Spectra of algebras of symmetric analytic functions on Cartesian products of Banach spaces of Lebesgue integrable functions

Abstract

The work is devoted to the study of algebras of entire symmetric functions on Cartesian products of real and complex Banach spaces of Lebesgue integrable functions.
For $$$p\in [1;+\infty)$$$, let $$$L_p^{(\mathbb{K})}$$$ be the Banach space over a field $$$\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$$$ of all $$$\mathbb{K}$$$-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 $$$L_{p_{1}}^{(\mathbb{K})} \times \ldots \times L_{p_{n}}^{(\mathbb{K})}$$$, 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 $$$(x_1;\ldots;x_n)\in L_{p_{1}}^{(\mathbb{K})} \times \ldots \times L_{p_{n}}^{(\mathbb{K})}$$$.
We describe spectra of Fréchet algebras of entire symmetric functions of bounded type on $$$L_{p_{1}}^{(\mathbb{K})} \times \ldots \times L_{p_{n}}^{(\mathbb{K})}$$$. Also we construct some isomorphisms of these algebras.