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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]
Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Tensor algebras of product systems and their C⁎-envelopes.
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  Data: <searchLink fieldCode="AR" term="%22Dor-On%2C+Adam%22">Dor-On, Adam</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> adoron@illinois.edu</i><br /><searchLink fieldCode="AR" term="%22Katsoulis%2C+Elias%22">Katsoulis, Elias</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> katsoulise@ecu.edu</i>
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  Data: <searchLink fieldCode="DE" term="%22Tensor+algebra%22">Tensor algebra</searchLink><br /><searchLink fieldCode="DE" term="%22Tensor+products%22">Tensor products</searchLink><br /><searchLink fieldCode="DE" term="%22Discrete+groups%22">Discrete groups</searchLink><br /><searchLink fieldCode="DE" term="%22Operator+algebras%22">Operator algebras</searchLink><br /><searchLink fieldCode="DE" term="%22Operator+theory%22">Operator theory</searchLink><br /><searchLink fieldCode="DE" term="%22Selfadjoint+operators%22">Selfadjoint operators</searchLink><br /><searchLink fieldCode="DE" term="%22Lattice+theory%22">Lattice theory</searchLink><br /><searchLink fieldCode="DE" term="%22Topological+algebras%22">Topological algebras</searchLink>
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  Data: 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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  Data: <i>Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=141237344
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      – Type: doi
        Value: 10.1016/j.jfa.2019.108416
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      – Code: eng
        Text: English
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        PageCount: 1
        StartPage: N.PAG
    Subjects:
      – SubjectFull: Tensor algebra
        Type: general
      – SubjectFull: Tensor products
        Type: general
      – SubjectFull: Discrete groups
        Type: general
      – SubjectFull: Operator algebras
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      – SubjectFull: Operator theory
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      – SubjectFull: Selfadjoint operators
        Type: general
      – SubjectFull: Lattice theory
        Type: general
      – SubjectFull: Topological algebras
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      – TitleFull: Tensor algebras of product systems and their C⁎-envelopes.
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            NameFull: Dor-On, Adam
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            NameFull: Katsoulis, Elias
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            – D: 15
              M: 04
              Text: Apr2020
              Type: published
              Y: 2020
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