English+GermanWikipedia - /page/entry.mu

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Diskalgebra</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 Diskalgebra (manchmal auch Discalgebra) ist eine in den mathematischen Teilgebieten <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a> und <a href="Funktionentheorie" title="Funktionentheorie">Funktionentheorie</a> betrachtete <a href="Algebra_%C3%BCber_einem_K%C3%B6rper" title="Algebra über einem Körper">Algebra</a>. Viele funktionalanalytische Eigenschaften der Diskalgebra sind direkte Folgen funktionentheoretischer Sätze.

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

Bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} :=\{z\in \mathbb {C} ;\,|z|\leq 1\}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/210ee0051888155c04bbe5a4f8e4125fbfa56a8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.419ex; height:2.843ex;" alt="{\displaystyle \mathbb {D} :=\{z\in \mathbb {C} ;\,|z|\leq 1\}}" loading="lazy"> die Kreisscheibe, so sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\mathbb {D} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a59a9c245e8f147c57ffcd908bf67ddd117260d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.23ex; height:2.843ex;" alt="{\displaystyle A(\mathbb {D} )}" loading="lazy"> die Menge aller

<a href="Stetige_Funktion" title="Stetige Funktion">stetigen</a> Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {D} \rightarrow \mathbb {C} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0351c81bb1e670ac09f08418309bf2a460804348.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {D} \rightarrow \mathbb {C} }" loading="lazy">, die im Inneren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} ^{\circ }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15803bbb2acebc7f78cb534069ffe9394fe51877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \mathbb {D} ^{\circ }}" loading="lazy"> <a href="Holomorphe_Funktion" title="Holomorphe Funktion">holomorph</a> sind.

Die Definitionen

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}(\lambda f)(z)&:=&\lambda f(z)\\(f+g)(z)&:=&f(z)+g(z)\\(fg)(z)&:=&f(z)g(z)\\(f^{*})(z)&:=&{\overline {f({\overline {z}})}}\\\end{array}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b0db251b38750bdb2a29445cc6776a6b9d65811.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:28.823ex; height:13.843ex;" alt="{\displaystyle {\begin{array}{rcl}(\lambda f)(z)&:=&\lambda f(z)\\(f+g)(z)&:=&f(z)+g(z)\\(fg)(z)&:=&f(z)g(z)\\(f^{*})(z)&:=&{\overline {f({\overline {z}})}}\\\end{array}}}" loading="lazy">,

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {C} ,z\in \mathbb {D} ,f,g\in A(\mathbb {D} )}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda \in \mathbb {C} ,z\in \mathbb {D} ,f,g\in A(\mathbb {D} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63e4dd72dccc04715c72f08f01f362ecc468581a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.048ex; height:2.843ex;" alt="{\displaystyle \lambda \in \mathbb {C} ,z\in \mathbb {D} ,f,g\in A(\mathbb {D} )}" loading="lazy">,

machen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\mathbb {D} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a59a9c245e8f147c57ffcd908bf67ddd117260d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.23ex; height:2.843ex;" alt="{\displaystyle A(\mathbb {D} )}" loading="lazy"> zu einer komplexen Algebra mit

<a href="Involution_(Mathematik)" title="Involution (Mathematik)">Involution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy">. Diese wird Diskalgebra genannt.<a href="#cite_note-1">[1]</a>

