In recent years, there have been several studies of various 'approximate' versions of the key notion of amenability, which is defined for all Banach algebras, these studies began with work of Ghahramam and Loy in 2004 The present memoir continues such work we shall define various notions of approximate amenability, and we shall discuss and extend the known background, which considers the relationships between different versions of approximate amenability There are a number of open questions on these relationships, these will be considered In Chapter 1, we shall give all the relevant definitions and a number of basic results, partly surveying existing work, we shall concentrate on the case of Banach function algebras In Chapter 2, we shall discuss these properties for the semigroup algebra l(1)(S) of a semigroup S In the case where S has only finitely many idempotents, l(1)(S) is approximately amenable if and only if it is amenable In Chapter 3, we shall consider the class of weighted semigroup algebras of the form l(1) (N-boolean AND omega), where omega Z -> [1, infinity) is an arbitrary function We shall determine necessary and sufficient conditions on omega for these Banach sequence algebras to have each of the various approximate amenability properties that interest us In this way we shall illuminate the implications between these properties In Chapter 4, we shall discuss Segal algebras on T and on lit It is a conjecture that every proper Segal algebra on T fails to be approximately amenable, we shall establish this conjecture for a wide class of Segal algebras
