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...

–a notion of amenability for topological semigroups is introduced. a topological semigroup s iscalled johnson amenable if for every banach s -bimodule e , every bounded crossed homomorphism froms to e* is principal. in this paper it is shown that a discrete semigroup s is johnson amenable if and only if1(s) is an amenable banach algebra. also, we show that if a topological semigroup s is johns...

Let S be a (discrete) semigroup, and let ` (S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ` (S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are ‘forbidden values’ for this constant. The second dual of ` (S) is the Banach ...

Let S be a semigroup, and let ` (S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ` (S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are ‘forbidden values’ for this constant. The second dual of ` (S) is the Banach algebraM(βS...

Let A be a unital separable amenable quasidiagonal simple C∗-algebra with real rank zero, stable rank one, weakly unperforated K0(A) and with a unique tracial state. We show that A must have tracial rank zero. Suppose also that A satisfies the Universal Coefficient Theorem. Then A can be classified by its (ordered) K-theory up to isomorphism. In particular, A must be a simple AH-algebra with no...

Let A be a Banach algebra. We study those closed ideals I of A for which the first cohomology group of A with coefficients in I is trivial; i.e. H(A, I) = {0}. We investigate such closed ideals when A is weakly amenable or biflat. Also we give some hereditary properties of ideal amenability.

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 study the semigroup C?-algebra of a positive cone P weakly quasi-lattice ordered group. That is, is subsemigroup discrete group G with P?P?1={e} and such that any two elements common upper bound in also have least bound. find sufficient conditions for to be nuclear. These involve idea generalised length function, called “controlled map”, into an amenable Here we give new definition controlle...