Geometry of Spaces of Orthogonally Additive Polynomials on C(K)

TL;DR

This paper investigates the geometric structure of the space of orthogonally additive polynomials on C(K), comparing two norms, characterizing extreme points, and establishing a Banach-Stone type theorem.

Contribution

It provides a detailed geometric analysis of orthogonally additive polynomials on C(K), including extreme points and a new Banach-Stone theorem, highlighting differences between norms.

Findings

Characterization of extreme points for both norms

Differences in geometric properties for even and odd degrees

A Banach-Stone theorem for these polynomial spaces

Abstract

We study the space of orthogonally additive $n$-homogeneous polynomials on $C(K)$. There are two natural norms on this space. First, there is the usual supremum norm of uniform convergence on the closed unit ball. As every orthogonally additive $n$-homogeneous polynomial is regular with respect to the Banach lattice structure, there is also the regular norm. These norms are equivalent, but have significantly different geometric properties. We characterise the extreme points of the unit ball for both norms, with different results for even and odd degrees. As an application, we prove a Banach-Stone theorem. We conclude with a classification of the exposed points.