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