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 infinit**e von Numann algebra M. Do $e\precsim f$ and $1-e\precsim 1-f$ imply $e\sim f$?



$e\precsim f$ and $1-e\precsim 1-f$ imply $e\sim f$?
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.


