Weak and (-1)-weak amenability of second dual of Banach algebras

For a banach algebra a, a'' is (—1)-weakly amenable if a' is a banach a''-bimodule and h (a, a') = {0}. in this paper we prove some important properties of this notion, for instance if a'' is (—1)- weakly amenable then a is essential and there is no non-zero point derivation on a. we also give some examples, namely, the second dual of every c*-algebras is (—1)-weakly amenable. finally, we study the relationships between the (—1)-weakly amenability of a'' and the weak amenability of a'' or a.