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

Open Access article
 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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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 KK-theory, UCT
MSC Classifications: 46L35 show english descriptions Classifications of $C^*$-algebras 46L35 - Classifications of $C^*$-algebras

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