type of excision we have in our bivariant $K$ -theory, where $\operatorname{se}$ stands for semi-split and $\operatorname{ex}$ stands for exact. In the remainder of this article, we use the superscript “sep” to denote separable versions of the categories described in Section 2.

In this section, we will first define a small stable $\infty$ -category $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ as a series of Dwyer-Kan localizations of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ . Then, we take ind-completions to get a presentable, stable $\infty$ -category $\mathrm{KK}^{G}_{!,\infty}$ together with a universal functor $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}\rightarrow\mathrm{KK}^{G}_{!,\infty}$ . We devote the next few subsections to analyzing each step of the localization. Before proceeding, we remind the readers that $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ is an essentially small category.

3.1 Inverting homotopy equivalences

In the first step, we perform the Dwyer-Kan localization of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ at the class of ( $G$ -equivariant) homotopy equivalences. We begin by recalling the relevant definitions.

Definition 3.0.

1.

Two maps $f_{0},f_{1}:A\rightarrow B$ in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ are said to be homotopic if there exists a morphism $F:A\rightarrow C([0,1],B)$ such that $\operatorname{ev}_{i}\circ F=f_{i}$ for $i=0,1$ .
2.

Let $f:A\rightarrow B$ be a morphism in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ . $f$ is said to be a homotopy equivalence if there exists a morphism $g:B\rightarrow A$ in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ such that the two composites are homotopic to the respective identity maps.

We denote by $\mathcal{W}_{h}$ the (essentially small) class of homotopy equivalences in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ .

Definition 3.0.

Denote by ${\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}}$ the Dwyer-Kan localization of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ at $\mathcal{W}_{h}$ , and denote by

L_{h}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}

the localization functor.

By definition, this means for any $\infty$ -category $\mathcal{D}$ , we have an equivalence of functor categories

\mathbf{Fun}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{h}^{*}}\mathbf{Fun}^{h}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}),

(3.1)

where the right-hand side denotes the full subcategory of $\mathbf{Fun}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D})$ spanned by functors which sends homotopy equivalences in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ to equivalences in $\mathcal{D}$ .

To phrase the relevant properties of the $\infty$ -category $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ , we need the notion of Schochet fibrations, defined first in [52]. Recall that the category $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ is topological enriched, as mentioned in Section 2. Although it is not necessary for us, the basic definitions and properties of Schochet fibrations also work verbatim in the category $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ .

Definition 3.1.

1.

A morphism $f:A\rightarrow B$ in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ is said to be a Schochet fibration if for all $D\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , the induced map of topological spaces

$\underline{\operatorname{Hom}}(D,A)\xrightarrow{f_{*}}\underline{\operatorname{Hom}}(D,B)$

is a Serre fibration.
2.

A cartesian square in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$

is called Schochet fibrant if $f$ is a Schochet fibration.
3.

For an $\infty$ -category $\mathcal{D}$ , a functor $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathcal{D}$ will be called Schochet exact if it maps Schochet fibrant cartesian squares to cartesian squares. We denote by

$\mathbf{Fun}^{Sch}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D})$

the full subcategory of $\mathbf{Fun}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D})$ spanned by functors which are Schochet exact.

Example 3.1.

We present a few important examples of Schochet fibrations which will be required later on. For proofs see the relevant sections in [52] and [17].

1.

For any $C^{*}$ -algebra $A$ , the canonical map $C([0,1],A)\xrightarrow{\operatorname{ev}_{0}}A$ is a Schochet fibration (and a homotopy equivalence). $C([0,1],A)$ is also called the path algebra of $A$ , and denoted $PA$ .
2.

Pullback of a Schochet fibration along any $*$ -homomorphism is a Schochet fibration. Given a $*$ -homomorphism $f:B\rightarrow C$ , denote by $Z(f)$ the pullback $B\times_{C}PC$ . $Z(f)$ is called the mapping cylinder of $f$ , and the canonical map $Z(f)\rightarrow B$ is a Schochet fibration (and a homotopy equivalence).

Concretely, we have

$Z(f)=\{(b,\gamma)\mid b\in B,\gamma\in PC,\gamma(0)=f(b)\}.$

There is also a canonical map $\operatorname{ev}_{1}:Z(f)\rightarrow C$ which is also a Schochet fibration. By definition, the mapping cone of $f$ fits in the short exact sequence

$0\rightarrow C(f)\rightarrow Z(f)\xrightarrow{\operatorname{ev}_{1}}C\rightarrow 0.$

This sequence is furthermore semi-split and is called the mapping cone sequence. It will be extremely important for us in Section 3.3.

In the special case that $f$ is an inclusion, $Z(f)$ is isomorphic to all paths in $C$ which start at $B$ .
3.

The maximal tensor product with an auxilliary algebra of a Schochet fibrant cartesian square remains a Schochet fibrant cartesian square. The minimal tensor product with an auxilliary algebra of of a mapping cone sequence remains a mapping cone sequence.

The following proposition discusses properties of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ and $L_{h}$ .

Proposition 3.1.

1.

The $\infty$ -category $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ is locally small, and is canonically equivalent to the $\infty$ -category presented by the Kan-enriched¹⁸¹⁸ 18 we implicitly always view a topologically enriched category as a Kan enriched category category $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , i.e., by taking its homotopy coherent nerve.
2.

$\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ is pointed and $L_{h}$ is reduced, i.e. maps the zero algebra to the zero object in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ .
3.

$\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ has finite limits and countable coproducts. $L_{h}$ preserves finite products and countable coproducts.

For every left-exact $\infty$ -category $\mathcal{D}$ , the equivalence in Equation 3.1 restricts to an equivalence

\mathbf{Fun}^{\operatorname{lex}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{h}^{*}}\mathbf{Fun}^{h,Sch}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}),

where the left-hand entity denotes the full subcategory of $\mathbf{Fun}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}},\mathcal{D})$ spanned by left-exact functors.

$L_{h}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ has a symmetric monoidal refinement for both the minimal and maximal tensor products on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ . Moreover, the functors

\otimes_{?}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\times\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\rightarrow\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}

are bi-left exact for $?\in\{\operatorname{min,max}\}$ .

For every left-exact and symmetric monoidal $\infty$ -category $\mathcal{D}$ and $?\in\{\min,\max\}$ , the pullback along the symmetric monoidal refinement of $L_{h}$ induces an equivalence

\mathbf{Fun}^{\operatorname{lex}}_{\otimes_{?}/\operatorname{lax}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{h}^{*}}\mathbf{Fun}^{h,Sch}_{\otimes_{?}/\operatorname{lax}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}),

where the subscript $\otimes_{?}/\operatorname{lax}$ indicates two separate formulas, one for symmetric monoidal functors and the other for lax monoidal functors.

Proof.

The proofs of [17, Section 3] can be readily adapted to this situation. The Real structure poses no extra difficulty. Note that the key point for the left-exact assertions is that the category of separable $G$ - $C^{*}$ -algebras becomes a category of fibrant objects with fibrations as the Schochet fibrations and weak equivalences as the homotopy equivalences. ∎

Remark 3.1.

For later use, we explicitly mention an easy corollary of Section 3.1.4: we have an equivalence of functors

L_{h}\circ(C_{0}(\mathbb{R}^{0,1},-))\simeq\Omega\circ L_{h}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}.

The details are exactly the same as in [17, Example 3.19].

We also mention at this juncture that all the subsequent localizations leading to $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ will be left-exact Dwyer-Kan localizations. Hence we can take $C_{0}(\mathbb{R}^{0,1},-)$ (descended to the category under consideration) as a model for the loops functor in each of the subsequent localizations.

We end this subsection by a casting the idea of the proof of Section 2.1 using the language of fiber sequences.

Proposition 3.1.

There exists a fiber sequence in $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ of the form

\mathcal{S}\xrightarrow{\operatorname{ev}_{0}}\mathbb{C}\xrightarrow{\operatorname{\iota}}Cl^{0,1},

(3.2)

where $\iota$ is the canonical inclusion of $\mathbb{C}\cong Cl^{0,0}$ into $Cl^{0,1}$ .

Proof.

This is immediate from Equation 2.1 and Section 3.1. ∎

3.2 Enforcing $\mathcal{K}_{G}$ -stability

At the second step, we want to enforce a equivariant form of stability in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ . The idea is to enforce that the map induced on the $G$ - $C^{*}$ -algebras of compact operators by isometric inclusions of $G$ -Hilbert spaces become equivalences. We begin with a couple of definitions.

Definition 3.2.

•

A morphism $f:A\rightarrow B$ in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ is said to be a ${\mathcal{K}_{G}}$ -equivalence if the induced map

$f\otimes 1_{{\mathcal{K}_{G}}}:A\otimes{\mathcal{K}_{G}}\rightarrow B\otimes{\mathcal{K}_{G}}$

is a homotopy equivalence.
•

For any $\infty$ -category $\mathcal{D}$ , a functor $F:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathcal{D}$ is said to be $G$ -stable if for every isometric embedding of separable $G$ -Hilbert spaces $V^{\prime}\hookrightarrow V$ , with $V^{\prime}\neq 0$ , and for all objects $A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , the induced map $F(A\otimes\mathcal{K}(V))\rightarrow F(A\otimes\mathcal{K}(V^{\prime}))$ is an equivalence.

Remark 3.2.

It is easy to see that a homotopy invariant functor on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ inverts ${\mathcal{K}_{G}}$ -equivalences if and only if is $G$ -stable. See [47, Proposition 2.2] for a proof. It so turns out that inverting the $\mathcal{K}_{G}$ -equivalences enables one to control the mapping spaces in the localization, whereas to verify that a homotopy invariant functor indeed inverts $\mathcal{K}_{G}$ -equivalences, one usually just checks $G$ -stability.

We denote by $\mathcal{W}_{{\mathcal{K}_{G}}}$ the (essentially small) class of $\mathcal{K}_{G}$ -equivalences.

Definition 3.2.

Denote by $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ the Dwyer-Kan localization of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ at $L_{h}(\mathcal{W}_{\mathcal{K}_{G}})$ , and denote by

L_{{\mathcal{K}_{G}}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\rightarrow\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}

the localization functor.

Note that this means that the canonical functor

L_{h,{\mathcal{K}_{G}}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\xrightarrow{L_{h}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\xrightarrow{L_{{\mathcal{K}_{G}}}}\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}

is a Dwyer-Kan localization of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ at $\mathcal{W}_{h}\cup\mathcal{W}_{{\mathcal{K}_{G}}}$ , whence we have an equivalence of functor categories

\mathbf{Fun}(\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{h,{\mathcal{K}_{G}}}^{*}}\mathbf{Fun}^{h,{\mathcal{K}_{G}}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}),

(3.3)

Proposition 3.3.

1.

The $\infty$ -category $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ is canonically equivalent to the full-subcategory of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ spanned by ${\mathcal{K}_{G}}$ -stable algebras. When $G$ is compact, the functor $L_{{\mathcal{K}_{G}}}$ is in fact the left-adjoint of a left Bousfield localization.
2.

$\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ has finite limits and countable coproducts. $L_{h,{\mathcal{K}_{G}}}$ preserves finite products and countable coproducts. $L_{\mathcal{K}_{G}}$ is left-exact.

For every left-exact $\infty$ -category $\mathcal{D}$ , the equivalence in Equation 3.3 restricts to an equivalence

\mathbf{Fun}^{\operatorname{lex}}(\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{h,{\mathcal{K}_{G}}}^{*}}\mathbf{Fun}^{h,{\mathcal{K}_{G}},Sch}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}).

(3.4)

4.

Countable coproducts in $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ are given by countable direct sums of the respective $C^{*}$ -algebras. The canonical map from a binary (resp. countable) coproduct to a binary (resp. countable) product in $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ is an equivalence, whence $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ is a semi-additive $\infty$ -category.

For every left-exact semi-additive $\infty$ -category $\mathcal{D}$ with countable coproducts, the equivalence in Equation 3.4 restricts to an equivalence

\mathbf{Fun}^{\operatorname{lex},\coprod_{\omega}}(\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{h,{\mathcal{K}_{G}}}^{*}}\mathbf{Fun}^{h,{\mathcal{K}_{G}},Sch,\oplus_{\omega}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}),

(3.5)

where the superscripts $\coprod_{\omega},\oplus_{\omega}$ denote countable coproduct and countable sum preserving functors, respectively.

$L_{\mathcal{K}_{G}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\rightarrow\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ has a symmetric monoidal refinement for both the tensor products on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ . Consequently the same assertion is also true for $L_{h,{\mathcal{K}_{G}}}$ . Moreover, the functors

\otimes_{?}:\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\times\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\rightarrow\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}

are bi-left exact and bi-countable coproduct preserving for $?\in\{\operatorname{min,max}\}$ .

For every symmetric monoidal left-exact semi-additive $\infty$ -category $\mathcal{D}$ with countable coproducts, the pullback along the symmetric monoidal refinement of $L_{h,{\mathcal{K}_{G}}}$ induces an equivalence

\mathbf{Fun}^{\operatorname{lex},\coprod_{\omega}}_{\otimes_{?}/\operatorname{lax}}(\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{h,{\mathcal{K}_{G}}}^{*}}\mathbf{Fun}^{h,{\mathcal{K}_{G}},Sch,\oplus_{\omega}}_{\otimes_{?}/\operatorname{lax}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}).

Proof.

This follows by combining the approaches in [17, Section 4], [11, Remark 3.34] and [25, Lemma 3.18, Corollary 3.19]. The Real structure poses no extra difficulty. ∎

3.3 Enforcing $!$ -excision

Let

be an extension of algebras in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , where the cpc-split $s$ exists if we are dealing with the case $!=\operatorname{se}$ . From this extension, we form the natural diagram

(3.6)

We now explain the algebras and the morphisms in the above diagram. The middle row is the mapping cone extension of $f$ as described in Section 3.1.2. $h_{f}$ is the map $b\mapsto(b,\mathrm{const}_{f(b)})$ , and $\iota_{f}$ is the map $a\mapsto(a,0)$ (without loss of generality, we are assuming that $A$ is an ideal in $B$ ). $\pi_{f}:C(f)\rightarrow Q(f)$ is the cokernel of $\iota_{f}$ . Note that we have an isomorphism

Q(f)\cong\{(c,\gamma)\in C\times PC\mid\gamma(0)=c,\gamma(1)=0\}\cong C_{0}([0,1),C),

and therefore $Q(f)$ is contractible.

We now note that if $s$ is a cpc-split of $f$ , then it induces natural cpc-splits $\tilde{s}$ and $\hat{s}$ of $\tilde{f}$ and $\pi_{f}$ , given by $\tilde{s}(c)=(s(c),\mathrm{const}_{c})$ and $\hat{s}(c,\gamma)=(s(c),\gamma)$ .

We denote by $\hat{W_{!}}$ the (essentially small) class of morphisms $L_{h,{\mathcal{K}_{G}}}(\iota_{f})$ for all $!$ -exact sequences. We then define $W_{!}$ to be the closure of $\hat{W_{!}}$ inside $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ under pullbacks, the 2-out-of-3 property and countable sums. Note that ${W_{!}}$ is also essentially small.

Definition 3.6.

Denote by $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ the Dwyer-Kan localization of $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ at $W_{!}$ , and denote by

L_{!,\oplus_{\omega}}:\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\rightarrow\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}

the localization functor.

Note that the composite

L_{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\xrightarrow{L_{h}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\xrightarrow{L_{\mathcal{K}_{G}}}\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\xrightarrow{L_{!,\oplus_{\omega}}}\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}

is a Dwyer-Kan localization of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , as can be seen from the proof of [11, Proposition 3.40]. Before describing the key properties of this localization, we mention the motivation for closing $\hat{W}_{!}$ under countable sums.

Remark 3.6.

It is a non-trivial property of classical $KK$ - and $E$ -categories for separable $C^{*}$ -algebras that they have countable coproducts, which are furthermore modeled by countable direct sums of the respective $C^{*}$ -algebras. If we do not close up $\hat{W}_{!}$ under countable sums, then there is no formal reason why our to-be-constructed $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ category will have countable coproducts (see the next paragraph too). The issue therefore arises when we want to extend $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ to the large category $\mathrm{KK}^{G}_{!,\infty}$ on all of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ : on the one hand, we can simply consider $\operatorname{Ind}_{\aleph_{0}}$ -completion of $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ , but then we would destroy the countable coproducts which already exist in $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ (since a posteori the homotopy category of $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ is equivalent to the classical bivariant $K$ -categories), and would therefore be an unnatural object to consider. On the other hand, our current approach (as done in Section 3.7) of taking $\operatorname{Ind}_{\aleph_{1}}(\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}})$ would a priori have no reason to have countable coproducts, but would have all uncountable filtered colimits, and would be yet another unnatural object! Of course, if we already know the comparison between the homotopy category of $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ and the classical bivariant $K$ -categories, then the $\operatorname{Ind}_{\aleph_{1}}$ -completion of $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ works out fine and becomes a presentable stable $\infty$ -category, but we want to postpone this comparison until the very end. Thus, it calls for enforcing that our $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ categories have countable coproducts.

Having said that, we mention that the step of closing up $\hat{W}_{!}$ under countable sums is only relevant for non-separable algebras (and even then, more for the aesthetic features of our large categories). In particular, if the reader is only interested in separable $C^{*}$ -algebras, then this step can be omitted, and one would still get a stable $\infty$ -categorical refinement of the classical bivariant $K$ -categories for separable $C^{*}$ -algebras which would a fortiori have countable coproducts.

Proposition 3.6.

1.

The $\infty$ -category $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ is left-exact, semi-additive and posseses countable coproducts.
2.

The localization functor $L_{!,\oplus_{\omega}}$ is left-exact and countable sum preserving. The functor $L_{h,{\mathcal{K}_{G}},!}$ is $!$ -exact.

For every left-exact semi-additive $\infty$ -category $\mathcal{D}$ admitting countable coproducts, we have equivalences

$\displaystyle\mathbf{Fun}^{\operatorname{lex},\coprod_{\omega}}(\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{!,\oplus}^{*}}\mathbf{Fun}^{\operatorname{lex},\coprod_{\omega},\hat{W}_{!}}(\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{h,{\mathcal{K}_{G}}}^{*}}\mathbf{Fun}^{{h,{\mathcal{K}_{G}},!+Sch,\oplus}_{\omega}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}),$

(3.7)

where the superscript $\hat{W}_{!}$ in the middle entity stands for functors on $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ which map morphisms in $\hat{W}_{!}$ to equivalences, and the superscript $!$ in the last entity stands for functors on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ which map $!$ -exact sequences to fiber sequences.

When $!=\operatorname{se}$ ,

L_{\operatorname{se},\oplus_{\omega}}:\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\rightarrow\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,\mathrm{se},\oplus_{\omega}}^{\mathrm{sep}}

has a symmetric monoidal refinement for both the tensor products on $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ . When $!=\operatorname{ex}$ ,

L_{\operatorname{ex},\oplus_{\omega}}:\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\rightarrow\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,\mathrm{ex},\oplus_{\omega}}^{\mathrm{sep}}

has a symmetric monoidal refinement for only the maximal tensor product on $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ . Consequently, the same assertions are true for $L_{h,{\mathcal{K}_{G}},\operatorname{se},\oplus_{\omega}}$ and $L_{h,{\mathcal{K}_{G}},\operatorname{ex},\oplus_{\omega}}$ .

Moreover, the functors

\otimes_{?}:\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}\times\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}\rightarrow\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}

are bi-left exact and bi-countable coproduct preserving (for every choice of $(?,!)$ except $(\operatorname{min,ex})$ ).

For every symmetric monoidal left-exact semi-additive $\infty$ -category $\mathcal{D}$ admitting countable coproducts, the pullback along the symmetric monoidal refinement of $L_{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}$ (as discussed in the previous assertion) induces an equivalence

\mathbf{Fun}^{\operatorname{lex},\coprod_{\omega}}_{\otimes_{?}/\operatorname{lax}}(\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}^{*}}\mathbf{Fun}^{{h,{\mathcal{K}_{G}},!+Sch,\oplus_{\omega}}}_{\otimes_{?}/\operatorname{lax}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}).

(3.8)

Proof.

This follows by combining the approaches in [17, Section 5] and [11, Remark 3.35]. Note that closing up $\hat{W}_{!}$ under countable sums has only been considered in [11], and an argument for the localization $L_{!,\oplus_{\omega}}$ to be countable coproduct preserving is omitted in loc. cit. This however follows from a dual version of [49, Theorem 2.6.4]¹⁹¹⁹ 19 note that loc. cit. is written with cofibration categories, whereas our current business is with fibration categories. However, loc. cit. uses [18, Proposition 7.7.1] in the proof, which is indeed written with fibration categories.. ∎

We end this subsection with an important lemma, which will be useful in formulating the universal property of $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ in the next section. The lemma can be considered as an automatic Schochet-exactness result, which is ubiquitous for applications since Schochet exactness of a functor can be very difficult to verify.

Lemma 3.8.

Any homotopy invaraint and $!$ -exact functor from $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ to a stable $\infty$ -category is automatically Schochet exact.

Proof.

The proof can be taken directly from [17, Lemma 5.2]. ∎

3.4 Inverting Bott

This subsection is going to somewhat different than the corresponding step in the complex theory as developed in [17, Sections 6,7]. Indeed, in the complex case, an analysis of the Toeplitz extensions

0\rightarrow\mathcal{K}(l^{2}\mathbb{N})\otimes B\rightarrow\mathcal{T}_{0}\otimes B\rightarrow C_{0}(\mathbb{R})\otimes B\rightarrow 0

²⁰²⁰ 20 That the Toeplitz algebra is nuclear can be deduced from nuclear

C^{*}

-algebras being closed under extensions, see [7, Section IV.3.1] for details.

yields a boundary map

C_{0}(\mathbb{R}^{2})\otimes B\xrightarrow{\beta_{!,B}}B

in (the complex analog of) $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ , which subsequently are inverted yielding the stable $\infty$ -category $\mathrm{KK}^{G,\mathrm{sep}}_{!,\infty,\mathbb{C}}$ (check loc. cit. for details). This also shows that $\Omega$ is a self-inverse endofunctor on $\mathrm{KK}^{G,\mathrm{sep}}_{!,\infty,\mathbb{C}}$ .

In the Real case, one has to work with the Real version of the Toeplitz extension. We discuss it next. Before proceeding, we setup some notation.

Notation 3.8.

Let $\mathbb{R}^{p,q}$ denote the Real space $\mathbb{R}^{p+q}$ , with the involution $\kappa_{p,q}$ given by

\kappa_{p,q}(x_{1},x_{2},...,x_{p+q})=(-x_{1},-x_{2},...,-x_{p},x_{p+1},x_{p+2},...,x_{p+q}).

Denote by $\mathbb{S}^{p,q}$ the unit sphere in $\mathbb{R}^{p,q}$ with the induced involution.

The following result is going to be our workhorse for the Real Bott periodicity.

Proposition 3.8.

There exists a semi-split extension of Real $C^{*}$ -algebras

0\rightarrow\mathcal{K}(l^{2}\mathbb{N})\rightarrow\mathcal{T}^{R}\rightarrow C(\mathbb{S}^{1,1}),

where $\mathcal{T}^{R}$ is the Real isometry classfier $C^{*}$ -algebra, i.e., there is a distinguised Real isometry $v$ in $\mathcal{T}^{R}$ such that for any unital (non-equivariant) Real $C^{*}$ -algebra $A$ , there is a bijection of sets

{\operatorname{Hom}}_{u}(\mathcal{T}^{R},A)\xrightarrow{\cong}\operatorname{Isom}(A),\penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \varphi\mapsto\varphi(v)

between the set of unital, Real homomorphisms from $\mathcal{T}^{R}$ to $A$ and the set of Real isometries in $A$ .

Before proving the above result, we begin with a quick justification that $\mathcal{T}^{R}$ indeed exists, and also set up some notation along the way.

Lemma 3.8.

The Real $C^{*}$ -algebra $\mathcal{T}^{R}$ as defined in Section 3.4 exists.

Proof.

Recall that the complex Toeplitz algebra $\mathcal{T^{\mathbb{C}}}$ , given by the $C^{*}$ -subalgebra generated by the unilateral shift (denoted $\boldsymbol{\iota}$ from now on) in $\mathcal{B}(l^{2}(\mathbb{N}))$ , is the universal (complex) isometry classifier ([19]). It therefore follows that one model for $\mathcal{T}^{R}$ is $\mathcal{T}^{\mathbb{C}}$ equipped with the unique Real structure such that the unilateral shift is fixed by it. It is also easily seen to be the same as the Real $C^{*}$ -subalgebra of $\mathcal{B}(l^{2}(\mathbb{N}))$ generated by the unilateral shift, where the Real structure on the Hilbert space $l^{2}(\mathbb{N})$ is by complex conjugation (as always). ∎

We now show that $C(\mathbb{S}^{1,1})$ is also a universal Real $C^{*}$ -algebra.

Lemma 3.8.

