Octahedral norms in spaces of operators (Q907715)

In [Stud. Math. 95, No. 1, 1--15 (1989; Zbl 0698.46011)], \textit{G. Godefroy} introduced octahedral norms in order to characterize Banach spaces containing an isomorphic copy of \(\ell_1\). A Banach space \(X\) has an octahedral norm if for every finite-dimensional subspace \(E\) of \(X\) and for every \(\varepsilon>0\) there is a \(y\in X\) with norm one such that \[ \| x+\lambda y\|\geq (1-\varepsilon)(\| x\|+|\lambda|) \] for every \(x\in E\) and \(\lambda\in \mathbb R\). Also, recall that a Banach space has the strong diameter \(2\) property if every convex combination of slices of its unit ball has diameter \(2\). It is known that a Banach space \(X\) has the strong diameter \(2\) property if and only if the dual space \(X^\ast\) has an octahedral norm. The authors study sufficient and necessary conditions for \(\mathcal{L}(X,Y)\), the space of all bounded linear operators from \(X\) to \(Y\), to have an octahedral norm. The main result of the paper under review says that if \(X^\ast\) and \(Y\) have an octahedral norm, then so does \(\mathcal{L}(X,Y)\). By the duality between octahedrality and the strong diameter \(2\) property, they are able to deduce that if \(X\) and \(Y\) have the strong diameter \(2\) property, then so does \(X\widehat{\otimes}_{\pi} Y\), the projective tensor product of \(X\) and \(Y\). As a necessary condition, the authors show that if \(\mathcal{L}(X,Y)\) has an octahedral norm and \(X^\ast\) is nonrough, then \(Y\) must have an octahedral norm.