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Factors subsection</span> </button> <ul id="toc-Factors-sublist" class="vector-toc-list"> <li id="toc-Type_I_factors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Type_I_factors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Type I factors</span> </div> </a> <ul id="toc-Type_I_factors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Type_II_factors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Type_II_factors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Type II factors</span> </div> </a> <ul id="toc-Type_II_factors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Type_III_factors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Type_III_factors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Type III factors</span> </div> </a> <ul id="toc-Type_III_factors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_predual" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_predual"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>The predual</span> </div> </a> <ul id="toc-The_predual-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weights,_states,_and_traces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Weights,_states,_and_traces"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Weights, states, and traces</span> </div> </a> <ul id="toc-Weights,_states,_and_traces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modules_over_a_factor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Modules_over_a_factor"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Modules over a factor</span> </div> </a> <ul id="toc-Modules_over_a_factor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Amenable_von_Neumann_algebras" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Amenable_von_Neumann_algebras"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Amenable von Neumann algebras</span> </div> </a> <ul id="toc-Amenable_von_Neumann_algebras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensor_products_of_von_Neumann_algebras" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tensor_products_of_von_Neumann_algebras"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Tensor products of von Neumann algebras</span> </div> </a> <ul id="toc-Tensor_products_of_von_Neumann_algebras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bimodules_and_subfactors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bimodules_and_subfactors"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Bimodules and subfactors</span> </div> </a> <ul id="toc-Bimodules_and_subfactors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-amenable_factors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-amenable_factors"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Non-amenable factors</span> </div> </a> <ul id="toc-Non-amenable_factors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Von Neumann algebra</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 15 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-15" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">15 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_de_von_Neumann" title="Àlgebra de von Neumann – Catalan" lang="ca" hreflang="ca" data-title="Àlgebra de von Neumann" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Von-Neumann-Algebra" title="Von-Neumann-Algebra – German" lang="de" hreflang="de" data-title="Von-Neumann-Algebra" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%86%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1_%CF%86%CE%BF%CE%BD_%CE%9D%CF%8C%CE%B9%CE%BC%CE%B1%CE%BD" title="Άλγεβρα φον Νόιμαν – Greek" lang="el" hreflang="el" data-title="Άλγεβρα φον Νόιμαν" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_de_von_Neumann" title="Álgebra de von Neumann – Spanish" lang="es" hreflang="es" data-title="Álgebra de von Neumann" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D9%81%D9%88%D9%86_%D9%86%DB%8C%D9%88%D9%85%D9%86" title="جبر فون نیومن – Persian" lang="fa" hreflang="fa" data-title="جبر فون نیومن" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_von_Neumann" title="Algèbre de von Neumann – French" lang="fr" hreflang="fr" data-title="Algèbre de von Neumann" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%B0_%EB%85%B8%EC%9D%B4%EB%A7%8C_%EB%8C%80%EC%88%98" title="폰 노이만 대수 – Korean" lang="ko" hreflang="ko" data-title="폰 노이만 대수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aljabar_von_Neumann" title="Aljabar von Neumann – Indonesian" lang="id" hreflang="id" data-title="Aljabar von Neumann" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Algebra_von_Neumann" title="Algebra von Neumann – Malay" lang="ms" hreflang="ms" data-title="Algebra von Neumann" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Von_Neumann-algebra" title="Von Neumann-algebra – Dutch" lang="nl" hreflang="nl" data-title="Von Neumann-algebra" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A9%E3%83%B3%E3%83%BB%E3%83%8E%E3%82%A4%E3%83%9E%E3%83%B3%E7%92%B0" title="フォン・ノイマン環 – Japanese" lang="ja" hreflang="ja" data-title="フォン・ノイマン環" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algebra_von_Neumanna" title="Algebra von Neumanna – Polish" lang="pl" hreflang="pl" data-title="Algebra von Neumanna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_de_von_Neumann" title="Álgebra de von Neumann – Portuguese" lang="pt" hreflang="pt" data-title="Álgebra de von Neumann" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D1%84%D0%BE%D0%BD_%D0%9D%D0%B5%D0%B9%D0%BC%D0%B0%D0%BD%D0%B0" title="Алгебра фон Неймана – Russian" lang="ru" hreflang="ru" data-title="Алгебра фон Неймана" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%86%AF%E8%AF%BA%E4%BE%9D%E6%9B%BC%E4%BB%A3%E6%95%B0" title="冯诺依曼代数 – Chinese" lang="zh" hreflang="zh" data-title="冯诺依曼代数" 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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">*-algebra of bounded operators on a Hilbert space</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"operator ring" redirects here. Not to be confused with <a href="/wiki/Ring_operator" class="mw-redirect" title="Ring operator">ring operator</a> or <a href="/wiki/Operator_assistance" title="Operator assistance">operator assistance</a>.</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>von Neumann algebra</b> or <b>W*-algebra</b> is a <a href="/wiki/*-algebra" title="*-algebra">*-algebra</a> of <a href="/wiki/Bounded_linear_operator" class="mw-redirect" title="Bounded linear operator">bounded operators</a> on a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> that is <a href="/wiki/Closed_set" title="Closed set">closed</a> in the <a href="/wiki/Weak_operator_topology" title="Weak operator topology">weak operator topology</a> and contains the <a href="/wiki/Identity_operator" class="mw-redirect" title="Identity operator">identity operator</a>. It is a special type of <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a>. </p><p>Von Neumann algebras were originally introduced by <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>, motivated by his study of <a href="/wiki/Operator_theory" title="Operator theory">single operators</a>, <a href="/wiki/Group_representation" title="Group representation">group representations</a>, <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a> and <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. His <a href="/wiki/Von_Neumann_double_commutant_theorem" class="mw-redirect" title="Von Neumann double commutant theorem">double commutant theorem</a> shows that the <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analytic</a> definition is equivalent to a purely <a href="/wiki/Abstract_algebra" title="Abstract algebra">algebraic</a> definition as an algebra of symmetries. </p><p>Two basic examples of von Neumann algebras are as follows: </p> <ul><li>The ring <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/016b7dd68f0a209dfb882c9779e5b1c4e690c46a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.946ex; height:2.843ex;" alt="{\displaystyle L^{\infty }(\mathbb {R} )}"></span> of <a href="/wiki/Essentially_bounded" class="mw-redirect" title="Essentially bounded">essentially bounded</a> <a href="/wiki/Measurable_function" title="Measurable function">measurable functions</a> on the real line is a commutative von Neumann algebra, whose elements act as <a href="/wiki/Multiplication_operator" title="Multiplication operator">multiplication operators</a> by pointwise multiplication on the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8722fb232f689925a4baa0e4ba478e43ee346672" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} )}"></span> of <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-integrable functions</a>.</li> <li>The algebra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}({\mathcal {H}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}({\mathcal {H}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/383351f34821120bcd710ba62aec678f54563179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.316ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}({\mathcal {H}})}"></span> of all <a href="/wiki/Bounded_operator" title="Bounded operator">bounded operators</a> on a Hilbert space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span> is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span>.</li></ul> <p>Von Neumann algebras were first studied by <a href="#CITEREFvon_Neumann1930">von Neumann (1930)</a> in 1929; he and <a href="/wiki/Francis_Joseph_Murray" title="Francis Joseph Murray">Francis Murray</a> developed the basic theory, under the original name of <b>rings of operators</b>, in a series of papers written in the 1930s and 1940s (F.J. Murray & J. von Neumann <a href="#CITEREFMurrayvon_Neumann1936">1936</a>, <a href="#CITEREFMurrayvon_Neumann1937">1937</a>, <a href="#CITEREFMurrayvon_Neumann1943">1943</a>; J. von Neumann <a href="#CITEREFvon_Neumann1938">1938</a>, <a href="#CITEREFvon_Neumann1940">1940</a>, <a href="#CITEREFvon_Neumann1943">1943</a>, <a href="#CITEREFvon_Neumann1949">1949</a>), reprinted in the collected works of <a href="#CITEREFvon_Neumann1961">von Neumann (1961)</a>. </p><p>Introductory accounts of von Neumann algebras are given in the online notes of <a href="#CITEREFJones2003">Jones (2003)</a> and <a href="#CITEREFWassermann1991">Wassermann (1991)</a> and the books by <a href="#CITEREFDixmier1981">Dixmier (1981)</a>, <a href="#CITEREFSchwartz1967">Schwartz (1967)</a>, <a href="#CITEREFBlackadar2005">Blackadar (2005)</a> and <a href="#CITEREFSakai1971">Sakai (1971)</a>. The three volume work by <a href="#CITEREFTakesaki1979">Takesaki (1979)</a> gives an encyclopedic account of the theory. The book by <a href="#CITEREFConnes1994">Connes (1994)</a> discusses more advanced topics. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are three common ways to define von Neumann algebras. </p><p>The first and most common way is to define them as <a href="/wiki/Weak_operator_topology" title="Weak operator topology">weakly closed</a> <a href="/wiki/*-algebra" title="*-algebra">*-algebras</a> of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by many other <a href="/wiki/Operator_topology" class="mw-redirect" title="Operator topology">common topologies</a> including the <a href="/wiki/Strong_operator_topology" title="Strong operator topology">strong</a>, <a href="/wiki/Ultrastrong_topology" title="Ultrastrong topology">ultrastrong</a> or <a href="/wiki/Ultraweak_topology" title="Ultraweak topology">ultraweak</a> operator topologies. The *-algebras of bounded operators that are closed in the <a href="/wiki/Norm_topology" class="mw-redirect" title="Norm topology">norm topology</a> are <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a>, so in particular any von Neumann algebra is a C*-algebra. </p><p>The second definition is that a von Neumann algebra is a subalgebra of the bounded operators closed under <a href="/wiki/Semigroup_with_involution" title="Semigroup with involution">involution</a> (the *-operation) and equal to its double <a href="/wiki/Commutant" class="mw-redirect" title="Commutant">commutant</a>, or equivalently the <a href="/wiki/Commutant" class="mw-redirect" title="Commutant">commutant</a> of some subalgebra closed under *. The <a href="/wiki/Von_Neumann_double_commutant_theorem" class="mw-redirect" title="Von Neumann double commutant theorem">von Neumann double commutant theorem</a> (<a href="#CITEREFvon_Neumann1930">von Neumann 1930</a>) says that the first two definitions are equivalent. </p><p>The first two definitions describe a von Neumann algebra concretely as a set of operators acting on some given Hilbert space. <a href="#CITEREFSakai1971">Sakai (1971)</a> showed that von Neumann algebras can also be defined abstractly as C*-algebras that have a <a href="/wiki/Predual" title="Predual">predual</a>; in other words the von Neumann algebra, considered as a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>, is the <a href="/wiki/Dual_space" title="Dual space">dual</a> of some other Banach space called the predual. The predual of a von Neumann algebra is in fact unique up to isomorphism. Some authors use "von Neumann algebra" for the algebras together with a Hilbert space action, and "W*-algebra" for the abstract concept, so a von Neumann algebra is a W*-algebra together with a Hilbert space and a suitable faithful unital action on the Hilbert space. The concrete and abstract definitions of a von Neumann algebra are similar to the concrete and abstract definitions of a C*-algebra, which can be defined either as norm-closed *-algebras of operators on a Hilbert space, or as <a href="/wiki/Banach_*-algebra" class="mw-redirect" title="Banach *-algebra">Banach *-algebras</a> such that ||<i>aa*</i>||=||<i>a</i>|| ||<i>a*</i>||. </p> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=2" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject. </p> <ul><li>A <b>factor</b> is a von Neumann algebra with trivial center, i.e. a center consisting only of scalar operators.