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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 $A \neq zI$?

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Why can't I take $A=I$? – Yemon Choi Oct 6 '12 at 8:41
Sorry, I forgot the restriction that A can not be a scalar. – heller Oct 7 '12 at 11:35
You probably want to exclude the case $\mathcal N = \mathcal M$ as well. – Jesse Peterson Oct 7 '12 at 12:02
Yemon and Jesse, thank you for pointing out my mistakes. Do you guys know any relative results about this question? – heller Oct 7 '12 at 14:07
Echoing Polya: what is the simplest case that you cannot do? Presumably you know examples of factors and subfactors. – Yemon Choi Oct 7 '12 at 20:51

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