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

Lifting obstructions for group actions, cocycle actions, and $\Gamma$-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 $C^*$-algebra $A$, we show that the classifying spaces $\mathcal{B}^D\mathcal{G}_A$ and $\mathcal{B}^DP\mathcal{G}_A$ of the respective crossed modules governing cocycle actions and $\Gamma$-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 $\Gamma$-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 $\Gamma$-spaces, and apply the May-Thomason infinite loop space machine.

Consequently, the natural transformation $H^1(\Gamma,\mathcal{G}) \to [\mathcal{B}\Gamma,\mathcal{B}^D\mathcal{G}]$ takes values in cohomology groups.