Anomalous symmetries of classifiable C*-algebras

TL;DR

This paper investigates the $H^3$ invariants associated with group homomorphisms into outer automorphism groups of classifiable C*-algebras, revealing obstructions and conditions for their vanishing, especially in the Jiang--Su algebra case.

Contribution

It establishes the existence of obstructions to $H^3$ invariants from the unitary algebraic $K_1$ group and proves that these invariants vanish for the Jiang--Su algebra, impacting the action of certain fusion categories.

Findings

Obstruction to $H^3$ invariants from $K_1$ group.

Vanishing of $H^3$ invariant for Jiang--Su algebra.

Non-trivial $H^3(G, ext{T})$ fusion categories cannot act on $ ext{Z}$.

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 $\mathcal{Z}$.