Generalizations of the Baum-Connes Conjecture

The "basic" Baum-Connes conjecture proposes a formula for the K-theory of reduced group C*-algebras. But from the beginning various generalizations have been formulated.

A useful generalization is to consider a C*-algebra A on which G acts by automorphisms, and propose a formula for the K-theory of the reduced crossed product of G by A. This is the Baum-Connes conjecture with coefficients. One if its features is that the conjecture with coefficients for G implies the conjecture with coefficients for all the closed subgroups of G.

The original work of Baum and Connes considered foliation C*-algebras. The C*-algebra of a foliation is the C*-algebra of a certain locally compact groupod (the holonomy groupoid of the foliation), and it is therefore natural to formulate a conjecture for more general groupoids.

There is a Baum-Connes conjecture for coarse geometric spaces that is inspired by research in controlled topology. Apart from the references below, see the page on coarse geometry for more information on this. The beautiful article of Skandalis, Yu and Tu shows that the coarse Baum-Connes conjecture is in fact a special case of the Baum-Connes conjecture for groupoids.

Unfortunately there are now counterexamples to all of these generalized conjectures. At the moment the counterexamples are rather exotic, and so there is hope that some realm of "reasonable" examples can be identified within which the generalized conjectures are true.

Selected References

Baum, Paul; Connes, Alain; Higson, Nigel. Classifying space for proper actions and K-theory of group C*-algebras. C*-algebras: 1943-1993 (San Antonio, TX, 1993), 240-291, Contemp. Math., 167, Amer. Math. Soc., Providence, RI, 1994.   MR
Baum, Paul; Karoubi, Max. On the Baum-Connes conjecture in the real case. Q. J. Math. 55 (2004), no. 3, 231-235.   MR
Dranishnikov, A. N.; Ferry, Steven C.; Weinberger, Shmuel. Large Riemannian manifolds which are flexible. Ann. of Math. (2) 157 (2003), no. 3, 919-938.   MR
Goswami, Debashish; Kuku, A. O. A complete formulation of the Baum-Connes conjecture for the action of discrete quantum groups. Special issue in honor of Hyman Bass on his seventieth birthday. Part IV. K-Theory 30 (2003), no. 4, 341-363.   MR
Higson, N.; Lafforgue, V.; Skandalis, G. Counterexamples to the Baum-Connes conjecture. Geom. Funct. Anal. 12 (2002), no. 2, 330-354.   MR
Roe, John. Warped cones and property A. Geom. Topol. 9 (2005), 163-178 (electronic).   MR
Skandalis, G.; Tu, J. L.; Yu, G. The coarse Baum-Connes conjecture and groupoids. Topology 41 (2002), no. 4, 807-834.   MR
Torpe, Anne Marie. K-theory for the leaf space of foliations by Reeb components. J. Funct. Anal. 61 (1985), no. 1, 15-71.   MR
Tu, Jean-Louis. The Baum-Connes conjecture for groupoids. C*-algebras (Muenster, 1999), 227-242, Springer, Berlin, 2000.   MR

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