Journal of Operator Theory

Volume 61, Issue 2, Spring 2009 pp. 381-417.

Limits and $C^*$-algebras of low rank or dimension

Authors: Lawrence G. Brown (1) and Gert K. Pedersen (2)
Author institution: (1) Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

Summary: 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 {\rm CCR} algebra has stable rank one if and only if it has topological dimension zero or one, and we characterize those $\sigma$-unital {\rm CCR} algebras whose multiplier algebras have stable rank one or extremal richness. (The real rank zero case was already known.)

Keywords: Extensions of $C^*$-algebras, extremally rich $C^*$-algebras, inverse limits, pullbacks, real rank zero, stable rank one, subhomogeneous $C^*$-algebras

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