Massoud Amini, Alireza Medghalchi
For a locally compact Hausdorff semigroup S, the $L^\infty$ representation algebra R(S) was extensively studied by Dunkl and Ramirez. The Fourier-Stieltjes algebra F(S) of a topological semigroup was studied by Lau. The aim of this paper is to investigate these two algebras and study the amenability of them with respect to the structure of S.
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We construct reduced and full semigroup C*-algebras for left cancellative semigroups. Our new construction covers particular cases already considered by A. Nica and also Toeplitz algebras attached to rings of integers in number fields due to J. Cuntz. Moreover, we show how (left) amenability of semigroups can be expressed in terms of these semigroup C*-algebras in analogy to the group case.
We investigate recent uniqueness theorems for reduced $C^*$-algebras of Hausdorff étale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full $C^*$-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.
We study C*-algebras associated with subsemigroups of groups. For a large class of such semigroups including positive cones in quasi-lattice ordered groups and left Ore semigroups, we describe the corresponding semigroup C*-algebras as C*-algebras of inverse semigroups, groupoid C*-algebras and full corners in associated group crossed products. These descriptions allow us to characterize nuclearity of semigroup C*-algebras in terms of faithfulness of left regular representations and amenability of group actions. Moreover, we also determine when boundary quotients of semigroup C*-algebras are U…
We argue that weak containment is an appropriate notion of amenability for inverse semigroups. Given an inverse semigroup $S$ and a homomorphism $ϕ$ of $S$ onto a group $G$, we show, under an assumption on $\ker(ϕ)$, that $S$ has weak containment if and only if $G$ is amenable and $\ker(ϕ)$ has weak containment. Using Fell bundle amenability, we find a related result for inverse semigroups with zero. We show that all graph inverse semigroups have weak containment and that Nica's inverse semigroup $\mcT_{G,P}$ of a quasi-lattice ordered group $(G,P)$ has weak containment if and only if $(G,P)$ …
We prove a functoriality result for the full C*-algebras of right-LCM monoids with respect to monoid inclusions that are closed under factorization and preserve orthogonality, and use this to show that if a right-LCM monoid is amenable in the sense of Nica, then so are its submonoids. As applications, we complete the classification of Artin monoids with respect to Nica amenability by showing that only the right-angled ones are amenable in the sense of Nica and we show that the Nica amenability of a graph product of right-LCM semigroups is inherited by the factors.
We consider the amenability of groupoids $G$ equipped with a group valued cocycle $c:G\to Q$ with amenable kernel $c^{-1}(e)$. We prove a general result which implies, in particular, that $G$ is amenable whenever $Q$ is amenable and if there is countable set $D\subset G$ such that $c(G^{u})D=Q$ for all $u\in G^{(0)}$. We show that our result is applicable to groupoids arising from partial semigroup actions. We explore these actions in detail and show that these groupoids include those arising from directed graphs, higher rank graphs and even topological higher rank graphs. We believe our metho…
We prove that if an infinite, discrete semigroup has the property that every right syndetic set is left syndetic, then the semigroup has a left invariant mean. We prove that the weak*-closed convex hull of the two-sided translates of every bounded function on an infinite discrete semigroup contains a constant function. Our proofs use the algebraic properties of the Stone-Cech compactification.
Amenability of locally compact quantum groups via the long-time behavior of convolution semigroups
该研究刻画了第二可数局部紧量子群的可和性，结合其对偶量子群的逼近性质，构造了受时间两端行为控制的卷积半群，相关结果在经典局部紧群情形下也属首次。
We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner's type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence. In the context of inverse semigroups $S$ we give a characterization of invariant measures on $S$ (in the sense of Day) in terms of two notions: $domain$ $measurability$ and $localization$. Given a unital representation of $S$ in terms of partial bijections on some set $X$ we define a n…
Essential groupoid amenability and nuclearity of groupoid C*-algebras
本文构造了étale群胚的本质C*-代数，引入其极大版本及本质顺从性条件（弱于拓扑顺从性），并将可消半群代数作用产生的Bruce-Li代数描述为奇异本质C*-代数。
A locally compact groupoid is said to have the weak containment property if its full $C^*$-algebra coincides with its reduced one. This property is strictly weaker than amenability and is known to be equivalent to amenability for transformation groupoids relative to actions of exact discrete groups. We believe that for general étale groupoids one should have the same equivalence of the two properties under some mild exactness assumption. In this paper we try to support this statement.
Generalizing Block and Weinberger's characterization of amenability we introduce the notion of uniformly finite homology for a group action on a compact space and use it to give a homological characterization of topological amenability for actions. By considering the case of the natural action of $G$ on its Stone-\vCech compactification we obtain a homological characterization of exactness of the group, answering a question of Nigel Higson.
