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 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}.
