A Bourgain-Pisier construction for general Banach spaces

TL;DR

This paper extends classical results by Bourgain and Pisier, showing that any Banach space can be embedded into an $\\mathcal{L}_{\infty}$-space with desirable properties, leading to new large-density spaces with the Schur and Radon-Nikodym properties.

Contribution

It introduces a Bourgain-Pisier type construction applicable to all Banach spaces, including non-separable ones, with specific geometric properties.

Findings

Every Banach space embeds into an $\mathcal{L}_\infty$-space with the Radon-Nikodym and Schur properties.

Constructs $\mathcal{L}_\infty$-spaces of arbitrary large densities with these properties.

Extends classical results to non-separable Banach spaces.

Abstract

We prove that every Banach space, not necessarily separable, can be isometrically embedded into a $\mathcal L_{\infty}$-space in a way that the corresponding quotient has the Radon-Nikodym and the Schur properties. As a consequence, we obtain $\mathcal L_\infty$ spaces of arbitrary large densities with the Schur and the Radon-Nikodym properties. This extents the a classical result by J. Bourgain and G. Pisier.