Convexity and smoothness of Banach spaces with numerical index one

Kadets, Vladimir; Martı́n, Miguel; Merí, Javier; Payá, Rafael

doi:10.1215/ijm/1264170844

Cited by 17 publications

(22 citation statements)

References 36 publications

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“…A Banach space E is said to have the Mazur-Ulam property (briefly MUP) provided that for every Banach space F , every surjective isometry T between the two unit spheres of E and F is the restriction of a linear isometry between the two spaces.Cheng and Dong attacked the problem for the class of CL-spaces admitting a smooth point and polyhedral spaces. Unfortunately their interesting attempt failed by a mistake at the very end of the proof (also see the introduction in [10,28]). In [10], Kadets and Martín proved that finite-dimensional polyhedral Banach spaces have the MUP.…”

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confidence: 99%

Generalized-lush spaces and the Mazur–Ulam property

2013

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We introduce a new class of Banach spaces, called generalized-lush spaces (GL-spaces for short), which contains almost-CL-spaces, separable lush spaces (specially, separable C-rich subspaces of C(K)), and even the two-dimensional space with hexagonal norm. We obtain that the space C(K, E) of the vector-valued continuous functions is a GL-space whenever E is, and show that the GL-space is stable under c 0 -, l 1 -and l ∞ -sums. As an application, we prove that the Mazur-Ulam property holds for a larger class of Banach spaces, called local-GL-spaces, including all lush spaces and GL-spaces. Furthermore, we generalize the stability properties of GL-spaces to local-GL-spaces. From this, we can obtain many examples of Banach spaces having the Mazur-Ulam property.2010 Mathematics Subject Classification. Primary 46B04; Secondary 46B20,46A22. 1 2 DONGNI TAN, XUJIAN HUANG, AND RUI LIU Recently in [3], Cheng and Dong considered the extension question of isometries between unit spheres of Banach space and introduced the Mazur-Ulam property:Definition 1.2. A Banach space E is said to have the Mazur-Ulam property (briefly MUP) provided that for every Banach space F , every surjective isometry T between the two unit spheres of E and F is the restriction of a linear isometry between the two spaces.Cheng and Dong attacked the problem for the class of CL-spaces admitting a smooth point and polyhedral spaces. Unfortunately their interesting attempt failed by a mistake at the very end of the proof (also see the introduction in [10,28]). In [10], Kadets and Martín proved that finite-dimensional polyhedral Banach spaces have the MUP. Notice that the problem is still open even in two dimension case. In [28], the authors Tan and Liu proved that every almost-CL-spaces admitting a smooth point (specially, separable almost-CL-spaces) has the MUP.Recall that R. Fullerton [8] first introduced the notion of CL-spaces. It was extended by Lima [14,15] who introduced the almost-CL-spaces and gave the examples of real CL-spaces which are L 1 (µ) and its isometric preduals, in particular C(K), where C(K) is a compact Hausdorff space. The infinite-dimensional complex L 1 (µ) spaces are proved by Martín and Payá [20] to be only almost-CL-spaces. Lush spaces were introduced recently in [1] and have been extensively studied recently in [2,10,11]. Such spaces are of importance to supply an example of a Banach space E with the numerical index n(E) < n(E * ). It thus gives a negative answer to a question which has been latent since the beginning of the theory of numerical indices in the seventies. Now, a natural and interesting question is: "Does every almost-CL-space, even every lush space, has the MUP? "In this paper, we introduce a natural concept of generalized-lush spaces (GL-spaces for short), which contains almost-CL-spaces, separable lush spaces (specially, separable C-rich subspaces of C(K)), and even the two-dimensional space with hexagonal norm. We obtain that the space C(K, E) of the vector-valued continuous functions is a generalized-lush s...

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confidence: 99%

Generalized-lush spaces and the Mazur–Ulam property

2013

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“…Proof. By Lemma 4.2 of [9], given y ∈ S X and ε > 0, there is a dense subset K y of K(X * ) (with the notation of section 3) such that y ∈ conv T S(S X , Re y * , ε) for every y * ∈ K y (observe that in [9, Lemma 4.2], K y is a subset of what is called K ′ (X * ) in the proof of Lemma 3.2, but since K(X * ) is also a Baire space, the same argument gives the result that we are using). Now, as the set of extreme points of B X * is contained in the closure of K y , there is y * ∈ K y such that Re y * (x) > 1 − ε, that is, x ∈ S(S X , Re y * , ε).…”

Section: Complex Lush Spaces and Lipschitz Numerical Indexmentioning

confidence: 85%

“…Our aim is to show that the same happens for complex spaces. We need a reformulation of lushness which follows from the results of [9]. Lemma 4.2.…”

Section: Complex Lush Spaces and Lipschitz Numerical Indexmentioning

confidence: 99%

Lipschitz slices and the Daugavet equation for Lipschitz operators

Kadets

Martı́n

Merí

et al. 2015

Proc. Amer. Math. Soc.

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“…Indeed, let X be a separable lush space. By [10,Theorem 4.3] and [3, Corollary 3.5], there exists a rounded subset C of S X * norming for X such that…”

Section: A Sufficient Condition For the Ahspmentioning

confidence: 99%

On Banach spaces with the approximate hyperplane series property

Choi¹,

Kim²,

Lee³

et al. 2015

Banach J. Math. Anal.

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Abstract. We present a sufficient condition for a Banach space to have the approximate hyperplane series property (AHSP) which actually covers all known examples. We use this property to get a stability result to vector-valued spaces of integrable functions. On the other hand, the study of a possible BishopPhelps-Bollobás version of a classical result of V. Zizler leads to a new characterization of the AHSP for dual spaces in terms of w * -continuous operators and other related results.

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Convexity and smoothness of Banach spaces with numerical index one

Cited by 17 publications

References 36 publications

Generalized-lush spaces and the Mazur–Ulam property

Generalized-lush spaces and the Mazur–Ulam property

Lipschitz slices and the Daugavet equation for Lipschitz operators

On Banach spaces with the approximate hyperplane series property

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