Generalized notions of character amenability

Abstract

Abstract. In this paper the concepts of character contractibility, approximate character amenability (contractibility) and uniform approximate character amenability (contractibility) are introduced. We are concerned with the relations among the generalized concepts of character amenability for Banach algebra. We prove that approximate character amenability and approximate character contractibility are the same properties, as are uniform approximate character amenability and character amenability, as are uniform approximate character contractibility and character contractibility. For commutative Banach algebra, we prove that character contractibility and contractibility are the same properties. Moreover, general theory for those concepts is developed.

Approximate essential character amenability of Banach algebras

“…According to [1,Theorem 3.8], A is approximately character amenable if and only if A # is so. Now, suppose that A is a Banach algebra which is approximately essentially character amenable, which it is not approximately charater amenable.…”

“…Meanwhile, A is called approximately (ϕ, A)amenable, if every derivation D : A −→ X * is approximately inner which X is a (ϕ, A)-bimodule. The concepts of approximate left and right character amenability for Banach algebras were introduced in [1] and also another versions of character amenability of Banach algebras, called module character amenability and module approximate character amenability of Banach algebras were introduced and studied in [2,3]. Meanwhile, the concept of essentially ϕ-amenable for a Banach algebra which ϕ ∈ σ(A) ∪ {0}, was introduced by Nasr-Isfahani and Nemati in [9].…”

On approximate left φ-biprojective Banach algebras

“…For more information about these concepts the reader referred to [7], [5] and [6]. Motivated by these considerations, in [1] the approximate notions of amenability have been introduced and studied.…”

Module amenability, module character biprojectivity and module character biflatness of lau product of two Banach algebras

“…Moreover, the notion of approximate character amenability of Banach algebras was introduced by Aghababa, Shi and Wu in [19]. They called a Banach algebra A is approximately right character amenable, if for every φ ∈ ∆(A) ∪ {0} and every A-bimodule X, that the left module action is a.x = φ(a)x (a ∈ A, x ∈ X), every derivation D : A −→ X * is approximately inner.…”