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 $A^{**}$.

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.

We consider Schatten ́s equation in connection with a Banach algebra and Banach modules. From this connection we infer a new proof of the fact that the second action of a bounded derivation is also a bounded derivation.

1. Introduction. Let (R") is the algebra Q1(Rn) of all real-valued functions of class C1. In other words, for every fEG1(Rn) there is a sequence {pn} of polynomials such that pn—*/ uniformly o...

In this paper we prove that for a commutative character amenable Banach algebra A, if T : A → A is a multiplier then T has closed range if and only if T = BP = PB, where B ∈ M(A) is invertible and p ∈ M(A) is idempotent. By this result we characterize each multiplier with closed range on such Banach algebra (proposition 3.7), and so we get a necessary condition for character amenability of alge...

We show that the structure of continuous and discontinuous homomorphisms from the Banach algebra Cn[0,1] of n times continuously differentiable functions on the unit interval [0,1] into finite dimensional Banach algebras is completely determined by higher point derivations.

The “spectral picture” of a bounded operator on a Banach space consists of its essential spectrum together with a mapping from its holes to the group of integers, obtained by taking the Fredholm index. In this note we abstract this from the Calkin algebra to a general Banach algebra, replacing the integers with the quotient of the group of invertibles by its connected component of the identity.

In this note, additive results are presented for the generalized Drazin inverse in Banach algebra. Necessary and sufficient conditions are given for the generalized Drazin invertibility of the sum of two commuting generalized Drazin invertible elements. These results are a generalization of the results from the paper [C.Y. Deng and Y. Wei. New additive results for the generalized Drazin inverse...

Let X be a Banach space and T be a bounded linear operator from X to itself (T ∈ B(X).) An operator S ∈ B(X) is a generalized inverse of T if TST = T . In this paper we look at several Banach algebras of operators and characterize when an operator in that algebra has a generalized inverse that is also in the algebra. Also, Drazin inverses will be related to generalized inverses and spectral pro...

Let A be a Banach algebra. The flip on A ⊗ A is defined through A ⊗ A ∋ a⊗ b 7→ b⊗ a. If A is ultraprime, El(A), the algebra of all elementary operators on A, can be algebraically identified with A ⊗ A, so that the flip is well defined on El(A). We show that the flip on El(A) is discontinuous if A = K(E) for a reflexive Banach space E with the approximation property.