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- Direct_integral abstract "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_integral wikiPageExternalLink sici?sici=0003-486X%28194904%292%3A50%3A2%3C401%3AOROORT%3E2.0.CO%3B2-H.
- Direct_integral wikiPageID "1256105".
- Direct_integral wikiPageLength "16387".
- Direct_integral wikiPageOutDegree "27".
- Direct_integral wikiPageRevisionID "681216335".
- Direct_integral wikiPageWikiLink Almost_everywhere.
- Direct_integral wikiPageWikiLink Artin–Wedderburn_theorem.
- Direct_integral wikiPageWikiLink Borel_equivalence_relation.
- Direct_integral wikiPageWikiLink C*-algebra.
- Direct_integral wikiPageWikiLink Category:Functional_analysis.
- Direct_integral wikiPageWikiLink Category:Measure_theory.
- Direct_integral wikiPageWikiLink Category:Von_Neumann_algebras.
- Direct_integral wikiPageWikiLink Center_(algebra).
- Direct_integral wikiPageWikiLink Direct_sum.
- Direct_integral wikiPageWikiLink Equivariant_map.
- Direct_integral wikiPageWikiLink Functional_analysis.
- Direct_integral wikiPageWikiLink George_Mackey.
- Direct_integral wikiPageWikiLink Hilbert_space.
- Direct_integral wikiPageWikiLink If_and_only_if.
- Direct_integral wikiPageWikiLink Induced_representation.
- Direct_integral wikiPageWikiLink Jacques_Dixmier.
- Direct_integral wikiPageWikiLink John_von_Neumann.
- Direct_integral wikiPageWikiLink Mathematics.
- Direct_integral wikiPageWikiLink Measure_(mathematics).
- Direct_integral wikiPageWikiLink Null_set.
- Direct_integral wikiPageWikiLink Polish_space.
- Direct_integral wikiPageWikiLink System_of_imprimitivity.
- Direct_integral wikiPageWikiLink Von_Neumann_algebra.
- Direct_integral wikiPageWikiLinkText "Direct integral".
- Direct_integral wikiPageWikiLinkText "Direct integral#Decomposable operators".
- Direct_integral wikiPageWikiLinkText "central decomposition of von Neumann algebras".
- Direct_integral wikiPageWikiLinkText "decomposition of abelian von Neumann algebras".
- Direct_integral wikiPageWikiLinkText "direct integral decomposition".
- Direct_integral wikiPageWikiLinkText "direct integral".
- Direct_integral date "September 2015".
- Direct_integral reason "Is this X assumed in advance to be a Borel space, or to be a standard Borel space?".
- Direct_integral wikiPageUsesTemplate Template:What.
- Direct_integral subject Category:Functional_analysis.
- Direct_integral subject Category:Measure_theory.
- Direct_integral subject Category:Von_Neumann_algebras.
- Direct_integral hypernym Generalization.
- Direct_integral type Algebra.
- Direct_integral type Function.
- Direct_integral comment "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.".
- Direct_integral label "Direct integral".
- Direct_integral sameAs Q5280347.
- Direct_integral sameAs m.04mn57.
- Direct_integral sameAs Q5280347.
- Direct_integral wasDerivedFrom Direct_integral?oldid=681216335.
- Direct_integral isPrimaryTopicOf Direct_integral.