Inner ideals, compact tripotents and \v{C}eby\v{s}\"ev subtriples of JB$^{*}$-triples and C$^*$-algebras

TL;DR

This paper classifies Chebyshev JB*-subtriples of JB*-triples, revealing that such subtriples are either rank-one Hilbert spaces, generated by a tripotent, rank-two triples, or the entire triple if rank is three or more.

Contribution

It provides a complete classification of Chebyshev JB*-subtriples in JB*-triples, detailing their structure and rank-related properties.

Findings

Chebyshev JB*-subtriples are either rank-one, generated by a tripotent, rank-two, or the whole triple for rank ≥ 3.

Characterization of Chebyshev subtriples based on rank and structure.

Identification of conditions under which subtriples are uniquely determined.

Abstract

The aim of this note is to study \v{C}eby\v{s}\"ev JB$^*$-subtriples of general JB$^*$-triples. It is established that if $F$ is a non-zero \v{C}eby\v{s}\"ev JB$^*$-subtriple of a JB$^*$-triple $E$, then exactly one of the following statements holds:\begin{enumerate}\item $F$ is a rank one JBW$^*$-triple with dim$(F)\geq 2$ (i.e. a complex Hilbert space regarded as a type 1 Cartan factor). Moreover, $F$ may be a closed subspace of arbitrary dimension and $E$ may have arbitrary rank, \item $F= \mathbb{C} e$, where $e$ is a complete tripotent in $E$, \item $E$ and $F$ are rank two JBW$^*$-triples, but $F$ may have arbitrary dimension, \item $F$ has rank greater or equal than three and $E=F$. \end{enumerate}