let $a_1$, $a_2$ be unital banach algebras and $x$ be an $a_1$-$a_2$- module. applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $a_i$ into the dual space $a^*_i$ (for$i=1,2$) and such derivations  from  the triangular banach algebraof t...

In this paper, we define the first topological (σ, τ)-cohomology group and examine vanishing of the first (σ, τ)-cohomology groups of certain triangular Banach algebras. We apply our results to study the (σ, τ)-weak amenability and (σ, τ)-amenability of triangular Banach algebras.

In the present paper, the Hyers-Ulam stability and also the superstability of double centralizers and multipliers on Banach algebras are established by using a fixed point method. With this method, the condition of without order on Banach algebras is no longer necessary.

1 Fundamentals of the Theory of Banach Algebras 2 1.1 Basic definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The spectrum and the resolvent set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Properties of the inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Characters and the Gelfand Transform . . . . . ....

In this paper we study the relation between module amenability of $theta$-Lau product $A×_theta B$ and that of Banach algebras $A, B$. We also discuss module biprojectivity of $A×theta B$. As a consequent we will see that for an inverse semigroup $S$, $l^1(S)×_theta l^1(S)$ is module amenable if and only if $S$ is amenable.

We survey the recent investigations on (bounded, sequential) approximate Connes amenability and pseudo-Connes amenability for dual Banach algebras. We will discuss the core problems concerning these notions and address the signicance of any solutions to them to the development of the eld.

We present a variety of open questions in automatic continuity theory, concentrating on homomorphisms between Banach algebras and derivations from a Banach algebra A into a Banach A-bimodule.

1. Topological Vector Spaces 1 1.1. The Krein-Milman theorem 7 2. Banach Algebras 11 2.1. Commutative Banach algebras 14 2.2. ∗–Algebras (over complexes) 17 2.3. Problems on Banach algebras 20 3. The Spectral Theorem 21 3.1. Problems on the Spectral Theorem (Multiplication Operator Form) 26 3.2. Integration with respect to a Projection Valued Measure 27 3.3. The Functional Calculus 34 4. Unboun...

Let $S$ be an inverse semigroup and let $E$ be its subsemigroup of idempotents. In this paper we define the $n$-th module cohomology group of Banach algebras and show that the first module cohomology group $HH^1_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is zero, for every odd $ninmathbb{N}$. Next, for a Clifford semigroup $S$ we show that $HH^2_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is a Banach sp...

In this article we study two different generalizations of von Neumann regularity, namely strong topological regularity and weak regularity, in the Banach algebra context. We show that both are hereditary properties and under certain assumptions, weak regularity implies strong topological regularity. Then we consider strong topological regularity of certain concrete algebras. Moreover we obtain ...