$C(\mathbb{S}^{1,1})$ is the Real unitary classifier $C^{*}$ -algebra.

Proof.

Again as before, the complex version of this result is well-known, see for example [56, Proposition 2.3.4]. It follows that the Real structure on the $C^{*}$ -algebra of continuous functions on the circle should be so defined that the identity function on the circle is a Real unitary. Since the Real structure on the codomain $\mathbb{C}$ is complex conjugation, it directly follows from the definitions that the Real structure stemming from the Real space $\mathbb{S}^{1,1}$ is indeed the right one. ∎

Proof (of Section 3.4).

The morphism $\mathcal{T}^{R}\xrightarrow{}C(\mathbb{S}^{1,1})$ is given by mapping $\boldsymbol{\iota}$ (in the concrete model of $\mathcal{T}^{R}$ mentioned in Section 3.4) to the inclusion $\mathbb{S}^{1,1}\hookrightarrow\mathbb{C}$ . The kernel of $\varphi$ contains

\boldsymbol{\iota}^{n}(1_{l^{2}\mathbb{N}}-\boldsymbol{\iota}\boldsymbol{\iota}^{*}){\boldsymbol{\iota}^{*}}^{m}\penalty\ \forall\penalty\ n,m\geq 0,

and therefore also contains the Real $C^{*}$ -subalgebra $\mathcal{K}(l^{2}\mathbb{N})$ of $\mathcal{B}(l^{2}(\mathbb{N}))$ . It is well-known that upon forgetting the Real structure, the kernel of $\varphi$ is precisely $\mathcal{K}(l^{2}\mathbb{N})$ , see for example [56, Section 2.4]. This shows the existence of the extension.

To show that it also admits a Real cpc-split, one directly checks that the cpc-split described in [17, Remark 6.7] is a Real map. ∎

Corollary 3.8.

There exists a semi-split extension of Real $C^{*}$ -algebras

0\rightarrow\mathcal{K}(l^{2}\mathbb{N})\rightarrow\mathcal{T}^{R}_{0}\rightarrow C_{0}(\mathbb{R}^{1,0})\rightarrow 0.

(3.9)

Given $A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ and a homotopy invariant, ${\mathcal{K}_{G}}$ -stable, split exact functor $F:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathcal{D}$ taking values in the group objects of a semi-additive $\infty$ -category $\mathcal{D}$ , $F(A\otimes\mathcal{T}^{R}_{0})\simeq 0$ .

Proof.

The existence of the semi-split extension readily follows from Section 3.4 and the isomorphism of Real $C^{*}$ -algebras

C_{0}(\mathbb{R}^{1,0})\cong\ker(C(\mathbb{S}^{1,1})\xrightarrow{\operatorname{ev}_{(0,1)}}\mathbb{C}).

The second part of the corollary is proven just as in the complex case, see [17, Section 6] and [53, Corollary 1.5.3]. ∎

Having Section 3.4 at our disposal, we can again emulate the treatment of [17] without the Real structure posing any extra difficulty.

For each $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , Equation 3.9 yields a semi-split extension

0\rightarrow\mathcal{K}(l^{2}\mathbb{N})\otimes B\rightarrow\mathcal{T}_{0}^{R}\otimes B\rightarrow C_{0}(\mathbb{R}^{1,0})\otimes B\rightarrow 0.

Applying the semi-split exact functor $L_{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}$ to these extensions, one obtains a Puppe sequence

$\displaystyle\cdots\rightarrow L_{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}\!\bigl(C_{0}(\mathbb{R}^{1,1})\otimes B\bigr)\xrightarrow{\beta_{!,B}}L_{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}\!\bigl(\mathcal{K}(l^{2}\mathbb{N})\otimes B\bigr)\rightarrow L_{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}\!\bigl(\mathcal{T}_{0}^{R}\otimes B\bigr)\rightarrow L_{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}\!\bigl(C_{0}(\mathbb{R}^{1,0})\otimes B\bigr)$

(3.10)

where $\beta_{!,B}$ are the relevant “boundary maps” in $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ . Using the stability of $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ , we obtain a natural transformation

\beta_{!}:=\beta_{!,\mathbb{C}}\otimes-:C_{0}(\mathbb{R}^{1,1})\otimes-\implies\operatorname{id}_{\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}}:\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}\rightarrow\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}.

The important feature of this natural transformation is the following.

Corollary 3.10.

If $A$ is a group object in $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ , then the component of $\beta_{!}$

\beta_{!,A}:C_{0}(\mathbb{R}^{1,1})\otimes_{!}A\rightarrow A

is an equivalence.

Proof.

The proof of [17, Corollary 6.11] will work with the obvious modifications. ∎

We are now in a position to define the small stable $\infty$ -category $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ . We denote by $W_{\operatorname{Bott}}$ the (essentially small) class of maps $\beta_{!,B}$ as in Equation 3.10.

Definition 3.10.

Define $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ to be the Dwyer-Kan localization of $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ at $W_{\operatorname{Bott}}$ , and denote by

L_{\operatorname{Bott}}:\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}\rightarrow\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}

the localization functor.

Note that the composite

\mathrm{kk}_{!,\infty}^{G,\mathrm{sep}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\xrightarrow{L_{h}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\xrightarrow{L_{\mathcal{K}_{G}}}\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\xrightarrow{L_{!,\oplus_{\omega}}}\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}\xrightarrow{L_{\operatorname{Bott}}}\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}

(3.11)

is a Dwyer-Kan localization of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , as can be deduced from the proof of [11, Proposition 3.40].

Proposition 3.11.

1.

$\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ is left-exact. $L_{\operatorname{Bott}}$ is also left-exact and has a fully faithful left-adjoint. $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ can be realized as the full subcategory of group objects of $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ .
2.

$\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ is a small stable $\infty$ -category.

$\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ is countably cocomplete. For a countable collection $(B_{n})_{n\in\mathbb{N}}$ of algebras in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , we have an equivalence

\coprod_{n\in\mathbb{N}}\mathrm{kk}_{!,\infty}^{G,\mathrm{sep}}(B_{n})\xrightarrow{\simeq}\mathrm{kk}_{!,\infty}^{G,\mathrm{sep}}(\oplus_{n\in\mathbb{N}}B_{n}).

When $!=\operatorname{ex}$ , $\mathrm{kk}_{!,\infty}^{G,\mathrm{sep}}$ even preserves general countable filtered colimits.

Every left-exact functor from $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ to a left-exact additive $\infty$ -category $\mathcal{D}$ automatically factors (essentially uniquely) through $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ . More precisely, for every left-exact $\infty$ -category $\mathcal{D}$ , we have an equivalence

\mathbf{Fun}^{\operatorname{lex}}(\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{\operatorname{Bott}}^{*}}\mathbf{Fun}^{\operatorname{lex}}(\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}},\mathcal{D}).

For every countably cocomplete stable $\infty$ -category $\mathcal{D}$ , we have an equivalence

\mathbf{Fun}^{\operatorname{colim}_{\omega}}(\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{{\mathrm{kk}_{!,\infty}^{G,\mathrm{sep}}}^{*}}\mathbf{Fun}^{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}),

where the superscript $\operatorname{colim}_{\omega}$ denotes the full subcategory of functors which are countable colimit preserving.

6.

$\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ has a symmetric monoidal refinement for both the tensor products when $!=\operatorname{se}$ , and only for the maximal tensor product when $!=\operatorname{ex}$ .
7.

$L_{\operatorname{Bott}}$ has a symmetric monoidal refinement for the tensor product(s) on $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ . Consequently, the same assertions are true for $\mathrm{kk}_{!,\infty}^{G,\mathrm{sep}}$ . Moreover, the functors

$\otimes_{?}:\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}\times\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}\rightarrow\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$

are bi-countable colimit preserving (for every choice of $(?,!)$ except $(\operatorname{min,ex})$ ).

For every left-exact symmetric monoidal additive $\infty$ -category $\mathcal{D}$ , pullback along the symmetric monoidal refinement(s) of $L_{\operatorname{Bott}}$ induces an equivalence

\mathbf{Fun}^{\operatorname{lex}}_{\otimes_{?}/\operatorname{lax}}(\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{\operatorname{Bott}}^{*}}\mathbf{Fun}^{\operatorname{lex}}_{\operatorname{\otimes_{?}/\operatorname{lax}}}(\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}},\mathcal{D}).

For every symmetric monoidal countably cocomplete stable $\infty$ -category $\mathcal{D}$ , pullback along the symmetric monoidal refinement(s) of $\mathrm{kk}_{!,\infty}^{G,\mathrm{sep}}$ induces an equivalence

\mathbf{Fun}^{\operatorname{colim}_{\omega}}_{\otimes_{?}/\operatorname{lax}}(\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{{\mathrm{kk}_{!,\infty}^{G,\mathrm{sep}}}^{*}}\mathbf{Fun}^{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}_{\otimes_{?}/\operatorname{lax}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}).

Proof.

We give some details in this proof as the result is ubiquitous for what we do later.

Assertion $1$ follows directly from Equation 3.9 and the dual version of [43, Proposition 5.2.7.4], as has been spelled out at the beginning of Section 7 in [17].

We now show Assertion 2. In light of [44, Proposition 1.4.2.27], it suffices to show $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ is small, pointed, left-exact and the loop functor on it is an equivalence. The first three properties are clear in light of Assertion 1 and the work done in the previous three subsections. It remains to see that loops is an equivalence. But this follows from Section 3.4 and the last part of Assertion 1.

We now move onto Assertion 3. By Section 3.3.1, $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}}$ has countable coproducts, whence so does $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ being a Bousfield localization of it (see [45, Subsection 7.1.4]). Any stable $\infty$ -category having countable coproducts is automatically countably cocomplete. That countable coproducts in $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ are modeled by countable direct sum of the algebras follows from Section 3.3.3 and because of general properties of (co)limits in a Bousfield localization; see again [45, Subsection 7.1.4]. The case of $!=\operatorname{ex}$ is the content of [11, Proposition 3.17] (the Real structure poses no additional difficulty).

Assertion 4 follows directly from Section 3.4 and Section 3.4.

Assertion 5 is arguably the most important result from this section for us. It follows from the series of equivalences

\mathbf{Fun}^{\operatorname{colim}_{\omega}}(\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{\operatorname{Bott}}^{*}}\mathbf{Fun}^{\operatorname{lex},\coprod_{\omega}}(\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus}^{\mathrm{sep}},\mathcal{D})\xrightarrow[\simeq]{L_{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}^{*}}\mathbf{Fun}^{{h,{\mathcal{K}_{G}},!+Sch,\oplus_{\omega}}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D}),

where the first equivalence follows from Assertion 4 and (the proof of) Assertion 3, and the second one is Equation 3.7. We now conclude using Section 3.3.

Assertion 6 follows from [35, Proposition 3.3.2] and the basic permeance properties of $!$ -exact sequences with respect to the minimal and maximal tensor products.

The first two parts of Assertion 7 follow immediately from the nuclearity of $\mathcal{K}(l^{2}\mathbb{N}),\mathcal{T}^{R},C(\mathbb{S}^{1,1})$ and [35, Proposition 3.3.2]. For the final part, we use the fact that a left-exact and countable coproduct preserving functor between stable $\infty$ -categories is automatically countable colimit preserving.

For Assertion 8, we build the diagram

The lower horizontal arrow is an equivalence from Section 3.4 and [35, Proposition 3.3.2]. The dashed arrow can be seen to exist using the fact that $L_{\operatorname{Bott}}$ is left-exact. It remains to show that it is an equivalence. For that, we use the following general fact about symmetric monoidal left-exact Dwyer-Kan localizations: for any symmetric monoidal $\infty$ -category $\mathcal{C}$ and an (essentially small) collection of morphisms $W\subseteq\mathcal{C}$ which are a tensor ideal in $\mathcal{C}$ , if the Dwyer-Kan localization of $\mathcal{C}$ at $W$ is left-exact, then we have an equivalence

\mathbf{Fun}^{\operatorname{lex}}_{\otimes_{?}/\operatorname{lax}}(\mathcal{C}[W^{-1}],\mathcal{E})\xrightarrow[\simeq]{L_{W}^{*}}\mathbf{Fun}^{\operatorname{lex},W}_{\otimes_{?}/\operatorname{lax}}(\mathcal{C},\mathcal{E})

for every left-exact symmetric monoidal $\infty$ -category $\mathcal{E}$ . See [18, Proposition 7.5.6] for details.

The proof of Assertion 9 follows in parallel to the proof of Assertion 5, in light of Assertion 8. ∎

We end this subsection by mentioning a couple of important corollaries to the construction of $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ .

Corollary 3.11.

1.

In $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ , we have an equivalence $qA\xrightarrow{\simeq}A$ for all $A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ .
2.

The fiber sequence

$\mathcal{S}\xrightarrow{\operatorname{ev}_{0}}\mathbb{C}\xrightarrow{\iota}Cl^{0,1}$

in $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ as discussed in Section 3.1 remains a fiber sequence in $\mathrm{KK}_{!,\infty}^{G_{2},\mathrm{sep}}$ .
3.

The composite

$(\mathcal{S}\hat{\otimes}Cl^{0,1})_{u}\xrightarrow{{\operatorname{ev}_{0}}_{*}}Cl^{0,1}_{u}\xrightarrow{\pi_{1}}\mathbb{C}$

is an equivalence in $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ .

Proof.

Assertion 1 is the content of [17, Lemma 11.4], the generalization to the equivariant case is routine. Assertion 2 follows directly from Section 3.1, since we have shown before that each of the localizations in Equation 3.11 after $L_{h}$ are left-exact. Assertion 3 follows from Assertion 1 in light of the function algebra model of $q\mathbb{C}$ as described in Section 2.1.4. ∎

Remark 3.11.

We present a more direct proof of Section 3.4.3 in this remark. Consider the ladder of semi-split exact sequences in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$

We have to show that the left vertical arrow is an equivalence in $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ . This will follow once the other two vertical arrows are shown to be equivalences in $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ . Only the case of the middle vertical arrow needs an argument. The fiber of

\{f\in C([0,1],M_{2}(\mathbb{C}))\mid f(0)=\begin{pmatrix}*&0\\ 0&0\end{pmatrix}\penalty\ \&\penalty\ f(1)\penalty\ \text{dia}\}\xrightarrow{(\operatorname{ev}_{0})_{1,1}\oplus(\operatorname{ev}_{1})_{1,1}}\mathbb{C}\oplus\mathbb{C}

is readily seen to be the mapping cone of the right lower corner inclusion $\mathbb{C}\rightarrow M_{2}(\mathbb{C})$ , and hence is contractible in $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ .

We end this subsection by mentioning a $1$ -categorical universal property of $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ . The nature of this universal property is different in flavor than the ones described earlier in this section, and is only needed so as to make the comparison with classical $KK$ - and $E$ -theory possible later. This technical result can be skipped until the reader reaches Section 6.

We need a definition before proceeding. See also [17, Section 13] for some definitions circling half exact additive $1$ -categories.

Definition 3.11.

Denote by $W_{\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}}$ the collection of morphisms inverted by the functor

\mathrm{kk}_{!,\infty}^{G,\mathrm{sep}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}.

Proposition 3.11.

(cf. [17, Proposition 13.13]) Let $\mathcal{D}$ be a half exact additive $1$ -category with countable coproducts. Let $F:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathcal{D}$ be a functor which is homotopy invariant, ${\mathcal{K}_{G}}$ -stable, half exact with respect to $!$ -exact sequences, takes countable direct sums in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ to countable coproducts in $\mathcal{D}$ , and suspension stable (i.e. for every morphism $f:A\rightarrow B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , $F(f)$ is an isomorphism if and only if $F(\Omega(f))$ is an isomorphism). Then $F$ inverts $W_{\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}}$ .

Proof.

Since $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ by construction is a series of Dwyer-Kan localizations of $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , we need to show that we can construct factorizations as indicated by the dashed arrows below

(3.12)

The first two hypothesis on $F$ yields the functor $F^{\prime}$ . Note that $F^{\prime}$ inverts $\hat{W}_{!}$ , as follows from [8, Theorem 21.4.4] (which states that $F$ yields natural long exact sequences) and a standard 5-lemma argument applied to Equation 3.6.

We now claim that $F^{\prime}$ also inverts the closure $W_{!}$ of $\hat{W}_{!}$ under 2-out-of-3, pullbacks, and countable sums. It suffices to show that the collection of morphisms inverted by $F^{\prime}$ is closed under pullbacks and countable sums. The argument for pullbacks can be extracted verbatim from the proof of the analogous assertion in [17, Proposition 13.13]. That the collection of morphisms inverted by $F^{\prime}$ are preserved under countable sums follows immediately from our hypothesis that the natural map

\coprod_{n\in\mathbb{N}}F(A_{n})\xrightarrow{}F(\oplus_{n\in\mathbb{N}}A_{n})

is an isomorphism. Thus, we deduce the existence of $F^{\prime\prime}$ .

The argument for $F^{\prime\prime}$ inverting the maps $\beta_{!,B}$ can be read off from the analogous step in the proof of [17, Proposition 13.13] (with the obvious changes to the Toeplitz extension in the Real case as discussed in Subsection 3.4). This shows the existence of $F^{\prime\prime\prime}$ and the proof is complete. ∎

3.5 Change of group functors

In this subsection, we discuss the several change of group functors connecting the $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ categories we defined earlier. The key features of these functors are that they are defined already at the level of algebras, and fit in useful adjunctions upon passing to the $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ categories. Note that it would also have been possible to develop the (dual) Green-Julg adjunctions in this framework, but we omit it since we do not have to put it to use.

We begin by defining the relevant functors at the level of algebras

Definition 3.12.

Let $H,G$ be locally compact, second countable groups.

Let $\varphi:H\rightarrow G$ be a continuous homomorphism²¹²¹ 21 not necessarily inclusion of a subgroup. Then we have the obvious restriction functor

\operatorname{res}^{G}_{H}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathsf{H}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}.

When $H\subseteq G$ is closed, we define the induction functor

\operatorname{ind}^{G}_{H}:\mathsf{H}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}

as follows:

•

On objects, $\operatorname{ind}^{G}_{H}$ sends $B\in\mathsf{H}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ to the invariant $C^{*}$ -subalgebra of $C_{b}(G,B)$ (the group $G$ acts on $C_{b}(G,B)$ by $(g,f)\longmapsto(g^{\prime}\mapsto f(g^{-1}g^{\prime}))$ ; note that this action is not neccesarily continuous, but it will be so on $\operatorname{ind}^{G}_{H}(B)$ ) generated by the functions $f:G\rightarrow B$ satisfying

f(gh)=h^{-1}f(g)\penalty\ \forall h\in H,\forall g\in G\penalty\ \penalty\ \&\penalty\ \penalty\ \text{projection of}\penalty\ \operatorname{supp}(f)\penalty\ \text{to}\penalty\ G/H\penalty\ \text{is compact}.

•

On morphisms, $\operatorname{ind}^{G}_{H}$ sends a morphism $\varphi:B\rightarrow B^{\prime}$ in $\mathsf{H}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ to

$\operatorname{ind}^{G}_{H}(\varphi):\operatorname{ind}^{G}_{H}(B)\rightarrow\operatorname{ind}^{G}_{H}(B^{\prime})$

given by $(\operatorname{ind}^{G}_{H}(\varphi)(f))(g):=\varphi(f(g))$ .

3.

When $H\subseteq G$ is closed and $G/H$ is compact, then we also call the associated induction functor as the coinduction functor, and is denoted by $\mathsf{coind}^{G}_{H}$ .²²²² 22 the coinduction functor can be defined under more general conditions, but we do not have any reason to do so.

Remark 3.12.

Note that $\operatorname{res}^{G}$ denotes the underlying non-equivariant $C^{*}$ -algebra functor, and $\operatorname{res}_{G}$ denotes the functor which equips a (non-equivariant) $C^{*}$ -algebra with the trivial $G$ -action (we usually omit the trivial group from the notation). Similar comment also applies to the functors $\operatorname{res}^{G_{2}}_{C_{2}}$ and $\operatorname{res}^{C_{2}}_{G_{2}}$ .

Proposition 3.12.

All the functors defined above descend essentially uniquely to the respective equivariant $KK_{!,\infty}^{\operatorname{sep}}$ -categories. Furthermore, the descended functors preserve countable colimits.

Proof.

This can be extracted from [13, Section 4] and [16, Section 2.1]; no comparison with classical bivariant $K$ -theories is made in loc. cit. to show that the functors descend, and the proof therein is based on universal properties which we have enforced by construction. Note that the last assertion follows from the universal property of the equivariant $KK_{!,\infty}^{\operatorname{sep}}$ -category as described in Section 3.4.5. ∎

Remark 3.12.

Note that the restriction functor has a symmetric monoidal refinement for every choice of $(?,!)$ except $(?,!)=(\min,\operatorname{ex})$ , see [13, Corollary 4.8]. The same is not true for the induction or coinduction functors, as can be seen by looking at the unit.

For the next result, we have to provide proofs without banking on classical bivariant $K$ -theories. Compare [16, Example 2.6]. Note that in all the adjunctions in this article, the left-adjoint is written on the left and the right-adjoint is written on the right.

Proposition 3.12.

If $H\subseteq G$ is open²³²³ 23 recall that open subgroups of topological groups are automatically closed, then we have an adjunction

\operatorname{ind}^{G}_{H}:\mathsf{S}_{\mathcal{K}_{H}}\mathsf{H}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\;\leftrightarrows\;\mathsf{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}:\operatorname{res}^{G}_{H}.

Proof.

We begin by the description of the unit and the co-unit.

The unit will be the natural transformation $\varepsilon:\operatorname{id}\rightarrow\mathsf{res}^{G}_{H}\circ\mathsf{ind}^{G}_{H}$ with

	$\displaystyle\varepsilon_{A}:A$	$\displaystyle\rightarrow\mathsf{res}^{G}_{H}\mathsf{ind}^{G}_{H}(A)$
	$\displaystyle a$	$\displaystyle\mapsto\chi_{H}(g)\alpha_{g^{-1}}a=\begin{cases}\alpha_{g^{-1}}a&\text{if}\penalty\ g\in H\\ 0&\text{otherwise}\end{cases}$

²⁴²⁴ 24 this is just notation.

A

is a

H

C^{*}

-algebra, so the action by a general element

g^{-1}

G

doesn’t make sense. However, this will only be considered when

g^{-1}\in H

The description of the co-unit $\eta:\mathsf{ind}^{G}_{H}\circ\mathsf{res}^{G}_{H}\rightarrow\operatorname{id}$ is more complicated. It is easy to see that

\mathsf{ind}^{G}_{H}(\mathsf{res}^{G}_{H}(B))\cong C_{0}(G/H)\otimes B

Upon representing $C_{0}(G/H)$ in $\mathcal{B}(L^{2}(G/H))$ as multiplication operators, $C_{0}(G/H)$ factors through $\mathcal{K}(L^{2}(G/H))$ since $G/H$ is discrete, courtesy $H$ being open in $G$ . We can now define the co-unit maps by

\eta_{B}:\mathsf{ind}^{G}_{H}(\mathsf{res}^{G}_{H}(B))\cong C_{0}(G/H)\otimes B\rightarrow\mathcal{K}(L^{2}(G/H))\otimes B\simeq B.

It remains to check the triangle identities. We have that the composite

\displaystyle\mathsf{res}^{G}_{H}

\displaystyle(B)\xrightarrow{\varepsilon_{\mathsf{res}^{G}_{H}(B)}}\mathsf{res}^{G}_{H}(\mathsf{ind}^{G}_{H}(\mathsf{res}^{G}_{H}(B)))\xrightarrow{\mathsf{res}^{G}_{H}(\eta_{B})}\mathsf{res}^{G}_{H}(B)

maps

	$\displaystyle\mathsf{res}^{G}_{H}(B)\ni b$	$\displaystyle\longmapsto g\mapsto\chi_{H}(g)\beta_{g^{-1}}(b)$
		$\displaystyle\longmapsto gH\mapsto\chi_{H}(g)\beta_{g}\beta_{g^{-1}}(b)$
		$\displaystyle=gH\mapsto\chi_{H}(g)b$
		$\displaystyle\longmapsto\chi_{H}\otimes b\in\mathsf{res}^{G}_{H}(\mathcal{K}(L^{2}(G/H))\otimes B)$
		$\displaystyle\longmapsfrom b,$

where the last map is a left-upper corner inclusion. Note that as a $H$ - $C^{*}$ -algebra, $\mathcal{K}(L^{2}(G/H))$ has the trivial action. It follows that

\mathsf{res}^{G}_{H}(\eta_{B})\circ\varepsilon_{\mathsf{res}^{G}_{H}(B)}\simeq\operatorname{id}_{\mathsf{res}^{G}_{H}(B)}.

