Infinite Loop Spaces and Group Actions on Strongly Self-Absorbing C*-Algebras

Abstract

Lifting obstructions for group actions, cocycle actions, and $\hat{I}“$-kernels admit a cohomological description via topological crossed modules, as recently developed by Izumi, Giron-Pacheco, and the first named author. For a strongly self-absorbing $C^{\ast }$-algebra $A$, we show that the classifying spaces ${\mathcal{B}}^D{\mathcal{G}}_A$ and ${\mathcal{B}}^{D}P{\mathcal{G}}_A$ of the respective crossed modules governing cocycle actions and $\hat{I}“$-kernels, respectively, carry infinite loop space structures induced by the tensor product; the same holds for the crossed module $B^D{\tilde{\mathcal{G}}}_A$ whenever $U(A)$ is connected. This confirms a conjecture from the aforementioned work and extends to $\hat{I}“$-kernels and cocycle actions the connection with stable homotopy theory. For the proof we construct $\mathbb{I}$-FCPs from the relevant crossed modules, pass to $\hat{I}“$-spaces, and apply the May-Thomason infinite loop space machine. Consequently, the natural transformation $H^1(\hat{I}“,\mathcal{G})\to [\mathcal{B}\hat{I}“,{\mathcal{B}}^D\mathcal{G}]$ takes values in cohomology groups.

29 pages