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Quadratic Conorm and extremally rich JB*-triples

We introduce and study the class of extremally rich JB$^*$-triples. We establish new results to determine the distance from an element $a$ in an extremally rich JB$^*$-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 $$\hbox{dist} (a,\partial_e (E_1)) =\max \{ 1, \|a\|-1\},$$ for every $a\in E$ which is not Brown-Pedersen quasi-invertible. As a consequence, we determine the form of the $λ$-function of Aron and Lohman on the open unit ball of an extremally rich JB$^*$-triple $E$, by showing that $λ(a)= \frac12$ for every non-BP quasi-invertible element $a$ in the open unit ball of $E$. We also prove that for an extremally rich JB$^*$-triple $E$, the quadratic connorm $γ^{q}(.)$ is continuous at a point $a\in E$ if, and only if, either $a$ is not von Neumann regular {\rm(}i.e. $γ^{q}(a)=0${\rm)} or $a$ is Brown-Pedersen quasi-invertible.