Determination of the distance from a projection to nilpotents

TL;DR

This paper calculates the exact distance from a projection to nilpotent elements in a factor algebra with a trace, resolving a conjecture and providing new insights even for matrix algebras.

Contribution

It provides a closed-form formula for the distance from a projection to nilpotents in a factor with a trace, including the first proof of a conjecture by Z. Cramer.

Findings

Distance formula: $(2 ext{cos} rac{ au(P) ext{pi}}{1+2 au(P)})^{-1}$

Result applies to matrix algebras as a special case

Settles a conjecture by Z. Cramer

Abstract

In this note, we study the distance from an arbitrary nonzero projection $P$ to the set of nilpotents in a factor $\mathcal{M}$ equipped with a normal faithful tracial state $\tau$. We prove that the distance equals $(2\cos \frac{\tau(P)\pi}{1+2\tau(P)})^{-1}$. This is new even in the case where $\mathcal{M}$ is the matrix algebra. The special case settles a conjecture posed by Z. Cramer.