Quasi-greedy bases in lp(0 < p < 1) are democratic

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), l(2), and all separable 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 non-locally convex spaces. We prove that all quasi-greedy bases in l(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)(infinity). The methods we develop allow us to obtain even more, namely that the same occurs in any separable L-p-space, 0 < p < 1, with the bounded approximation property. (C) 2020 Elsevier Inc. All rights reserved.