Schröder–Bernstein Theorems For Operator Algebras - For Von Neumann Algebras

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 UM 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

By induction,

It is clear that

Let

So


M = oplus_{i geq 0} ( M_i ominus N_i ) quad oplus quad oplus_{j geq 0} ( N_j ominus M_{j+1}) quad oplus R

and


N_0 = oplus_{i geq 1} ( M_i ominus N_i ) quad oplus quad oplus_{j geq 0} ( N_j ominus M_{j+1}) quad oplus R.

Notice

The theorem now follows from the countable additivity of ~.

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