$varphi$-Connes amenability of dual Banach algebras

‎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 ‎following are equivalent for a unital weakly cancellative‎ ‎semigroup algebra $l^1(S)$‎: (i) $S$ is $chi$-amenable‎. (ii) $l^1(S)$ is $hat{chi}$-Connes amenable‎. (iii) $l^1(S)$ has a $hat{chi}$-normal‎, ‎virtual diagonal‎.