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A generalization of the weak amenability of Banach algebras. (English) Zbl 1163.46034

Summary: Let A be a Banach algebra and let φ and ψ be continuous homomorphisms on A. We consider the following module actions on A,

a·x=φ(a)x,x·a=xψ(a)(a,xA)·

We denote by A (φ,ψ) the above A-module. We call the Banach algebra A (φ,ψ)-weakly amenable if every derivation from A into (A (φ,ψ) ) * is inner. In this paper, among other things, we investigate the relations between weak amenability and (φ,ψ)-weak amenability of A. Some conditions can be imposed on A such that the (φ '' ,ψ '' )-weak amenability of A ** implies the (φ,ψ)-weak amenability of A.

MSC:
46H25Normed modules and Banach modules, topological modules