**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 *P _{M}* and

*P*, onto

_{N}*M*and

*N*respectively, are elements of

**M**,

*M*«

*N*if

*P*«

_{M}*P*.

_{N}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 *N*_{0} is an isometric copy of *N* in *M*. By assumption, it is also true that, *N*, therefore *N*_{0}, contains an isometric copy *M*_{1} of *M*. Therefore one can write

By induction,

It is clear that

Let

So

and

Notice

The theorem now follows from the countable additivity of ~.

Read more about this topic: Schröder–Bernstein Theorems For Operator Algebras

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