Let $M$ be a von Neumann algebra and $A,B,C,D\in M$ be self-adjoint. My question is about the following equation: $$ AXB=CX+XD, ~\forall X\in M.$$
In the case when $M$ is $B(H)$, it is known that the above equation implies that either $A,C$ are scalars or else $B,D$ are scalars (see Lemma 2). For the case when $M$ is a semifinite factor (see Theorem 3.1), the same result holds true.
I am wondering what can we say about the case when $M$ is not a factor (say, $M$ is semifinite or just finite). My conjecture is: there exists a central projection $z\in M$ such that either $Az,Cz, B(1-z), D(1-z)$ are in the center of $M$.
I am not sure whether there are any known results on such an equation.