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

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∈ [1;+∞), let Lₚ(K) be the Banach space over a field K∈ ,C\ of all K-valued functions on $[0;1]$, the p-th 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(K) × … × L_pₙ(K), where p₁,…,pₙ ∈ [1;+∞), is called symmetric if f((x₁∘σ;…;xