{"title":"定位\\(C^*-\\)代数和索引配对","authors":"Hang Wang, Chaohua Zhang, Dapeng Zhou","doi":"10.1007/s40062-022-00320-z","DOIUrl":null,"url":null,"abstract":"<div><p>Kasparov <i>KK</i>-theory for a pair of <span>\\(C^*\\)</span>-algebras <span>\\((A,\\,B)\\)</span> can be formulated equivalently in terms of the <i>K</i>-theory of Yu’s localization algebra by Dadarlat-Willett-Wu. We investigate the pairings between <i>K</i>-theory <span>\\(K_j(A)\\)</span> and the two notions of <i>KK</i>-theory which are Kasparov <i>KK</i>-theory <span>\\(KK_i(A,B)\\)</span> and the localization algebra description of <span>\\(KK_i(A,B)\\)</span> and show that the two pairings are compatible.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Localization \\\\(C^*-\\\\)algebras and index pairing\",\"authors\":\"Hang Wang, Chaohua Zhang, Dapeng Zhou\",\"doi\":\"10.1007/s40062-022-00320-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Kasparov <i>KK</i>-theory for a pair of <span>\\\\(C^*\\\\)</span>-algebras <span>\\\\((A,\\\\,B)\\\\)</span> can be formulated equivalently in terms of the <i>K</i>-theory of Yu’s localization algebra by Dadarlat-Willett-Wu. We investigate the pairings between <i>K</i>-theory <span>\\\\(K_j(A)\\\\)</span> and the two notions of <i>KK</i>-theory which are Kasparov <i>KK</i>-theory <span>\\\\(KK_i(A,B)\\\\)</span> and the localization algebra description of <span>\\\\(KK_i(A,B)\\\\)</span> and show that the two pairings are compatible.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-11-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40062-022-00320-z\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40062-022-00320-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Kasparov KK-theory for a pair of \(C^*\)-algebras \((A,\,B)\) can be formulated equivalently in terms of the K-theory of Yu’s localization algebra by Dadarlat-Willett-Wu. We investigate the pairings between K-theory \(K_j(A)\) and the two notions of KK-theory which are Kasparov KK-theory \(KK_i(A,B)\) and the localization algebra description of \(KK_i(A,B)\) and show that the two pairings are compatible.
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