Inner ideals, compact tripotents and Čebyšëv subtriples of JB\(^{}\)-triples and C\(^\)-algebras

The aim of this note is to study Čebyš\"ev JB\(^*\)-subtriples of general JB\(^*\)-triples. It is established that if \(F\) is a non-zero Čebyš\"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}