Offenbar ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\mathbb {D} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a59a9c245e8f147c57ffcd908bf67ddd117260d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.23ex; height:2.843ex;" alt="{\displaystyle A(\mathbb {D} )}" loading="lazy"> eine Unteralgebra der <a href="Funktionenraum" title="Funktionenraum">Funktionenalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\mathbb {D} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7038956b84f2fce408d6ab006da3160172fe82b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.254ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {D} )}" loading="lazy"> der stetigen Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} \rightarrow \mathbb {C} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4488171cab39bc0d575a49dcd44fe4ee47056b3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.97ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} \rightarrow \mathbb {C} }" loading="lazy">. Die Diskalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\mathbb {D} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a59a9c245e8f147c57ffcd908bf67ddd117260d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.23ex; height:2.843ex;" alt="{\displaystyle A(\mathbb {D} )}" loading="lazy"> ist bezüglich der <a href="Maximumsnorm" title="Maximumsnorm">Maximumsnorm</a>, die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\mathbb {D} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7038956b84f2fce408d6ab006da3160172fe82b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.254ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {D} )}" loading="lazy"> zu einer <a href="Banachalgebra" title="Banachalgebra">Banachalgebra</a> macht, <a href="Abgeschlossene_Menge" title="Abgeschlossene Menge">abgeschlossen</a>, denn nach dem <a href="Weierstra%C3%9Fscher_Konvergenzsatz" title="Weierstraßscher Konvergenzsatz">weierstraßschen Konvergenzsatz</a> sind <a href="Gleichm%C3%A4%C3%9Fige_Konvergenz" title="Gleichmäßige Konvergenz">gleichmäßige Limiten</a> holomorpher Funktionen ebenfalls holomorph.

Der <a href="Funktionenraum" title="Funktionenraum">Funktionenraum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\mathbb {D} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a59a9c245e8f147c57ffcd908bf67ddd117260d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.23ex; height:2.843ex;" alt="{\displaystyle A(\mathbb {D} )}" loading="lazy"> ist daher selbst eine Banachalgebra, sogar mit <a href="Isometrie" title="Isometrie">isometrischer Involution</a>, das heißt, es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f^{*}\|=\|f\|}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51d6a4eefd7d43e05c3ae9ac527bb4b7f871dfa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.402ex; height:2.843ex;" alt="{\displaystyle \|f^{*}\|=\|f\|}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in A(\mathbb {D} )}">

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<annotation encoding="application/x-tex">{\displaystyle f\in A(\mathbb {D} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/036a0e6bfdcb0d62d4e772fd2afa2522dbd134a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.35ex; height:2.843ex;" alt="{\displaystyle f\in A(\mathbb {D} )}" loading="lazy">. Die Diskalgebra ist auch Unterbanachalgebra von

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3203ee49321f9169b09ebbb42cba91aecffd7934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.979ex; height:2.343ex;" alt="{\displaystyle H^{\infty }}" loading="lazy">, der Banachalgebra aller auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} ^{\circ }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15803bbb2acebc7f78cb534069ffe9394fe51877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \mathbb {D} ^{\circ }}" loading="lazy"> holomorphen und beschränkten Funktionen mit der <a href="Supremumsnorm" title="Supremumsnorm">Supremumsnorm</a>.

Mittels Einschränkung auf den Rand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \mathbb {D} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f776ef0ce72b5d5d984bd77de36b970ad4c50dc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle \partial \mathbb {D} }" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b932b553742ca27776057f1262527014ebbb46a0.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 {D} }" loading="lazy"> erhält man eine Abbildung

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<annotation encoding="application/x-tex">{\displaystyle A(\mathbb {D} )\rightarrow C(\partial \mathbb {D} ),\,f\mapsto f|_{\partial \mathbb {D} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5be699bbd5f7ee61bfc93b9eb43772cb179377d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.006ex; height:3.009ex;" alt="{\displaystyle A(\mathbb {D} )\rightarrow C(\partial \mathbb {D} ),\,f\mapsto f|_{\partial \mathbb {D} }}" loading="lazy">.

Diese Abbildung ist nach dem <a href="Maximumprinzip_(Mathematik)" title="Maximumprinzip (Mathematik)">Maximumprinzip</a> für holomorphe Funktionen ein isometrischer <a href="Homomorphismus" title="Homomorphismus">Homomorphismus</a>.

