Hilbert C*-modules over a commutative C*-algebra

TL;DR

This paper investigates embedding and isomorphism problems for countably generated Hilbert C*-modules over commutative C*-algebras, providing conditions under which modules can be embedded or are isomorphic, and computing the Cuntz semigroup in low-dimensional cases.

Contribution

It introduces new criteria for embedding and isomorphism of Hilbert C*-modules over commutative algebras, extending results to recursive subhomogeneous algebras and computing the Cuntz semigroup for low-dimensional spaces.

Findings

Embedding occurs when fibre dimensions differ sufficiently relative to the spectrum.

Isomorphism and embedding are determined by restrictions to constant fibre dimension sets for certain modules.

Cuntz semigroup for $C_0(X)$ with $dim X \,\leq 3$ is computed using cohomological data.

Abstract

This paper studies the problems of embedding and isomorphism for countably generated Hilbert C*-modules over commutative C*-algebras. When the fibre dimensions differ sufficiently, relative to the dimension of the spectrum, we show that there is an embedding between the modules. This result continues to hold over recursive subhomogeneous C*-algebras. For certain modules, including all modules over $C_0(X)$ when $dim X \leq 3$, isomorphism and embedding are determined by the restrictions to the sets where the fibre dimensions are constant. These considerations yield results for the Cuntz semigroup, including a computation of the Cuntz semigroup for $C_0(X)$ when $dim X \leq 3$, in terms of cohomological data about $X$.