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

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.