algebra neprerivnix funktsiy

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$algebra neprerivnix funktsiy plan: 1. algebra neprerivnix funktsiy 2. nekotorie elementarnie funktsii 3. spektralnaya teoriya material iz vikipedii — svobodnoy entsiklopedii pereyti k navigatsiipereyti k poisku bánaxovoy algebroy nad kompleksnim ili deystvitelnim polem nazivaetsya assotsiativnaya algebra, yavlyayushayasya pri etom banaxovim prostranstvom. pri etom umnojenie v ney doljno bit soglasovano s normoy: {\displaystyle \forall x,y\in a,\|x\,y\|\ \leq \|x\|\,\|y\|}eto svoystvo trebuetsya dlya neprerivnosti operatsii umnojeniya otnositelno normi. banaxova algebra nazivaetsya unitalnoy ili banaxovoy algebroy s edinitsey, esli ona obladaet edinitsey (to est takim elementom {\displaystyle \mathbf {1} }, chto dlya vsex {\displaystyle x\in a} spravedlivo {\displaystyle x\mathbf {1} =\mathbf {1} x=x}). pri etom obichno trebuyut, chtobi norma edinitsi bila ravna 1. esli edinitsa sushestvuet, to ona edinstvenna. vsyakuyu banaxovu algebru {\displaystyle a} mojno izometricheski vlojit v sootvetstvuyushuyu ey unitalnuyu banaxovu algebru {\displaystyle a_{e}} v kachestve zamknutogo dvustoronnego ideala. banaxova algebra nazivaetsya kommutativnoy, esli operatsiya umnojeniya v ney kommutativna. primeri · polya kompleksnix …$

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$banaxovom prostranstve, otnositelno operatornoy normi i kompozitsii v kachestve umnojeniya. mnojestvo kompaktnix operatorov otnositelno tex je operatsiy yavlyaetsya zamknutim idealom v etoy algebre. · esli {\displaystyle g} — lokalno kompaktnaya xausdorfova topologicheskaya gruppa s meroy xaara {\displaystyle \mu }, to banaxovo prostranstvo {\displaystyle l_{1}(g)} integriruemix otnositelno meri {\displaystyle \mu } kompleksnoznachnix funktsiy na {\displaystyle g} yavlyaetsya banaxovoy algebroy otnositelno umnojeniya-svyortki, opredelyaemoy po formule {\displaystyle (xy)(g)=\int _{g}x(h)y(h^{-1}g)\,\mathrm {d} \mu (h),\;g\in g}{\displaystyle l_{1}(\mathbb {r} )} — algebra summiruemix na pryamoy funktsiy so svertkoy v kachestve umnojeniya. eto chastniy sluchay predidushego primera. · c*-algebra — algebra s *-involyutsiey, soglasovannoy s normoy: {\displaystyle \forall a\ ||a^{*}a||=||a||^{2}} svoystva nekotorie elementarnie funktsii mojno pri pomoshi stepennix ryadov opredelit dlya elementov banaxovoy algebri. v chastnosti, mojno opredelit eksponentu elementa banaxovoy algebri, trigonometricheskie funktsii, i, v obshem sluchae, lyubuyu tseluyu funktsiyu. dlya elementov banaxovoy algebri ostayotsya spravedlivoy formula summi beskonechno ubivayushey geometricheskoy progressii (ryad neymana). mnojestvo obratimix elementov …$

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$imi, izomorfna {\displaystyle \mathbb {c} }. spektralnaya teoriya v unitalnix banaxovix algebrax vvoditsya ponyatie spektra, kotoroe rasshiryaet ponyatie spektra operatora na bolee obshiy klass ob'ektov. element {\displaystyle a\in a} algebri {\displaystyle a} nazivaetsya obratimim, esli naydetsya takoy element {\displaystyle a^{-1}\in a}, chto {\displaystyle aa^{-1}=a^{-1}a=\mathbf {1} }. spektrom {\displaystyle \sigma (a)} elementa {\displaystyle a} nazivaetsya mnojestvo takix {\displaystyle \lambda \in \mathbb {c} ,} chto element {\displaystyle a-\lambda \mathbf {1} } neobratim. spektr vsyakogo elementa unitalnoy kompleksnoy banaxovoy algebri — nepustoy kompakt. s drugoy storoni, dlya lyubogo kompakta {\displaystyle k\subset \mathbb {c} } spektr elementa {\displaystyle w} iz algebri {\displaystyle c(k)}, opredelyaemogo po formule {\displaystyle w(z)=z}, sovpadaet s {\displaystyle k}, poetomu drugix ogranicheniy na spektr elementa v proizvolnoy banaxovoy algebre net. spektralnim radiusom {\displaystyle \mathrm {r} (x)} elementa {\displaystyle x\in a} nazivaetsya velichina {\displaystyle \mathrm {r} (x)=\sup\{|\lambda |:\lambda \in \sigma (x)\}}. spravedliva formula byorlinga-gelfanda dlya spektralnogo radiusa: {\displaystyle \mathrm {r} (x)=\lim _{n\to …$

