Daniel Galicer

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.

Extending Polynomials in Maximal and Minimal Ideals

Given an homogeneous polynomial on a Banach space E belonging to some maximal or minimal polynomial ideal, we consider its iterated extension to an ultrapower of E and prove that this extension remains in the ideal and has the same ideal norm. As a consequence, we show that the Aron-Berner extension is a well defined isometry for any maximal or minimal ideal of homogeneous polynomials. This allow us to obtain symmetric versions of some basic results of the metric theory of tensor products.

Natural symmetric tensor norms

Abstract In the spirit of the work of Grothendieck, we introduce and study natural symmetric n-fold tensor norms. These are norms obtained from the projective norm by some natural operations. We prove that there are exactly six natural symmetric tensor norms for n ⩾ 3 , a noteworthy difference with the 2-fold case in which there are four. We also describe the polynomial ideals associated to these natural symmetric tensor norms. Using a symmetric version of a result of Carne, we establish which natural symmetric tensor norms preserve the Banach algebra structure.

UNCONDITIONALITY IN TENSOR PRODUCTS AND IDEALS OF POLYNOMIALS, MULTILINEAR FORMS AND OPERATORS

We study tensor norms that destroy unconditionality in the following sense: for every Banach space

L q dimensions and projections of random measures

E

Geometry of integral polynomials, M-ideals and unique norm preserving extensions

with unconditional basis, the

The minimal volume of simplices containing a convex body

n

COINCIDENCE OF EXTENDIBLE VECTOR-VALUED IDEALS WITH THEIR MINIMAL KERNEL

-fold tensor product of

The sup-norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials

E

FIVE BASIC LEMMAS FOR SYMMETRIC TENSOR PRODUCTS OF NORMED SPACES

(with the corresponding tensor norm) does not have unconditional basis. We establish an easy criterion to check weather a tensor norm destroys unconditionality or not. Using this test we get that all injective and projective tensor norms different from