</li> <li>A <a href="/wiki/Finite_von_Neumann_algebra" title="Finite von Neumann algebra"><b>finite</b> von Neumann algebra</a> is one which is the <a href="/wiki/Direct_integral#Direct_integrals_of_von_Neumann_algebras" title="Direct integral">direct integral</a> of finite factors (meaning the von Neumann algebra has a faithful normal tracial state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau :M\rightarrow \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau :M\rightarrow \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82aa3b8f42160e1cd42f59328147d6c6ac1f51b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.873ex; height:2.176ex;" alt="{\displaystyle \tau :M\rightarrow \mathbb {C} }"></span><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>). Similarly, <b>properly infinite</b> von Neumann algebras are the direct integral of properly infinite factors.</li> <li>A von Neumann algebra that acts on a separable Hilbert space is called <b>separable</b>. Note that such algebras are rarely <a href="/wiki/Separable_space" title="Separable space">separable</a> in the norm topology.</li> <li>The von Neumann algebra <b>generated</b> by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators.</li> <li>The <b>tensor product</b> of two von Neumann algebras acting on two Hilbert spaces is defined to be the von Neumann algebra generated by their algebraic tensor product, considered as operators on the Hilbert space tensor product of the Hilbert spaces.</li></ul> <p>By <a href="/wiki/Forgetting_(mathematics)" class="mw-redirect" title="Forgetting (mathematics)">forgetting</a> about the topology on a von Neumann algebra, we can consider it a (unital) <a href="/wiki/Star-algebra" class="mw-redirect" title="Star-algebra">*-algebra</a>, or just a ring. Von Neumann algebras are <a href="/wiki/Semihereditary_ring" class="mw-redirect" title="Semihereditary ring">semihereditary</a>: every finitely generated submodule of a <a href="/wiki/Projective_module" title="Projective module">projective module</a> is itself projective. There have been several attempts to axiomatize the underlying rings of von Neumann algebras, including <a href="/wiki/Baer_*-ring" class="mw-redirect" title="Baer *-ring">Baer *-rings</a> and <a href="/wiki/AW*-algebra" title="AW*-algebra">AW*-algebras</a>. The <a href="/wiki/*-algebra" title="*-algebra">*-algebra</a> of <a href="/wiki/Affiliated_operator" title="Affiliated operator">affiliated operators</a> of a finite von Neumann algebra is a <a href="/wiki/Von_Neumann_regular_ring" title="Von Neumann regular ring">von Neumann regular ring</a>. (The von Neumann algebra itself is in general not von Neumann regular.) </p> <div class="mw-heading mw-heading2"><h2 id="Commutative_von_Neumann_algebras">Commutative von Neumann algebras</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=3" title="Edit section: Commutative von Neumann algebras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Abelian_von_Neumann_algebra" title="Abelian von Neumann algebra">Abelian von Neumann algebra</a></div> <p>The relationship between <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> von Neumann algebras and <a href="/wiki/Measure_space" title="Measure space">measure spaces</a> is analogous to that between commutative <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a> and <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff spaces</a>. Every commutative von Neumann algebra is isomorphic to <a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup>∞</sup></a>(<i>X</i>) for some measure space (<i>X</i>, μ) and conversely, for every σ-finite measure space <i>X</i>, the *-algebra <i>L</i><sup>∞</sup>(<i>X</i>) is a von Neumann algebra. </p><p>Due to this analogy, the theory of von Neumann algebras has been called noncommutative measure theory, while the theory of <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a> is sometimes called <a href="/wiki/Noncommutative_topology" title="Noncommutative topology">noncommutative topology</a> (<a href="#CITEREFConnes1994">Connes 1994</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Projections">Projections</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=4" title="Edit section: Projections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Operators <i>E</i> in a von Neumann algebra for which <i>E</i> = <i>EE</i> = <i>E*</i> are called <b>projections</b>; they are exactly the operators which give an orthogonal projection of <i>H</i> onto some closed subspace. A subspace of the Hilbert space <i>H</i> is said to <b>belong to</b> the von Neumann algebra <i>M</i> if it is the image of some projection in <i>M</i>. This establishes a 1:1 correspondence between projections of <i>M</i> and subspaces that belong to <i>M</i>. Informally these are the closed subspaces that can be described using elements of <i>M</i>, or that <i>M</i> "knows" about. </p><p>It can be shown that the closure of the image of any operator in <i>M</i> and the kernel of any operator in <i>M</i> belongs to <i>M</i>. Also, the closure of the image under an operator of <i>M</i> of any subspace belonging to <i>M</i> also belongs to <i>M</i>. (These results are a consequence of the <a href="/wiki/Polar_decomposition" title="Polar decomposition">polar decomposition</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Comparison_theory_of_projections">Comparison theory of projections</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=5" title="Edit section: Comparison theory of projections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The basic theory of projections was worked out by <a href="#CITEREFMurrayvon_Neumann1936">Murray & von Neumann (1936)</a>. Two subspaces belonging to <i>M</i> are called (<b>Murray–von Neumann</b>) <b>equivalent</b> if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, if <i>M</i> "knows" that the subspaces are isomorphic). This induces a natural <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> on projections by defining <i>E</i> to be equivalent to <i>F</i> if the corresponding subspaces are equivalent, or in other words if there is a <a href="/wiki/Partial_isometry" title="Partial isometry">partial isometry</a> of <i>H</i> that maps the image of <i>E</i> isometrically to the image of <i>F</i> and is an element of the von Neumann algebra. Another way of stating this is that <i>E</i> is equivalent to <i>F</i> if <i>E=uu*</i> and <i>F=u*u</i> for some partial isometry <i>u</i> in <i>M</i>. </p><p>The equivalence relation ~ thus defined is additive in the following sense: Suppose <i>E</i><sub>1</sub> ~ <i>F</i><sub>1</sub> and <i>E</i><sub>2</sub> ~ <i>F</i><sub>2</sub>. If <i>E</i><sub>1</sub> ⊥ <i>E</i><sub>2</sub> and <i>F</i><sub>1</sub> ⊥ <i>F</i><sub>2</sub>, then <i>E</i><sub>1</sub> + <i>E</i><sub>2</sub> ~ <i>F</i><sub>1</sub> + <i>F</i><sub>2</sub>. Additivity would <i>not</i> generally hold if one were to require unitary equivalence in the definition of ~, i.e. if we say <i>E</i> is equivalent to <i>F</i> if <i>u*Eu</i> = <i>F</i> for some unitary <i>u</i>. The <a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorems_for_operator_algebras" title="Schröder–Bernstein theorems for operator algebras">Schröder–Bernstein theorems for operator algebras</a> gives a sufficient condition for Murray-von Neumann equivalence. </p><p>The subspaces belonging to <i>M</i> are partially ordered by inclusion, and this induces a partial order ≤ of projections. There is also a natural partial order on the set of <i>equivalence classes</i> of projections, induced by the partial order ≤ of projections. If <i>M</i> is a factor, ≤ is a total order on equivalence classes of projections, described in the section on traces below. </p><p>A projection (or subspace belonging to <i>M</i>) <i>E</i> is said to be a <b>finite projection</b> if there is no projection <i>F</i> < <i>E</i> (meaning <i>F</i> ≤ <i>E</i> and <i>F</i> ≠ <i>E</i>) that is equivalent to <i>E</i>. For example, all finite-dimensional projections (or subspaces) are finite (since isometries between Hilbert spaces leave the dimension fixed), but the identity operator on an infinite-dimensional Hilbert space is not finite in the von Neumann algebra of all bounded operators on it, since it is isometrically isomorphic to a proper subset of itself. However it is possible for infinite dimensional subspaces to be finite. </p><p>Orthogonal projections are noncommutative analogues of indicator functions in <i>L</i><sup>∞</sup>(<b>R</b>). <i>L</i><sup>∞</sup>(<b>R</b>) is the ||·||<sub>∞</sub>-closure of the subspace generated by the indicator functions. Similarly, a von Neumann algebra is generated by its projections; this is a consequence of the <a href="/wiki/Self-adjoint_operator#Spectral_theorem" title="Self-adjoint operator">spectral theorem for self-adjoint operators</a>. </p><p>The projections of a finite factor form a <a href="/wiki/Continuous_geometry" title="Continuous geometry">continuous geometry</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Factors">Factors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=6" title="Edit section: Factors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A von Neumann algebra <i>N</i> whose <a href="/wiki/Center_(algebra)" title="Center (algebra)">center</a> consists only of multiples of the identity operator is called a <b>factor</b>. As <a href="#CITEREFvon_Neumann1949">von Neumann (1949)</a> showed, every von Neumann algebra on a separable Hilbert space is isomorphic to a <a href="/wiki/Direct_integral" title="Direct integral">direct integral</a> of factors. This decomposition is essentially unique. Thus, the problem of classifying isomorphism classes of von Neumann algebras on separable Hilbert spaces can be reduced to that of classifying isomorphism classes of factors. </p><p><a href="#CITEREFMurrayvon_Neumann1936">Murray & von Neumann (1936)</a> showed that every factor has one of 3 types as described below. The type classification can be extended to von Neumann algebras that are not factors, and a von Neumann algebra is of type X if it can be decomposed as a direct integral of type X factors; for example, every commutative von Neumann algebra has type I<sub>1</sub>. Every von Neumann algebra can be written uniquely as a sum of von Neumann algebras of types I, II, and III. </p><p>There are several other ways to divide factors into classes that are sometimes used: </p> <ul><li>A factor is called <b>discrete</b> (or occasionally <b>tame</b>) if it has type I, and <b>continuous</b> (or occasionally <b>wild</b>) if it has type II or III.</li> <li>A factor is called <b>semifinite</b> if it has type I or II, and <b>purely infinite</b> if it has type III.</li> <li>A factor is called <b>finite</b> if the projection 1 is finite and <b>properly infinite</b> otherwise. Factors of types I and II may be either finite or properly infinite, but factors of type III are always properly infinite.