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# Non-splitting in Kirchberg's Ideal-related $KK$-Theory

Published:2010-08-03
Printed: Mar 2011
• Søren Eilers,
Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark
• Gunnar Restorff,
Faculty of Science and Technology, University of Faroe Islands, Tórshavn, Faroe Islands
• Efren Ruiz,
Department of Mathematics, University of Hawaii Hilo, Hilo, Hawaii, U.S.A
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## Abstract

A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's ideal-related $KK$-theory in the fundamental case of a $C^*$-algebra with one specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain conditions. Employing certain $K$-theoretical information derivable from the given operator algebras using a method introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the $K$-theory which must be used to classify $*$-isomorphisms for purely infinite $C^*$-algebras with one non-trivial ideal.
 Keywords: KK-theory, UCT
 MSC Classifications: 46L35 - Classifications of $C^*$-algebras