$varphi$-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.