#### Algebraic and Geometric Topology 4 (2004),
paper no. 18, pages 333-346.

## 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.
**Keywords**.
Real K-theory, complex K-theory, bivariant K-theory

**AMS subject classification**.
Primary: 19K35, 55N15.

**DOI:** 10.2140/agt.2004.4.333

**E-print:** `arXiv:math.KT/0311295`

Submitted: 24 November 2003.
Accepted: 29 May 2004.
Published: 29 May 2004.

Notes on file formats
Thomas Schick

Fachbereich Mathematik, Georg-August-Universitaet

Goettingen, Germany

Email: schick@uni-math.gwdg.de

URL: http://www.uni-math.gwdg.de/schick

AGT home page

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