A generalization of the weak amenability of Banach algebras

Generalization of the weak amenability on various Banach algebras

“…Here we use the technique of the proof in [8] to show that L 1 (G) is (ϕ, ψ)-weakly amenable. In fact, we generalize the result of [4], Example 4.2 which asserts that for any locally compact group G, L 1 (G) is (ϕ, ψ)-weakly amenable in which either ϕ or ψ is zero homomorphism. Theorem 2.6.…”

“…Then A is called (ϕ, ψ)-weakly amenable if H 1 (A, (A (ϕ,ψ) ) * ) = {0}. These concepts are introduced and investigated in [4] and [15] (for the generalization of n-weak amenability refer to [5]). It is also proved in [4], Example 4.2 that for any locally compact group algebra G, L 1 (G) is (ϕ, 0) and (0, ψ)-weakly amenable.…”

“…These concepts are introduced and investigated in [4] and [15] (for the generalization of n-weak amenability refer to [5]). It is also proved in [4], Example 4.2 that for any locally compact group algebra G, L 1 (G) is (ϕ, 0) and (0, ψ)-weakly amenable. For the module versions of these notions refer to [3] and [2].…”

“…Let A and B be Banach algebras. Similarly to [4], we denote by Hom(A, B) the space of all bounded homomorphisms from A into B and denote Hom(A, A) by Hom(A). Suppose that A is a Banach algebra, X is a Banach A-module and ϕ, ψ ∈ Hom(A).…”

Generalized biprojectivity and biflatness of abstract Segal algebras

“…In fact, A is called φ−contractible if there exists a φ−diagonal for A; that is, an element m ∈ A ⊗A such that φ(π A (m)) = 1 and a.m = φ(a)m, for each a ∈ A. Also in [3], it was introduced a new definition of amenability which was related to homomorphisms of Banach algebras, and then weak amenability of Banach algebras was generalized.…”