Concept Of Amenability Research Articles

Amenability and profinite completions of finitely generated groups

This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group A and an uncountable family of finitely generated, residually finite non-amenable groups, all of which are profinitely isomorphic to A . All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group A embeds in these groups such that the inclusion induces an isomorphism of profinite completions. In addition, we review the concept of uniform amenability, a strengthening of amenability introduced in the 70s, and we prove that uniform amenability is indeed detectable from the profinite completion.

Jordan amenability of banach algebras

AbstractIn this paper, we introduce and study the concepts of Jordan amenability and Jordan biflatness of Banach algebras and compare those with the classical notions of amenability and biflatness. We also find some relations between Jordan amenability and the existence of Jordan approximate and virtual diagonals. These could be considered as Jordan versions of the classical results due to Johnson and Helemskii. We show that, for allC*-algebras, the concepts of amenability and Jordan amenability coincide.

The coexistence of amenity and biodiversity in urban landscapes

Amenity is a long-standing component of town planning and municipal governance. Biodiversity is a far more recent concept, yet interpreting the conservation mandate in a local context is a significant challenge for landscape and urban planners. This article explores the concepts of amenity and biodiversity and investigates their compatibility in an urbanising world. Their historical expression in law and urban planning is considered, and empirical research on the links between human well-being, green environments and biodiversity is reviewed. We argue that amenity is an underutilised vehicle for achieving biodiversity goals in line with new urban greening paradigms because of its long-standing currency with planning professionals. However, conflict between biodiversity and amenity can arise in practice, depending on a city’s social–ecological context. These challenges can be overcome through setting clear objectives, utilising scientific evidence, engaging with local communities and ensuring landscape policy is sufficiently flexible to accommodate local needs and characteristics.

A generalization of the $$n$$ n -weak module amenability of banach algebras

In this paper, we generalize the notion of \(n\)-weak module amenability of a Banach algebra \(A\) which is a Banach module over another Banach algebra \(U\) with compatible actions to that of \((\sigma )-n\)-weak module amenability for \(n\in \mathbb {N}\) and \(\sigma \in Hom_{U}(A)\). We also investigate the relation between this new concept of amenability of \(A\) and the quotient Banach algebra \(A/J\) where \(J\) is the closed ideal of \(A\) generated by elements of the form \((a\cdot \alpha )b-a(\alpha \cdot b)\) for \(a,b\in A\) and \(\alpha \in U\). As a consequence, we show that the semigroup algebra \(l^{1}(S)\) is \((\sigma )\)-\((2n+1)\)-weakly module amenable as an \(l^{1}(E)\)-module for each \(n\in \mathbb {N}\) and \(\sigma \in Hom_{l^{1}(E)}(l^{1}(S))\), where \(S\) is an inverse semigroup with the set of idempotents \(E\).

Γ-amenability of locally compact groups

Let G be a locally compact group and let Γ be a closed subgroup of G × G. Pier introduced the notion of Γ-amenability which gives a new classification of groups. This concept generalizes the concept of amenability and inner amenability for locally compact groups. In this paper, among other things, we extend some standard results for amenable groups to Γ-amenable groups and give various characterizations for Γ-amenable groups. A sequence of characterizations of Γ-amenable groups is given here by developing the well-known Folner’s conditions for amenable locally compact groups. Several characterizations of inner amenability are also given.

Cohomology on hypergroup algebras

There are concepts which are related to or can be formulated by homological techniques, such as derivations, multipliers and lifting problems. Moreover, a Banach algebra A is said to be amenable if H1(A,X*)=0 for every A-dual module X*. Another concept related to the theory is the concept of amenability in the sense of Johnson. A topological group G is said to be amenable if there is an invariant mean on L 8(G). Johnson has shown that a topological group is amenable if and only if the group algebra L1(G) is amenable. The aim of this research is to define the cohomology on a hypergroup algebra L(K) and extend the results of L1(G) over to L(K). At first stage it is viewed that Johnson's theorem is not valid so more. If A is a Banach algebra and h is a multiplicative linear functional on A, then (A,h) is called left amenable if for any Banach two-sided A-module X with ax=h(a)x(a? A, x? X),H1(A,X*)=0. We prove that (L(K),h) is left amenable if and only if K is left amenable. Where, the latter means that there is a left invariant mean m on C(K), i. e., m(lf)=m(f)x, where lxf(µ)=f(dx*µ). In this case we briefly say that L(K) is left amenable. Johnson also showed that L1(G) is amenable if and only if the augmentation ideal I={f? L1(G)|∫Gf=0}0 has abounded right approximate identity. We extend this result to hypergroups.

Amenable equivalence relations and Turing degrees

In [12] Slaman and Steel posed the following problem:Assume ZF + DC + AD. Suppose we have a function assigning to each Turing degree d a linear order <d of d. Then must the rationals embed order preservingly in <d for a cone of d's?They had already obtained a partial answer to this question by showing that there is no such d ↦ <d with <d of order type ζ = ω* + ω on a cone. Already the possibility that <d has order type ζ · ζ was left open.We use here, ideas and methods associated with the concept of amenability (of groups, actions, equivalence relations, etc.) to prove some general results from which one can obtain a positive answer to the above problem.

Amenable pairs of groups and ergodic actions and the associated von Neumann algebras

If X and Y are ergodic G-spaces, where G is a locally compact group, and X is an extension of Y, we study a notion of amenability for the pair ( X , Y ) (X,Y) . This simultaneously generalizes and expands upon previous work of the author concerning the notion of amenability in ergodic theory based upon fixed point properties of affine cocycles, and the work of Eymard on the conditional fixed point property for groups. We study the relations between this concept of amenability, properties of the von Neumann algebras associated to the actions by the Murray-von Neumann construction, and the existence of relatively invariant measures and conditional invariant means.

A Preliminary Survey of the Public's Concepts of Amenity in British Forestry

A Preliminary Survey of the Public's Concepts of Amenity in British Forestry