Topological equivariant coarse K-homology

L‐theory of C∗$C^*$‐algebras

“…In [40], it was then observed that the $$\infty$$‐categorical localization of $$\mathrm{C^*Alg}^\mathrm{sep}$$ at the KK‐equivalences is a stably symmetric monoidal $$\infty$$‐category whose homotopy category is canonically equivalent to the tensor triangulated category KK of Kasparov. This observation has also been taken up in [4] (including extensions of these results to possibly nonseparable $$C^*$$‐algebras) in the equivariant case and was used in [5] in a proof of an equivariant form of Paschke duality. Definition We denote by $$\mathrm{KK}= \mathrm{C^*Alg}^\mathrm{sep}[\mathrm{KK}^{-1}]$$ the $$\infty$$‐categorical localization of $$\mathrm{C^*Alg}^\mathrm{sep}$$ at the KK‐equivalences.…”

“…In his seminal work on the Novikov conjecture [29], Kasparov invented (equivariant) bivariant topological K‐theory, known as KK‐theory. Phrased in categorical language, Kasparov's machine allowed to construct a tensor triangulated category $$\mathrm{KK}$$ and a functor $$$$\begin{equation*} \mathrm{C^*Alg}^\mathrm{sep}\longrightarrow \mathrm{KK} \end{equation*}$$$$that was later shown to be a localization (necessarily at the KK‐equivalences, that is, those $$*$$‐homomorphisms whose induced map in the KK‐category is an isomorphism) [15] and to be the initial functor to an additive category that is split exact and stable [18], see, for example, [4] for more precise statements and a guide through (parts of) the literature. In [40], it was then observed that the $$\infty$$‐categorical localization of $$\mathrm{C^*Alg}^\mathrm{sep}$$ at the KK‐equivalences is a stably symmetric monoidal $$\infty$$‐category whose homotopy category is canonically equivalent to the tensor triangulated category KK of Kasparov.…”

“…These more classical definitions are in fact given for possibly nonseparable algebras and satisfy $$$$\begin{equation*} \mathrm{K}(A) \simeq \operatornamewithlimits{colim}\limits _{A^{\prime } \subseteq _{\mathrm{sep}} A} \mathrm{K}(A^{\prime }) \end{equation*}$$$$so we may also view the above definition as describing K‐theory of possibly nonseparable algebras. In [4], this was formalized by considering the ind‐completion of $$\mathrm{KK}$$ and again considering the functor corepresented by $$\mathbb {R}$$. This definition, however, does not give all structure that topological K‐theory has: For instance, it is a purely formal consequence of the definitions that K‐theory sends certain short exact sequences (e.g., where the surjection is a Schochet fibration or admits a cpc split) to fiber sequences, but it is not a priori clear that it sends all short exact sequences of $$C^*$$‐algebras to fiber sequences.…”

“…This definition, however, does not give all structure that topological K‐theory has: For instance, it is a purely formal consequence of the definitions that K‐theory sends certain short exact sequences (e.g., where the surjection is a Schochet fibration or admits a cpc split) to fiber sequences, but it is not a priori clear that it sends all short exact sequences of $$C^*$$‐algebras to fiber sequences. However, this is known to be true, for example, as a special case of [4, Theorem 1.15] (for $$X=\ast$$ in the notation of [4]).…”

“…We now observe that $$\tau _{\geqslant 0}\mathrm{k}^{hC_2}$$ is excisive, so it suffices to show that the fiber of the right vertical map is excisive as well. For this, let us consider a pullback diagram of $$C^*$$‐algebras whose vertical maps are surjections with a completely positive contractive split (any pullback diagram in $$\mathrm{KK}$$ is the image of such a diagram, e.g., by [4, Theorem 1.4(2) and Lemma 2.14]). We need to show that the fiber of $$\tau _{\geqslant 0} \mathrm{L}\rightarrow \tau _{\geqslant 0}\mathrm{k}^{tC_2}$$ sends this square to a pullback diagram of connective spectra.…”