Algebraic & Geometric Topology

Real versus complex K–theory using Kasparov's bivariant KK–theory

Thomas Schick

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In this paper, we use the KK–theory of Kasparov to prove exactness of sequences relating the K–theory of a real C–algebra and of its complexification (generalizing results of Boersema). We use this to relate the real version of the Baum-Connes conjecture for a discrete group to its complex counterpart. In particular, the complex Baum–Connes assembly map is an isomorphism if and only if the real one is, thus reproving a result of Baum and Karoubi. After inverting 2, the same is true for the injectivity or surjectivity part alone.

Article information

Algebr. Geom. Topol., Volume 4, Number 1 (2004), 333-346.

Received: 24 November 2003
Accepted: 29 May 2004
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19K35: Kasparov theory ($KK$-theory) [See also 58J22] 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX}

real $K$–theory complex $K$–theory bivariant $K$–theory


Schick, Thomas. Real versus complex K–theory using Kasparov's bivariant KK–theory. Algebr. Geom. Topol. 4 (2004), no. 1, 333--346. doi:10.2140/agt.2004.4.333.

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