φ-Connes amenability of dual Banach algebras

‎Generalizing the notion of character amenability for Banach‎ ‎algebras‎, ‎we study the concept of φ-Connes amenability of‎ ‎a dual Banach algebra A with predual A∗‎, ‎where φ is a homomorphism from A onto C‎ ‎that lies in A∗‎. ‎Several characterizations of‎ ‎φ-Connes amenability are given‎. ‎We also prove that the‎ ‎following are equivalent for a unital weakly cancellative‎ ‎semigroup algebra l1(S)‎:(i) S is χ-amenable‎.(ii) l1(S) is χ^-Connes amenable‎.(iii) l1(S) has a χ^-normal‎, ‎virtual diagonal‎.