Central sequence C*-algebras and tensorial absorption of the Jiang–Su algebra

Eberhard Kirchberg; Mikael Rørdam

doi:10.1515/crelle-2012-0118

Article

Central sequence C^*-algebras and tensorial absorption of the Jiang–Su algebra

Published/Copyright: March 9, 2013

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2014 Issue 695

Abstract

We study properties of the central sequence algebra Aω∩A' of a C^*-algebra A, and we present an alternative approach to a recent result of Matui and Sato. They prove that every unital separable simple nuclear C^*-algebra, whose trace simplex is finite-dimensional, tensorially absorbs the Jiang–Su algebra if and only if it has the strict comparison property. We extend their result to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension. We are also able to relax the assumption on the C^*-algebra of having the strict comparison property to a weaker property that we call local weak comparison. Namely, we prove that a unital separable simple nuclear C^*-algebra, whose trace simplex has finite-dimensional closed extreme boundary, tensorially absorbs the Jiang–Su algebra if and only if it has the local weak comparison property.

We can also eliminate the nuclearity assumption if instead we assume the (SI) property of Matui and Sato, and, moreover, that each II₁-factor representation of the C^*-algebra is a McDuff factor.

Funding source: Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation at University of Copenhagen

Received: 2012-10-11

Published Online: 2013-3-9

Published in Print: 2014-10-1

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https://doi.org/10.1515/crelle-2012-0118