Inner ideals, compact tripotents and Čebyšëv subtriples of JB$^{*}$-triples and C$^*$-algebras

Abstract

The aim of this note is to study Čebyšëv JB$^*$-subtriples of general JB$^*$-triples. It is established that if $F$ is a non-zero Čebyšëv 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}