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

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 $X$.