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?