Uffe Haagerup, Kristian Knudsen Olesen
In this paper we prove that the Thompson groups $T$ and $V$ are not inner amenable. In particular, their group von Neumann algebras do not have property $Γ$. Moreover, we prove that if the reduced group $C^\ast$-algebra of $T$ is simple, then the Thompson group $F$ is non-amenable. Furthermore, we give a few new equivalent characterizations of amenability of $F$.
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In this paper we prove that the general version, F(N) of the Thompson group is inner amenable. As a consequence we generalize a result of P.Jolissaint. To do so, we prove first that F(N) together with a normal subgroup are i.c.c (infinite conjugacy classes) groups. Then, we investigate the relative McDuff property out of which we extract property $Γ$ for the group von Neumann algebras involved. By a result of E.G.Effros, F(N) follows inner amenable.
In this paper we study the non-amenability question of the Thompson group $F$ from the $C^{*}$ algebra side. Using a characterization of amenability in this framework we set about evaluating the reduced norm of the averages $\frac{1}{n}\sum x_0^ix_1x_0^{-i}$, where $x_0$ and $x_1$ are the generators of $F$ in its finite presentation. We prove that when $n$ is sufficiently large the above norm concentrates on a specific subset of $F$, easy to describe using the new normal form for elements in $F$, found by Guba and Sapir. We view this subset as the only obstruction against non-amenability.
It is a long standing open problem whether the Thompson group $F$ is an amenable group. In this paper we show that if $A$, $B$, $C$ denote the standard generators of Thompson group $T$ and $D:=C B A^{-1}$ then $$\sqrt2+\sqrt3\,<\,\frac1{\sqrt{12}}||(I+C+C^2)(I+D+D^2+D^3)||\,\le\, 2+\sqrt2.$$ Moreover, the upper bound is attained if the Thompson group $F$ is amenable. Here, the norm of an element in the group ring $\mathbb{C} T$ is computed in $B(\ell^2(T))$ via the regular representation of $T$. Using the "cyclic reduced" numbers $τ(((C+C^2)(D+D^2+D^3))^n)$, $n\in\mathbb{N}$, and some methods …
Let $G$ be a subgroup of a discrete (countable) group $Γ$. We introduce a notion of relative inner amenability of $G$ in $Γ$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss briefly the corresponding relative property gamma for pairs of finite factors and we deduce from this a characterization of discrete, icc groups which do not have Kazhdan's property $T$.
Let $H$ be a proper subgroup of a discrete group $G$. We introduce a notion of relative inner amenability of $H$ in $G$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss the corresponding relative property gamma for pairs of finite factors $N\subset M$ and we deduce from this a characterization of discrete, icc groups which do not have Kazhdan's property (T).
We find new presentations for the Thompson's groups $F$, the derived group $F^{'}$ and the intermediate group $D$. These presentations have a common ground in that their relators are the same and only the generating sets differ. As an application of these presentations we extract the following consequences: the cost of the group $F^{'}$ is 1 hence the cost cannot decide the (non)amenability question of $F$; the $II_1$ factor $L(F^{'})$ is inner asymptotically abelian and the reduced $C^*$-algebra of $F$ is not residually finite dimensional.
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…
Let $F$ denote the Thompson group with standard generators $A=x_0$, $B=x_1$. It is a long standing open problem whether $F$ is an amenable group. By a result of Kesten from 1959, amenability of $F$ is equivalent to $$(i)\qquad ||I+A+B||=3$$ and to $$(ii)\qquad ||A+A^{-1}+B+B^{-1}||=4,$$ where in both cases the norm of an element in the group ring $\mathbb{C} F$ is computed in $B(\ell^2(F))$ via the regular representation of $F$. By extensive numerical computations, we obtain precise lower bounds for the norms in $(i)$ and $(ii)$, as well as good estimates of the spectral distributions of $(I+A…
We show that a large class of i.c.c., countable, discrete groups satisfying a weak negative curvature condition are not inner amenable. By recent work of Hull and Osin, our result recovers that mapping class groups and Out(F_n) are not inner amenable. We also show that the group-measure space constructions associated to free, strongly ergodic p.m.p. actions of such groups do not have property Gamma of Murray and von Neumann.
A note on inner amenability for FLC point sets
本文证明第二可数局部紧群中近似格关联的群胚是内顺从的，并将结果推广到有限局部复杂性点集，部分回应了Ananantharman-Delaroche关于étale群胚内顺从性的公开问题。
On inner-amenability and boundary actions
研究离散可数群\(\Gamma\)，证明若\(\Gamma\)是 ICC 内顺从非顺从的则不满足(AO)性质，推广该现象；还证明若\(\Gamma\)是特定离散子群则不可能内顺从，推广了相关已知事实。
We here consider inner amenability from a geometric and group theoretical perspective. We prove that for every non-elementary action of a group $G$ on a finite dimensional irreducible CAT(0) cube complex, there is a nonempty $G$-invariant closed convex subset such that every conjugation invariant mean on $G$ gives full measure to the stabilizer of each point of this subset. Specializing our result to trees leads to a complete characterization of inner amenability for HNN-extensions and amalgamated free products. One novelty of the proof is that it makes use of the existence of certain idempote…
We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff II_1 factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non implications were obtained in [Ef73,JS85,Va09,Ki12a,Ki12b]. We complete the picture with the remaining implications and counterexamples.
