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In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. 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. Results on direct integrals can be viewed as generalizations of results about finite-dimensional C *-algebras of matrices; in this case the results are easy to prove directly. The infinite-dimensional case is complicated by measure-theoretic technicalities. Direct integral theory was also used by George Mackey in his analysis of systems of imprimitivity and his general theory of induced representations of locally compact separable groups. == Direct integrals of Hilbert spaces == The simplest example of a direct integral are the ''L''2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space ''X''. Somewhat more generally one can consider a separable Hilbert space ''H'' and the space of square-integrable ''H''-valued functions : Terminological note: The terminology adopted by the literature on the subject is followed here, according to which a measurable space ''X'' is referred to as a ''Borel space'' and the elements of the distinguished σ-algebra of ''X'' as Borel sets, regardless of whether or not the underlying σ-algebra comes from a topological space (in most examples it does). A Borel space is ''standard'' if and only if it is isomorphic to the underlying Borel space of a Polish space; all Polish spaces of a given cardinality are isomorphicto each other (as Borel spaces). Given a countably additive measure μ on ''X'', a measurable set is one that differs from a Borel set by a null set. The measure μ on ''X'' is a ''standard'' measure if and only if there is a null set ''E'' such that its complement ''X'' − ''E'' is a standard Borel space. All measures considered here are σ-finite. Definition. Let ''X'' be a Borel space equipped with a countably additive measure μ. A ''measurable family of Hilbert spaces'' on (''X'', μ) is a family ''x''∈ ''X'', which is locally equivalent to a trivial family in the following sense: There is a countable partition : by measurable subsets of ''X'' such that : where H''n'' is the canonical ''n''-dimensional Hilbert space, that is : A ''cross-section'' of ''x''∈ ''X'' is a family ''x'' ∈ ''X'' such that ''s''''x'' ∈ ''H''''x'' for all ''x'' ∈ ''X''. A cross-section is measurable if and only if its restriction to each partition element ''X''''n'' is measurable. We will identify measurable cross-sections ''s'', ''t'' that are equal almost everywhere. 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 ''x''∈ ''X''. This is a Hilbert space under the inner product : Given the local nature of our definition, many definitions applicable to single Hilbert spaces apply to measurable families of Hilbert spaces as well. Remark. This definition is apparently more restrictive than the one given by von Neumann and discussed in Dixmier's classic treatise on von Neumann algebras. In the more general definition, the Hilbert space ''fibers'' ''H''''x'' 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 can be reduced to the simpler one given here. Note that the direct integral of a measurable family of Hilbert spaces depends only on the measure class of the measure μ; more precisely: Theorem. Suppose μ, ν are σ-finite countably additive measures on ''X'' that have the same sets of measure 0. Then the mapping : is a unitary operator : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「direct integral」の詳細全文を読む スポンサード リンク
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