Unconditionality for \(m\)-homogeneous polynomials on \(\ell_{\infty}^{n}\) (Q2804308)

\({\mathcal L}(^m\ell_\infty^n)\) is used to denote the space of \(m\)-linear mappings on \(\ell_\infty^n\) (\({\mathbb C}^n\) endowed with the supremum norm) and \({\mathcal P}(^n\ell_\infty)\) is used to denote the space of \(m\)-homogeneous polynomials on \(\ell_\infty^n\). In \({\mathcal L}(^m\ell_\infty^n)\), let \((e_{\mathbf i})_{\mathbf i}\) denote the basis given by \(e_{\mathbf i}(x_1,\dots, x_m)= e_{i_1}(x_1) \cdots e_{i_m}(x_m)\), where \({\mathbf i}=(i_1,\dots,i_m)\) for \(1\leq i_j\leq n\), while in \({\mathcal P}(^m\ell_\infty^n)\), \((z^\alpha)_\alpha\) denotes the monomial basis. The unconditional basis constants for these bases are denoted by \(\chi((e_{\mathbf i} )_{\mathbf i}, {\mathcal L}(^m\ell_\infty^n))\) and \(\chi((z^\alpha)_\alpha,{\mathcal P}(^m \ell_\infty^n))\), respectively. The main result of the paper is that NEWLINE\[NEWLINE\lim_{n\to\infty}{{\sup_m\root m\of{\chi((e_{\mathbf i})_{\mathbf i}, {\mathcal L} (^m\ell_\infty^n))}}\over{\sqrt{n}}}=1NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\lim_{n\to\infty}{{\sup_m\root m\of{\chi((z^{\alpha})_{\alpha}, {\mathcal P} (^m\ell_\infty^n))}}\over{\sqrt{n/\log n}}}=1.NEWLINE\]NEWLINE Two applications are presented. The first concerns the growth of the unconditional basis constant for spaces of injective and symmetric injective tensor products on \(\ell_1^n\). The second concerns the growth of the \(n^{th}\) Bohr radius.