In diesem Sinne kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\mathbb {D} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a59a9c245e8f147c57ffcd908bf67ddd117260d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.23ex; height:2.843ex;" alt="{\displaystyle A(\mathbb {D} )}" loading="lazy"> auch als Unterbanachalgebra von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\partial \mathbb {D} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/167d67a4c95ea07a751f3cd97ff4e65afbc7d3ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.572ex; height:2.843ex;" alt="{\displaystyle C(\partial \mathbb {D} )}" loading="lazy">

auffassen, das heißt die Diskalgebra wird zu einer <a href="Uniforme_Algebra" title="Uniforme Algebra">uniformen Algebra</a> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial \mathbb {D} }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45ad28bf2b342b5a263a69cc69044ad32ec2586f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle {\partial \mathbb {D} }}" loading="lazy">.

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a59a9c245e8f147c57ffcd908bf67ddd117260d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.23ex; height:2.843ex;" alt="{\displaystyle A(\mathbb {D} )}" loading="lazy"> ist dann die Menge aller stetigen Funktionen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \mathbb {D} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f776ef0ce72b5d5d984bd77de36b970ad4c50dc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle \partial \mathbb {D} }" loading="lazy">, die sich holomorph nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} ^{\circ }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15803bbb2acebc7f78cb534069ffe9394fe51877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.343ex;" alt="{\displaystyle \mathbb {D} ^{\circ }}" loading="lazy"> fortsetzen lassen.

Dies wäre eine alternative Definition der Diskalgebra.

Die Diskalgebra wird von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} _{\mathbb {D} }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {id} _{\mathbb {D} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27b43c11a6d6720e30a774d698781a131c018ba3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.358ex; height:2.509ex;" alt="{\displaystyle \mathrm {id} _{\mathbb {D} }}" loading="lazy"> erzeugt, das heißt, die kleinste Unterbanachalgebra, die diese Funktion enthält, ist die Diskalgebra selbst.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Der_Gelfandraum">Der Gelfandraum</h2></div>

Für jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {D} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2725e37fb4755d6597c8bfdd58edeb6de88a027.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 {D} }" loading="lazy"> ist die Punktauswertung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z}:A(\mathbb {D} )\rightarrow \mathbb {C} ,\,f\mapsto f(z)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ec3aa9ec004de1f0d668fba4d2124e31b412761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.984ex; height:2.843ex;" alt="{\displaystyle \delta _{z}:A(\mathbb {D} )\rightarrow \mathbb {C} ,\,f\mapsto f(z)}" loading="lazy"> ein Homomorphismus und damit ein Element des <a href="Gelfand-Raum" class="mw-redirect" title="Gelfand-Raum">Gelfand-Raums</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A(\mathbb {D} )}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48dd09ec89f857674e6b76c90528cdd0b3759a78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.855ex; height:3.009ex;" alt="{\displaystyle X_{A(\mathbb {D} )}}" loading="lazy"> der Diskalgebra.

Man kann zeigen, dass mit den <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{z}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fee6a037fd2068de909aed167e1f41ea8775169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.034ex; height:2.676ex;" alt="{\displaystyle \delta _{z}}" loading="lazy"> bereits alle Homomorphismen der Diskalgebra mit Werten in den komplexen Zahlen

gefunden sind, und dass die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \colon \mathbb {D} \rightarrow X_{A(\mathbb {D} )},\,z\mapsto \delta _{z}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta \colon \mathbb {D} \rightarrow X_{A(\mathbb {D} )},\,z\mapsto \delta _{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f78bc8991c6e9fb7725b9ef39b3cd091649412b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.387ex; height:3.176ex;" alt="{\displaystyle \delta \colon \mathbb {D} \rightarrow X_{A(\mathbb {D} )},\,z\mapsto \delta _{z}}" loading="lazy"> ein <a href="Hom%C3%B6omorphismus" title="Homöomorphismus">Homöomorphismus</a> ist, wobei die sogenannte Gelfandtopologie durch die <a href="Relativtopologie" class="mw-redirect" title="Relativtopologie">relative</a> <a href="Schwach-*-Topologie" title="Schwach-*-Topologie">schwach-*-Topologie</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{a}\subset A^{\prime }}">

<msub>

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</msub>

<msup>

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</msup>

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dcaef82c345a4b948573d47a0d23880626e6465b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.552ex; height:2.843ex;" alt="{\displaystyle X_{a}\subset A^{\prime }}" loading="lazy"> gegeben ist.

