Remarks on anomalous symmetries of C*-algebras

Abstract

For a group $G$ and $\omega\in Z^{3}(G, \text{U}(1))$, an $\omega$-anomalous action on a C*-algebra $B$ is a $\text{U}(1)$-linear monoidal functor between 2-groups $\text{2-Gr}(G, \text{U}(1), \omega)\rightarrow \underline{\text{Aut}}(B)$, where the latter denotes the 2-group of $*$-automorphisms of $B$. The class $[\omega]\in H^{3}(G, \text{U}(1))$ is called the anomaly of the action. We show for every $n\ge 2$ and every finite group $G$, every anomaly can be realized on the stabilization of a commutative C*-algebra $C(M)\otimes \mathcal{K}$ for some closed connected $n$-manifold $M$. We also show that although there are no anomalous symmetries of Roe C*-algebras of coarse spaces, for every finite group $G$, every anomaly can be realized on the Roe corona $C^{*}(X)/\mathcal{K}$ of some bounded geometry metric space $X$ with property $A$.