TITLE: Integrality of the canonical trace and topology
QUESTION [7 upvotes]: Let $G$ be a discrete group and consider the reduced group C* algebra $C_r^\ast(G)$, viewed as an algebra of bounded operators on $\ell^2(G)$ by the regular representation.  The canonical trace on $C_r^\ast(G)$ is simply defined to be: $\tau(T) = (T \delta_e, \delta_e)$ where $\delta_e$ is the basis vector of $\ell^2(G)$ corresponding to the identity.  This trace gives rise to a homomorphism $\tau_\ast: K_0(C_r^\ast(G)) \to \mathbb{R}$ on K-theory.
It is a very deep topological fact that the image of $\tau_\ast$ consists of integers for a huge class of groups, specifically those groups for which the Baum-Connes conjecture is true.  To see this, let $M$ be a closed manifold with fundamental group $G$, let $\mu: K_0(M) \to K_0(C_r^\ast(G))$ be the assembly map on K-homology, and note that $\tau_\ast \circ \mu$ agrees with the (integer valued) index map $K_0(M) \to \mathbb{R}$.  Thus $\tau_\ast$ is integer-valued so long as the assembly map is surjective, and the Baum-Connes conjecture asserts that it is an isomorphism.  
Here are my questions:

Are there easy proofs of the integrality of $\tau_\ast$ in (nontrivial) special cases, or at least proofs that don't resort to heavy-duty topological machinery?

Even better,

Are there groups for which the Baum-Connes conjecture is not known but $\tau_\ast$ is known to be integer-valued?

These two questions stem from my intuition that this should be a problem in analysis or possibly representation theory rather than topology.  But in the likely event that this intuition is wrong...

Is the integrality of $\tau_\ast$ related (or equivalent) to any more transparently topological phenomena?

REPLY [4 votes]: For abelian groups $G$ (maybe a trivial case?), the canonical trace is given by integration over the Pontryagin dual $\hat{G}$ (with respect to normalized Haar measure), so we have equivalences:
(1) $\tau_*$ is integer-valued on $K_0(C(\hat{G}))$;
(2) the Kaplansky-Kadison conjecture holds for $C^*_r(G)$;
(3) $\hat{G}$ is connected;
(4) $G$ is torsion-free.
In general, for non-abelian $G$, proving the integrality of the trace is done by means of an index theorem, so that it is deeply related to surjectivity of the Baum-Connes map. This is to say that, to my knowledge, there are no examples of groups where integrality of the trace is known, but Baum-Connes is not. There are a few cases however, where the Kaplansky-Kadison conjecture has been established independently of the Baum-Connes conjecture. This is the case for torsion-free hyperbolic groups, see Michael Puschnigg, The Kadison-Kaplansky conjecture for word-hyperbolic groups. 
Invent. Math. 149, no. 1, 153--194, 2002.