Derivations from Totally Ordered Semigroup Algebras into their Duals

Abstract

Abstract For a well-behaved measure μ , on a locally compact totally ordered set X , with continuous part μ c , we make L p ( X , μ c ) into a commutative Banach bimodule over the totally ordered semigroup algebra L p ( X , μ ), in such a way that the natural surjection from the algebra to the module is a bounded derivation. This gives rise to bounded derivations from L p ( X , μ ) into its dual module and in particular shows that if μ c is not identically zero then L p ( X , μ ) is not weakly amenable. We show that all bounded derivations from L 1 ( X , μ ) into its dual module arise in this way and also describe all bounded derivations from L p ( X , μ ) into its dual for 1 < p < ∞ the case that X is compact and μ continuous.