# von Neumann Algebras

Let be a magma and . The **commutant** of is defined as:

for all . We have that and for and that iff is abelian. We can also refer to the commutant of the whole structure as the **center** of (i.e. ).

**Definition 1**. A **von Neumann algebra** is a C*-algebra such that .

A **projection** in a *-algebra is an element such that . A **partial isometry** is an element such that is a projection. Recall an element is positive, denoted , if and , and that if . We will also say that two elements are **orthogonal**, denoted , if .

**Definition 2.** Two projections are **Murray-von Neumann equivalent**, denoted , if there is a partial isometry such that and . We say is **subordinate** to , denoted , if there is a projection such that and .

**Proposition 3**. Let be a *-algebra and be a sequence of pairs of projections such that and for and for all . Then Also, if for all then

*Proof. *If , then and Hence and . Now define and Then we have

where the last equality follows from orthogonality. Hence

Now suppose Then there are such that and Then since by the previous claim, it remains to show that But since for all So we have the result.

**Proposition 4 (Schroder-Bernstein).** Let and be projections in a *-algebra such that and Then .

It turns out that if a unital von Neumann algebra is a **factor** (), then is a total order on the projections.

**Definition 5.** A projection is

**abelian**if is commutative;**finite**if where implies that ;**infinite**if it is not finite;**properly infinite**if and where and

**Lemma 6.** If is an infinite projection in a von Neumann algebra , then there is a projection such that is nonzero and properly infinite.

This allows for a somewhat complicated decomposition (see [1] for details) of

where is a *discrete central projection*, is the largest *finite continuous central projection*, is the largest *properly infinite semifinite continuous projection*, and is a purely infinite projection. The algebra is said to be of **pure type** if for all

[1] Blackadar, Bruce. *Operator Algebras*. Encyclopedia of Mathematical Sciences. Springer-Verlag. 2006.