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<title>Gelfand-Transformation</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Gelfand-Transformation</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Die Gelfand-Transformation (nach <a href="Israel_Gelfand" class="mw-redirect" title="Israel Gelfand">Israel Gelfand</a>) ist das wichtigste Instrument in der Theorie der kommutativen Banach-Algebren. Sie bildet eine <a href="Kommutativgesetz" title="Kommutativgesetz">kommutative</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">-<a href="Banachalgebra" title="Banachalgebra">Banachalgebra</a> A in eine Algebra stetiger Funktionen ab. Jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> wird eine stetige Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}\colon X\to {\mathbb {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {a}}\colon X\to {\mathbb {C} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d837494b26d6734bd69f5468ecf51e879efc3e08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.536ex; height:2.176ex;" alt="{\displaystyle {\hat {a}}\colon X\to {\mathbb {C} }}" loading="lazy"> zugeordnet, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ein geeigneter <a href="Lokalkompakter_Raum" title="Lokalkompakter Raum">lokalkompakter</a> <a href="Hausdorff-Raum" title="Hausdorff-Raum">Hausdorff-Raum</a> ist. Die Zuordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mapsto {\hat {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle a\mapsto {\hat {a}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4836f8d8cc8695355419a6d13bfa32d65560c372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.074ex; height:2.176ex;" alt="{\displaystyle a\mapsto {\hat {a}}}" loading="lazy"> ist dabei ein stetiger Algebren-<a href="Homomorphismus" title="Homomorphismus">Homomorphismus</a>.

<div class="mw-heading mw-heading2"><h2 id="Motivation,_Gelfand-Raum">Motivation, Gelfand-Raum</h2></div>

Betrachtet man eine kommutative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">-<a href="Banachalgebra" title="Banachalgebra">Banachalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> nur als <a href="Normierter_Raum" title="Normierter Raum">normierten Raum</a> mit <a href="Dualraum" title="Dualraum">Dualraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}">

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<annotation encoding="application/x-tex">{\displaystyle A'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98a12527148d6ed68adc91d9b419eb4b92d58ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}" loading="lazy"> und Bidualraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A''=(A')'}">

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<annotation encoding="application/x-tex">{\displaystyle A''=(A')'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e566fdb92ea02d1e9fcd8a0fbd6422a7f6a1c7e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.901ex; height:3.009ex;" alt="{\displaystyle A''=(A')'}" loading="lazy">, so lassen sich die Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> folgendermaßen auf stetige Funktionen abbilden: Man ordne jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}\colon A'\rightarrow {\mathbb {C} },{\hat {a}}(\varphi ):=\varphi (a)}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {a}}\colon A'\rightarrow {\mathbb {C} },{\hat {a}}(\varphi ):=\varphi (a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4f622e6178d210bddbfc0ebb380c9aa4cd2f1e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.881ex; height:3.009ex;" alt="{\displaystyle {\hat {a}}\colon A'\rightarrow {\mathbb {C} },{\hat {a}}(\varphi ):=\varphi (a)}" loading="lazy"> zu. Dabei handelt es sich um die bekannte isometrische Einbettung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> in den Bidualraum, denn jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {a}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/233a5bda7c263f804b049be11c03d12e3d65103a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.176ex;" alt="{\displaystyle {\hat {a}}}" loading="lazy"> ist ein Element aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A''}">

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<annotation encoding="application/x-tex">{\displaystyle A''}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72d00703267032c414d92a0473cad02eefbb8125.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle A''}" loading="lazy">. Jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {a}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/233a5bda7c263f804b049be11c03d12e3d65103a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.176ex;" alt="{\displaystyle {\hat {a}}}" loading="lazy"> ist auch stetig. Dabei erweist sich die <a href="Normtopologie" title="Normtopologie">Normtopologie</a> als unnötig stark. Aus diesem Grunde betrachtet man auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}">

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<annotation encoding="application/x-tex">{\displaystyle A'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98a12527148d6ed68adc91d9b419eb4b92d58ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}" loading="lazy"> die <a href="Schwach-*-Topologie" title="Schwach-*-Topologie">schwach-*-Topologie</a>, diese ist gerade definiert als die gröbste Topologie, die alle Abbildungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {a}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/233a5bda7c263f804b049be11c03d12e3d65103a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.176ex;" alt="{\displaystyle {\hat {a}}}" loading="lazy"> stetig macht.

