Characterization of module Connes amenability for certain Banach algebras by homomorphisms

Let [Formula: see text] and [Formula: see text] be dual Banach algebras and [Formula: see text] be a Banach algebra. Let also [Formula: see text] be a normal Banach [Formula: see text]-bimodule and [Formula: see text] be an [Formula: see text]-bimodule algebraic homomorphism. In this paper, we investigate the notion of [Formula: see text]-bimodule Connes amenability for the projective tensor product [Formula: see text] pertinent to its closed two-sided ideals. Moreover, we characterize the [Formula: see text]-bimodule [Formula: see text]-Connes amenability of [Formula: see text]-Lau product [Formula: see text], where [Formula: see text] is a [Formula: see text]-continuous [Formula: see text]-bimodule algebraic homomorphism, [Formula: see text] is a bounded linear functional such that [Formula: see text] (the predual of [Formula: see text]) and [Formula: see text] is Banach [Formula: see text]-bimodule. Furthermore, we obtain a characterization for [Formula: see text]-bimodule [Formula: see text]-Connes amenability of [Formula: see text]-bimodule extension of Banach algebra [Formula: see text]. Lastly, for more clarity of our results, we present some concrete examples in the setting of matrices algebra.