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

Abstract

We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral polynomials over a real Banach space X is {±φ k : φ ∈ X * , φ = 1}. With this description we show that, for real Banach spaces X and Y , if X is a non trivial Mideal in Y , then k,s ε k,s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M -ideal in k,s ε k,s Y . This result marks up a difference with the behavior of non-symmetric tensors since, when X is an M -ideal in Y , it is known that k ε k X (the k-th tensor product of X endowed with the injective tensor norm) is an M -ideal in k ε k Y . Nevertheless, if X is Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. We explicitly describe this extension.We also give necessary and sufficient conditions (related with the continuity of the Aron-Berner extension morphism) for a fixed k-homogeneous polynomial P belonging to a maximal polynomial ideal Q( k X) to have a unique norm preserving extension to Q( k X * * ). To this end, we study the relationship between the bidual of the symmetric tensor product of a Banach space and the symmetric tensor product of its bidual and show (in the presence of the BAP) that both spaces have 'the same local structure'. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.2010 Mathematics Subject Classification. 46G25, 46M05, 46B28.

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“…We refer to ([1]- [9], [11]- [32] and references therein) for some recent work about extremal properties of multilinear mappings and homogeneous polynomials on some classical Banach spaces.…”

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