Symmetric polynomials on the Cartesian power of the real Banach space $L_\infty[0,1]$

  • T. Vasylyshyn Vasyl Stefanyk Precarpathian National University, Ukraine
  • A. Zagorodnyuk Vasyl Stefanyk Precarpathian National University, Ukraine
Keywords: Symmetric polynomial, Spectrum of the Fr\'{e}chet algebra

Abstract

We construct an algebraic basis of the algebra of symmetric (invariant under composition of the variable with any measure preserving bijection of $[0,1]$) continuous polynomials on the $n$th Cartesian power of the real Banach
space $L_^{(\mathbb{R})}\infty[0,1]$ of Lebesgue measurable essentially bounded real valued functions on $[0,1].$ Also we describe the spectrum of the Fr\'{e}chet algebra $A_s(L_^{(\mathbb{R})}\infty[0,1])$ of symmetric real-valued functions on the space $L_^{(\mathbb{R})}\infty[0,1]$, which is the completion of the algebra of symmetric continuous real-valued polynomials on  $L_^{(\mathbb{R})}\infty[0,1]$ with respect to the family of norms of uniform convergence of complexifications of polynomials. We show that $A_s(L_^{(\mathbb{R})}\infty[0,1])$ contains not only analytic functions. Results of the paper can be used for investigations of algebras of symmetric functions on the $n$th Cartesian power of the Banach space $L_^{(\mathbb{R})}\infty[0,1]$.

Author Biographies

T. Vasylyshyn, Vasyl Stefanyk Precarpathian National University, Ukraine

Vasyl Stefanyk Precarpathian National University, Ukraine

A. Zagorodnyuk, Vasyl Stefanyk Precarpathian National University, Ukraine

Vasyl Stefanyk Precarpathian National University, Ukraine

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Published
2020-06-24
How to Cite
1.
Vasylyshyn T, Zagorodnyuk A. Symmetric polynomials on the Cartesian power of the real Banach space $L_\infty[0,1]$. Mat. Stud. [Internet]. 2020Jun.24 [cited 2020Jul.6];53(2):192-05. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/41
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