Relatively weakly open sets in closed balls of Banach spaces, and real \(JB^*\)-triples of finite rank (Q1882601)

This important paper continues the study of Banach spaces with the following property (P) ``opposite'' to the RNP: Any relatively weakly open subset of the closed unit ball has diameter 2. It was proven by \textit{O. Nygaard} and \textit{D. Werner} [Arch. Math. 76, 441--444 (2001; Zbl 0994.46006)] that uniform algebras share this property. Later, in [J. Lond. Math. Soc. (2) 68, 753--761 (2003; Zbl 1060.46007)], \textit{J. Becerra Guerrero, G. López Pérez} and \textit{A. Rodrígues-Palacios} showed that if a complex \(JB^\ast\)-triple \(X\) enjoys property (P), then the Banach space of \(X\) is Hilbertizable, implying that \(C^\ast\)-algebras share (P). In the paper under review, the authors improve this in various ways. They turn to a wider class, the real \(JB^\ast\)-triples, and prove that such a triple fails (P) if and only if its Banach space is a Hilbert space if and only if the triple itself has the RNP, thus proving a strong dichotomy for real \(JB^\ast\)-triples. The key observation, of its own interest, is that every non-reflexive Banach space \(X\) such that \(X^\ast\) is an L-summand in \(X^{\ast\ast\ast}\) has (P). This gives a \(JB^\ast\)-free proof of the result for \(C^\ast\)-algebras. In a final section, two different purely algebraic characterizations of those real \(JB^\ast\)-triples having (P) are given.