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 mapping ⊗ ˜ β n , s E ′ → ( ⊗ ˜ β ′ n , s E ) ′ is a metric surjection. This can be rephrased as the isometric isomorphism Q min ( E ) = Q ( E ) for some 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. 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. An analogous study is carried out for full tensor products.