Connes-biprojective dual Banach algebra

In this paper, we introduce a new notion of biprojectivity, called Connes-biprojective, for dual Banach algebras. We study the relation between this new notion to Connes-amenability and we show that, for a given dual Banach algebra $\mathcal{A}$, it is Connes-amenable if and only if $\mathcal{A}$ is Connes-biprojective and has a bounded approximate identity. Also, for an Arens regular Banach algebra $\mathcal{A}$, we show that if $\mathcal{A}$ is biprojective, then the dual Banach algebra ${\mathcal{A}}^{\ast \ast }$ is Connes-biprojective.