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.
Categories: math.OA math.RA