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 module homomorphism from A to A that is wk∗ -continuous. We are mainly concerned with the study of φ-module normal, virtual diagonals. We show that if S is a weakly cancellative and S is an inverse semigroup with subsemigroup E of idempotents, χ is a bounded module homomorphism from l(S) to l(S) that is wk∗ -continuous and l(S) as a Banach module over l(E) is χ-Connes module amenable, then it has a χ-module normal, virtual diagonal. In the case χ = id, the converse also holds.