Inverse Semigroup <i>C</i>*-Algebras Associated with Left Cancellative Semigroups

Norling, Magnus Dahler

doi:10.1017/s0013091513000540

Cited by 30 publications

(35 citation statements)

References 25 publications

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“…for s, t ∈ S. It follows that C * (S) admits a dense spanning set v s v * t with s, t ∈ S, see [Nor14,BLS18]. We let e sS denote the projection v s v * s in C * (S), for each s ∈ S.…”

Section: Preliminariesmentioning

confidence: 99%

“…and tractable properties. A general construction of full and reduced C * -algebras for left cancellative monoids was introduced by Li, [Li12], and soon fundamental questions about the interplay between nuclearity of the C * -algebra and amenability of the monoid or its left inverse hull were raised, see [Li13,Nor14]. Subsequent work on semigroup C *algebras that centered on computing K-theory revealed impressively rich connections to number theory, see [Li14,KT], and geometric group theory, see [ELR16].…”

Section: Introductionmentioning

confidence: 99%

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𝐶*-algebras of right LCM monoids and their equilibrium states

Brownlowe

Larsen

Ramagge

et al. 2020

Trans. Amer. Math. Soc.

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We study the internal structure of C * -algebras of right LCM monoids by means of isolating the core semigroup C * -algebra as the coefficient algebra of a Fock-type module on which the full semigroup C * -algebra admits a left action. If the semigroup has a generalised scale, we classify the KMS-states for the associated time evolution on the semigroup C * -algebra, and provide sufficient conditions for uniqueness of the KMS β -state at inverse temperature β in a critical interval.

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“…for s, t ∈ S. It follows that C * (S) admits a dense spanning set v s v * t with s, t ∈ S, see [Nor14,BLS18]. We let e sS denote the projection v s v * s in C * (S), for each s ∈ S.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

𝐶*-algebras of right LCM monoids and their equilibrium states

Brownlowe

Larsen

Ramagge

et al. 2020

Trans. Amer. Math. Soc.

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“…See [26] for the abstract construction of C * (S) valid for arbitrary left cancellative semigroups, and [1] or [29] for the case of right LCM semigroups. When S is a right LCM semigroup, [2,Corollary 7.11] implies that C * (S) is isomorphic to the Nica-Toeplitz algebra N T (X) for the compactly aligned product system X over S with fibers X s ∼ = C for all s ∈ S. It is therefore natural to ask if Theorem 2.19 can be applied.…”

Section: * -Algebras Associated To Right Lcm Semigroupsmentioning

confidence: 99%

Nica–Toeplitz algebras associated with product systems over right LCM semigroups

Kwaśniewski

Larsen

2019

Journal of Mathematical Analysis and Applications

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We prove uniqueness of representations of Nica-Toeplitz algebras associated to product systems of C * -correspondences over right LCM semigroups by applying our previous abstract uniqueness results developed for C * -precategories. Our results provide an interpretation of conditions identified in work of Fowler and Fowler-Raeburn, and apply also to their crossed product twisted by a product system, in the new context of right LCM semigroups, as well as to a new, Doplicher-Roberts type C * -algebra associated to the Nica-Toeplitz algebra. As a derived construction we develop Nica-Toeplitz crossed products by actions with completely positive maps. This provides a unified framework for Nica-Toeplitz semigroup crossed products by endomorphisms and by transfer operators. We illustrate these two classes of examples with semigroup C * -algebras of right and left semidirect products. A x, y · z = x · y, z A for all x, y, z ∈ X. An equivalence A-bimodule is a Hilbert bimodule which is full over left and right. We say that two Hilbert A-bimodules X, Y are Morita equivalent if there is an equivalence A-bimodule E such that X ⊗ A E ∼ = Y ⊗ A E. Remark 1.3. A product system X is (left) essential if each C

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“…Note that the statement in the last corollary was also obtained in [31]. Now set ∂Ω P := ∂ E, where E is the semilattice of idempotents of S = I l (P ).…”

Section: Corollary 316 If P Is a Subsemigroup Of An Amenable Group mentioning

confidence: 82%

“…The full inverse semigroup C*-algebra C * (S) of an inverse semigroup S is the universal C*-algebra for *-representations of S by partial isometries. We mod out 0 if S has a zero, as in [31].…”

Section: Without the Assumption Of Topological Freeness)mentioning

confidence: 99%

Partial Transformation Groupoids Attached to Graphs and Semigroups

2016

Int Math Res Notices

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Abstract. We introduce the notion of continuous orbit equivalence for partial dynamical systems, and give an equivalent characterization in terms of Cartan-isomorphisms for partial C*-crossed products. Both graph C*-algebras and semigroup C*-algebras can be described as C*-algebras attached to partial dynamical systems. As applications, for graphs, we generalize and explain a result of Matsumoto and Matui relating orbit equivalence and Cartan-isomorphism, and for semigroups, we strengthen several structural results for semigroup C*-algebras concerning amenability, nuclearity as well as simplicity of boundary quotients. We also discuss pure infiniteness for partial transformation groupoids arising from graphs and semigroups.

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Inverse Semigroup C*-Algebras Associated with Left Cancellative Semigroups

Cited by 30 publications

References 25 publications

𝐶*-algebras of right LCM monoids and their equilibrium states

𝐶*-algebras of right LCM monoids and their equilibrium states

Nica–Toeplitz algebras associated with product systems over right LCM semigroups

Partial Transformation Groupoids Attached to Graphs and Semigroups

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