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Type_I_factors">Type I factors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=7" title="Edit section: Type I factors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A factor is said to be of <b>type I</b> if there is a minimal projection <i>E ≠ 0</i>, i.e. a projection <i>E</i> such that there is no other projection <i>F</i> with 0 < <i>F</i> < <i>E</i>. Any factor of type I is isomorphic to the von Neumann algebra of <i>all</i> bounded operators on some Hilbert space; since there is one Hilbert space for every <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal number</a>, isomorphism classes of factors of type I correspond exactly to the cardinal numbers. Since many authors consider von Neumann algebras only on separable Hilbert spaces, it is customary to call the bounded operators on a Hilbert space of finite dimension <i>n</i> a factor of type I<sub><i>n</i></sub>, and the bounded operators on a separable infinite-dimensional Hilbert space, a factor of type I<sub>∞</sub>. </p> <div class="mw-heading mw-heading3"><h3 id="Type_II_factors">Type II factors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=8" title="Edit section: Type II factors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A factor is said to be of <b>type II</b> if there are no minimal projections but there are non-zero <a class="mw-selflink-fragment" href="#Comparison_theory_of_projections">finite projections</a>. This implies that every projection <i>E</i> can be "halved" in the sense that there are two projections <i>F</i> and <i>G</i> that are <a class="mw-selflink-fragment" href="#Comparison_theory_of_projections">Murray–von Neumann equivalent</a> and satisfy <i>E</i> = <i>F</i> + <i>G</i>. If the identity operator in a type II factor is finite, the factor is said to be of type II<sub>1</sub>; otherwise, it is said to be of type II<sub>∞</sub>. The best understood factors of type II are the <a href="/wiki/Hyperfinite_type_II-1_factor" class="mw-redirect" title="Hyperfinite type II-1 factor">hyperfinite type II<sub>1</sub> factor</a> and the <a href="/wiki/Hyperfinite_type_II-infinity_factor" class="mw-redirect" title="Hyperfinite type II-infinity factor">hyperfinite type II<sub>∞</sub> factor</a>, found by <a href="#CITEREFMurrayvon_Neumann1936">Murray & von Neumann (1936)</a>. These are the unique hyperfinite factors of types II<sub>1</sub> and II<sub>∞</sub>; there are an uncountable number of other factors of these types that are the subject of intensive study. <a href="#CITEREFMurrayvon_Neumann1937">Murray & von Neumann (1937)</a> proved the fundamental result that a factor of type II<sub>1</sub> has a unique finite tracial state, and the set of traces of projections is [0,1]. </p><p>A factor of type II<sub>∞</sub> has a semifinite trace, unique up to rescaling, and the set of traces of projections is [0,∞]. The set of real numbers λ such that there is an automorphism rescaling the trace by a factor of λ is called the <b>fundamental group</b> of the type II<sub>∞</sub> factor. </p><p>The tensor product of a factor of type II<sub>1</sub> and an infinite type I factor has type II<sub>∞</sub>, and conversely any factor of type II<sub>∞</sub> can be constructed like this. The <b>fundamental group</b> of a type II<sub>1</sub> factor is defined to be the fundamental group of its tensor product with the infinite (separable) factor of type I. For many years it was an open problem to find a type II factor whose fundamental group was not the group of <a href="/wiki/Positive_reals" class="mw-redirect" title="Positive reals">positive reals</a>, but <a href="/wiki/Alain_Connes" title="Alain Connes">Connes</a> then showed that the von Neumann group algebra of a countable discrete group with <a href="/wiki/Kazhdan%27s_property_(T)" title="Kazhdan's property (T)">Kazhdan's property (T)</a> (the trivial representation is isolated in the dual space), such as SL(3,<b>Z</b>), has a countable fundamental group. Subsequently, <a href="/wiki/Sorin_Popa" title="Sorin Popa">Sorin Popa</a> showed that the fundamental group can be trivial for certain groups, including the <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of <b>Z</b><sup>2</sup> by SL(2,<b>Z</b>). </p><p>An example of a type II<sub>1</sub> factor is the von Neumann group algebra of a countable infinite discrete group such that every non-trivial conjugacy class is infinite. <a href="#CITEREFMcDuff1969">McDuff (1969)</a> found an uncountable family of such groups with non-isomorphic von Neumann group algebras, thus showing the existence of uncountably many different separable type II<sub>1</sub> factors. </p> <div class="mw-heading mw-heading3"><h3 id="Type_III_factors">Type III factors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=9" title="Edit section: Type III factors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lastly, <b>type III</b> factors are factors that do not contain any nonzero finite projections at all. In their first paper <a href="#CITEREFMurrayvon_Neumann1936">Murray & von Neumann (1936)</a> were unable to decide whether or not they existed; the first examples were later found by <a href="#CITEREFvon_Neumann1940">von Neumann (1940)</a>. Since the identity operator is always infinite in those factors, they were sometimes called type III<sub>∞</sub> in the past, but recently that notation has been superseded by the notation III<sub>λ</sub>, where λ is a real number in the interval [0,1]. More precisely, if the Connes spectrum (of its modular group) is 1 then the factor is of type III<sub>0</sub>, if the Connes spectrum is all integral powers of λ for 0 < λ < 1, then the type is III<sub>λ</sub>, and if the Connes spectrum is all positive reals then the type is III<sub>1</sub>. (The Connes spectrum is a closed subgroup of the positive reals, so these are the only possibilities.) The only trace on type III factors takes value ∞ on all non-zero positive elements, and any two non-zero projections are equivalent. At one time type III factors were considered to be intractable objects, but <a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a> has led to a good structure theory. In particular, any type III factor can be written in a canonical way as the <a href="/wiki/Crossed_product" title="Crossed product">crossed product</a> of a type II<sub>∞</sub> factor and the real numbers. </p> <div class="mw-heading mw-heading2"><h2 id="The_predual">The predual</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=10" title="Edit section: The predual"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any von Neumann algebra <i>M</i> has a <b>predual</b> <i>M</i><sub>∗</sub>, which is the Banach space of all ultraweakly continuous linear functionals on <i>M</i>. As the name suggests, <i>M</i> is (as a Banach space) the dual of its predual. The predual is unique in the sense that any other Banach space whose dual is <i>M</i> is canonically isomorphic to <i>M</i><sub>∗</sub>. <a href="#CITEREFSakai1971">Sakai (1971)</a> showed that the existence of a predual characterizes von Neumann algebras among C* algebras. </p><p>The definition of the predual given above seems to depend on the choice of Hilbert space that <i>M</i> acts on, as this determines the ultraweak topology. However the predual can also be defined without using the Hilbert space that <i>M</i> acts on, by defining it to be the space generated by all positive <b>normal</b> linear functionals on <i>M</i>. (Here "normal" means that it preserves suprema when applied to increasing nets of self adjoint operators; or equivalently to increasing sequences of projections.) </p><p>The predual <i>M</i><sub>∗</sub> is a closed subspace of the dual <i>M*</i> (which consists of all norm-continuous linear functionals on <i>M</i>) but is generally smaller. The proof that <i>M</i><sub>∗</sub> is (usually) not the same as <i>M*</i> is nonconstructive and uses the axiom of choice in an essential way; it is very hard to exhibit explicit elements of <i>M*</i> that are not in <i>M</i><sub>∗</sub>. For example, exotic positive linear forms on the von Neumann algebra <i>l</i><sup>∞</sup>(<i>Z</i>) are given by <a href="/wiki/Ultrafilter" title="Ultrafilter">free ultrafilters</a>; they correspond to exotic *-homomorphisms into <i>C</i> and describe the <a href="/wiki/Stone%E2%80%93%C4%8Cech_compactification" title="Stone–Čech compactification">Stone–Čech compactification</a> of <i>Z</i>. </p><p>Examples: </p> <ol><li>The predual of the von Neumann algebra <i>L</i><sup>∞</sup>(<b>R</b>) of essentially bounded functions on <b>R</b> is the Banach space <i>L</i><sup>1</sup>(<b>R</b>) of integrable functions. The dual of <i>L</i><sup>∞</sup>(<b>R</b>) is strictly larger than <i>L</i><sup>1</sup>(<b>R</b>) For example, a functional on <i>L</i><sup>∞</sup>(<b>R</b>) that extends the <a href="/wiki/Dirac_measure" title="Dirac measure">Dirac measure</a> δ<sub>0</sub> on the closed subspace of bounded continuous functions <i>C</i><sup>0</sup><sub>b</sub>(<b>R</b>) cannot be represented as a function in <i>L</i><sup>1</sup>(<b>R</b>).</li> <li>The predual of the von Neumann algebra <i>B</i>(<i>H</i>) of bounded operators on a Hilbert space <i>H</i> is the Banach space of all <a href="/wiki/Trace_class" title="Trace class">trace class</a> operators with the trace norm ||<i>A</i>||= Tr(|<i>A</i>|). The Banach space of trace class operators is itself the dual of the C*-algebra of compact operators (which is not a von Neumann algebra).</li></ol> <div class="mw-heading mw-heading2"><h2 id="Weights,_states,_and_traces"><span id="Weights.2C_states.2C_and_traces"></span>Weights, states, and traces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=11" title="Edit section: Weights, states, and traces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Noncommutative_measure_and_integration" title="Noncommutative measure and integration">Noncommutative measure and integration</a></div> <p>Weights and their special cases states and traces are discussed in detail in (<a href="#CITEREFTakesaki1979">Takesaki 1979</a>). </p> <ul><li>A <b>weight</b> ω on a von Neumann algebra is a linear map from the set of <a href="/wiki/C*-algebra#Self-adjoint_elements" title="C*-algebra">positive elements</a> (those of the form <i>a*a</i>) to [0,∞].</li> <li>A <b>positive linear functional</b> is a weight with ω(1) finite (or rather the extension of ω to the whole algebra by linearity).</li> <li>A <b><a href="/wiki/State_(functional_analysis)" title="State (functional analysis)">state</a></b> is a weight with ω(1) = 1.</li> <li>A <b>trace</b> is a weight with ω(<i>aa*</i>) = ω(<i>a*a</i>) for all <i>a</i>.</li> <li>A <b>tracial state</b> is a trace with ω(1) = 1.</li></ul> <p>Any factor has a trace such that the trace of a non-zero projection is non-zero and the trace of a projection is infinite if and only if the projection is infinite. Such a trace is unique up to rescaling. For factors that are separable or finite, two projections are equivalent if and only if they have the same trace. The type of a factor can be read off from the possible values of this trace over the projections of the factor, as follows: </p> <ul><li>Type I<sub><i>n</i></sub>: 0, <i>x</i>, 2<i>x</i>, ....,<i>nx</i> for some positive <i>x</i> (usually normalized to be 1/<i>n</i> or 1).</li> <li>Type I<sub>∞</sub>: 0, <i>x</i>, 2<i>x</i>, ....,∞ for some positive <i>x</i> (usually normalized to be 1).</li> <li>Type II<sub>1</sub>: [0,<i>x</i>] for some positive <i>x</i> (usually normalized to be 1).</li> <li>Type II<sub>∞</sub>: [0,∞].</li> <li>Type III: {0,∞}.</li></ul> <p>If a von Neumann algebra acts on a Hilbert space containing a norm 1 vector <i>v</i>, then the functional <i>a</i> → (<i>av</i>,<i>v</i>) is a normal state. This construction can be reversed to give an action on a Hilbert space from a normal state: this is the <a href="/wiki/GNS_construction" class="mw-redirect" title="GNS construction">GNS construction</a> for normal states. </p> <div class="mw-heading mw-heading2"><h2 id="Modules_over_a_factor">Modules over a factor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=12" title="Edit section: Modules over a factor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given an abstract separable factor, one can ask for a classification of its modules, meaning the separable Hilbert spaces that it acts on. The answer is given as follows: every such module <i>H</i> can be given an <i>M</i>-dimension dim<sub><i>M</i></sub>(<i>H</i>) (not its dimension as a complex vector space) such that modules are isomorphic if and only if they have the same <i>M</i>-dimension. The <i>M</i>-dimension is additive, and a module is isomorphic to a subspace of another module if and only if it has smaller or equal <i>M</i>-dimension. </p><p>A module is called <b>standard</b> if it has a cyclic separating vector. Each factor has a standard representation, which is unique up to isomorphism. The standard representation has an antilinear involution <i>J</i> such that <i>JMJ</i> = <i>M<span class="nowrap" style="padding-left:0.05em;">′</span></i>. For finite factors the standard module is given by the <a href="/wiki/GNS_construction" class="mw-redirect" title="GNS construction">GNS construction</a> applied to the unique normal tracial state and the <i>M</i>-dimension is normalized so that the standard module has <i>M</i>-dimension 1, while for infinite factors the standard module is the module with <i>M</i>-dimension equal to ∞. </p><p>The possible <i>M</i>-dimensions of modules are given as follows: </p> <ul><li>Type I<sub><i>n</i></sub> (<i>n</i> finite): The <i>M</i>-dimension can be any of 0/<i>n</i>, 1/<i>n</i>, 2/<i>n</i>, 3/<i>n</i>, ..., ∞. The standard module has <i>M</i>-dimension 1 (and complex dimension <i>n</i><sup>2</sup>.)</li> <li>Type I<sub>∞</sub> The <i>M</i>-dimension can be any of 0, 1, 2, 3, ..., ∞. The standard representation of <i>B</i>(<i>H</i>) is <i>H</i>⊗<i>H</i>; its <i>M</i>-dimension is ∞.</li> <li>Type II<sub>1</sub>: The <i>M</i>-dimension can be anything in [0, ∞]. It is normalized so that the standard module has <i>M</i>-dimension 1. The <i>M</i>-dimension is also called the <b>coupling constant</b> of the module <i>H</i>.</li> <li>Type II<sub>∞</sub>: The <i>M</i>-dimension can be anything in [0, ∞]. There is in general no canonical way to normalize it; the factor may have outer automorphisms multiplying the <i>M</i>-dimension by constants. The standard representation is the one with <i>M</i>-dimension ∞.</li> <li>Type III: The <i>M</i>-dimension can be 0 or ∞. Any two non-zero modules are isomorphic, and all non-zero modules are standard.