APPROXIMATION AND CONVEX DECOMPOSITION BY EXTREMALS AND THE λ-FUNCTION IN JBW*-TRIPLES

We establish new estimates to compute the lambda-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, not subset of(E-1)) = max{1 - m(q) (a), parallel to a parallel to - 1}, where not subset of(E-1) denotes the set of extreme points of the closed unit ball E-1 of E. It is proved that lambda(a) = (1 + m(q) (a))/2, for every Brown-Pedersen quasi-invertible element a in E-1, where mq (a) is the square root of the quadratic conorm of a. For an element a in E-1 which is not Brown-Pedersen quasi-invertible, we can only estimate that lambda(a) <= 1/2 (1 - alpha(q) (a)). A complete description of the lambda-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 lambda-property.