Abstract
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 derivative(e)(E-1) of all extreme points of the closed unit ball of E. More concretely, we prove that
dist (a, partial derivative e(E-1)) = max{1, parallel to a parallel to - 1},
for every a is an element of 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*-triple E by showing that lambda(a) = 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*-triple E, the quadratic conorm gamma(q)(.) is continuous at a point a is an element of E if and only if either a is not von Neumann regular (i.e., gamma(q)(a) = 0) or a is Brown-Pedersen quasi-invertible.