Approximate weak amenability of Banach algebras

In this paper we deal with four generalized notions of amenability which are called approximate, approximate weak, approximate cyclic and approximate n-weak amenability. The first two were introduced and studied by Ghahramani and Loy in [9]. We introduce the third and fourth ones and we show by means of some examples, their distinction with their classic analogs. Our main result is that under some mild conditions on a given Banach algebra A, if its second dual A** is (2n -- 1)-weakly [respectively approximately/ approximately weakly/ approximately n-weakly] amenable, then so is A. Also if A is approximately (n + 2)-weakly amenable, then it is approximately n-weakly amenable. Moreover we show the relationship between approximate trace extension property and approximate weak [respectively cyclic] amenability. This answers question 9.1 of [9] for approximate weak and cyclic amenability. Key words and phrases : Approximately inner derivation, Approximately weakly amenable, Approximately n-weakly amenable, Approximately amenable, Approximate trace extension property.