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?



derivation between two $C^{*}$ algebras
The answer is no. Let $A$ be commutative von Neumann algebra. Let $T\in B(H)$ be such that $Ta-aT\neq 0$ for some $a\in A$. Then $\delta(a)=Ta-aT$ is bounded derivation but there is no element $h\in A^{-}=A$, such that $\delta(a)=ha-ah$ for all $a\in A$.


