Abstract
We describe the one-dimensional. Cebysev subspaces of a JBW*-triple M by showing that for a non-zero element x in M, Cx is a. Cebysev subspace of M if and only if x is a Brown-Pedersen quasi-invertible element in M. We study the. Cebysev JBW*-subtriples of a JBW*-triple M. We prove that for each non-zero. Cebysev JBW*-subtriple N of M, exactly one of the following statements holds:
(a) N is a rank-one JBW*-triple with dim(N) >= 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 = Ce, where e is a complete tripotent in M;
(c) N and M have rank two, but N may have arbitrary dimension >= 2;
(d) N has rank greater than or equal to three, and N = M.
We also provide new examples of. Cebysev subspaces of classic Banach spaces in connection with ternary rings of operators.