Wenden wir uns wieder der Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> zu, so müssen wir feststellen, dass die Zuordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mapsto {\hat {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle a\mapsto {\hat {a}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4836f8d8cc8695355419a6d13bfa32d65560c372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.074ex; height:2.176ex;" alt="{\displaystyle a\mapsto {\hat {a}}}" loading="lazy"> kein <a href="Homomorphismus" title="Homomorphismus">Homomorphismus</a> ist; sie ist nicht multiplikativ, d. h. es gilt nicht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {ab}}={\hat {a}}{\hat {b}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {ab}}={\hat {a}}{\hat {b}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/620e0f303cadcdeb9157f507fdd1f0a92bd9e9eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.01ex; width:7.825ex; height:3.009ex;" alt="{\displaystyle {\widehat {ab}}={\hat {a}}{\hat {b}}}" loading="lazy">. Dazu müsste nämlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {ab}}(\varphi )={\hat {a}}(\varphi ){\hat {b}}(\varphi )}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {ab}}(\varphi )={\hat {a}}(\varphi ){\hat {b}}(\varphi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c4203707935b089f567edbbca4346740b9cd381.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.01ex; width:17.813ex; height:3.509ex;" alt="{\displaystyle {\widehat {ab}}(\varphi )={\hat {a}}(\varphi ){\hat {b}}(\varphi )}" loading="lazy"> und damit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (ab)=\varphi (a)\varphi (b)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (ab)=\varphi (a)\varphi (b)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e453ad6ee6b5667984794ea900655da8cfc698b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.541ex; height:2.843ex;" alt="{\displaystyle \varphi (ab)=\varphi (a)\varphi (b)}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \in A'}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi \in A'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/465a611d521a9fcc782b537e8a138cd9ee27bcc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.789ex; height:3.009ex;" alt="{\displaystyle \varphi \in A'}" loading="lazy"> gelten, aber ein lineares Funktional ist in der Regel nicht multiplikativ. Diese Beobachtung gibt aber einen Hinweis, wie man einen Homomorphismus der gewünschten Art konstruieren kann. Man verwendet statt ganz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}">

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<annotation encoding="application/x-tex">{\displaystyle A'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98a12527148d6ed68adc91d9b419eb4b92d58ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}" loading="lazy"> nur die multiplikativen Funktionale in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}">

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<annotation encoding="application/x-tex">{\displaystyle A'}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98a12527148d6ed68adc91d9b419eb4b92d58ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}" loading="lazy">, und genau das ist die Gelfand-Transformation.

Wir setzen daher <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}:=\{\varphi \in A';\varphi \,{\text{multiplikativ}},\varphi \not =0\}}">

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<mtext>multiplikativ</mtext>

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<annotation encoding="application/x-tex">{\displaystyle X_{A}:=\{\varphi \in A';\varphi \,{\text{multiplikativ}},\varphi \not =0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/522ce7ecfcda8360576bd6d64defcfbfc1aeb73d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.186ex; height:3.009ex;" alt="{\displaystyle X_{A}:=\{\varphi \in A';\varphi \,{\text{multiplikativ}},\varphi \not =0\}}" loading="lazy">. Diese Menge nennt man das Spektrum (Gelfand-Spektrum) von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> oder auch den Gelfand-Raum von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">. Man beachte, dass der Nullhomomorphismus herausgenommen wurde.