We prove that the alternating group of a topologically free action of a countably infinite group $Γ$ on the Cantor set has the property that all of its $\ell^2$-Betti numbers vanish and, in the case that $Γ$ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable $Γ$ that there are many such alternating groups which are simple, finitely generated, and C$^*$-simple. The device for establishing nonisomorphism among these examples is a topological version of Austin's result on the invariance o…
Group von Neumann algebras, inner amenability, and unit groups of continuous rings
本文证明非内顺从离散群 $G$ 的群冯诺依曼代数相伴算子环的单位群在 rank 度量拓扑下非顺从，由此构造出非顺从单位群的非离散不可约连续环实例，并建立与 $\mathrm{II}_{1}$ 因子酉群 Eymard--Greenleaf 顺从性的联系。
We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction. We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau--Paterson, resolve a recent conjecture of Ng--Viselter, and prove that, for inner amenable quantum groups $\mathbb{G}$, approximation properties of the dual operator algebras can be averaged to approximation properties $\mathbb{G}$. Similar homological techniques are …
We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^\ast$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^\ast$-simple…
We prove a variety of results about the group von Neumann algebras associated to Thompson-like groups arising from so called $d$-ary cloning systems. Cloning systems are a framework developed by Witzel and the second author, with a $d$-ary version subsequently developed by Skipper and the second author, which can be used to construct generalizations of the classical Thompson's groups $F$, $T$, and $V$. Given a family of groups $(G_n)_{n\in\mathbb{N}}$ with a $d$-ary cloning system, we get a Thompson-like group $\mathscr{T}_d(G_*)$, and in this paper we find some mild, natural conditions under …
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。
As is well known, the equivalence between amenability of a locally compact group $G$ and injectivity of its von Neumann algebra $\mathcal{L}(G)$ does not hold in general beyond inner amenable groups. In this paper, we show that the equivalence persists for all locally compact groups if $\mathcal{L}(G)$ is considered as a $\mathcal{T}(L_2(G))$-module with respect to a natural action. In fact, we prove an appropriate version of this result for every locally compact quantum group.
On groupoids beyond partial actions, inner amenability, and models for Kirchberg algebras
本文构造了首个非内可amenable且非部分作用变换群oid的例子，解决了Anantharaman--Delaroche和Exel的问题，并展示了Deaconu--Renault群oid不来源于离散群的部分作用，同时提供了变换群oid模型。
We define a Banach algebra A to be dual if $A = (A_\ast)^\ast$ for a closed submodule $A_\ast$ of $A^\ast$. The class of dual Banach algebras includes all $W^\ast$-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus finite-dimensional. We…
Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure of the crossed product over the Fourier algebra of the acting group. The resulting characterisation of injectivity for crossed products generalises a result of Anantharaman-Delaroche on discrete groups. Amenable actions of locally compact groups on $C^*$-algebras are investigated in the same way, and amenability of the action is related to nuclearity of the corresponding crossed product. A survey is given to show that this notion of amenable action for $C^*$-algebras s…
A finitely generated ICC non property (T) group that is neither inner amenable nor properly proximal
本文构造了Caprace非内可均群的有限生成扩张，该扩张为GL_4(F_2(t))的ICC子群，无性质(T)，且非适当近端，从而解决了Boutonnet等人提出的问题。
Suppose that ${\mathcal A}$ is an algebra, $σ,τ:{\mathcal A}\to{\mathcal A}$ are two linear mappings such that both $σ({\mathcal A})$ and $τ({\mathcal A})$ are subalgebras of ${\mathcal A}$ and ${\mathcal X}$ is a $\big(τ({\mathcal A}),σ({\mathcal A})\big)$-bimodule. A linear mapping $D:{\mathcal A}\to {\mathcal X}$ is called a $(σ,τ)$-derivation if $D(ab)=D(a)\cdotσ(b)+τ(a)\cdot D(b) (a,b\in {\mathcal A})$. A $(σ,τ)$-derivation $D$ is called a $(σ,τ)$-inner derivation if there exists an $x\in{\mathcal X}$ such that $D$ is of the form either $D_x^-(a)=x\cdot σ(a)-τ(a)\cdot x (a\in {\mathcal A}…
Groupoid exactness and the weak containment problem
研究局部紧群体中精确性的类比，证明六个精确性概念的等价性，并探讨群体可接受性与C*-代数关系。
We introduce inner amenability for discrete p.m.p. groupoids and investigate its basic properties, examples, and the connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid. Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives rise to an inner amenable orbit equivalence relation. We also obtain an analogous result for compact extensions of equivalence relations which either are stable or have a non-trivial central sequence in their full group.
