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.