Character amenability and character inner amenability of morphism product of Banach algebras

Let $T$ be a Banach algebra homomorphism from a Banach algebra $\mathcal{B}$ to a Banach algebra $\mathcal{A}$ with $\mathrm{\parallel }T\mathrm{\parallel }\le 1$. Recently it has been obtained some results about Arens regularity and also various notions of amenability of $\mathcal{A}{\times }_T\mathcal{B}$, in the case where $\mathcal{A}$ is commutative. In the present paper, most of these results have been generalized and proved for an arbitrary Banach algebra $\mathcal{A}$.