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Amenable_von_Neumann_algebras">Amenable von Neumann algebras</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=13" title="Edit section: Amenable von Neumann algebras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="#CITEREFConnes1976">Connes (1976)</a> and others proved that the following conditions on a von Neumann algebra <i>M</i> on a separable Hilbert space <i>H</i> are all <b>equivalent</b>: </p> <ul><li><i>M</i> is <b>hyperfinite</b> or <b>AFD</b> or <b>approximately finite dimensional</b> or <b>approximately finite</b>: this means the algebra contains an ascending sequence of finite dimensional subalgebras with dense union. (Warning: some authors use "hyperfinite" to mean "AFD and finite".)</li> <li><i>M</i> is <b>amenable</b>: this means that the <a href="/wiki/Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivations</a> of <i>M</i> with values in a normal dual Banach bimodule are all inner.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li><i>M</i> has Schwartz's <b>property P</b>: for any bounded operator <i>T</i> on <i>H</i> the weak operator closed convex hull of the elements <i>uTu*</i> contains an element commuting with <i>M</i>.</li> <li><i>M</i> is <b>semidiscrete</b>: this means the identity map from <i>M</i> to <i>M</i> is a weak pointwise limit of completely positive maps of finite rank.</li> <li><i>M</i> has <b>property E</b> or the <b>Hakeda–Tomiyama extension property</b>: this means that there is a projection of norm 1 from bounded operators on <i>H</i> to <i>M</i> '.</li> <li><i>M</i> is <b>injective</b>: any completely positive linear map from any self adjoint closed subspace containing 1 of any unital C*-algebra <i>A</i> to <i>M</i> can be extended to a completely positive map from <i>A</i> to <i>M</i>.</li></ul> <p>There is no generally accepted term for the class of algebras above; Connes has suggested that <b>amenable</b> should be the standard term. </p><p>The amenable factors have been classified: there is a unique one of each of the types I<sub><i>n</i></sub>, I<sub>∞</sub>, II<sub>1</sub>, II<sub>∞</sub>, III<sub>λ</sub>, for 0 < λ ≤ 1, and the ones of type III<sub>0</sub> correspond to certain ergodic flows. (For type III<sub>0</sub> calling this a classification is a little misleading, as it is known that there is no easy way to classify the corresponding ergodic flows.) The ones of type I and II<sub>1</sub> were classified by <a href="#CITEREFMurrayvon_Neumann1943">Murray & von Neumann (1943)</a>, and the remaining ones were classified by <a href="#CITEREFConnes1976">Connes (1976)</a>, except for the type III<sub>1</sub> case which was completed by Haagerup. </p><p>All amenable factors can be constructed using the <b><a href="/wiki/Crossed_product" title="Crossed product">group-measure space construction</a></b> of <a href="/wiki/Francis_Joseph_Murray" title="Francis Joseph Murray">Murray</a> and <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a> for a single <a href="/wiki/Ergodic" class="mw-redirect" title="Ergodic">ergodic</a> transformation. In fact they are precisely the factors arising as <a href="/wiki/Crossed_product" title="Crossed product">crossed products</a> by free ergodic actions of <i>Z</i> or <i>Z/nZ</i> on abelian von Neumann algebras <i>L</i><sup>∞</sup>(<i>X</i>). Type I factors occur when the <a href="/wiki/Measure_space" title="Measure space">measure space</a> <i>X</i> is <a href="/wiki/Atom_(measure_theory)" title="Atom (measure theory)">atomic</a> and the action transitive. When <i>X</i> is diffuse or <a href="/wiki/Atom_(measure_theory)" title="Atom (measure theory)">non-atomic</a>, it is <a href="/wiki/Equivalence_(measure_theory)" title="Equivalence (measure theory)">equivalent</a> to [0,1] as a <a href="/wiki/Measure_space" title="Measure space">measure space</a>. Type II factors occur when <i>X</i> admits an <a href="/wiki/Equivalence_(measure_theory)" title="Equivalence (measure theory)">equivalent</a> finite (II<sub>1</sub>) or infinite (II<sub>∞</sub>) measure, invariant under an action of <i>Z</i>. Type III factors occur in the remaining cases where there is no invariant measure, but only an <a href="/wiki/Quasi-invariant_measure" title="Quasi-invariant measure">invariant measure class</a>: these factors are called <b>Krieger factors</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Tensor_products_of_von_Neumann_algebras">Tensor products of von Neumann algebras</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=14" title="Edit section: Tensor products of von Neumann algebras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Hilbert space tensor product of two Hilbert spaces is the completion of their algebraic tensor product. One can define a tensor product of von Neumann algebras (a completion of the algebraic tensor product of the algebras considered as rings), which is again a von Neumann algebra, and act on the tensor product of the corresponding Hilbert spaces. The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The type of the tensor product of two von Neumann algebras (I, II, or III) is the maximum of their types. The <b>commutation theorem for tensor products</b> states that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M\otimes N)^{\prime }=M^{\prime }\otimes N^{\prime },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>⊗</mo> <mi>N</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>⊗</mo> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M\otimes N)^{\prime }=M^{\prime }\otimes N^{\prime },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc02cb893c0ab4cfab27e9a4acbddae118e88428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.417ex; height:3.009ex;" alt="{\displaystyle (M\otimes N)^{\prime }=M^{\prime }\otimes N^{\prime },}"></span></dd></dl> <p>where <i>M<span class="nowrap" style="padding-left:0.05em;">′</span></i> denotes the <a href="/wiki/Commutant" class="mw-redirect" title="Commutant">commutant</a> of <i>M</i>. </p><p>The tensor product of an infinite number of von Neumann algebras, if done naively, is usually a ridiculously large non-separable algebra. Instead <a href="#CITEREFvon_Neumann1938">von Neumann (1938)</a> showed that one should choose a state on each of the von Neumann algebras, use this to define a state on the algebraic tensor product, which can be used to produce a Hilbert space and a (reasonably small) von Neumann algebra. <a href="#CITEREFArakiWoods1968">Araki & Woods (1968)</a> studied the case where all the factors are finite matrix algebras; these factors are called <b>Araki–Woods</b> factors or <b>ITPFI factors</b> (ITPFI stands for "infinite tensor product of finite type I factors"). The type of the infinite tensor product can vary dramatically as the states are changed; for example, the infinite tensor product of an infinite number of type I<sub>2</sub> factors can have any type depending on the choice of states. In particular <a href="#CITEREFPowers1967">Powers (1967)</a> found an uncountable family of non-isomorphic hyperfinite type III<sub>λ</sub> factors for 0 < λ < 1, called <b>Powers factors</b>, by taking an infinite tensor product of type I<sub>2</sub> factors, each with the state given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto {\rm {Tr}}{\begin{pmatrix}{1 \over \lambda +1}&0\\0&{\lambda \over \lambda +1}\\\end{pmatrix}}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>λ</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mi>λ</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto {\rm {Tr}}{\begin{pmatrix}{1 \over \lambda +1}&0\\0&{\lambda \over \lambda +1}\\\end{pmatrix}}x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba846548d43d0a0b3d0bba107d1994f7e9ae02e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.055ex; height:8.176ex;" alt="{\displaystyle x\mapsto {\rm {Tr}}{\begin{pmatrix}{1 \over \lambda +1}&0\\0&{\lambda \over \lambda +1}\\\end{pmatrix}}x.}"></span></dd></dl> <p>All hyperfinite von Neumann algebras not of type III<sub>0</sub> are isomorphic to Araki–Woods factors, but there are uncountably many of type III<sub>0</sub> that are not. </p> <div class="mw-heading mw-heading2"><h2 id="Bimodules_and_subfactors">Bimodules and subfactors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=15" title="Edit section: Bimodules and subfactors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>bimodule</b> (or correspondence) is a Hilbert space <i>H</i> with module actions of two commuting von Neumann algebras. Bimodules have a much richer structure than that of modules. Any bimodule over two factors always gives a <a href="/wiki/Subfactor" title="Subfactor">subfactor</a> since one of the factors is always contained in the commutant of the other. There is also a subtle relative tensor product operation due to <a href="/wiki/Alain_Connes" title="Alain Connes">Connes</a> on bimodules. The theory of subfactors, initiated by <a href="/wiki/Vaughan_Jones" title="Vaughan Jones">Vaughan Jones</a>, reconciles these two seemingly different points of view. </p><p>Bimodules are also important for the von Neumann group algebra <i>M</i> of a discrete group Γ. Indeed, if <i>V</i> is any <a href="/wiki/Unitary_representation" title="Unitary representation">unitary representation</a> of Γ, then, regarding Γ as the diagonal subgroup of Γ × Γ, the corresponding <a href="/wiki/Induced_representation" title="Induced representation">induced representation</a> on <i>l</i><sup>2 </sup>(Γ, <i>V</i>) is naturally a bimodule for two commuting copies of <i>M</i>. Important <a href="/wiki/Representation_theory" title="Representation theory">representation theoretic</a> properties of Γ can be formulated entirely in terms of bimodules and therefore make sense for the von Neumann algebra itself. For example, Connes and Jones gave a definition of an analogue of <a href="/wiki/Kazhdan%27s_property_(T)" title="Kazhdan's property (T)">Kazhdan's property (T)</a> for von Neumann algebras in this way. </p> <div class="mw-heading mw-heading2"><h2 id="Non-amenable_factors">Non-amenable factors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=16" title="Edit section: Non-amenable factors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Von Neumann algebras of type I are always amenable, but for the other types there are an uncountable number of different non-amenable factors, which seem very hard to classify, or even distinguish from each other. Nevertheless, <a href="/wiki/Dan-Virgil_Voiculescu" title="Dan-Virgil Voiculescu">Voiculescu</a> has shown that the class of non-amenable factors coming from the group-measure space construction is <b>disjoint</b> from the class coming from group von Neumann algebras of free groups. Later <a href="/wiki/Narutaka_Ozawa" title="Narutaka Ozawa">Narutaka Ozawa</a> proved that group von Neumann algebras of <a href="/wiki/Hyperbolic_group" title="Hyperbolic group">hyperbolic groups</a> yield <a href="/wiki/Prime_number" title="Prime number">prime</a> type II<sub>1</sub> factors, i.e. ones that cannot be factored as tensor products of type II<sub>1</sub> factors, a result first proved by Leeming Ge for free group factors using Voiculescu's <a href="/wiki/Free_probability_theory" class="mw-redirect" title="Free probability theory">free entropy</a>. Popa's work on fundamental groups of non-amenable factors represents another significant advance. The theory of factors "beyond the hyperfinite" is rapidly expanding at present, with many new and surprising results; it has close links with <a href="/wiki/Grigory_Margulis" title="Grigory Margulis">rigidity phenomena</a> in <a href="/wiki/Geometric_group_theory" title="Geometric group theory">geometric group theory</a> and <a href="/wiki/Ergodic_theory" title="Ergodic theory">ergodic theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=17" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The essentially bounded functions on a σ-finite measure space form a commutative (type I<sub>1</sub>) von Neumann algebra acting on the <i>L</i><sup>2</sup> functions. For certain non-σ-finite measure spaces, usually considered <a href="/wiki/Pathological_(mathematics)" title="Pathological (mathematics)">pathological</a>, <i>L</i><sup>∞</sup>(<i>X</i>) is not a von Neumann algebra; for example, the σ-algebra of measurable sets might be the <a href="/wiki/Countable-cocountable_algebra" class="mw-redirect" title="Countable-cocountable algebra">countable-cocountable algebra</a> on an uncountable set. A fundamental approximation theorem can be represented by the <a href="/wiki/Kaplansky_density_theorem" title="Kaplansky density theorem">Kaplansky density theorem</a>.</li> <li>The bounded operators on any Hilbert space form a von Neumann algebra, indeed a factor, of type I.</li> <li>If we have any <a href="/wiki/Unitary_representation" title="Unitary representation">unitary representation</a> of a group <i>G</i> on a Hilbert space <i>H</i> then the bounded operators commuting with <i>G</i> form a von Neumann algebra <i>G<span class="nowrap" style="padding-left:0.05em;">′</span></i>, whose projections correspond exactly to the closed subspaces of <i>H</i> invariant under <i>G</i>. Equivalent subrepresentations correspond to equivalent projections in <i>G<span class="nowrap" style="padding-left:0.05em;">′</span></i>. The double commutant <i>G<span class="nowrap" style="padding-left:0.05em;">′</span></i><span class="nowrap" style="padding-left:0.15em;">′</span> of <i>G</i> is also a von Neumann algebra.