# Frobenius structures over Hilbert C*-modules

## TL;DR

This paper explores the structure of Hilbert C*-modules over commutative C*-algebras using categorical quantum mechanics, characterizing Frobenius structures and their relation to algebra bundles and spectral properties.

## Contribution

It provides a categorical characterization of Frobenius structures over Hilbert C*-modules, linking them to bundles, coverings, and spectral subobjects, with new insights into their algebraic and topological properties.

## Key findings

- Frobenius structures correspond to bundles of finite-dimensional C*-algebras.
- Characterization of commutative Frobenius structures as finite coverings.
- Subobjects of the tensor unit relate to clopen subsets of the Gelfand spectrum.

## Abstract

We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobenius over the base if and only if it is dagger Frobenius over its centre and the centre is dagger Frobenius over the base. We characterise the commutative dagger Frobenius structures as finite coverings, and give nontrivial examples of both commutative and central dagger Frobenius structures. Subobjects of the tensor unit correspond to clopen subsets of the Gelfand spectrum of the C*-algebra, and we discuss dagger kernels.