On real cartan factors

Kaup, Wilhelm

doi:10.1007/bf02678189

Cited by 60 publications

(74 citation statements)

References 20 publications

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“…It should be noticed that when X is a complex Banach space with a conjugation τ and x is a norm-one element in X τ satisfying that X τ is smooth at x then X does not need to be smooth at x. For example let X τ denote the real spin factor of type IV n,0 n in the terminology of [19,Theorem 4.1], where we consider X, the complexification of X τ , equipped with triple product and norm given by {xyz} := (x|y)z + (z|y)x − (x|σ(z))σ(y) and…”

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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“…Also, the corresponding spaces of all symmetric, S(H), and skew, A(H), bounded linear operators on H can be considered real JB * -triples. The above examples become particular cases of those arising by considering either the so-called complex Cartan factors (regarded as real JB * -triples) or real forms of complex Cartan factors [16]. We recall that real forms of a complex Banach space X are defined as the real closed subspaces of X of the form X τ := {x ∈ X : τ (x) = x}, for some conjugation (i.e., conjugate-linear isometry of period two) on X.…”

Section: Corollary 210 the Following Banach Spaces Are Extremely Romentioning

confidence: 99%

“…We denote byÂ = A ⊕ iA the complexification of A. By [21],Â is a purely atomic complex JBW * triple and then, by [11],Â is the ∞ sum of type I Cartan factors, that is, the ∞ sum of w * -closed simple ideals which are either finite-dimensional, infinite-dimensional complex spin factors or of the form L(H, K), S(H) or A(H) for some complex Hilbert spaces H and K. Taking into account that the conjugation τ preserves the triple product and is w * -continuous, it is enough to apply [16] to deduce that A is the ∞ sum of w * -closed simple ideals which are either finite-dimensional, infinite-dimensional generalized real spin factors or of the form L(H, K), S(H) or A(H) for some real, complex or quaternionnic Hilbert spaces H and K. Finally, as the Radon-Nikodym property is stable by 1 sums and the preduals of the above spaces satisfy the Radon-Nikodym property (see [7]) we deduce that A * verifies the Radon-Nikodym property.…”

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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“…The above examples become particular cases of those arising by considering either the so-called complex Cartan factors (regarded as real J B * -triples) or real forms of complex Cartan factors [20]. We recall that real forms of a complex Banach space X are defined as the real closed subspaces of X of the form X τ := {x ∈ X : τ (x) = x}, for some conjugation (i.e., conjugate-linear isometry of period two) on X.…”

Section: Proposition 22 the Predual Of Every Real J Bw * -Triple Ismentioning

confidence: 99%

“…Since, for j ∈ B, the mapping y j → y j + τ (y j ) from Y j (regarded as a real J B * -triple) to Z τ j is a surjective linear isometry preserving triple products, we can write For the determination of finite-dimensional simple real J B * -triples (including finite-dimensional simple complex J B * -triples) and of real forms of complex spin factors, the reader is referred to [22] and [20], respectively.…”

Section: Proposition 22 the Predual Of Every Real J Bw * -Triple Ismentioning

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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On real cartan factors

Cited by 60 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

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

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

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