Symmetric module and Connes amenability

In this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric Connes amenability. We determine symmetric module amenability and symmetric Connes amenability of some concrete Banach algebras. Indeed, it is shown that ℓ1(S) is a symmetric ℓ1(E)-module amenable if and only if S is amenable, where S is an inverse semigroup with subsemigroup E(S) of idempotents. In symmetric connes amenability, we have proved that M(G) is symmetric connes amenable if and only if G is amenable.