Unique Extension of Atomic Functionals of JB*-Triples

This thesis initiates and proceeds to develop a theory of unique norm preserving extensions of extreme dual ball functionals and their σ-convex sums, in the category of JB*-triples. All such functionals are completely determined by Cartan factors and `∞-sums of Cartan factors residing as weak* closed ideals in the second dual of a JB*-triple. Implications for structure, particularly involving inner ideals, is a theme running throughout the thesis. This thesis makes an analysis of inclusions C ⊂ D of Cartan factors for which there exist an element in ∂e(C∗,1) with unique extension in ∂e(D∗,1). A number of abstract characterisations are given and special cases examined in detail. We prove that, for C to be an inner ideal it is sufficient for a single functional in S(C∗,1) \ ∂e(C∗,1) to have unique norm one extension. Information gathered on Cartan factors is used to develop a more general theory of unique extension of dual ball extreme points, of Cartan functionals and other atomic functionals, and culminates with an investigation of the extreme, Cartan and atomic extension properties of a separable JB*subtriple in a JBW*-triple.