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

Kostyantyn Boyko ¹   ; Vladimir Kadets ¹   ; Miguel Martín ²   ; Javier Merí ²

¹ Department of Mechanics and Mathematics Kharkov National University Pl. Svobody 4 61077 Kharkov, Ukraine
² Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain

Studia Mathematica, Tome 190 (2009) no. 2, pp. 117-133

Résumé

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 $c_0$ 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 $X =(\mathbb R^n, \|\cdot\|)$ with absolute norm such that $X$-sum preserves lushness of summands, showing that this property is equivalent to lushness of $X$.

Détail

DOI : 10.4064/sm190-2-2

Keywords: concept lushness introduced recently banach space property which ensures space has numerical index nbsp prove asplund spaces lushness actually equivalent having numerical index nbsp prove every separable banach space containing isomorphic copy renormed equivalently lush have numerical index nbsp rest paper devoted study lushness just property banach spaces prove lushness separably determined stable under ultraproducts characterize those spaces form mathbb cdot absolute norm x sum preserves lushness summands showing property equivalent lushness