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

Dor-On, Adam; Kakariadis, Evgenios T. A.; Katsoulis, Elias G.; Laca, Marcelo; Li, Xin

doi:10.1016/j.aim.2022.108286

Cited by 12 publications

(16 citation statements)

References 37 publications

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“…If (, 𝐺, 𝛿) is a cosystem, then  𝑟 ⋅  𝑠 ⊆  𝑟𝑠 for all 𝑟, 𝑠 ∈ 𝐺, since 𝛿 is a homomorphism. Remark 2.2 [12]. A coaction 𝛿 of 𝐺 on  is automatically non-degenerate, in the sense that…”

Section: Coactions On Operator Algebrasmentioning

confidence: 99%

Boundary quotient C*‐algebras of semigroups

Kakariadis

Katsoulis

Laca

et al. 2022

Journal of London Math Soc

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We study two classes of operator algebras associated with a unital subsemigroup P$P$ of a discrete group G$G$: one related to universal structures and one related to co‐universal structures. First we provide connections between universal C*‐algebras that arise variously from isometric representations of P$P$ that reflect the space J$\mathcal {J}$ of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation Cλ∗(P)$\mathrm{C}^*_\lambda (P)$, and we consider the boundary quotient ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ related to the minimal boundary space. We show that ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ is co‐universal in two different classes: (a) with respect to the equivariant constructible isometric representations of P$P$; and (b) with respect to the equivariant C*‐covers of the reduced non‐selfadjoint semigroup algebra scriptAfalse(Pfalse)$\mathcal {A}(P)$. If P$P$ is an Ore semigroup, or if G$G$ acts topologically freely on the minimal boundary space, then ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ coincides with the usual C*‐envelope Cenv∗(Afalse(Pfalse))$\mathrm{C}^*_{\text{env}}(\mathcal {A}(P))$ in the sense of Arveson. This covers total orders, finite type and right‐angled Artin monoids, the Thompson monoid, multiplicative semigroups of non‐zero algebraic integers, and the ax+b$ax+b$‐semigroups over integral domains that are not a field. In particular, we show that P$P$ is an Ore semigroup if and only if there exists a canonical ∗$*$‐isomorphism from ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$, or from Cenv∗(Afalse(Pfalse))$\mathrm{C}^*_{\text{env}}(\mathcal {A}(P))$, onto Cλ∗(G)$\mathrm{C}^*_\lambda (G)$. If any of the above holds, then scriptAfalse(Pfalse)$\mathcal {A}(P)$ is shown to be hyperrigid.

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Section: Coactions On Operator Algebrasmentioning

confidence: 99%

Boundary quotient C*‐algebras of semigroups

Kakariadis

Katsoulis

Laca

et al. 2022

Journal of London Math Soc

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“…Coactions of discrete groups on general operator algebras were introduced in [34] as a generalization of coactions of discrete groups on C*-algebras [53]. The following is a semigroup variant of [34, Definition 3.1], adapted to our context.…”

Section: Constructing Exel-loring Approximations Via Semigroup Coactionsmentioning

confidence: 99%

“…Then, C * max (A) is RFD. Coactions of groups on general operator algebras were introduced in [34] for the purpose of showing the existence of C * -algebras satisfying co-universality with respect to representations of product systems over right LCM monoids. In the C * -algebra literature, coactions by a discrete group G are interpreted as actions of the quantum dual group of G on the C * -algebra, and are useful in many instances [17,37,53,?Seh19].…”

Section: Introductionmentioning

confidence: 99%

“…In the C * -algebra literature, coactions by a discrete group G are interpreted as actions of the quantum dual group of G on the C * -algebra, and are useful in many instances [17,37,53,?Seh19]. Since our semigroups are assumed to be FDP, they will not contain infinite groups as subsemigroups, and hence our development of coactions by semigroups differs from the one for group coactions in [34].…”

Section: Introductionmentioning

confidence: 99%

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Finite-dimensional approximations and semigroup coactions for operator algebras

Clouâtre¹,

Dor-On²

2021

Preprint

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The residual finite-dimensionality of a C * -algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this paradigm to general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge in this greater generality, we exhibit novel tools for constructing finite-dimensional approximations. One such tool is a notion of a residually finite-dimensional coaction of a semigroup on an operator algebra, which allows us to construct finite-dimensional approximations for operator algebras of functions and operator algebras of semigroups. Our investigation is intimately related to the question of whether residual finite-dimensionality of an operator algebra is inherited by its maximal C * -cover, which we resolve in many cases of interest.

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“…In [53,Section 6, Question 1] Viselter asked if his C*-algebras have a universal property in the spirit of a gauge-invariant uniqueness theorem. Gauge-invariant uniqueness theorems have a plethora of applications in the structure and representation theory of operator algebras, and have been extended significantly to various scenarios [44,33,45,34,7,18,21]. Hence, it is natural to ask for such uniqueness theorems in the context of subproduct systems.…”

Section: Introductionmentioning

confidence: 99%

Toeplitz quotient C*-algebras and ratio limits for random walks

Dor-On¹

2021

Preprint

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We study quotients of the Toeplitz C*-algebra of a random walk, similar to those studied by the author and Markiewicz for finite stochastic matrices. We introduce a new Cuntz-type quotient C*algebra for random walks that have convergent ratios of transition probabilities. These C*-algebras give rise to new notions of ratio limit space and boundary for such random walks, which are computed by appealing to a companion paper by Woess. Our combined results are leveraged to identify a unique symmetry-equivariant quotient C*-algebra for any symmetric random walk on a hyperbolic group, shedding light on a question of Viselter on C*-algebras of subproduct systems.

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

Cited by 12 publications

References 37 publications

Boundary quotient C*‐algebras of semigroups

Boundary quotient C*‐algebras of semigroups

Finite-dimensional approximations and semigroup coactions for operator algebras

Toeplitz quotient C*-algebras and ratio limits for random walks

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