Approximation and convex decomposition by extremals and the λ-function in JBW∗-triples

Abstract

We establish new estimates to compute the λ-function of Aron and Lohman on the unit ball of a JB∗-triple. It is established that for every Brown-Pedersen quasi-invertible element a in a JB∗-triple E we have dist(a,E(E1)) = max{1 - mq (a), ||a|| - 1}, where E(E1) denotes the set of extreme points of the closed unit ball E1 of E. It is proved that λ(a) = (1 + mq (a))/2, for every Brown-Pedersen quasi-invertible element a in E1, where mq (a) is the square root of the quadratic conorm of a. For an element a inE1 which is not Brown-Pedersen quasi-invertible, we can only estimate that λ(a) < 1 2 (1 - αq (a)). A complete description of the λ-function on the closed unit ball of every JBW∗-triple is also provided, and as a consequence, we prove that every JBW∗-triple satisfies the uniform λ-property.