The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of Operators.
One of von Neumann's goals in this paper was to reduce the classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors.
Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings.
Let X be a Borel space equipped with a countably additive measure μ.
Given a measurable family of Hilbert spaces, the direct integral consists of equivalence classes (with respect to almost everywhere equality) of measurable square integrable cross-sections of {Hx}x∈ X.
In the more general definition, the Hilbert space fibers Hx are allowed to vary from point to point without having a local triviality requirement (local in a measure-theoretic sense).
One of the main theorems of the von Neumann theory is to show that in fact the more general definition is equivalent to the simpler one given here.
Then the mapping is a unitary operator The simplest example occurs when X is a countable set and μ is a discrete measure.
Decomposable operators can be characterized as those which commute with diagonal matrices: The above example motivates the general definition: A family of bounded operators {Tx}x∈ X with Tx ∈ L(Hx) is said to be strongly measurable if and only if its restriction to each Xn is strongly measurable.
Examples of decomposable operators are those defined by scalar-valued (i.e. C-valued) measurable functions λ on X.
This version of the spectral theorem does not explicitly state how the underlying standard Borel space X is obtained.
If the Abelian von Neumann algebra A is unitarily equivalent to both L∞μ(X) and L∞ν(Y) acting on the direct integral spaces and μ, ν are standard measures, then there is a Borel isomorphism where E, F are null sets such that The isomorphism φ is a measure class isomorphism, in that φ and its inverse preserve sets of measure 0.
The previous two theorems provide a complete classification of Abelian von Neumann algebras on separable Hilbert spaces.
A family of von Neumann algebras {Ax}x ∈ X with is measurable if and only if there is a countable set D of measurable operator families that pointwise generate {Ax} x ∈ X as a von Neumann algebra in the following sense: For almost all x ∈ X, where W*(S) denotes the von Neumann algebra generated by the set S. If {Ax}x ∈ X is a measurable family of von Neumann algebras, the direct integral of von Neumann algebras consists of all operators of the form for Tx ∈ Ax.
One of the main theorems of von Neumann and Murray in their original series of papers is a proof of the decomposition theorem: Any von Neumann algebra is a direct integral of factors.
If {Ax}x ∈ X is a measurable family of von Neumann algebras and μ is standard, then the family of operator commutants is also measurable and Suppose A is a von Neumann algebra.
This is a special case of the central decomposition theorem of von Neumann.
One can show that the direct integral can be indexed on the so-called quasi-spectrum Q of A, consisting of quasi-equivalence classes of factor representations of A.