The analysis of the other composite is more subtle. We have that the composite

\mathsf{ind}^{G}_{H}(A)\xrightarrow{\mathsf{ind}^{G}_{H}(\varepsilon_{A})}\mathsf{ind}^{G}_{H}(\mathsf{res}^{G}_{H}(\mathsf{ind}^{G}_{H}(A)))\xrightarrow{\eta_{\mathsf{ind}^{G}_{H}(A)}}\mathsf{ind}^{G}_{H}(A)

maps

	$\displaystyle\mathsf{ind}^{G}_{H}(A)\ni(g\mapsto f(g))$	$\displaystyle\quad\longmapsto(g\mapsto(l\mapsto\chi_{H}(l)\alpha_{l^{-1}}f(g)))\in\mathsf{ind}^{G}_{H}(\mathsf{res}^{G}_{H}(\mathsf{ind}^{G}_{H}(A)))$
		$\displaystyle\quad\longmapsto(gH\mapsto(l\mapsto\chi_{H}(g^{-1}l)\alpha_{(g^{-1}l)^{-1}}f(g)))\in C_{0}(G/H,\operatorname{ind}^{G}_{H}(A))$
		$\displaystyle\quad=(gH\mapsto(l\mapsto\chi_{H}(g^{-1}l)f(l)))\in C_{0}(G/H,\operatorname{ind}^{G}_{H}(A))$
		$\displaystyle\quad\xmapsto{(\dagger)}\sum_{kH\in G/H}\chi_{kH}\otimes\chi_{kH}f\in\mathcal{K}(L^{2}(G/H))\otimes\mathsf{ind}^{G}_{H}(A)\penalty\ \text{(see \lx@cref{creftypecap~refnum}{lem: technical manipulation} below)}$

It remains to analyse the fate of $\sum_{kH\in G/H}\chi_{kH}\otimes\chi_{kH}f$ under the equivalences

\mathcal{K}(L^{2}(G/H))\otimes\mathsf{ind}^{G}_{H}(A)\xrightarrow{\simeq}\mathcal{K}(\mathbb{C}\oplus L^{2}(G/H))\otimes\mathsf{ind}^{G}_{H}(A)\xleftarrow{\simeq}\mathsf{ind}^{G}_{H}(A).

We will be done if we can prove the following

Claim.

The two homomorphisms $\mathsf{ind}^{G}_{H}(A)\rightarrow\mathcal{K}(\mathbb{C}\oplus L^{2}(G/H))\otimes\mathsf{ind}^{G}_{H}(A)$ given by

f\mapsto\sum_{kH\in G/H}\chi_{kH}\otimes\chi_{kH}f

and

f\mapsto p\otimes f

are homotopic, where $p\in\mathcal{K}(\mathbb{C}\oplus L^{2}(G/H))$ is the projection onto the summand $\mathbb{C}$ .

Proof.

Let $i:\mathcal{K}(L^{2}(G/H))\xrightarrow{}\mathcal{K}(\mathbb{C}\oplus L^{2}(G/H))$ denote the embedding. Then, $i(\chi_{kH})\in\mathcal{K}(\mathbb{C}\oplus L^{2}(G/H))$ is a (not $G$ -equivariant) one-dimensional projection. It is elementary to see that there is a rank-one partial isometry $u\in\mathcal{K}(\mathbb{C}\oplus L^{2}(G/H))$ such that $upu^{*}=i(\chi_{eH})$ . Defining $u_{k}:=ku$ , we also have $u_{k}pu_{k}^{*}=i(\chi_{kH})$ . Our sought after homotopy is then given by the following one-parameter family of $G$ -equivariant homomorphisms

	$\displaystyle\varphi_{t}:\mathsf{ind}^{G}_{H}(A)$	$\displaystyle\rightarrow\mathcal{K}(L^{2}(G/H))\otimes\mathsf{ind}^{G}_{H}(A)$
	$\displaystyle\quad f$	$\displaystyle\mapsto\underset{kH\in G/H}{\Sigma}(\cos(\frac{\pi}{2}t)^{2}i(\chi_{kH})+\sin(\frac{\pi}{2}t)^{2}p+\cos(\frac{\pi}{2}t)\sin(\frac{\pi}{2}t)(u_{k}+u_{k}^{*}))\otimes\chi_{kH}f.$

∎

This finishes the proof of the proposition. ∎

Remark 3.12.

We give here details of the map labelled $(\dagger)$ in the last proof.

On the one hand, we have

\displaystyle gH\longmapsto

\displaystyle(l\mapsto\chi_{H}(g^{-1}l)f(l))=(l\mapsto\chi_{gH}(l)f(l))

On the other hand, under the identification $C_{0}(G/H)\otimes\operatorname{ind}^{G}_{H}(A)\cong C_{0}(G/H,\operatorname{ind}^{G}_{H}(A))$ , we have

\underset{{kH\in G/H}}{\sum}\chi_{kH}\otimes\chi_{kH}f

corresponds to the map

\displaystyle gH\longmapsto(l\mapsto\underset{kH\in G/H}{\sum}\chi_{kH}(g).\chi_{kH}(l)f(l)).

On a fixed $gH\in G/H$ , both the functions take the value $f(l)$ if $l\in gH$ , and $0$ otherwise.

Theorem 3.13.

Let $H\subseteq G$ be open. Then we have an adjunction

\operatorname{ind}^{G}_{H}:\mathrm{KK}^{H,\mathrm{sep}}_{!,\infty}\;\leftrightarrows\;\mathrm{KK}^{G,\mathrm{sep}}_{!,\infty}:\operatorname{res}^{G}_{H}.

Let $H\subseteq G$ be closed and $G/H$ be compact. Then we have an adjunction

\operatorname{res}^{G}_{H}:\mathrm{KK}^{G,\mathrm{sep}}_{!,\infty}\;\leftrightarrows\;\mathrm{KK}^{H,\mathrm{sep}}_{!,\infty}:\mathsf{coind}^{G}_{H}.

Proof.

The first statement follows directly from Section 3.5 and Section 3.5, since adjunctions descend to a Dwyer-Kan localization if functors do.

The second statement also follows from Section 3.5 since we have already have an adjunction

\operatorname{res}^{G}_{H}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\;\leftrightarrows\;\mathsf{H}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}:\mathsf{coind}^{G}_{H}.

at the level of algebras (see [6, Lemma 4.2]). For our ease of reference later, we mention the unit and counit of this adjunction here:

	$\displaystyle\eta_{A}:$	$\displaystyle A\rightarrow\mathsf{coind}^{G}_{H}\mathsf{res}^{G}_{H}(A),\penalty\ \penalty\ a\mapsto\eta(a):=(G\ni t\mapsto ta\in A)$
	$\displaystyle\epsilon_{B}:$	$\displaystyle\mathsf{res}^{G}_{H}\mathsf{coind}^{G}_{H}(B)\rightarrow B,\penalty\ \penalty\ f\mapsto\epsilon_{B}(f):=f(1).$

∎

3.6 Connection to $K$ -theories

In this subsection, we show that the mapping spaces in $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ can be understood completely when in the first variable we plug in certain “special” algebras. Consequently, we recover the classical as well as graded, $G$ -equivariant $K$ -theory spaces as described in Section 2 as the infinite loop spaces of the mapping spectrum out of $\mathbb{C}$ and $\mathcal{S}$ in $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ respectively; see Theorem 3.14.

Since we do not yet know that $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ is an $\infty$ -categorical refinement of the classical bivariant $K$ -categories, the results of these subsection are crucial for nevertheless being able to make connections to $K$ -theory, which we use heavily in Section 5.

We now introduce the notion of a $!$ -special $C^{*}$ -algebra. In what follows, $G$ is a compact group.

Definition 3.13.

$A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ is called $!$ -special if it satisfies the following three conditions²⁵²⁵ 25 The H in the enumeration below stands for Hypothesis!.

For any $!$ -exact sequence $0\rightarrow B\rightarrow C\rightarrow D\rightarrow 0$ in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , the induced map

\operatorname{Map}_{\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}}(A,B\otimes{\mathcal{K}_{G}})\rightarrow\operatorname{Map}_{\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}}(A,C\otimes{\mathcal{K}_{G}})\rightarrow\operatorname{Map}_{\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}}(A,D\otimes{\mathcal{K}_{G}})

is a fiber sequence in Anima.

2.

The natural map of abelian monoids

$\oplus_{n\in\mathbb{N}}[A,B_{n}\otimes{\mathcal{K}_{G}}]_{G}\rightarrow[A,\oplus_{n\in\mathbb{N}}(B_{n}\otimes{\mathcal{K}_{G}})]_{G}$

is an isomorphism for all countable collection of algebras $\{B_{n}\}_{n\in\mathbb{N}}$ in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ .
3.

The abelian monoid of stable $G$ -homotopy classes of maps $[A,B\otimes{\mathcal{K}_{G}}]_{G}$ is a group for all $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ .

Remark 3.13.

Clearly, $\operatorname{ex-special}\implies\operatorname{se-special}$ . In the algebras we consider, we will be showing that they are $\operatorname{ex-special}$ .

H1 will be satisfied if the induced map of topological spaces

\underline{\operatorname{Hom}}_{G}(A,B)\rightarrow\underline{\operatorname{Hom}}_{G}(A,C)\rightarrow\underline{\operatorname{Hom}}_{G}(A,D)

is a Serre fibration of topological spaces for every $!$ -exact sequence $0\rightarrow B\rightarrow C\rightarrow D\rightarrow 0$ . Indeed, this follows since $-\otimes{\mathcal{K}_{G}}$ is an exact functor on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , and because the $\infty$ -category $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ is presented by the topologically enriched category $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ ; see Section 3.1.1.

Similarly, a sufficient criterion to check H2 is that the functor

[A,-]_{G}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\operatorname{Set}

preserves countable filtered colimits. This is because countable direct sums can be written as a filtered colimit of finite direct sums, and the functor $[A,-]_{G}$ clearly preserves finite direct sums of algebras in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ .

The reason we care about $!$ -special algebras is the following result.

Proposition 3.13.

If $A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ is $\operatorname{!-special}$ , then

The functor

\operatorname{map}_{\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}}(A,-):\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\rightarrow\mathbf{CMon}(\operatorname{An})

takes values in $\mathbf{CGrp}(\operatorname{An})$ .

The functor

F_{A}:=\operatorname{map}_{\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}}(A,-):\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\rightarrow\mathbf{CGrp}(\operatorname{An})

factors through $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ (necessarily essentially uniquely) as the functor

\tau_{\geq 0}\operatorname{map}_{\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}}(A,-):\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}\rightarrow\mathbf{CGrp}(\operatorname{An})

For each $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , there exists a natural equivalence in $\mathbf{CGrp}(\operatorname{An})$

\operatorname{map}_{\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}}(A,B)\xrightarrow{\simeq}\tau_{\geq 0}\operatorname{map}_{\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}}(A,B)

Proof.

Assertion 1 is an immediate consequence of H3 and the fact that being a commutative group is a property of a commutative monoid which is furthermore tested on $\pi_{0}$ .

To show Assertion 2, we first show that we have a factorization of $F_{A}$ as indicated by the dashed arrows below.

To see that $F_{A}^{\prime}$ exists, we bring Section 3.3.3 into play. We need to check that $F_{A}$ is left-exact, inverts $\hat{W}_{!}$ and preserves countable coproducts. Being a corepresentable functor, $F_{A}$ is clearly left-exact. That it inverts $\hat{W}_{!}$ follows directly from H1 in presence of Equation 3.6. That $F_{A}$ also preserves countable coproducts is a consequence of H2, as can be seen from the Whitehead’s theorem for spectra together with the fact that filtered colimits of spectra commute with finite limits. To now see that $F_{A}^{\prime\prime}$ exists, we simply use Section 3.4.4.

That the factorizations $F_{A}^{\prime}$ and $F_{A}^{\prime\prime}$ are essentially unique follows from the fact that $L_{!,\oplus_{\omega}}$ and $L_{\operatorname{Bott}}$ are Dwyer-Kan localizations. That $F_{A}^{\prime}$ and $F_{A}^{\prime\prime}$ are also corepresentable by $A$ follows since the left-Kan extension of a corepresentable functor is corepresentable ([45, Corollary 8.3.1.9]), and in our case, $F_{A}^{\prime}$ and $F_{A}^{\prime\prime}$ are also left-Kan extensions of $F_{A}$ along the localization maps.

Assertion 3 also follows from the previous paragraph. ∎

As an immediate consequence of Assertion 3 of Section 3.6 and the Eckmann-Hilton argument we have the following corollary (cf. [25, Remark 3.24]).

Corollary 3.13.

For any $!$ -special algebra $A$ and any $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , we have an weak homotopy equivalence of spaces

\underline{\operatorname{Hom}}_{G}(A,B\otimes\mathcal{K}_{G})\simeq\Omega^{\infty}\operatorname{map}_{\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}}(A,B).

Moreover, under the above equivalence, the block-sum addition on $[A,B\otimes\mathcal{K}_{G}]_{G}$ coincides with the additive structure on $\pi_{0}(\operatorname{map}_{\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}}(A,B))$ .

The next two lemmas show that $\mathcal{S}$ and $q\mathbb{C}$ are ex-special.

Lemma 3.13.

$\mathcal{S}\in\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ is $\operatorname{ex-special}$ .

Proof.

The necessary verifications are the content of the first three assertions of Theorem 2.3. Note that we are implicitly using that the $C^{*}$ -algebra of compact operators on a Hilbert space is inner graded. ∎

Lemma 3.13.

$q\mathbb{C}\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ is $\operatorname{ex-special}$ .

Proof.

That $q\mathbb{C}$ satisfies H3 follows from the fact that $[q\mathbb{C},B\otimes{\mathcal{K}_{G}}]_{G}$ is naturally isomorphic to the abelian group $K^{G}_{0}(B)$ . Since $K^{G}_{0}(-)$ is known to preserve filtered colimits (for example, by combining Assertion 3 and Assertion 6 of Theorem 2.3), $q\mathbb{C}$ also satisfies H2.

That $q\mathbb{C}$ satisfies H1 can be shown directly, but we instead deduce it from Section 3.6. To that end, we leverage the isomorphism of $C^{*}$ -algebras $q\mathbb{C}\cong(\mathcal{S}\hat{\otimes}Cl^{0,1})_{u}$ as mentioned in Section 2.1.5. The rest follows from the fact that $\mathcal{S}$ satisfies H1 for the group $G_{2}$ and the restriction-coinduction adjunction. In more detail, for any $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , we have natural equivalences

	$\displaystyle\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}(q\mathbb{C},B)$	$\displaystyle\simeq\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\!\left(\mathsf{res}^{G_{2}}_{G}(\mathcal{S}\hat{\otimes}Cl^{0,1}),B\right)$
		$\displaystyle\simeq\mathsf{S}_{\mathcal{K}_{G_{2}}}\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\!\left(\mathcal{S}\hat{\otimes}Cl^{0,1},B\hat{\otimes}Cl^{0,1}\right)$
		$\displaystyle\xrightarrow[\simeq]{\hat{\otimes}Cl^{1,0}}\mathsf{S}_{\mathcal{K}_{G_{2}}}\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}(\mathcal{S},B).$

Note that for the second equivalence, we are using the identification²⁶²⁶ 26 the more well known isomorphism is with the ungraded tensor product, but since $B$ has the trivial $C_{2}$ -action, we can also use the graded tensor product. $\mathsf{coind}^{G_{2}}_{G}B\cong B\hat{\otimes}Cl^{0,1}$ . For the final equivalence, we are also using that the graded tensor products $\hat{\otimes}_{!}$ on $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ also descend to a symmetric monoidal structure on $\mathsf{S}_{\mathcal{K}_{G_{2}}}\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ , which is easy to see since the algebras of continuous functions on the unit interval and compact operators on a Hilbert space are inner graded. ∎

Remark 3.13.

We note that $\mathbb{C}$ is not $!$ -special, as H3 fails for it. We mention without proof that $C_{0}(\mathbb{R}^{0,1})$ is $!$ -special.

Finally, we obtain the following important connection between classical models of equivariant and graded $K$ -theories and the infinite loop spaces of the mapping spectrum in the stable $\infty$ -categories we constructed, as mentioned at the beginning of this subsection.

Theorem 3.14.

We have natural equivalences of spaces for any $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ and $\hat{B}\in\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ :

\mathbb{K}^{G}(B)\simeq\Omega^{\infty}\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}(\mathbb{C},B),\qquad\mathbb{\hat{K}}^{G}(\hat{B})\simeq\Omega^{\infty}\mathrm{KK}_{!,\infty}^{G_{2},\mathrm{sep}}(\mathcal{S},\hat{B}).

Moreover, $K_{0}^{G}(-),\mathrm{KK}_{!,0}^{G,\mathrm{sep}}(\mathbb{C},-):\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\operatorname{Ab}$ are equivalent as lax monoidal functors, where $\mathrm{KK}_{!,0}^{G,\mathrm{sep}}(\mathbb{C},-)$ denotes the homotopy category of $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ .

Proof.

The first two assertions follow from Section 3.6 and the last two lemmas. For the first assertion we are also using that $q\mathbb{C}$ and $\mathbb{C}$ are equivalent in $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ , as follows from Section 3.4.1. The last assertion follows from the lax case of Section 3.4.9. ∎

3.7 Extension to all $C^{*}$ -algebras

In this subsection, we construct presentable stable $\infty$ -categories $\mathrm{KK}^{G}_{!,\infty}$ which contain $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ as its full subcategory of $\aleph_{1}$ -compact objects.

Definition 3.14.

Denote by

\mathrm{KK}^{G}_{!,\infty}:=\operatorname{Ind}_{\aleph_{1}}(\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}),

where $\operatorname{Ind}_{\aleph_{1}}(\mathcal{C})$ denotes the $\aleph_{1}$ -ind completion of $\mathcal{C}$ , i.e. the category obtained by freely adding²⁷²⁷ 27 see [43, Proposition 5.3.5.10] for the precise universal property $\aleph_{1}$ -filtered colimits to $\mathcal{C}$ . We define functors

\mathrm{kk}^{G}_{!,\infty}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}\rightarrow\mathrm{KK}^{G}_{!,\infty}

as left Kan-extensions

where $y:\mathcal{C}\rightarrow\operatorname{Ind}_{\aleph_{1}}(\mathcal{C})$ denotes the canonical embedding (which preserves countable filtered colimits by [43, Proposition 5.3.5.14]).

Before studying the general properties of the above construction, we need the following definition and an accompanying lemma.

Definition 3.14.

A functor $F$ defined on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ is called s-finitary if for every $A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ , the canonical map

\operatorname{colim}_{A^{\prime}\subseteq_{\operatorname{sep}}A}F(A^{\prime})\rightarrow F(A)

is an equivalence, where $A^{\prime}$ runs through the separable $G$ -invariant $C^{*}$ -subalgebras of $A$ .

Lemma 3.14.

Let $F:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}\rightarrow\mathcal{D}$ be a functor to an $\infty$ -category $\mathcal{D}$ having all small filtered colimits. We have a canonical natural transformation $c:\hat{F}\implies F$ , where $\hat{F}$ is the left Kan extension of $F\mid_{\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}}$ . Then the functor $F$ is s-finitary if and only if $c$ is an equivalence.

Proof.

This follows immediately from the pointwise formula for left Kan extension, and the fact that every morphism $A\rightarrow B$ in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ with $A$ separable factors through a $G$ -invariant separable $C^{*}$ -subalgebra of $B$ . See [13, Lemma 3.1] for more details. ∎

Proposition 3.14.

1.

The category $\mathrm{KK}^{G}_{!,\infty}$ is a presentably symmetric monoidal (for every choice of $(?,!)$ except $(\min,\operatorname{ex})$ ), stable $\infty$ -category, and $\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}}$ can be identified with the full subcategory of $\mathrm{KK}^{G}_{!,\infty}$ comprising of $\aleph_{1}$ -compact objects.
2.

The functor $\mathrm{kk}^{G}_{!,\infty}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}\rightarrow\mathrm{KK}^{G}_{!,\infty}$ is symmetric monoidal (for every choice of $(?,!)$ except $(\min,\operatorname{ex})$ ), homotopy invariant, ${\mathcal{K}_{G}}$ -stable, $!$ -exact, preserves countable direct sums and is s-finitary²⁸²⁸ 28 note that any functor on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ which preserves countable direct sums and is s-finitary automatically preserves all direct sums..

Pullback along $\mathrm{kk}^{G}_{!,\infty}$ induces for every cocomplete symmetric monoidal stable $\infty$ -category $\mathcal{D}$ with bi-cocontinous tensor product an equivalence

(\mathrm{kk}^{G}_{!,\infty})^{*}:\mathbf{Fun}^{\operatorname{colim}}_{\otimes_{?}\operatorname{/lax}}(\mathrm{KK}^{G}_{!,\infty},\mathcal{D})\xrightarrow{\simeq}\mathbf{Fun}^{h,{\mathcal{K}_{G}},\operatorname{!},\oplus_{\omega},\operatorname{sfin}}_{\otimes_{?}\operatorname{/lax}}(\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg},\mathcal{D}),

where the superscript $\operatorname{sfin}$ denotes s-finitary functors.

Proof.

Assertion 1 follows from the general fact that if $\mathcal{C}$ is a small, symmetric monoidal, $\kappa$ -cocomplete (for some regular cardinal $\kappa$ ), stable $\infty$ -category such that the tensor product preserves $\kappa$ -small colimits in each variable, then $\operatorname{Ind}_{\kappa}(\mathcal{C})$ is a presentably symmetric monoidal stable $\infty$ -category and $\mathcal{C}$ can be identified with the full subcategory of $\operatorname{Ind}_{\kappa}(\mathcal{C})$ comprising of $\aleph_{1}$ -compact objects. See [43, Theorem 5.5.1.1], [44, Proposition 1.1.3.6] and [44, Corollary 4.8.1.14].

The proof of Assertion 2 can be taken from [13, Theorem 1.18] and [13, Proposition 3.8]²⁹²⁹ 29 the proof of lax monoidality of $\mathrm{kk}^{G}_{!,\infty}$ is formal, but to show its (strong) monoidality, one has to look into the structure of the minimal and maximal tensor products; see loc. cit. for details. Note that although [13] is written with $\operatorname{Ind}_{\aleph_{0}}$ -completion, all the filtered diagrams considered in Section 3 of loc. cit. are indeed $\aleph_{1}$ -filtered, and so the proofs go through in our context too. The Real and the (potentially) non-discrete group case do not pose any difficulties.

Assertion 3 follows from the series of equivalences

	$\displaystyle\mathbf{Fun}^{\operatorname{colim}}_{\otimes_{?}\operatorname{/lax}}(\mathrm{KK}^{G}_{!,\infty},\mathcal{D})$	$\displaystyle\xrightarrow[\simeq]{y^{*}}\mathbf{Fun}^{\operatorname{colim}_{\omega}}_{\otimes_{?}\operatorname{/lax}}(\mathrm{KK}_{!,\infty}^{G,\mathrm{sep}},\mathcal{D})$
		$\displaystyle\xrightarrow[\simeq]{{\mathrm{kk}_{!,\infty}^{G,\mathrm{sep}}}^{}}\mathbf{Fun}^{h,{\mathcal{K}_{G}},!,\oplus_{\omega}}_{\otimes_{?}\operatorname{/lax}}(\mathsf{G}\mathsf{C}^{}\!\mathsf{Alg}^{\mathrm{sep}},\mathcal{D})$
		$\displaystyle\xleftarrow[\simeq]{\operatorname{incl}^{}}\mathbf{Fun}^{h,{\mathcal{K}_{G}},!,\oplus_{\omega},\operatorname{sfin}}_{\otimes_{?}\operatorname{/lax}}(\mathsf{G}\mathsf{C}^{}\!\mathsf{Alg},\mathcal{D}).$

where the first equivalence follows from [43, Proposition 5.3.5.10] and [44, Corollary 4.8.1.14], the second equivalence is the content of Section 3.4.9, and the last equivalence follows from Section 3.7 and Assertion 2. Note that the inverse to the last equivalence is precisely the left Kan-extension along the inclusion $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\xrightarrow{\operatorname{incl.}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ . ∎

The next result is a generalization of Theorem 3.14 to the non-separable case.

Theorem 3.15.

We have natural equivalences of spaces for any $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ and $\hat{B}\in\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ :

\mathbb{K}^{G}(B)\simeq\Omega^{\infty}\mathrm{KK}^{G}_{!,\infty}(\mathbb{C},B),\qquad\mathbb{\hat{K}}^{G}(\hat{B})\simeq\Omega^{\infty}\mathrm{KK}^{G_{2}}_{!,\infty}(\mathcal{S},\hat{B}).

Moreover, $K_{0}^{G}(-),\mathrm{KK}^{G}_{!,0}(\mathbb{C},-):\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}\rightarrow\operatorname{Ab}$ are equivalent as lax monoidal functors, where $\mathrm{KK}^{G}_{!,0}$ denotes the homotopy category of $\mathrm{KK}^{G}_{!,\infty}$ .

Proof.

