Localization \(C^*-\)algebras and index pairing

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.