The symmetric Radon-Nikodým property for tensor norms

Abstract

We introduce the symmetric-Radon-Nikodým 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 ${\tilde{\otimes }}_{\beta }^{n,s}{E}^{\prime }\to ({\tilde{\otimes }}_{{\beta }^{\prime }}^{n,s}E{)}^{\prime }$ 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.

17 pages

References in corpus (3)

• Natural symmetric tensor norms
• Extending polynomials in maximal and minimal ideals
• Unconditionality in tensor products and ideals of polynomials, multilinear forms and operators

Cited by in corpus (3)

• Holomorphic Functions and polynomial ideals on Banach spaces
• Geometry of integral polynomials, $M$-ideals and unique norm preserving extensions
• Five basic lemmas for symmetric tensor products of normed spaces