Amalgamations of classes of Banach spaces with a monotone basis

TL;DR

This paper introduces a new amalgamation technique to construct isometrically universal Banach spaces with a monotone basis, extending previous results to a broader class of spaces with specific structural properties.

Contribution

It develops a variant of the amalgamation method that guarantees isometric universality for classes of Banach spaces with a monotone Schauder basis, including reflexive spaces.

Findings

Constructed an isometrically universal space for classes with a monotone basis.

Extended amalgamation techniques to ensure isometric rather than just isomorphic universality.

Applied the method to classes of reflexive Banach spaces with analytic structure.

Abstract

It was proved by Argyros and Dodos that, for many classes $ C $ of separable Banach spaces which share some property $ P $, there exists an isomorphically universal space that satisfies $ P $ as well. We introduce a variant of their amalgamation technique which provides an isometrically universal space in the case that $ C $ consists of spaces with a monotone Schauder basis. For example, we prove that if $ C $ is a set of separable Banach spaces which is analytic with respect to the Effros-Borel structure and every $ X \in C $ is reflexive and has a monotone Schauder basis, then there exists a separable reflexive Banach space that is isometrically universal for $ C $.