Weak amenability of certain classes of Banach algebras without bounded approximate identities. Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 2, 357–371. In this paper the authors investigate derivations and multipliers from a Segal algebra into itself and into its dual module. In particular, they prove that every Segal algebra on a locally compact abelian group is weakly amenable (this generalises a result of H. G. Dales and S. S. Pandey [Proc. Amer. Math. Soc. 128 (2000), no. 5, 1419–1425; MR1641681 (2000j:46096)]). More generally still, if A is a commutative weakly amenable Banach algebra, then any abstract Segal subalgebra of A having an approximate identity is weakly amenable. The authors introduce the interesting Lebesgue-Fourier algebra LA(G) = L 1 (G) ∩ A(G) of a locally compact group G. This may be given either the convolution product or the pointwise product. In the former case, LA(G) is a Segal algebra, while in the latter case it is a commutative abstract Segal algebra with respect to A(G). The authors show that LA(G) with convolution product is amenable if and only if G is discrete and amenable. When given the pointwise product, LA(G) is amenable if and only if G is compact and A(G) is amenable. They also show that for a compact group G, LA(G) is Arens regular when given the convolution product and observe that the same is true if G is discrete and LA(G) is given the pointwise product. The converses of these last two results remain open
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