Approximate identities in approximate amenability

Abstract

We answer several open questions in the theory of approximate amenability for Banach algebras. First we give examples of Banach algebras which are boundedly approximately amenable but which do not have bounded approximate identities. This answers a question open since the year 2000 when Ghahramani and Loy founded the notion of approximate amenability. We give a nice condition for a co-direct-sum of amenable Banach algebras to be approximately amenable, which gives us a reasonably large and varied class of such examples. Then we examine our examples in some detail, and thereby find answers to other open questions: the two notions of bounded approximate amenability and bounded approximate contractibility are not the same; the direct-sum of two approximately amenable Banach algebras does not have to be approximately amenable; and a 1-codimensional closed ideal in a boundedly approximately amenable Banach algebra need not be approximately amenable.