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Point-evaluation functionals on algebras of symmetric functions on \((L_\infty)^2\). (English) Zbl 1441.46032

The author considers the Cartesian product of the space of essentially bounded functions on the closed interval \([0,1]\) with itself, \((L_\infty)^2\), endowed with the norm \[\|x_1,x_2\|_{\infty,2}=\max\{\|x_1\| _\infty, \|x_2\|_\infty\}.\] A function \(f\) on \((L_\infty)^2\) is said to be symmetric if \(f((x_1\circ\sigma,x_2\circ\sigma))=f((x_1,x_2))\) for all \((x_1, x_2)\) in \((L_\infty)^2\) and all measurable bijections \(\sigma\colon [0,1]\to [0, 1]\) which have a measurable inverse. Given a pair of natural numbers \(k=(k_1, k_2)\), \(R_k:(L_\infty)^2\to \mathbb{C}\), define the symmetric \((k_1+k_2)\)-homogeneous polynomial by \[ R_k(x)=\int_{[0,1]}\prod_{s=1,k_s>0}^2(x_s(t))^{k_s}\,dt. \] The main result of the paper is that, if \(c=(c(z))_{n\in \mathbb{Z}^2\setminus \{(0,0)\}}\) is a complex-valued function on \(\mathbb{Z}^2\setminus\{(0,0)\}\) such that \(\sup_k|c(k)|^{\frac{1}{k_1+k_2}}<\infty\), then there is \(x_c\) in \((L_\infty)^2 \) such that \(R_k(x_c)=c(k)\) for every \(k\in \mathbb{Z}^2\setminus\{(0,0)\}\). Moreover, it is shown that \(\|x_c\|\le \frac{24}{M^3}\sup_k|c( k)|^{\frac{1}{k_1+k_2}}\), where \(M=\prod_{n=1}^\infty\cos\left(\frac{\pi}{2}\frac{1}{n+1}\right)\).
The proof of the theorem uses Rademacher functions and the analogous result for symmetric polynomials on \(L_\infty\). Using this result, the author shows that, if \(A\) is topological algebra on \((L_\infty)^2\) which contains all symmetric polynomials as a dense subalgebra and if all evaluations at points of \((L_\infty)^2\) are continuous, then a continuous multiplicative linear functional \(\varphi: A\to \mathbb C\) is a point evaluation if and only if \(\sup_k|\varphi(R_k)|^{\frac{1}{k_1+k_2}}\) is finite.

MSC:

46G25 (Spaces of) multilinear mappings, polynomials
46G20 Infinite-dimensional holomorphy
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