Perturbation of $\ell_1$-copies in Preduals of JBW$^*$-triples

TL;DR

This paper demonstrates that the orthogonality of normal functionals in JBW*-triples is stable under small perturbations, extending known results from C*-algebras to a broader class of operator algebras.

Contribution

It extends the stability of orthogonality of functionals from C*-algebras to JBW*-triples, including non-normal functionals, under small norm perturbations.

Findings

Orthogonality is stable under small perturbations in JBW*-triples.

Results apply to finitely and infinitely many functionals.

Analogous stability results hold for non-normal functionals.

Abstract

Two normal functionals on a JBW$^*$-triple are known to be orthogonal if and only if they are $L$-orthogonal (meaning that they span an isometric copy of $\ell_1(2)$). This is shown to be stable under small norm perturbations in the following sense: if the linear span of the two functionals is isometric up to $\delta>0$ to $\ell_1(2)$, then the functionals are less far (in norm) than $\eps>0$ from two orthogonal functionals, where $\eps\to0$ as $\delta\to0$. Analogous statements for finitely and even infinitely many functionals hold as well. And so does a corresponding statement for non-normal functionals. Our results have been known for C$^*$-algebras.