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

Abstract

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 $SU_q(2).$