Es gibt Banach-Algebren mit leerem Spektrum, z. B. eine Banachalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> mit der Nullmultiplikation, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b=0}">

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<annotation encoding="application/x-tex">{\displaystyle a\cdot b=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4071e9cfdc140c0e0c92a814cd4f758d2a7ed3de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.167ex; height:2.176ex;" alt="{\displaystyle a\cdot b=0}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in A}">

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<annotation encoding="application/x-tex">{\displaystyle a,b\in A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/923af80638e57592b33fa715fb6c23651729fb73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.845ex; height:2.509ex;" alt="{\displaystyle a,b\in A}" loading="lazy">. Ist aber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}\not =\emptyset }">

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<annotation encoding="application/x-tex">{\displaystyle X_{A}\not =\emptyset }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fe14b5e1e9778560e3a477bab8f45f4e357a0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.65ex; height:2.843ex;" alt="{\displaystyle X_{A}\not =\emptyset }" loading="lazy">, so kann man zeigen, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{A}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c3b750756d33b2e60efd90c59a059d0dae129cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle X_{A}}" loading="lazy"> mit der <a href="Teilraumtopologie" title="Teilraumtopologie">relativen</a> schwach-*-Topologie ein <a href="Lokalkompakter_Raum" title="Lokalkompakter Raum">lokalkompakter</a> <a href="Hausdorff-Raum" title="Hausdorff-Raum">Hausdorff-Raum</a> ist.

Nach obigen Ausführungen ist

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}:A\rightarrow C_{0}(X_{A}),\,a\mapsto {\hat {a}},\,{\hat {a}}(\varphi )=\varphi (a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57794687c42be34df605f4c9345ae1ce722cff7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.734ex; height:2.843ex;" alt="{\displaystyle {\mathcal {G}}:A\rightarrow C_{0}(X_{A}),\,a\mapsto {\hat {a}},\,{\hat {a}}(\varphi )=\varphi (a)}" loading="lazy"></dd></dl>

ein <a href="Stetige_Funktion" title="Stetige Funktion">stetiger</a> <a href="Homomorphismus" title="Homomorphismus">Homomorphismus</a> mit <a href="Norm_(Mathematik)" title="Norm (Mathematik)">Norm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle \leq 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6198e3fc967f992f3ee59bf4b37f401e4283c3db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.616ex; height:2.343ex;" alt="{\displaystyle \leq 1}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}(X_{A})}">

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<annotation encoding="application/x-tex">{\displaystyle C_{0}(X_{A})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/346db1f70537dde3ae20fe4c9d95906492c2b595.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.914ex; height:2.843ex;" alt="{\displaystyle C_{0}(X_{A})}" loading="lazy"> ist dabei die Algebra der stetigen, <a href="Komplexwertige_Funktion" title="Komplexwertige Funktion">komplexwertigen Funktionen</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{A}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c3b750756d33b2e60efd90c59a059d0dae129cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle X_{A}}" loading="lazy">, die <a href="Lokalkompakter_Raum#Verschwinden_im_Unendlichen" title="Lokalkompakter Raum">im Unendlichen verschwinden</a>. Dieser Homomorphismus heißt Gelfand-Transformation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {a}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {a}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/233a5bda7c263f804b049be11c03d12e3d65103a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.176ex;" alt="{\displaystyle {\hat {a}}}" loading="lazy"> nennt man die Gelfand-Transformierte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Beispiel_C0(Z)">Beispiel C0(Z)</h2></div>

Sei Z ein lokalkompakter Hausdorffraum und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=C_{0}(Z)}">

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<annotation encoding="application/x-tex">{\displaystyle A=C_{0}(Z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea682bb817af65b259b569e41a38f7cfb476fdc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.047ex; height:2.843ex;" alt="{\displaystyle A=C_{0}(Z)}" loading="lazy">, so ist A bereits eine Algebra von der Art, auf die die Gelfand-Transformation abbildet. Um die Gelfand-Transformation für diesen Fall zu bestimmen, müssen wir uns einen Überblick über die multiplikativen Funktionale auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> verschaffen. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in Z}">

