A characterization of KK-theory

Research output: Contribution to journalArticle

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Abstract

We characterize the KK-groups of G. G. Kasparov, along with the Kasparov product KK(A, B) × KK(B, C) → KK(A, C), from the point of view of category theory (in a very elementary sense): the product is regarded as a law of composition in a category and we show that this category is the universal one with "homotopy invariance", "stability" and "split exactness". The third property is a weakened type of half-exactness: it amounts to the fact that the KK-groups transform split exact sequences of C*-algebras to split exact sequences of abelian groups. The method is borrowed from Joachim Cuntz’s approach to KK-theory, in which cycles for KK(A, B) are regarded as generalized homomorphisms from A to B: the results follow from an analysis of the Kasparov product in this light.

Original languageEnglish (US)
Pages (from-to)253-276
Number of pages24
JournalPacific Journal of Mathematics
Volume126
Issue number2
DOIs
StatePublished - Jan 1 1987

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KK-theory
Exactness
Exact Sequence
Homotopy Invariance
Category Theory
Homomorphisms
C*-algebra
Abelian group
Transform
Cycle

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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A characterization of KK-theory. / Higson, Nigel.

In: Pacific Journal of Mathematics, Vol. 126, No. 2, 01.01.1987, p. 253-276.

Research output: Contribution to journalArticle

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