$varphi$-CONNES MODULE AMENABILITY OF DUAL BANACH ALGEBRAS

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