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Real versus complex K-theory using Kasparov’s bivariant KK-theory. (English) Zbl 1050.19003
Generalizing a result of J. L. Boersema [K-Theory 26, 345–402 (2002; Zbl 1024.46021)], the author proves the exactness of the sequences of type \[ \dots \to KKO^\Gamma_{q-1}(A;B)@>\chi>> KKO^\Gamma_q(A;B)@>c>> KK_q^\Gamma(A_\mathbb C;B_\mathbb C) @>\delta>> KKO^\Gamma_{q-2}(A;B) \to \dots \] relating KK-theory of a real C*-algebra to its complexification (Theorem 2.16) for an arbitrary discrete group \(\Gamma\) acting on a \(\sigma\)-unital C*-algebra \(B\), and a C*-algebra \(A\). The particular cases of this relation let the author to compare the real version with the complex version of the Baum-Connes conjecture (Corollary 2.13): The real Baum-Connes conjecture is true if and only if the complex Baum-Connes conjecture is true.

MSC:
19K35 Kasparov theory (\(KK\)-theory)
55N15 Topological \(K\)-theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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References:
[1] M F Atiyah, \(K\)-theory and reality, Quart. J. Math. Oxford Ser. \((2)\) 17 (1966) 367 · Zbl 0146.19101 · doi:10.1093/qmath/17.1.367
[2] B Blackadar, \(K\)-theory for operator algebras, Mathematical Sciences Research Institute Publications 5, Cambridge University Press (1998) · Zbl 0913.46054
[3] J L Boersema, Real \(C^*\)-algebras, united \(KK\)-theory, and the universal coefficient theorem, \(K\)-Theory 33 (2004) 107 · Zbl 1075.19001 · doi:10.1007/s10977-004-1961-1
[4] J L Boersema, The range of united \(K\)-theory, J. Funct. Anal. 235 (2006) 701 · Zbl 1102.46046 · doi:10.1016/j.jfa.2005.12.012
[5] J L Boersema, Real \(C^*\)-algebras, united \(K\)-theory, and the Künneth formula, \(K\)-Theory 26 (2002) 345 · Zbl 1024.46021 · doi:10.1023/A:1020671031447
[6] A K Bousfield, A classification of \(K\)-local spectra, J. Pure Appl. Algebra 66 (1990) 121 · Zbl 0713.55007 · doi:10.1016/0022-4049(90)90082-S
[7] H Cartan, S Eilenberg, Homological algebra, Princeton University Press (1956) · Zbl 0075.24305
[8] N Hitchin, Harmonic spinors, Advances in Math. 14 (1974) 1 · Zbl 0284.58016 · doi:10.1016/0001-8708(74)90021-8
[9] M Karoubi, A descent theorem in topological \(K\)-theory, \(K\)-Theory 24 (2001) 109 · Zbl 1005.19003 · doi:10.1023/A:1012785711074
[10] G G Kasparov, The operator \(K\)-functor and extensions of \(C^*\)-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980) 571, 719
[11] G G Kasparov, Equivariant \(KK\)-theory and the Novikov conjecture, Invent. Math. 91 (1988) 147 · Zbl 0647.46053 · doi:10.1007/BF01404917 · eudml:143536
[12] J McCleary, User’s guide to spectral sequences, Mathematics Lecture Series 12, Publish or Perish (1985) · Zbl 0577.55001
[13] P Baum, M Karoubi, On the Baum-Connes conjecture in the real case, Q. J. Math. 55 (2004) 231 · Zbl 1064.19003 · doi:10.1093/qjmath/55.3.231
[14] P Piazza, T Schick, Bordism, rho-invariants and the Baum-Connes conjecture, J. Noncommut. Geom. 1 (2007) 27 · Zbl 1158.58012 · doi:10.4171/JNCG/2
[15] J Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics 90, Published for the Conference Board of the Mathematical Sciences, Washington, DC (1996) · Zbl 0853.58003
[16] H Schröder, \(K\)-theory for real \(C^*\)-algebras and applications, Pitman Research Notes in Mathematics Series 290, Longman Scientific & Technical (1993) · Zbl 0785.46056
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