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

Martínez, Juan Carlos; Peralta, Antonio M.

doi:10.1007/s002090050002

Cited by 43 publications

(41 citation statements)

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“…By [20] and [1] we know that the assumption of the separate weak * -continuity is redundant. The bidual E * * of every real or complex JB * -triple is a JBW * -triple with triple product extending the product of E (cf.…”

Section: Resultsmentioning

confidence: 99%

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

Guerrero

Peralta

2004

manuscripta math.

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Section: Resultsmentioning

confidence: 99%

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

Guerrero

Peralta

2004

manuscripta math.

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“…Real J BW * -triples were first introduced as those real J B * -triples which are dual Banach spaces in such a way that the triple product becomes separately w * -continuous (see [15,Definition 4.1 and Theorem 4.4]). Later, it was shown in [23] that the requirement of separate w * -continuity of the triple product is superabundant. We will apply without notice that the bidual of every real J B * -triple X is a J BW * -triple under a suitable triple product which extends the one of…”

Section: The Main Resultsmentioning

confidence: 99%

Relatively weakly open sets in closed balls of Banach spaces, and real JB*-triples of finite rank

et al. 2004

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Abstract. We prove that, given a real J B * -triple X, there exists a nonempty relatively weakly open subset of the closed unit ball of X with diameter less than 2 (if and) only if the Banach space of X is isomorphic to a Hilbert space. Moreover we give the structure of real J B * -triples whose Banach spaces are isomorphic to Hilbert spaces. Such real J B * -triples are also characterized in two different purely algebraic ways.

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“…Real JBW * -triples where first introduced as those real JB * -triples which are dual Banach spaces in such a way that the triple product becomes separately w * -continuous (see [14, Definition 4.1 and Theorem 4.4]). Later, it was shown in [18] that the requirement of separate w * -continuity of the triple product is superabundant. Finally, we recall that an element x of a real JB * -triple E is said to be a tripotent if {xxx} = x.…”

Section: Corollary 210 the Following Banach Spaces Are Extremely Romentioning

confidence: 99%

The big slice phenomena in M-embedded and L-embedded spaces

López-Pérez¹

2005

Proc. Amer. Math. Soc.

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Abstract. We obtain sufficient conditions on an M-embedded or L-embedded space so that every nonempty relatively weakly open subset of its unit ball has norm diameter 2. We prove that, up to renorming, this holds for every Banach space containing c 0 and, as a consequence, for every proper M-ideal. The result obtained for L-embedded spaces can be applied to show that the above property is satisfied for every predual of an atomless real JBW*-triple. As a consequence, a characterization of the Radon-Nikodym property is obtained in this setting, showing that a predual of a real JBW*-triple E verifies the Radon-Nikodym property if, and only if, E is the l ∞ -sum of real type I triple factors. IntroductionThe nonexistence of denting points in the unit ball of some function spaces has been the subject of several recent papers [13], [23]. A point x 0 in the sphere of a Banach space X, S X , is a denting point of the unit ball in X, B X , if there are slices, that is, subsets defined ascontaining x 0 , with diameter arbitrarily small. From [6], x 0 is a denting point of the unit ball of X if, and only if, x 0 is an extreme point in B X and x 0 is a point of weak-norm continuity, that is, a point of continuity for the identity map from (B X , w) onto (B X , n), where w and n denote the weak and the norm topology, respectively. In particular, the existence of denting points in the unit ball of a Banach space X implies the existence of nonempty relatively weakly open subsets of the unit ball in X with diameter arbitrarily small. Then the extreme opposite property to the existence of denting points in the unit ball of a Banach space is that every nonempty relatively weakly open subset of the unit ball has diameter 2. This is the case, for example, for infinite-dimensional C * -algebras [5], uniform algebras [20], non-hilbertizable real JB*-triples [4] and for some Banach spaces of vector valued functions and some spaces of operators [3].The aim of this note is to study when every nonempty relatively weakly open subset of the unit ball of an M-embedded or L-embedded space has diameter 2. In Theorem 2.4, we obtain sufficient conditions in order to assure the above property in the M-ideals case, when only the original norms are considered, by improving the results in [23]. After this, it is shown in Proposition 2.6 that every Banach space containing c 0 can be equivalently renormed so that every nonempty relatively weakly open subset of its unit ball has diameter 2, and then the same is true for proper M-ideals.The result for the L-embedded case is Theorem 2.8, where a sufficient condition to have diameter 2 for all nonempty relatively weakly open subsets of the unit ball of an L-embedded space is obtained. This condition works in the setting of preduals of real JBW*-triples and, as a consequence, we prove in Theorem 2.12 that every nonempty relatively weakly open subset of the unit ball of the predual of an atomless real JBW*-triple has diameter 2. Then the same holds for preduals of atomless Von Neumann algebras. Finally an...

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

Abstract: Abstract. We prove that, if E is a real JB-triple having a predual E , then E * is the unique predual of E and the triple product on E is separately σ(E, E * )−continuous. Mathematics Subject Classification (1991):17C65, 46K70, 46L05, 46L10, 46L70

Cited by 43 publications

References 20 publications

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

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

Relatively weakly open sets in closed balls of Banach spaces, and real JB*-triples of finite rank

The big slice phenomena in M-embedded and L-embedded spaces

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

Abstract: Abstract. We prove that, if E is a real JB*-triple having a predual E * , then E * is the unique predual of E and the triple product on E is separately σ(E, E * )−continuous. Mathematics Subject Classification (1991):17C65, 46K70, 46L05, 46L10, 46L70

Cited by 43 publications

References 20 publications

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

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

Relatively weakly open sets in closed balls of Banach spaces, and real JB*-triples of finite rank

The big slice phenomena in M-embedded and L-embedded spaces

Contact Info

Product

Resources

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

Abstract: Abstract. We prove that, if E is a real JB-triple having a predual E , then E * is the unique predual of E and the triple product on E is separately σ(E, E * )−continuous. Mathematics Subject Classification (1991):17C65, 46K70, 46L05, 46L10, 46L70