Abstract
We start by discussing general necessary and sufficient conditions for a module extension Banach algebra to benn-weakly amenable, forn=0,1,2,⋯n = 0,1,2,\cdots. Then we investigate various special cases. All these case studies finally provide us with a way to construct an example of a weakly amenable Banach algebra which is not33-weakly amenable. This answers an open question raised by H. G. Dales, F. Ghahramani and N. Grønbæk.
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