Extensions of Hilbert C*-modules: classification in simple cases

TL;DR

This paper explores the classification of extensions of Hilbert C*-modules, especially in simple cases involving commutative algebras and free modules, linking algebraic properties to vector bundle isometries.

Contribution

It provides a topological evaluation of extensions for rank-one free modules and connects algebraic invariants to geometric structures.

Findings

Extensions correspond to isometric maps of vector bundles in commutative cases

Topological classification of rank-one free module extensions

Algebraic properties of Busby invariants facilitate this identification

Abstract

Theory of extensions of Hilbert C*-modules was developed by D. Bakic and B. Guljas. An easy observation shows that in the case, when the underlying C*-algebra extension is commutative and the Hilbert C*-modules are projective of finite type, the algebraic properties of the corresponding Busby invariant allow to identify extensions with isometric maps of the corresponding vector bundles. When the Hilbert C*-modules are free of rank one, we evaluate the set of extensions in topological terms.