C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems

Abstract

A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify to the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.

C*-envelopes of tensor algebras of product systems

“…Definition 2.2. Following [6], we will refer to the C * -algebra generated by the range of ψ + as the Fock algebra of E, and will denote it by T λ (E). The tensor algebra of E, denoted by T λ (E) + , is the closed subalgebra of T λ (E) generated by the range of ψ + .…”

“…Dor-On, Kakariadis, Katsoulis, Laca and Li introduced in [6] the notion of a C * -envelope for a cosystem. A cosystem consists of an operator algebra equipped with an appropriately defined coaction of a discrete group G, and the C * -envelope of a cosystem takes the coaction on the operator algebra into account.…”

“…Hence the resulting C * -algebra is the smallest C * -algebra carrying a coaction of G that is generated by an equivariant completely isometric copy of the underlying operator algebra. For P a submonoid of a group G, the canonical gauge coaction of G on T λ (E) restricts to a coaction on the tensor algebra T λ (E) + (see [6,Proposition 4.1]). The C * -envelope of the corresponding cosystem has the C * -envelope C * env (T λ (E) + ) as a canonical quotient C * -algebra.…”

“…The C * -envelope of the corresponding cosystem has the C * -envelope C * env (T λ (E) + ) as a canonical quotient C * -algebra. However, the question of whether the C * -envelope of T λ (E) + automatically carries a coaction of G for which the quotient map from T λ (E) is gaugeequivariant was left open in [6]. When E is a compactly aligned product system over a right LCM submonoid of G, the C * -envelope of this canonical cosystem associated to the tensor algebra T λ (E) + is canonically isomorphic to the reduced analogue of the covariance algebra of E [6, Theorem 5.3].…”

“…In this paper our main goal is to show that all of the above notions of boundary quotients for a product system E = (E p ) p∈P coincide. We were motivated by the recent analysis of the connection between these C * -algebras in particular cases, see [5,6,12,13]. Precisely, we show that the C * -envelope of the tensor algebra T λ (E) + is canonically isomorphic to the reduced cross sectional C * -algebra of the Fell bundle associated to the canonical coaction of G on the covariance algebra of E, for E = (E p ) p∈P a product system over a submonoid P of G. In particular, the C * -envelope of T λ (E) + necessarily carries a coaction of G, for which the quotient map from T λ (E) is gauge-equivariant.…”