In Polyrakis (1983; Math. Proc. Cambridge Phil. Soc. 94, 519) it is proved that each infinite-dimensional, closed lattice-subspace of ℓ1 is order-isomorphic to ℓ1 and in Polyrakis (1987; Math. Anal. Appl. 184, 1) that each separable Banach lattice is order isomorphic to a closed lattice-subspace of C[0,1]. Therefore ℓ1 contains only one lattice-subspace but C[0,1] 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 C[0,1]. 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.
Positivity – Springer Journals
Published: Oct 17, 2004
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