A survey on χ-module Connes Amenability of Semigroup Algebras

We shall study the χ-module Connes amenability of a semigroup algebra l 1 (S) , where χ is a bounded module homomorphism from l 1 (S) to l 1 (S) that is w ∗ -continuous and S is an inverse weakly cancellative semigroup with subsemigroup E of idempotents. We are mainly concerned with the study of χ-module normal, virtual diagonals. We characterize the χ- module Connes amenability of a semigroup algebra l 1 (S) . Also, we show that if l 1 (S) as a Banach module over l 1 (E) has an id-module normal, virtual diagonal then it is id-module Connes amenable. It is obtained other characterizations for χ- module Connes amenability of l 1 (S).