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arXiv · 1407.4713

Invariant Basis Number for $C^*$-Algebras

Philip M. Gipson
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Abstract

We develop the ring-theoretic notion of Invariant Basis Number in the context of unital $C^*$-algebras and their Hilbert $C^*$-modules. Characterization of $C^*$-algebras with Invariant Basis Number is given in $K$-theoretic terms, closure properties of the class of $C^*$-algebras with Invariant Basis Number are investigated, and examples of $C^*$-algebras both with and without the property are explored. For $C^*$-algebras without Invariant Basis Number we determine structure in terms of a "Basis Type" and describe a class of $C^*$-algebras which are universal in an appropriate sense. We conclude by investigating properties which are strictly stronger than Invariant Basis Number.

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Philip M. Gipson. 2015-09-12. Invariant Basis Number for $C^*$-Algebras. https://arxiv.org/abs/1407.4713

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