C*-envelopes of tensor algebras of product systems

Abstract

Let $P$ be a submonoid of a group $G$ and let $\mathcal{E}=({\mathcal{E}}_p{)}_{p\in P}$ 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 ${\mathcal{T}}_{\lambda }(\mathcal{E}{)}^{+}$ of $\mathcal{E}$; the reduced cross sectional C*-algebra of the Fell bundle associated to the canonical coaction of $G$ on the covariance algebra $A{\times }_{\mathcal{E}}P$ of $\mathcal{E}$; and the C*-envelope of the cosystem obtained by restricting the canonical gauge coaction on ${\mathcal{T}}_{\lambda }(\mathcal{E})$ to the tensor algebra. As a consequence, for every submonoid $P$ of a group $G$ and every product system $\mathcal{E}=({\mathcal{E}}_p{)}_{p\in P}$ over $P$, the C*-envelope ${\mathcal{C}}_{\mathrm{env}}^{\ast }({\mathcal{T}}_{\lambda }(\mathcal{E}{)}^{+})$ automatically carries a coaction of $G$ that is compatible with the canonical gauge coaction on ${\mathcal{T}}_{\lambda }(\mathcal{E})$. This answers a question left open by Dor-On, Kakariadis, Katsoulis, Laca and Li. We also analyse co-universal properties of ${\mathcal{C}}_{\mathrm{env}}^{\ast }({\mathcal{T}}_{\lambda }(\mathcal{E}{)}^{+})$ with respect to injective gauge-compatible representations of $\mathcal{E}$. When $\mathcal{E}={\mathbb{C}}^P$ is the canonical product system over $P$ with one-dimensional fibres, our main result implies that the boundary quotient $\partial {\mathcal{T}}_{\lambda }(P)$ is canonically isomorphic to the C*-envelope of the closed non-selfadjoint subalgebra spanned by the canonical generating isometries of ${\mathcal{T}}_{\lambda }(P)$. Our results on co-universality imply that $\partial {\mathcal{T}}_{\lambda }(P)$ 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 $P$.

arXiv comment22 pages; a change of terminology from previous version: 'Fock algebra' was replaced by 'Toeplitz algebra' in v2; this is the accepted version

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