Idempotents in complex group rings: theorems of Zalesskii and Bass revisited

Abstract: Let $\Gamma$ be a group, and let ${\scriptstyle\C}\Gamma$ be the group ring of $\Gamma$ over ${\scriptstyle\C}$. We first give a simplified and self-contained proof of Zalesskii's theorem \cite{Zal} that the canonical trace on ${\scriptstyle\C}\Gamma$ takes rational values on idempotents. Next, we contribute to the conjecture of idempotents by proving the following result: for a group $\Gamma$, denote by $P_{\Gamma}$ the set of primes $p$ such that $\Gamma$ embeds in a finite extension of a pro-$p$-group; if $\Gamma$ is torsion-free and $P_{\Gamma}$ is infinite, then the only idempotents in ${\scriptstyle\C}\Gamma$ are 0 and 1. This implies Bass' theorem \cite{Bas} asserting that the conjecture of idempotents holds for torsion-free subgroups of $ GL_{n}({\scriptstyle\C})$.

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