ON THE AMENABILITY OF A CLASS OF BANACH ALGEBRAS WITH APPLICATION TO MEASURE ALGEBRA

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 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 < L-2 > = 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. (C) 2019 Mathematical Institute Slovak Academy of Sciences