Isomorphism of Hilbert modules over stably finite C*-algebras

TL;DR

This paper characterizes when countably generated Hilbert modules over stably finite C*-algebras are algebraically finitely generated and projective, linking this to their Cuntz semigroup elements and module isomorphism classes.

Contribution

It establishes a precise criterion for Hilbert modules over stably finite C*-algebras to be algebraically finitely generated and projective, and explores their isomorphism classes under CEI equivalence.

Findings

A Hilbert module gives rise to a compact element of the Cuntz semigroup iff it is algebraically finitely generated and projective.

CEI-equivalent modules that are algebraically finitely generated and projective are isomorphic.

Counterexamples show CEI-equivalence does not imply isomorphism in general.

Abstract

It is shown that if A is a stably finite C*-algebra and E is a countably generated Hilbert A-module, then E gives rise to a compact element of the Cuntz semigroup if and only if E is algebraically finitely generated and projective. It follows that if E and F are equivalent in the sense of Coward, Elliott and Ivanescu (CEI) and E is algebraically finitely generated and projective, then E and F are isomorphic. In contrast to this, we exhibit two CEI-equivalent Hilbert modules over a stably finite C*-algebra that are not isomorphic.