CONICET | Buscador de Institutos y Recursos Humanos

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