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.