Editing Direct integral (section)

== Decomposable operators ==

For the example of a discrete measure on a countable set, any bounded linear operator ''T'' on

: <math> H = \bigoplus_{k \in \mathbb{N}} H_k </math>

is given by an infinite matrix

:<math> \begin{bmatrix} T_{1 1} & T_{1 2} & \cdots & T_{1 n} & \cdots \\ T_{2 1} & T_{2 2} & \cdots & T_{2 n} & \cdots \\
\vdots & \vdots & \ddots & \vdots & \cdots \\
T_{n 1} & T_{n 2} & \cdots & T_{n n} & \cdots \\
\vdots & \vdots & \cdots & \vdots & \ddots
\end{bmatrix}. </math>

For this example, of a discrete measure on a countable set, ''decomposable operators'' are defined as the operators that are [[block diagonal]], having zero for all non-diagonal entries. Decomposable operators can be characterized as those which commute with diagonal matrices:

:<math> \begin{bmatrix} \lambda_{1} & 0 & \cdots & 0 & \cdots \\ 0 & \lambda_{2} & \cdots & 0 & \cdots \\
\vdots & \vdots & \ddots & \vdots & \cdots \\
0 & 0 & \cdots & \lambda_{n} & \cdots \\
\vdots & \vdots & \cdots & \vdots & \ddots
\end{bmatrix}. </math>

The above example motivates the general definition: A family of bounded operators {''T''<sub>''x''</sub>}<sub>''x''∈ ''X''</sub> with ''T''<sub>''x''</sub> ∈ L(''H''<sub>''x''</sub>) is said to be ''[[strongly measurable]]'' if and only if its restriction to each ''X''<sub>''n''</sub> is strongly measurable. This makes sense because ''H''<sub>''x''</sub> is constant on ''X''<sub>''n''</sub>.

Measurable families of operators with an [[essential infimum and essential supremum|essentially bounded norm]], that is

:<math> \operatorname{ess-sup}_{x \in X} \|T_x\| < \infty </math>

define bounded linear operators

:<math> \int^\oplus_X \ T_x d \mu(x)  \in \operatorname{L}\bigg(\int^\oplus_X H_x \ d \mu(x)\bigg) </math>

acting in a pointwise fashion, that is

:<math> \bigg[\int^\oplus_X \ T_x d \mu(x) \bigg] \bigg(\int^\oplus_X \ s_x d \mu(x) \bigg) =  \int^\oplus_X \ T_x(s_x) d \mu(x). </math>

Such operators are said to be ''decomposable''.

Examples of decomposable operators are those defined by scalar-valued (i.e. '''C'''-valued) measurable functions λ on ''X''. In fact,

'''Theorem'''. The mapping

: <math> \phi: L^\infty_\mu(X) \rightarrow \operatorname{L}\bigg(\int^\oplus_X H_x \ d \mu(x)\bigg) </math>

given by

:<math> \lambda \mapsto \int^\oplus_X \ \lambda_x d \mu(x) </math>

is an involutive algebraic isomorphism onto its image.

This allows ''L''<sup>∞</sup><sub>μ</sub>(''X'') to be identified with the image of φ.

'''Theorem'''<ref>{{citation | last = Takesaki | first = Masamichi | authorlink = Masamichi Takesaki | title=Theory of Operator Algebras I  | publisher=[[Springer-Verlag]] | year=2001 | isbn=3-540-42248-X }}, Chapter IV, Theorem 7.10, p. 259</ref> Decomposable operators are precisely those that are in the operator commutant of the abelian algebra ''L''<sup>∞</sup><sub>μ</sub>(''X'').

=== Decomposition of Abelian von Neumann algebras ===

The spectral theorem has many variants. A particularly powerful version is as follows:

'''Theorem'''. For any [[Abelian von Neumann algebra]] '''A''' on a separable Hilbert space ''H'', there is a standard Borel space ''X'' and a measure μ on ''X'' such that it is unitarily equivalent as an operator algebra to ''L''<sup>∞</sup><sub>μ</sub>(''X'') acting on a direct integral of Hilbert spaces

: <math> \int_X^\oplus H_x d \mu(x). \quad</math>

To assert '''A''' is unitarily equivalent to ''L''<sup>∞</sup><sub>μ</sub>(''X'') as an operator algebra means that there is a unitary

:<math> U: H \rightarrow \int_X^\oplus H_x d\mu(x) </math>

such that ''U'' '''A''' ''U''* is the algebra of diagonal operators ''L''<sup>∞</sup><sub>μ</sub>(''X''). Note that this asserts more than just the algebraic equivalence of '''A''' with the algebra of diagonal operators.

This version of the spectral theorem does not explicitly state how the underlying standard Borel space ''X'' is obtained. There is a uniqueness result for the above decomposition.

'''Theorem'''. If the Abelian von Neumann algebra '''A''' is unitarily equivalent to both ''L''<sup>∞</sup><sub>μ</sub>(''X'') and ''L''<sup>∞</sup><sub>ν</sub>(''Y'') acting on the direct integral spaces

: <math> \int_X^\oplus H_x d \mu(x), \quad  \int_Y^\oplus K_y d \nu(y) </math>

and μ, ν are standard measures, then there is a [[Borel isomorphism]]

:<math>\varphi: X - E \rightarrow Y - F </math>

where ''E'', ''F'' are null sets such that

:<math> K_{\phi(x)} = H_x \quad \mbox{almost everywhere} </math>

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. This classification takes into account the realization of the von Neumann algebra as an algebra of operators. If one considers the underlying von Neumann algebra independently of its realization (as a von Neumann algebra), then its structure is determined by very simple measure-theoretic invariants.