Let X,Y be locally compact Hausdorff spaces and M,N be Banach algebras. Let θ : C0(X,M) → C0(Y,N ) be a zero-product preserving bounded linear map with dense range. We show that θ is given by a continuous field of algebra homomorphisms from M into N if N is irreducible. As corollaries, such a surjective θ arises from an algebra homomorphism, provided thatM is a W*-algebra and N is a semi-simple...

Let $(X,d)$ be an infinite compact metric space, let $(B,parallel . parallel)$ be a unital Banach space, and take $alpha in (0,1).$ In this work, at first we define the big and little $alpha$-Lipschitz vector-valued (B-valued) operator algebras, and consider the little $alpha$-lipschitz $B$-valued operator algebra, $lip_{alpha}(X,B)$. Then we characterize its second dual space.

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.

0. Introduction. Though spectral measures — i.e. countably additive idempotent-valued set functions defined on a cr-algebra of sets with values in =S?(£) — form the nominal topic of this paper, it can be looked on as having for its twin objects of study the spectral operators and the Boolean algebras of projections on locally convex spaces, since both are intimately connected with spectral meas...

Let $A$ be an arbitrary Banach algebra and $varphi$ a homomorphism from $A$ onto $Bbb C$. Our first purpose in this paper is to give some equivalent conditions under which guarantees a $varphi$-mean of norm one. Then we find some conditions under which there exists a $varphi$-mean in the weak$^*$ cluster of ${ain A; |a|=varphi(a)=1}$ in $A^{**}$.

We discuss two facets of the interaction between geometry and algebra in Banach algebras. In class unital algebras, there is essentially one known example which also strictly convex as a space. recall this example, finite-dimensional, consider open question generalising it to infinite dimensions. C*-algebras, we exhibit striking tighter relationship that exists there.

Let 0 → I → E → B → 0 be a short exact sequence of C*-algebras whereE is separable, I is quasidiagonal (QD) andB is nuclear, QD and satisfies the UCT. It is shown that if the boundary map ∂ : K1(B) → K0(I) vanishes then E must be QD also. A Hahn-Banach type property for K0 of QD C ∗-algebras is also formulated. It is shown that every nuclear QD C∗-algebra has this K0Hahn-Banach property if and ...

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.