Kaplansky Idempotent Conjecture

Kaplansky's idempotent conjecture asserts that for a torsion-free group $G$, the complex group algebra $\mathbb C[G]$ has no idempotents besides the trivial ones: if $e^2=e$, then $e=0$ or $e=1$. Like Kaplansky's other group-ring problems it circulated informally, around 1950, before appearing in his 1957 problem list, so the date is conventional [Kaplansky1957Problems].

The conjecture sits below several stronger statements. It follows from the zero-divisor conjecture, since a nontrivial idempotent satisfies $e(1-e)=0$, and in appropriate settings it is implied by the Baum-Connes conjecture, which yields the idempotent statement for the many groups for which that conjecture is known [Valette2002BaumConnes]. Analytic and $L^2$-methods, surveyed by Lück, confirm it for many further geometric classes of groups [Lueck2002L2]. The Kadison-Kaplansky conjecture, asserting the same triviality of idempotents in the reduced group $C^*$-algebra, is a stronger analytic statement.

For an arbitrary torsion-free group the question remains open; a resolution requires either a torsion-free group whose complex group algebra contains a nontrivial idempotent, or an argument free of geometric and analytic hypotheses.