Transitivity of the norm on Banach spaces having a Jordan structure (Q1575061)

Let \(X\) be a Banach space, \(S=S(X)\) and \(B=B(X)\) the unit sphere and the unit ball of \(X\), respectively, and let \(\mathcal G ={\mathcal G}(X) \) be the group of all surjective linear isometries of \(X\). The norm of \(X\) is called transitive, almost transitive, convex transitive, if \({\mathcal G}(x) = S, {\mathcal G}(x)\) is dense in \(S\), or \(\overline{co}({\mathcal G}(x))= B\), respectively, for every \(x\in S\). The famous Banach-Mazur problem asks whether a separable Banach space with transitive norm must be a Hilbert space. For finite dimensional spaces the answer is ``yes'', but there are known examples of non-separable Banach spaces and of separable incomplete normed spaces with transitive norms [see, e.g., the book by \textit{S. Rolewicz}, ``Metric Linear Spaces'', Warszawa (1972; Zbl 0226.46001), and (2nd ed.) Dordrecht (1985; 573.46001), and the survey paper by \textit{F. Cabello Sánchez}, Extr. Math. 12, No.~2, 97-116 (1997; Zbl 0906.46006)]. The present paper is concerned with the Banach-Mazur problem in the context of \(JB^*\)- and \(JBW^*\)-triples in the sense of \textit{W. Kaup,} Math. Ann. 228, 39-64 (1977; Zbl 0335.58005). The cases of \(C^*\)-algebras and of Jordan-Banach algebras are also considered. As samples of the results obtained by the authors we quote: If \(X\) is a separable predual of a \(JBW^*\)-triple and the norm of \(X\) is transitive then \(X\) is a Hilbert space (Corollary 2.5); If \(X\) is a \(JBW^*\)-triple with almost transitive norm then \(X\) is a Hilbert space (Corollary 2.6); If \(X\) is a predual of a \(JBW^*\)-triple, its unit ball has an extreme point, and the norm of \(X\) is convex transitive then \(X\) is a Hilbert space (Theorem 3.1).