Determination of the distance from a projection to nilpotents

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 $\m...

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Bibliographic Details
Main Authors Izumi, Masaki, Mori, Michiya
Format Journal Article
LanguageEnglish
Published 13.06.2024
Subjects
Mathematics - Functional Analysis
Mathematics - Operator Algebras
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Summary: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.
DOI:10.48550/arxiv.2406.09234