Approximation and Convex Decomposition by Extremals and the λ-Function in Jbw∗-Triples

jamjoom, fatmah b ;

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.