On the bounded approximation property on subspaces of $\ell_p$ when $0<p<1$ and related issues

TL;DR

This paper investigates the bounded approximation property in quasi Banach spaces, particularly subspaces of  spaces for 0<p<1, extending classical results and exploring universal space constructions.

Contribution

It extends the BAP results to kernels of surjective operators on  spaces for 0<p1, and develops nonlocally convex universal spaces related to the BAP.

Findings

Kernel of surjective -to-X operator has BAP if X has BAP

Constructs nonlocally convex universal spaces for BAP

Generalizes Lusky's Banach space results to quasi Banach spaces

Abstract

This paper studies the bounded approximation property (BAP) in quasi Banach spaces. In the first part of the paper we show that the kernel of any surjective operator $\ell_p\to X$ has the BAP when $X$ has it and $0<p\leq 1$, which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec-Pe\l czy\'nski-Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.