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 - 2022/5/14

Y1 - 2022/5/14

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 - Article

AN - SCOPUS:85125300501

SN - 0001-8708

VL - 400

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108286

ER -