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 $\|T\|\leq 1$. 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 $\mathcal A$.