
oru sii *-beejaganithatthinte spectrum

ganithashaasthrathil, oru c *-beejaganithatthinte spectrum allengil oru c *-beejaganithatthinte eratta "a", "Â" ennu soochippikkunnathu, "a" nte irreducible *-praathinidhyangalude unitory equivalence klaasukalude koottamaanu. oru hilbert space "h" ll "a" yude a *-praathinidhyam π ennathu "h" ll ninnu vyathyasthamaaya clossed subpace "k" illengil maathrame irreducible aakoo, koodaathe ellah oopparettarmaarkkum keezhil π ("x") "x" ω "a" ullathum inverentaayirikkum. irreducible praathinidhyam ennaal "noonn-nall" irreducible praathinidhyam ennaanu ardhamaakkunnathennu njangal parokshamaayi anumaanikkunnu, athinaal nissaramaaya (i.iii. samaanamaayi 0) ekamaana edangalile praathinidyangal. chuvade vishadeekarichirikkunnathu, spectrum "-" swaabhaavikamaayum oru topological space aanu; ithu oru valayathinte spectrum enna aashayathin samaanamaan.
eee aashayathinte ettavum pradhanappetta prayogangalilonnu praadeshikamaayi othungunna eathoru groupinum eratta vasthuvinekurichulla oru ashayam nalkuka ennathaanu. eee eratta vasthu oru forrier parivartthanatthinum type I-nte unimodular verthirikkaavunna praadeshikamaayi othukkamulla grouppukalkkulla oru plancherel sidhaanthathinum type I-yude verthirikkaavunna praadeshikamaayi othukkamulla grouppukalude ekapaksheeyamaaya praathinidyangalkkulla oru vighadana sidhaanthathinum anuyojyamaanu. ennirunnaalum, praadeshikamaayi compact grouppukalkkulla thathfalamaayundaakunna duvality sidhaantham compact topological grouppukalkkulla thannaka-kreen duvality sidhaanthathekkaalum allengil praadeshikamaayi compact "abelian" grouppukalkkulla pontriyaajin deuvalityekkaalum valare durbalamaanu, iva randum poornnamaaya maattangalalla. duval oru samboornna inevarant allennu eluppathil kaanaan kazhiyum, eathenkilum parimitha-dimensional phul matricks aljibrayude duval m "n" (c) orotta pooyint ulkkollunnu.
aadima spectrum.
Â" ennathinte topology niravadhi thulyamaaya reethikalil nirvachikkaam. aadima spektrathinte adisthaanathil nammal aadyam athine nirvachikkunnu.
a" yude aadima spectrum "a" yude aadima idealukalude prima ("a") koottamaanu, ivide oru aadima ideal poojyamallaatha irreducible *-praathinidhyatthinte kernalaanu. hull-kernel topology (allengil jacobson topology) ulla oru topological spessaanu premitteevu idealukalude koottam. ithine thaazhepparayunna reethiyil nirvachikkunnu "x" ennathu aadima idealukalude oru koottamaanengil, athinte hull-kernel closher
hull-kernel closser eluppathil oru idempottant ooppareshanaayi kaanikkunnu, athaayath
ithu kuratovsky closser aksiyangal thripthippeduthunnuvennu kaanikkam. thalphalamaayi, primil ("a") oru savisheshamaaya topology τ undennu kaanikkan kazhiyum, athaayath τ-maayi bandhappett oru settu "x" adaykkunnathu "x"-nte hull-kernel closserinu samaanamaan.
ekathaanamaayi thulyamaaya praathinidyangalkku ore kernel ullathinaal, oru surjective maappiloode mapp π kerr (π) gadakangal
topology nirvachikkaan nammal "k" enna bhoopadam upayogikkunnu
nirvachanam. Â"-nte ooppan settukal praiminte ("a") ooppan subsettukalude "k"-1 ("u") enna vipareeta chithrangalaanu. ithu sarikkum oru topologyaanu.
commutative ringukalkkaayulla sariski topology noonn-commutative ringukalkkulla oru analogaanu hull-kernel topology.
hull-kernel topologyil ninnu prachodanam ulkkonda topologyil "a" yude avasthakalude adisthaanathil mattu swabhaavasavisheshathakalundu.
udaaharanagal.
commutative sii *-algibras.
oru commutative sii *-aljibra "eiii" yude spectrum "eiii" yude jelfand duvalumaayi poruthappedunnu (banach space "eiii" yude duval "eiii" yumayi thettidharikkaruthu). prathyekichum, "x" oru compact hosdorff space aanennu karuthuka. appol oru swaabhaavika homomorphisamundu.
eee mapping nirvachichirikkunnathu
c ("x")-ll i ("x") oru clossed maximal idealaanu, athinaal ithu yadhaarthathil premitteevu aanu. thelivukalude vishadaamshangalkku dixmear reference kaanuka. oru commutative sii *-algibraikku,
parimithamaaya oopparettarmaarude sii *-beejaganitam.
h" verthirikkaavunna ananthamaaya alavilulla hilbert space ayirikkatte. "l" ("" h "") nuu rund norm-clossed *-idealukalundu "" i "" 0 = {0}, compact oopparettarmaarude ideal "" k "" = "" k "" ("" h "") ". athinaal oru koottamaayi, prime ("l" ("h")) = {"i" 0 ", k"}. ippol
athinaal prime ("l" ("h")) oru noonn-hosdorff space aanu.
maruvashathu "l" ("h")-nte spectrum valare valuthaanu. kernel "k" ("h") allengil kernel {0} upayogichu asamamaya irreducible praathinidyangalundu.
parimitha-dimensional sii *-algibras.
