Extending polynomials in maximal and minimal ideals

TL;DR

This paper proves that extensions of homogeneous polynomials in maximal or minimal ideals to ultrapowers preserve the ideal norm, establishing the Aron-Berner extension as an isometry and enabling symmetric tensor product results.

Contribution

It demonstrates that polynomial extensions to ultrapowers preserve ideal membership and norm, and establishes the Aron-Berner extension as an isometry for these ideals.

Findings

Extension of polynomials to ultrapowers remains in the ideal with the same norm.

The Aron-Berner extension is an isometry for maximal and minimal polynomial ideals.

Results enable symmetric versions of metric tensor product theorems.

Abstract

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.