Let $mathcal{R}$ be a  commutative ring with identity, let $A$ and $B$ be two $mathcal{R}$-algebras and $varphi:Blongrightarrow A$ be an $mathcal{R}$-additive algebra homomorphism. We introduce a new algebra $Atimes_varphi B$, and give some basic properties of this algebra. Generalized $2$-cocycle derivations on $Atimes_varphi B$ are studied. Accordingly, $Atimes_varphi B$ is considered from th...

In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra B(E) of all bounded linear operators on a Banach space E could ever be amenable if dimE = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros– Haydon result that solves the “scalar plus compact problem”: there is a...

Let G be a locally compact group, and let WAP(G) denote the space of weakly almost periodic functions on G. We show that, if G is a [SIN]-group, but not compact, then the dual Banach algebra WAP(G)∗ does not have a normal, virtual diagonal. Consequently, whenever G is an amenable, non-compact [SIN]-group, WAP(G)∗ is an example of a Connes-amenable, dual Banach algebra without a normal, virtual ...

Let G be a locally compact group, and let WAP(G) denote the space of weakly almost periodic functions on G. We show that, if G is a [SIN]-group, but not compact, then the dual Banach algebra WAP(G)∗ does not have a normal, virtual diagonal. Consequently, whenever G is an amenable, non-compact [SIN]-group, WAP(G)∗ is an example of a Connes-amenable, dual Banach algebra without a normal, virtual ...

Let G be a locally compact group, and let WAP(G) denote the space of weakly almost periodic functions on G. We show that, if G is a [SIN]-group, but not compact, then the dual Banach algebra WAP(G)∗ does not have a normal, virtual diagonal. Consequently, whenever G is an amenable, non-compact [SIN]-group, WAP(G)∗ is an example of a Connes-amenable, dual Banach algebra without a normal, virtual ...

In this paper we study the concept of ph-biatness ofa Banach algebra A, where ph is a continuous homomorphism on A.We prove that if ph is a continuous epimorphism on A and A hasa bounded approximate identity and A is ph- biat, then A is ph-amenable. In the case where ph is an isomorphism on A we showthat the ph- amenability of A implies its ph-biatness.

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...

We show that, if E is a Banach space with a basis satisfying a certain condition, then the Banach algebra l∞(K(l2 ⊕ E)) is not amenable; in particular, this is true for E = l with p ∈ (1,∞). As a consequence, l∞(K(E)) is not amenable for any infinite-dimensional Lp-space. This, in turn, entails the non-amenability of B(lp(E)) for any Lp-space E, so that, in particular, B(lp) and B(Lp[0, 1]) are...

Let Lω(G) be a Beurling algebra on a locally compact abelian group G. We look for general conditions on the weight which allows the vanishing of continuous derivations of Lω(G). This leads us to introducing vector-valued Beurling algebras and considering the translation of operators on them. This is then used to connect the augmentation ideal to the behavior of derivation space. We apply these ...