Cebysev subspaces of JBW*-triples

We describe the one-dimensional \v{C}eby\v{s}"{e}v subspaces of a JBW${}^{\ast }$-triple $M,$ by showing that for a non-zero element $x$ in $M$, $\mathbb{C}x$ is a \v{C}eby\v{s}"{e}v subspace of $M$ if, and only if, $x$ is a Brown-Pedersen quasi-invertible element in $M$. We study the \v{C}eby\v{s}"{e}v JBW${}^{\ast }$-subtriples of a JBW${}^{\ast }$-triple $M$. We prove that, for each non-zero \v{C}eby\v{s}"{e}v JBW${}^{\ast }$-subtriple $N$ of $M$, then exactly one of the following statements holds: $(a)$ $N$ is a rank one JBW${}^{\ast }$-triple with dim$(N)\ge 2$ (i.e. a complex Hilbert space regarded as a type 1 Cartan factor). Moreover, $N$ may be a closed subspace of arbitrary dimension and $M$ may have arbitrary rank; $(b)$ $N=\mathbb{C}e$, where $e$ is a complete tripotent in $M$; $(c)$ $N$ and $M$ have rank two, but $N$ may have arbitrary dimension; $(d)$ $N$ has rank greater or equal than three and $N=M$. We also provide new examples of \v{C}eby\v{s}"{e}v subspaces of classic Banach spaces in connection with ternary rings of operators.