We establish that a second countable locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable. We give some examples and applications.
We give a new definition of the semigroup C*-algebra of a left cancellative semigroup, which resolves problems of the construction by X. Li. Namely, the new construction is functorial, and the independence of ideals in the semigroup does not influence the independence of the generators. It has a group C*-algebra as a natural quotient. The C*-algebra of the old construction is a quotient of the new one. All this applies both to the full and reduced C*-algebras. The construction is based on the universal inverse semigroup generated by a left cancellative semigroup. We apply this approach to conn…
In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson's group $\mathcal{F}$. We introduce the notion of unique factorization semigroup which contains Thompson's semigroup $\mathcal{S}$ and the free semigroup $\mathcal{F}_n$ on $n$ generators ($\geq2$). Let $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal{S}$ and $\mathcal{F}_n$, respectively. It is proved that all derivations on $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ are automatically c…
Amenability for unitary groups of simple monotracial C*-algebras
本文证明单位正则无限核C*-代数的等距半群右可 Amenability，且迹 GNS 表示超有限的单位单迹C*-代数的酉群在弱拓扑下斜可 Amenability，部分解决相关猜想与问题。
We fix a path model for the space of filters of the inverse semigroup $\mathcal{S}_Λ$ associated to a left cancellative small category $Λ$. Then, we compute its tight groupoid, thus giving a representation of its $C^*$-algebra as a (full) groupoid algebra. Using it, we characterize when these algebras are simple. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.
Let G be a locally compact topological group and X a compact space with continuous G-action. The main result of this essay states that the following statements are equivalent : 1) The action of G on X is topologically amenable ; 2) Every dual (G, X)-module of type C is a relatively injective Banach G-module ; 3) There is a G-invariant element in the double dual of C(X, L1(G)).
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 bou…
A Nica-amenable submonoid of a group which fails to embed in an amenable group
构造了一个群的弱拟格序子幺半群，其半群C*-代数是核的，但该幺半群不能嵌入任何可amenable群，从而表明Nica amenable性和核性不刻画可嵌入性。
Groupoid strict comparison for Ample Groupoids: Fiberwise Supramenability and Topological Amenability
本文为局部紧Hausdorff étale群胚引入纤维上超可支配性，证明在最小充足群胚中，若满足纤维上超可支配性或拓扑可amenable性，则其闭开型半群几乎无穿孔，进而σ-紧时具有群胚严格比较，并应用于Matui的AH猜想及纯无穷性。
We introduce the notion of the Fourier and Fouier-Stieltjes algebra of a topological *-semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras.
We provide groupoid models for Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs. Extending the groupoid models used in the theory of graph algebras and topological dynamical systems to our setting, we prove results on essential freeness and amenability of the groupoids which capture the existing theory, and extend results involving group crossed products of graph algebras.
Tracially amenable actions and purely infinite crossed products
本文引入迹顺从作用概念并给出等价刻画，证明含自由群F2的群在简单、单位、Z-稳定C*-代数上的外迹顺从作用具有纯无限交叉积，并提供自由群在AF-代数上的具体例子。
We introduce and analyze the full $\mathcal{NT}_{\mathcal{L}}(\mathcal{K})$ and the reduced $\mathcal{NT}_{\mathcal{L}}^r(\mathcal{K})$ Nica-Toeplitz algebra associated to an ideal $\mathcal{K}$ in a right tensor $C^*$-precategory $\mathcal{L}$ over a right LCM semigroup $P$. Our main results are uniqueness theorems in the spirit of classical Coburn's theorem, generalizing uniqueness results for Toeplitz-type $C^*$-algebras associated to single $C^*$-correspondences, quasi-lattice ordered semigroups, and crossed products twisted by product systems of $C^*$-correspondences obtained by Fowler, L…
We prove a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type. To this end, we introduce Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product system of 'regular' C*-correspondences. Under a certain aperiodicity condition on the latter, we obtain the uniqueness theorem and a simplicity criterion for the algebras in question. These results generalize the corresponding ones for crossed products by discrete groups, due to Archbold and Spielberg, and for Exel's crossed products, due to Exel and Vershik…
In this paper, we consider topological semigroup actions on compact topological spaces. Under mild assumptions on the semigroup and the action, we construct a semi-direct product groupoid with a Haar system. We also show that it is equivalent to a transformation groupoid. We apply this construction to the Wiener-Hopf $C^{*}$-algebras.
