Linear versus lattice embeddings between Banach lattices

A well-known classical result states that c0 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 c0. Furthermore, we show that any infinite-dimensional closed sublattice of C[0, 1] is either lattice isomorphic to c0 or contains a closed 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]. (C) 2022 The Authors. Published by Elsevier Inc.