A complete formulation of the Baum-Connes conjecture for the action of discrete quantum groups

We formulate a version of the Baum-Connes conjecture for a discrete quantum group, building on our earlier work. Given such a quantum group A, we construct a directed family {E-F} of C*- algebras (F varying over some suitable index set), borrowing the ideas of Cuntz such that there is a natural action of A on each E-F satisfying the assumptions of Goswami and Kuku which makes it possible to define the 'analytical assembly map', say mu(i)(r,F), i = 0, 1, as in our previous work, from the A-equivariant K-homolgy groups of E-F to the K-theory groups of the 'reduced' dual (A) over cap (r) (c.f. [9] and the references therein for more details). As a result, we can define the Baum - Connes maps mu(i)(r) : lim KKiA (E-F, C) --> K-i ((A) over cap (r)), and in the classical case, i. e. when A is C-0(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).