Kaplansky idempotent conjecture

Remarks

The conjecture is one of Kaplansky's conjectures on group rings, and it is open in general. It follows from the zero divisor conjecture for $K[G]$, since an idempotent $e$ satisfies $e(1 - e) = 0$. Torsion-freeness is needed: if $g \in G$ has finite order $n > 1$ and $n$ is invertible in $K$, then $e = \frac{1}{n}(1 + g + \cdots + g^{n-1})$ is an idempotent other than $0$ and $1$. For $K = \C$ it is implied by the Kadison–Kaplansky conjecture for the reduced group C*-algebra, which follows from the Baum–Connes conjecture. If $K$ has characteristic zero, it holds for every torsion-free group $G$ for which the K-theoretic Farrell–Jones conjecture holds for $K[G]$; this includes all torsion-free hyperbolic groups. For $K = \C$, Kaplansky proved that the coefficient of the identity element in an idempotent $e$ is a real number between $0$ and $1$. He also proved that this coefficient is $0$ only when $e = 0$.