Vol. 7, No. 3, 2014

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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
Abstract

We show that ${C}^{\ast }$-algebras of the form $C\left(X\right)\otimes \mathsc{Z}$, where $X$ is compact and Hausdorff and $\mathsc{Z}$ denotes the Jiang–Su algebra, have decomposition rank at most $2$. This amounts to a dimension reduction result for ${C}^{\ast }$-bundles with sufficiently regular fibres. It establishes an important case of a conjecture on the fine structure of nuclear ${C}^{\ast }$-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