Denote by $\precsim$ the order comes from "Murray-von Neumann" equivalence in the projection lattice of a von Numann algebra. Let e and f be two projections in a properly infinite von Numann algebra M. Do $e\precsim f$ and $1-e\precsim 1-f$ imply $e\sim f$?
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No. Take e=0 and 0 < f < 1 such that both f and 1−f are infinite, with (1−f)~1. Then e≾f because 0≾f for any projection f. Also 1−e≾1−f because 1≾1−f, which holds by definition of f.
answered Mar 25, 2018 at 7:55
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$\begingroup$ How about non-zero projections $e$ and $f$? $\endgroup$ Commented Mar 25, 2018 at 8:30
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1$\begingroup$ You can also take 0 < e < f < 1, with the same properties. The two properties will continue to hold: the first one by definition, the second one because 1~(1−f). $\endgroup$ Commented Mar 25, 2018 at 8:52