We formulate a version of Baum-Connes' conjecture for a discrete quantum group, building on our earlier work (\cite{ GK}). Given such a quantum group $\cla$, we construct a directed family $\{ \cle_F \}$ of $C^*$-algebras ($F$ varying over some suitable index set), borrowing the ideas of \cite{ cuntz}, such that there is a natural action of $\cla$ on each $\cle_F$ satisfying the assumptions of \cite{ GK}, which makes it possible to define the "analytical assembly map", say $\mu^{ r,F}_i$, $i=0,1,$ as in \cite{ GK}, from the $\cla$-equivariant $K$-homolgy groups of $\cle_F$ to the $K$-theory groups of the "reduced" dual $\hat{ \cla_r}$ (c.f. \cite{ GK} and the references therein for more details). As a result, we can define the Baum-Connes' maps $\mu^r_i : \stackrel{ \rm lim}{ \longrightarrow} KK_i^\cla(\cle_F,\IC) \raro K_i(\hat{ \cla_r}),$ and in the classical case, i.e. when $\cla$ 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 $SU_q(2).$
Categories: math.KT math.OA