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<annotation encoding="application/x-tex">{\displaystyle z\in Z}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3472db536f24864ab268d42ae277fcfc90d2998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle z\in Z}" loading="lazy">, so ist die Punktauswertung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z}:A\rightarrow {\mathbb {C} },\,\delta _{z}(f):=f(z)}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{z}:A\rightarrow {\mathbb {C} },\,\delta _{z}(f):=f(z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19578212034dc4cf070b16f874c89c65e85797d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.471ex; height:2.843ex;" alt="{\displaystyle \delta _{z}:A\rightarrow {\mathbb {C} },\,\delta _{z}(f):=f(z)}" loading="lazy"> offenbar ein multiplikatives Funktional, und man kann zeigen, dass dies bereits alle sind, d. h., dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}=\{\delta _{z};z\in Z\}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{A}=\{\delta _{z};z\in Z\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaeb2de7f206857cc26b770bde6350e3bd4d2ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.49ex; height:2.843ex;" alt="{\displaystyle X_{A}=\{\delta _{z};z\in Z\}}" loading="lazy"> gilt. Z kann also mittels der Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto \delta _{z}}">

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<annotation encoding="application/x-tex">{\displaystyle z\mapsto \delta _{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd01d31cee9265b0654256c5934d50d00c7d0b61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.736ex; height:2.676ex;" alt="{\displaystyle z\mapsto \delta _{z}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{A}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c3b750756d33b2e60efd90c59a059d0dae129cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle X_{A}}" loading="lazy"> identifiziert werden, zumindest als Menge. Man kann zeigen, dass diese Abbildung sogar ein <a href="Hom%C3%B6omorphismus" title="Homöomorphismus">Homöomorphismus</a> ist, so dass man Z und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{A}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c3b750756d33b2e60efd90c59a059d0dae129cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle X_{A}}" loading="lazy"> auch als <a href="Topologischer_Raum" title="Topologischer Raum">topologische Räume</a> identifizieren kann. In diesem Fall ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}:A\rightarrow C_{0}(X_{A})=C_{0}(Z)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}:A\rightarrow C_{0}(X_{A})=C_{0}(Z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1bf885e7d9c670aa6165fb4fa818dc75a8493d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.905ex; height:2.843ex;" alt="{\displaystyle {\mathcal {G}}:A\rightarrow C_{0}(X_{A})=C_{0}(Z)}" loading="lazy"> nichts weiter als die Identität. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=C_{0}(Z)}">

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<annotation encoding="application/x-tex">{\displaystyle A=C_{0}(Z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea682bb817af65b259b569e41a38f7cfb476fdc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.047ex; height:2.843ex;" alt="{\displaystyle A=C_{0}(Z)}" loading="lazy"> bietet die Gelfand-Transformation nichts Neues.

<div class="mw-heading mw-heading2"><h2 id="Beispiel_L1(ℝ)">Beispiel L1(ℝ)</h2></div>

Der Banachraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=L^{1}(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle A=L^{1}(\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fe08728f54ddbd8876300567102b38c6daf6d53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.966ex; height:3.176ex;" alt="{\displaystyle A=L^{1}(\mathbb {R} )}" loading="lazy"> ist mit der <a href="Faltung_(Mathematik)" title="Faltung (Mathematik)">Faltung</a> als Multiplikation und der <a href="Lp-Raum" title="Lp-Raum">1-Norm</a> eine kommutative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">-Banachalgebra. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\in L^{1}(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle f,g\in L^{1}(\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b80f9acc3deaf03a4fa2fca88f56a44ec5aeedc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.394ex; height:3.176ex;" alt="{\displaystyle f,g\in L^{1}(\mathbb {R} )}" loading="lazy"> gilt dabei

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<annotation encoding="application/x-tex">{\displaystyle f*g(t):=\int _{-\infty }^{\infty }f(s)g(t-s)\mathrm {d} s}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/098c592153a86e5ef622ccddfa7ef4eb90c5f78c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.074ex; height:6.009ex;" alt="{\displaystyle f*g(t):=\int _{-\infty }^{\infty }f(s)g(t-s)\mathrm {d} s}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \|f\|_{1}=\int _{-\infty }^{\infty }|f(s)|\mathrm {d} s}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/106ab6acf0f5efe0d4ffaf74c4ef38f18a14d917.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.445ex; height:6.009ex;" alt="{\displaystyle \|f\|_{1}=\int _{-\infty }^{\infty }|f(s)|\mathrm {d} s}" loading="lazy"></dd></dl>

