M-ideals of homogeneous polynomials

TL;DR

This paper investigates conditions under which the space of weakly continuous homogeneous polynomials forms an M-ideal within the space of all continuous homogeneous polynomials, providing new criteria and examples.

Contribution

It introduces a polynomial version of property (M), establishes conditions for M-ideal status, and explores the structure of block diagonal polynomials.

Findings

If the space of weakly continuous polynomials is an M-ideal, it coincides with those weakly continuous at zero.

The paper provides conditions involving the M-ideal property of compact operators in Banach spaces.

It shows the set of polynomials not attaining their norm via Aron-Berner extension is nowhere dense.

Abstract

We study the problem of whether $\mathcal{P}_w(^nE)$, the space of $n$-homogeneous polynomials which are weakly continuous on bounded sets, is an $M$-ideal in the space of continuous $n$-homogeneous polynomials $\mathcal{P}(^nE)$. We obtain conditions that assure this fact and present some examples. We prove that if $\mathcal{P}_w(^nE)$ is an $M$-ideal in $\mathcal{P}(^nE)$, then $\mathcal{P}_w(^nE)$ coincides with $\mathcal{P}_{w0}(^nE)$ ($n$-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property $(M)$ and derive that if $\mathcal{P}_w(^nE)=\mathcal{P}_{w0}(^nE)$ and $\mathcal{K}(E)$ is an $M$-ideal in $\mathcal{L}(E)$, then $\mathcal{P}_w(^nE)$ is an $M$-ideal in $\mathcal{P}(^nE)$. We also show that if $\mathcal{P}_w(^nE)$ is an $M$-ideal in $\mathcal{P}(^nE)$, then the set of $n$-homogeneous polynomials whose Aron-Berner extension do not attain the norm is nowhere dense in $\mathcal{P}(^nE)$. Finally, we face an analogous $M$-ideal problem for block diagonal polynomials.