Tensor algebras of product systems and their C^⁎-envelopes

Let (G,P) be an abelian, lattice ordered group and let X be a compactly aligned product system over P with coefficients in A. We show that the C*-envelope of the Nica tensor algebra NT_X ⁺ coincides with both Sehnem's covariance algebra A×_XP and the co-universal C^⁎-algebra NO_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 NO_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)=(Z,N).