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arXiv · 2004.05206

Quasi-greedy bases in the spaces $\ell_p$ ($0<p<1$) are democratic

Fernando Albiac·Jose L. Ansorena·Przemyslaw Wojtaszczyk
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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.

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Fernando Albiac, Jose L. Ansorena, Przemyslaw Wojtaszczyk. 2020-04-10. Quasi-greedy bases in the spaces $\ell_p$ ($0<p<1$) are democratic. https://arxiv.org/abs/2004.05206

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