On JB\(^*\)-triples defined by fibre bundles
JB\(^*\)-triples are defined as natural generalizations of the models obtained by adding to the Banach space of a \(C^*\)-algebra the triple product \(\{abc\}:= {1\over 2} (ab^* c+ cb^* a)\) [the author, Math. Ann. 228, 39-64 (1977; Zbl 0344.58006)]. As in the case of \(C^*\)- algebras [\textit{R. V. Kadison}, Ann. Math. II. Ser. 54, 325-338 (1951; Zbl 0045.062)], the triple product of a general JB\(^*\)-triple can be recovered from the underlying Banach space structure. More even, JB\(^*\)-triples can be intrinsically characterized as those complex Banach spaces \(E\) for which the group of all biholomorphic transformations of the open unit ball \(B\) of \(E\) acts transitively on \(B\) [the author, Math. Z. 183, 503-529 (1983; Zbl 0519.32024)]. In the present paper, the author studies in deep the JB\(^*\)-triples associated to principal fibre bundles over a locally compact base space. Under reasonable conditions on such a JB\(^*\)-triple \(E\), the real forms of \(E\) [in the sense of \textit{M. Isidro}, the author and \textit{A. Rodríguez}, Manuscripta Math. 86, 311-335 (1995)] are described. Moreover, topological obstructions for the existence of real forms are given. This allows the author to provide the first known example of a JB\(^*\)-triple without real forms.
group of all biholomorphic transformations
JB\(^*\)- triples associated to principal fibre bundles over a locally compact base space
topological obstructions for the existence of real forms
scientific article; zbMATH DE number 784082
scientific article