C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems
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C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems. / Dor-On, A.; Kakariadis, E. T.A.; Katsoulis, E.; Laca, M.; Li, X.
In: Advances in Mathematics, Vol. 400, 108286, 2022.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems
AU - Dor-On, A.
AU - Kakariadis, E. T.A.
AU - Katsoulis, E.
AU - Laca, M.
AU - Li, X.
N1 - Publisher Copyright: © 2022 Elsevier Inc.
PY - 2022
Y1 - 2022
N2 - A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify with the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.
AB - A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify with the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.
KW - C-envelope
KW - Co-universal algebra
KW - Coaction
KW - Covariance algebra
KW - Nica-Pimsner algebras
KW - Product systems
UR - http://www.scopus.com/inward/record.url?scp=85125300501&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2022.108286
DO - 10.1016/j.aim.2022.108286
M3 - Journal article
AN - SCOPUS:85125300501
VL - 400
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
M1 - 108286
ER -
ID: 310973613