This is an announcement for the paper "M-ideals of homogeneous polynomials" by Veronica Dimant.

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 $E^*$ has the approximation property and $\mathcal{P}_w(^nE)$ 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.

Archive classification: math.FA

Mathematics Subject Classification: 46G25, 46B04, 47L22, 46B20.

Submitted from: vero@udesa.edu.ar

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1005.1260

or

http://arXiv.org/abs/1005.1260