This follows immediately from Theorem 3.14 upon noting that all the relevant functors preserve $\aleph_{1}$ -filtered colimits. ∎

Corollary 3.15.

$\mathrm{KK}^{C_{2}}_{!,0}(\mathcal{S},\mathcal{S})\cong\frac{\mathbb{Z}[x]}{x^{2}=1}$ as rings, with the preferred isomorphism mapping $[\operatorname{1}_{\mathcal{S}}]$ to $1$ , and $[\alpha_{\mathcal{S}}]$ to $x$ . For any graded $C^{*}$ -algebra $A$ , the action of $\alpha_{A}$ on $\hat{K}_{0}(A)$ coincides with the action of $\alpha_{\mathcal{S}}\in\mathrm{KK}^{C_{2}}_{!,0}(\mathcal{S},\mathcal{S})$ on $\hat{K}_{0}(A)$ stemming from the product in the $\mathrm{KK}^{C_{2}}_{!,0}$ category.

Proof.

This is immediate from Assertion 8 of Section 2.2 and Theorem 3.15. ∎

Remark 3.15.

The second perspective in Section 3.7 directly shows why even the boundary maps in the long exact sequences of graded $K$ -theory groups preserve the action of the grading (and this is because of the associativity of composition in the $\mathrm{KK}^{C_{2}}_{!,0}$ category), and hence the associated long exact sequence can be viewed as a long exact sequence of $C_{2}$ -abelian groups.

The final result of this section discusses the change of group functors on the categories of all $C^{*}$ -algebras.

Theorem 3.16.

There are obvious analogues of the change of group functors of Section 3.5 to the categories of all $C^{*}$ -algebras. These functors then descend to cocontinuous functors between the respective equivariant $KK_{!,\infty}$ -categories. The restriction functors have symmetric monoidal refinement for every choice of $(?,!)$ except for $(\min,\operatorname{ex})$ . If $H\subseteq G$ is open, then we have an adjunction

\operatorname{ind}^{G}_{H}:\mathrm{KK}^{H}_{!,\infty}\leftrightarrows\mathrm{KK}^{G}_{!,\infty}:\operatorname{res}^{G}_{H}.

If $H\subseteq G$ is closed and $G/H$ is compact, then we have an adjunction

\operatorname{res}^{G}_{H}:\mathrm{KK}^{G}_{!,\infty}\leftrightarrows\mathrm{KK}^{H}_{!,\infty}:\mathsf{coind}^{G}_{H}.

Proof.

In light of the results of Section 3.5 and Section 3.7.3, we have to show that the functors $\mathsf{ind}^{G}_{H}(-),\mathsf{res}^{G}_{H}(-),\mathsf{coind}^{G}_{H}(-)$ between the relevant categories of all $C^{*}$ -algebras are s-finitary. But this is contained in [13, Section 4], see also [16, Section 2.1]. ∎

4 Graded, Real $KK^{G}_{!,\infty}$ -theory

In this section, we define the graded $KK^{G}_{!,\infty}$ -categories. We follow some ideas from [15] and an unpublished note of Bunke-Duenzinger ([12]). Note that we make our definitions on the large category $\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ from the get go.

It will be ubiquitous for our approach to graded bivariant $K$ -theories that $\mathcal{S}$ is a cocommutative coalgebra (also called comonoid) in $(\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg},\widehat{\otimes}_{?})$ . In what follows, we stop decorating the tensor product because $\mathcal{S}$ is nuclear, see [63, Corollary E.2.19]. We first come with a diagonal map $\mathcal{S}\rightarrow\mathcal{S}\hat{\otimes}\mathcal{S}$ . The following result is proven in detail in [26, Proposition 4.3].

Proposition 4.0.

There is a unique graded $*$ -homomorphism $\mathcal{S}\rightarrow\mathcal{S}\hat{\otimes}\mathcal{S}$ such that

u\mapsto u\hat{\otimes}u,\penalty\ \penalty\ \penalty\ \penalty\ v\mapsto u\hat{\otimes}v+v\hat{\otimes}u,

where $u:=e^{-x^{2}}$ , $v=xe^{-x^{2}}$ .

We also have a graded $*$ -homomorphism $\eta:\mathcal{S}\rightarrow\mathbb{C}$ , given by $\eta(f)=f(0)$ .

Corollary 4.0.

The $*$ -homomorphisms $\eta$ and $\triangle$ equip $\mathcal{S}$ with a cocommutative coalgebra structure; in the sense that the following diagrams commute

Proof.

This is immediate by considering the elements $u$ and $v$ in $\mathcal{S}$ . ∎

Lemma 4.0.

$\mathrm{KK}^{G_{2}}_{!,\infty}$ has a presentably symmetric monoidal structure for the graded tensor products $\hat{\otimes}_{?}$ for every choice of $(?,!)$ except $(?,!)=(\min,\operatorname{ex})$ , and the canonical functor $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}\rightarrow\mathrm{KK}^{G_{2}}_{!,\infty}$ has a symmetric monoidal refinement too.

Proof.

We just discuss the separable analogue of the statement; the generalization to the large categories is a routine argument in light of the results of Section 3.7.

One checks directly that each of the steps in the localizations

\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\xrightarrow{L_{h}}\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\xrightarrow{L_{\mathcal{K}_{G_{2}}}}\mathrm{S}_{\mathcal{K}_{G_{2}}}\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}\xrightarrow{L_{!,\oplus}}\mathrm{S}_{\mathcal{K}_{G_{2}}}\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}_{h,!,\oplus_{\omega}}^{\mathrm{sep}}\xrightarrow{L_{\operatorname{Bott}}}\mathrm{KK}_{!,\infty}^{G_{2},\mathrm{sep}}.

has a symmetric monoidal refinement for the graded tensor products, since under the restrictions imposed on $(?,!)$ , it is clear that at each step $A\hat{\otimes}_{!}-$ preserves the class of morphisms that are inverted at that step (to see this for the excision step, we use that the underlying Banach spaces of the graded and ungraded tensor products coincide, see [63, Lemma E.2.18]). The rest of the details are exactly as in the ungraded case. ∎

Notation 4.0.

Henceforth whenever we mention symmetric monoidal structures on equivariant $KK_{!,\infty}$ categories, we consider all choices of $(?,!)$ except $(?,!)=(\min,\operatorname{ex})$ . We also stop decorating the tensor product every time; the decoration is implicitly assumed to be there abiding by the exception mentioned in the previous sentence.

We now proceed with the task of defining symmetric monoidal bivariant $K$ -categories for $\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ . We achieve this by defining $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ as a full subcategory of the category of comodules over $\mathcal{S}$ in $\mathrm{KK}^{G_{2}}_{!,\infty}$ . In general, if $(\cal{C},\otimes)$ is any presentably symmetric monoidal category with a cocommutative coalgebra object $\mathcal{S}$ , then we have an adjunction

\operatorname{forget}:\mathbf{Comod}_{\mathcal{C}}(\mathcal{S})\leftrightarrows\mathcal{C}:\mathcal{S}\otimes-

where $\mathbf{Comod}_{\mathcal{C}}(\mathcal{S})$ denotes the presentable $\infty$ -category of comodules over $\mathcal{S}$ , and $\mathcal{S}\otimes-$ denotes the cofree comodule functor. The details of this can be found in [44, Section 4.2.4]. ³⁰³⁰ 30 Note that use of the word (co)free is not analogous to the use of the word free as in “free abelian group on a set”, and should not create any confusion! We mention that the proof of presentability of the category of comodules is subtle (see [50, Proposition 3.8]), and was explained to the author by Maxime Ramzi.

The next definition should be read together with the remarks following it.

Definition 4.0.

The graded bivariant $KK^{G}_{!}$ -category, denoted $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ , is the full subcategory spanned by cofree comodules in $\mathbf{Comod}_{\mathrm{KK}^{G_{2}}_{!,\infty}}(\mathcal{S})$ . Note that $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ becomes a symmetric monoidal category with respect to the balanced tensor product of comodules ([44, Section 4.4]).

Denote by $\widehat{\mathrm{kk}}^{G}_{!,\infty}$ the composition of the symmetric monoidal functors

\widehat{\mathrm{kk}}^{G}_{!,\infty}:\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}\xrightarrow{\mathrm{kk}^{G_{2}}_{!,\infty}}\mathrm{KK}^{G_{2}}_{!,\infty}\xrightarrow{\mathcal{S}\hat{\otimes}-}\widehat{\mathrm{KK}}^{G}_{!,\infty}\

Remark 4.0.

The following remarks about Section 4 are in order.

1.

Explicitly written out, the relative tensor product of $\mathcal{S}\hat{\otimes}A$ and $\mathcal{S}\hat{\otimes}B$ is $\mathcal{S}\hat{\otimes}A\hat{\otimes}B$ , with the codiagonal map $\mathcal{S}\xrightarrow{\Delta}\mathcal{S}\hat{\otimes}\mathcal{S}$ being used to cancel out one action of $\mathcal{S}$ . This also roughly shows why $\mathrm{KK}^{G_{2}}_{!,\infty}\xrightarrow{\mathcal{S}\hat{\otimes}-}\widehat{\mathrm{KK}}^{G}_{!,\infty}$ is symmetric monoidal.
2.

It is not clear if the category $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ is stable, since cofree comodules need not be closed under the formation of finite limits or colimits. Nevertheless, it is enriched over Spectra being the full subcategory of a stable $\infty$ -category, as follows from the next point.
3.

At first glance, a natural solution to the above problem could seem to simply work with the category of all $\mathcal{S}$ -comodules. But then, there is no reason for this category to be symmetric monoidal, since one does not have appropriate commutation of the tensor product with cosimplicial limits in general (necessary for the dual of [44, Theorem 4.5.2.1] to be applicable). Note that the monoidality assertions in [15, Definition 2.45] are incorrect.
4.

In applications, it is more important to have symmetric monoidality of the graded bivariant $K$ -categories than having presentability. Therefore we choose the definition above.

Denote by $\widehat{\mathrm{KK}}^{G}_{!,0}$ the homotopy category of $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ . From the forgetful-cofree adjunction, we immediately get the hom groups in $\widehat{\mathrm{KK}}^{G}_{!,0}$ are given by

{\widehat{\mathrm{KK}}^{G}_{!,0}}(A,B)\cong{\mathrm{KK}^{G_{2}}_{!,0}}(\mathcal{S}\hat{\otimes}A,B).

The composite of $\varphi\in\widehat{\mathrm{KK}}^{G}_{!,0}(A,B)\cong\mathrm{KK}^{G_{2}}_{!,0}(\mathcal{S}\hat{\otimes}A,B)$ and $\psi\in\widehat{\mathrm{KK}}^{G}_{!,0}(B,C)\cong\mathrm{KK}^{G_{2}}_{!,0}(\mathcal{S}\hat{\otimes}B,C)$ is given by the composite in $\mathrm{KK}^{G_{2}}_{!,0}$

\mathcal{S}\hat{\otimes}A\xrightarrow{\Delta\hat{\otimes}1_{A}}\mathcal{S}\hat{\otimes}\mathcal{S}\hat{\otimes}A\xrightarrow{1_{\mathcal{S}}\hat{\otimes}\varphi}\mathcal{S}\hat{\otimes}B\xrightarrow{\psi}C.

The functor

\widehat{\mathrm{kk}}^{G}_{!,0}:\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}\rightarrow\widehat{\mathrm{KK}}^{G}_{!,0}

is identity on objects and precomposes with $\mathcal{S}\hat{\otimes}A\xrightarrow{{\operatorname{ev}_{0}}_{*}}A$ on morphisms emanating out of $A$ .

We now mention important by-products of our definitions. These will be crucially used in the next section.

Proposition 4.0.

If $A,B\in\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ are equivalent in $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ , then $\mathcal{S}\hat{\otimes}A$ and $\mathcal{S}\hat{\otimes}B$ are equivalent in $\mathrm{KK}^{G_{2}}_{!,\infty}$ .

Proof.

By Section 4, $A$ and $B$ are equivalent in $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ means that we have an equivalence

\mathrm{kk}^{G_{2}}_{!,\infty}(\mathcal{S})\hat{\otimes}\mathrm{kk}^{G_{2}}_{!,\infty}(A)\simeq\mathrm{kk}^{G_{2}}_{!,\infty}(\mathcal{S})\hat{\otimes}\mathrm{kk}^{G_{2}}_{!,\infty}(B)

of cofree comodules. Invoking the functoriality of the forgetful functor $\mathbf{Comod}_{\mathrm{KK}^{G_{2}}_{!,\infty}}(\mathcal{S})\rightarrow\mathrm{KK}^{G_{2}}_{!,\infty}$ now finishes the argument. ∎

Theorem 4.1.

For any $\hat{B}\in\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ , we have a natural equivalence of spaces

\mathbb{\hat{K}}^{G}(\hat{B})\simeq\Omega^{\infty}\widehat{\mathrm{KK}}^{G}_{!,\infty}(\mathbb{C},\hat{B}).

Moreover, $\hat{K}_{0}^{G}(-),\;\widehat{\mathrm{KK}}^{G}_{!,0}(\mathbb{C},-):\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}\longrightarrow\operatorname{Ab}$ are equivalent as lax monoidal functors.

Proof.

This is an immediate consequence of Section 4, Theorem 3.15, and Theorem 2.3.4. The lax monoidality assertion follows upon observing that the lax monoidality of $\mathrm{KK}^{G_{2}}_{!,\infty}(\mathcal{S},-)$ stems from the codiagonal map $\mathcal{S}\xrightarrow{\Delta}\mathcal{S}\hat{\otimes}\mathcal{S}$ too, as mentioned in Section 4.1. ∎

We now show that graded $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ -theories recover ungraded $\mathrm{KK}^{G}_{!,\infty}$ -theories when the algebras are trivially graded. More precisely,

Theorem 4.2.

In the commutative square

the right functor is fully faithful.

Proof.

We have to show that the composite

\mathrm{KK}^{G}_{!,\infty}(A,B)\xrightarrow{\mathsf{res}^{G}_{G_{2}}(-)}\mathrm{KK}^{G_{2}}_{!,\infty}(\mathsf{res}^{G}_{G_{2}}(A),\mathsf{res}^{G}_{G_{2}}(B))\xrightarrow{\operatorname{ev}_{0}^{*}}\mathrm{KK}^{G_{2}}_{!,\infty}(\mathcal{S}\hat{\otimes}\mathsf{res}^{G}_{G_{2}}(A),\mathsf{res}^{G}_{G_{2}}(B))

(4.3)

is an equivalence. To that end, we accommodate Equation 4.3 into the commutative diagram (note that the left part of the diagram commutes can be seen by explicitly calculating the fate of a $G$ -equivariant homomorphism from $A$ to $B$ using the definition of the unit of the restriction-coinduction adjunction as described in Section 3.5. The formulae of loc. cit. also work in the non-separable case)

Note that we implicitly have used that $A$ and $B$ have the trivial grading so as to replace some graded tensor products with the ungraded ones. In light of the above diagram, our thesis would follow once we have shown that the composite

(\mathcal{S}\hat{\otimes}Cl^{0,1})_{u}\xrightarrow{{\operatorname{ev}_{0}}_{*}}Cl^{0,1}_{u}\xrightarrow{\pi_{1}}\mathbb{C}

is an equivalence in $\mathrm{KK}^{G}_{!,\infty}$ . But this readily follows from Section 3.4.3.

∎

4.1 The Bott map in $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ and consequences

We can now export results from [26, Section 9] in our setup. We begin with the definition of the Bott class. For $V$ any Euclidean vector space equipped with a self-adjoint involution ( $V$ was called a pseudo Euclidean vector space in [26, Section 9.1]), we denote by $\mathcal{C}(V)$ the graded $C^{*}$ -algebra $C_{0}(V,Cl(V))$ . For the next few results, we do not consider the external group $G$ .

Definition 4.3.

Let $V$ be a pseudo Euclidean vector space. Let $\iota:V\rightarrow Cl(V)$ denote the canonical inclusion of $V$ into $Cl(V)$ . Then the functional calculus of $\iota$ yields a graded homomorphism

b_{V}:\mathcal{S}\rightarrow\mathcal{C}(V),

which we call the Bott map. It naturally determines a point in the spectral $K$ -theory space $\mathbb{\hat{K}}\bigl(\mathcal{C}(V)\bigr),$ and hence also a point in $\Omega^{\infty}\widehat{\mathrm{KK}}_{!,\infty}\bigl(\mathbb{C},\mathcal{C}(V)\bigr).$ In light of Theorem 4.1, we denote either of them by $b_{V}$ .

Remark 4.3.

[26, Section 9] contains a detailed proof that $\pi_{0}\hat{\mathbb{K}}\bigl(\mathcal{C}(V)\bigr)\cong\mathbb{Z}$ , and that $b_{V}$ is a generator of this group. Consequently, $b_{V}$ is also a generator of $\pi_{0}\operatorname{map}_{\widehat{\mathrm{KK}}_{!,\infty}}\bigl(\mathbb{C},\mathcal{C}(V)\bigr)\cong\mathbb{Z}$ . Furthermore, the proof of Section 2.2.4 shows that

b_{V}\times b_{\overline{V}}\in\pi_{0}\hat{\mathbb{K}}\bigl(\mathcal{C}(V)\hat{\otimes}\mathcal{C}(\overline{V})\bigr)\cong\pi_{0}\hat{\mathbb{K}}(\mathcal{C}(V\oplus\overline{V}))\cong\mathbb{Z}

is a generator of the group.

We end this subsection with a technical lemma. Although the result is an immediate corollary of Theorem 4.4, we need the lemma in the proof of it, and so must have it a priori.

Lemma 4.3.

For any pseudo-Euclidean vector space $V$ , the morphism

b_{V}\times b_{\overline{V}}\in\widehat{\mathrm{KK}}_{!,0}\bigl(\mathbb{C},\mathcal{C}(V)\hat{\otimes}\mathcal{C}(\overline{V})\bigr)

is an isomorphism in $\widehat{\mathrm{KK}}_{!,0}$ .

Proof.

Section 4.1 shows that $b_{V}\times b_{\overline{V}}$ is a unit in $\widehat{\mathrm{KK}}_{!,0}(\mathbb{C},\mathcal{C}(V)\hat{\otimes}\mathcal{C}(\overline{V}))$ . On the other hand, the Toeplitz extension theory shows that there is a $\widehat{\mathrm{KK}}_{!,0}$ -equivalence between $\mathbb{C}$ and $\mathcal{C}(V)\hat{\otimes}\mathcal{C}(\overline{V})$ , given by the image of the boundary map of the Toeplitz extension under the canonical functor $\mathrm{KK}^{C_{2}}_{!,0}\rightarrow\widehat{\mathrm{KK}}_{!,0}$ . This also has to be a unit in $\mathbb{Z}$ because of the following general fact: in any additive $1$ -category $\mathcal{D}$ , if there are isomorphic objects $x$ and $y$ such that $\operatorname{hom}_{\mathcal{D}}(x,y)\cong\mathbb{Z}$ as abelian groups, then the isomorphism necessarily corresponds to $\pm 1$ .³¹³¹ 31 a proof of this fact can be deduced from the elementary algebra fact that any commutative unital ring structure on the abelian group $\mathbb{Z}$ must be isomorphic to the standard ring structure on $\mathbb{Z}$ ([1, Exercise 2.16]). To prove our thesis, we simply use that if $f$ is an isomorphism in an additive category, then so is $-f$ . ∎

We next prove the following result, which is well known in the classical frameworks of $KK$ - and $E$ -theory; see [39], [40], [32]. We remind the reader that we do not yet know that our $\widehat{\mathrm{KK}}_{!,0}$ categories are equivalent to classical graded $KK$ - or $E$ -categories, and hence need to give an independent proof.

Theorem 4.4.

For any pseudo Euclidean vector space $V$ , the Bott element $b_{V}$ is an equivalence in $\widehat{\mathrm{KK}}_{!,0}$ .

We now build up to the proof of Theorem 4.4. We start off by spelling out the interchange law in a monoidal ( $\infty$ )-categories.

Lemma 4.4.

Let $\mathcal{C}$ be a monoidal $(\infty)$ -category, and let $A,B,C,A^{\prime},B^{\prime},C^{\prime}$ be objects of $\mathcal{C}$ . For composable morphisms

A\xrightarrow{f}B\xrightarrow{g}C,\qquad A^{\prime}\xrightarrow{h}B^{\prime}\xrightarrow{k}C^{\prime},

the functoriality of the tensor product $\otimes:\mathcal{C}\times\mathcal{C}\longrightarrow\mathcal{C}$ gives a canonical homotopy

(g\circ f)\otimes(k\circ h)\simeq(g\otimes k)\circ(f\otimes h)

in the mapping space $\operatorname{Map}_{\mathcal{C}}(A\otimes A^{\prime},\,C\otimes C^{\prime}).$

Lemma 4.4.

For any $X\in\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ , the following diagram of abelian groups commutes

(4.5)

Furthermore, all the arrows in the above diagram are isomorphisms.

Proof.

That all the morphisms in the diagram are additive is clear. That it also commutes is an application of Section 4.1. In the notation of loc. cit. we take

\mathcal{C}:=\widehat{\mathrm{KK}}_{!,0},A=A^{\prime}=\mathbb{C},B=\mathcal{C}(V),B^{\prime}=C^{\prime}=\mathcal{C}(\overline{V}),C=X.

As for the morphisms, we can take any $g\in\widehat{\mathrm{KK}}_{!,0}(\mathcal{C}(V),X)$ , and

f=b_{V},h=b_{\overline{V}},k=\operatorname{id}_{\mathcal{C}(\overline{V})}.

Since Equation 4.5 commutes, it suffices to show that the two left vertical arrows and the bottom horizontal arrows are isomorphisms. That

-\hat{\otimes}\mathcal{C}(\overline{V}):\widehat{\mathrm{KK}}_{!,0}(\mathcal{C}(V),X)\rightarrow\widehat{\mathrm{KK}}_{!,0}(\mathcal{C}(V)\hat{\otimes}\mathcal{C}(\overline{V}),X\hat{\otimes}\mathcal{C}(\overline{V}))

is an isomorphism follows because tensoring with $\mathcal{C}(\overline{V})$ yields an isomorphism in $\mathrm{KK}^{C_{2}}_{!,0}$ (with the inverse being tensoring with $\mathcal{C}(V)$ ), and because the cofree comodule functor $\mathrm{KK}^{C_{2}}_{!,\infty}\xrightarrow{\mathcal{S}\hat{\otimes}-}\widehat{\mathrm{KK}}_{!,\infty}$ is symmetric monoidal. That $b_{V}\times b_{\overline{V}}$ is an isomorphism follows from Section 4.1, and the bottom horizontal arrow being an isomorphism is the content of [26, Theorem 9.13]. ∎

Remark 4.5.

We mention that the proof of [26, Theorem 9.13] uses non-trivial functional analysis and features of classical graded $K$ -theory (such as graded asymptotic morphisms between graded $C^{*}$ -algebras induce morphisms between the graded $K$ -groups). However, the “ $K$ -theoretic” result is enough for the proof of Section 4.1, and we do not need to know that there is any relation between the $\widehat{\mathrm{KK}}_{!,0}$ categories and asymptotic morphisms. And indeed, when $!=\operatorname{se}$ , a general asymptotic morphism would not induce a morphism in $\widehat{\mathrm{KK}}_{\mathrm{se},0}$ .

Proof of Theorem 4.4.

This readily follows from the Yoneda lemma and Section 4.1. ∎

We next deduce an important corollary of Theorem 4.4. We can now bring back our external compact group $G$ .

Corollary 4.5.

In $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ , we have an equivalence $C_{0}(V)\simeq Cl(\overline{V})$ . In particular, the functor

-\hat{\otimes}Cl^{1,0}:\widehat{\mathrm{KK}}^{G}_{!,\infty}\rightarrow\widehat{\mathrm{KK}}^{G}_{!,\infty}

is equivalent to the loops functor on $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ .

Proof.

We begin by first observing that Theorem 4.4 also shows that the Bott element is an equivalence in $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ . This is because $\mathbb{C}$ and $\mathcal{C}(V)$ have the trivial $G$ -action, and we have an obvious “equipping with the trivial $G$ -action” functor $\widehat{\mathrm{KK}}_{!,\infty}\rightarrow\widehat{\mathrm{KK}}^{G}_{!,\infty}$ .³²³² 32 More formally, we could have developed a graded variant of the change of group functors, and deduced this from the functoriality of the restriction functor. But because we would not have the occassion to use it anywhere else, we omit the details.

It suffices to show that $C_{0}(V)$ and ${Cl}(\overline{V})$ are isomorphic in $\widehat{\mathrm{KK}}^{G}_{!,0}$ . But this follows from the implications

\mathbb{C}\cong\mathcal{C}(V)\implies Cl(\overline{V})\cong C_{0}(V)\hat{\otimes}Cl(V)\hat{\otimes}Cl(\overline{V})\implies Cl(\overline{V})\cong C_{0}(V),

where we used that tensoring with $Cl(\overline{V})$ is already an isomorphism in $\mathrm{KK}^{G_{2}}_{!,0}$ , and because $\mathbb{C}\rightarrow Cl(V\oplus\overline{V})$ is a $\mathrm{KK}^{G_{2}}_{!,0}$ -equivalence.

