Anomalous symmetries of classifiable C*-algebras

We study the $H^3$ invariant of a group homomorphism $ϕ:G \rightarrow \mathrm{Out}(A)$, 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 $\mathrm{Hilb}(G, ω)$ for non-trivial $ω\in H^3(G, \mathbb{T})$ cannot act on $\mathcal{Z}$.

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