Let A be a Banach algebra, and X a Banach A-bimodule. A bounded linear mapping D : A → X is approximately semi-inner derivation if there eixist nets (ξ α ) α and (µ α ) α in X such that, for each a ∈ A, D(a) = lim α (a.ξ α − µ α .a). A is called approximately semi-amenable if for every Banach A-bimodule X , every D ∈ Z 1 (A, X * ) is approximtely semi-inner. There are some Banach algebras which are approximately semi-amenable, but not approximately amenable. In this manuscript, we investigate some properties of approximate semi-amenability of Banach algebras. Also in Theorem 2.7 we prove the approximate semi-amenability of Segal algebras on a locally compact group G.

In this paper, we study approximate identity properties, some propositions from Baker, Dales, Lau in general situations and we establish some relationships between the topological centers of module actions and factorization properties with some results in group algebras. We consider under which sufficient and necessary conditions the Banach algebra $A\widehat{\otimes}B$ is Arens regular.