For Von Neumann Algebras
Suppose M is a von Neumann algebra and E, F are projections in M. Let ~ denote the Murray-von Neumann equivalence relation on M. Define a partial order « on the family of projections by E « F if E ~ F' ≤ F. In other words, E « F if there exists a partial isometry U ∈ M such that U*U = E and UU* ≤ F.
For closed subspaces M and N where projections PM and PN, onto M and N respectively, are elements of M, M « N if PM « PN.
The Schröder–Bernstein theorem states that if M « N and N « M, then M ~ N.
A proof, one that is similar to a set-theoretic argument, can be sketched as follows. Colloquially, N « M means that N can be isometrically embedded in M. So
where N0 is an isometric copy of N in M. By assumption, it is also true that, N, therefore N0, contains an isometric copy M1 of M. Therefore one can write
It is clear that
The theorem now follows from the countable additivity of ~.
Read more about this topic: Schröder–Bernstein Theorems For Operator Algebras
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