Derivations on Lau product of Banach algebras

Extension of Banach algebras defined by Cartesian product by a linear functional such as are called -Lau Banach algebras. Recently, this type of Banach algebras are interested by many researchers. Derivations play an important role in algebraic structures. By using this role, one can discover some of the properties of algebraic structures on which a derivation is defined, such as their semi simplicity. In this paper, we consider derivations that are defined on Lau product of Banach algebras and we characterize these derivations in some various cases. As a main characterization, for two Banach algebra , and , we show that is a derivation if and only if there are derivations , and a -derivation such that for all . We investigate the converse case of the above obtained result in some various cases and according to the obtained results related to characterization of derivations, we investigate and characterize the first cohomology of Banach algebras obtained by this product.