The symmetric Radon-Nikodým property for tensor norms

Abstract

We introduce the symmetric-Radon-Nikodým property (sRN property) for finitely generated s-tensor norms $β$ of order $n$ and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if $β$ is a projective s-tensor norm with the sRN property, then for every Asplund space $E$, the canonical map $\widetilde{\otimes}_β^{n,s} E' \to \Big(\widetilde{\otimes}_{β'}^{n,s} E \Big)'$ 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-Nikodým 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.