Semigroup C∗-Algebras

Li, Xin

doi:10.1007/978-3-319-39286-8_9

Cited by 11 publications

(9 citation statements)

References 43 publications

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“…They are constructed from weakly regular branch groups [16], that we then make act over an infinite alphabet. Moreover, the groupoids in the third item are groupoids associated to a left cancellative lcm monoid and give the first examples where the tight groupoid (which in this case is the universal groupoid) of the inverse hull has a non-simple reduced C * -algebra, but the C * -algebra associated to Li's regular representation [25] is simple (although the latter result requires an unpublished result of Exel and the first author).…”

Section: Theorem Bmentioning

confidence: 99%

“…The reduced C * -algebras of these examples have a non-trivial gray ideal since their complex Steinberg algebras have non-trivial singular ideal, and hence are not simple by the results of [32]. However, it will follow from forthcoming work of Exel and Steinberg that the gray ideal is the kernel of the action of the reduced C * -algebra under Li's regular representation [25] and so Li's algebra associated to the left cancellative lcm monoid M = Y * G will be simple. This seems to be the first example of such a phenomenon (since for these monoids the boundary quotient is the same as the reduced C * -algebra of the inverse hull).…”

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confidence: 99%

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Simplicity of inverse semigroup and étale groupoid algebras

Steinberg¹,

Szakács²

2020

Preprint

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In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups.The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies.Using Galois descent, we show that simplicity of étale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic 0. We also provide examples of inverse semigroups and étale groupoids with simple algebras outside of a prescribed set of prime characteristics.

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Section: Theorem Bmentioning

confidence: 99%

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confidence: 99%

Simplicity of inverse semigroup and étale groupoid algebras

Steinberg¹,

Szakács²

2020

Preprint

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“…Here, the set of constructible right ideals of left-cancellative semigroups is an important concept introduced by Xin Li in his work on semigroup C * -algebras. One may refer to [20] for more detail. (1) Baumslag-Solitar monoids form another class of quasi-lattice ordered groups recently studied in [31,5].…”

Section: Introductionmentioning

confidence: 99%

Regular Dilation and Nica-covariant Representation on Right LCM Semigroups

2019

Integr. Equ. Oper. Theory

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Regular dilation has recently been extended to graph product of N, where having a * -regular dilation is equivalent to having a minimal isometric Nica-covariant dilation. In this paper, we first extend the result to right LCM semigroups, and establish a similar equivalence among * -regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result can be applied to various semigroups to establish conditions for * -regular dilation.

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“…We recall that the reduced semigroup C * -algebra can be constructed for an arbitrary left cancellative semigroup. This algebra is a very natural object because it is generated by the left regular representation of the left cancellative semigroup.The study of such semigroup C * -algebras goes back to L. A. Coburn [18,19], R. G. Douglas [20], G. J. Murphy [21,22] and is continued at the present time (see, for example, [23,24] and references there in).…”

Section: Introductionmentioning

confidence: 99%

Inductive Limits for Systems of Toeplitz Algebras

Gumerov

2019

Lobachevskii J Math

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This article deals with inductive systems of Toeplitz algebras over arbitrary directed sets. For such a system the family of its connecting injective * -homomorphisms is defined by a set of natural numbers satisfying a factorization property. The motivation for the study of those inductive systems comes from our previous work on the inductive sequences of Toeplitz algebras defined by sequences of numbers and the limit automorphisms for the inductive limits of such sequences. We show that there exists an isomorphism in the category of unital C * -algebras and unital * -homomorphisms between the inductive limit of an inductive system of Toeplitz algebras over a directed set defined by a set of natural numbers and a reduced semigroup C * -algebra for a semigroup in the group of all rational numbers. The inductive systems of Toeplitz algebras over arbitrary partially ordered sets defined by sets of natural numbers are also studied.2010 Mathematics Subject Classification. 46L05, 47L40, 81T05 .

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Semigroup C∗-Algebras

Cited by 11 publications

References 43 publications

Simplicity of inverse semigroup and étale groupoid algebras

Simplicity of inverse semigroup and étale groupoid algebras

Regular Dilation and Nica-covariant Representation on Right LCM Semigroups

Inductive Limits for Systems of Toeplitz Algebras

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