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

We provide a concrete and an abstract definition of a product system over any unital discrete left-cancellative semigroup, that extends Fowler's setting. The motivation arises from a Fock representation realization, which we use to model covariant representations. The strongly covariant representations of Sehnem, for product systems over group embeddable semigroups, find their analogue here. Moreover, we give two independent proofs of the existence of the co-universal C*-algebra with respect to equivariant injective Fock-covariant representations. The first realizes it as the C*-envelope of the normal cosystem on the Fock tensor algebra in the sense of Dor-On, Kakariadis, Katsoulis, Laca and Li. The second realizes it as the reduced C*-algebra of Sehnem's strong covariance relations by using tools of Exel, of Raeburn, and of Carlsen, Larsen, Sims and Vittadello. We note that this C*-algebra is boundary for a larger class of equivariant injective representations.