on φ-connes module amenability for dual banach algebra and φ-module normal virtual diagonals

In this paper, we define φ-connes module amenability of a dual banach algebra a, where φ is a bounded module homomorphism from a to itself that is ω*-continuous. we give a characterization of φ-connes module amenability for a via modul φ-derivation. also, by considering that s is an inverse weakly cancellative semigroup with subsemigroup e of idempotents, we define χ-connes module amenability of a semigroup algebra l^1(s), where χ is a boundedmodule homomorphism from l^1(s) to l^1(s) that is ω*-continuous. we are mainly concerned with the study of χ-module normal, virtual diagonals. in the other word, we show that if l^1(s) as a banach module over l^1(e) is χ-connes module amenable, then it has a χ-module normal, virtual diagonal. other results in this direction are also obtained.