Narrow Results

The notion of the amenability of a locally compact group has been extended in various ways. Two weaker versions of amenability, weak amenability and the approximation property, have been defined for locally compact groups (by Haagerup and Haagerup and Kraus, respectively) and Bekka has defined a notion of amenability for representations of locally compact groups. Correspondences can be viewed as a generalization of representations of such groups. Using this viewpoint, Ananthraman–Delaroche has defined a notion of (left) amenability for correspondences. In this paper, we define notions of weak amenability and the approximation property for correspondences (and representations of locally compact groups), and obtain various results concerning these notions. Ananthraman–Delaroche showed that if N ⊂ M is an inclusion of von Neumann algebras, and if the associated inclusion correspondence is left amenable, then various approximation properties of N (semidiscreteness, the weak* completely bounded approximation property, and the weak* operator approximation property) are shared by M. We show that if this correspondence has the (weaker) approximation property, then if N has the weak* operator approximation property, so does M. An application of this result to crossed products is also given.

We define concepts of amenability and co-amenability for locally compact quantum groups in the sense of J. Kustermans and S. Vaes. Co-amenability of a lcqg (locally compact quantum group) is proved to be equivalent to a series of statements, all of which imply amenability of the dual lcqg. Further, it is shown that if a lcqg is amenable, then its universal dual lcqg is nuclear. We also define and study amenability and weak containment concepts for representations and corepresentations of lcqg's.

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a noncommutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SU_q(2) arising from measures on its spectrum.

For a locally compact quantum group 𝔾, we generalize some notions of amenability such as amenability of locally compact quantum groups and inner amenability of locally compact groups to the case of right Banach L¹(𝔾)-modules. Also, we investigate the concept of harmonic functionals over right Banach L¹(𝔾)-modules and use these devices to study, among others, amenability of 𝔾.

Consider an inclusion of diffuse von Neumann algebras $A\subset M$. We say that $A\subset M$ has the absorbing amenability property (AAP) if for any diffuse subalgebra $B\subset A$ and any amenable intermediate algebra $B\subset D\subset M$ we have that $D$ is contained in $A.$ We prove that the cup subalgebra associated to any subfactor planar algebra has the AAP.

Let $𝔾$ be a locally compact quantum group. Then the space $T(L^2(𝔾))$ of trace class operators on $L^2(𝔾)$ is a Banach algebra with the convolution induced by the right fundamental unitary of $𝔾$. We show that properties of $𝔾$ such as amenability, triviality and compactness are equivalent to the existence of left or right invariant means on the convolution Banach algebra $T(L^2(𝔾))$. We also investigate the relation between the existence of certain (weakly) compact right and left multipliers of $T{(L^2(𝔾))}^{\ast \ast }$ and some properties of $𝔾$.

We study some properties of the Cayley graph of R. Thompson's group F in generators x₀, x₁. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x₀, x₁. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x₀, x₁ and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of .

We study the subgroups of R. J. Thompson's group F and PL_o(I), the group of orientation preserving, piecewise linear self homeomorphisms of [0, 1]. We exhibit, for each non-limit ordinal α ≤ ω² + 1, an elementary amenable group of elementary class α (under Chou's stratification of elementary amenable groups) that is a subgroup of F and thus of PL_o(I). We also give examples that negatively answer a question of Sapir about non-solvable groups in F and PL_o(I).

The structure of a self-similar group G naturally gives rise to a transformation which assigns to any probability measure μ on G and any vertex w in the action tree of the group a new probability measure μ^w. If the measure μ is self-similar in the sense that μ^w is a convex combination of μ and the δ-measure at the group identity, then the asymptotic entropy of the random walk (G, μ) vanishes; therefore, the random walk is Liouville and the group G is amenable. We construct self-similar measures on several classes of self-similar groups, including the Grigorchuk group of intermediate growth.

We construct an example of a right amenable semigroup which is the ascending union of semigroups, none of which are right amenable.

Richard Thompson's group F is nonamenable if and only if there exists a ruinous set. It is known that no three element set is ruinous. Here we explicitly construct a family of thin sets for any three element set each of whose trees has three carets. We then show where this construction fails when each of the trees has four or more carets.

Let F denote the Thompson group with standard generators A = x₀, B = x₁. 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

By the density of a finite graph we mean its average vertex degree. For an $m$-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with $m$ generators is amenable if and only if the density of the corresponding Cayley graph equals $2m$.

