The Schröder–Bernstein theorem from set theory has analogs in the context operator 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.
In this setting, the Schröder–Bernstein theorem reads: A proof that resembles the previous argument can be outlined.
One has In turn, By induction, and Now each additional summand in the direct sum expression is obtained using one of the two fixed partial isometries, so This proves the theorem.