Invariant Basis Number for $C^{\ast }$-Algebras

Abstract

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

arXiv comment14 pages, various minor changes, to appear in Illinois J Math