On certain subspaces of ${\ell }_p$ for $0<p\le 1$ and their applications to conditional quasi-greedy bases in $p$-Banach spaces

Abstract

We construct for each $0<p\le 1$ an infinite collection of subspaces of ${\ell }_p$ that extend the example from [J. Lindenstrauss, On a certain subspace of ${\ell }_1$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 539-542] of a subspace of ${\ell }_1$ with no unconditional basis. The structure of this new class of $p$-Banach spaces is analyzed and some applications to the general theory of ${\mathcal{L}}_p$-spaces for $0<p<1$ are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in $p$-Banach spaces for $p<1$. Among the topics we consider are the existence of infinitely many conditional quasi-greedy bases in the spaces ${\ell }_p$ for $p\le 1$ and the careful examination of the conditionality constants of the "natural basis" of these spaces.