</li> <li>The <b>von Neumann group algebra</b> of a discrete group <i>G</i> is the algebra of all bounded operators on <i>H</i> = <i>l</i><sup>2</sup>(<i>G</i>) commuting with the action of <i>G</i> on <i>H</i> through right multiplication. One can show that this is the von Neumann algebra generated by the operators corresponding to multiplication from the left with an element <i>g</i> ∈ <i>G</i>. It is a factor (of type II<sub>1</sub>) if every non-trivial conjugacy class of <i>G</i> is infinite (for example, a non-abelian free group), and is the hyperfinite factor of type II<sub>1</sub> if in addition <i>G</i> is a union of finite subgroups (for example, the group of all permutations of the integers fixing all but a finite number of elements).</li> <li>The tensor product of two von Neumann algebras, or of a countable number with states, is a von Neumann algebra as described in the section above.</li> <li>The <a href="/wiki/Crossed_product" title="Crossed product">crossed product</a> of a von Neumann algebra by a discrete (or more generally locally compact) group can be defined, and is a von Neumann algebra. Special cases are the <b>group-measure space construction</b> of Murray and <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a> and <b>Krieger factors</b>.</li> <li>The von Neumann algebras of a measurable <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> and a measurable <a href="/wiki/Groupoid" title="Groupoid">groupoid</a> can be defined. These examples generalise von Neumann group algebras and the group-measure space construction.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=18" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Von Neumann algebras have found applications in diverse areas of mathematics like <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a>, <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, <a href="/wiki/Local_quantum_physics" class="mw-redirect" title="Local quantum physics">local quantum physics</a>, <a href="/wiki/Free_probability" title="Free probability">free probability</a>, <a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">noncommutative geometry</a>, <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a>, <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, and <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>. </p><p>For instance, <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a> provides an alternative axiomatization to probability theory. In this case the method goes by the name of <a href="/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction" title="Gelfand–Naimark–Segal construction">Gelfand–Naimark–Segal construction</a>. This is analogous to the two approaches to measure and integration, where one has the choice to construct measures of sets first and define integrals later, or construct integrals first and define set measures as integrals of characteristic functions. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/AW*-algebra" title="AW*-algebra">AW*-algebra</a> – algebraic generalization of a W*-algebra<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Central_carrier" title="Central carrier">Central carrier</a></li> <li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a> – Mathematical method in functional analysis</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Von_Neumann_algebra&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://perso.ens-lyon.fr/gaboriau/evenements/IHP-trimester/IHP-CIRM/Notes=Cyril=finite-vonNeumann.pdf">An Introduction To II1 Factors</a> ens-lyon.fr</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFConnes1978" class="citation journal cs1">Connes, A (May 1978). "On the cohomology of operator algebras". <i>Journal of Functional Analysis</i>. <b>28</b> (2): 248–253. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-1236%2878%2990088-5">10.1016/0022-1236(78)90088-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Functional+Analysis&rft.atitle=On+the+cohomology+of+operator+algebras&rft.volume=28&rft.issue=2&rft.pages=248-253&rft.date=1978-05&rft_id=info%3Adoi%2F10.1016%2F0022-1236%2878%2990088-5&rft.aulast=Connes&rft.aufirst=A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArakiWoods1968" class="citation cs2">Araki, H.; Woods, E. J. (1968), "A classification of factors", <i>Publ. Res. Inst. Math. Sci. Ser. A</i>, <b>4</b> (1): 51–130, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2977%2Fprims%2F1195195263">10.2977/prims/1195195263</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Publ.+Res.+Inst.+Math.+Sci.+Ser.+A&rft.atitle=A+classification+of+factors&rft.volume=4&rft.issue=1&rft.pages=51-130&rft.date=1968&rft_id=info%3Adoi%2F10.2977%2Fprims%2F1195195263&rft.aulast=Araki&rft.aufirst=H.&rft.au=Woods%2C+E.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span><a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0244773">0244773</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlackadar2005" class="citation cs2">Blackadar, B. (2005), <i>Operator algebras</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-28486-9" title="Special:BookSources/3-540-28486-9"><bdi>3-540-28486-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Operator+algebras&rft.pub=Springer&rft.date=2005&rft.isbn=3-540-28486-9&rft.aulast=Blackadar&rft.aufirst=B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://wolfweb.unr.edu/homepage/bruceb/Cycr.pdf"><i>corrected manuscript</i></a> <span class="cs1-format">(PDF)</span>, 2013</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=corrected+manuscript&rft.date=2013&rft_id=http%3A%2F%2Fwolfweb.unr.edu%2Fhomepage%2Fbruceb%2FCycr.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConnes1976" class="citation cs2">Connes, A. (1976), "Classification of Injective Factors", <i>Annals of Mathematics</i>, Second Series, <b>104</b> (1): 73–115, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1971057">10.2307/1971057</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1971057">1971057</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Classification+of+Injective+Factors&rft.volume=104&rft.issue=1&rft.pages=73-115&rft.date=1976&rft_id=info%3Adoi%2F10.2307%2F1971057&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1971057%23id-name%3DJSTOR&rft.aulast=Connes&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConnes1994" class="citation cs2">Connes, A. (1994), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/noncommutativege0000conn"><i>Non-commutative geometry</i></a></span>, Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-185860-X" title="Special:BookSources/0-12-185860-X"><bdi>0-12-185860-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-commutative+geometry&rft.pub=Academic+Press&rft.date=1994&rft.isbn=0-12-185860-X&rft.aulast=Connes&rft.aufirst=A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnoncommutativege0000conn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDixmier1981" class="citation cs2">Dixmier, J. (1981), <i>Von Neumann algebras</i>, 凡異出版社, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-86308-7" title="Special:BookSources/0-444-86308-7"><bdi>0-444-86308-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Von+Neumann+algebras&rft.pub=%E5%87%A1%E7%95%B0%E5%87%BA%E7%89%88%E7%A4%BE&rft.date=1981&rft.isbn=0-444-86308-7&rft.aulast=Dixmier&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span> (A translation of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDixmier1957" class="citation cs2">Dixmier, J. (1957), <i>Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann</i>, Gauthier-Villars</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Les+alg%C3%A8bres+d%27op%C3%A9rateurs+dans+l%27espace+hilbertien%3A+alg%C3%A8bres+de+von+Neumann&rft.pub=Gauthier-Villars&rft.date=1957&rft.aulast=Dixmier&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>, the first book about von Neumann algebras.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJones2003" class="citation cs2">Jones, V.F.R. (2003), <a rel="nofollow" class="external text" href="http://www.math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf"><i>von Neumann algebras</i></a> <span class="cs1-format">(PDF)</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=von+Neumann+algebras&rft.date=2003&rft.aulast=Jones&rft.aufirst=V.F.R.&rft_id=http%3A%2F%2Fwww.math.berkeley.edu%2F~vfr%2FMATH20909%2FVonNeumann2009.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>; incomplete notes from a course.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKostecki2013" class="citation cs2">Kostecki, R.P. (2013), <i>W*-algebras and noncommutative integration</i>, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1307.4818">1307.4818</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013arXiv1307.4818P">2013arXiv1307.4818P</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=W%2A-algebras+and+noncommutative+integration&rft.date=2013&rft_id=info%3Aarxiv%2F1307.4818&rft_id=info%3Abibcode%2F2013arXiv1307.4818P&rft.aulast=Kostecki&rft.aufirst=R.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcDuff1969" class="citation cs2"><a href="/wiki/Dusa_McDuff" title="Dusa McDuff">McDuff, Dusa</a> (1969), "Uncountably many II<sub>1</sub> factors", <i>Annals of Mathematics</i>, Second Series, <b>90</b> (2): 372–377, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970730">10.2307/1970730</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970730">1970730</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Uncountably+many+II%3Csub%3E1%3C%2Fsub%3E+factors&rft.volume=90&rft.issue=2&rft.pages=372-377&rft.date=1969&rft_id=info%3Adoi%2F10.2307%2F1970730&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970730%23id-name%3DJSTOR&rft.aulast=McDuff&rft.aufirst=Dusa&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurray2006" class="citation cs2">Murray, F. J. (2006), "The rings of operators papers", <i>The legacy of John von Neumann (Hempstead, NY, 1988)</i>, Proc. Sympos. Pure Math., vol. 50, Providence, RI.: Amer. Math. Soc., pp. 57–60, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-4219-6" title="Special:BookSources/0-8218-4219-6"><bdi>0-8218-4219-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+rings+of+operators+papers&rft.btitle=The+legacy+of+John+von+Neumann+%28Hempstead%2C+NY%2C+1988%29&rft.place=Providence%2C+RI.&rft.series=Proc.+Sympos.+Pure+Math.&rft.pages=57-60&rft.pub=Amer.+Math.+Soc.&rft.date=2006&rft.isbn=0-8218-4219-6&rft.aulast=Murray&rft.aufirst=F.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span> A historical account of the discovery of von Neumann algebras.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurrayvon_Neumann1936" class="citation cs2">Murray, F.J.; von Neumann, J. (1936), "On rings of operators", <i>Annals of Mathematics</i>, Second Series, <b>37</b> (1): 116–229, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968693">10.2307/1968693</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968693">1968693</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=On+rings+of+operators&rft.volume=37&rft.issue=1&rft.pages=116-229&rft.date=1936&rft_id=info%3Adoi%2F10.2307%2F1968693&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1968693%23id-name%3DJSTOR&rft.aulast=Murray&rft.aufirst=F.J.&rft.au=von+Neumann%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>. This paper gives their basic properties and the division into types I, II, and III, and in particular finds factors not of type I.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurrayvon_Neumann1937" class="citation cs2">Murray, F.J.; von Neumann, J. (1937), "On rings of operators II", <i>Trans. Amer. Math. Soc.</i>, <b>41</b> (2), American Mathematical Society: 208–248, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1989620">10.2307/1989620</a></span>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1989620">1989620</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Trans.+Amer.+Math.+Soc.&rft.atitle=On+rings+of+operators+II&rft.volume=41&rft.issue=2&rft.pages=208-248&rft.date=1937&rft_id=info%3Adoi%2F10.2307%2F1989620&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1989620%23id-name%3DJSTOR&rft.aulast=Murray&rft.aufirst=F.J.&rft.au=von+Neumann%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>. This is a continuation of the previous paper, that studies properties of the trace of a factor.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurrayvon_Neumann1943" class="citation cs2">Murray, F.J.; von Neumann, J. (1943), "On rings of operators IV", <i>Annals of Mathematics</i>, Second Series, <b>44</b> (4): 716–808, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969107">10.2307/1969107</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969107">1969107</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=On+rings+of+operators+IV&rft.volume=44&rft.issue=4&rft.pages=716-808&rft.date=1943&rft_id=info%3Adoi%2F10.2307%2F1969107&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969107%23id-name%3DJSTOR&rft.aulast=Murray&rft.aufirst=F.J.