Kaplansky's idempotent conjecture for torsion-free groups in characteristic zero

Corollary 1.2(ii): for every torsion-free group $G$ and every commutative unital domain $R$ of characteristic zero, the only idempotents of $RG$ are $0$ and $1$. Part (i): for torsion-free $G$ the canonical trace of an idempotent matrix over $\mathbb CG$ is the rank of its augmentation, so $\tau_*(K_0(\mathbb CG))=\mathbb Z$. Both are deduced from the Bass trace theorem. Nothing is claimed in positive characteristic or about projections in $C^*_r(G)$; the release separately claims a torsion-free counterexample to the Kadison-Kaplansky projection conjecture there, which is consistent with this result.

Corollary 1.2(ii): for every torsion-free group $G$ and every commutative unital domain $R$ of characteristic zero, the only idempotents of $RG$ are $0$ and $1$. Part (i): for torsion-free $G$ the canonical trace of an idempotent matrix over $\mathbb CG$ is the rank of its augmentation, so $\tau_*(K_0(\mathbb CG))=\mathbb Z$. Both are deduced from the Bass trace theorem. Nothing is claimed in positive characteristic or about projections in $C^*_r(G)$; the release separately claims a torsion-free counterexample to the Kadison-Kaplansky projection conjecture there, which is consistent with this result.