Invariant Basis Number for $C^*$-Algebras

TL;DR

This paper extends the concept of Invariant Basis Number to unital $C^*$-algebras, characterizing them via $K$-theory, exploring their structural properties, and identifying classes with or without this property.

Contribution

It introduces the notion of Invariant Basis Number for $C^*$-algebras, provides $K$-theoretic characterizations, and investigates structural and universal properties related to this concept.

Findings

Characterization of $C^*$-algebras with Invariant Basis Number via $K$-theory

Identification of structural properties of $C^*$-algebras lacking Invariant Basis Number

Description of a universal class of $C^*$-algebras with or without the property

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.