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

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Title: Tensor algebras of product systems and their C⁎-envelopes.
Authors: Dor-On, Adam1 (AUTHOR) adoron@illinois.edu, Katsoulis, Elias1,2 (AUTHOR) katsoulise@ecu.edu
Source: Journal of Functional Analysis. Apr2020, Vol. 278 Issue 7, pN.PAG-N.PAG. 1p.
Subjects: Tensor algebra, Tensor products, Discrete groups, Operator algebras, Operator theory, Selfadjoint operators, Lattice theory, Topological algebras
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 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 N 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) = (Z , N). [ABSTRACT FROM AUTHOR]
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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 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 N 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) = (Z , N). [ABSTRACT FROM AUTHOR]
ISSN:00221236
DOI:10.1016/j.jfa.2019.108416