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

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 Lₚ, where p∈ [1;+∞), be the complex Banach space of all complex-valued functions on $[0;1]$, the pth powers of absolute values of which are Lebesgue integrable. Let Ξ[0;1] be the set of all bijections σ:[0;1] → [0;1] such that both σ and σ⁻¹ are measurable and preserve Lebesgue measure, i.e. μ(σ(E)) = μ(σ⁻¹(E)) = μ(E) for every Lebesgue measurable set E⊂ [0;1], where μ is Lebesgue measure. A function f on the Cartesian product L_p₁ × L_ p₂ × … × L_pₙ, where p₁,p₂, …, pₙ ∈ [1;+∞), is called symmetric if f((x₁∘σ;x₂∘σ;…;xₙ∘σ))=f((x₁;x₂;…;xₙ)) for every σ∈ Ξ[0;1] and (x₁;x₂;…;xₙ)∈ L_p₁ × L_ p₂ × … × L_pₙ. We construct an algebraic basis of the algebra of all complex-valued continuous symmetric polynomials on L_p₁ × L_ p₂ × … × L_pₙ. Also we construct some isomorphisms of Fréchet algebras of complex-valued entire symmetric functions of bounded type on L_p₁ × L_ p₂ × … × L_pₙ.