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1
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The alternative Dunford-Pettis Property in the predual of a von Neumann algebra

100%
Martín M., Peralta A.
Studia Mathematica
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2001
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tom 147
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nr 2
197-200
EN
Let A be a type II von Neumann algebra with predual A⁎. We prove that A⁎ does not have the alternative Dunford-Pettis property introduced by W. Freedman [7], i.e., there is a sequence (φₙ) converging weakly to φ in A⁎ with ||φₙ|| = ||φ|| = 1 for all n ∈ ℕ and a weakly null sequence (xₙ) in A such that φₙ(xₙ) ↛ 0. This answers a question posed in [7].
2
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Numerical index of vector-valued function spaces

100%
Martín M., Payá R.
Studia Mathematica
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2000
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tom 142
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nr 3
269-280
EN
We show that the numerical index of a $c_0$-, $l_1$-, or $l_∞$-sum of Banach spaces is the infimum of the numerical indices of the summands. Moreover, we prove that the spaces C(K,X) and $L_1(μ,X)$ (K any compact Hausdorff space, μ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.
3
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Extremely non-complex Banach spaces

100%
Martín M., Merí J.
Open Mathematics
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2011
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tom 9
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nr 4
797-802
EN
A Banach space X is said to be an extremely non-complex space if the norm equality ∥Id +T 2∥ = 1+∥T 2∥ holds for every bounded linear operator T on X. We show that every extremely non-complex Banach space has positive numerical index, it does not have an unconditional basis and that the infimum of diameters of the slices of its unit ball is positive.
4
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Properties of lush spaces and applications to Banach spaces with numerical index 1

81%
Boyko K., Kadets V., Martín M., Merí J.
Studia Mathematica
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2009
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tom 190
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nr 2
117-133
EN
The concept of lushness, introduced recently, is a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to having numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c₀ 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 = (ℝⁿ,||·||) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.
5
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The Daugavet equation for polynomials

81%
Choi Y., García D., Maestre M., Martín M.
Studia Mathematica
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2007
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tom 178
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nr 1
63-84
EN
We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ||Id + P|| = 1 + ||P|| is satisfied for all weakly compact polynomials P: X → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation $max_{|ω|=1} ||Id + ωP|| = 1 + ||P||$ for polynomials P: X → X. We show that this equation holds for every polynomial on the complex space X = C(K) (K arbitrary) with values in X. This result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for k-homogeneous polynomials.
6
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On the Bishop-Phelps-Bollobás theorem for operators and numerical radius

81%
Kim S., Lee H., Martín M.
Studia Mathematica
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2016
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tom 233
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nr 2
141-151
EN
We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ₁-sums and $ℓ_{∞}$-sums of Banach spaces. More precisely, we introduce a property of Banach spaces, which we call strongly lush. We find that if X is strongly lush and X ⊕₁ Y has the weak BPBp-nu, then (X,Y) has the Bishop-Phelps-Bollobás property (BPBp). On the other hand, if Y is strongly lush and $X ⊕_{∞} Y$ has the weak BPBp-nu, then (X,Y) has the BPBp. Examples of strongly lush spaces are C(K) spaces, L₁(μ) spaces, and finite-codimensional subspaces of C[0,1].
7
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On order structure and operators in L ∞(μ)

71%
Krasikova I., Martín M., Merí J., Mykhaylyuk V., Popov M.
Open Mathematics
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2009
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tom 7
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nr 4
683-693
EN
It is known that there is a continuous linear functional on L ∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L ∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L ∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L ∞(μ) to c 0(Γ) is narrow while not every such an operator is AM-compact.
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