Lushness, numerical index one and duality (Q1025807)

Given a Banach space \(X\), the symbols \(S_X\) and \(B_X\) mean the unit sphere and the closed unit ball of \(X\), respectively. For any subset \(B\) of \(X\) the authors denote by \(\text{aconv} \, (B)\) the absolute convex hull of \(B\) in \(X\). A Banach space \(X\) is said to be \textit{lush} if, for every \(x,y \in S_X\) and every \(\varepsilon > 0\), there is a slice \(S = S(B_X,x^*,\varepsilon) = \{u \in S_X: \text{Re} \, x^*(u) > \sup \,\text{Re} \, x^*(B_X) - \varepsilon \}\) containing \(x\) such that \(\text{dist} (y, \text{aconv} \, (S)) < \varepsilon\). A Banach space \(X\) is said to have numerical index one if, for every bounded linear operator \(T\) on \(X\), one has \(\sup\{ | x^*(Tx)|: x \in S_X\), \(x^* \in S_{X^*}\), \(x^*(x) = 1\}=\|T\|\). Lushness was introduced several years ago, in particular, to show that there is a Banach space having numerical index one and whose dual does not. The authors study relationships between the notions of lushness and numerical index one. Among some related results, it is shown that Banach spaces with numerical index one need not be lush, answering a question of the first and second-named authors and \textit{R.\,Payá} [RACSAM, Rev.\ R.\ Acad.\ Cienc.\ Exactas Fís.\ Nat., Ser.\,A Mat.\ 100, No.\,1--2, 155--182 (2006; Zbl 1111.46007)].