Relatively Weakly Open Sets in Closed Balls of $C^*$-Algebras

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

Guerrero

López-Pérez

Peralta

et al. 2004

Math. Ann.

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Relatively weakly open sets in closed balls of Banach spaces, and the centralizer

Guerrero

Rodríguez-Palacios

2008

Math. Z.

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We prove that, if the centralizer of a Banach space X is infinite-dimensional, then every nonempty relatively weakly open subset of the closed unit ball of X has diameter equal to 2. This result, together with a suitable refinement also proven in the paper, contains (and improves in some cases) previously known facts for C * -algebras, J B * -triples, spaces of vector valued continuous functions, and spaces of operators.

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Slices in the unit ball of the symmetric tensor product of $\mathcal{C}(K)$ and L1(μ)

Acosta

Guerrero

2009

Ark. Mat.

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We prove that for the cases X =C(K) (K infinite) and X =L 1 (µ) (µ σ-finite and atomless) it holds that every slice of the unit ball of the N -fold symmetric tensor product of X has diameter two. In fact, we prove more general results for weak neighborhoods relative to the unit ball. As a consequence, we deduce that the spaces of N -homogeneous polynomials on those classical Banach spaces have no points of Fréchet differentiability.

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Relatively Weakly Open Sets in Closed Balls of $C^*$-Algebras

Cited by 17 publications

References 20 publications

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

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

Relatively weakly open sets in closed balls of Banach spaces, and the centralizer

Slices in the unit ball of the symmetric tensor product of $\mathcal{C}(K)$ and L1(μ)

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