$varphi$-CONNES MODULE AMENABILITY OF DUAL BANACH ALGEBRAS

Abstract

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