We study the problem of whether , the space of n-homogeneous polynomials which are weakly continuous on bounded sets, is an M-ideal in the space (ⁿE) of continuous n-homogeneous polynomials. We obtain conditions that ensure this fact and present some examples. We prove that if is an M-ideal in (ⁿE), then coincides with (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if and (E) is an M-ideal in (E), then is an M-ideal in (ⁿE). We also show that if is an M-ideal in (ⁿE), then the set of n-homogeneous polynomials whose Aron-Berner extension does not attain its norm is nowhere dense in (ⁿE). Finally, we discuss an analogous M-ideal problem for block diagonal polynomials.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm202-1-5,
author = {Ver\'onica Dimant},
title = {M-ideals of homogeneous polynomials},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {81-104},
zbl = {1237.46033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm202-1-5}
}
Verónica Dimant. M-ideals of homogeneous polynomials. Studia Mathematica, Tome 204 (2011) pp. 81-104. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm202-1-5/