keey. keey.-siddhant

ganitat, 'keey. keey.'-siddhant heey keey-homology aanee keey-siddhant yaa donhinche eka samanya samanyikaran aahe, jeey vegale karata yenyajoge see *-algebrasvar eka yogatmak dwibhinna ghatak mhanoon aahe. hii kalpana 1980 madhye russian ganitshastradnya gennadi kasparov yanni mandali.
aatiyah-singer nirdeshank prameyasathi fredholm moduleschya atiyahchya sankalpanecha aanee laurence jee dware see *-algebraschya vistaranche vargikaran yacha prabhav hota. braun, ronald jee. 1977 madhye dugglus aanee pieter arther fillmore. parinami, nirdeshank siddhant aanee anoo see *-algebraschya vargikaranakade operator bijganitiy aupchariktet tyala mothe yash midale aahe, kaaran operator keey-siddhantatil anek samasyanche nirakaran karanyachi tee gurukilli hotee, udaharanarth, keval "keey"-gatanchi ganana. yavyatirikt, bomm-cons anuman vikasit karnyasathi heey aavashyak hote aanee gair-vinimayatmak topologymadhye mahattvapoorn bhoomika bajavate.
"kk"-siddhantanantar "e"-siddhant aanee dwibhinna aavartak chakriy siddhant yasarkhya samaan dvi-ghatak sanrachananchi malika aalee, tyapaiki bahutekanmadhye adhik shreni-saiddhantik swad aahet, kinwa vegale karata yenyajogya "c" *-aljebrapeksha algebranchya dusarya vargasandarbhat kinwa gut krutincha samavesh aahe.
vyakhya.
khaleel vyakhya moolatah kasparovne dilelya vyakhyechya agadi jawal aahe. heey asey swaroop aahe jyamadhye anuprayoganmadhye bahutek keey. keey.-ghatak udbhavtat.
a" aanee "b" heey vegale karata yenyajoge "c" *-algebras asoo dya, jethe "b" dekhil σ-ekatma asalyache gruheet dharale jaate. chakrancha sanch haah tihericha sanch aahe ("h", "ρ", f "), jethe" h "haah" b "varr gananiyapane tayar kelela shrenibaddha hilbert module aahe, ρ haah" a "chey" h "varr sambaddh operator mhanoon pratinidhitva aahe jeey" b "saha pravas karatat aanee" f "haah 1 vya degreechya" h "varr eka baddh operator aahe joou punha" b "saha pravas karato. tyanni hii aut purn karane aavashyak aahe qee
aaa" madheel "aaa" sathee serve "buub"-compact operator aahet. jarr tinhi abhivyakti serve 'aaa' sathee 0 asateel tarr chakr adhogatishil mhatale jaate.
done chakranna samaroop kinwa samaroop mhatale jaate, jarr "aaa" aanee "aay. bii.". darmyan eka chakr asel, jethe "aay. bii."............................................................................................................................................................................................................................
tyanantar a aanee b darmyancha kk-gut kk (a, b) haah chakrancha sanch modulo samarupi mhanoon paribhashit keela jaato. jode mhanoon bimoduleschya thet berries operation antargat aanee tyache tatasth ghatak mhanoon degenerate modulescha varg mhanoon too eka abelian gut banato.
keey. keey.-siddhantachya vividh, parantu samatulya vyakhya aahet, visheshtah joachim kuntzamule jyaa chitratoon bimodule aanee 'fredholm' operator eff kadhoon taktat aanee uchchar purnapane samarupatevar ρ thevatat. adhik tantotantpane tyachi vyakhya samarupi vargancha sanch mhanoon kelee jaau shakate.
ardh-samarupatechya vargikaran bijganit 'qa' pasoon 'b' neey tanalelya anant aayami vegale karata yenyajogya hilbert jagechya sankshipt prachalakanchya 'c' *-bijganitaparyant samarupateche vargikaran. yethe, 'qa' chee vyakhya, 'qa' chya 'aaa' chya 'aaa' * 'aaa' yaa bijganitiy mukt utpadanatun donhi ghatakanchya olakhadvare paribhashit kelelya 'aaa' paryantchya nakashatil curnell mhanoon kelee jaate.
malmatta.
jevha ekhadi vyakti jatil sankhyanche 'c' *-bijganit c heey 'kk' (c, b) madhye 'kk' chey pahile yuktivad mhanoon ghete, tevha haah yogatmak gut naisargikaritya naisargikaritya yuktivad 'b' chya 'k' 0-gut 'k' 0 ('b') sathee samaroop asato. kuntzchya drushtikonatun, 'buub' chaa 'keey' 0-varg haah aanakhi kahee nasoon jatil sankhyanpasun 'buub' chya sthirikaranaparyantchya 'samarupata' chaa samarupata varg aahe. tyachpramane jevha konee pahila yuktivad mhanoon anantatevar kshay hott asalelya vastavik reshevaril satatchya karyanche bijganit "c" 0 (r) gheto, tevha prapt gut "kk" ("c" 0 (r), "b") naisargikaritya "k" 1 ("b") sathee samaroop aahe.
keey. keey". chaa eka mahattvacha gunadharm-tathakathit kasparov utpadan kinwa rachana utpadan haah siddhant aahe,
jeey mishrit gut sanrachanechya sandarbhat dvirekheey aahe. visheshtah "kk" ("a", "b") chaa pratyek ghatak "k" * ("a") → "k" * ("b") aanee dusra samarupata "k" * ("b") → "k" * ("a") haah samarupata deto.
qa" teey "a" aanee "b" teey "k" ("h") Â "b" paryantche naisargik nakashe aahet jeey "kk"-samatulyates pravrutt karatat, heey lakshat gheta utpadan adhik sahajpane kuntz chitrat paribhashit kele jaau shakate.
rachana utpadan eka naveen shreni sootra dete, jyachya vastoo vegalya karata yenyajogya "see" *-aljebradware dilya jaatat tarr tyanchyamadhil aakarman sambandhit keey. keey.-gatanchya ghatkandware dile jaatat. yavyatirikt, 'aaa' chaa 'buub' madhye konatahi 'samarupata' keey. keey. '(' aii ',' bii ') chaa ghatak prerit karate aanee haah patravyavahar vegale karata yenyajogya' see '*-algebranchya mool shrenitil eka functor sootra 4 madhye deto. algebranche andaje antargat automorphism sootra 4 madhye olakh swaroop banatat.
heey functor sootra _ 7 heey vibhajya 'see' *-algebraschya shrenitil vibhajan-achook, samarupi aparivartaniy aanee sthir yojak functormadhye sarvatrik aahe. assa konatahi siddhant yogya arthane bott aavartakta purn karato kaaran sootra _ 4 karate.
kasparov utpadanache pudhil swarupat samanyikaran kele jaau shakateh
yaat keval keey-saiddhantik cup utpadanach navhe tarr keey-saiddhantik topi, cross aanee tirkas utpadane aanee vistaranche utpadan dekhil vishesh prakarane mhanoon samavisht aahe.