The last assertion follows from the observation that the canonical functor $\mathrm{KK}^{G_{2}}_{!,\infty}\rightarrow\widehat{\mathrm{KK}}^{G}_{!,\infty}$ is left-exact, and loops in $\mathrm{KK}^{G_{2}}_{!,\infty}$ is modeled by $C_{0}(\mathbb{R}^{0,1})$ . ∎

We end this section by showing a fundamental feature (or defect!) of graded $\widehat{\mathrm{KK}}_{!,\infty}$ -categories. Recall that (equivariant) $KK_{!,\infty}$ -categories for ungraded, separable $C^{*}$ algebras enjoy the universal property of being a Dwyer-Kan localization at certain class of morphisms. We show that the analogous statement for $\widehat{\mathrm{KK}}_{!,\infty}$ categories is false. This issue seems to not have been discussed anywhere in the literature, and Bunke-Duenzinger left it as an open question (in the complex case) in [12]. The author learnt the crux of the proof from Johannes Ebert. The proof in its present form was suggested by Maxime Ramzi.

Proposition 4.5.

The full subcategory of $\widehat{\mathrm{KK}}_{!,\infty}$ spanned by separable, graded $C^{*}$ -algebras (denoted $\widehat{\mathrm{KK}}_{!,\infty}^{\operatorname{sep}}$ ) is not the Dwyer-Kan localization of the category of separable, graded $C^{*}$ -algebras at any class of morphisms.

Proof.

Since the grading on a graded $C^{*}$ -algebra induces a $C_{2}$ -action on the graded $K$ -groups, we have a commutative diagram of functors

Note that the bottom triangle commutes from Theorem 4.1. But since the forgetful functor $C_{2}\!\operatorname{-Ab}\rightarrow\operatorname{Ab}$ reflects isomorphisms, had $\widehat{KK}_{!,\infty}^{\operatorname{sep}}$ been a Dwyer-Kan localization, we would obtain a lift in the above diagram $\widehat{\mathrm{KK}}_{!,\infty}^{\operatorname{sep}}\rightarrow C_{2}\!\operatorname{-Ab}$ , and thereby conclude that the $\hat{K}_{0}$ -groups of two $\widehat{\mathrm{KK}}_{!,\infty}^{\operatorname{sep}}$ -equivalent $C^{*}$ -algebras are in fact isomorphic as $C_{2}$ -abelian groups. In light of Theorem 4.4, this a contradiction, since $\hat{K}_{0}(\mathbb{C})\cong\mathbb{Z}$ with the trivial $C_{2}$ -action, whereas $\hat{K}_{0}(C_{0}(\mathbb{R},Cl^{0,1}))\cong\mathbb{Z}$ with the action $n\mapsto-n$ , as shown in Section 2.2.4. ∎

Remark 4.5.

Note that there is an obvious equivariant version of Section 4.1 too.

5 Applications: certain fiber sequences connecting different K-theories

We now use the technology developed in the previous section in order to study the relationships between graded $K$ -theory, $C_{2}$ -equivariant $K$ -theory and ungraded (classical) $K$ -theory. All the results can be deduced in the presence of an external compact group $G$ , as we will be doing. The following remarks are in order.

Remark 5.0.

We will be using the following elementary observations in this section even without mentioning.

1.

The datum of a $G\times H$ -action on a $C^{*}$ -algebra $A$ is same as the datum of a $G$ -action and a $H$ -action on $A$ which commute. Under this identification, it makes sense to talk about the $G$ - and $H$ -actions on a $(G\times H)$ - $C^{*}$ -algebra. Similar comment also applies to equivariant projective modules. For $G_{2}$ -equivariant projective modules, we will be omitting the $G$ -action from the notation, and just keep track of the $C_{2}$ -action. For example, when we write $(A,\alpha)\in\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ , we mean the $C_{2}$ -action is given by $\alpha$ .
2.

Suppose $A,B,C\in\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ have trivial $G$ -action, and fit in a fiber sequence in $\mathrm{KK}^{C_{2}}_{!,\infty}$ . Then, since $\operatorname{res}^{C_{2}}_{G_{2}}$ is left-exact, they also fit in a fiber sequence in $\mathrm{KK}^{G_{2}}_{!,\infty}$ .
3.

With Theorem 3.15 and Theorem 4.1 at our disposal, we will henceforth be (abusing notation and) denoting by $\mathbb{K}^{G}$ (resp. $\hat{\mathbb{K}}^{G}$ ) the $G$ -equivariant (resp. graded $G$ -equivariant) $K$ -theory spectra, given by $KK_{!,\infty}^{G}(\mathbb{C},-)$ (resp. $\widehat{\mathrm{KK}}^{G}_{!,\infty}(\mathbb{C},-)\simeq KK_{!,\infty}^{G_{2}}(\mathcal{S},-)$ ). As before, we will be denoting their homotopy groups by $K_{*}^{G}(-)$ (resp. $\hat{K}_{*}^{G}(-)$ ).
4.

In the remainder of this section, we fix a compact group $G$ . For concreteness, we work with $\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}$ in this section. The same calculations can also be done with $KK_{\operatorname{se},\infty}^{G_{2}}$ .

5.1 A fiber sequence connecting the different $K$ -theory spectra

The low hanging fruit that we can now pluck off is the following. We will be using the shorthand $(-)_{u}$ to denote the functor $\operatorname{res}^{G_{2}}_{G}(-)$ ( $u$ stands for the “underlying ungraded $C^{*}$ -algebra”).

Proposition 5.0.

We have a natural fiber sequence of spectra

\mathbb{K}^{G}(A_{u})\rightarrow\mathbb{K}^{G_{2}}(A)\rightarrow\hat{\mathbb{K}}^{G}(A)

(5.1)

for every $A\in\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ .

Proof.

Upon applying the exact functor

\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(-,A):\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}\longrightarrow\operatorname{Sp}

to the fiber sequence of Section 3.4.2, we are done once we justify the maps $(1)$ , $(2)$ , and $(3)$ in the following diagram:

The map labeled $(1)$ arises from the induction-restriction adjunction (Theorem 3.16), since we can identify $\mathsf{ind}^{G_{2}}_{G}\mathbb{C}$ with $Cl^{0,1}$ . The second and third equivalence follow from Theorem 3.15. ∎

In the next two remarks, we explain a couple of different formulations of the fiber sequence in Equation 5.1. Though the above sequence will be our main workhorse, these other formulations come handy in certain cases.

Remark 5.1.

We claim that there is a fiber sequence

\mathbb{K}^{G}(A_{u})\xrightarrow{(A\rightarrow A\hat{\otimes}Cl^{0,1})_{*}}\mathbb{K}^{G}((A\hat{\otimes}Cl^{0,1})_{u})\rightarrow\hat{\mathbb{K}}^{G}(A)

(5.2)

which is equivalent to the fiber sequence in Equation 5.1. Indeed, this will follow from the following commutative diagram

(5.3)

Note that the top triangle commutes because the diagram below commutes in $\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}$ , where $f:Cl^{0,1}\rightarrow A$ is any morphism in $\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}$ . The left triangle below does not commute at the level of algebras, but only up to homotopy.

(5.4)

Remark 5.4.

We claim there is a fiber sequence

\mathbb{K}^{G_{2}}(A\hat{\otimes}Cl^{1,0})\xrightarrow{(A\rightarrow A\hat{\otimes}Cl^{1,0})^{*}}\mathbb{K}^{G_{2}}(A)\rightarrow\hat{\mathbb{K}}^{G}(A)

(5.5)

which is equivalent to the fiber sequence in Equation 5.1. The map labeled $A\rightarrow(A\hat{\otimes}Cl^{1,0})^{*}$ should be taken with a pinch of salt as $\mathbb{K}^{G_{2}}(-)$ is naturally covariantly functorial; what we mean is the map obtained from the zig-zag

A\hat{\otimes}Cl^{1,0}\rightarrow A\hat{\otimes}Cl^{1,1}\xleftarrow{\simeq}A.

When $A$ is unital, it can be shown ([3, Section 4.5.2]) that on $\pi_{0}$ , this has the same effect as precomposing the structure map of a $G_{2}$ -equivariant $A\hat{\otimes}Cl^{1,0}$ projective module with the homomorphism $A\rightarrow A\hat{\otimes}Cl^{1,0}$ . The equivalence of the fiber sequences in Equation 5.1 and Equation 5.5 can be shown by a similar treatment as in the last remark.

This formulation of the fiber sequence is directly related to the classical Atiyah-Bott-Shapiro description of $K$ -groups of a point, see Section 5.1 for more details.

Proposition 5.5.

When $A$ is inner graded, the fiber sequence in Equation 5.1 (and hence also the ones in Equation 5.2 and Equation 5.5) splits.

Proof.

In light of Equation 5.2, we have to show that the map of spectra

\mathrm{KK}^{G}_{\mathrm{ex},\infty}(\mathbb{C},A_{u})\xrightarrow{(A\to A\hat{\otimes}Cl^{0,1})_{*}}\mathrm{KK}^{G}_{\mathrm{ex},\infty}\bigl(\mathbb{C},(A\hat{\otimes}Cl^{0,1})_{u}\bigr).

has a retraction when $A$ is inner graded. In this case, since $A\hat{\otimes}Cl^{0,1}\cong A\otimes Cl^{0,1}$ in $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ , we can extend the aforementioned map to

	$\displaystyle\mathrm{KK}^{G}_{\mathrm{ex},\infty}(\mathbb{C},A_{u})$	$\displaystyle\xrightarrow{(A\to A\hat{\otimes}Cl^{0,1})_{*}}\mathrm{KK}^{G}_{\mathrm{ex},\infty}\bigl(\mathbb{C},(A\hat{\otimes}Cl^{0,1})_{u}\bigr)$
		$\displaystyle\xrightarrow{\simeq}\mathrm{KK}^{G}_{\mathrm{ex},\infty}\bigl(\mathbb{C},(A\otimes Cl^{0,1})_{u}\bigr)\xrightarrow{\simeq}\mathrm{KK}^{G}_{\mathrm{ex},\infty}(\mathbb{C},A_{u}\oplus A_{u})$
		$\displaystyle\xrightarrow{\simeq}\mathrm{KK}^{G}_{\mathrm{ex},\infty}(\mathbb{C},A_{u})\oplus\mathrm{KK}^{G}_{\mathrm{ex},\infty}(\mathbb{C},A_{u}).$

It can be directly checked that this composite is the diagonal map of $KK_{\operatorname{ex},\infty}^{G}(\mathbb{C},A_{u})$ . Since the diagonal map of any spectrum has two canonical retractions, we are done. ∎

Remark 5.5.

At this stage, the reader may want to see an example where the fiber sequence in Equation 5.1 does not split. This is intimately related to a folklore question (Section 5.1), and we discuss it in Section 5.1.

One next want to analyze the fiber sequence in Equation 5.1 when $A$ is not inner graded. To that end, we give an explicit description of the maps in the sequence at level of $\pi_{0}$ .

Lemma 5.5.

Let $(A,\alpha)$ be a unital algebra in $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ . We have a natural group homomorphism

F_{A}:K_{0}^{G_{2}}(A)\rightarrow{K}_{0}^{G,\operatorname{Fr}}(A)

given by taking a finitely generated $G_{2}$ -equivariant projective $A$ -module $M$ to the Fredholm cycle $(M,0)\in K_{0}^{G,\operatorname{Fr}}(A)$ .

Proof.

The assertion is clear, modulo the verification that every finitely generated $G_{2}$ -equivariant projective $A$ -module is canonically a graded $G$ -equivariant Hilbert $A$ -module. This follows by an easy case by case analysis with signs in light of [8, Proposition 11.2.3]. ∎

Corollary 5.5.

Following the notation of Section 5.1, the diagram

commutes.

Proof.

Given a $G_{2}$ -equivariant projective $A$ -module $M$ , we study the fate of $[M]\in K_{0}^{G_{2}}(A)$ along the up-right-down itinerary.

The classifying map of $M$ is a $G_{2}$ -equivariant $*$ -homomorphism

\varphi:\mathbb{C}\rightarrow\mathcal{K}_{A}(V\otimes A),

where $V$ is a finite dimensional $G_{2}$ -vector space. $\varphi$ is the image of $[M]$ in $\pi_{0}(KK_{\operatorname{ex},\infty}^{G_{2}}(\mathbb{C},A))$ under the left vertical map, as can be seen by unraveling the maps in Theorem 3.15 and the equivariant Kasparov stabilization theorem. Also note that $\mathcal{K}_{A}(V\otimes A)=\mathcal{B}_{A}(V\otimes A)$ since $\mathcal{K}_{A}(V\otimes A)$ is unital, see [62, Theorem 15.4.2]. It remains to determine the image of $\operatorname{\varphi\circ ev_{0}}$ under the right vertical map. From Theorem 2.3.5, one gets that this image is the Fredholm cycle (we are also using the notation of the proof of Theorem 2.3.5)

(\overline{\varphi\circ\operatorname{ev}_{0}(\mathcal{S})(V\otimes A)},\tilde{\varphi}(x))=(M,0)

Indeed, this follows from the commutative diagram

The dotted extension of $(\varphi\circ\operatorname{ev}_{0})^{\operatorname{nd}}$ is unique. However, $\varphi^{\operatorname{nd}}\circ\operatorname{ev}_{0}$ is yet another extension, whence they must agree. Our thesis is deduced readily from the latter. ∎

Before stating the next result, we need to setup some notation

Notation 5.5.

Let $(A,\alpha)\in\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ . Denote by $A_{\alpha}$ the $G$ -equivariant $A_{u}$ -module $\underline{A}$ (the underlying abelian group of $A$ ) with the $A_{u}$ -action given by $\underline{a}.b:=\underline{a\alpha(b)}$ and the $G$ -action given by $g.\underline{a}:=\underline{ga}$ . Note that $A_{\alpha}$ is a $A_{u}$ - $A_{u}$ bimodule in a canonical way.

Lemma 5.5.

Let $(A,\alpha)$ be a unital algebra in $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ . We have a natural group homomorphism

G_{A}:K_{0}^{G}(A_{u})\rightarrow K_{0}^{G_{2}}(A)

given by taking a finitely generated $G$ -equivariant projective $A_{u}$ -module $M$ to the $G_{2}$ -equivariant finitely generated projective $A$ -module $M\oplus M_{\alpha}$ , where $M_{\alpha}:=M\otimes_{A_{u}}A_{\alpha}$ ³³³³ 33 explicitly, $M_{\alpha}$ is the abelian group $\underline{M}$ with the $A_{u}$ -action given by $\underline{m}.a:=\underline{m\alpha(a)}$ . $M$ and $M_{\alpha}$ are not isomorphic as $A_{u}$ -modules in general. and the $C_{2}$ -action on $M\oplus M_{\tau}$ is flip, i.e. given by $(m,\underline{n})\mapsto(n,\underline{m})$ .

Proof.

Obvious. ∎

Corollary 5.5.

Following the notation of Section 5.1, the diagram

commutes.

Proof.

This follows by chasing through the maps in the diagram. Given a $G$ -equivariant finitely generated projective $A_{u}$ -module $M$ , the classifying map of $[M]\in K_{0}(A_{u})$ is a $G$ -equivariant $*$ -homomorphism

p:\mathbb{C}\rightarrow\mathcal{K}_{A_{u}}(V\otimes A_{u})

where $V$ is some finite dimensional $G$ -vector space. We therefore obtain

\operatorname{ind}^{G_{2}}_{G}\mathbb{C}\xrightarrow{\operatorname{ind}^{G_{2}}_{G}p}\operatorname{ind}^{G_{2}}_{G}(\mathcal{K}_{A_{u}}(V\otimes A_{u}))\hookrightarrow\mathcal{K}_{A}(V\otimes A_{u}\oplus V\otimes A_{u},\operatorname{twisted\penalty\ flip}),

where for the last inclusion we used elementary properties of induction for group representations on vector spaces and $C^{*}$ -algebras (see [13, Lemma 4.10]), as well as the description of the co-unit of the induction-restriction adjunction. Precomposing with the inclusion $\mathbb{C}\rightarrow\operatorname{ind}^{G_{2}}_{G}\mathbb{C}$ , we conclude that the up-right-down itinerary maps $p$ to the homomorphism

\tilde{p}:\mathbb{C}\rightarrow\mathcal{K}_{A}(V\otimes A_{u}\oplus V\otimes A_{u},\operatorname{twisted\penalty\ flip}),\penalty\ \penalty\ \penalty\ \penalty\ \tilde{p}(1)=(p(1)\oplus\alpha_{*}p(1)).

It remains to check that the $G_{2}$ -equivariant $A$ -modules $(p(1)(V\otimes A_{u})\oplus\alpha_{*}p(1)(V\otimes A_{u}),\operatorname{twisted\penalty\ flip})\penalty\$ and $(M\oplus M_{\alpha},\operatorname{flip})$ are isomorphic. We have $p(1)(V\otimes A_{u})\cong M$ as $G$ -equivariant $A_{u}$ -modules by definition. To see that $\alpha_{*}p(1)(V\otimes A_{u})\cong M_{\alpha}$ as $G$ -equivariant $A_{u}$ -modules, note that we have

\alpha_{*}p(1)(V\otimes A_{u})=\{v\otimes\alpha a\mid v\otimes a\in p(1)(V\otimes A_{u})\}.

The map

	$\displaystyle\varphi:(p(1)(V\otimes A_{u}))_{\alpha}$	$\displaystyle\rightarrow\alpha_{*}p(1)(V\otimes A_{u})$
	$\displaystyle\underline{v\otimes a}$	$\displaystyle\mapsto v\otimes\alpha(a)$

now readily produces an isomorphism between the $G$ -equivariant $A_{u}$ -modules $M_{\alpha}$ and $\alpha_{*}p(1)(V\otimes A_{u})$ . It is now also direct to check that juxtaposing the two isomorphisms yields a $G_{2}$ -equivariant isomorphism between $(M\oplus M_{\alpha},\operatorname{flip})$ and $(p(1)(V\otimes A_{u})\oplus\alpha_{*}p(1)(V\otimes A_{u}),\operatorname{twisted\penalty\ flip}).$ ∎

We end this subsection with the following remark, which connects our results to the classical result of Atiyah-Bott-Shapiro as mentioned in Theorem 1.2. For simplicity, we ignore the external group $G$ in this remark.

Remark 5.5.

When $A=Cl^{n,0}$ , Atiyah-Bott-Shapiro asserts that there are short exact sequences of abelian groups

0\rightarrow K_{0}^{C_{2}}(Cl^{n+1,0})\xrightarrow{(Cl^{n,0}\rightarrow Cl^{n+1,0})^{*}}K_{0}^{C_{2}}(Cl^{n,0})\rightarrow\hat{K}_{0}(Cl^{n,0})\rightarrow 0,

(5.6)

where the first map is as described in Section 5.1. Since $\hat{K}_{0}(Cl^{n,0})\cong K_{n}(\mathbb{C})$ , one obtains a convenient (and perhaps miraculous!) description of higher $K$ -groups of a point as certain Clifford module quotients. This immediately makes one ask the following question.

Question 5.6.

Can the higher $K$ -groups of any (graded), (Real) unital $C^{*}$ -algebra $A$ be expressed in terms of Clifford module quotients a’la Atiyah-Bott-Shapiro? More precisely, does there exist a short exact sequence of abelian groups of the form

0\rightarrow K_{0}^{C_{2}}(A\hat{\otimes}Cl^{n+1,0})\xrightarrow{(A\hat{\otimes}Cl^{n,0}\rightarrow A\hat{\otimes}Cl^{n+1,0})^{*}}K_{0}^{C_{2}}(A\hat{\otimes}Cl^{n,0})\rightarrow\hat{K}_{n}(A)\rightarrow 0\penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ \penalty\ ?

We explain that the answer to this question is in the negative. To that end, we recall that to show exactness of Equation 5.6 at $\hat{K}_{0}(Cl^{n,0})$ , one explicitly constructs a section of the map $K_{0}^{C_{2}}(Cl^{n,0})\rightarrow\hat{K}_{0}(Cl^{n,0})$ using the notion of index of a Fredholm operator on a Hilbert space with Clifford symmetries. The construction of this index map crucially needs the analytic fact that the kernel of a Fredholm operator on a Hilbert space is automatically finitely generated projective. Since the same is not true for a general Hilbert module over a $C^{*}$ -algebra, the index of a Fredholm operator over an ungraded unital $C^{*}$ -algebra $B$ is typically defined using the notion of taming (as described in detail in [26, Section 8.2]). Roughly speaking, taming works by adding a dummy finite dimensional free $B$ -module to the game, and then subtracting it later in the group $K_{0}(B)$ . When $B$ is graded, the taming construction can fail; for example when $B:=A\hat{\otimes}Cl^{n,0}$ the taming construction can be seen to not necessarily preserve the relevant Clifford symmetries. So in general, one cannot expect to construct the crucial index map witnessing a section of the map $K_{0}^{C_{2}}(A\hat{\otimes}Cl^{n,0})\rightarrow\hat{K}_{0}(A\hat{\otimes}Cl^{n,0})$ .

And indeed, concrete counterexamples exist. Taking $A$ to be the complex $C^{*}$ -algebra $C(S^{1},\mathbb{C})$ and $n=1$ , we know that $\hat{K}_{1}(C(S^{1},\mathbb{C}))\cong K_{1}(C(S^{1},\mathbb{C}))\cong\mathbb{Z}$ , but the relevant Clifford module quotient can be shown to be $0$ . In context of the fiber sequence in Equation 5.5, this failure can be explained to be due to the corresponding boundary map being injective. More elaborations on this can be found in [3, Example 4.51]. See also [10, Appendix A] for other examples and discussions.

5.2 The dual fiber sequence and endomorphisms of $K$ -groups

In this subsection, we obtain a dual of the fiber sequence described in Section 5.1. The key to the existence of this fiber sequence is the equivalence between $Cl^{1,0}$ and $C_{0}(\mathbb{R}^{0,1})$ in $\mathrm{KK}^{G_{2}}_{!,\infty}$ , as shown in Section 4.1.

Proposition 5.6.

We have a fiber sequence

Cl^{1,0}\xrightarrow{j}\mathbb{C}\xrightarrow{p}\mathcal{S}

in $\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}$ , where $j:Cl^{1,0}\rightarrow\mathbb{C}$ is obtained from the zig-zag $Cl^{1,0}\rightarrow Cl^{1,1}\leftarrow\mathbb{C},$ and $p\in\pi_{0}(\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(\mathbb{C},\mathcal{S}))$ is the image under the $\operatorname{res}^{C_{2}}_{G_{2}}$ homomorphism of a generator of $K_{0}^{C_{2}}(\mathcal{S})\cong\mathbb{Z}$ (Section 2.1).

Proof.

We begin the proof by mentioning that the assertion of the proposition is indeed just: in any stable $\infty$ -category $\mathcal{C}$ , a fiber sequence $A\xrightarrow{f}B\xrightarrow{g}C$ naturally yields a equivalent fiber sequence $A\xrightarrow{f}B\xrightarrow{-g}C$ . In light of Section 5.2, it suffices to demonstrate the existence of the fiber sequence in $\mathrm{KK}^{C_{2}}_{\mathrm{ex},\infty}$ .

We begin by applying the functor $-\hat{\otimes}Cl^{1,0}$ to Equation 3.2, thereby obtaining the fiber sequence

\mathcal{S}\hat{\otimes}Cl^{1,0}\xrightarrow{\operatorname{ev}_{0}{{}_{*}}}Cl^{1,0}\xrightarrow{\operatorname{\iota}_{*}}Cl^{1,1}.

Now, Section 4 and Section 4.1 show that we have a fiber sequence

\Omega\mathcal{S}\rightarrow Cl^{1,0}\xrightarrow{\iota_{*}}Cl^{1,1}

in $\mathrm{KK}^{C_{2}}_{\mathrm{ex},\infty}$ . Taking cofibering in $\mathrm{KK}^{C_{2}}_{\mathrm{ex},\infty}$ and using the equivalence $\mathbb{C}\simeq Cl^{1,1}$ we obtain a fiber sequence

Cl^{1,0}\xrightarrow{j}\mathbb{C}\rightarrow\mathcal{S}.

To prove our thesis, it will suffice to show that the second morphism above is a generator of $K_{0}^{C_{2}}(\mathcal{S})\cong\mathbb{Z}$ . Consider the long exact sequence of $K_{*}^{C_{2}}(-)(=\pi_{*}\mathrm{KK}^{C_{2}}_{\mathrm{ex},\infty}(\mathbb{C},-))$ groups associated to this fiber sequence

...\rightarrow K_{0}^{C_{2}}(Cl^{1,0})\xrightarrow{j_{*}}K_{0}^{C_{2}}(\mathbb{C})\rightarrow K_{0}^{C_{2}}(\mathcal{S})\xrightarrow{\partial}K_{-1}^{C_{2}}(Cl^{1,0})\rightarrow...

