The symmetric Radon-Nikodým property for tensor norms

Abstract

We introduce the symmetric-Radon-Nikodým property (sRN pr operty) 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 prop- erty, then for every Asplund space E , the canonical map e ⊗ n,s β E ′ → e ⊗ n,s β ′ E ′ is a metric surjection. This can be rephrased as the isometric isomorph ism Q min ( E ) = Q ( E ) for certain polynomial ideal Q . We also relate the sRN property of an s-tensor norm with the A splund or Radon-Nikodým properties of different tensor products. S imilar results for full tensor products are also given. As an application, results concern ing the ideal of n -homogeneous extendible polynomials are obtained, as well as a new proof o f the well known isometric isomorphism between nuclear and integral polynomials on As plund spaces.