Real Banach Spaces with Numerical Index 1

López, Ginés; Martı́n, Miguel; Payá, Rafael

doi:10.1112/s002460939800513x

Cited by 43 publications

(43 citation statements)

References 15 publications

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“…Recent results can be found in [7,12,14,15]. Let us mention here some facts concerning the numerical index which will be relevant to our discussion.…”

Section: N(x) = Max{k ≥ 0 : K T ≤ V(t ) ∀ T ∈ L(x)}mentioning

confidence: 99%

Banach spaces having the Radon-Nikodym property and numerical index 1

Martı́n

2003

Proc. Amer. Math. Soc.

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“…Recent results can be found in [7,12,14,15]. Let us mention here some facts concerning the numerical index which will be relevant to our discussion.…”

Section: N(x) = Max{k ≥ 0 : K T ≤ V(t ) ∀ T ∈ L(x)}mentioning

confidence: 99%

Banach spaces having the Radon-Nikodym property and numerical index 1

Martı́n

2003

Proc. Amer. Math. Soc.

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“…Indeed, let us consider the real space Y given in [4] such that Y * is isomorphic to 1 and Y has the Radon-Nikodým property. Then, Y is an infinitedimensional real Banach space having the Radon-Nikodým property and it is also Asplund, so [12,Corollary 4] shows that it does not admit an equivalent norm with numerical index 1.…”

Section: Lush Renormingsmentioning

confidence: 99%

“…any number in [0, 1[ in the real case and any number in [1/e, 1[ in the complex case). On the other hand, some necessary conditions for a Banach space to be renormed with numerical index 1 were given in [12].…”

Section: Introductionmentioning

confidence: 99%

Properties of lush spaces and applications to Banach spaces with numerical index 1

et al. 2009

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Abstract. The concept of lushness was introduced recently as a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c 0 can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably determined, is stable under ultraproducts, and we characterize those spaces of the form X = (R n , · ) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.

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“…Classical references on this topic are the monographs by F. Bonsall and J. Duncan [3,4]. For recent results we refer the reader to [6,10,11,15,16,18] and the survey paper [13].…”

Section: It Is Clear That V Is a Seminorm On L(x) Satisfying V(t )mentioning

confidence: 99%

Finite-dimensional Banach spaces with numerical index zero

Martı́n

Merí

Rodríguez-Palacios

2004

Indiana Univ. Math. J.

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Abstract. We prove that a finite-dimensional Banach space X has numerical index 0 if and only if it is the direct sum of a real space X 0 and nonzero complex spaces X 1 , . . . , Xn in such a way that the equalityholds for suitable positive integers q 1 , . . . , qn, and every ρ ∈ R and every x j ∈ X j (j = 0, 1, . . . , n). If the dimension of X is two, then the above result gives X = C, whereas dim(X) = 3 implies that X is an absolute sum of R and C. We also give an example showing that, in general, the number of complex spaces cannot be reduced to one.

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Real Banach Spaces with Numerical Index 1

Abstract: We show that an infinite-dimensional real Banach space with numerical index 1 satisfying the RadonNikodỳm property contains l 1 . It follows that a reflexive or quasi-reflexive real Banach space cannot be re-normed to have numerical index 1, unless it is finite-dimensional.

Cited by 43 publications

References 15 publications

Banach spaces having the Radon-Nikodym property and numerical index 1

Banach spaces having the Radon-Nikodym property and numerical index 1

Properties of lush spaces and applications to Banach spaces with numerical index 1

Finite-dimensional Banach spaces with numerical index zero

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