Uniqueness of extensions of homogeneous polynomials on \(c_{0}\)-sum of Hilbert spaces (Q935934)

The authors study homogeneous polynomials on the \(c_0\)-sum \(X\) of complex Hilbert spaces \(\{H_i\}_{i\in I}\). They show that if \(P\) is a 2-homogeneous polynomial on \(X''\) which attains its norm at a point \(x=(x_j)_j\) of \(X\), then the set of \(j\) in \(I\) where \(\| x_j\| _{H_j}=1\) is finite. From this it follows that every 2-homogeneous polynomial on \(X\) which attains its norm has a unique norm-preserving extension to \(X''\). When \(n\geq 3\), the authors provide examples of \(n\)-homogeneous polynomials on \(X\) which have two distinct norm-preserving extensions to \(X''\). \textit{P.--K.\thinspace Lin} [Acta Math.\ Sin.\ 24, No.\,5, 877--880 (2008; Zbl 1158.46032), see the preceding review] has recently shown that every \(2\)-homogeneous polynomial on complex \(c_0\) has a unique norm-preserving extension to \(\ell_\infty\). Thus it is reasonable to conjecture that every \(2\)-homogeneous polynomial on \(X\) has a unique norm-preserving extension to \(X''\)