&rft.au=von+Neumann%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>. This studies when factors are isomorphic, and in particular shows that all approximately finite factors of type II<sub>1</sub> are isomorphic.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPowers1967" class="citation cs2"><a href="/wiki/Robert_T._Powers" title="Robert T. Powers">Powers, Robert T.</a> (1967), "Representations of Uniformly Hyperfinite Algebras and Their Associated von Neumann Rings", <i>Annals of Mathematics</i>, Second Series, <b>86</b> (1): 138–171, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970364">10.2307/1970364</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970364">1970364</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Representations+of+Uniformly+Hyperfinite+Algebras+and+Their+Associated+von+Neumann+Rings&rft.volume=86&rft.issue=1&rft.pages=138-171&rft.date=1967&rft_id=info%3Adoi%2F10.2307%2F1970364&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970364%23id-name%3DJSTOR&rft.aulast=Powers&rft.aufirst=Robert+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSakai1971" class="citation cs2"><a href="/wiki/Shoichiro_Sakai" title="Shoichiro Sakai">Sakai, S.</a> (1971), <i>C*-algebras and W*-algebras</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-63633-1" title="Special:BookSources/3-540-63633-1"><bdi>3-540-63633-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=C%2A-algebras+and+W%2A-algebras&rft.pub=Springer&rft.date=1971&rft.isbn=3-540-63633-1&rft.aulast=Sakai&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartz1967" class="citation cs2"><a href="/wiki/Jacob_T._Schwartz" title="Jacob T. Schwartz">Schwartz, Jacob</a> (1967), <i>W-* Algebras</i>, Gordon & Breach Publishing, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-677-00670-5" title="Special:BookSources/0-677-00670-5"><bdi>0-677-00670-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=W-%2A+Algebras&rft.pub=Gordon+%26+Breach+Publishing&rft.date=1967&rft.isbn=0-677-00670-5&rft.aulast=Schwartz&rft.aufirst=Jacob&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShtern2001" class="citation cs2">Shtern, A.I. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=von_Neumann_algebra">"von Neumann algebra"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=von+Neumann+algebra&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Shtern&rft.aufirst=A.I.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3Dvon_Neumann_algebra&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTakesaki1979" class="citation cs2">Takesaki, M. (1979), <i>Theory of Operator Algebras I, II, III</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-42248-X" title="Special:BookSources/3-540-42248-X"><bdi>3-540-42248-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Operator+Algebras+I%2C+II%2C+III&rft.pub=Springer&rft.date=1979&rft.isbn=3-540-42248-X&rft.aulast=Takesaki&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1930" class="citation cs2">von Neumann, J. (1930), "Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren", <i>Math. Ann.</i>, <b>102</b> (1): 370–427, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1930MatAn.102..685E">1930MatAn.102..685E</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01782352">10.1007/BF01782352</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121141866">121141866</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Ann.&rft.atitle=Zur+Algebra+der+Funktionaloperationen+und+Theorie+der+normalen+Operatoren&rft.volume=102&rft.issue=1&rft.pages=370-427&rft.date=1930&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121141866%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01782352&rft_id=info%3Abibcode%2F1930MatAn.102..685E&rft.aulast=von+Neumann&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>. The original paper on von Neumann algebras.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1936" class="citation cs2">von Neumann, J. (1936), "On a Certain Topology for Rings of Operators", <i>Annals of Mathematics</i>, Second Series, <b>37</b> (1): 111–115, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968692">10.2307/1968692</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968692">1968692</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=On+a+Certain+Topology+for+Rings+of+Operators&rft.volume=37&rft.issue=1&rft.pages=111-115&rft.date=1936&rft_id=info%3Adoi%2F10.2307%2F1968692&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1968692%23id-name%3DJSTOR&rft.aulast=von+Neumann&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>. This defines the ultrastrong topology.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1938" class="citation cs2">von Neumann, J. (1938), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=CM_1939__6__1_0">"On infinite direct products"</a>, <i>Compos. Math.</i>, <b>6</b>: 1–77</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Compos.+Math.&rft.atitle=On+infinite+direct+products&rft.volume=6&rft.pages=1-77&rft.date=1938&rft.aulast=von+Neumann&rft.aufirst=J.&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DCM_1939__6__1_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>. This discusses infinite tensor products of Hilbert spaces and the algebras acting on them.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1940" class="citation cs2">von Neumann, J. (1940), "On rings of operators III", <i>Annals of Mathematics</i>, Second Series, <b>41</b> (1): 94–161, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968823">10.2307/1968823</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968823">1968823</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=On+rings+of+operators+III&rft.volume=41&rft.issue=1&rft.pages=94-161&rft.date=1940&rft_id=info%3Adoi%2F10.2307%2F1968823&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1968823%23id-name%3DJSTOR&rft.aulast=von+Neumann&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>. This shows the existence of factors of type III.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1943" class="citation cs2">von Neumann, J. (1943), "On Some Algebraical Properties of Operator Rings", <i>Annals of Mathematics</i>, Second Series, <b>44</b> (4): 709–715, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969106">10.2307/1969106</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969106">1969106</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=On+Some+Algebraical+Properties+of+Operator+Rings&rft.volume=44&rft.issue=4&rft.pages=709-715&rft.date=1943&rft_id=info%3Adoi%2F10.2307%2F1969106&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969106%23id-name%3DJSTOR&rft.aulast=von+Neumann&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>. This shows that some apparently topological properties in von Neumann algebras can be defined purely algebraically.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1949" class="citation cs2">von Neumann, J. (1949), "On Rings of Operators. Reduction Theory", <i>Annals of Mathematics</i>, Second Series, <b>50</b> (2): 401–485, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969463">10.2307/1969463</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969463">1969463</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=On+Rings+of+Operators.+Reduction+Theory&rft.volume=50&rft.issue=2&rft.pages=401-485&rft.date=1949&rft_id=info%3Adoi%2F10.2307%2F1969463&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969463%23id-name%3DJSTOR&rft.aulast=von+Neumann&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>. This discusses how to write a von Neumann algebra as a sum or integral of factors.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann1961" class="citation cs2">von Neumann, John (1961), Taub, A.H. (ed.), <i>Collected Works, Volume III: Rings of Operators</i>, NY: Pergamon Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collected+Works%2C+Volume+III%3A+Rings+of+Operators&rft.place=NY&rft.pub=Pergamon+Press&rft.date=1961&rft.aulast=von+Neumann&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span>. Reprints von Neumann's papers on von Neumann algebras.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWassermann1991" class="citation cs2"><a href="/wiki/Antony_Wassermann" title="Antony Wassermann">Wassermann, A. J.</a> (1991), <a rel="nofollow" class="external text" href="http://iml.univ-mrs.fr/~wasserm/OHS.ps"><i>Operators on Hilbert space</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Operators+on+Hilbert+space&rft.date=1991&rft.aulast=Wassermann&rft.aufirst=A.+J.&rft_id=http%3A%2F%2Fiml.univ-mrs.fr%2F~wasserm%2FOHS.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVon+Neumann+algebra" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist 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title="Gelfand–Naimark theorem">Gelfand–Naimark theorem</a></li> <li><a href="/wiki/Gelfand_representation" title="Gelfand representation">Gelfand representation</a></li> <li><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li> <li><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li> <li><a href="/wiki/Spectral_theory_of_normal_C*-algebras" title="Spectral theory of normal C*-algebras">Spectral theory of normal C*-algebras</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special Elements/Operators</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isospectral" title="Isospectral">Isospectral</a></li> <li><a href="/wiki/Normal_element" title="Normal element">Normal</a> <a href="/wiki/Normal_operator" title="Normal operator">operator</a></li> <li><a href="/wiki/Self-adjoint" title="Self-adjoint">Hermitian/Self-adjoint</a> <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">operator</a></li> <li><a href="/wiki/Unitary_element" title="Unitary element">Unitary</a> <a href="/wiki/Unitary_operator" title="Unitary operator">operator</a></li> <li><a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">Unit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Krein%E2%80%93Rutman_theorem" title="Krein–Rutman theorem">Krein–Rutman theorem</a></li> <li><a href="/wiki/Normal_eigenvalue" title="Normal eigenvalue">Normal eigenvalue</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">Spectral radius</a></li> <li><a href="/wiki/Spectral_asymmetry" title="Spectral asymmetry">Spectral asymmetry</a></li> <li><a href="/wiki/Spectral_gap" title="Spectral gap">Spectral gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decomposition</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_spectrum_(functional_analysis)" title="Decomposition of spectrum (functional analysis)">Decomposition of a spectrum</a> <ul><li><a href="/wiki/Continuous_spectrum_(functional_analysis)" class="mw-redirect" title="Continuous spectrum (functional analysis)">Continuous</a></li> <li><a href="/wiki/Point_spectrum" class="mw-redirect" title="Point spectrum">Point</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Residual_spectrum" title="Spectrum (functional analysis)">Residual</a></li></ul></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Approximate_point_spectrum" title="Spectrum (functional analysis)">Approximate point</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)#Compression_spectrum" title="Spectrum (functional analysis)">Compression</a></li> <li><a href="/wiki/Direct_integral" title="Direct integral">Direct integral</a></li> <li><a href="/wiki/Discrete_spectrum_(mathematics)" title="Discrete spectrum (mathematics)">Discrete</a></li> <li><a href="/wiki/Spectral_abscissa" title="Spectral abscissa">Spectral abscissa</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spectral Theorem</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min-max theorem</a></li> <li><a href="/wiki/Positive_operator-valued_measure" class="mw-redirect" title="Positive operator-valued measure">Positive operator-valued measure</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued measure</a></li> <li><a href="/wiki/Riesz_projector" title="Riesz projector">Riesz projector</a></li> <li><a href="/wiki/Rigged_Hilbert_space" title="Rigged Hilbert space">Rigged Hilbert space</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li> <li><a href="/wiki/Spectral_theory_of_compact_operators" title="Spectral theory of compact operators">Spectral theory of compact operators</a></li> <li><a href="/wiki/Spectral_theory_of_normal_C*-algebras" title="Spectral theory of normal C*-algebras">Spectral theory of normal C*-algebras</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amenable_Banach_algebra" title="Amenable Banach algebra">Amenable Banach algebra</a></li> <li>With an <a href="/wiki/Approximate_identity" title="Approximate identity">Approximate identity</a></li> <li><a href="/wiki/Banach_function_algebra" title="Banach function algebra">Banach function algebra</a></li> <li><a href="/wiki/Disk_algebra" title="Disk algebra">Disk algebra</a></li> <li><a href="/wiki/Nuclear_C*-algebra" title="Nuclear C*-algebra">Nuclear C*-algebra</a></li> <li><a href="/wiki/Uniform_algebra" title="Uniform algebra">Uniform algebra</a></li> <li><a class="mw-selflink selflink">Von Neumann algebra</a> <ul><li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Finite-Dimensional</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alon%E2%80%93Boppana_bound" title="Alon–Boppana