On unconditionally saturated Banach spaces

TL;DR

This paper establishes a structural property of unconditionally saturated Banach spaces, showing that for any analytic set of such spaces, there exists a universal space with a Schauder basis containing isomorphic copies of all spaces in the set.

Contribution

It proves the existence of a universal unconditionally saturated Banach space with a Schauder basis for any analytic class of such spaces.

Findings

Existence of a universal unconditionally saturated Banach space with a Schauder basis.

Construction of a space containing isomorphic copies of all spaces in a given analytic set.

Structural insight into the class of unconditionally saturated Banach spaces.

Abstract

We prove a structural property of the class of unconditionally saturated separable Banach spaces. We show, in particular, that for every analytic set $\aaa$, in the Effros-Borel space of subspaces of $C[0,1]$, of unconditionally saturated separable Banach spaces, there exists an unconditionally saturated Banach space $Y$, with a Schauder basis, that contains isomorphic copies of every space $X$ in the class $\aaa$.