Non-Commutative Vector Bundles for Non-Unital Algebras

TL;DR

This paper explores the structure of modules over non-unital $C^*$-algebras, characterizing them via multiplier modules and extending the theory to bi-Hilbertian bimodules with finite indices.

Contribution

It introduces a fullness condition that characterizes modules similar to sections of vector bundles over non-unital algebras and analyzes the multiplier-module construction in this context.

Findings

Fullness condition characterizes modules akin to vector bundle sections.

Multiplier-module construction is extended to bi-Hilbertian bimodules.

Analysis includes modules with finite numerical and Watatani indices.

Abstract

We revisit the characterisation of modules over non-unital $C^*$-algebras analogous to modules of sections of vector bundles. A fullness condition on the associated multiplier module characterises a class of modules which closely mirror the commutative case. We also investigate the multiplier-module construction in the context of bi-Hilbertian bimodules, particularly those of finite numerical index and finite Watatani index.