arXiv:1905.10719v1 [math.FA] 26 May 2019 SEMI-AMENABILITY OF BANACH ALGEBRAS M. SHAMS KOJANAGHI AND K. HAGHNEJAD AZAR * Abstract. Let A be a Banach algebra, and X be a Banach A-bimodule. A derivation D : A→X from Banach algebra into Banach space is called semi-inner if there are η, ξ ∈X such that D(a)= a.η - ξ.a = δ η,ξ (a), (a ∈A). A Banach algebra A is semi-amenable (resp. semi-contractible) if, for each Banach A-bimodule X , every derivation D from A into X * (resp. into X ) is semi-inner. In this paper, we study some problems in semi- amenability of Banach algebras which have been studied in amenability case. We extend some definitions and concepts for semi-amenability, that is, we introduce approximately semi-amenability, semi-contractibility with solving some problems which former have been studied for amenabil- ity case. Keywords: Amenability, Semi-amenability, Approximately semi-amenability. MSC(2010): Primary 46H25, 46H20; Secondary 46H35. 1. Introduction The cohomological notion of an amenable Banach algebra was introduced by Barry E. Johnson in 1972 in [11] and has proved to be of enormous importance in Banach algebra theory, and has widely application in modern analysis. The concept of amenability originated from the measure theoretic problems, it moves from there to abstract harmonic analysis where amenable locally compact groups were considered. A locally compact group G is called amenable if it possesses a translation invariant mean. That is, if there is a linear functional m : L ∞ (G) → C satisfying m(1) =‖ m ‖=1 and m(L x ∗ f )= m(f ), (x ∈ G, f ∈ L ∞ (G)). Johnson, initiated the theory of amenable Banach algebras by showing that, for a locally compact group G, it’s group algebra, L 1 (G) is amenable if and only if G is amenable group in the classical sence (which is a famous and motivating theorem of B. E. Johnson.) [11]. Let A be a Banach algebra. We say X is a Banach A-bimodule, that is, X is a Banach space and an A-module such that, the module operations (a,x) → ax and (x,a) → xa Date : Received: , Accepted: . * Corresponding author. 1