Let $\mathcal{M}$ be a II_1 factor. If $\mathcal{N}$ is a subfactor of $\mathcal{M}$ ($\mathcal{N} \neq \mathcal{M}$), does there always exist a projection in $\mathcal{M}$ such that $(I-P)AP \neq 0$ for any $A$ in $\mathcal{N}$ such that $A \neq zI$?

simplestcase that you cannot do? Presumably you know examples of factors and subfactors. – Yemon Choi Oct 7 '12 at 20:51