C⁎-envelopes of tensor algebras of product systems

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