Slices in the unit ball of the symmetric tensor product of \(\mathcal{C}(K)\) and \(L_{1}(\mu )\)
The authors consider the symmetric projective tensor product of \(N\) copies of \(C_0(\Omega)\), with \(\Omega\) locally compact, and of \(L_1(\mu)\). The main results are: {\parindent7mm \begin{itemize}\item[(1)] If \(\Omega\) is infinite, then every relatively weakly open subset of the unit ball of \(\bigotimes_{\pi,s}^N C_0(\Omega)\) has diameter 2. \item[(2)] If \(\mu\) is \(\sigma\)-finite and atomless, then every slice of the unit ball of \(\bigotimes_{\pi,s}^N L_1(\mu)\) has diameter 2. \item[(3)] If, in addition, \(N\) is odd, then every relatively weakly open subset of the unit ball of \(\bigotimes_{\pi,s}^N L_1(\mu)\) has diameter 2. \end{itemize}} The results have consequences concerning the roughness of the norm in the dual of \(\bigotimes_{\pi,s}^N X\), the space of \(N\)-homogeneous polynomials on \(X\), for \(X=C_0(\Omega)\) and \(X=L_1(\mu)\).
scientific article; zbMATH DE number 5603926