Tensor algebras of product systems and their C*-envelopes

TL;DR

This paper establishes the C*-envelope of Nica tensor algebras of product systems over abelian, lattice ordered groups, linking it to covariance algebras and resolving several open problems in operator algebra theory.

Contribution

It proves the C*-envelope coincides with Sehnem's covariance algebra and the co-universal algebra, solving open problems and extending results to higher-rank and topological graphs.

Findings

Confirmed the existence of the co-universal algebra al O^r_X.

Characterized the C*-envelope of tensor algebras for higher-rank graphs.

Proved reduced Hao-Ng isomorphisms for generalized gauge actions.

Abstract

Let $(G, P)$ be an abelian, lattice ordered group and let $X$ be a compactly aligned product system over $P$. We show that the C*-envelope of the Nica tensor algebra $\mathcal{N}\mathcal{T}^+_X$ coincides with both Sehnem's covariance algebra $\mathcal{A} \times_X P$ and the co-universal C*-algebra $\mathcal{N}\mathcal{O}^r_X$ for injective, gauge compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of $\mathcal{N}\mathcal{O}^r_X$, thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C*-envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C*-algebras of product systems. This generalizes recent results that were obtained by various authors in the case where $(G, P) =(\mathbb{Z},\mathbb{N})$.