Cartan subalgebras in C*-algebras. Existence and uniqueness

2017/03/30 by Li, Xin, Renault, Jean · 1 citation
#22A22 #FOS: Mathematics #Operator Algebras (math.OA) #Primary 46L05 #Secondary 46L85

paper · doi:10.48550/arxiv.1703.10505

Abstract

We initiate the study of Cartan subalgebras in C*-algebras, with a particular focus on existence and uniqueness questions. For homogeneous C*-algebras, these questions can be analysed systematically using the theory of fibre bundles. For group C*-algebras, while we are able to find Cartan subalgebras in C*-algebras of many connected Lie groups, there are classes of (discrete) groups, for instance non-abelian free groups, whose reduced group C*-algebras do not have any Cartan subalgebras. Moreover, we show that uniqueness of Cartan subalgebras usually fails for classifiable C*-algebras. However, distinguished Cartan subalgebras exist in some cases, for instance in nuclear Roe algebras.

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