M-ideals of homogeneous polynomials

Abstract

We study the problem of whether Pw(nE), the space of n-homogeneous polynomials which are weakly continuous on bounded sets, is an M-ideal in the space P(nE) of continuous n-homogeneous polynomials. We obtain conditions that ensure this fact and present some examples. We prove that if Pw(nE) is an M-ideal in P(nE), then Pw(nE) coincides with Pwo(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 P w(nE) = Pw0(nE) and κ(E) is an M-ideal in (E), then Pw(nE) is an Mideal in P( nE). We also show that if Pw(nE) is an M-ideal in P(nE), then the set of n-homogeneous polynomials whose Aron-Berner extension does not attain its norm is nowhere dense in P(nE). Finally, we discuss an analogous M-ideal problem for block diagonal polynomials.