A Complete Formulation of Baum-Conens' Conjecture for the Action of Discrete Quantum Groups

الأصل · EN

We formulate a version of Baum-Connes' conjecture for a discrete quantum group, building on our earlier work (GK). Given such a quantum group, we construct a directed family { } of C*-algebras (F varying over some suitable index set), borrowing the ideas of cuntz, such that there is a natural action of on each satisfying the assumptions of GK, which makes it possible to define the "analytical assembly map", say μʳ,ᶠᵢ, i=0,1, as in GK, from the -equivariant K-homolgy groups of to the K-theory groups of the "reduced" dual ᵣ (c.f. GK and the references therein for more details). As a result, we can define the Baum-Connes' maps μʳᵢ: lim KKᵢ(,) Kᵢ(ᵣ), and in the classical case, i.e. when is C₀(G) for a discrete group, the isomorphism of the above maps for i=0,1 is equivalent to the Baum-Connes' conjecture. Furthermore, we verify its truth for an arbitrary finite dimensional quantum group and obtain partial results for the dual of SUq(2).