Linear versus lattice embeddings between Banach lattices

TL;DR

This paper investigates the relationship between linear and lattice embeddings in Banach lattices, showing that for certain spaces like C[0,1], linear and lattice embeddability are equivalent, and characterizing sublattices of C[0,1].

Contribution

It extends classical results by proving that C[0,1] shares the linear-lattice embeddability equivalence with c_0, and characterizes sublattices of C[0,1].

Findings

C[0,1] shares the linear and lattice embeddability property with c_0.

Any infinite-dimensional sublattice of C[0,1] is either isomorphic to c_0 or contains a sublattice isomorphic to C[0,1].

For separable Banach lattices, linear and lattice embeddability are equivalent if and only if they embed into C[0,1].

Abstract

A well-known classical result states that $c_0$ is linearly embeddable in a Banach lattice if and only if it is lattice embeddable. Improving results of H.P.~Lotz, H.P.~Rosenthal and N.~Ghoussoub, we prove that $C[0,1]$ shares this property with $c_0$. Furthermore, we show that any infinite-dimensional sublattice of $C[0,1]$ is either lattice isomorphic to $c_0$ or contains a sublattice isomorphic to $C[0,1]$. As a consequence, it is proved that for a separable Banach lattice $X$ the following conditions are equivalent: (1) $X$ is linearly embeddable in a Banach lattice if and only if it is lattice embeddable; (2) $X$ is lattice embeddable into $C[0,1]$.