Remarks on anomalous symmetries of C*-algebras

TL;DR

This paper explores the realization of anomalies in symmetries of C*-algebras, showing that all anomalies can be represented on certain stabilized commutative C*-algebras and Roe corona algebras, with implications for coarse geometry.

Contribution

It demonstrates that any anomaly associated with a finite group can be realized on stabilized commutative C*-algebras of manifolds and on Roe corona algebras of spaces with property A.

Findings

All anomalies can be realized on stabilization of commutative C*-algebras of manifolds.

No anomalous symmetries exist for Roe C*-algebras of coarse spaces.

Every anomaly can be realized on Roe corona algebras of spaces with property A.

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 $A$.