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$yle \gamma } — spryamlyaemiy jordanov kontur, lejashiy v {\displaystyle d}, soderjashiy spektr elementa {\displaystyle x} i orientirovanniy polojitelno, a {\displaystyle r_{a}} — rezolventa elementa {\displaystyle a}. v chastnosti, pri pomoshi etoy formuli mojno opredelit eksponentu elementa iz banaxovoy algebri. ideali i xarakteri pust a — unitalnaya kommutativnaya banaxova algebra nad polem kompleksnix chisel. xarakterom χ algebri a nazivaetsya nenulevoy lineyniy funktsional, obladayushiy svoystvom multiplikativnosti: dlya lyubix a, b ∈ a spravedlivo χ(ab) = χ(a)χ(b) i χ(1) = 1. to est xarakter — eto nenulevoy gomomorfizm algebr a i {\displaystyle \mathbb {c} }. mojno proverit, chto vsyakiy xarakter v banaxovoy algebre nepreriven i ego norma ravna 1. yadro xaraktera predstavlyaet soboy maksimalniy ideal v a. esli {\displaystyle {\mathfrak {m}}} — maksimalniy ideal, to faktoralgebra {\displaystyle a/{\mathfrak {m}}} yavlyaetsya polem i banaxovoy algebroy, togda, po teoreme gelfanda-mazura, ona izomorfna {\displaystyle \mathbb {c} }. poetomu kajdomu maksimalnomu idealu {\displaystyle {\mathfrak {m}}} mojno …$

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$,a\to \mathbb {c} }, opredelyaemaya po formule {\displaystyle {\hat {a}}(\chi )=\chi (a)} dlya vsex xarakterov χ. preobrazovanie gelfanda osushestvlyaet sjimayushiy gomomorfizm algebri a v algebru c(spec a) neprerivnix funktsiy na kompakte. radikalom algebri a nazivaetsya peresechenie vsex eyo maksimalnix idealov. esli radikal sostoit tolko iz nulya, algebra a nazivaetsya poluprostoy. yadro preobrazovaniya gelfanda sovpadaet s radikalom algebri, poetomu preobrazovanie gelfanda in'ektivno togda i tolko togda, kogda algebra a poluprosta. takim obrazom, vsyakaya poluprostaya kommutativnaya banaxova algebra s edinitsey sovpadaet s tochnostyu do izomorfizma s nekotoroy algebroy funktsiy, neprerivnix na kompakte — s obrazom preobrazovaniya gelfanda. algebra funktsiy - poluprostaya kommutativnaya banaxova algebra a, realizovannaya v vide algebri neprerivnix funktsiy na prostranstve maksimalnix idealov. esli a ∈ a i f - nek-raya funktsiya, opredelennaya na spektre elementa a (t. e. na mnojestve znacheniy funktsii â = a), to f(a) est nek-raya funktsiya na . uslovie f(a) ∈ a, konechno, ne obyazano …$

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algebra neprerivnix funktsiy plan: 1. algebra neprerivnix funktsiy 2. nekotorie elementarnie funktsii 3. spektralnaya teoriya material iz vikipedii — svobodnoy entsiklopedii pereyti k navigatsiipereyti k poisku bánaxovoy algebroy nad kompleksnim ili deystvitelnim polem nazivaetsya assotsiativnaya algebra, yavlyayushayasya pri etom banaxovim prostranstvom. pri etom umnojenie v ney doljno bit soglasovano s normoy: {\displaystyle \forall x,y\in a,\|x\,y\|\ \leq \|x\|\,\|y\|}eto svoystvo trebuetsya dlya neprerivnosti operatsii umnojeniya otnositelno normi. banaxova algebra nazivaetsya unitalnoy ili banaxovoy algebroy s edinitsey, esli ona obladaet edinitsey (to est takim elementom {\displaystyle \mathbf {1} }, chto dlya vsex {\displaystyle x\in a} spravedlivo {\displaystyle x\mathbf {1} =\mat...

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