By tensor invertibility of $Cl^{0,1}$ in $\mathrm{KK}^{C_{2}}_{\mathrm{ex},\infty}$ and the induction-restriction adjunction, we have

K_{-1}^{C_{2}}(Cl^{1,0})\cong K_{-1}(\mathbb{C}_{u})\cong 0,

whence the map

K_{0}^{C_{2}}(\mathbb{C})\rightarrow K_{0}^{C_{2}}(\mathcal{S})

above is surjective. We now consider the square

where the vertical maps are isomorphisms as mentioned in Section 2.1 and Section 2.1, and the map $\dagger$ is so defined to make the diagram commute. Now, since the morphisms in the square are also $C_{2}$ -equivariant and the $C_{2}$ -abelian group $\mathbb{Z}\oplus\mathbb{Z}$ is generated by $(1,0)$ , we conclude from the surjectivity of ${\dagger}$ that it maps $(1,0)$ to $\pm 1$ . ∎

Proposition 5.6.

We have a natural fiber sequence

\hat{\mathbb{K}}^{G}(A)\xrightarrow{}\mathbb{K}^{G_{2}}(A)\xrightarrow{}\mathbb{K}^{G}((A\hat{\otimes}Cl^{0,2})_{u})

(5.7)

for every $A\in\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ .

Proof.

We apply the exact functor

\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(-,A):\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}\longrightarrow\operatorname{Sp}.

to the fiber sequence in Section 5.2. In light of the proof of Section 5.1, we only need to identify $\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,0},A)$ with $\mathbb{K}^{G}((A\hat{\otimes}Cl^{0,2})_{u})$ . For that we use the route

\resizebox{22609920}{}{$\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,0},A)\xrightarrow[\simeq]{-\hat{\otimes}Cl^{0,2}}\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,2},A\hat{\otimes}Cl^{0,2})\xrightarrow[\simeq]{(Cl^{0,1}\rightarrow Cl^{1,2})^{*}}\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{0,1},A\hat{\otimes}Cl^{0,2})\xrightarrow[\simeq]{\mathsf{ind}\dashv\mathsf{res}}\mathbb{K}^{G}\bigl((A\hat{\otimes}Cl^{0,2})_{u}\bigr)$}.

∎

At this juncture, we record an interpretation of the map $j^{*}:\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(\mathbb{C},A)\rightarrow\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,0},A)$ which will come handy later.

Lemma 5.7.

The diagram

commutes.

Proof.

We expand the diagram to

It suffices to show that each of the four shapes with one vertex $\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,1},A)$ commutes. But all of them are clear (for the right-angled triangle shaped diagram, use an argument similar to the one for Equation 5.4). ∎

Unlike in the case of Section 5.1, the induced maps at the level of $\pi_{0}$ groups of the fiber sequence in Equation 5.7 are not easy to write down explicitly. We instead discuss the various endomorphisms of the $K$ -theory groups that arise from the fiber sequences in Equation 5.1 and Equation 5.7.

Consider the following diagram in $\mathrm{KK}^{C_{2}}_{\mathrm{ex},\infty}$ :

Lemma 5.7.

$\operatorname{ev}_{0}\circ p=1-x\in\mathrm{KK}^{C_{2}}_{\mathrm{ex},0}(\mathbb{C},\mathbb{C})\cong\frac{Z[x]}{x^{2}=1}$ , with the isomorphism being as in Section 2.1.1 (keeping in mind Theorem 3.15³⁴³⁴ 34 the two natural ring structures on the endomorphisms of the unit in any monoidal category coincide thanks to the Eckmann-Hilton argument. See also [14, Lemma 5.1].).

Proof.

From Section 2.1 and Section 2.1, we have a short exact sequence

0\rightarrow K_{0}^{C_{2}}(\mathcal{S})\xrightarrow{{\operatorname{ev}_{0}}_{*}}K_{0}^{C_{2}}(\mathbb{C})\xrightarrow{\iota_{*}}K_{0}^{C_{2}}(Cl^{0,1})\rightarrow 0.

The composition in the $\mathrm{KK}^{C_{2}}_{\mathrm{ex},0}$ -category will be the image of $p$ under $\operatorname{ev_{0_{*}}}$ . The rest follows from Section 2.1. ∎

We now record a corollary of the above calculation which will come handy later.

Corollary 5.7.

The diagram

commutes.

Proof.

It suffices to show that the diagram commutes on $[\operatorname{ev}_{0}]$ . The bottom-right itinerary maps it to $2\in\mathbb{Z}$ . To analyse its fate under the right-down itinerary, we bring Section 5.2 into play, which shows that the upper horizontal arrow is product in $\mathrm{KK}^{C_{2}}_{!,\infty}$ -product with $([\mathbb{C}^{\operatorname{triv}}]-[\mathbb{C}^{\operatorname{sign}}])$ .

Postcomposition with $[\mathbb{C}]^{\operatorname{triv}}$ maps $[\operatorname{ev}_{0}]$ to itself, since $[\mathbb{C}^{\operatorname{triv}}]$ is the identity element of $\mathrm{KK}^{C_{2}}_{\mathrm{ex},0}(\mathbb{C},\mathbb{C})$ . To prove our thesis, we have to show that postcomposition with $[\mathbb{C}^{\operatorname{sign}}]$ maps $[\operatorname{ev}_{0}]$ to $-[\operatorname{ev}_{0}]$ . Unraveling the definition of postcomposition with $[\mathbb{C}^{\operatorname{sign}}]$ , this follows directly from the proof of Theorem 2.3.1. ∎

Corollary 5.7.

Let $A$ be a unital algebra in $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ . The composite

K_{0}^{G_{2}}(A)\xrightarrow{\operatorname{ev}_{0}^{*}}\hat{K}^{G}_{0}(A)\xrightarrow{p^{*}}K^{G_{2}}_{0}(A)

maps a finitely generated $G_{2}$ -equivariant projective $A$ -module $M$ to $[M]-[M^{\operatorname{op}}]\in K_{0}^{G_{2}}(A)$ , where $M^{\operatorname{op}}$ is $M$ as a $G$ -equivariant $A_{u}$ -module, with the $C_{2}$ -action given as the negative of the $C_{2}$ -action on $M$ .

Proof.

Unraveling the definitions and from Section 5.2, we conclude that the composite is nothing but the composition with the image under $\operatorname{res}^{C_{2}}_{G_{2}}$ of $1-x\in\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(\mathbb{C},\mathbb{C})\cong\frac{Z[x]}{x^{2}=1}$ . Identifying $\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(\mathbb{C},\mathbb{C})$ with the classical $G_{2}$ -equivariant $K$ -theory of a point, it is clear that $1$ corresponds to the trivial representation, and $x$ corresponds to the sign representation of $G_{2}$ on $\mathbb{C}$ .³⁵³⁵ 35 more precisely, $G$ acts trivially on $\mathbb{C}$ in both cases, and $C_{2}$ acts either trivially or via the sign representation. Since the internal and external products

\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(\mathbb{C},\mathbb{C})\times\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(\mathbb{C},A)\rightarrow\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(\mathbb{C},A)

are the same (this is true in any monoidal category), our thesis follows since the external multiplication of $[M]$ with $x$ yields $[M^{\operatorname{op}}]$ (keeping in mind the lax monoidality assertion of Theorem 3.15). ∎

The proof of the next result requires slightly more work than needed in the proof of Section 5.2, but is similar in spirit. Somewhat surprisingly Section 5.2 is needed in the proof!

Lemma 5.7.

$p\circ\operatorname{ev}_{0}=1+x\in\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(\mathcal{S},\mathcal{S})\cong\frac{Z[x]}{x^{2}=1}$ , with the isomorphism being as in Section 3.7.

Proof.

From the fiber sequence of Section 5.2, we obtain a long exact sequence of $\hat{K}_{*}(-)(=\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(\mathcal{S},-))$ groups

\cdots\longrightarrow\hat{K}_{0}(Cl^{1,0})\xrightarrow{j_{*}}\hat{K}_{0}(\mathbb{C})\xrightarrow{p_{*}}\hat{K}_{0}(\mathcal{S})\xrightarrow{\partial}\cdots

The composition $p\circ\operatorname{ev}_{0}$ will be the image of $[\operatorname{ev}_{0}]$ under $p_{*}$ . Since $p_{*}$ is a $C_{2}$ -equivariant map, there exists $k\in\mathbb{Z}$ rendering the diagram

(5.8)

commutative, where the left isomorphism is the standard one (mapping $[\operatorname{ev}_{0}]$ to $1$ ), and the right one is the one mentioned in Section 2.2.8.

Claim.

$k=1$ in the above diagram.

Proof.

We expand the diagram to

(5.9)

Observe that the right square commutes, as can be explicitly checked on the two generators $[\operatorname{id_{\mathcal{S}}}]$ and $[\alpha_{\mathcal{S}}]$ . Furthermore, the rectangle

commutes too, as seen in Section 5.2. The claim follows. ∎

This also finishes the proof of the lemma. ∎

Corollary 5.9.

Let $A$ be a unital algebra in $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ . The composite

{K}_{0}^{G,\operatorname{Fr}}(A)\xrightarrow{p^{*}}K_{0}^{G_{2}}(A)\xrightarrow{\operatorname{ev}_{0}^{*}}{K}^{G,\operatorname{Fr}}_{0}(A)

maps the Fredholm cycle $(\mathcal{E},F)\in{K}_{0}^{G,\operatorname{Fr}}(A)$ to $(\mathcal{E}\oplus\mathcal{E},F\oplus(-F))\in{K}_{0}^{G,\operatorname{Fr}}(A)$ .

Proof.

This follows from Section 5.2 and the formula of the explicit isomorphism $\pi_{0}{\mathbb{K}}^{\operatorname{G,sp}}(A)\cong{K}_{0}^{G,\operatorname{Fr}}(A)$ as mentioned in Theorem 2.3.6. ∎

Remark 5.9.

Note that $(\mathcal{E},-F)$ is not the inverse of $(\mathcal{E},F)\in{{K}}^{G,\operatorname{Fr}}_{0}(A)$ ; the usual formula of the inverse also requires to flip the grading on $\mathcal{E}$ (see [26, Lemma 8.24] for details). In fact, when $A$ is inner graded, $(\mathcal{E},-F)=(\mathcal{E},F)$ , as can be seen from Theorem 2.3.6. Thus, for inner graded $A$ , ${K}_{0}^{G,\operatorname{Fr}}(A)\xrightarrow{\operatorname{ev}_{0}^{*}\circ p^{*}}{K}_{0}^{G,\operatorname{Fr}}(A)$ is the multiplication by $2$ map.

We now consider the composite

\mathbb{K}^{G}(A_{u})\xrightarrow{\iota^{*}}\mathbb{K}^{G_{2}}(A)\xrightarrow{j^{*}}\mathbb{K}^{G}((A\hat{\otimes}Cl^{0,2})_{u}).

For a general unital $A\in\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ . the composite does not have an illuminating description, since we do not have a good understanding of the algebra $(A\hat{\otimes}Cl^{0,2})_{u}$ . However, the situation is better in the following cases

1.

$A$ is inner graded (and potentially Real).
2.

$A$ is complex (with potentially non-inner grading).

We begin with the first case. So let $A$ be a unital, inner graded algebra in $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ . Then, we assert that the map of spectra $\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(\mathbb{C},A)\xrightarrow{j^{*}}\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,0},A)$ splits. To see this, we identify $\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,0},A)$ with $\mathbb{K}^{G}(A_{u})$ via the route:

\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,0},A)\xrightarrow[\simeq]{-\hat{\otimes}Cl^{0,1}}\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,1},A\hat{\otimes}Cl^{0,1})\xrightarrow[A\penalty\ \text{inner graded},\simeq]{(\mathbb{C}\rightarrow Cl^{1,1})^{*}}\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(\mathbb{C},A\otimes Cl^{0,1})\xrightarrow[\simeq]{\operatorname{res}\dashv\mathsf{coind}}\mathbb{K}^{G}(A_{u}).

(5.10)

We want to understand the induced map $\mathbb{K}^{G_{2}}(A)\rightarrow\mathbb{K}^{G}(A)$ under the identification above. To that end, we need the following lemma.

Lemma 5.10.

The diagram

commutes, where the left vertical equivalence is induced from Theorem 3.15, and the right vertical equivalence is induced by Equation 5.10. In particular, for unital $A$ , the induced map $K_{0}^{G_{2}}(A)\rightarrow K_{0}^{G}(A)$ forgets the $C_{2}$ -action on a $G_{2}$ -equivariant projective $A$ -module in $K_{0}^{G_{2}}(A)$ .

Proof.

In light of Section 5.2, we just need to show that the composite

\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(\mathbb{C},A)\xrightarrow{(A\rightarrow A\otimes Cl^{0,1})_{*}}\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(\mathbb{C},A\otimes Cl^{0,1})\xrightarrow[\cong]{\text{res}\dashv\text{coind}}\mathrm{KK}^{G}_{\mathrm{ex},\infty}(\mathbb{C},A_{u})

identifies with the restriction map. But this is clear in light of description of the counit of the restriction-coinduction adjunction as mentioned in Theorem 3.13 (the adjunction holds good for non-separable algebras too). ∎

Proposition 5.10.

Let $A$ be an inner graded algebra in $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ . Then, the fiber sequence in Equation 5.7 splits.

Proof.

In light of Section 5.2, it suffices to construct a section of the map $\mathbb{K}^{G_{2}}(A)\xrightarrow{\mathsf{res}^{G_{2}}_{G}(-)}\mathbb{K}^{G}(A_{u})$ . If $A$ were trivially graded, this section could have been taken to be $\mathsf{res}^{G}_{G_{2}}(-)$ . In the general inner graded case, we use the fact that $A\simeq\operatorname{res}^{G}_{G_{2}}\operatorname{res}^{G_{2}}_{G}A$ in $\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}$ as follows by an application of [63, Example E.2.13] (see also [16, Lemma 2.1]). ∎

As an immediate consequence of Section 5.2, we have the following corollary.

Corollary 5.10.

Let $(A,\alpha)$ be a unital, inner graded algebra in $\mathsf{G}_{2}\mathsf{C}^{*}\!\mathsf{Alg}$ .

1.

The composite $K_{0}^{G}(A_{u})\xrightarrow{\iota^{*}}K_{0}^{G_{2}}(A)\xrightarrow{j^{*}}K_{0}^{G}(A_{u})$ is given by

$\displaystyle K_{0}^{G}(A_{u})$ $\displaystyle\rightarrow K_{0}^{G}(A_{u})$

$\displaystyle[M]$ $\displaystyle\mapsto[M\oplus M_{\alpha}]=2[M]$

The composite $K_{0}^{G_{2}}(A)\xrightarrow{j^{*}}K_{0}^{G}(A_{u})\xrightarrow{\iota^{*}}K_{0}^{G_{2}}(A)$ is given by

	$\displaystyle K_{0}^{G_{2}}(A)$	$\displaystyle\rightarrow K_{0}^{G_{2}}(A)$
	$\displaystyle[M,\alpha_{M}]$	$\displaystyle\mapsto[M\oplus M_{\alpha},\operatorname{flip}]=[M\oplus M,(m,n)\mapsto(\alpha_{M}(n),\alpha_{M}(m))]$

Proof.

Everything follows immediately from Section 5.1 and Section 5.2, except that we have to justify the equalities in Assertions 1 and 2.

For Assertion 1, it suffices to show that for $M$ a $G$ -equivariant $A_{u}$ -module over an inner graded $A$ , then $M_{\alpha}\cong M$ as $G$ -equivariant $A_{u}$ -modules. If $u$ is the symmetry implementing the grading on $A$ , then

M_{\alpha}\rightarrow M,\penalty\ \penalty\ \penalty\ \underline{m}\mapsto mu

is such an isomorphism.

For Assertion 2, we first show that when $M$ is a $G_{2}$ -equivariant projective $A$ -module, then $M\cong M_{\alpha}$ as $G$ -equivariant $A_{u}$ -modules. Indeed, an isomorphism is given by

	$\displaystyle M$	$\displaystyle\rightarrow M_{\alpha}$
	$\displaystyle m$	$\displaystyle\mapsto\underline{\alpha_{M}(m)}$

Under this isomorphism, $[M\oplus M_{\alpha},\operatorname{flip}]=[M\oplus M,(m,n)\mapsto(\alpha_{M}(n),\alpha_{M}(m))]\in K_{0}^{G_{2}}(A)$ . Also compare the proof of Section 5.1.³⁶³⁶ 36 actually this argument did not use that $A$ is inner graded, see also Section 5.2. ∎

We now tackle the case when $A$ is a (potentially non-inner graded) complex $C^{*}$ -algebra. In this case, we recall that there is a preferred isomorphism $Cl^{1,0}\cong Cl^{0,1}$ , mapping the canonical generator of $Cl^{1,0}$ to $i$ times the canonical generator of $Cl^{0,1}$ . Furthermore, the diagram

(5.11)

commutes up to homotopy, as can be easily constructed via the usual sine-cosine formula. We can therefore identify $\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,0},A)$ with $\mathbb{K}^{G}(A_{u})$ via the route

\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{1,0},A)\simeq\mathrm{KK}^{G_{2}}_{\mathrm{ex},\infty}(Cl^{0,1},A)\simeq\mathbb{K}^{G}(A_{u}),

(5.12)

where the last equivalence stems from the induction-restriction adjunction.

Lemma 5.12.

The diagram

commutes, where the left vertical isomorphism is induced from Theorem 3.15, and the other maps are as described above.³⁷³⁷ 37 we use the same notations as in the Real case; this should not create any confusion.

Proof.

The commutativity of the upper right triangle follows from the commutativity up to homotopy of Equation 5.11. The commutativity of the trapezoid follows from explicitly chasing the definitions, similar to the proof of Section 5.1. ∎

Corollary 5.12.

Let $(A,\alpha)$ be a unital, complex $G\times C_{2}$ -equivariant complex $C^{*}$ -algebra.

1.

The composite $K_{0}^{G}(A_{u})\xrightarrow{\iota^{*}}K_{0}^{G_{2}}(A)\xrightarrow{j^{*}}K_{0}^{G}(A_{u})$ is given by

$\displaystyle K_{0}^{G}(A_{u})$ $\displaystyle\rightarrow K_{0}^{G}(A_{u})$

$\displaystyle[M]$ $\displaystyle\mapsto[M\oplus M_{\alpha}]$

The composite $K_{0}^{G_{2}}(A)\xrightarrow{j^{*}}K_{0}^{G}(A_{u})\xrightarrow{\iota^{*}}K_{0}^{G_{2}}(A)$ is given by

	$\displaystyle K_{0}^{G_{2}}(A)$	$\displaystyle\rightarrow K_{0}^{G_{2}}(A)$
	$\displaystyle[M,\alpha_{M}]$	$\displaystyle\mapsto[M\oplus M_{\alpha},\operatorname{flip}]=[M\oplus M,(m,n)\mapsto(\alpha_{M}(n),\alpha_{M}(m))]$

Proof.

Exactly the same proof as Section 5.2. ∎

Remark 5.12.

We also mention another description of the map $K_{0}^{G_{2}}(A)\rightarrow K_{0}^{G_{2}}(A)$ as in Section 5.2.2. Using the isomorphism $Cl^{1,0}\cong Cl^{0,1}$ , we can treat the composition

\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(\mathbb{C},A)\xrightarrow{j^{*}}\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(Cl^{1,0},A)\simeq\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(Cl^{0,1},A)\xrightarrow{\iota^{*}}\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}(\mathbb{C},A)

as the $\mathrm{KK}^{G_{2}}_{\mathrm{ex},0}$ product by the image under $\operatorname{res}^{C_{2}}_{G_{2}}$ of a certain element in $\mathrm{KK}^{C_{2}}_{\mathrm{ex},0}(\mathbb{C},\mathbb{C})$ . It turns out that this element is³⁸³⁸ 38 we omit the details since it is not really needed anywhere $(1,1)$ , whence the induced map must be

	$\displaystyle K_{0}^{G_{2}}(A)$	$\displaystyle\rightarrow K_{0}^{G_{2}}(A)$
	$\displaystyle[M,\alpha_{M}]$	$\displaystyle\mapsto[M\oplus M^{\operatorname{op}},\alpha_{M}\oplus(-\alpha_{M})]$

And indeed, we have an isomorphism of $G_{2}$ -equivariant $A$ -modules

	$\displaystyle M\oplus M_{\alpha}$	$\displaystyle\rightarrow M\oplus M^{\operatorname{op}}$
	$\displaystyle(m,\underline{n})$	$\displaystyle\mapsto(\frac{m+\alpha_{M}(n)}{2},\frac{m-\alpha_{M}(n)}{2}).$

Thus, we actually have isomorphisms of $G_{2}$ -equivariant $A$ -modules

(M\oplus M^{\operatorname{op}},\alpha_{M}\oplus(-\alpha_{M}))\cong(M\oplus M_{\alpha},\operatorname{flip})\cong(M\oplus M,(m,n)\mapsto(\alpha_{M}(n),\alpha_{M}(n))).

Remark 5.12.

Note that when $A$ is a non-inner graded $C^{*}$ -algebra, the fiber sequence in Equation 5.7 need not split. A concrete counterexample is $A=\mathbb{C}l^{1}$ , since $K_{0}^{C_{2}}(\mathbb{C}l^{1})\cong\mathbb{Z}$ (Section 2.1) and $K_{0}(\mathbb{C}l^{3}_{u})\cong K_{0}(\mathbb{C}l^{1}_{u})\cong\mathbb{Z}\oplus\mathbb{Z}$ .

6 Connections with classical bivariant $K$ -theories

We finally want to show that the homotopy categories of the full subcategories of the graded $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ categories spanned by separable algebras can be identified with classical graded (Kasparov’s) $KK^{G}$ ([39], [40])-category or (Connes-Higson’s) $E^{G}$ -category ([20],[29],[32]), according as $!=\operatorname{se}$ or $!=\operatorname{ex}$ , respectively. We give a self contained treatment for the case of $E$ -theory; see Section 6.4 for a discussion on the graded $KK$ -theoretic analogue.

As mentioned in the introduction, the results of this section are not relevant for our study of graded/equivariant/classical $K$ -theory as in Section 5.

6.1 Recapitulation of classical ungraded $KK^{G}$ -theory

We begin by recalling the classical definitions of the ungraded (Real) $KK^{G}$ -categories. Kasparov in his seminal work defined them in terms of now-called Kasparov modules: given $G$ - $C^{*}$ -algebras $A$ and $B$ , $\mathrm{KK}^{G}_{\mathrm{Kas}}(A,B)$ was built out of certain equivalence classes of triples $(\mathcal{E},\varphi,F)$ comprising of a Hilbert $G$ - $B$ -module $\mathcal{E}$ , a equivariant representation $\varphi$ of $A$ on the adjointable operators on $\mathcal{E}$ , and an adjointable operator $F$ on $\mathcal{E}$ satisfying certain compactness and equivariance conditions³⁹³⁹ 39 Kasparov worked with graded algebras and modules from the get-go.. It then takes a truckload of work to prove that upon restricting to separable algebras there exists a composition law, now known as the Kasparov product

\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}(A,B)\times\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}(B,C)\rightarrow\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}(A,C)

which is unital and associative, and one concludes that $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ is a category.

Since every equivariant $*$ -homomorphism naturally gives a Kasparov module, we have a canonical functor

\mathrm{kk}_{\mathrm{Kas}}^{G,\mathrm{sep}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}.

The question whether the category $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ has a universal property was settled in [58, Theorem 2.2],[46, Theorem 6.6] (with non-equivariantly the issue being first tackled in [33, Theorem 4.5]). Note that Meyer also mentions (as can be readily checked too) that the results also work in the Real case.

Theorem 6.1.

(Higson-Meyer-Thomsen) The functor $\mathrm{kk}_{\mathrm{Kas}}^{G,\mathrm{sep}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ is initial among functors to an additive $1$ -category which are reduced, homotopy invariant, ${\mathcal{K}_{G}}$ -stable and split exact.

Remark 6.1.

The following remarks about Kasparov’s $KK^{G}$ -theory are in order

1.

It is a deep result of Kasparov ([40, Theorem 2.9]) that $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ has countable coproducts, and the functor $\mathrm{kk}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ takes countable direct sums in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ to countable coproducts in $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ . An alternative proof using the Cuntz-Meyer picture of $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ can be given following [17, Corollary 11.16].
2.

One can view the additive $1$ -category $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ as a half exact additive category by taking the maximal marking and taking the distinguished half exact sequences to be all short exact sequences in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ admitting a equivariant cpc-split; the verification that this indeed is a half exact structure follows from [5, Section 7] (see also [28, Theorem 3.28] and the discussion above it).
3.

