C*-Algebras of Right LCM One-Relator Monoids and Artin–Tits Monoids of Finite Type

Li, Xin; Omland, Tron; Spielberg, Jack

doi:10.1007/s00220-020-03758-5

Cited by 12 publications

(32 citation statements)

References 47 publications

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“…As explained in [37,Remark 3.6], our K-theory computation for C * λ (P) then yields the following K-theory formula for ∂C * λ (P):…”

Section: Semigroup C*-algebras Of One-relator Monoidsmentioning

confidence: 98%

“…The semigroup C*-algebras of such one-relator monoids have been studied in [37]. As observed in [37, § 2.1.4], P is right LCM if * (u) = * (v) or if * (u) < * (v) and there exists a ∈ S with a (u) > a (v).…”

Section: Semigroup C*-algebras Of One-relator Monoidsmentioning

confidence: 99%

“…Now assume that |S| ≥ 3. Then by [37,Corollar 3.5], the boundary quotient ∂C * λ (P) (see for instance [15, § 5.7] for an introduction) is purely infinite simple, and there is an exact sequence 0…”

Section: Semigroup C*-algebras Of One-relator Monoidsmentioning

confidence: 99%

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K-Theory for Semigroup C*-Algebras and Partial Crossed Products

2021

Commun. Math. Phys.

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Using the Baum–Connes conjecture with coefficients, we develop a K-theory formula for reduced C*-algebras of strongly 0-E-unitary inverse semigroups, or equivalently, for a class of reduced partial crossed products. This generalizes and gives a new proof of previous K-theory results of Cuntz, Echterhoff and the author. Our K-theory formula applies to a rich class of C*-algebras which are generated by partial isometries. For instance, as new applications which could not be treated using previous results, we discuss semigroup C*-algebras of Artin monoids, Baumslag-Solitar monoids and one-relator monoids, as well as C*-algebras generated by right regular representations of semigroups of number-theoretic origin, and C*-algebras attached to tilings.

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“…As explained in [37,Remark 3.6], our K-theory computation for C * λ (P) then yields the following K-theory formula for ∂C * λ (P):…”

Section: Semigroup C*-algebras Of One-relator Monoidsmentioning

confidence: 98%

Section: Semigroup C*-algebras Of One-relator Monoidsmentioning

confidence: 99%

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K-Theory for Semigroup C*-Algebras and Partial Crossed Products

2021

Commun. Math. Phys.

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“…It is well known that χ is surjective (cf. [5,Lemma 2.3]), and thus χ is surjective. By Proposition 3.9, χ is injective, and hence χ is a bijection.…”

Section: The Laca-boundary Of a Monoidmentioning

confidence: 99%

“…This topological space plays an essential role for de ning boundary quotients of C * -algebras associated to monoids (cf. [4, § 7], [5]). Indeed one has the following (cf.…”

Section: Introductionmentioning

confidence: 99%

Monoids, their boundaries, fractals and C*-algebras

Verme

Weigel

2020

Topological Algebra and Its Applications

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In this note we establish some connections between the theory of self-similar fractals in the sense of John E. Hutchinson (cf. [3]), and the theory of boundary quotients of C*-algebras associated to monoids. Although we must leave several important questions open, we show that the existence of self-similar ℳ-fractals for a given monoid ℳ, gives rise to examples of C*-algebras (1.9) generalizing the boundary quotients \partial C_\lambda ^*(\mathcal{M}) discussed by X. Li in [4, §7, p. 71]. The starting point for our investigations is the observation that the universal boundary of a finitely 1-generated monoid carries naturally two topologies. The fine topology plays a prominent role in the construction of these boundary quotients, while the cone topology can be used to define canonical measures on the attractor of an ℳ-fractal for a finitely 1-generated monoid ℳ.

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