TITLE: Kaplansky's idempotent conjecture for Thompson's group F
QUESTION [8 upvotes]: Let $K$ be a field and $G$ be a torsion-free group. Kaplansky's idempotent conjecture states that the group ring $K[G]$ does not contain any non-trivial idempotent, i.e. if $x^2=x$ then $x=0$ or $x=1$.
Is Kaplansky's idempotent conjecture known for Thompson's group $F$?

REPLY [6 votes]: They satisfy Baum-Connes conjecture, so by surjevtivity of the assembly map, the Thompson's group $F$ satisfies even the stronger conjecture of Kaplansky and Kadison: the reduced group $C^*$-algebra has no idempotents or projections.