The Structure of Lattice-Subspaces

In Polyrakis (1983; Math. Proc. Cambridge Phil. Soc. 94, 519) it is proved that each infinite-dimensional, closed lattice-subspace of . . .₁ is order-isomorphic to . . .₁ and in Polyrakis (1987; Math. Anal. Appl. 184, 1) that each separable Banach lattice is order isomorphic to a closed lattice-subspace of C0,1]. Therefore . . .₁ contains only one lattice-subspace but C0,1] contains all the separable Banach lattices. In the first section of this article we study the kind of the order embeddability of a separable Banach lattice in C0,1]. We show that the AM spaces have the ``best' behavior and the AL-spaces the ``worst'. In the second section we prove that the closure of a lattice-subspace is not necessarily a lattice-subspace and in the least one we study lattice-subspaces with positive bases.