The Character and Approximate Identity with Approximate Amenability of Banach Algebras

Abstract

It is shown that approximate amenability and approximate contractibility are the same properties, as are uniform approximate amenability and amenability. Bounded approximate contractibility and bounded approximate amenability are characterized by the existence of suitable operator bounded approximate identities for the diagonal ideal. We continue the study on -amenability of Banach algebras. In the study of amenability of a Banach algebra defined with respect to agvien character. We give examples of Banach algebras which are boundedly approximately amenable but which do not have bounded approximate identities. We show that if a Banach space is “fairly close" a Hilbert space, then he Banach algebra of all compact operators on is approximately amenable.