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 $\mathcal{C}^*_{\mathrm{env}}(\mathcal{T}_\lambda(\mathcal{E})^+)$ 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 $\mathcal{C}^*_{\mathrm{env}}(\mathcal{T}_\lambda(\mathcal{E})^+)$ 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 $P$.