on φ-connes amenability of dual banach algebras

Let φ be a w∗-continuous homomorphism from a dual banach algebra to c. the notion of ϕ-connes amenability is studied and some characterizations is given. a type of diagonal for dual banach algebras is defined. it is proved that the existence of such a diagonal is equivalent to ϕ-connes amenability. it is also shown that ϕ-connes amenability is equivalent to so-called ϕ-splitting of a certain short exact sequence.