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Extremely non-complex Banach spaces

100%
Martín M., Merí J.
Open Mathematics
|
2011
|
tom 9
|
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.
2
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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
|
2009
|
tom 190
|
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.
3
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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
|
2009
|
tom 7
|
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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