On JB\(^*\)-triples defined by fibre bundles
type: Item
instance of: scholarly article
title: On JB\(^*\)-triples defined by fibre bundles
zbMATH Open document ID: 0832.46044
DOI: 10.1007/BF02570482
author: Wilhelm Kaup
published in: Manuscripta Mathematica
publication date: 1996-02-15T00:00:00Z
full work available at URL: https://eudml.org/doc/156095
review text: 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.
Mathematics Subject Classification ID: 46H70
Mathematics Subject Classification ID: 17C65
zbMATH DE Number: 784082
zbMATH Keywords: triple product
zbMATH Keywords: group of all biholomorphic transformations
zbMATH Keywords: JB\(^*\)- triples associated to principal fibre bundles over a locally compact base space
zbMATH Keywords: topological obstructions for the existence of real forms
reviewed by: Ángel Rodríguez Palacios
MaRDI profile type: Publication
cites work: Weak*-continuity of Jordan triple products and its applications.
cites work: M-structure and the Banach-Stone theorem
cites work: Real contractive projections on commutative C\(^*\)-algebras
cites work: Representation of rings by sections
cites work: The centroid of a JB*-triple system.
cites work: Champs continus d'espaces hilbertiens et de $C^*$-algèbres
cites work: Q3320942
cites work: The structure of algebras of operator fields
cites work: Function Representation of Commutative Operator Triple Systems
cites work: Q5186181
cites work: Linear algebraic groups in infinite dimensions
cites work: On real forms of JB\(^*\)-triples
cites work: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces
cites work: On the classification of the Banach spaces whose duals are \(L_1\) spaces
cites work: On the Classification of Complex Lindenstrauss Spaces.
cites work: Q5825483
cites work: Applications of fibre bundles to the certain class of \(C^*\)-algebras
OpenAlex ID: W2018326097
Recommended article: Q4313492
Recommended article: On the manifold of tripotents in \(JB^*\)-triples
Recommended article: Q3615614
Recommended article: Manifolds of tripotents in \(JB^*\)-triples
Recommended article: ON THE STRONG * TOPOLOGY OF A JBW*-TRIPLE
Recommended article: On Grassmannians associated with JB\(^*\)-triples
Recommended article: Compact-like manifolds associated to \(JB^*\)-triples
Recommended article: Q4896284
Recommended article: On real forms of JB\(^*\)-triples
Recommended article: HOMOTOPES OF JB*-TRIPLES AND A RUSSO—DYE THEOREM
description: scientific article; zbMATH DE number 784082