Murray-von Neumann Ordering

Recall that

The Partial Order on

Let be the set of self-adjoint elements in . For , we write if . This defines a partial order on .

Link to original

Projections are self-adjoint, so they are partially ordered by the order on self-adjoint elements. Under this usual ordering , we have

Lemma

Suppose are projections, then iff .

However, there is a more refined notion of ordering for projections in von Neumann algebras, known as the Murray-von Neumann ordering.

Murray-von Neumann Ordering

Suppose is a von Neumann algebra. We define a preorder on the set of projections in by declaring if there exists a partial isometry such that and .
Consider the equivalence relation induced by this preorder, the corresponding equivalence classes are called Murray-von Neumann equivalence classes. The order type of the quotient poset is called the Murray-von Neumann order type of .

Definition

A projection is said to be infinite if for some (that is, such that but ). Otherwise, we say is finite.
A von Neumann algebra is said to be finite if its identity is a finite projection, and infinite otherwise.