Symmetric polynomials on Cartesian products of Banach spaces of Lebesgue integrable functions

Abstract

The work is devoted to the study of complex-valued continuous symmetric polynomials on Cartesian products of complex Banach spaces of Lebesgue integrable functions. Let $L_p$, where $p\in [1;+\infty)$, be the complex Banach space of all complex-valued functions on $[0;1]$, the $p$th powers of absolute values of which are Lebesgue integrable. Let $\Xi_{[0;1]}$ be the set of all bijections $\sigma:[0;1] \to [0;1]$ such that both $\sigma$ and $\sigma^{-1}$ are measurable and preserve Lebesgue measure, i.e.~$\mu(\sigma(E)) = \mu(\sigma^{-1}(E)) = \mu(E)$ for every Lebesgue measurable set $E\subset [0;1]$, where $\mu$ is Lebesgue measure. A function $f$ on the Cartesian product $L_{p_{1}} \times \dots \times L_{p_{n}}$, where $p_1,\ldots, p_n \in [1;+\infty)$, is called symmetric if $$f((x_1\circ\sigma,\ldots,x_n\circ\sigma))=f((x_1,\ldots,x_n))$$ for every $\sigma\in \Xi_{[0;1]}$ and $(x_1,\ldots,x_n)\in L_{p_{1}} \times \dots \times L_{p_{n}}$. We construct an algebraic basis of the algebra of all complex-valued continuous symmetric polynomials on $L_{p_{1}} \times \dots \times L_{p_{n}}$. Also we construct some isomorphisms of Fréchet algebras of complex-valued entire symmetric functions of bounded type on $L_{p_{1}} \times \dots \times L_{p_{n}}$.