Wie sehen die multiplikativen Funktionale auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> aus? Die Punktauswertungen des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}(Z)}">

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<annotation encoding="application/x-tex">{\displaystyle C_{0}(Z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5253d56afd187b677425f27fae58df25257a524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.206ex; height:2.843ex;" alt="{\displaystyle C_{0}(Z)}" loading="lazy">-Beispiels kommen nicht in Frage, denn für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74c288d1089f1ec85b01b4de25c441fc792bd2d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}" loading="lazy">-Funktionen ist der Funktionswert an einer Stelle gar nicht definiert. Man kann zeigen, dass für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in {\mathbb {R} }}">

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<annotation encoding="application/x-tex">{\displaystyle z\in {\mathbb {R} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4fbf579c137c083acded8b79b0e3869d3f8b885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in {\mathbb {R} }}" loading="lazy"> durch

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{z}(f):=\int _{-\infty }^{\infty }f(t)e^{-itz}\mathrm {d} t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9680978bbca4a759ec7eb15e0dc5f572ba331bcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.773ex; height:6.009ex;" alt="{\displaystyle \varphi _{z}(f):=\int _{-\infty }^{\infty }f(t)e^{-itz}\mathrm {d} t}" loading="lazy"></dd></dl>

ein multiplikatives Funktional auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=L^{1}(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle A=L^{1}(\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fe08728f54ddbd8876300567102b38c6daf6d53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.966ex; height:3.176ex;" alt="{\displaystyle A=L^{1}(\mathbb {R} )}" loading="lazy"> erklärt ist, und dass umgekehrt jedes multiplikative Funktional von dieser Form ist. Es gilt also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}=\{\varphi _{z};z\in \mathbb {R} \}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{A}=\{\varphi _{z};z\in \mathbb {R} \}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a533bc14e58dd4e917611fb6616aa3c97b73cc0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.975ex; height:2.843ex;" alt="{\displaystyle X_{A}=\{\varphi _{z};z\in \mathbb {R} \}}" loading="lazy"> und man kann weiter zeigen, dass die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto \varphi _{z}}">

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<annotation encoding="application/x-tex">{\displaystyle z\mapsto \varphi _{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eaeb1b73c4618942ddd664988ee3210cc9ee40d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.224ex; height:2.343ex;" alt="{\displaystyle z\mapsto \varphi _{z}}" loading="lazy"> ein <a href="Hom%C3%B6omorphismus" title="Homöomorphismus">Homöomorphismus</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c3b750756d33b2e60efd90c59a059d0dae129cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle X_{A}}" loading="lazy"> ist. Identifiziert man daher <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {R} }}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathbb {R} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c35d4fb24ff006be5c264f4a3cf7760653a06b30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle {\mathbb {R} }}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c3b750756d33b2e60efd90c59a059d0dae129cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle X_{A}}" loading="lazy"> mittels dieser Abbildung, so hat die Gelfand-Transformation die Gestalt:

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

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</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}:L^{1}(\mathbb {R} )\rightarrow C_{0}(\mathbb {R} ),\,f\mapsto {\hat {f}},\,{\hat {f}}(z)=\int _{-\infty }^{\infty }f(t)e^{-itz}\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c08acd05b18a76b3255e5dd6573ac5945524525.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.818ex; height:6.009ex;" alt="{\displaystyle {\mathcal {G}}:L^{1}(\mathbb {R} )\rightarrow C_{0}(\mathbb {R} ),\,f\mapsto {\hat {f}},\,{\hat {f}}(z)=\int _{-\infty }^{\infty }f(t)e^{-itz}\mathrm {d} t}" loading="lazy">.</dd></dl>

Die Gelfand-Transformation erweist sich damit als eine Abstraktion der <a href="Fourier-Transformation" title="Fourier-Transformation">Fourier-Transformation</a>.

