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

For a Banach algebra A, Ȁ is (-1)-weakly amenable if A0 is a Banach A00-bimodule and H1(Ȁ;Ȁ) = f0g. In this paper we prove some important properties of this notion, for instance if Ȁ 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 A00 and the weak amenability of Ȁ or A.