Weak limits of the extreme dual ball in \(JB\ast\)-triples (Q1598148)

Given a \(\text{JB}^*\)-triple \(E\), denote by \(\delta_e(E^*_1)\) the extreme points of the closed unit ball of the dual \(E^*\) of \(E\). In the present note, the authors study the weak closure of \(\delta_e(E^*_1)\). Among other things, they show that the set of weak limits and the set of weak sequential limits of \(\delta_e(E^*_1)\) can differ by no more than the zero functional, and that \(\delta_e(E^*_1)\) is closed under weak sequential limits if and only if \(E\) has only finite-dimensional primitive quotients. Further implications for algebraic structure are discussed.