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 with coefficients in A. We show that the C*-envelope of the Nica tensor algebra N T + X coincides with both Sehnem's covariance algebra A ×X P and the co-universal C * -algebra N 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 N 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 ) = (Z, N).2010 Mathematics Subject Classification. Primary: 46L08, 46L55, 47B49, 47L40, 47L65, 46L05.

Classification of irreversible and reversible Pimsner operator algebras

“…The non-commutative generalization of the Shilov boundary is called the -envelope, and was first shown to exist through Hamana's injective envelope [Ham79]. The -envelope is defined to be the smallest -algebra containing the given operator algebra in a reasonable sense and provides a fruitful connection between -algebras and non-self-adjoint operator algebras (see [DK20, Kat17]). In this paper, we apply ideas from Arveson's non-commutative boundary theory to uncover a precise hierarchy between classification of irreversible algebras and classification of reversible algebras with additional structure.…”

Co-universal $C^{\ast}$-algebras for product systems over quasi-lattice ordered groupoids

“…In [5], it is proved that the algebras satisfying such co-universal property do exist for many cases. In further work, Dor-On and Katsoulis showed that the C * -envelope of the Fock tensor algebra (T r X ) + has the co-universal property for every compactly aligned product system X over an abelian lattice ordered group ( [8]). Motivated by this result, Dor-On, Kakariadis, Katsoulis, Laca, and Li characterized the co-universal algebras of compactly aligned product systems over group-embeddable right LCM semigroups by using C * -envelopes for cosystems in [7].…”

C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems

“…In that paper dilation theoretic techniques merged with uniqueness theorems for the images of equivariant homomorphisms and gave strong motivating evidence that one can use C*-envelope techniques in order to prove the existence of a co-universal object for more general product systems over abelian orders. This approach was fully materialized by Dor-On and Katsoulis [20] who proved that for a compactly aligned product system X over any abelian lattice ordered semigroup, C * env (T λ (X) + ) has the co-universal property proposed in [9], thus showing in particular that the co-universal algebra N O r (X) of [9] exists without the injectivity assumption when the semigroup is abelian lattice ordered. This result strengthened the important connection between nonselfadjoint and selfadjoint operator algebra theory and raised the tantalising possibility of proving directly the existence of an appropriate notion of C*-envelope that satisfies the desired co-universal property automatically beyond abelian orders.…”

“…This result strengthened the important connection between nonselfadjoint and selfadjoint operator algebra theory and raised the tantalising possibility of proving directly the existence of an appropriate notion of C*-envelope that satisfies the desired co-universal property automatically beyond abelian orders. Even though some of the techniques of [20] are indeed applicable to more general settings, it soon became clear that significant progress would require new ideas. The purpose of the present paper is to realize this possibility through the use of an equivariant version of the C*-envelope.…”