The Kadison-Kaplansky Projection Conjecture

There is a finitely generated torsion-free group $G=G_0*\mathbb{F}_N$ and a projection $e\in C_r^*(G)$ with $0<\tau(e)<1/2$, so $e\ne 0,1$. It is built from a matrix projection with small unnormalized trace over a graphical small-cancellation group, compressed to a scalar projection by free products. Not shown: a counterexample to the algebraic Kaplansky idempotent conjecture (the group ring has no nontrivial idempotents), or a finitely presented group.

There is a finitely generated torsion-free group $G=G_0*\mathbb{F}_N$ and a projection $e\in C_r^*(G)$ with $0<\tau(e)<1/2$, so $e\ne 0,1$. It is built from a matrix projection with small unnormalized trace over a graphical small-cancellation group, compressed to a scalar projection by free products. Not shown: a counterexample to the algebraic Kaplansky idempotent conjecture (the group ring has no nontrivial idempotents), or a finitely presented group.