Tensor algebras of product systems and their C*-envelopes

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 $\mathcal{N}{\mathcal{O}}_X^r$ 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 $\mathcal{N}{\mathcal{O}}_X^r$, 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 $(G,P)=(\mathbb{Z},\mathbb{N})$.

arXiv comment30 pages, final version, accepted for publication in the Journal of Functional Analysis