Quasi-greedy bases in the spaces ${\ell }_p$ ($0<p<1$) are democratic

Abstract

The list of known Banach spaces whose linear geometry determines the (nonlinear) democracy functions of their quasi-greedy bases to the extent that they end up being democratic, reduces to $c_0$, ${\ell }_2$, and all separable ${\mathcal{L}}_1$-spaces. Oddly enough, these are the only Banach spaces that, when they have an unconditional basis, it is unique. Our aim in this paper is to study the connection between quasi-greediness and democracy of bases in nonlocally convex spaces. We prove that all quasi-greedy bases in ${\ell }_p$ for $0<p<1$ (which also has a unique unconditional basis) are democratic with fundamental function of the same order as $(m^{1/p}{)}_{m=1}^{\infty }$. The methods we develop allow us to obtain even more, namely that the same occurs in any separable ${\mathcal{L}}_p$-space, $0<p<1$, with the bounded approximation property.