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

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.