Amalgamations of classes of Banach spaces with a monotone basis

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 is an element of C is reflexive and has a monotone Schauder basis, then there exists a separable reflexive Banach space that is isometrically universal for C.

Keywords

• isometrically universal Banach space
• Effros Borel structure
• analytic set
• monotone basis
• tree space