Quadratic Conorm and extremally rich JB*-triples

Abstract

We introduce and study the class of extremally rich JB${}^{\ast }$-triples. We establish new results to determine the distance from an element $a$ in an extremally rich JB${}^{\ast }$-triple $E$ to the set ${\partial }_e(E_1)$ of all extreme points of the closed unit ball of $E$. More concretely, we prove that $$dist(a,{\partial }_e(E_1))=\max \{1,\Vert a\Vert -1\},$$ for every $a\in E$ which is not Brown-Pedersen quasi-invertible. As a consequence, we determine the form of the $\lambda$-function of Aron and Lohman on the open unit ball of an extremally rich JB${}^{\ast }$-triple $E$, by showing that $\lambda (a)=\frac{1}{2}$ for every non-BP quasi-invertible element $a$ in the open unit ball of $E$. We also prove that for an extremally rich JB${}^{\ast }$-triple $E$, the quadratic connorm ${\gamma }^q(.)$ is continuous at a point $a\in E$ if, and only if, either $a$ is not von Neumann regular {\rm(}i.e. ${\gamma }^q(a)=0${\rm)} or $a$ is Brown-Pedersen quasi-invertible.