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 T λ (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×E P of E; and the C * -envelope of the cosystem obtained by restricting the canonical gauge coaction on T λ (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 C * env (T λ (E) + ) automatically carries a coaction of G that is compatible with the canonical gauge coaction on T λ (E). This answers a question left open by Dor-On, Kakariadis, Katsoulis, Laca and Li. We also analyse co-universal properties of C * env (T λ (E) + ) with respect to injective gaugecompatible representations of E. When E = C P is the canonical product system over P with one-dimensional fibres, our main result implies that the boundary quotient ∂T λ (P ) is canonically isomorphic to the C * -envelope of the closed non-selfadjoint subalgebra spanned by the canonical generating isometries of T λ (P ). Our results on co-universality imply that ∂T λ (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 .