  In recent work of the authors the notion of a derivation being approximately
semi-inner arose as a tool for investigating (approximate) amenability
questions for Banach algebras. Here we investigate this property in its own
right, together with the consequent one of approximately semi-amenability.
Under certain hypotheses regarding approximate identities this new notion is
the same as approximate amenability, but more generally it covers some
important classes of algebras which are not approximately amenable, in
particular Segal algebras on amenable SIN-groups.
