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 \(m\geq 1\).


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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