<div class="mw-heading mw-heading2"><h2 id="Beispiel_'holomorphe_Fortsetzung'">Beispiel 'holomorphe Fortsetzung'</h2></div>

Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"> die Kreislinie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{z\in \mathbb {C} ;|z|=1\}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{z\in \mathbb {C} ;|z|=1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cad71f303ac2ca50f70cda97ebc4786b45ff0dcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.608ex; height:2.843ex;" alt="{\displaystyle \{z\in \mathbb {C} ;|z|=1\}}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}(Z)}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{0}(Z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5253d56afd187b677425f27fae58df25257a524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.206ex; height:2.843ex;" alt="{\displaystyle C_{0}(Z)}" loading="lazy"> eine kommutative Banachalgebra mit 1.

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> die <a href="Diskalgebra" title="Diskalgebra">Diskalgebra</a>, das heißt die Unteralgebra aller Funktionen, die eine <a href="Analytische_Fortsetzung" title="Analytische Fortsetzung">holomorphe Fortsetzung</a> ins Innere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{z\in \mathbb {C} ;|z|<1\}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{z\in \mathbb {C} ;|z|<1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/607a2b09021766bde7ffd679422bb54bfc99535f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.608ex; height:2.843ex;" alt="{\displaystyle \{z\in \mathbb {C} ;|z|<1\}}" loading="lazy"> besitzen. Mit ein wenig <a href="Funktionentheorie" title="Funktionentheorie">Funktionentheorie</a> (<a href="Maximumprinzip_(Mathematik)" title="Maximumprinzip (Mathematik)">Maximumprinzip</a>) zeigt man, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> eine Unter-Banachalgebra von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}(Z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{0}(Z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5253d56afd187b677425f27fae58df25257a524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.206ex; height:2.843ex;" alt="{\displaystyle C_{0}(Z)}" loading="lazy"> ist. Wie sehen die multiplikativen Funktionale auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> aus? Zunächst sind die Punktauswertungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z},|z|=1}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{z},|z|=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b476155b1fe0ad1e83b8104a9e7f62a9df9ac61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.711ex; height:2.843ex;" alt="{\displaystyle \delta _{z},|z|=1}" loading="lazy">, die ja schon multiplikative Funktionale auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0}(Z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{0}(Z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5253d56afd187b677425f27fae58df25257a524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.206ex; height:2.843ex;" alt="{\displaystyle C_{0}(Z)}" loading="lazy"> sind, natürlich auch multiplikative Funktionale auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">. Es gibt aber weitere. Da die holomorphe Fortsetzung einer Funktion ins Innere eindeutig ist, sind auch alle Punktauswertungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z},|z|<1}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{z},|z|<1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebf9947e5c2dff0653511ac7a37a9065c116cd7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.711ex; height:2.843ex;" alt="{\displaystyle \delta _{z},|z|<1}" loading="lazy">, multiplikative Funktionale auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">. Man zeigt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}=\{\delta _{z};|z|\leq 1\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{A}=\{\delta _{z};|z|\leq 1\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5aed63bda614ac954dd3796cacd66093cc4b2203.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.523ex; height:2.843ex;" alt="{\displaystyle X_{A}=\{\delta _{z};|z|\leq 1\}}" loading="lazy"> und dass man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{A}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c3b750756d33b2e60efd90c59a059d0dae129cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle X_{A}}" loading="lazy"> mittels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto \delta _{z}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\mapsto \delta _{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd01d31cee9265b0654256c5934d50d00c7d0b61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.736ex; height:2.676ex;" alt="{\displaystyle z\mapsto \delta _{z}}" loading="lazy"> auch topologisch mit der Kreisfläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\{z\in \mathbb {C} ;|z|\leq 1\}}">

</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\{z\in \mathbb {C} ;|z|\leq 1\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c57bdeee253c89318fb30ae30e98dc20ef22a0e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.773ex; height:2.843ex;" alt="{\displaystyle K=\{z\in \mathbb {C} ;|z|\leq 1\}}" loading="lazy"> identifizieren kann. In diesem Beispiel ist daher

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<msub>

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</msub>

<mover>

</mover>

</mrow>

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}\colon A\rightarrow C_{0}(K),\,f\mapsto {\hat {f}},\,{\hat {f}}=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1682d71ceb9d057e8ed397e9de124a3121c5845a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.961ex; height:3.343ex;" alt="{\displaystyle {\mathcal {G}}\colon A\rightarrow C_{0}(K),\,f\mapsto {\hat {f}},\,{\hat {f}}=}" loading="lazy"> holomorphe Fortsetzung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">,</dd></dl>

d. h. die Gelfand-Transformation spielt hier die Rolle eines Fortsetzungsoperators.