bound">Alon–Boppana bound</a></li> <li><a href="/wiki/Bauer%E2%80%93Fike_theorem" title="Bauer–Fike theorem">Bauer–Fike theorem</a></li> <li><a href="/wiki/Numerical_range" title="Numerical range">Numerical range</a></li> <li><a href="/wiki/Schur%E2%80%93Horn_theorem" title="Schur–Horn theorem">Schur–Horn theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dirac_spectrum" title="Dirac spectrum">Dirac spectrum</a></li> <li><a href="/wiki/Essential_spectrum" title="Essential spectrum">Essential spectrum</a></li> <li><a href="/wiki/Pseudospectrum" title="Pseudospectrum">Pseudospectrum</a></li> <li><a href="/wiki/Structure_space" class="mw-redirect" title="Structure space">Structure space</a> (<a href="/wiki/Shilov_boundary" title="Shilov boundary">Shilov boundary</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_index_group" class="mw-redirect" title="Abstract index group">Abstract index group</a></li> <li><a href="/wiki/Banach_algebra_cohomology" title="Banach algebra cohomology">Banach algebra cohomology</a></li> <li><a href="/wiki/Cohen%E2%80%93Hewitt_factorization_theorem" title="Cohen–Hewitt factorization theorem">Cohen–Hewitt factorization theorem</a></li> <li><a href="/wiki/Extensions_of_symmetric_operators" title="Extensions of symmetric operators">Extensions of symmetric operators</a></li> <li><a href="/wiki/Fredholm_theory" title="Fredholm theory">Fredholm theory</a></li> <li><a href="/wiki/Limiting_absorption_principle" title="Limiting absorption principle">Limiting absorption principle</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorems_for_operator_algebras" title="Schröder–Bernstein theorems for operator algebras">Schröder–Bernstein theorems for operator algebras</a></li> <li><a href="/wiki/Sherman%E2%80%93Takeda_theorem" title="Sherman–Takeda theorem">Sherman–Takeda theorem</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded operator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wiener_algebra" title="Wiener algebra">Wiener algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_Mathieu_operator" title="Almost Mathieu operator">Almost Mathieu operator</a></li> <li><a href="/wiki/Corona_theorem" title="Corona theorem">Corona theorem</a></li> <li><a href="/wiki/Hearing_the_shape_of_a_drum" title="Hearing the shape of a drum">Hearing the shape of a drum</a> (<a href="/wiki/Dirichlet_eigenvalue" title="Dirichlet eigenvalue">Dirichlet eigenvalue</a>)</li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Kuznetsov_trace_formula" title="Kuznetsov trace formula">Kuznetsov trace formula</a></li> <li><a href="/wiki/Lax_pair" title="Lax pair">Lax pair</a></li> <li><a href="/wiki/Proto-value_function" title="Proto-value function">Proto-value function</a></li> <li><a href="/wiki/Ramanujan_graph" title="Ramanujan graph">Ramanujan graph</a></li> <li><a href="/wiki/Rayleigh%E2%80%93Faber%E2%80%93Krahn_inequality" title="Rayleigh–Faber–Krahn inequality">Rayleigh–Faber–Krahn inequality</a></li> <li><a href="/wiki/Spectral_geometry" title="Spectral geometry">Spectral geometry</a></li> <li><a href="/wiki/Spectral_method" title="Spectral method">Spectral method</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a></li> <li><a href="/wiki/Superstrong_approximation" title="Superstrong approximation">Superstrong approximation</a></li> <li><a href="/wiki/Transfer_operator" title="Transfer operator">Transfer operator</a></li> <li><a href="/wiki/Transform_theory" title="Transform theory">Transform theory</a></li> <li><a href="/wiki/Weyl_law" title="Weyl law">Weyl law</a></li> <li><a href="/wiki/Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Hilbert_spaces" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Hilbert_space" title="Template:Hilbert space"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Hilbert_space" title="Template talk:Hilbert space"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Hilbert_space" title="Special:EditPage/Template:Hilbert space"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Hilbert_spaces" style="font-size:114%;margin:0 4em"><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product</a> and <a href="/wiki/L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> and <a href="/wiki/Prehilbert_space" class="mw-redirect" title="Prehilbert space">Prehilbert space</a></li> <li><a href="/wiki/Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a></li> <li><a href="/wiki/Orthonormal_basis" title="Orthonormal basis">Orthonormal basis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's inequality</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other results</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hilbert_projection_theorem" title="Hilbert projection theorem">Hilbert projection theorem</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a> (<a href="/wiki/Parallelogram_law#The_parallelogram_law_in_inner_product_spaces" title="Parallelogram law">Parallelogram law</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">Compact operator on Hilbert space</a></li> <li><a href="/wiki/Densely_defined_operator" title="Densely defined operator">Densely defined</a></li> <li><a href="/wiki/Sesquilinear_form#Hermitian_form" title="Sesquilinear form">Hermitian form</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">Sesquilinear form</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)"><i>C</i><sup><i>n</i></sup>(<i>K</i>) with <i>K</i> compact & <i>n</i><∞</a></li> <li><a href="/wiki/Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann <i>F</i></a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Banach_space_topics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Banach_spaces" title="Template:Banach spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Banach_spaces" title="Template talk:Banach spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Banach_spaces" title="Special:EditPage/Template:Banach spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Banach_space_topics" style="font-size:114%;margin:0 4em"><a href="/wiki/Banach_space" title="Banach space">Banach space</a> topics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of Banach spaces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Asplund_space" title="Asplund space">Asplund</a></li> <li><a href="/wiki/Banach_space" title="Banach space">Banach</a> <ul><li><a href="/wiki/List_of_Banach_spaces" title="List of Banach spaces">list</a></li></ul></li> <li><a href="/wiki/Banach_lattice" title="Banach lattice">Banach lattice</a></li> <li><a href="/wiki/Grothendieck_space" title="Grothendieck space">Grothendieck </a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a> <ul><li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></li></ul></li> <li>(<a href="/wiki/Polynomially_reflexive_space" title="Polynomially reflexive space">Polynomially</a>) <a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Riesz_space" title="Riesz space">Riesz</a></li> <li><a href="/wiki/L-semi-inner_product" title="L-semi-inner product">L-semi-inner product</a></li> <li>(<a href="/wiki/B-convex_space" title="B-convex space">B</a></li> <li><a href="/wiki/Strictly_convex_space" title="Strictly convex space">Strictly</a></li> <li><a href="/wiki/Uniformly_convex_space" title="Uniformly convex space">Uniformly</a>) convex</li> <li><a href="/wiki/Uniformly_smooth_space" title="Uniformly smooth space">Uniformly smooth</a></li> <li>(<a href="/wiki/Injective_tensor_product" title="Injective tensor product">Injective</a></li> <li><a href="/wiki/Projective_tensor_product" title="Projective tensor product">Projective</a>) <a href="/wiki/Topological_tensor_product" title="Topological tensor product">Tensor product</a> (<a href="/wiki/Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">of Hilbert spaces</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Banach spaces are:</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/F-space" title="F-space">F-space</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a> <ul><li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces#Tame_Fréchet_spaces" title="Differentiation in Fréchet spaces">tame</a></li></ul></li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a> <ul><li><a href="/wiki/Locally_convex_topological_vector_space#Definition_via_seminorms" title="Locally convex topological vector space">Seminorms</a>/<a href="/wiki/Minkowski_functional" title="Minkowski functional">Minkowski functionals</a></li></ul></li> <li><a href="/wiki/Mackey_space" title="Mackey space">Mackey</a></li> <li><a href="/wiki/Metrizable_topological_vector_space" title="Metrizable topological vector space">Metrizable</a></li> <li><a href="/wiki/Normed_space" class="mw-redirect" title="Normed space">Normed</a> <ul><li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a></li></ul></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinormed</a></li> <li><a href="/wiki/Stereotype_space" class="mw-redirect" title="Stereotype space">Stereotype</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Function space Topologies</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach%E2%80%93Mazur_compactum" title="Banach–Mazur compactum">Banach–Mazur compactum</a></li> <li><a href="/wiki/Dual_topology" title="Dual topology">Dual</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual space</a> <ul><li><a href="/wiki/Dual_norm" title="Dual norm">Dual norm</a></li></ul></li> <li><a href="/wiki/Operator_topologies" title="Operator topologies">Operator</a></li> <li><a href="/wiki/Ultraweak_topology" title="Ultraweak topology">Ultraweak</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak</a> <ul><li><a href="/wiki/Weak_topology_(polar_topology)" class="mw-redirect" title="Weak topology (polar topology)">polar</a></li> <li><a href="/wiki/Weak_operator_topology" title="Weak operator topology">operator</a></li></ul></li> <li><a href="/wiki/Strong_topology" title="Strong topology">Strong</a> <ul><li><a href="/wiki/Strong_topology_(polar_topology)" class="mw-redirect" title="Strong topology (polar topology)">polar</a></li> <li><a href="/wiki/Strong_operator_topology" title="Strong operator topology">operator</a></li></ul></li> <li><a href="/wiki/Ultrastrong_topology" title="Ultrastrong topology">Ultrastrong</a></li> <li><a href="/wiki/Topology_of_uniform_convergence" class="mw-redirect" title="Topology of uniform convergence">Uniform convergence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">Linear operators</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">Adjoint</a></li> <li><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</a> <ul><li><a href="/wiki/Bilinear_form" title="Bilinear form">form</a></li> <li><a href="/wiki/Bilinear_map" title="Bilinear map">operator</a></li> <li><a href="/wiki/Sesquilinear_form" title="Sesquilinear form">sesquilinear</a></li></ul></li> <li>(<a href="/wiki/Unbounded_operator" title="Unbounded operator">Un</a>)<a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Closed_linear_operator" title="Closed linear operator">Closed</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a> <ul><li><a href="/wiki/Compact_operator_on_Hilbert_space" title="Compact operator on Hilbert space">on Hilbert spaces</a></li></ul></li> <li>(<a href="/wiki/Discontinuous_linear_map" title="Discontinuous linear map">Dis</a>)<a href="/wiki/Continuous_linear_operator" title="Continuous linear operator">Continuous</a></li> <li><a href="/wiki/Densely_defined" class="mw-redirect" title="Densely defined">Densely defined</a></li> <li>Fredholm <ul><li><a href="/wiki/Fredholm_kernel" title="Fredholm kernel">kernel</a></li> <li><a href="/wiki/Fredholm_operator" title="Fredholm operator">operator</a></li></ul></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Linear_form" title="Linear form">Functionals</a> <ul><li><a href="/wiki/Positive_linear_functional" title="Positive linear functional">positive</a></li></ul></li> <li><a href="/wiki/Pseudo-monotone_operator" title="Pseudo-monotone operator">Pseudo-monotone</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint</a></li> <li><a href="/wiki/Strictly_singular_operator" title="Strictly singular operator">Strictly singular</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operator_theory" title="Operator theory">Operator theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebras</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a></li> <li><a href="/wiki/Operator_space" title="Operator space">Operator space</a></li> <li><a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">Spectrum</a> <ul><li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectral_radius" title="Spectral radius">radius</a></li></ul></li> <li><a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a> <ul><li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">of ODEs</a></li> <li><a href="/wiki/Spectral_theorem" title="Spectral theorem">Spectral theorem</a></li></ul></li> <li><a href="/wiki/Polar_decomposition" title="Polar decomposition">Polar decomposition</a></li> <li><a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">Singular value decomposition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anderson%E2%80%93Kadec_theorem" title="Anderson–Kadec