Algebraic & Geometric Topology

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

Thomas Schick

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

Permanent link to this document
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

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

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

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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References

  • M F Atiyah, $K$-theory and reality, Quart. J. Math. Oxford Ser. (2) 17 (1966) 367–386,
  • Bruce Blackadar, $K$-theory for operator algebras, volume 5 of Mathematical Sciences Research Institute Publications, second edition, Cambridge University Press, Cambridge (1998)
  • Jeffrey L Boersema, Real C*-algebras, United KK-theory, and the Universal Coefficient Theorem.
  • Jeffrey L Boersema, The Range of United K-Theory, preprint (2003) \arxivmath.OA/0310209
  • Jeffrey L Boersema, Real $C\sp *$-algebras, united $K$-theory, and the Künneth formula, $K$-Theory 26 (2002) 345–402,
  • A K Bousfield, A classification of $K$-local spectra, J. Pure Appl. Algebra 66 (1990) 121–163,
  • Henri Cartan, Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J. (1956)
  • Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974) 1–55,
  • Max Karoubi, A descent theorem in topological $K$-theory, $K$-Theory 24 (2001) 109–114
  • G G Kasparov, The operator $K$-functor and extensions of $C\sp{\ast} $-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980) 571–636, 719,
  • G G Kasparov, Equivariant $KK$-theory and the Novikov conjecture, Invent. Math. 91 (1988) 147–201,
  • John McCleary, User's guide to spectral sequences, volume 12 of Mathematics Lecture Series, Publish or Perish Inc., Wilmington, DE (1985)
  • Paul Baum and Max Karoubi, On the Baum-Connes conjecture in the real case, Preprint, October 9, 2003, K-theory Preprint Archives, available from: http://www.math.uiuc.edu/K-theory/0658/
  • Paolo Piazza, Thomas Schick, Bordism and rho invariants, in preparation
  • John Roe, Index theory, coarse geometry, and topology of manifolds, volume 90 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC (1996),
  • Herbert Schr öder, $K$-theory for real $C\sp *$-algebras and applications, volume 290 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow (1993)