Gauge-equivariant Hilbert bimodules and crossed products by endomorphisms

TL;DR

This paper explores the structure of gauge-equivariant Hilbert bimodules and their crossed products by endomorphisms, linking noncommutative geometry with duality theories for tensor categories and group bundles.

Contribution

It introduces a framework for understanding crossed products by endomorphisms with non-trivial Chern classes, including the existence and uniqueness of gauge group bundles.

Findings

Describes the moduli space associated with endomorphisms.

Connects noncommutative bimodules to gauge-equivariant vector bundles.

Provides conditions for the existence and uniqueness of gauge group bundles.

Abstract

C*-endomorphisms arising from superselection structures with non-trivial centre define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz-Pimsner algebra of a vector bundle having the above-mentioned rank and first Chern class, and can be used to construct a duality for abstract (nonsymmetric) tensor categories vs. group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e., the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with the given endomorphism. The above-mentioned Hilbert bimodules are noncommutative analogues of gauge-equivariant vector bundles in the sense of Nistor-Troitsky.