L 2-Betti Numbers of C*-Tensor Categories Associated with Totally Disconnected Groups

Author:

Valvekens Matthias1

Affiliation:

1. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium

Abstract

Abstract We prove that the $L^2$-Betti numbers of a rigid $C^*$-tensor category vanish in the presence of an almost-normal subcategory with vanishing $L^2$-Betti numbers, generalising a result of [ 7]. We apply this criterion to show that the categories constructed from totally disconnected groups in [ 6] have vanishing $L^2$-Betti numbers. Given an almost-normal inclusion of discrete groups $\Lambda <\Gamma $, with $\Gamma $ acting on a type $\textrm{II}_1$ factor $P$ by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion $P\rtimes \Lambda \subset P\rtimes \Gamma $ to that of the Schlichting completion $G$ of $\Lambda <\Gamma $. If $\Lambda <\Gamma $ is unimodular, this correspondence allows us to prove that the $L^2$-Betti numbers of $P\rtimes \Lambda \subset P\rtimes \Gamma $ are equal to those of $G$.

Funder

European Research Council

Flemish Government

Polish Academy of Sciences

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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