On the theory of \(L_p (L_q)\)-Banach lattices
It is well-known that \(L_p\)-spaces, \(1 \leq p < \infty\), are characterized as the class of Banach lattices \(X\) satisfying \(\| x + y \| ^p = \| x \| ^p + \| y \| ^p\) for every pair of disjoint elements in \(X\). The goal of the paper under review is to give such a characterization for \(L_p(L_q)\)-spaces, in terms of first order statements, as defined in [\textit{C.\,Henson} and \textit{J.\,Iovino}, Lond.\ Math.\ Soc.\ Lect.\ Note Ser.\ 262, 1--110 (2002; Zbl 1026.46007)]. Actually, such a characterization cannot exist because the class of \(L_p(L_q)\)-spaces is not closed under ultraproducts. The authors replace this class by a larger class, the class of \(BL_pL_q\)-spaces, which is the class of bands in \(L_p(L_q)\)-spaces. In order to get the desired axioms, they first prove that for \(1 \leq p,q < \infty\), the range of any positive contractive projection in a \(BL_pL_q\)-Banach lattice is isometrically lattice isomorphic to another \(BL_pL_q\)-Banach lattice. They then define the analog of the \({\mathcal L}_p\)-spaces, called the \({\mathcal L}_p{\mathcal L}_q\)-lattices, and show that, for \(1 \leq p,q < \infty\), a Banach lattice is a \(({\mathcal L}_p{\mathcal L}_q)_1\)-lattice if and only if it is isometrically lattice isomorphic to a \(BL_pL_q\)-Banach lattice. In the last section of their paper, the authors give a characterization, in terms of first order statements, for the sublattices of \(L_p(L_q)\)-Banach lattices. When \(1 \leq p < q < \infty\), this characterization is that the \(p^{th}\) power of the norm of \(X\) is a negative definite function on the semigroup \((X_+, +_q)\), where \(X_+\) is the positive cone of \(X\) and \(x +_q y = (x^q + y^q)^{1/q}\). When \(1 \leq q \leq p < \infty\) and \(p/q\) is not an integer, this characterization is more complicated. A characterization of the sublattices of \(L_p \{L_\infty\}\), a bigger space than \(L_p(L_\infty)\), is also given for \(1\leq p < \infty\).
scientific article; zbMATH DE number 5162441