Invariant Basis Number for $C^{*}$-algebras

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 given, 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.