$\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ is a symmetric monoidal category and the functor $\mathrm{kk}_{\mathrm{Kas}}^{G,\mathrm{sep}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ has a symmetric monoidal refinement for both the minimal and the maximal tensor products on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ ; this is explicitly stated in [23, Theorem 4.a]

Using Theorem 6.1, we can prove the comparison result between $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ and $\mathrm{KK}^{G,\mathrm{sep}}_{\mathrm{se},0}$ .

Theorem 6.2.

We have a natural equivalence of additive, symmetric monoidal (for both symmetric monoidal refinements) categories

\mathrm{KK}^{G,\mathrm{sep}}_{\mathrm{se},0}\simeq\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}.

Proof.

By construction, the canonical functor

\mathrm{kk}^{G,\mathrm{sep}}_{\mathrm{se},0}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathrm{KK}^{G,\mathrm{sep}}_{\mathrm{se},0}

is reduced, homotopy invariant, ${\mathcal{K}_{G}}$ -stable, and additive. Thus, Theorem 6.1 says that there exists a unique additive functor $\mathcal{F}:\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}\rightarrow\mathrm{KK}^{G,\mathrm{sep}}_{\mathrm{se},0}$ such that $\mathcal{F}\circ\mathrm{kk}_{\mathrm{Kas}}^{G,\mathrm{sep}}=\mathrm{kk}^{G,\mathrm{sep}}_{\mathrm{se},0}$ .

On the other hand, the canonical functor

\mathrm{kk}_{\mathrm{Kas}}^{G,\mathrm{sep}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}

is clearly reduced, homotopy invariant and ${\mathcal{K}_{G}}$ -stable, as can be read off from Theorem 6.1. That it also satisfies the other hypothesis asked for in Section 3.4 is the content of Section 6.1. We thus obtain a unique functor $\mathcal{G}:\mathrm{KK}^{G,\mathrm{sep}}_{\mathrm{se},0}\rightarrow\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ such that $\mathcal{G}\circ\mathrm{kk}^{G,\mathrm{sep}}_{\mathrm{se},0}=\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ . $\mathcal{G}$ is furthermore symmetric monoidal, as follows from the proof of Section 3.4 and [35, Proposition 3.3.2]. Our thesis now follows from a standard yoga with universal functors. ∎

Remark 6.2.

We implicitly used in the above proof the following basic facts: If $\mathcal{C},\mathcal{D}$ are additive (resp. symmetric monoidal) categories, and $F:\mathcal{C}\rightarrow\mathcal{D}$ is an additive (resp. symmetric monoidal) functor which is an equivalence of the underlying categories, then the inverse is automatically additive (resp. symmetric monoidal).

We finally compare $\mathrm{KK}^{G}_{\mathrm{se},0}$ with the classical non-separable extension of $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ , as described in [55, Definition 5.5] and [2, Section 2.1].

Proposition 6.2.

For $A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ and $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}$ , we have a natural isomorphism of abelian groups

\mathrm{KK}^{G}_{\mathrm{Kas}}(A,B)\cong\mathrm{KK}^{G}_{\mathrm{se},0}(A,B).

Proof.

In light of Section 3.7, we have for $A$ and $B$ as in the statement of this proposition, an equivalence (with notation as in Section 3.7) of spectra

\displaystyle\operatorname{colim}_{B^{\prime}\subseteq_{\operatorname{sep}}B}\mathrm{KK}^{G}_{\mathrm{se},\infty}\bigl(\mathrm{kk}^{G}_{\mathrm{se},\infty}(A),\mathrm{kk}^{G}_{\mathrm{se},\infty}(B^{\prime})\bigr)\xrightarrow{\simeq}\mathrm{KK}^{G}_{\mathrm{se},\infty}\bigl(\mathrm{kk}^{G}_{\mathrm{se},\infty}(A),\mathrm{kk}^{G}_{\mathrm{se},\infty}(B)\bigr)

Since homotopy groups of spectra commute with filtered colimits, our thesis follows from the classical result that

\operatorname{colim}_{B^{\prime}\subseteq_{\operatorname{sep}}B}\mathrm{KK}^{G}_{\mathrm{Kas}}(A,B^{\prime})\xrightarrow{\cong}\mathrm{KK}^{G}_{\mathrm{Kas}}(A,B),

see for example [2, Lemma 2.1]. ∎

Remark 6.2.

When the $C^{*}$ -algebra $A$ in the first variable is not separable, $\mathrm{KK}^{G}_{\mathrm{se},0}(A,-)$ need not coincide with $\mathrm{KK}^{G}_{\mathrm{Kas}}(A,-)$ a’la Skandalis; it boils down to the issue that homotopy groups of homotopy limits need not be limits of homotopy groups.

6.2 Recapitulation of classical ungraded $E^{G}$ -theory (with some details)

Just as in the last subsection, we begin by recalling the classical definitions of the ungraded (Real) $E^{G}$ -categories.

Classically, ungraded $E^{G}$ -theory was defined in terms of asymptotic morphisms by Connes-Higson in [20]. The foundational works on (equivariant) $E$ -theory ([20], [29], [34]) did not discuss the Real structure, and the author (of this article) is only aware of Boersema’s work [9] on the same ⁴⁰⁴⁰ 40 which was pointed out to him by Yuezhao Li. However, loc. cit. does not discuss equivariant matters. Note that although [9] is written with real $C^{*}$ -algebras, the modifications to the Real case are obvious.

In the remainder of this subsection, we will therefore be making a comparison of $\mathrm{KK}^{G,\mathrm{sep}}_{\mathrm{ex},0}$ with a equivariant version of Boersema’s (Real) $E$ -category. The complex situation can be dealt similarly, and with lesser difficulties. Note that we omit a discussion of non-separable $E$ -theory (the analogues of Section 6.1 and Section 6.1 hold in this context too). Therefore, we use the notation $\mathrm{E}^{G}_{\mathrm{class}}$ to always denote the separable theory.

We begin with the key definitions.

Definition 6.2.

Let $A,B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ . An asymptotic morphism from $A$ to $B$ is a one-parameter family of functions $\varphi_{t}:A\to B$ with $t\in[1,\infty)$ , satisfying the continuity condition that for all $a\in A$ , the map

t\longmapsto\varphi_{t}(a):[1,\infty)\to B

is bounded and continuous, and the asymptotic conditions that for all $a,a_{1},a_{2}\in A$ and $\lambda\in\mathbb{C}$

\begin{cases}\varphi_{t}(a_{1}a_{2})-\varphi_{t}(a_{1})\varphi_{t}(a_{2})\\ \varphi_{t}(a_{1}+a_{2})-\varphi_{t}(a_{1})-\varphi_{t}(a_{2})\\ \varphi_{t}(\lambda a)-\lambda\varphi_{t}(a)\\ \varphi_{t}(a^{*})-\varphi_{t}(a)^{*}\\ \varphi_{t}(g.a)-g.\varphi_{t}(a)\end{cases}\to 0,\qquad\text{as }t\to\infty

When $A$ and $B$ are Real, we shall require that in addition

\tau(\varphi_{t}(a))-\varphi_{t}(\tau(a))\to 0\qquad\text{as }t\to\infty.

We shall denote an asymptotic morphism $\varphi:A\dashrightarrow B$ with a dashed arrow.

2.

There is a natural notion of homotopy between asymptotic morphisms which is an equivalence relation (see [29, Chapter 2]), and we denote by $[[A,B]]_{G}$ the $G$ -equivariant asymptotic homotopy classes of maps⁴¹⁴¹ 41 as in the ordinary homotopy case, if the external group $G$ is clear from the context, we omit it from the notation. from $A$ to $B$ . This forms a category $\Asy^{G}$ , whose objects are separable $G$ - $C^{*}$ -algebras, and morphisms are $G$ -equivariant asymptotic homotopy classes of maps.
3.

Just as the passage from $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ to $\mathrm{S}_{{\mathcal{K}_{G}}}\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}_{h}^{\mathrm{sep}}$ (see Section 3.2.1), one can form the category $\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}$ via left and right Bousfield localizations of $\Asy^{G}$ , whose hom monoids would be ${\mathcal{K}_{G}}$ -stabilized asymptotic homotopy classes of maps.

Note that $\Asy^{G}$ and $\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}$ are $1$ -categories; an $\infty$ -categorical refinement of these categories has been studied in [11] when $G$ is a discrete countable group. The methods of loc. cit. would not generalize to general locally compact second countable groups. We anyway work only with the $1$ -categorical variants in this section.

The classical $E^{G}$ -groups are given by

\mathrm{E}^{G}_{\mathrm{class}}(A,B):=\operatorname{hom}_{\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}}(\Omega A,\Omega B)=[[\Omega A\otimes{\mathcal{K}_{G}},\Omega B\otimes{\mathcal{K}_{G}}]],

and yield a symmetric monoidal additive category structure on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ with respect to the maximal tensor product on $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ . That it indeed becomes a symmetric monoidal additive category can be seen by combining the approaches and relevant results of [29] and [9]. There is an obvious functor

\mathrm{e}^{G}_{\mathrm{class}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathrm{E}^{G}_{\mathrm{class}}.

5.

One can view the additive $1$ -category $\mathrm{E}^{G}_{\mathrm{class}}$ as a half exact additive category by taking the maximal marking and taking the distinguished half exact sequences to be all short exact sequences in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ ; see Section 6.2.3 for details. In the remainder of this subsection, we by default work with this half exact structure.

For later use, we mention a few technical results about asymptotic morphisms now. In the non-equivariant case, some of these results are also discussed in the PhD thesis of Andreas Thom ([57, Section 2.2]) with a little more elaborations.

Lemma 6.2.

Let $A,B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ . Denote by $\widetilde{[[A,B]]}$ the set of homotopy classes of asymptotic morphisms $\varphi:A\dashrightarrow B$ which have the property that $\varphi_{1}(a)=0\penalty\ \forall\penalty\ a\in A$ . Homotopies are also required to satisfy this condition. Then the natural map

\Phi_{A,B}:\widetilde{[[A,B]]}\rightarrow[[A,B]]

is an isomorphism.

Proof.

We just write a map in the other direction; that it is well-defined and inverse to $\Phi_{A,B}$ is a routine check.

Fix a function $f:[1,\infty)\rightarrow[0,1]$ with the property that $f(1)=0,f(t)=1\penalty\ \forall\penalty\ t\geq 2$ . Then the pointwise multiplication of $f$ with an element of $[[A,B]]$ does the job. ∎

The above lemma has the following interesting consequence. For $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , denote by $C_{0b}([1,\infty),B)$ the (non-separable) $G$ - $C^{*}$ -algebra of $G$ -continuous, continuous, bounded functions $f:[1,\infty)\rightarrow B$ which further satisfy $f(1)=0$ . Section 6.2 shows that the datum of an (equivalence class) of an asymptotic morphism $A\dashrightarrow B$ is the same datum as a $G$ -equivariant homomorphism $A\rightarrow\frac{C_{0b}([1,\infty),B)}{C_{0}((1,\infty),B)}$ . Thus, we can naturally associate to an asymptotic morphism $\varphi:A\dashrightarrow B$ the extension given in the top row of the diagram

(6.3)

where the right square is a pullback. Note that $P_{\varphi}\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ courtesy both $C_{0}((1,\infty),B)$ and $A$ being separable.

Lemma 6.3.

In the notation of the above discussion, let $\psi:\Omega A\dashrightarrow\Omega B$ be the asymptotic morphism obtained by applying the equivariant (Real) Connes-Higson construction ([29, Proposition 5.5], [9, Proposition 4.3]) to the top extension in Equation 6.3. Then $[\psi]=-[\Omega\varphi]$ in $[[\Omega A,\Omega B]]$ .

Proof.

This is the content of [37, Lemma 1.3], see also [59, Lemma 3.4]. Note that there is a sign error in the analogous assertion [29, Proposition 5.12]. ∎

Our first important result of this subsection is to prove the following theorem, which is folklore even in the complex case: to the best of the author’s knowledge, the classical literature on $E$ -theory just state the result without a proof; see also [11, Remark 3.33].

Theorem 6.4.

(folklore) The functor $\mathrm{e}^{G}_{\mathrm{class}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathrm{E}^{G}_{\mathrm{class}}$ is initial among functors to an additive $1$ -category which are reduced, homotopy invariant, ${\mathcal{K}_{G}}$ -stable and half exact.

A (detailed) proof of the above result will take us quite some time. We begin with some preparations. The first check one needs to make to prove Theorem 6.4 is that $\mathrm{e}^{G}_{\mathrm{class}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathrm{E}^{G}_{\mathrm{class}}$ is homotopy invariant, ${\mathcal{K}_{G}}$ -stable and exact. While the first two are obvious from the definition, half exactness is not so. The first crucial step to that end is demonstrating for each $A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , the functor

\mathrm{E}^{G}_{\mathrm{class}}(A,-):\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\operatorname{Ab}

is split exact. This follows directly from the equivariant, Real analogue of [8, Corollary 25.5.7]. However, it can also be seen as a consequence of the following stronger result, which is also going to come handy in the next subsection. The complex, non-equivariant case of the result is due to Dadarlat-Loring ([24, Proposition 3.2]).

Proposition 6.4.

For each $D\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , the functor $[[D,-]]:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\operatorname{Set}_{*}$ is split exact.

Proof.

Let $0\rightarrow A\rightarrow B\xrightarrow{f}C\rightarrow 0$ be a short exact sequence in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ which is split by a homomorphism $\sigma:C\rightarrow B$ . Without loss of generality, we can assume that the map $A\rightarrow B$ is the inclusion of an ideal. From the standard excision properties in the asymptotic category and Puppe sequences in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ ,([29, Proposition 5.16], [32, Proposition 2.5]), we get a long exact sequence of pointed sets

...\rightarrow[[D,\Omega B]]\xrightarrow{\Omega f_{*}}[[D,\Omega C]]\xrightarrow{\partial}[[D,C(f)]]\rightarrow[[D,B]]\xrightarrow{f_{*}}[[D,C]]\rightarrow 0

(6.5)

where $C(f)$ denotes the mapping cone of $f$ . Since $f$ is split by $\sigma$ , $\Omega f$ is split by $\Omega\sigma$ , and we can extract the split exact sequence

0\rightarrow[[D,C(f)]]\rightarrow[[D,B]]\xrightarrow{f_{*}}[[D,C]]\rightarrow 0

from Equation 6.5. To prove our thesis, it therefore suffices to show that the inclusion $A\rightarrow C(f)$ is an isomorphism in $\Asy^{G}$ .

Using the split $\sigma$ , we can re-write $C(f)$ as

C(f)=\{(a+\sigma(\gamma(0)),\gamma)\mid a\in A\}\subseteq B\oplus PC.

Let $u_{t}$ be a (Real) quasicentral approximate unit in $A$ ([29, Lemma 5.3]). In light of [29, Lemma 5.6.(iii)], we can now define a $1$ -parameter family of homomorphisms

	$\displaystyle\varphi_{t}:C_{0}(0,1]$	$\displaystyle\rightarrow A$
	$\displaystyle f$	$\displaystyle\mapsto f(u_{t})$

given by functional calculi of the individual $u_{t}$ . We can now define an asymptotic morphism given by the family of maps

	$\displaystyle\rho_{t}:C_{0}(0,1]\otimes C$	$\displaystyle\rightarrow A$
	$\displaystyle f\otimes c$	$\displaystyle\mapsto f(u_{t})\sigma(c)$

Note that in defining it, we used the universal property of the maximal tensor product, see [8, Section 25.2.3]. We can now define an asymptotic morphism $\varphi:C(f)\dashrightarrow A$ given by

\varphi_{t}(a+\sigma(\gamma(0)),\gamma)=a+\rho_{t}(f).

The verification that it indeed is an inverse to the inclusion $A\rightarrow C(f)$ in $\Asy^{G}$ can now be taken from the proof of [24, Proposition 3.2]. The equivariance and Real structure do not pose any additional difficulties. ∎

As the reader may guess, the reason we care about the split exactness result is to bring $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ (or equivalently, $\mathrm{KK}_{!,0}^{G,\mathrm{sep}}$ ) into the game. From results of the previous subsection and playing around a bit with the uniqueness of universal functors ([9, Theorem 3.2]), we conclude that there is a unique functor $c:\mathrm{KK}_{!,0}^{G,\mathrm{sep}}\rightarrow\mathrm{E}^{G}_{\mathrm{class}}$ such that $c\circ\mathrm{kk}_{!,0}^{G,\mathrm{sep}}=\mathrm{e}^{G}_{\mathrm{class}}$ ⁴²⁴² 42 c stands for comparison functor, which is furthermore symmetric monoidal with respect to the maximal tensor product.

Since we already have knowledge of periodicity in $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ (equivalently, in $\mathrm{KK}_{!,0}^{G,\mathrm{sep}}$ ), we can now get periodicity results in $\mathrm{E}^{G}_{\mathrm{class}}$ too. All this (and a little more) is the content of the next lemma.

Lemma 6.5.

1.

Let $F:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathcal{D}$ be a functor to a half exact additive $1$ -category which is homotopy invariant, ${\mathcal{K}_{G}}$ -stable, half exact with respect to exact sequences. Then there is an isomorphism

$F(C_{0}(\mathbb{R}^{2,2})\otimes A)\xrightarrow[\cong]{\operatorname{Bott}^{F}}F(A)$

which is natural with respect to morphisms in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ and also with respect to the boundary maps of the long exact sequence associated to $F$ , as described in [8, Theorem 21.4.4].
2.

The natural map

$C_{0}(\mathbb{R}^{0,1})\otimes-:\mathrm{E}^{G}_{\mathrm{class}}(A,B)\rightarrow\mathrm{E}^{G}_{\mathrm{class}}(\Omega A,\Omega B)$

is an isomorphism.
3.

The functor $\mathrm{e}^{G}_{\mathrm{class}}:\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\mathrm{E}^{G}_{\mathrm{class}}$ is homotopy invariant, ${\mathcal{K}_{G}}$ -stable, and half exact.

The natural map

C_{0}(\mathbb{R}^{1,0})\otimes-:\mathrm{E}^{G}_{\mathrm{class}}(A,B)\rightarrow\mathrm{E}^{G}_{\mathrm{class}}(C_{0}(\mathbb{R}^{1,0})\otimes A,C_{0}(\mathbb{R}^{1,0})\otimes B)

is an isomorphism.

The isomorphisms of Assertion 1 applied to the functors $\mathrm{E}^{G}_{\mathrm{class}}(A,-)$ and $\mathrm{E}^{G}_{\mathrm{class}}(-,C_{0}(\mathbb{R}^{2,2})\otimes B)$ fit into a commutative diagram

Proof.

We begin with Assertion 1. In light of the given properties of $F$ , the Real analog of the proof of [29, Proposition 6.16] shows that there exists an asymptotic morphism

\alpha:C_{0}(\mathbb{R}^{1,1})\dashrightarrow\mathcal{K}(l^{2}\mathbb{N})

such that for each $A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , the asymptotic morphism

\alpha\otimes 1_{A}:C_{0}(\mathbb{R}^{1,1})\otimes A\dashrightarrow\mathcal{K}(l^{2}\mathbb{N})\otimes A

yields a natural isomorphism

F(C_{0}(\mathbb{R}^{1,1})\otimes A)\xrightarrow[\cong]{\operatorname{Bott}_{A}^{F}}F(A).

Our sought after natural map is now obtained by applying the above isomorphism twice

F(C_{0}(\mathbb{R}^{2,2})\otimes A)\xrightarrow[\cong]{\operatorname{Bott}_{C_{0}(\mathbb{R}^{1,1})\otimes A}^{F}}F(C_{0}(\mathbb{R}^{1,1})\otimes A)\xrightarrow[\cong]{\operatorname{Bott}_{A}^{F}}F(A).

The map in Assertion 2 being an isomorphism can be readily taken from [9, Lemma 4.5], with the obvious modifications to tackle the equivariant case.

We now prove Assertion 3 (note that it settles the easy part of Theorem 6.4). That $\mathrm{e}^{G}_{\mathrm{class}}$ is homotopy invariant and ${\mathcal{K}_{G}}$ -stable is obvious. That it is also half exact follows by combining [9, Theorem 4.4] and the excision theory of equivariant $E$ -theory developed in chapters 5 and 6 of [29]. The careful reader will observe that (variants of) Assertions 1 and 2 go as an input to proving the exactness in the first variable in [9] and [29].

For Assertion 4, we can accommodate the map in the commutative diagram

The discussion preceding the lemma shows that the horizontal arrow is an isomorphism, since

C_{0}(\mathbb{R}^{1,1})\simeq_{\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}}\mathbb{C}.

The right arrow is an isomorphism from Assertion 2.

Finally we show Assertion 5. In light of Assertion 1, it essentially follows from the commutativity of the diagram

for every asympototic morphism $f:A\dashrightarrow B$ , as can be seen from the symmetric monoidal properties of $\Asy^{G}$ . See also [29, Theorem 4.6] and the proof of [29, Proposition 6.16]. ∎

Before continuing with the proof of Theorem 6.4, we mention a corollary to some results of the last lemma, which will come handy in the next section.

Corollary 6.5.

There exists an asymptotic morphism $\alpha:C_{0}(\mathbb{R}^{1,1})\dashrightarrow\mathcal{K}(l^{2})$ such that

\alpha\otimes 1_{B}:C_{0}(\mathbb{R}^{1,1})\otimes B\dashrightarrow\mathcal{K}(l^{2})\otimes B

determines an invertible class in $\mathrm{E}^{G}_{\mathrm{class}}(C_{0}(\mathbb{R}^{1,1})\otimes B,B)$ . Furthermore, the functor

\Omega:\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}\rightarrow\mathrm{E}^{G}_{\mathrm{class}}

exhibits $\mathrm{E}^{G}_{\mathrm{class}}$ as the $1$ -categorical localization of $\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}$ at the classes of maps $\alpha\otimes 1_{B}$ .

Proof.

The first statement follows readily from the proof of Assertion 1 of Section 6.2 by taking $\mathcal{D}=\mathrm{E}^{G}_{\mathrm{class}}$ . The second statement follows from the following general fact about $(\infty-)$ categories (it is the dual version of [43, Proposition 5.2.7.4]) : Let $\mathcal{C}$ be a category with an endofunctor $R:\mathcal{C}\rightarrow\mathcal{C}$ and a natural transformation $\mathfrak{T}:R\implies\operatorname{id}_{\mathcal{C}}$ . If for every object $C\in\mathcal{C}$ the morphims

\mathfrak{T}_{R(C)},R(\mathfrak{T}_{C}):R(R(C))\rightarrow R(C)

are equivalences, then $R:\mathcal{C}\rightarrow R\mathcal{C}$ is the right-adjoint of a right Bousfield localization with counit $\mathfrak{T}$ . It is also the Dwyer-Kan localization at the set of morphisms $W_{R}:=\{\mathfrak{T}_{C}\mid C\in\mathcal{C}\}$ .

Indeed, in the above notation, take $\mathcal{C}:=\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}$ , and $R:=C_{0}(\mathbb{R}^{1,1})\otimes-:\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}\rightarrow\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}$ . The rest follows from the results of Section 6.2 and the definition of $\mathrm{E}^{G}_{\mathrm{class}}$ . ∎

The heart of the proof of the uniqueness assertion in Theorem 6.4 is contained in the following result.

Proposition 6.5.

(cf. [59, Theorem 4.6]) Let $F$ be a homotopy invariant, ${\mathcal{K}_{G}}$ -stable and half exact functor from $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ to $\operatorname{Ab}$ . Fix $A$ in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ and consider an arbitrary element $d\in F(A)$ . Then there is a unique natural transformation

T:\mathrm{E}^{G}_{\mathrm{class}}(A,-)\implies F(-)

such that $T_{A}([\operatorname{id}_{A}])=d$ .

Proof.

Firstly, we massage the hom groups of $\mathrm{E}^{G}_{\mathrm{class}}$ a bit, and write them as

\mathrm{E}^{G}_{\mathrm{class}}(A,B)=[[C_{0}(\mathbb{R}^{2,1})\otimes A\otimes{\mathcal{K}_{G}},C_{0}(\mathbb{R}^{2,1})\otimes B\otimes{\mathcal{K}_{G}}]].

Indeed this can be done from Assertions 2 and 4 of Section 6.2 (the reason we do this maneuver will be clear from the upcoming proof, but see also Section 6.2).

To avoid notational clutter, we subsequently denote $C_{0}(\mathbb{R}^{p,q})\otimes X$ as $\mathbb{R}^{p,q}X$ for any $X\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ . Same comment applies to functors involving these algebras.

Now consider $x\in[[\mathbb{R}^{2,1}A\otimes{\mathcal{K}_{G}},\mathbb{R}^{2,1}B\otimes{\mathcal{K}_{G}}]]$ . Let it be represented by the equivariant asymptotic morphism

\varphi:\mathbb{R}^{2,1}A\otimes{\mathcal{K}_{G}}\dashrightarrow\mathbb{R}^{2,1}B\otimes{\mathcal{K}_{G}}.

