Symmetric analytic functions on the Cartesian power of the complex Banach space of Lebesgue measurable essentially bounded functions on \([0,1]\)
The author characterises the spectrum of the algebra of bounded symmetric holomorphic functions on the \(n\)-fold Cartesian product of \(L_\infty[0,1]\). The author uses \(\Xi\) to denote the group of all measurable bijections \(\sigma \colon [0,1]\to [0,1]\) such that both \(\sigma\) and \(\sigma^{-1}\) preserve Lebesgue measure and \(\mathcal{H}_{bs}((L_\infty[0,1])^n)\) to denote the space of all holomorphic functions \(f\) on \((L_\infty[0,1])^n\) which are bounded on bounded subsets of \((L_\infty[0,1])^n\) and which have the property that \[ f(y_1,\ldots,y_n)=f(y_1\circ\sigma,\ldots,y_n\circ\sigma) \] for all \((y_1,\ldots, y_n)\) in \((L_\infty[0,1])^n\) and all \(\sigma\) in \(\Xi\). Using the result [\textit{T.~V. Vasylyshyn}, Mat. Stud. 52, No.~1, 71--85 (2019; Zbl 1441.46033)] that every \(N\)-homogeneous polynomial \(P\) in \(\mathcal{H}_{bs}((L_\infty[0,1])^n)\) has a unique representation of the form \[ P(y)=\sum_{\substack{l\colon M_N\to \mathbb{Z}^+\\ \sum_{k\in M_N}|\mathbf{k}|l(\mathbf{k})=N}} \alpha_l\prod_{\substack{\mathbf{k}\in M_N,\\ l(k)>0}}(R_{\mathbf{k}}(y))^{l(\mathbf{k})} , \] where \(M_N=\{\mathbf{k}\in \mathbb Z_+^n:1\le |\mathbf{k}|\le N\}\) and \[ R_{\mathbf{k}}(y)=\int_{[0,1]}\prod_{s=1,k_s>0}^n(y_s(t))^{k_s}\, dt , \] and the corresponding result for the algebra of symmetric functions \(\mathcal{H}_ {bs}(L_\infty[0,1])\), the author shows that every non-zero, multiplicative linear functional on \(\mathcal{H}_{bs}((L_\infty[0,1])^n)\) is an evaluation at a point of \((L_\infty[0, 1])^n\). The main result used in establishing this characterisation of the spectrum is that for each \(n\) in \(\mathbb{N}\) there is \(K_n>0\) such that for every function \(c\colon \mathbb{Z}^n _+\setminus\{(0,\ldots,0)\}\to \mathbb{C}\) with \(\sup_{\mathbf{k}\in \mathbf{Z}^n _+\setminus\{(0,\ldots,0)\}}|c(\mathbf{k})|^{1/|\mathbf{k}|}<\infty\) there is \(x_c\) in \((L_\infty[0,1])^n\) such that \(R_{\mathbf{k}}(x_c)=c(\mathbf{k})\) for all \(\mathbf{k} \) in \(\mathbb{Z}^n_+\setminus\{(0,\ldots,0)\}\).
Lebesgue measurable essentially bounded function
scientific article; zbMATH DE number 7472996
scientific article