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 mappingsurjection. 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.A result of Boyd and Ryan [5] and also of Carando and Dimant [9] implies that, for an Asplund space E, the space P n I (E) of integral polynomials is isometric to the space P n N (E) of nuclear polynomials (the isomorphism between these spaces was previously obtained by Alencar in [1,2]). In other words, if E is Asplund, the space of integral polynomials on E coincides isometrically with its minimal hull (P n I ) min (E) = P n N (E). This fact was used, for example, in [5,6,17] to study geometric properties of spaces of polynomials and tensor products (e.g., extreme and exposed points of their unit balls), and in [7,8] to characterize isometries between spaces of polynomials and centralizers of symmetric tensor products. When the above mentioned isometry is stated as the isometric coincidence between a maximal ideal and its minimal hull, it resembles the Lewis theorem for operator ideals and (2-fold) tensor norms (see [24] and [15, 33.3]). The Radon-Nikodým property for tensor norms is a key ingredient for Lewis theorem.The aim of this article is to find conditions under which the equality Q(E) = Q min (E) holds isometrically for a maximal polynomial ideal Q. In terms of symmetric tensor products, we want conditions on an s-tensor norms ensuring the isometry n,s β E = ( n,s β E) . To this end, we introduce the symmetric Radon-Nikodým property for s-tensor norms and show our main result, a Lewis-type theorem (Theorem 2.2): if an s-tensor norm has the symmetric Radon-Nikodým property (sRN property), then the canonical mapping n,s \β/ E → ( n,s /β \ E) is a metric surjection for every Asplund space E (see the notation below). As a consequence, if Q is the maximal ideal (of n-homogeneous polynomials) associated to a projective s-tensor norm β with the sRN property, then Q min (E) = Q(E) isometrically.

Coincidence of extendible vector-valued ideals with their minimal kernel

“…To study when the mapping ̺ is actually a quotient mapping a condition on the tensor norm is needed. A fundamental ingredient both in [Lew77] and [CG11b], where coincidence results are studied (in the operator frame and multilinear/polynomial context, respectively), is the Radon-Nikodým property for tensor norms. Based on the definitions therein, we give a vector-valued version of this notion not for tensor norms but for ideals of multilinear operators.…”

“…Other properties (such as separability, Asplund or the Radon-Nikodým properties), in many cases are also preserved by the tensor product (see for example [Bu03,BB06,BDDO03,CG11b,RS82] and also the references therein). A tensorial representation of the ideal and these kind of transference results, permit to deduce many attributes of the space A(E 1 , .…”

“…Lewis ([Lew77] and [DF93,33.3]) obtained many results of the form A min (E; F ′ ) = A(E; F ′ ) if A is a maximal operator ideal or, in other words, coincidences of the form E ′⊗ α F ′ = (E⊗ α ′ F ) ′ . Based on Lewis' work, the first author and Carando tackled in [CG11b] an analogous problem for scalar ideals of multilinear operators and polynomials (i.e., where the target space is the scalar field). As in Lewis' theorem, the Radon-Nikodým property becomes a key ingredient.…”

“…As in Lewis' theorem, the Radon-Nikodým property becomes a key ingredient. In this article we follow the lines of [CG11b] to address the vector-valued case. We stress that the vector-valued problem adds some technical difficulties.…”

“…For a given ideal of multilinear operators, we introduce in Definition 2.1 a vector-valued Radon-Nikodým property in the sense of [Lew77,CG11b] (where a similar property is given for tensor norms). This definition is related with a coincidence result in c 0 -spaces.…”