Čebyšëv subspaces of JBW ∗ -triples

Abstract

We describe the one-dimensional Čebyšëv subspaces of a JBW ∗ -triple M by showing that for a non-zero element x in M, Cx is a Čebyšëv subspace of M if and only if x is a Brown-Pedersen quasi-invertible element in M. We study the Čebyšëv JBW ∗ -subtriples of a JBW ∗ -triple M. We prove that for each non-zero Čebyšëv 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 Čebyšëv subspaces of classic Banach spaces in connection with ternary rings of operators.