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 $a\in A$?



Derivation of von Neumann algebra which is zero on MASA
Let $M= L \mathbb F\_2$ and $H = \ell^2 \mathbb F\_2$, where $\mathbb F\_2 = \langle a,b \rangle$ is the free group on two generators and $B(H)$ is a bimodule via the left and right multiplication with the left-regular representation $\lambda \colon L \mathbb F\_2 \to B(\ell^2 \mathbb F\_2)$.


Define $\delta(x) = [x,\lambda(a)]$. Then, $\lambda(a^{\pm1})'' \subset L \mathbb F\_2$ is a MASA (as can be shown) and $\delta$ vanishes on it. However, $\delta(\lambda(b)) \neq 0$ so that $\delta$ does not vanish on $L \mathbb F\_2$.


