Heh, you've picked an open problem: this is the Kadison-Kaplansky conjecture... I would answer it, but first I have to find a sufficiently big margin in which to write the proof.
To be less flippant, it is known to follow (but I don't understand exactly how) from the Baum-Connes conjecture: thus, if a torsion-free discrete group satisfies BC, then its reduced group C*-algebra contains no non-trivial projections.
Trying to answer this question was, I think, one of the original motivations of Connes and others in some of the older work on cyclic cohomology and souped-up versions thereof. See e.g.
M. Puschnigg, The Kadison-Kaplansky conjecture for word-hyperbolic groups. Invent. Math. 149 (2002), no. 1, 153--194.
for some relatively recent work on those lines. Since I'm not an expert, I'd suggest Googling some combination of Kadison-Kaplansky and Baum-Connes and going from there.