a" oru parimitha-dimensional c *-beejaganithamaanennu karuthuka. poornna matricks aaljibrakalude parimithamaaya nerittulla thukayude isomorphic aanu "eiii" ennu ariyappedunnu
minimum ("a") ennathu "a" yude ettavum kuranja kendra pravachanangalaan. a" yude spectrum discreet topologyumaayi miinii ("a") lake canonic aayi isomorphic aanu. parimitha-dimensional sii *-algebrakalkku, namukk isomorphisavum undu.
spektrathinte mattu swabhaavasavisheshathakal.
hull-kernel topology amoorthamaayi vivarikkan eluppamaanu, ennaal praadeshikamaayi othukkamulla topological grouppukalumaayi bandhapetta sii *-aaljibrakalkk praayogikamaayi, positteev nischitha pravartthanangalude kaaryatthil spectrathile topology mattu swabhaavasavisheshathakal abhikaamyamaanu.
vaasthavathil, "Â" ennathile topology inipparayunnava kaanikkunnathupole praathinidhyangalude durbalamaaya niyanthranam enna ashayavumaayi adutha bandhamullathaann
randaamathe vyavastha ardhamaakkunnathu π "s" ll durbalamaayi adangiyirikkunnu ennaanu.
oru sii *-aljibra "eiii" yude avasthakale "eiii" yude praathinidyangalumaayi bandhappeduthunnathinulla oru paachakakkurippaan jii. enn. ezz nirmmaanam. jii. enn. ezz nirmmaanavumaayi bandhapetta adisthaana sidhaanthangalilonnil, bandhapetta praathinidhyam π "f" irreducible aanenkil maathrame oru avastha "f" shudhamaakoo. koodaathe, mapping κ: purestate ("a") → "-" f "π" f "nirvachichirikkunnathu oru surjective maappaan.
muun sidhaanthathil ninnu oraalkk inipparayunnava eluppathil theliyikkaan kazhiyum;
space irr "enn" ("eiii").
uchithamaaya pooyint vais kanverjans topologyulla oru topological space aayi praathinidhyangalude idam pariganikkunnathiloode uyarnnuvarunna topology mattoru swabhaava savisheshathayundu. kooduthal krithyamaayi paranjaal, "n" oru cardinal samkhyayaayirikkatte, "hn" ennathu "n" enna alavilulla canonical hilbert space ayirikkatte.
irr "enn" ("eiii") ennathu irreducible *-nte space aanu-pooyint durbalamaaya topologyumaayi "echen" ll "eiii" yude praathinidhyam. valakalude samyojanathinte kaaryatthil, eee topology nirvachichirikkunnathu π "i" → π; aanenkil maathram
irr "enn" ("eiii")-leeii eee topology pooyint-strong topologic thulyamaanennu ithu maarunnu, i.iii. π "i" → π aanenkil maathram
paramarsham. vividha "aan" kall orumichu koottunnathu valare sankeernamaayirikkum.
mackie-boral ghadana.
"-" oru topological sthalamaanu, athinaal ithu oru boral sthalamaayum kanakkakkaam. jii enna oru prashasthamaaya anumaanam. boral space staanderd aanenkil maathram praadeshikamaayi verthirikkaavunna compact group type I aanennu mackie nirddeshichu.iii. poornnamaayum verthirikkaavunna metrik spacinte adisthaana boral spacinu isomorphic (boral spaceukal vibhaagathil) aanu. eee savisheshatha minusamarnna boral speysukal ennu mackie vilikkunnu. 1961-leeii prabandhatthil thaazhepparayunna paraamarsangalil listt cheythirikkunna verthirikkaavunna sii *-algibras enna jaims glim eee anumaanam theliyichu.
nirvachanam. verthirikkaavunna c *-beejaganitam "a" yude oru degenerate cheyyaatha *-praathinidhyam π ennathu π ("a") srishtikkunna voonn neuman beejaganitam oru dimensional aanenkil maathram oru ghadaka praathinidyamaanu. a" yude eathenkilum verthirikkaavunna ghadaka praathinidhyam irterducible orennathinte parimithamo kanakkakkaavunnatho aaya gunithamaanengil maathrame c *-beejaganitam "a" type i aakoo.
sii * (jii) type aii aaya praadeshikamaayi verthirikkaavunna compact grouppukalaaya "jii" yude udaaharanagal bandhippicha (yadhaartha) nilpottant lye grouppukalum bandhippicha yadhaartha ardha-lalithamaaya lye grouppukalum aanu. athinaal heisenberg grouppukalellaam type i aanu. compact, abelian grouppukalum type I aanu.
verthirikkaavunna tharam i c *-aaljibrakalude praathinidhyangaludeyum athinanusarichu verthirikkaavunna praadeshikamaayi compact grouppukaludeyum gadanayude deerghadoora saamaanyavalkkaranathe phalam soochippikkunnu.
beejaganita aadima spectra.
oru c *-beejaganitam "a" oru valayamayathinaal, "a" ennathinte aadima aadarshangalude koottavum namukk pariganikkaam, avide "a" beejaganithaparamaayi kanakkaakkappedunnu. oru valayathinu oru ideal oru lalithamaaya modyoolinte unmoolanam aanenkil maathrame athu aadimamaakoo. mukalil nirvachicha ardhathil aadimamaanengil maathrame oru sii *-beejaganita "eiii" kku oru ideal beejaganithaparamaayi aadimamaakoo.
dixmear referencinte 2.9.5 sidhaanthathinte anantharafalamaanithu.
jii" praadeshikamaayi othungunna oru grouppaanengil, "jii" yude sii *-aljibra sii * ("jii") grouppinte eratta spacile topology jeey enna peril peritta phaal topology ennu vilikkunnu. emm. jii. veenu.

