Document Type: Original Manuscript


1 Department of Mathematics, University of Semnan, P.O. Box 35195-363, Semnan, Iran.

2 Faculty of Engineering- East Guilan, University of Guilan, P.O. Box 44891-63157, Rudsar, Iran.


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