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Anomalous symmetries of classifiable C*-algebras

Samuel Evington, Sergio Girón Pacheco Studia Mathematica 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 $\mathcal {Z}$.

Authors

  • Samuel EvingtonMathematical Institute
    University of Münster
    Einsteinstraße 62
    48149 Münster, Germany
    e-mail
  • Sergio Girón PachecoMathematical Institute
    University of Oxford
    Oxford, OX2 6GG, UK
    e-mail

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