Abstract
We describe the one-dimensional Cebysev subspaces of a JBW∗-triple M by showing that for a non-zero element x in M, $\mathbb{C}x$ 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: We also provide new examples of Cebysev subspaces of classic Banach spaces in connection with ternary rings of operators.
Highlights
Let V be a subspace of a Banach space X
A classical theorem due to Haar establishes that an n-dimensional subspace V of the space C(K), of all continuous complex-valued functions on K, is a Čebyšëv subspace of C(K) if and only if any non-zero f ∈ V admits at most n – zeros
In this note we present the first results about Čebyšëv subspaces and Čebyšëv subtriples in Jordan structures
Summary
Let V be a subspace of a Banach space X. The first main difference in the setting of JB∗-triples is the existence of Čebyšëv JB∗-subtriples with arbitrary dimensions; complex Hilbert spaces and spin factors give a complete list of examples (compare Remark and comments before it). The complete tripotents of a JB∗-triple E coincide with the real and complex extreme points of its closed unit ball E
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