Decomposition rank of 𝒵-stable C∗-algebras

Tikuisis, Aaron; Winter, Wilhelm

doi:10.2140/apde.2014.7.673

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Decomposition rank of $\mathcal{Z}$-stable $\mathrm{C}^*$-algebras

Aaron Tikuisis and Wilhelm Winter

Vol. 7 (2014), No. 3, 673–700

DOI: 10.2140/apde.2014.7.673

Abstract

We show that $C^{*}$ -algebras of the form $C (X) \otimes Z$ , where $X$ is compact and Hausdorff and $Z$ denotes the Jiang–Su algebra, have decomposition rank at most $2$ . This amounts to a dimension reduction result for $C^{*}$ -bundles with sufficiently regular fibres. It establishes an important case of a conjecture on the fine structure of nuclear $C^{*}$ -algebras of Toms and Winter, even in a nonsimple setting, and gives evidence that the topological dimension of noncommutative spaces is governed by fibres rather than base spaces.

Keywords

nuclear $\mathrm{C}^*$-algebras, decomposition rank, nuclear dimension, Jiang–Su algebra, classification, $C(X)$-algebras

Mathematical Subject Classification 2010

Primary: 46L35, 46L85

Milestones

Received: 30 April 2013

Revised: 5 September 2013

Accepted: 4 October 2013

Published: 18 June 2014

Authors

Aaron Tikuisis
Institute of Mathematics University of Aberdeen Fraser Noble Building Aberdeen AB24 3UE United Kingdom

Wilhelm Winter
Mathematisches Institut Universität Münster Einsteinstraße 62 D-48149 Münster Germany

Vol. 7, No. 3, 2014