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‚Äď700
Abstract

We show that C-algebras of the form C(X) 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‚ÄďSu 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
Universit√§t M√ľnster
Einsteinstraße 62
D-48149 M√ľnster
Germany