Diagonals of projective tensor products and orthogonally additive polynomials
Volume 221 / 2014
Abstract
Let $E$ be a Banach space with $1$-unconditional basis. Denote by $\varDelta(\mathop{\hat{\otimes}_{n,\pi}}E)\!$ (resp. $\varDelta(\mathop{\hat{\otimes}_{n,s,\pi}}E)$) the main diagonal space of the $n$-fold full (resp. symmetric) projective Banach space tensor product, and denote by $\varDelta(\mathop{\hat{\otimes}_{n,|\pi|}}E)$ (resp. $\varDelta(\mathop{\hat{\otimes}_{n,s,|\pi|}}E)$) the main diagonal space of the $n$-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to $E_{[n]}$, the completion of the $n$-concavification of $E$. Using these isometries, we also show that the norm of any (vector valued) continuous orthogonally additive homogeneous polynomial on $E$ equals the norm of its associated symmetric linear operator.
