Editing Direct integral (section)

=== 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.