Unique Hahn-Banach theorems for spaces of homogeneous polynomials (Q2747267)

Let \(E\) be a Banach space. Let \(P(^nE)\) denote the spaces of all continuous \(n\)-homogeneous polynomials on \(E.\) It is known that every continuous \(n\)-homogeneous polynomial \(P\) on \(E\) has a norm-preserving extension \(\tilde{P}\) on the bidual \(E''.\) In this paper, the authors examine under what conditions they have a unique norm-preserving extension for spaces of homogeneous polynomials. Indeed, they show that there is a unique norm-preserving extension to \(l_{\infty}\) for every \(2\)-homogeneous norm-attaining polynomial on \(c_0,\) but there is no unique norm-preserving extension from \(P(^nc_0)\) to \( P(^nl_\infty)\) for \(n>2.\) The authors also study norm-preserving extensions of nuclear polynomials from \(M\)-ideals to their biduals.