As in Equation 6.3, this yields an extension

0\rightarrow\mathbb{R}^{2,2}B\otimes{\mathcal{K}_{G}}\rightarrow P_{\varphi}\rightarrow\mathbb{R}^{2,1}A\otimes{\mathcal{K}_{G}}\rightarrow 0.

(6.6)

Consequently, we obtain a morphism $F(\mathbb{R}^{2,2}A\otimes{\mathcal{K}_{G}})\rightarrow F(\mathbb{R}^{2,2}B\otimes{\mathcal{K}_{G}})$ in $\operatorname{Ab}$ via the connecting map of the long exact sequence associated to $F$ as in [8, Theorem 21.4.4]. The ${\mathcal{K}_{G}}$ -stability of $F$ together with Assertion 1 of Section 6.2 naturally produces a morphism $T_{x}:F(A)\rightarrow F(B)$ . Now define

	$\displaystyle T_{B}:[[\mathbb{R}^{2,1}A\otimes{\mathcal{K}_{G}},\mathbb{R}^{2,1}B\otimes{\mathcal{K}_{G}}]]\rightarrow F(B)$
	$\displaystyle x\mapsto T_{x}(d).$

In light of the machinery we developed in this subsection and functoriality properties of the Connes-Higson construction, the well-definedness and naturality of $T_{B}$ , as well as the verification $T_{A}([\operatorname{id}_{A}])=d$ is a routine check, as has been done in the proof of [59, Theorem 4.6].

We now prove the uniqueness part. Let

T^{\prime}:[[\mathbb{R}^{2,1}A\otimes{\mathcal{K}_{G}},\mathbb{R}^{2,1}-\otimes{\mathcal{K}_{G}}]]\rightarrow F(-)

be a natural transformation such that $T^{\prime}([\operatorname{id}_{A}])=d$ . Consider $x=[\varphi]\in[[\mathbb{R}^{2,1}A\otimes{\mathcal{K}_{G}},\mathbb{R}^{2,1}B\otimes{\mathcal{K}_{G}}]]$ . Let $\partial_{1},\partial_{2},\partial_{3},\partial_{4}$ denote the connecting maps obtained by applying the functors

[[\mathbb{R}^{4,3}A\otimes{\mathcal{K}_{G}},\mathbb{R}^{2,1}-]],[[\mathbb{R}^{2,1}A\otimes{\mathcal{K}_{G}},\mathbb{R}^{2,1}-]],F(\mathbb{R}^{2,2}-),F(-)

respectively to the extension in Equation 6.6⁴³⁴³ 43 denoting by $\mathcal{G}$ any one of these four functors, we are implicitly using the natural identification of $\mathcal{G}(-)$ with $\mathcal{G}(\mathbb{R}^{2,2}\otimes{\mathcal{K}_{G}}\otimes-)$ in what follows.. We get a commutative diagram

(6.7)

where the undirected edges are induced by the maps described in Section 6.2.1. Note that the fact that the diagram indeed commutes needs a lot of the machinery we developed earlier in this subsection.

We now claim the following.

Claim.

The composite

\partial_{1}\circ(\mathbb{R}^{2,2}-):\mathrm{E}^{G}_{\mathrm{class}}(A,A)\rightarrow\mathrm{E}^{G}_{\mathrm{class}}(\mathbb{R}^{2,2}A,\mathbb{R}^{2,2}B)

equals the composite

(\mathbb{R}^{2,2}-)\circ\varphi_{*}:\mathrm{E}^{G}_{\mathrm{class}}(A,A)\rightarrow\mathrm{E}^{G}_{\mathrm{class}}(\mathbb{R}^{2,2}A,\mathbb{R}^{2,2}B).

Proof.

We build the diagram

We need to show that it commutes. The bottom rectangle commutes by the symmetric monoidal properties of $\mathrm{E}^{G}_{\mathrm{class}}$ . The middle rectangle commutes by an application of Section 6.2, together with the fact that composition with the Connes-Higson morphism yields the boundary map in $\mathrm{E}^{G}_{\mathrm{class}}$ , as described in [29, Theorem 6.15]. The top square commuting is a consequence of the general fact that for any $A^{\prime},B^{\prime},C^{\prime}\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ and $\psi\in\mathrm{E}^{G}_{\mathrm{class}}(B^{\prime},C^{\prime})$ , the diagram

commutes. This boils down to the fact that the map $S^{1}\wedge S^{1}\xrightarrow{\operatorname{flip}}S^{1}\wedge S^{1}$ has degree $-1$ .

∎

The above claim therefore shows that the composite of the purple arrows (we are reversing the direction of the last arrow so as to compose) maps $[\operatorname{id}_{A}]\in\mathrm{E}^{G}_{\mathrm{class}}(A,A)$ to $[\varphi]\in\mathrm{E}^{G}_{\mathrm{class}}(A,B)$ . The commutativity of Equation 6.7 now tells that

\partial_{4}(T_{A}^{\prime}([\operatorname{id}_{A}]))=T_{B}^{\prime}([\varphi]).

Since $T_{A}^{\prime}([\operatorname{id}_{A}])$ is given to be $d$ , the proof of the proposition follows. ∎

Remark 6.7.

In the above proof, $(2,2)$ -periodicity played a crucial role. It would indeed be natural to expect that the same could be done by the more standard $(1,1)$ -periodicity. Unraveling the proof of Section 6.2, one would then want to take

\mathrm{E}^{G}_{\mathrm{class}}(A,B)=[[\mathbb{R}^{1,0}A\otimes{\mathcal{K}_{G}},\mathbb{R}^{1,0}B\otimes{\mathcal{K}_{G}}]].

But it is not clear why this definition coincides with $[[\mathbb{R}^{0,1}A\otimes{\mathcal{K}_{G}},\mathbb{R}^{0,1}B\otimes{\mathcal{K}_{G}}]]$ : the results of Section 6.2 do not say this! Tracing further back, the reader may wonder “why not define $\mathrm{E}^{G}_{\mathrm{class}}(A,B):=[[\mathbb{R}^{1,0}A\otimes{\mathcal{K}_{G}},\mathbb{R}^{1,0}B\otimes{\mathcal{K}_{G}}]]$ ?” The reason is that then it would not be clear why this category has excision; the development of the excision results in $E$ -theory and the Connes-Higson morphism depend on Puppe sequences, where the loops are $C_{0}(\mathbb{R}^{0,1})$ and not $C_{0}(\mathbb{R}^{1,0})$ .

Proof of Theorem 6.4.

With Section 6.2 at our disposal, the argument now follows by standard yoga with the uniqueness of the natural transformations described in loc. cit. See for example [33, Theorem 4.5]. A few more details on a similar line of ideas is in [59, Theorem 4.9]. ∎

We can now compare $\mathrm{KK}^{G,\mathrm{sep}}_{\mathrm{ex},0}$ with $E^{G}_{\operatorname{class}}$ .

Theorem 6.8.

We have a natural equivalence of additive, symmetric monoidal categories

\mathrm{KK}^{G,\mathrm{sep}}_{\mathrm{ex},0}\simeq\mathrm{E}^{G}_{\mathrm{class}}.

Proof.

The proof of Theorem 6.2 can be readily adapted to this case. The main check one needs to make is that $\mathrm{E}^{G}_{\mathrm{class}}$ has countable coproducts, and the functor $\mathrm{E}^{G}_{\mathrm{class}}$ takes countable direct sums in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ to countable coproducts in $\mathrm{E}^{G}_{\mathrm{class}}$ . But this is the content of [29, Proposition 7.1]. The Real structure poses no additional difficulty. ∎

6.3 A technical result of Dadarlat-Loring

In this subsection, we prove the following technical result, which will be handy in the later subsections when we study graded $E$ -theory. The complex, non-equivariant case of the result is due to Dadarlat-Loring ([24])

Theorem 6.9.

(Generalized Dadarlat-Loring) Let $A\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ . Suppose that for all $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ the abelian monoid

\operatorname{hom}_{\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}}(A,B)=[[A\otimes{\mathcal{K}_{G}},B\otimes{\mathcal{K}_{G}}]]

is a group. Then, the map

\Omega:[[A\otimes{\mathcal{K}_{G}},B\otimes{\mathcal{K}_{G}}]]\rightarrow\mathrm{E}^{G}_{\mathrm{class}}(A,B)

is an isomorphism of abelian groups for all $B\in\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ .

Remark 6.9.

An $A$ in $\mathsf{G}\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ satisfying the hypothesis of the above theorem is called homotopy symmetric. Note that it is equivalent to asking that the abelian monoid $[[A\otimes{\mathcal{K}_{G}},A\otimes{\mathcal{K}_{G}}]]$ is a group.

Our proof strategy will be very different from [24], except that we too use Section 6.2 in a crucial way. We begin by proving the following result, wherein we use the notation from the last subsection, especially Section 6.2.

Proposition 6.9.

For each homotopy symmetric $A\in\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}$ , the functor

\operatorname{hom}_{\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}}(A,-):\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}\rightarrow\operatorname{Ab}

inverts $\alpha:C_{0}(\mathbb{R}^{1,1})\dashrightarrow\mathcal{K}(l^{2})$ .

Proof.

In light of Section 6.2 and the universal property of $KK^{G}_{\operatorname{Kas}}$ , there is a unique dashed arrow rendering the below diagram commutative.

(6.10)

We now momentarily ignore the external group $G$ . Since $\mathrm{KK}_{\mathrm{Kas}}^{\mathrm{sep}}$ satisfies $(1,1)$ -periodicity, we know in light of the Cuntz picture of $\mathrm{KK}_{\mathrm{Kas}}^{\mathrm{sep}}$ that there is a morphism $q\mathbb{C}\xrightarrow{\beta}C_{0}(\mathbb{R}^{1,1})\otimes\mathcal{K}(l^{2})$ in $\mathsf{C}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ which implements the $(1,1)$ -periodicity. There is also a homomorphism $q\mathbb{C}\xrightarrow{\gamma}\mathcal{K}$ given by the composite

q\mathbb{C}\xrightarrow{\pi_{1}}\mathbb{C}\xrightarrow{\iota}\mathcal{K}(l^{2}).

We now claim the following.

Claim.

The diagram

(6.11)

commutes upto a sign in the $\mathcal{K}(l^{2}\mathbb{N})$ -stabilized asymptotic homotopy category $L_{\mathcal{K}}\!\operatorname{Asy}$ .

Proof.

We consider the following diagram, where the vertical maps are induced by the localization functor $L_{\mathcal{K}}\!\operatorname{Asy}\rightarrow E_{\operatorname{class}}$ (i.e. the trivial group case of Section 6.2).

$\begin{array}[]{ccccc}[[q\mathbb{C},C_{0}(\mathbb{R}^{1,1})\otimes\mathcal{K}(l^{2}\mathbb{N})]]&\times&[[C_{0}(\mathbb{R}^{1,1})\otimes\mathcal{K}(l^{2}\mathbb{N}),\mathcal{K}(l^{2}\mathbb{N})]]&\longrightarrow&[[q\mathbb{C},\mathcal{K}(l^{2}\mathbb{N})]]\cong\mathbb{Z}\\[5.16663pt] \downarrow&&\downarrow&&\downarrow\scriptstyle{\cong}\\[5.16663pt] E_{\operatorname{class}}\bigl(q\mathbb{C},C_{0}(\mathbb{R}^{1,1})\otimes\mathcal{K}(l^{2}\mathbb{N})\bigr)&\times&E_{\operatorname{class}}\bigl(C_{0}(\mathbb{R}^{1,1})\otimes\mathcal{K}(l^{2}\mathbb{N}),\mathcal{K}(l^{2}\mathbb{N})\bigr)&\longrightarrow&E_{\operatorname{class}}\bigl(q\mathbb{C},\mathcal{K}(l^{2}\mathbb{N})\bigr)\cong\mathbb{Z}\\[2.15277pt] \cong\mathbb{Z}&&\cong\mathbb{Z}&&\end{array}$

We now justify the isomorphisms appearing on the diagram. That $[[q\mathbb{C},\mathcal{K}(l^{2}\mathbb{N})]]\cong\mathbb{Z}$ follows from

\mathbb{Z}\xrightarrow[\cong]{\lx@cref{creftypecap~refnum}{prop: Cuntz picture k-theory}}[q\mathbb{C},\mathcal{K}(l^{2}\mathbb{N})]\xrightarrow[\cong]{\lx@cref{creftypecap~refnum}{rem: qA properties}.6}[[q\mathbb{C},\mathcal{K}(l^{2}\mathbb{N})]].

We furthermore note that $\gamma$ corresponds to $1$ under this isomorphism; see Section 2.1. The isomorphisms in the bottom row follow since $q\mathbb{C},\mathcal{K}(l^{2}\mathbb{N}),C_{0}(\mathbb{R}^{1,1})$ are all $KK_{\operatorname{Kas}}^{\operatorname{sep}}$ -equivalent to $\mathbb{C}$ , and $K_{0}(\mathbb{C})\cong\mathbb{Z}$ . That the right vertical map is an isomorphism follows from the composite map

[q\mathbb{C},\mathcal{K}(l^{2}\mathbb{N})]\xrightarrow{\cong}[[q\mathbb{C},\mathcal{K}(l^{2}\mathbb{N})]]\rightarrow E_{\operatorname{class}}(\mathbb{C},\mathbb{C})

being an isomorphism from Section 3.6 (we are implicitly also using the universal property of $\mathcal{K}(l^{2})$ -stabilized homotopy category of separable $C^{*}$ -algebras here). From the discussion preceding the claim, $\beta$ and $\alpha$ map to a generator in $\mathbb{Z}$ under the two left vertical arrows. An argument similar to the one in Section 4.1 shows that the bottom bilinear map must send $1\times 1$ to $\pm 1$ . The conclusion follows. ∎

We are now in a position to bring back the group $G$ . Since all the algebras in Equation 6.11 have the trivial $G$ -action, we conclude that Equation 6.11 also commutes in $\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}$ . Furthermore, $\gamma$ and $\beta$ remain equivalences in $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ , whence so is $\alpha$ . The proof of the proposition follows in light of Equation 6.10. ∎

Proof of Theorem 6.9.

In light of Section 6.2 and general facts about mapping spaces (in our context, mapping abelian groups) out of colocal objects ([17, Remark 9.9]), it suffices to show that the functor

\operatorname{hom}_{\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}}(A,-):\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}\rightarrow\operatorname{Ab}

inverts the morphisms $\alpha\otimes 1_{B}:C_{0}(\mathbb{R}^{1,1})\otimes B\dashrightarrow\mathcal{K}(l^{2}\mathbb{N})\otimes B$ . But this follows upon tensoring Equation 6.11 with $B$ ; the diagram continues to commute up to a sign (which is independent of $B$ ), and the morphisms $q\mathbb{C}\otimes B\xrightarrow{\beta\otimes 1_{B}}C_{0}(\mathbb{R}^{1,1})\otimes\mathcal{K}(l^{2}\mathbb{N})\otimes B$ and $q\mathbb{C}\otimes B\xrightarrow{\gamma\otimes 1_{B}}\mathcal{K}(l^{2}\mathbb{N})\otimes B$ are isomorphisms in $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ . ∎

Remark 6.11.

The following remarks about Theorem 6.9 are in order.

1.

There is a non-technical way to think about the result. Note that Section 6.2 roughly says that one obtains $\mathrm{E}^{G}_{\mathrm{class}}$ from $\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}$ by “inverting Bott”. When $A$ is homotopy symmetric, Equation 6.10 shows that the functor

$\operatorname{hom}_{\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}}(A,-):\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}\rightarrow\operatorname{Ab}$

indeed “inverts Bott”, since it factors through $\mathrm{KK}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ . Section 6.3 basically makes this line of argument rigorous. And probably this is also the argument which was hinted at the end of Section 4 in [24].
2.

A treatment analogous to Section 3.4 would now show that $\mathrm{E}^{G}_{\mathrm{class}}$ is the full subcategory of group objects of $\mathrm{S}_{{\mathcal{K}_{G}}}\Asy^{G}$ . This seems to not have been mentioned in the literature.

6.4 Comparison with classical, graded $\widehat{E}^{G}$ -theory

We are finally in a position to discuss classical, graded $\widehat{\mathrm{E}}^{G}_{\mathrm{class}}$ -categories. The main reference for the same is [32, Section 2] (in the complex case). Following the formulation given there, we introduce its Real analogue. In what follows, $[[A,B]]_{\operatorname{gr}}$ denotes the homotopy classes of graded, $G$ -equivariant asymptotic morphisms from $A$ to $B$ . Recall that $\widehat{\mathrm{E}}^{G}_{\mathrm{class}}$ still denotes the separable theory.

Definition 6.11.

(cf. [32, Chapter 2]) For $A,B\in\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , define a category $\widehat{\mathrm{E}}^{G}_{\mathrm{class}}$ with objects the objects of $\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}$ , and morphims from $A$ to $B$ being given by

\widehat{\mathrm{E}}^{G}_{\mathrm{class}}(A,B):=[[\mathcal{S}\hat{\otimes}A\hat{\otimes}\hat{\mathcal{K}}_{G},B\hat{\otimes}\hat{\mathcal{K}}_{G}]]_{\operatorname{gr}}.

The composition law in $\widehat{\mathrm{E}}^{G}_{\mathrm{class}}$ is given by the usual formula (see [32, Definition 2.9]) using the codiagonal map $\mathcal{S}\xrightarrow{\Delta}\mathcal{S}\hat{\otimes}\mathcal{S}$ . Denote by

\widehat{\mathrm{e}}^{G}_{\mathrm{class}}:\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}^{\mathrm{sep}}\rightarrow\widehat{\mathrm{E}}^{G}_{\mathrm{class}}

the canonical functor induced by precomposition with the homomorphism $\mathcal{S}\xrightarrow{\operatorname{ev}_{0}}\mathbb{C}$ .

To connect $\widehat{\mathrm{E}}^{G}_{\mathrm{class}}$ and $\widehat{\mathrm{KK}}_{\mathrm{ex},0}^{G,\mathrm{sep}}$ , we need the following lemma.

Lemma 6.11.

The abelian monoid $[[\mathcal{S}\hat{\otimes}A\hat{\otimes}\hat{\mathcal{K}}_{G},B\hat{\otimes}\hat{\mathcal{K}}_{G}]]_{\operatorname{gr}}$ is a group for all $A,B\in\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ .

Proof.

This follows from an argument very similar to the argument given in the proof of Theorem 2.3.1. See [32, Lemma 2.1] for more details. ∎

Theorem 6.12.

1.

For all $A,B\in\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , we have a natural isomorphism $\mathrm{E}^{G_{2}}_{\mathrm{class}}(\mathcal{S}\hat{\otimes}A,B)\cong\widehat{\mathrm{E}}^{G}_{\mathrm{class}}(A,B)$ .
2.

We have an equivalence of additive, symmetric monoidal (for the graded maximal tensor product) categories

$\widehat{\mathrm{KK}}_{\mathrm{ex},0}^{G,\mathrm{sep}}\simeq\widehat{\mathrm{E}}^{G}_{\mathrm{class}}.$
3.

For ungraded $C^{*}$ -algebras $A$ and $B$ in $\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ , we have a natural isomorphism

$\widehat{\mathrm{E}}^{G}_{\mathrm{class}}(A,B)\cong\mathrm{E}^{G}_{\mathrm{class}}(A,B).$

Proof.

Assertion 1 follows immediately from Theorem 6.9 and Section 6.4, keeping in mind that $\hat{\mathcal{K}}_{G}$ is inner graded. Assertion 2 follows from Theorem 6.8 for the group $G\times C_{2}$ (with the monoidal structure being the graded maximal tensor product), Assertion 1, and Section 4.5. Assertion 3 was shown in the context of our stable $\infty$ -categories in Theorem 4.2. We are therefore done by Assertion 2. ∎

We end the article with a remark circling the analogous comparison between $\widehat{KK}_{\operatorname{Kas}}^{G,\operatorname{sep}}$ and $\widehat{\mathrm{KK}}_{\mathrm{se},0}^{G,\mathrm{sep}}$ .

Remark 6.12.

In light of the main result of [59], we can rework the last two subsections with completely positive asymptotic morphisms, and obtain that for all $A,B\in\mathsf{G}\widehat{\mathsf{C}}^{*}\!\mathsf{Alg}^{\mathrm{sep}}$ ,

\widehat{\mathrm{KK}}_{\mathrm{se},0}^{G,\mathrm{sep}}(A,B)\cong[[\mathcal{S}\hat{\otimes}A\hat{\otimes}\hat{\mathcal{K}}_{G},B\hat{\otimes}\hat{\mathcal{K}}_{G}]]_{\operatorname{gr}}^{\operatorname{cp}}.

However, unlike the definition of $\widehat{\mathrm{E}}^{G}_{\mathrm{class}}$ (which on the face of it is by asymptotic morphisms), the definition of $\widehat{\mathrm{KK}}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ is in terms of Kasparov cycles. Therefore to complete the circle of ideas, we would need to obtain a natural isomorphism

\widehat{\mathrm{KK}}_{\mathrm{Kas}}^{G,\mathrm{sep}}(A,B)\cong[[\mathcal{S}\hat{\otimes}A\hat{\otimes}\hat{\mathcal{K}}_{G},B\hat{\otimes}\hat{\mathcal{K}}_{G}]]_{\operatorname{gr}}^{\operatorname{cp}}.

It is not clear to the author how to get a (clean) natural map in either direction. The problem seems to be that completely positive asymptotic morphisms do not naturally arise in practical situations.

Nevertheless, Meyer has proven using different methods in [46, Section 7] the analogue of the first two assertions of Theorem 6.12 for $\widehat{KK}_{\operatorname{Kas}}^{G,\operatorname{sep}}$ , whence the analogue of Assertion 3 of Theorem 6.12 for $\widehat{\mathrm{KK}}_{\mathrm{Kas}}^{G,\mathrm{sep}}$ also follows from Theorem 4.2.

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	$\displaystyle K_{0}^{G}(A_{u})$	$\displaystyle\rightarrow K_{0}^{G}(A_{u})$
	$\displaystyle[M]$	$\displaystyle\mapsto[M\oplus M_{\alpha}]=2[M]$

A higher categorical approach to graded bivariant KK-theories for Real C∗C^{*}-algebras

Abstract

1 Introduction

Theorem 1.1.

Theorem 1.2.

1.1 Organization of the article

2 Background on KK-theory spaces, groups, and some calculations

Definition 2.0.

Notation 2.0.

Example 2.0.

Notation 2.0.

Remark 2.0.

2.1 Miscellaneous results on equivariant (ungraded) KK-theory

Remark 2.0.

Proposition 2.0.

Proof.

Corollary 2.0.

Proof.

Definition 2.1.

Remark 2.1.

Proposition 2.1.

Proof.

Remark 2.1.

Proposition 2.2.

Proof.

Corollary 2.2.

Proof.

2.2 The spectral model of graded, GG-equivariant KK-theory

Definition 2.2.

Definition 2.2.

Theorem 2.3.

Proof sketch.

Notation 2.3.

Proposition 2.3.

Proof.

3 Equivariant (ungraded) Real KK!KK_{!}-theory

3.1 Inverting homotopy equivalences

Definition 3.0.

Definition 3.0.

Definition 3.1.

Example 3.1.

Proposition 3.1.

Proof.

Remark 3.1.

Proposition 3.1.

Proof.

3.2 Enforcing 𝒦G\mathcal{K}_{G}-stability

Definition 3.2.

Remark 3.2.

Definition 3.2.

Proposition 3.3.

Proof.

3.3 Enforcing !!-excision

Definition 3.6.

Remark 3.6.

Proposition 3.6.

Proof.

Lemma 3.8.

Proof.

3.4 Inverting Bott

Notation 3.8.

Proposition 3.8.

Lemma 3.8.

Proof.

Lemma 3.8.

Proof.

Proof (of Section 3.4).

Corollary 3.8.

Proof.

Corollary 3.10.

Proof.

Definition 3.10.

Proposition 3.11.

Proof.

Corollary 3.11.

Proof.

Remark 3.11.

Definition 3.11.

Proposition 3.11.

Proof.

A higher categorical approach to graded bivariant $K$ -theories for Real $C^{*}$ -algebras

2 Background on $K$ -theory spaces, groups, and some calculations

2.1 Miscellaneous results on equivariant (ungraded) $K$ -theory

2.2 The spectral model of graded, $G$ -equivariant $K$ -theory

3 Equivariant (ungraded) Real $KK_{!}$ -theory

3.2 Enforcing $\mathcal{K}_{G}$ -stability

3.3 Enforcing $!$ -excision

3.6 Connection to $K$ -theories

3.7 Extension to all $C^{*}$ -algebras

4 Graded, Real $KK^{G}_{!,\infty}$ -theory

4.1 The Bott map in $\widehat{\mathrm{KK}}^{G}_{!,\infty}$ and consequences