Norm attaining operators from \(L_1 (\mu)\) into \(L_{\infty} (\nu)\) (Q1592803)

An old problem in Banach space geometry is to identify the Banach spaces \(X\) and \(Y\) such that the norm attaining operators are dense in \(L(X,Y)\), the space of bounded linear operators from \(X\) to \(Y\). This paper proves that the classical spaces \(X=L_1\) and \(Y=L_{\infty}\) have this property. There are many known spaces with this property. However, the best previous result for \(L_1\) and \(L_{\infty}\) spaces was that the norm attaining operators are dense in \(L(L_1(\mu), L_{\infty}[0,1])\) if \(\mu\) is \(\sigma\)-finite [\textit{C. Finet} and \textit{R Payá}, Isr. J. Math 108, 139-143 (1998; Zbl 0929.47037)]. Here, the authors prove that the norm attaining operators are dense in \(L(L_1(\mu), L_{\infty}(\nu))\) for arbitrary measures \(\mu\) and localizable measures \(\nu\).