{"title":"Anomalous symmetries of classifiable C*-algebras","authors":"Samuel Evington, Sergio Gir'on Pacheco","doi":"10.4064/sm220117-25-6","DOIUrl":null,"url":null,"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 $\\mathcal{Z}$.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studia Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4064/sm220117-25-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 5
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 $\mathcal{Z}$.
期刊介绍:
The journal publishes original papers in English, French, German and Russian, mainly in functional analysis, abstract methods of mathematical analysis and probability theory.