Complete C*-categories and a topos theoretic Green-Julg theorem

TL;DR

This paper develops a notion of categorical completeness for C*-categories, characterizes categories of Hilbert modules, and proves a topos-theoretic Green-Julg theorem, all within a constructive framework.

Contribution

It introduces several variants of categorical completeness for C*-categories and applies these to establish a topos-theoretic Green-Julg theorem.

Findings

Categories of Hilbert modules are characterized as complete C*-categories with enough absolutely compact morphisms.

The category of Hilbert spaces over a topos is an example of a complete C*-category.

A topos-theoretic Green-Julg theorem is proved constructively, linking topos theory and C*-algebraic structures.

Abstract

We investigate what would be a correct definition of categorical completeness for C*-categories and propose several variants of such a definition that make the category of Hilbert modules over a C*-algebra a free (co)completion. We extend results about generators and comparison theory known for W*-categories with direct sums and splitting of symmetric projections to our "complete C*-categories" and we give an abstract characterization of categories of Hilbert modules over a C*-algebra or a C*-category as "complete C*-category having enough absolutely compact morphisms (and a generator)". We then apply this to study the category of Hilbert spaces over a topos showing that this is an example of a complete C*-category. We prove a topos theoretic Green-Julg theorem: The category of Hilbert spaces over a topos which is locally decidable, separated and whose localic reflection is locally compact and completely regular is a category of Hilbert modules over a C*-algebras attached to the topos. All the results in this paper are proved constructively and hence can be applied themselves internally to a topos. Moreover we give constructive proof of some known classical results about C*-algebras and Hilbert modules.