Banach spaces of almost universal complemented disposition

TL;DR

This paper introduces the concept of almost universal complemented disposition (a.u.c.d.) in Banach spaces, demonstrating their universality properties and relationships with other Banach space classes, expanding the understanding of Banach space structure.

Contribution

It defines a.u.c.d. spaces, proves their universality for separable dual spaces, and explores their isometric properties and relations to spaces of universal complemented disposition.

Findings

Every Banach space with separable dual embeds into a separable a.u.c.d. space.

All a.u.c.d. spaces with 1-FDD are isometric and contain copies of all separable Banach spaces with 1-FDD.

Constructs and analyzes spaces of universal complemented disposition (u.c.d.) and their variants.

Abstract

We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and 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. 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.