## Algebraic & Geometric Topology

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

Thomas Schick

#### Abstract

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

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

Dates
Accepted: 29 May 2004
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882480

Digital Object Identifier
doi:10.2140/agt.2004.4.333

Mathematical Reviews number (MathSciNet)
MR2077669

Zentralblatt MATH identifier
1050.19003

#### Citation

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. https://projecteuclid.org/euclid.agt/1513882480

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