TITLE: Derivation of von Neumann algebra which is zero on MASA
QUESTION [5 upvotes]: Are there any example of $II_1$-factor $M$ with maximal abelian von Neumann subalgebra $A$ and non-zero derivation $\delta:M\rightarrow B(H)$ such that $\delta(a)=0$ for every $a\in A$?

REPLY [2 votes]: Sorry to be naive but why can't you just represent $M$ on a separable Hilbert space and take the inner derivation induced by any non-central element in the masa? This maps into $M\subset B(H)$, vanishes on the masa, but is non-trivial on $M$ because the element is non-central.