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Weak amenability of a class of Banach algebras. (English) Zbl 1156.46306

From the text: In a recent paper [Stud. Math. 128, No. 1, 19–54 (1998; Zbl 0903.46045)] H. G. Dales, F. Ghahramani and N. Grønbæk have introduced the concept of \(n\)-weak amenability for Banach algebras. They point out the fact that, for \(n\geq 1\), \((n+2)\)-weak amenability always implies \(n\)-weak amenability, and prove further that if a Banach algebra \(\mathfrak A\) is an ideal in \(\mathfrak A^{\ast\ast}\), then the weak amenability of \(\mathfrak A\) also implies the \((2m+1)\)-weak amenability of \(\mathfrak A\) for all \(m > 0\). As to the general case, they have raised an open question: Does weak amenability imply 3-weak amenability? This question has been answered in negative by the author in [Trans. Am. Math. Soc. 354, No. 10, 4131–4151 (2002; Zbl 1008.46019)].
In this note we consider the Banach algebras which are one sided ideals in their second dual algebras, and discuss sufficient conditions under which weak amenability will imply \((2m+1)\)-weak amenability for \(m > 0\). We also consider an example to show the use of our result.
We show that, if a Banach algebra \(\mathfrak A\) is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of \(\mathfrak A\) implies the (\(2m+1\))-weak amenability of \(\mathfrak A\) for all \(m\geq 1\).

MSC:

46H20 Structure, classification of topological algebras
46H10 Ideals and subalgebras
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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