Denote by $\sim$ the "Murray-von Neumann" equivalence in the projection lattice of a von Numann algebra. Let $e$ be a projection in a properly infinite factor. Is it always true that $e\sim 1$ or $1-e\sim 1$?
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1$\begingroup$ Would you recall the meaning of $\sim$? (I'm not sure the downvote is appropriate, the language of von Neumann algebras always sounds weird since it uses words that are more or less meaningful in other contexts) $\endgroup$– YCorMay 24, 2018 at 20:48
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$\begingroup$ @YCor Thank you, I added the meaning of $\sim$. $\endgroup$– Meisam Soleimani MalekanMay 24, 2018 at 21:00
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