C⁎-envelopes of tensor algebras of product systems

Abstract

Let P be a submonoid of a group G and let E=(Ep)p∈P be a product system over P with coefficient C⁎-algebra A. We show that the following C⁎-algebras are canonically isomorphic: the C⁎-envelope of the tensor algebra Tλ(E)+ of E; the reduced cross sectional C⁎-algebra of the Fell bundle associated to the canonical coaction of G on the covariance algebra A×EP of E; and the C⁎-envelope of the cosystem obtained by restricting the canonical gauge coaction on Tλ(E) to the tensor algebra. As a consequence, for every submonoid P of a group G and every product system E=(Ep)p∈P over P, the C⁎-envelope Cenv⁎(Tλ(E)+) automatically carries a coaction of G that is compatible with the canonical gauge coaction on Tλ(E). This answers a question left open by Dor-On, Kakariadis, Katsoulis, Laca and Li. We also analyse co-universal properties of Cenv⁎(Tλ(E)+) with respect to injective gauge-compatible representations of E. When E=CP is the canonical product system over P with one-dimensional fibres, our main result implies that the boundary quotient ∂Tλ(P) is canonically isomorphic to the C⁎-envelope of the closed non-selfadjoint subalgebra spanned by the canonical generating isometries of Tλ(P). Our results on co-universality imply that ∂Tλ(P) is a quotient of every nonzero C⁎-algebra generated by a gauge-compatible isometric representation of P that in an appropriate sense respects the zero element of the semilattice of constructible right ideals of P.