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.