φ-Connes module amenability of dual Banach algebras

In this paper we define φ-Connes module amenability of a dual Banach algebra A where φ is a bounded wk*-module homomorphism from A to A. We are mainly concerned with the study of φ-module normal virtual diagonals. We show that if S is a weakly cancellative inverse semigroup with subsemigroup E of idempotents, $\xi$ is a bounded wk*-module homomorphism from $l^{1}(S)$ to $l^{1}(S)$ and $l^{1}(S)$ as a Banach module over $l^{1}(E)$ is $\xi$-Connes module amenable, then it has a $\xi$-module normal virtual diagonal. In the .case $\xi$= id, the converse holds