Kida and Tucker-Drob recently extended the notion of inner amenability from countable groups to discrete p.m.p. groupoids. In this article, we show that inner amenable groupoids have "fixed priced 1" in the sense that every principal extension of an inner amenable groupoid has cost 1. This simultaneously generalizes and unifies two well known results on cost from the literature, namely, (1) a theorem of Kechris stating that every ergodic p.m.p. equivalence relation admitting a nontrivial asymptotically central sequence in its full group has cost 1, and (2) a theorem of Tucker-Drob stating that…
Decomposable Fourier Multipliers and an Operator-Algebraic Characterization of Amenability
本文研究局部紧群上可分解傅里叶乘子代数与傅里叶-斯蒂尔杰斯代数的关系，证明离散群情形等距同构，并利用压缩投影给出顺从性的新算子代数刻画，揭示调和分析、算子代数与群几何的结构联系。
It is known that exactness for a discrete group is equivalent to C*-exactness, i.e., the exactness of its reduced C*-algebra. The problem of whether this equivalence holds for general locally compact groups has recently been reduced by Cave and Zacharias to the case of totally disconnected unimodular groups. We prove that the equivalence does hold for a class of groups that includes all locally compact groups with reduced C*-algebra admitting a tracial state. As a consequence, we present original proofs that totally disconnected locally compact (tdlc) invariant-neighbourhood (IN) groups and a …
Amenable actions of compact and discrete quantum groups on von Neumann algebras
本文利用非交换 Lp-空间技术证明紧量子群动力系统中强等变顺从性与等变顺从性等价，并通过对偶推广到离散量子群，同时首次给出顺从离散量子群非顺从作用于冯·诺依曼代数的显式例子。
A well known result of Haagerup from 1983 states that every C*-algebra A is weakly amenable, that is, every (associative) derivation from A into its dual is inner. A Banach algebra B is said to be ternary weakly amenable if every continuous Jordan triple derivation from B into its dual is inner. We show that commutative C*-algebras are ternary weakly amenable, but that B(H) and K(H) are not, unless H is finite dimensional. More generally, we inaugurate the study of weak amenability for Jordan Banach triples, focussing on commutative JB*-triples and some Cartan factors.
We study various operator homological properties of the Fourier algebra $A(G)$ of a locally compact group $G$. Establishing the converse of two results of Ruan and Xu, we show that $A(G)$ is relatively operator 1-projective if and only if $G$ is IN, and that $A(G)$ is relatively operator 1-flat if and only if $G$ is inner amenable. We also exhibit the first known class of groups for which $A(G)$ is not relatively operator $C$-flat for any $C\geq1$. As applications of our techniques, we establish a hereditary property of inner amenability, answer an open question of Lau and Paterson, and answer…
Cohomology of Amenable Traces
本文构造了检测单位C*-代数上迹态可迹性的Hochschild上循环，并给出了嵌入超有限II1因子超幂的等价刻画，证明了可迹性使Christensen-Sinclair二上循环成为上边界。
In this paper, the notion of proper proximality (introduced in [BIP18]) is studied and classified in various families of groups. We show that if a group acts non-elementarily by isometries on a tree such that for any two edges, the intersection of their edge stabilizers is finite, then G is properly proximal. We show that the wreath product G\wr H is properly proximal if and only if H is non-amenable. We then completely classify proper proximality among graph products of non-trivial groups. Our results generalize recent work of Duchesne, Tucker-Drob and Wesolek classifying inner amenability fo…
On Similarity Structure Groups and their W$^*$ and C$^*$-Algebras
本文研究可数相似结构群的一个子类CSS$^*$群，证明许多CSS$^*$群生成素群von Neumann代数，并探讨其C$^*$-简单性和非内可安性等性质。
We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G. Assuming that G is nonamenable, we obtain structural results for the stabilizer subgroups of amenable actions which allow us to relate the first l2-Betti number of G with that of the stabilizer subgroups. An analogous relationship is also shown to hold for cost. This relationship becomes even more pronounced for transitive amenable actions, leading to a simple criterion for vanishing of the first l2-Betti number and triviality of cost. Moreover, …
Stationary boundaries on the space of amenable subgroups and C*-simplicity
本文研究了可数群 G 存在非平凡 μ-边界条件，探讨了 amenable 子群空间的 μ-stationary 性质，并讨论了 Thompson 群 F 的 C*-简单性问题。
On weakly amenable groupoids
本研究推广群的相关结果，引入广群的弱顺从性与拟Cartan对概念，证明弱顺从广群的内正合性及相关Cowling-Haagerup常数不等式，并给出等号成立的几类广群。
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