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 ${\partial }_e\left(E_1\right)$ of all extreme points of the closed unit ball of $E$. More concretely, we prove that

$$dist(a,{\partial }_e(E_1\left)\right)=max\hspace{0.17em}\{1,\Vert a\Vert -1\},$$ 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 $\lambda \left(a\right)=1/2$ 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 conorm ${\gamma }^q(\cdot )$ is continuous at a point $a\in E$ if and only if either $a$ is not von Neumann regular (i.e., ${\gamma }^q\left(a\right)=0$) or $a$ is Brown–Pedersen quasi-invertible.