Boundedness of the averaging projections in nonlocally convex Lorentz sequence spaces and applications to basis theory

Abstract

We study the boundedness of averaging projections associated with symmetric Schauder bases in quasi-Banach spaces. Although this property is standard in the Banach setting, it is far from clear in the absence of local convexity and, indeed, fails for a broad class of quasi-Banach spaces with a symmetric basis, including ${\ell }_p$ for $0<p<1$. Our main result shows that, nevertheless, the canonical basis of an entire class of weighted Lorentz sequence spaces, including the spaces ${\ell }_{p,q}$ for $0<q<1<p<\infty$, has uniformly bounded averaging projections. Thus, bounded averaging projections do not characterize local convexity among quasi-Banach spaces with symmetric bases. As applications, we obtain new consequences for the structure of special bases. In particular, as a byproduct of our approach, we derive new examples of conditional and almost greedy bases in nonlocally convex spaces.