The symmetric Radon-Nikod\'ym property for tensor norms

TL;DR

This paper introduces the symmetric-Radon-Nikodým property for s-tensor norms, establishing new isometric isomorphisms and relations with Asplund and Radon-Nikodým properties, impacting polynomial ideals and tensor product theory.

Contribution

It defines the sRN property for s-tensor norms and proves a Lewis type theorem, linking tensor norm properties with classical Banach space properties and polynomial ideals.

Findings

Establishes isometric isomorphisms for tensor products on Asplund spaces.

Connects the sRN property with Asplund and Radon-Nikodým properties.

Provides new proofs and results for polynomial ideals and tensor product relations.

Abstract

We introduce the symmetric-Radon-Nikod\'ym property (sRN property) for finitely generated s-tensor norms $\beta$ of order $n$ and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if $\beta$ is a projective s-tensor norm with the sRN property, then for every Asplund space $E$, the canonical map $\widetilde{\otimes}_{\beta}^{n,s} E' \to \Big(\widetilde{\otimes}_{\beta'}^{n,s} E \Big)'$ is a metric surjection. This can be rephrased as the isometric isomorphism $\mathcal{Q}^{min}(E) = \mathcal{Q}(E)$ for certain polynomial ideal $\Q$. We also relate the sRN property of an s-tensor norm with the Asplund or Radon-Nikod\'{y}m properties of different tensor products. Similar results for full tensor products are also given. As an application, results concerning the ideal of $n$-homogeneous extendible polynomials are obtained, as well as a new proof of the well known isometric isomorphism between nuclear and integral polynomials on Asplund spaces.