The lattice Schäffer constant
Michael Alexánder Rincón Villamizar, Timur Oikhberg
TL;DR
The paper studies the lattice Schäffer constant $\lambda^+(X)$ and its companion $\beta(X)$ for Banach lattices, revealing deep connections to global lattice structure such as KB-spaces, abstract $L$-spaces, and lattice copies of $\ell_\infty^2$. It develops stability results showing $\lambda^+(X)$ and $\beta(X)$ are determined by finite-dimensional sublattices and are preserved under duals and ultrapowers, with explicit computations in classical spaces like $L_p$ spaces. It further links these constants to monotonicity moduli $\sigma_X$ and $\delta_{m,X}$, deriving inequalities such as $\tilde{\varepsilon}_{0,m}(X)\leq 2-\lambda^+(X)$ and characterizations like $\tilde{\varepsilon}_{0,m}(X)<1$ iff $\lambda^+(X)>1$, and discusses implications for superreflexivity. The results offer a comprehensive geometric framework for understanding lattice non-squareness and its impact on global lattice properties.
Abstract
For a Banach lattice $X$, its lattice Schäffer constant is defined by: \begin{gather*} λ^+(X)=\inf\{\max\{\|x+y\|,\|x-y\|\}\,\colon\,\|x\|=\|y\|=1,x,y\geq{\bf0}\}. \end{gather*} In this paper, we investigate this constant, as well as the companion parameter \begin{gather*} β(X)=\inf\{\|x\vee y\|\,\colon\,\mbox{$\|x\|=\|y\|=1$, $x,y\geq{\bf0}$ and $x\wedge y={\bf0}$}\}. \end{gather*} Our main results fall into two groups. (1) We link the behavior of the parameters $λ^+$ and $β$ to the global properties of the lattice $X$. For instance, we prove that (i) if $λ^+(X)>1$, then the Banach lattice $X$ is a KB-space, and moreover, it satisfies a lower $q$-estimate for some $q\in(1,\infty)$; (ii) $λ^+(X)=1$ if and only if $X$ contains lattice-almost isometric copies of $\ell_\infty^2$; and (iii) that $λ^+(X)=2$ if and only if $X$ is an abstract $L$-space. (2) We establish inequalities relating $λ^+(X)$ to the characteristics of monotonicity, $\varepsilon_{0,m}(X)$ and $\tilde\varepsilon_{0,m}(X)$. Along the way, we compute $λ^+(X)$ and $β(X)$ for various Banach lattices $X$.