Anomalous symmetries of classifiable C*-algebras

Abstract

We study the $H^3$ invariant of a group homomorphism $\varphi :G\to \mathrm{Out}(A)$, where $A$ is a classifiable C${}^{\ast }$-algebra. We show the existence of an obstruction to possible $H^3$ invariants arising from considering the unitary algebraic $K_1$ group. In particular, we prove that when $A$ is the Jiang--Su algebra $\mathcal{Z}$ this invariant must vanish. We deduce that the unitary fusion categories $\mathrm{Hilb}(G,\omega )$ for non-trivial $\omega \in H^3(G,\mathbb{T})$ cannot act on $\mathcal{Z}$.

arXiv comment29 pages; accepted version; Studia Math., to appear