Limits and C*-algebras of low rank or dimension

TL;DR

This paper investigates how various limit constructions affect the low rank and dimension properties of C*-algebras, providing new characterizations and conditions for stability of these properties under extensions and limits.

Contribution

It introduces conditions under which limit constructions preserve low rank and dimension in C*-algebras, extending existing results and applying them to CCR algebras.

Findings

CCR algebra has stable rank one iff topological dimension is zero or one

Characterization of sigma-unital CCR algebras with stable rank one

Conditions for preservation of low rank under extensions and limits

Abstract

We explore various limit constructions for C*-algebras, such as composition series and inverse limits, in relation to the notions of real rank, stable rank, and extremal richness. We also consider extensions and pullbacks. We identify some conditions under which the constructions preserve low rank for the C*-algebras or their multiplier algebras. We also discuss the version of topological dimension theory appropriate for primitive ideal spaces of C*-algebras and provide an analogue for rank of the countable sum theorem of dimension theory. As an illustration of how the main results can be applied, we show that a CCR algebra has stable rank one if and only if it has topological dimension zero or one, and we characterize those sigma-unital CCR algebras whose multiplier algebras have stable rank one or extremal richness. (The real rank zero case was already known.)