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Weak amenability of certain classes of Banach algebras without bounded approximate identities. (English) Zbl 1010.46048

A Banach algebra \(A\) is called weakly amenable if each continuous derivation from \(A\) into the dual space \(A^*\) treated as an \(A\)-bimodule is an inner one. Here the authors deal with weak amenability of various types of Segal algebras. Among other results, they show that every Segal algebra on a locally compact abelian group is weakly amenable and that an abstract Segal sub-algebra (with an approximate identity) of a commutative weakly amenable Banach algebra is weakly amenable. They also introduce the Lebesgue-Fourier algebra of a locally compact group and obtain several results concerning Arens regularity, derivations and multipliers.

MSC:

46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
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