Towards the Baum-Connes' Analytical Assembly Map for the Actions of Discrete Quantum Groups

Abstract

Given an action of a discrete quantum group (in the sense of Van Daele, Kustermans and Effros-Ruan) ${\cal A}$ on a $C^*$-algebra ${\cal C}$, satisfying some regularity assumptions resembling the proper $\Gamma$-compact action for a classical discrete group $\Gamma$ on some space, we are able to construct canonical maps $\mu^i_r$ (\$\mu_i$ respectively) ($i=0,1$) from the ${\cal A}$-equivariant K-homology groups $KK_i^{{\cal A}}({\cal C}, C |)$ to the K-theory groups $K_i(\hat{{\cal A}_r})$ ($K_i(\hat{{\cal A}})$ respectively), where $\hat{{\cal A}_r}$ and $\hat{{\cal A}}$ stand for the quantum analogues of the reduced and full group $C^*$-algebras (c.f. [11],[6]). We follow the steps of the construction of the classical Baum-Connes map, although in the context of quantum group the nontrivial modular property of the invariant weights (and the related fact that the square of the antipode is not identity) has to be taken into serious consideration, making it somewhat tricky to guess and prove the correct definitions of relevant Hilbert module structures.