On the amenability of a class of Banach algebras with application to measure algebra

Abstract

Let L be a Lau algebra and X be a topologically invariant subspace of L containing UC(L). We prove that if L has a bounded approximate identity, then strict inner amenability of L is equivalent to the existence of a strictly inner invariant mean on X. We also show that when L is inner amenable the cardinality of the set of topologically left invariant means on L is equal to the cardinality of the set of topologically left invariant means on RUC(L). Applying this result, we prove that if L is inner amenable and hL2i = L, then the essential left amenability of L is equivalent to the left amenability of L. Finally, for a locally compact group G, we consider the measure algebra M(G) to study strict inner amenability of M(G) and its relation with inner amenability of G. © 2019 Mathematical Institute Slovak Academy of Sciences.