Product systems and their representations: an approach using Fock spaces and Fell bundles

TL;DR

This paper explores product systems over semigroups using Fock spaces and Fell bundles, extending definitions and establishing covariant representation results with new constructions of co-universal C*-algebras.

Contribution

It extends Fowler's product systems to unital semigroups, introduces new methods for covariant representations, and constructs co-universal C*-algebras via Fell bundle techniques.

Findings

Extended product systems over semigroups using Fock spaces.

Established covariant representation results for these systems.

Constructed co-universal C*-algebras with two different techniques.

Abstract

In this exposition we highlight product systems as the semigroup analogue of Fell bundles. Motivated by Fock creation operators we extend the definition of Fowler's product systems over unital discrete left-cancellative semigroups, via both a concrete and an abstract characterization. We next underline the importance of Fock spaces and Fell bundle methods for obtaining covariant representation results. First, the strongly covariant representations of Sehnem, for product systems over group embeddable semigroups, find their analogue here. Secondly, we give two solutions for the existence of the co-universal C*-algebra with respect to equivariant injective Fock-covariant representations in this context. The first one uses an intertwining isometry technique to realize it as the C*-envelope of the normal cosystem on the Fock tensor algebra. The second uses a Fell bundle quotient technique to realize it as the reduced C*-algebra of Sehnem's strong covariance relations. This C*-algebra is boundary for a larger class of equivariant injective representations.