M-ideals of homogeneous polynomials

Verónica Dimant. "M-ideals of homogeneous polynomials." Studia Mathematica 202.1 (2011): 81-104. <http://eudml.org/doc/285491>.

@article{VerónicaDimant2011,
abstract = {We study the problem of whether $_\{w\}(ⁿE)$, 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 $_\{w\}(ⁿE)$ is an M-ideal in (ⁿE), then $_\{w\}(ⁿE)$ coincides with $_\{w0\}(ⁿE)$ (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if $_\{w\}(ⁿE) = _\{w0\}(ⁿE)$ and (E) is an M-ideal in (E), then $_\{w\}(ⁿE)$ is an M-ideal in (ⁿE). We also show that if $_\{w\}(ⁿE)$ 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.},
author = {Verónica Dimant},
journal = {Studia Mathematica},
keywords = {M-ideals; homogeneous polynomials},
language = {eng},
number = {1},
pages = {81-104},
title = {M-ideals of homogeneous polynomials},
url = {http://eudml.org/doc/285491},
volume = {202},
year = {2011},
}

TY - JOUR
AU - Verónica Dimant
TI - M-ideals of homogeneous polynomials
JO - Studia Mathematica
PY - 2011
VL - 202
IS - 1
SP - 81
EP - 104
AB - We study the problem of whether $_{w}(ⁿE)$, 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 $_{w}(ⁿE)$ is an M-ideal in (ⁿE), then $_{w}(ⁿE)$ coincides with $_{w0}(ⁿE)$ (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if $_{w}(ⁿE) = _{w0}(ⁿE)$ and (E) is an M-ideal in (E), then $_{w}(ⁿE)$ is an M-ideal in (ⁿE). We also show that if $_{w}(ⁿE)$ 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.
LA - eng
KW - M-ideals; homogeneous polynomials
UR - http://eudml.org/doc/285491
ER -