Perturbation of ${\ell }_1$-copies in Preduals of JBW${}^{\ast }$-triples

Abstract

Two normal functionals on a JBW${}^{\ast }$-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 $\delta \to 0$. 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${}^{\ast }$-algebras.