TY - JOUR AU - Vasylyshyn, T. AU - Zagorodnyuk, A. PY - 2020/06/24 Y2 - 2024/03/29 TI - Symmetric polynomials on the Cartesian power of the real Banach space $L_\infty[0,1]$ JF - Matematychni Studii JA - Mat. Stud. VL - 53 IS - 2 SE - Articles DO - 10.30970/ms.53.2.192-205 UR - http://matstud.org.ua/ojs/index.php/matstud/article/view/41 SP - 192-205 AB - 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 Banachspace $L_^{(\mathbb{R})}\infty[0,1]$ 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 $L_^{(\mathbb{R})}\infty[0,1]$. ER -