$\varphi$-CONNES MODULE AMENABILITY OF DUAL BANACH ALGEBRAS

Abstract

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

Keywords

• banach algebras
• module amenability
• derivation
• semigroup algebra