Let $\mathcal{G}$ be a locally compact $σ$-compact Hausdorff ample groupoid on a compact space. In this paper, we further examine the (ubiquitous) fiberwise amenability introduced by the author and Jianchao Wu for $\mathcal{G}$. We define the corresponding concepts of Følner sequences and Banach densities for $\mathcal{G}$, based on which, we establish a topological groupoid version of the Ornstein-Weiss quasi-tilling theorem. This leads to the notion of almost finiteness in measure for ample groupoids as a weaker version of Matui's almost finiteness. As applications, we first show that $C^*_r…
Amenability for unitary groups of C*-algebras
本文提出刻画幺正C*-代数酉群顺从性的猜想，证明了其对特定类C*-代数成立，并基于Kirchberg工作给出Ng相关问题的反例。
This is a survey of some aspects of the subject of approximation properties for locally compact quantum groups, based on lectures given at the {\it Topological Quantum Groups} Graduate School, 28 June - 11 July, 2015 in Bedłewo, Poland. We begin with a study of the dual notions of amenability and co-amenability, and then consider weakenings of these properties in the form of the Haagerup property and weak amenability. For discrete quantum groups, the interaction between these properties and various operator algebra approximation properties are investigated. We also study the connection between…
Amenability of group actions on compact spaces and the associated Banach algebras
本文将拓扑群顺从性的经典刻画推广至紧 Hausdorff 空间上的离散群作用，通过引入自然关联的 Banach 代数和弱化的顺从性概念，获得了顺从作用的类似刻画，并证明了加强 Dong 和 Wang (2015) 定理的不动点性质。
In this paper, we define the first topological $(σ,τ)$-cohomology group and examine vanishing of the first $(σ,τ)$-cohomology groups of certain triangular Banach algebras. We apply our results to study the $(σ,τ)$-weak amenability and $(σ,τ)$-amenability of triangular Banach algebras.
In this work we introduce and study a new notion of amenability for actions of locally compact groups on $C^*$-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and permanence properties analogous to those known in the discrete case. For example, for actions on commutative $C^*$-algebras, we show that our notion of amenability is equivalent to measurewise amenability. Combined with a recent result of Alex Bearden and Jason Crann, this also settles a long s…
Given a partial action $π$ of an inverse semigroup $S$ on a ring $\mathcal{A}$ one may construct its associated skew inverse semigroup ring $\mathcal{A} \rtimes_πS$. Our main result asserts that, when $\mathcal{A}$ is commutative, the ring $\mathcal{A} \rtimes_πS$ is simple if, and only if, $\mathcal{A}$ is a maximal commutative subring of $\mathcal{A} \rtimes_πS$ and $\mathcal{A}$ is $S$-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore,…
Amenable unitary representations of locally compact groupoids
本文研究局部紧群体的可约酉表示，通过Hilbert束理论和算子值向量测度定义拓扑不变均值，将Bekka对群的可约性刻画推广到群体。
We associate a boundary $\mathcal B_{π,u}$ to each covariant representation $(π,u,H)$ of $C^*$-dynamical system $(G,A,α)$ and study the action of $G$ on $\mathcal B_{π,u}$ and its amenability properties. We relate rigidity properties of traces on the associated crossed product C*-algebra to faithfulness of action of the group on this boundary.
We establish several new characterizations of amenable $W^*$- and $C^*$-dynamical systems over arbitrary locally compact groups. In the $W^*$-setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz-Schur multipliers of $(M,G,α)$ converging point weak* to the identity of $G\bar{\ltimes}M$. In the $C^*$-setting, we prove that amenability of $(A,G,α)$ is equivalent to an analogous Herz-Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$, as well as a particular case of the pos…
We present some generalizations of the well-known correspondence, found by R. Exel, between partial actions of a group $G$ on a set $X$ and semigroup homomorphism of $S(G)$ on the semigroup $I(X)$ of partial bijections of $X,$ being $S(G)$ an inverse monoid introduced by Exel. We show that any unital premorphism $θ:G\to S$, where $S$ is an inverse monoid, can be extended to a semigroup homomorphism $θ^*:T\to S$ for any inverse semigroup $T$ with $S(G)\subseteq T\subseteq P^*(G)\times G,$ being $P^*(G)$ the semigroup of non-empty subset of $G$, and such that $E(S)$ satisfies some lattice theore…
Given a discrete and countable inverse semigroup $S$ one can study, in analogy to the group case, its geometric aspects. In particular, we can equip $S$ with a natural metric, given by the path metric in the disjoint union of its Schützenberger graphs. This graph, which we denote by $Λ_S$, inherits much of the structure of $S$. In this article we compare the C*-algebra $\mathcal{R}_S$, generated by the left regular representation of $S$ on $\ell^2(S)$ and $\ell^\infty(S)$, with the uniform Roe algebra over the metric space, namely $C^*_u(Λ_S)$. This yields a chacterization of when $\mathcal{R}…
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