<div class="mw-heading mw-heading2"><h2 id="Bedeutung">Bedeutung</h2></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> eine <a href="Kommutativgesetz" title="Kommutativgesetz">kommutative</a> <a href="C*-Algebra" title="C*-Algebra">C*-Algebra</a>, so ist die Gelfand-Transformation der <a href="Isometrie" title="Isometrie">isometrische</a> <a href="Isomorphismus" title="Isomorphismus">Isomorphismus</a> aus dem <a href="Satz_von_Gelfand-Neumark" title="Satz von Gelfand-Neumark">Satz von Gelfand-Neumark</a> für kommutative C*-Algebren.

Das ist der Ausgangspunkt der <a href="Spektralma%C3%9F" title="Spektralmaß">Spektraltheorie</a>.

Das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}({\mathbb {R} })}">

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle L^{1}({\mathbb {R} })}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3488e531b9d25a46b737dc41af00b8de723f363a.svg" 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^{1}({\mathbb {R} })}" loading="lazy">-Beispiel verallgemeinert sich auf lokalkompakte, <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. Der Gelfand-Raum von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(G)}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle L^{1}(G)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e10e36f941f2b8fec8f9c058ebefd8ac69609b97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.273ex; height:3.176ex;" alt="{\displaystyle L^{1}(G)}" loading="lazy"> wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {G}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {G}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/481aa9ebb6efdec53b01da30a4f1ca2ce77873c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.843ex;" alt="{\displaystyle {\hat {G}}}" loading="lazy"> bezeichnet und kann wieder mit einer Gruppenstruktur versehen werden. Man nennt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {G}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {G}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/481aa9ebb6efdec53b01da30a4f1ca2ce77873c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.843ex;" alt="{\displaystyle {\hat {G}}}" loading="lazy"> dann die <a href="Pontrjagin-Dualit%C3%A4t" title="Pontrjagin-Dualität">Dualgruppe</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. Das ist ein Ausgangspunkt der <a href="Harmonische_Analyse" title="Harmonische Analyse">abstrakten harmonischen Analyse</a>.

Die <a href="Stone-%C4%8Cech-Kompaktifizierung" title="Stone-Čech-Kompaktifizierung">Stone-Čech-Kompaktifizierung</a> eines <a href="Vollst%C3%A4ndig_regul%C3%A4rer_Raum" title="Vollständig regulärer Raum">vollständig regulären Hausdorffraums</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> kann als Anwendung der Gelfand-Transformation auf die kommutative C*-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{b}(X)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{b}(X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94a41c16b65caf4f015df339f971b64c4c9c2fbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.389ex; height:2.843ex;" alt="{\displaystyle C_{b}(X)}" loading="lazy"> der stetigen und <a href="Beschr%C3%A4nkte_Abbildung" title="Beschränkte Abbildung">beschränkten Funktionen</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> erhalten werden.

Der Kern der Gelfand-Transformation ist im Falle einer kommutativen Banachalgebra das <a href="Jacobson-Radikal" title="Jacobson-Radikal">Jacobson-Radikal</a>, insbesondere ist das Jacobson-Radikal stets abgeschlossen. Hier zeigt sich wieder, wie algebraische und topologische Begriffe in der Theorie der Banachalgebren ineinandergreifen.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Richard_Kadison" title="Richard Kadison">R.V. Kadison</a>, <a href="John_Ringrose" title="John Ringrose">J. R. Ringrose</a>: Fundamentals of the Theory of Operator Algebras, 1983</li>

<li><a href="Masamichi_Takesaki" title="Masamichi Takesaki">M. Takesaki</a>: Theory of Operator Algebras I (Springer 1979, 2002)</li></ul></div><div><div class="zim-footer">

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