Connes amenability for certain product of Banach algebras

In this paper we develop the notions of Connes amenability for certain product of Banach algebras. We give necessary and sufficient conditions for the existence of an invariant mean on the predual of $\Theta$-Lau product $\mathcal{A}\times_{\Theta}\mathcal{B}$, module extension Banach algebra $\mathcal{A}\oplus\mathcal{X}$ and projective tensor product $\mathcal{A} \widehat{{\otimes}} \mathcal{B}$, where $\mathcal{A}$ and $\mathcal{B}$ are dual Banach algebras with preduals $\mathcal{A}_*$ and $\mathcal{B}_*$ respectively and $\mathcal{X}$ is a normal Banach $\mathcal{A}$-bimodule with predual $\mathcal{X}_*$.