Little Grothendieck`s theorem for real JB*-triples

We prove that given a real JB*-triple E, and a real Hilbert space H, then the set of those bounded linear operators T from E to H, such that there exists a norm one functional and corresponding pre-Hilbertian semi-norm on E such that for all , is norm dense in the set of all bounded linear operators from E to H. As a tool for the above result, we show that if A is a JB-algebra and is a bounded linear operator then there exists a state such that for all . Received June 28, 1999; in final form January 28, 2000 / Published online March 12, 2001     

On summing operators on JB*-triples

In this paper we introduce 2-JB^*-triple-summing operators on real and complex JB^*-triples. These operators generalize 2-C^*-summing operators on C^*-algebras. We also obtain a Pietsch's factorization theorem in the setting of 2-JB^*-triple-summing operators on JB^*-triples. Received: 18 December 2000     

Isometries between JB*-triples

Let Z and W be JB*-triples and let T be a linear isometry from Z into W. For any z ∈ Z with ||z||<1, we show that Open image in new window if the Möbius transform induced by T(z) preserves the unit ball of T(Z). We show further that T is, locally, a triple homomorphism via a tripotent: for any z ∈ Z, there is a tripotent u in W** such that Open image in new window for all a, b, c in the smallest subtriple Z _z of Z containing z, and also, {u,T(·),u}:Z _z →W** is an isometry.     

On JB*-triples defined by fibre bundles

We associate JB^*-triples to certain fibre bundles and describe their automorphisms, conjugations and real forms. In particular, we construct a JB^*-triple that does not admit a single conjugation.     

Separate weak*-continuity of the triple product in dual real JB*-triples

We prove that, if E is a real JB*-triple having a predual then is the unique predual of E and the triple product on E is separately $sigma (E,E_{*_{}})-$continuous. Received February 1, 1999; in final form March 29, 1999 / Published online May 8, 2000     

Manifolds of tripotents in JB*-triples

Abstract. We describe the affine connections, geodesics and symmetries of various Banach manifolds of tripotents in JB*-triples which include the C*-algebras and Hilbert spaces where the nonzero tripotents are respectively the partial isometries and the extreme points of the closed unit ball. Received July 7, 1998; in final form November 16, 1998     

Compact-like manifolds associated to JB*-triples

Finite dimensional JB ^\ast;-triples have a uniquely associated simply connected compact symmetric manifold. In infinite dimensions, different constructions have been used to yield three possibly different Banach manifolds from a JB ^\ast; all of which display compact like behaviour. We clarify the relationship between these three manifolds and characterise a case when all three of these manifolds coincide as in the finite dimensional case. These results have been announced in [11]. Received: 29 May 2001     

Subdifferentiability of the norm and the Banach-Stone theorem for real and complex JB*-triples

We study the points of strong subdifferentiability for the norm of a real JB*-triple. As a consequence we describe weakly compact real JB*-triples and rediscover the Banach-Stone Theorem for complex JB*-triples.Authors Partially supported by I+D MCYT projects no. BFM2002-01529 and BFM2002-01810, and Junta de Andalucía grant FQM 0199Mathematics Subject Classification (2000): 46B04, 46L05, 46L70Revised version: 2 May 2004Acknowledgements. The authors would like to thank A. Rodr{\i}guez Palacios for fruitful comments and discussions during the preparation of this paper and to the referee for his or her interesting suggestions.     

Weak continuity of holomorphic automorphisms in JB*-triples

Supported by DAAD grant 313, programm II (1991)     

Weak limits of the extreme dual ball in JB*-triples

We study weak limits of the extreme points, ∂ _e (E ^* ₁), of the dual ball of a JB*-triple, E. We show that all such weak limits, except possibly the zero functional, are weak sequential limits and we discuss implications for the structure of E. Received: 9 April 2001     

Von Neumann Regularity and Quadratic Conorms in JB^＊-triples and C^＊-algebras

We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-triple as the minimum reduced modulus of the mapping Q（a）. It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW^＊-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C^＊-algebras and von Neumann algebras are also studied.     

On the axiomatic definition of real JB*–triples

In the last twenty years, a theory of real Jordan triples has been developed. In 1994 T. Dang and B. Russo introduced the concept of J*B–triple. These J*B–triples include real C*–algebras and complex JB*–triples. However, concerning J*B–triples, an important problem was left open. Indeed, the question was whether the complexification of a J*B–triple is a complex JB*–triple in some norm extending the original norm. T. Dang and B. Russo solved this problem for commutative J*B–triples. In this paper we characterize those J*B–triples with a unitary element whose complexifications are complex JB*–triples in some norm extending the original one. We actually find a necessary and sufficient new axiom to characterize those J*B–triples with a unitary element which are J*B–algebras in the sense of [1] or real JB*–triples in the sense of [4].     

On range tripotents in JBW*-triples

Let B be a JBW*-triple, let A be a JB*-subtriple of B and let be the set of range tripotents relative to A. It is shown that, under certain conditions, the supremum of a family of range tripotents in coincides with that in the complete lattice of all tripotents in B. As a consequence, a sufficient condition for a tripotent to be a range tripotent relative to A is obtained. The action of isomorphisms on range tripotents is investigated, and an analysis of the suprema of families of spectral range tripotents leads to a generalization of a result known for open projections in W*-algebras. Received: 8 July 2008