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 k-homogeneous 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 nontrivial M-ideal in Y, then ⊗ ˆ ε k , s k , s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in ⊗ ˆ ε k , s k , s Y . This result marks up a difference with the behavior of nonsymmetric tensors since, when X is an M-ideal in Y, it is known that ⊗ ˆ ε k k X (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in ⊗ ˆ ε k k Y . Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.