$M$-ideals of homogeneous polynomials
Tom 202 / 2011
Streszczenie
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 $\mathcal{P}(^nE)$ of continuous $n$-homogeneous polynomials. We obtain conditions that ensure 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 does not attain its norm is nowhere dense in $\mathcal{P}(^nE)$. Finally, we discuss an analogous $M$-ideal problem for block diagonal polynomials.