Der Gelfandraum der Diskalgebra kann daher mit der Kreisscheibe identifiziert werden.

Bei dieser Identifikation ist die <a href="Gelfand-Transformation" title="Gelfand-Transformation">Gelfand-Transformation</a> die Identität auf der Diskalgebra.

<div class="mw-heading mw-heading2"><h2 id="Die_Nicht-Regularität_der_Diskalgebra">Die Nicht-Regularität der Diskalgebra</h2></div>

Auf dem Gelfandraum <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>

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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"> einer kommutativen Banachalgebra betrachtet man die sogenannte Hülle-Kern-Topologie, die durch die Abschlussoperation

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {E}}:=\{\delta \in X_{A};\,\ker(\delta )\supset \bigcap _{\varphi \in E}\ker(\varphi )\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f230119fbb4a87aeb66928346d53fa1b92354e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.142ex; height:6.009ex;" alt="{\displaystyle {\overline {E}}:=\{\delta \in X_{A};\,\ker(\delta )\supset \bigcap _{\varphi \in E}\ker(\varphi )\}}" loading="lazy">

gegeben ist. Fällt diese mit der Gelfandtopologie zusammen, so nennt man die Banachalgebra regulär.

Die Diskalgebra ist ein Beispiel für eine nicht-reguläre Banachalgebra.<a href="#cite_note-3">[3]</a>

In der Tat ist bei der Identifikation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A(\mathbb {D} )}=\mathbb {D} }">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcb5fd2466346c445a31d56cffcae097938c6d6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.632ex; height:3.009ex;" alt="{\displaystyle X_{A(\mathbb {D} )}=\mathbb {D} }" loading="lazy"> die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E:=\{0\}\cup \{{\tfrac {1}{n}};\,n\in \mathbb {N} \}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E:=\{0\}\cup \{{\tfrac {1}{n}};\,n\in \mathbb {N} \}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7b76a858fc726bf07c64d7c432fde83ff95ae8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.073ex; height:3.343ex;" alt="{\displaystyle E:=\{0\}\cup \{{\tfrac {1}{n}};\,n\in \mathbb {N} \}}" loading="lazy"> abgeschlossen in der Gelfandtopologie.

Ist nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle f\in \bigcap _{n\in \mathbb {N} }\ker(\delta _{\frac {1}{n}})}">

</mrow>

</munder>

<msub>

<mfrac>

</mfrac>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle f\in \bigcap _{n\in \mathbb {N} }\ker(\delta _{\frac {1}{n}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57b0c918d844e0ea9734a9e1daa8e9ff703ff9e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.826ex; height:3.843ex;" alt="{\displaystyle \textstyle f\in \bigcap _{n\in \mathbb {N} }\ker(\delta _{\frac {1}{n}})}" loading="lazy">, so folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\tfrac {1}{n}})=0}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f({\tfrac {1}{n}})=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bbc4cb7253306a907c6d4477b8e9540656e415e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.171ex; height:3.343ex;" alt="{\displaystyle f({\tfrac {1}{n}})=0}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, und aus dem <a href="Identit%C3%A4tssatz_f%C3%BCr_holomorphe_Funktionen" title="Identitätssatz für holomorphe Funktionen">Identitätssatz für holomorphe Funktionen</a> folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0}">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ee0fdf0f50fcba5afe3e856fcc7dc6acfa61014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle f=0}" loading="lazy">. Daher ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \bigcap _{\varphi \in E}\ker(\varphi )=\{0\}}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \bigcap _{\varphi \in E}\ker(\varphi )=\{0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95bfe403c5cdb41b347b863688425cff5665c6d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.069ex; height:3.343ex;" alt="{\displaystyle \textstyle \bigcap _{\varphi \in E}\ker(\varphi )=\{0\}}" loading="lazy"> und es folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {E}}=X_{A}}">

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f84dab851d6d24bc01a0e3b947f9ca1ab6fc938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.417ex; height:3.343ex;" alt="{\displaystyle {\overline {E}}=X_{A}}" loading="lazy"> bezüglich der Hülle-Kern-Topologie, letztere kann daher nicht mit der Gelfandtopologie übereinstimmen.

