Prime JB*‐Triples and Extreme Dual Ball Density

Properties of the extreme points ∂_e(E*₁) of the closed dual ball E*₁ of a JB*‐triple E are studied. It is shown that the canonical mapping from ∂_e(E*₁) onto the structure space, Prim (E), of primitive M‐ideals of E is an open mapping. This property is utilised to show that ∂_e(E*₁) is weak* dense in E*₁ if and only if E is an infinite dimensional Hilbert space, an infinite dimensional spin factor or E is prime with zero socle.