Let $M$ be an injective von Neumann subalgebra of $B(H)$. For a completely positive map $\phi:B(H)\to B(H)$, let $Mult(\phi)$ denote be the multiplicative domain of $\phi$. For any conditional expectation (CE) $E:B(H)\to M$, the range of $E$ is contained in $Mult(E)$ because $E$ is a bimodule map. Indeed, the image of $E$ is contained in $\cap\{Mult(E): E \text{ is a CE from }B(H)\text{ onto }M\}$.

Question: Is $M=\cap\{Mult(E): E \text{ is a CE from }B(H)\text{ onto }M\}$? I am mainly interested in the answer to this question when $M$ is Type I, or even when $M$ is a masa.

Evidence: not much. If there is a faithful CE $E$ onto $M$ then $M=Mult(E)$; in this case the answer is "yes". So, for example, the claim is true when $M$ is a purely atomic masa. I believe the answer is also "yes" when $M$ is purely atomic (not necessarily abelian). Beyond that I have no idea.