<div class="mw-heading mw-heading2"><h2 id="Der_Schilowrand">Der Schilowrand</h2></div>

Identifiziert man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{A(\mathbb {D} )}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48dd09ec89f857674e6b76c90528cdd0b3759a78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.855ex; height:3.009ex;" alt="{\displaystyle X_{A(\mathbb {D} )}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} }">

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b932b553742ca27776057f1262527014ebbb46a0.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 {D} }" loading="lazy">, so fällt der topologische Rand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \mathbb {D} =\{z\in \mathbb {C} ;\,|z|=1\}}">

</mrow>

</mrow>

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</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5960cc28f385934c12d21da7f621a8781f543c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.09ex; height:2.843ex;" alt="{\displaystyle \partial \mathbb {D} =\{z\in \mathbb {C} ;\,|z|=1\}}" loading="lazy"> mit dem <a href="Schilow-Rand" title="Schilow-Rand">Schilow-Rand</a> zusammen. Dazu ist zu zeigen, dass jede Funktion der Diskalgebra, die wegen der vorgenommenen Identifikation ja mit ihrer Gelfand-Transformierten übereinstimmt, ihr Betragsmaximum auf dem Rand der Kreisscheibe annimmt, aber das ist genau die Aussage des <a href="Maximumprinzip_(Mathematik)" title="Maximumprinzip (Mathematik)">Maximumprinzips</a> für holomorphe Funktionen.<a href="#cite_note-4">[4]</a>

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

Wie oben erwähnt kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\mathbb {D} )}">

</mrow>

</mstyle>

</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a59a9c245e8f147c57ffcd908bf67ddd117260d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.23ex; height:2.843ex;" alt="{\displaystyle A(\mathbb {D} )}" loading="lazy"> mittels der Einschränkungsabbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto f|_{\partial \mathbb {D} }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\mapsto f|_{\partial \mathbb {D} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b85cceb024b328d59bd5739fd384c75d9c6079d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.169ex; height:3.009ex;" alt="{\displaystyle f\mapsto f|_{\partial \mathbb {D} }}" loading="lazy"> als Unterbanachalgebra von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\partial \mathbb {D} )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/167d67a4c95ea07a751f3cd97ff4e65afbc7d3ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.572ex; height:2.843ex;" alt="{\displaystyle C(\partial \mathbb {D} )}" loading="lazy"> auffassen. Der <a href="Maximalit%C3%A4tssatz_von_Wermer" title="Maximalitätssatz von Wermer">Maximalitätssatz von Wermer</a> sagt aus, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(\mathbb {D} )\subset C(\partial \mathbb {D} )}">

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(\mathbb {D} )\subset C(\partial \mathbb {D} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b34a8f243da0e6b6f999afe4ad6dd671ffa93229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.901ex; height:2.843ex;" alt="{\displaystyle A(\mathbb {D} )\subset C(\partial \mathbb {D} )}" loading="lazy"> eine maximale Unterbanachalgebra ist.

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> F. F. Bonsall, J. Duncan: Complete Normed Algebras. Springer-Verlag 1973, ISBN 3-540-06386-2, §1.16

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> F. F. Bonsall, J. Duncan: Complete Normed Algebras. Springer-Verlag 1973, ISBN 3-540-06386-2, §19.3

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> F. F. Bonsall, J. Duncan: Complete Normed Algebras. Springer-Verlag 1973, ISBN 3-540-06386-2, §23.9

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> F. F. Bonsall, J. Duncan: Complete Normed Algebras. Springer-Verlag 1973, ISBN 3-540-06386-2, §22.5 für n=1

</li>

</ol></div><div><div class="zim-footer">

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