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The aim of this note is to study \v{C}eby\v{s}\"ev JB$^*$-subtriples ofgeneral 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 ofthe following statements holds:\begin{enumerate}\item $F$ is a rank oneJBW$^*$-triple with dim$(F)\geq 2$ (i.e. a complex Hilbert space regarded as atype 1 Cartan factor). Moreover, $F$ may be a closed subspace of arbitrarydimension 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 thanthree and $E=F$. \end{enumerate}

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

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