This is an announcement for the paper “Banach and quasi-Banach spaces of almost universal complemented disposition” by Jesús M. F. Castillohttps://arxiv.org/find/math/1/au:+Castillo_J/0/1/0/all/0/1, Yolanda Morenohttps://arxiv.org/find/math/1/au:+Moreno_Y/0/1/0/all/0/1.

Abstract: We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) and show the existence of separable a.u.c.d. spaces with and without a Finite Dimensional Decomposition. We show that all a.u.c.d. spaces with $1$-FDD are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. Both assertions fail without the FDD assumption. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separable spaces. In the last section we consider $p$-Banach versions of all previous constructions showing that there are striking differences with either the Banach case or the classical case of simple universal disposition.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1708.02431