Equivalence of the Bochner norm and its transpose characterizes \(L^p\)-spaces (Q2484189)

Let \(E\) be a Banach lattice. A well-known result of \textit{J. L. Krivine} [Séminaire Maurey-Schwartz 1973--1974: Espaces \(L^{p}\), Appl. radonif., Géom. Espaces de Banach, Exposé XXII et XXIII (1974; Zbl 0295.47024)] characterizes those \(E\) that are Riesz and topologically or isometrically isomorphic to an \(L^p(\mu)\) using Krivine's functional calculus. This interesting paper uses this to give a new characterization of Banach lattices that are Riesz and topologically or isometrically isomorphic to an \(L^p(\mu)\)-space for \(1 \leq p < \infty\), where \(\mu\) is a finite or \(\sigma\)-finite measure. Let \(\Delta_p\) denote the usual Bochner norm on the tensor product space \(L^p(\mu) \otimes E\). A key observation is that the conditions considered by Krivine are equivalent to the norms \(\Delta_p\) and its transpose being equivalent (equal) on the tensor product space \(L^p(\mu) \otimes E\).