Arens regularity of module actions

Let A be a Banach algebra, A’’ a Banach A-module. In this paper, we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions,and them investigate those conditions under which the second adjoint of a derivation into a dual Banach algebra module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included. A^(4) is a banach algebra with four Arens products. The bilinear map T is Arens regular when the equality T*** = T^( r***r ) . If T: A × A’’ → A’’ is multiplication left module on A , the following statements are equivalent , i:T is regular ii : T**** = T^(r****r) iii : T****( A’’’, A’’) ⊆ A’’’ iv : the linear map a → T*( a’’’, a) : A → A’’’ is weakly compact for every a’’’ ∈ A’’’. Also If module actions are regular, then every inner derivation D : A → A’’’ is weakly compact; moreover, D** : (A’’, □ ) → A^(5) and D** : (A’’, ⋄ ) → A^(5) are also inner derivation.