# Direct Integral Decomposition

Let be a collection of Hilbert spaces such that is a measure space. Now define

where (the th component of ). Then

defines a pre inner product on . Now let . Then is an inner product space, and its completion is a Hilbert space called the **direct integral of** . We denote the direct integral by

Now suppose is a collection of linear operators where such that is uniformly bounded. Then there is an operator where and where we can define

This gives a representation defined by

which is essentially just a component-wise left action of on . is called the **algebra of diagonalizable operators** of , which we will denote .

**Theorem 1**. Let be a representation of von Neumann algebra on a separable Hilbert space such that is a von Neumann subalgebra of . Then there exists a measure space , a collection of Hilbert spaces , and a unitary map such that for all and corresponding and

for all .

Thus if we let from above, then is a factor and we also write

This is called the **central decomposition** of . This also gives a representation of on defined by .

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

# 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.