Anomalous symmetries of classifiable C*-algebras
Volume 270 / 2023
Studia Mathematica 270 (2023), 73-101
MSC: Primary 46L35; Secondary 46L37.
DOI: 10.4064/sm220117-25-6
Published online: 13 October 2022
Abstract
We study the $H^3$ invariant of a group homomorphism $\phi :G \rightarrow \mathrm {Out}(A)$, where $A$ is a classifiable C$^*$-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 $\mathcal {Z}$.
