Classical subspaces of non-atomic Banach lattices

Tsirelson's space was the first example of a Banach space which does not have a subspace isomoprhic to any of the classical spaces $\ell_p$, $1\leqslant p<\infty$, or $c_0$. As this space has a $1$-unconditional basis, it can also be viewed as a Banach lattice. Therefore Banach lattices need not contain isomorphic copy of any of the classical spaces $\ell_p$ or $c_0$.

What about in the case of a Banach lattice which is not discrete? Is there some positive result about the classical spaces which must be subspaces of a Banach lattice which is not discrete?

• Do you wish your non-discrete Banach lattice to contain the classical sequence spaces as closed sublattices, or just as closed subspaces? – Jochen Glueck Feb 23 '18 at 22:23
• By the way, two remarks concerning the spaces $c_0$ and $\ell^1$: (i) A Banach lattice $E$ is a KB-space if and only if $E$ does not contain $c_0$ as a closed subspace if and only if $E$ does not contain $c_0$ as a closed sublattice [1, Theorems 2.4.12 and 2.5.6]; (ii) A Banach lattice $E$ is reflexive if and only $E$ does not contain $c_0$ nor $\ell^1$ as a closed subspace if and only if $E$ does not contain $c_0$ nor $\ell^1$ as a closed sublattice [1, Theorem 2.4.15]. Reference: [1] "Peter Meyer-Nieberg: Banach Lattices (1991)" – Jochen Glueck Feb 23 '18 at 22:28
• I only care that it contains the classical spaces as subspaces. I do not require that they be sublattices. – user114263 Feb 23 '18 at 22:49
• You can try to look at the paper Tokarev, E. V. A symmetric Banach function space that does not contain $\ell_p$ $(1\le p<\infty)$ and $c_0$ (Russian) Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 75-76 (it was translated into English). I should warn you that this author is known for making errors in proofs. – Mikhail Ostrovskii Feb 24 '18 at 6:12
• For an example illustrating @MikhailOstrovskii 's claim, see mathoverflow.net/questions/171680/… – Tomasz Kania Feb 24 '18 at 10:58