Point-evaluation functionals on algebras of symmetric functions on $(L_\infty)^2$

Abstract

It is known that every continuous symmetric (invariant under the composition of its argument with each Lebesgue measurable bijection of $[0,1]$ that preserve the Lebesgue measure of measurable sets) polynomial on the Cartesian power of the complex Banach space $L_\infty$ of all Lebesgue measurable essentially bounded complex-valued functions on $[0,1]$ can be uniquely represented as an algebraic combination, i.e., a linear combination of products, of the so-called elementary symmetric polynomials. Consequently, every continuous complex-valued linear multiplicative functional (character) of an arbitrary topological algebra of the functions on the Cartesian power of $L_\infty,$ which contains the algebra of continuous symmetric polynomials on the Cartesian power of $L_\infty$ as a dense subalgebra, is uniquely determined by its values on elementary symmetric polynomials. Therefore, the problem of the description of the spectrum (the set of all characters) of such an algebra is equivalent to the problem of the description of sets of the above-mentioned values of characters on elementary symmetric polynomials.
 In this work, the problem of the description of sets of values of characters, which are point-evaluation functionals, on elementary symmetric polynomials on the Cartesian square of $L_\infty$ is completely solved. We show that sets of values of point-evaluation functionals on elementary symmetric polynomials satisfy some natural condition. Also, we show that for any set $c$ of complex numbers, which satisfies the above-mentioned condition, there exists an element $x$ of the Cartesian square of $L_\infty$ such that values of the point-evaluation functional at $x$ on elementary symmetric polynomials coincide with the respective elements of the set $c.$