Basic sequences and subspaces in Lorentz sequence spaces without local convexity

Abstract:

After some preliminary results $(\S 1)$, we give in $\S 2$ another proof of the result of N. J. Kalton [5] concerning the unicity of the unconditional bases of ${l_p}$, $0 < p < 1$. Using this result we prove in §3 the unicity of certain bounded symmetric block bases of the subspaces of the Lorentz sequence spaces $d(w,p)$, $0 < p < 1$. In $\S 4$ we show that every infinite dimensional subspace of $d(w,p)$ contains a subspace linearly homeomorphic to ${l_p}$, $0 < p < 1$. Unlike the case $p \geqslant 1$ there are subspaces of $d(w,p)$, $0 < p < 1$, which contain no complemented subspaces of $d(w,p)$ linearly homeomorphic to ${l_p}$. In fact there are spaces $d(w,p)$, $0 < p < 1$, which contain no complemented subspaces linearly homeomorphic to ${l_p}$. We conjecture that this is true for every $d(w,p)$, $0 < p < 1$. The answer to the previous question seems to be important: for example we can prove that a positive complemented sublattice $E$ of $d(w,p)$, $0 < p < 1$, with a symmetric basis is linearly homeomorphic either to ${l_p}$ or to $d(w,p)$; consequently, a positive answer to this question implies that $E$ is linearly homeomorphic to $d(w,p)$. In $\S 5$ we are able to characterise the sublattices of $d(w,p)$, $p = {k^{ - 1}}$ (however under a supplementary restriction concerning the sequence $({w_n})_{n = 1}^\infty )$, which are positive and contractive complemented, as being the order ideals of $d(w,p)$. Finally, in $\S 6$, we characterise the Mackey completion of $d(w,p)$ also in the case $p = {k^{ - 1}}$, $k \in {\mathbf {N}}$.