theorem">Anderson–Kadec</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_theorem" title="Banach–Mazur theorem">Banach–Mazur</a></li> <li><a href="/wiki/Banach%E2%80%93Saks_theorem" class="mw-redirect" title="Banach–Saks theorem">Banach–Saks</a></li> <li><a href="/wiki/Open_mapping_theorem_(functional_analysis)" title="Open mapping theorem (functional analysis)">Banach–Schauder (open mapping)</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Banach–Steinhaus (Uniform boundedness)</a></li> <li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's inequality</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a></li> <li><a href="/wiki/Closed_graph_theorem" title="Closed graph theorem">Closed graph</a></li> <li><a href="/wiki/Closed_range_theorem" title="Closed range theorem">Closed range</a></li> <li><a href="/wiki/Eberlein%E2%80%93%C5%A0mulian_theorem" title="Eberlein–Šmulian theorem">Eberlein–Šmulian</a></li> <li><a href="/wiki/Freudenthal_spectral_theorem" title="Freudenthal spectral theorem">Freudenthal spectral</a></li> <li><a href="/wiki/Gelfand%E2%80%93Mazur_theorem" title="Gelfand–Mazur theorem">Gelfand–Mazur</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Goldstine_theorem" title="Goldstine theorem">Goldstine</a></li> <li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a> <ul><li><a href="/wiki/Hyperplane_separation_theorem" title="Hyperplane separation theorem">hyperplane separation</a></li></ul></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Invariant_subspace_problem#Known_special_cases" title="Invariant subspace problem">Lomonosov's invariant subspace</a></li> <li><a href="/wiki/Mackey%E2%80%93Arens_theorem" title="Mackey–Arens theorem">Mackey–Arens</a></li> <li><a href="/wiki/Mazur%27s_lemma" title="Mazur's lemma">Mazur's lemma</a></li> <li><a href="/wiki/M._Riesz_extension_theorem" title="M. Riesz extension theorem">M. Riesz extension</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Riesz%27s_lemma" title="Riesz's lemma">Riesz's lemma</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson-Ursescu</a></li> <li><a href="/wiki/Schauder_fixed-point_theorem" title="Schauder fixed-point theorem">Schauder fixed-point</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Analysis</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a></li> <li><a href="/wiki/Banach_manifold" title="Banach manifold">Banach manifold</a> <ul><li><a href="/wiki/Banach_bundle" title="Banach bundle">bundle</a></li></ul></li> <li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Convex_series" title="Convex series">Convex series</a></li> <li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li> <li><a href="/wiki/Derivative" title="Derivative">Derivatives</a> <ul><li><a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet</a></li> <li><a href="/wiki/Gateaux_derivative" title="Gateaux derivative">Gateaux</a></li> <li><a href="/wiki/Functional_derivative" title="Functional derivative">functional</a></li> <li><a href="/wiki/Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">holomorphic</a></li> <li><a href="/wiki/Quasi-derivative" title="Quasi-derivative">quasi</a></li></ul></li> <li><a href="/wiki/Integral" title="Integral">Integrals</a> <ul><li><a href="/wiki/Bochner_integral" title="Bochner integral">Bochner</a></li> <li><a href="/wiki/Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">Gelfand–Pettis</a></li> <li><a href="/wiki/Regulated_integral" title="Regulated integral">regulated</a></li> <li><a href="/wiki/Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">weak</a></li></ul></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a> <ul><li><a href="/wiki/Borel_functional_calculus" title="Borel functional calculus">Borel</a></li> <li><a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">continuous</a></li> <li><a href="/wiki/Holomorphic_functional_calculus" title="Holomorphic functional calculus">holomorphic</a></li></ul></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measures</a> <ul><li><a href="/wiki/Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">Lebesgue</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li> <li><a href="/wiki/Vector_measure" title="Vector measure">Vector</a></li></ul></li> <li><a href="/wiki/Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="/wiki/Strongly_measurable_functions" class="mw-redirect" title="Strongly measurable functions">Strongly</a> measurable function</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of sets</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolutely_convex_set" title="Absolutely convex set">Absolutely convex</a></li> <li><a href="/wiki/Absorbing_set" title="Absorbing set">Absorbing</a></li> <li><a href="/wiki/Affine_space" title="Affine space">Affine</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced/Circled</a></li> <li><a href="/wiki/Bounded_set_(topological_vector_space)" title="Bounded set (topological vector space)">Bounded</a></li> <li><a href="/wiki/Convex_set" title="Convex set">Convex</a></li> <li><a href="/wiki/Convex_cone" title="Convex cone">Convex cone <span style="font-size:85%;">(subset)</span></a></li> <li><a href="/wiki/Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H<i>x</i>), and (Hw<i>x</i>))</li> <li><a href="/wiki/Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Linear cone <span style="font-size:85%;">(subset)</span></a></li> <li><a href="/wiki/Radial_set" title="Radial set">Radial</a></li> <li><a href="/wiki/Star_domain" title="Star domain">Radially convex/Star-shaped</a></li> <li><a href="/wiki/Symmetric_set" title="Symmetric set">Symmetric</a></li> <li><a href="/wiki/Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subsets / set operations</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_hull" title="Affine hull">Affine hull</a></li> <li>(<a href="/wiki/Algebraic_interior#Relative_algebraic_interior" title="Algebraic interior">Relative</a>) <a href="/wiki/Algebraic_interior" title="Algebraic interior">Algebraic interior (core)</a></li> <li><a href="/wiki/Bounding_point" title="Bounding point">Bounding points</a></li> <li><a href="/wiki/Convex_hull" title="Convex hull">Convex hull</a></li> <li><a href="/wiki/Extreme_point" title="Extreme point">Extreme point</a></li> <li><a href="/wiki/Interior_(topology)" title="Interior (topology)">Interior</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Minkowski_addition" title="Minkowski addition">Minkowski addition</a></li> <li><a href="/wiki/Polar_set" title="Polar set">Polar</a></li> <li>(<a href="/wiki/Algebraic_interior#Quasi_relative_interior" title="Algebraic interior">Quasi</a>) <a href="/wiki/Algebraic_interior#Relative_interior" title="Algebraic interior">Relative interior</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Template:ListOfBanachSpaces" class="mw-redirect" title="Template:ListOfBanachSpaces">Examples</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_continuity" title="Absolute continuity">Absolute continuity <i>AC</i></a></li> <li><a href="/wiki/Ba_space" title="Ba space"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ba(\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>a</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ba(\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58fe61351e3531b14043fa2d09e98c2437bd1a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.715ex; height:2.843ex;" alt="{\displaystyle ba(\Sigma )}"></span></a></li> <li><a href="/wiki/C_space" title="C space">c space</a></li> <li><a href="/wiki/BK-space" title="BK-space">Banach coordinate <i>BK</i></a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{p,q}^{s}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9919cf78ad095c237169772d2b27a37bfbef1b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.524ex; height:3.009ex;" alt="{\displaystyle B_{p,q}^{s}(\mathbb {R} )}"></span></a></li> <li><a href="/wiki/Birnbaum%E2%80%93Orlicz_space" class="mw-redirect" title="Birnbaum–Orlicz space">Birnbaum–Orlicz</a></li> <li><a href="/wiki/Bounded_variation" title="Bounded variation">Bounded variation <i>BV</i></a></li> <li><a href="/wiki/Bs_space" title="Bs space">Bs space</a></li> <li><a href="/wiki/Continuous_functions_on_a_compact_Hausdorff_space" title="Continuous functions on a compact Hausdorff space">Continuous <i>C(K)</i> with <i>K</i> compact Hausdorff</a></li> <li><a href="/wiki/Hardy_space" title="Hardy space">Hardy H<sup><i>p</i></sup></a></li> <li><a href="/wiki/Hilbert_space#Definition" title="Hilbert space">Hilbert <i>H</i></a></li> <li><a href="/wiki/Morrey%E2%80%93Campanato_space" title="Morrey–Campanato space">Morrey–Campanato <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\lambda ,p}(\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\lambda ,p}(\Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8af58fa038369c3ec6386c6656aab82825e372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.545ex; height:3.176ex;" alt="{\displaystyle L^{\lambda ,p}(\Omega )}"></span></a></li> <li><a href="/wiki/Sequence_space#ℓp_spaces" title="Sequence space"><i>ℓ<sup>p</sup></i></a> <ul><li><a href="/wiki/L-infinity#Sequence_space" title="L-infinity"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8348195cf09473662c6f59e6717722a6fc01d0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.845ex; height:2.343ex;" alt="{\displaystyle \ell ^{\infty }}"></span></a></li></ul></li> <li><a href="/wiki/Lp_space" title="Lp space"><i>L<sup>p</sup></i></a> <ul><li><a href="/wiki/L-infinity#Function_space" title="L-infinity"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"></span></a></li> <li><a href="/wiki/Lp_space#Weighted_Lp_spaces" title="Lp space">weighted</a></li></ul></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\left(\mathbb {R} ^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\left(\mathbb {R} ^{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0465acd58a0f31e32b095aed742d9ccc6331369c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.592ex; height:2.843ex;" alt="{\displaystyle S\left(\mathbb {R} ^{n}\right)}"></span></a></li> <li><a href="/wiki/Segal%E2%80%93Bargmann_space" title="Segal–Bargmann space">Segal–Bargmann <i>F</i></a></li> <li><a href="/wiki/Sequence_space" title="Sequence space">Sequence space</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev W<sup><i>k,p</i></sup></a> <ul><li><a href="/wiki/Sobolev_inequality" title="Sobolev inequality">Sobolev inequality</a></li></ul></li> <li><a href="/wiki/Triebel%E2%80%93Lizorkin_space" title="Triebel–Lizorkin space">Triebel–Lizorkin</a></li> <li><a href="/wiki/Wiener_amalgam_space" title="Wiener amalgam space">Wiener amalgam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(X,L^{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(X,L^{p})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b37b1dc9714960c525cb561a4828f41feb5844ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:2.843ex;" alt="{\displaystyle W(X,L^{p})}"></span></a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Mathematical formulation of quantum mechanics</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Ordinary Differential Equations (ODEs)</a></li> <li><a href="/wiki/Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Functional_analysis_(topics_–_glossary)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functional_analysis" title="Template:Functional analysis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functional_analysis" title="Template talk:Functional analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functional_analysis" title="Special:EditPage/Template:Functional analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Functional_analysis_(topics_–_glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="/wiki/List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="/wiki/Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li> <li><a href="/wiki/Nuclear_space" title="Nuclear space">Nuclear</a></li> <li><a href="/wiki/Orlicz_space" title="Orlicz space">Orlicz</a></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev</a></li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual</a> (<a href="/wiki/Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="/wiki/Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li> <li><a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Separable_space" title="Separable space">Separable</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min–max</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li> <li><a class="mw-selflink selflink">Von Neumann algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li> <li><a href="/wiki/Mahler%27s_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hardy_space" title="Hardy space">Hardy space</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a></li> <li><a href="/wiki/Integral_operator" title="Integral operator">Integral operator</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_property" title="Approximation property">Approximation property</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced set</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak topology</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur 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