We introduce the notion of Johnson pseudo-Connes amenability for dual Banach algebras. study relation between this new with various notions Connes like amenability, approximate and pseudo amenability. also investigate some hereditary properties notion. prove that a locally compact group G,M(G) is amenable if only G amenable. Also we show every non-empty set I,MI(C) under forced to have finite i...

In the present paper for two $mathfrak{A}$-module Banach algebras $A$ and $B$, we investigate relations between $varphi$-$mathfrak{A}$-module approximate amenability of $A$, $psi$-$mathfrak{A}$-module approximate amenability of $B$, and $varphioplus psi$-$mathfrak{A}$-module approximate amenability of $Aoplus B$ ($l^1$-direct sum of $A$ and $B$), where $varphiin$ Hom$_{mathfrak{A}}(A)$ and $psi...

For a Banach algebra $A$, $A''$ is $(-1)$-Weakly amenable if $A'$ is a Banach $A''$-bimodule and $H^1(A'',A')={0}$. In this paper, among other things,  we study the relationships between the $(-1)$-Weakly amenability of $A''$ and the weak amenability of $A''$ or $A$. Moreover, we show that the second dual of every $C^ast$-algebra is $(-1)$-Weakly amenable.

Abstract We prove that for a Banach algebra A having bounded $\mathcal {Z}(A)$ -approximate identity and every $\mathbf {[IN]}$ group G with weight w which is either constant on conjugacy classes or satisfies $w \geq 1$ , {Z}(L^{1}_{w}(G) \otimes ^{\gamma } A) \cong \mathcal {Z}(L^{1}_{w}(G)) . As an application, we discuss the conditions under {Z}(L^{1}_{\omega }(G,A))$ enjoys certain algebrai...

in 2005, a. abdollahi and a. rejali, studied the relations betweenparadoxical decomposition and configuration for semigroups. in thepresent paper, we introduce one other concept of amenability typeon semigroups and groups which includes amenability of semigroupsand inner-amenability of groups. we have extend the previous knownresults to semigroups and groups satisfying this concept.

we study the notion of bounded approximate connes-amenability for‎ ‎dual banach algebras and characterize this type of algebras in terms‎ ‎of approximate diagonals‎. ‎we show that bounded approximate‎ ‎connes-amenability of dual banach algebras forces them to be unital‎. ‎for a separable dual banach algebra‎, ‎we prove that bounded‎ ‎approximate connes-amenability implies sequential approximate...

We present a new approach to the amenability of groupoids (both in the measure theoretical and the topological setups) based on using Markov operators. We introduce the notion of an invariant Markov operator on a groupoid and show that the Liouville property (absence of non-trivial bounded harmonic functions) for such an operator implies amenability of the groupoid. Moreover, the groupoid actio...