Generalizing the notion of character amenability for Banach algebras, we study the concept of $varphi$-Connes amenability of a dual Banach algebra $mathcal{A}$ with predual $mathcal{A}_*$, where $varphi$ is a homomorphism from $mathcal{A}$ onto $Bbb C$ that lies in $mathcal{A}_*$. Several characterizations of $varphi$-Connes amenability are given. We also prove that the follo...