A famous problem on the amenability of R. Thompson’s group $F$ is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators $\{x_0,x_1\}$, is at least $3.5$. This estimate has not been exceeded so far.

For the set of symmetric generators $S=\{x_1,{\bar{x}}_1\}$, where ${\bar{x}}_1=x_1{x}_0^{-1}$, the same example only gave an estimate of $3$. There was a conjecture that for this generating set equality holds. If so, $F$ would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set $X\subset F$, the inequality $\left|S^{\pm 1}X\right|\ge 2\left|X\right|$ holds.

In this paper, we disprove this conjecture showing that the density of the Cayley graph of $F$ in symmetric generators $S$ strictly exceeds $3$. Moreover, we show that even larger generating set $S_0=\{x_0,x_1,{\bar{x}}_1\}$ does not have doubling property.

We prove a computable version of Hall’s Harem theorem and apply it to computable versions of Tarski’s alternative theorem.

In this paper, we introduce a concept of an evacuation scheme on the Cayley graph of an infinite finitely generated group. This is a collection of infinite simple paths bringing all vertices to infinity. We impose a restriction that every edge can be used a uniformly bounded number of times in this scheme. An easy observation shows that the existence of such a scheme is equivalent to non-amenability of the group. A special case happens if every edge can be used only once. These schemes are called pure. We obtain a criterion for the existence of such a scheme in terms of isoperimetric constant of the graph. We analyze R. Thompson’s group $F$, for which the amenability property is a famous open problem. We show that pure evacuation schemes do not exist for the set of generators $\{x_0,x_1,{\bar{x}}_1\}$, where ${\bar{x}}_1=x_1{x}_0^{-1}$. However, the question becomes open if edges with labels $x_0^{\pm 1}$ can be used twice. The existence of pure evacuation schemes for this version is implied by some natural conjectures.

Amenability of some discrete subgroups of the group of diffeomorphisms of interval is proved. As a consequence, a solution of the problem of amenability of the Thompson group F is given.

We generalise Savchenko's definition of topological entropy for special flows over countable Markov shifts by considering the corresponding notion of topological pressure. For a large class of Hölder continuous height functions not necessarily bounded away from zero, this pressure can be expressed by our new notion of induced topological pressure for countable state Markov shifts with respect to a non-negative scaling function and an arbitrary subset of finite words, and we are able to set up a variational principle in this context. Investigating the dependence of induced pressure on the subset of words, we give interesting new results connecting the Gurevič and the classical pressure with exhaustion principles for a large class of Markov shifts. In this context we consider dynamical group extensions to demonstrate that our new approach provides a useful tool to characterise amenability of the underlying group structure.

In this paper, we introduce the concept of configuration graph and show how one can use this notion to simplify the theorem proved by Rejali and Yousofzadeh [Configuration of groups and paradoxical decompositions, Bull. Belg. Math. Soc. Simon Stevin18 (2011) 157–172].

Let ${𝒜}$ and $𝔄$ be two Banach algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with the left and right compatible action of $𝔄$ on ${𝒜}$. Let ${𝒜}⋊𝔄$ be a strongly splitting Banach algebra extension of $𝔄$ by ${𝒜}$. We show that (super) amenability of ${𝒜}⋊𝔄$ implies (super) module amenability of ${𝒜}$ and (super) amenability $𝔄$. We investigate biprojectivity and biflatness of ${𝒜}⋊𝔄$ in the some especial cases. We also give some results related to module biprojectivity and module biflatness of ${𝒜}$, when ${𝒜}⋊𝔄$ is biprojective or biflat.

Let $\mathrm{\Lambda }$ be a direct set, ${\{A_{\alpha }\}}_{\alpha \in \mathrm{\Lambda }}$ be a family of Banach algebras with bounded approximate identity (or unital) and $I$ be a set. We consider the Banach algebra ${\ell }^{\infty }({\ell }_{A_{\beta }}(I,A_{\alpha }))$. We show that this algebra has a bounded approximate identity (or is unital) if and only if $I$ is finite. We also characterize the left multipliers of these algebras and investigate their amenability of them. Moreover, we characterize the character spaces (Gelfand spaces) of these algebras in a special case.