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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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