given two $C^{*}$ algebras $A\subset B $, acting on the same Hilbert space $H$, and $\delta $ is a derivation from $A$ into $B$,(in this case, a derivation is a linear mapping such that $\delta(ab)=a\delta(b)+\delta(a)b,\forall a,b \in A$) and assume it is bounded, then, is there an element $h$ of $A^{-}$(the weak operator closure of $A$), such that $\delta(a)=ha-ah, \forall a\in A$? \

Especially, I want to consider the case when $A,B$ are all commucative $C^{*}$ algebras.In other words, is there no nontrivial bounded derivation?

oldversion of this question that thenewquestion is answered by an old paper of Kamowitz (Trans. AMS, 1962): ams.org/mathscinet-getitem?mr=170219 This paper was one of the ingredients that motivated Johnson to define amenability for Banach algebras, and the result I refer to can be generalized to the case of amenable commutative Banach algebras (of which $C(X)$ is a key example). – Yemon Choi Mar 3 '11 at 19:31