The aim of this note is to study Cebyšëv JB ˇ ∗-subtriples of general JB∗-triples. It
is established that if F is a non-zero Cebyšëv JB ˇ ∗-subtriple of a JB∗-triple E, then exactly
one of the following statements holds:
(a) F is a rank one JBW∗-triple with dim (F) ≥ 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;
(b) F = Ce, where e is a complete tripotent in E;
(c) E and F are rank two JBW∗-triples, but F may have arbitrary dimension;
(d) F has rank greater or equal than three and E = F.

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• Inner ideals, compact tripotents and Cebyšëv subtriples ˇ of JB∗-triples and C∗-algebras