Centralisers of spaces of symmetric tensor products and applications

Abstract

We show that the centraliser of the space of n-fold symmetric injective tensors, n ≥ 2, on a real Banach space is trivial. With a geometric condition on the set of extreme points of its dual, the space of integral polynomials we obtain the same result for complex Banach spaces. We give some applications of this results to centralisers of spaces of homogeneous polynomials and complex Banach spaces. In addition, we derive a Banach-Stone Theorem for spaces of vector-valued approximable polynomials.

The symmetric Radon–Nikodým property for tensor norms

“…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 also [15,Section 33.3]).…”