In this paper we prove that for a commutative character amenable Banach algebra A, if T : A → A is a multiplier then T has closed range if and only if T = BP = PB, where B ∈ M(A) is invertible and p ∈ M(A) is idempotent. By this result we characterize each multiplier with closed range on such Banach algebra (proposition 3.7), and so we get a necessary condition for character amenability of alge...

We show that amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This applies to a wide class of groups, amenability of which was an open problem, as well as unifies many known examples to one general proof. In particular, this includes Grigorchuk’s group, Basilica group, groups associated with...

In this paper we define $varphi$-Connes module amenability of a dual Banach algebra $mathcal{A}$ where $varphi$ is a bounded $w_{k^*}$-module homomorphism from $mathcal{A}$ to $mathcal{A}$. We are mainly concerned with the study of $varphi$-module normal virtual diagonals. We show that if $S$ is a weakly cancellative inverse semigroup with subsemigroup $E$ of idemp...

in this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric connes amenability. we determine symmetric module amenability and symmetric connes amenability of some concrete banach algebras. indeed, it is shown that $ell^1(s)$ is  a symmetric $ell^1(e)$-module amenable if and only if $s$ is amenable, where $s$ is an inverse semigroup with subsemigr...

We characterize groups with Guoliang Yu’s property A (i.e., exact groups) by the existence of a family of uniformly bounded representations which approximate the trivial representation. Property A is a large scale geometric property that can be viewed as a weak counterpart of amenability. It was shown in [12], that for a finitely generated group property A implies the Novikov conjecture. It was...

In this paper, we introduce a concept of an evacuation scheme on the Cayley graph infinite finitely generated group. This is collection simple paths bringing all vertices to infinity. We impose restriction that every edge can be used uniformly bounded number times in scheme. An easy observation shows existence such equivalent non-amenability A special case happens if only once. These schemes ar...

We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in sense Bourbaki. prove characterizations skew-amenability for groups isometries and automorphisms, clarify connection with extensive amenability group actions, establish Følner-type characterization, discuss closure properties class gro...

We define the isoperimetric constant for any locally finite metric space and we study the property of having isoperimetric constant equal to zero. This property, called Small Neighborhood property, clearly extends amenability to any locally finite space. Therefore, we start making a comparison between this property and other notions of amenability for locally finite metric spaces that have been...

We combine the fundamental results of Breuillard, Green, and Tao (Publ Math Inst Hautes Études Sci 116:115–221, 2012) on structure approximate groups, together with “tame” arithmetic regularity methods based work authors Terry (J Eur Soc (JEMS) 24(2):583–621, 2022), to give a theorem for finite subsets A arbitrary groups G where has “small tripling” bounded VC-dimension: Roughly speaking, up sm...