{"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}
引用次数: 0
Abstract
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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