Decomposition rank of Z-stable C*-algebras

TL;DR

This paper proves that C*-algebras formed by tensoring continuous functions on a compact space with the Jiang--Su algebra have a decomposition rank of at most 2, indicating a dimension reduction in their structure.

Contribution

It establishes a key case of a conjecture on the structure of nuclear C*-algebras, showing the importance of fiber structure over base space topology.

Findings

C*-algebras of the form C(X) ⊗ Z have decomposition rank ≤ 2

Supports the conjecture relating noncommutative dimension to fiber structure

Provides a dimension reduction result for C*-bundles with regular fibers

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 the second named author, even in a nonsimple setting, and gives evidence that the topological dimension of noncommutative spaces is governed by fibres rather than base spaces.