Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism

Let A and B be Banach algebras and T: B . A be a continuous homomorphism. n-weak amenability of the Banach algebra A . T B (defined in Bade, W. G., Curtis, P. C., Dales, H. G.: Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc., 55(2), 359.377 (1987)) is studied. The new version of a Banach algebra defined with a continuous homomorphism is introduced and Arens regularity and various notions of amenability of this algebra are studied.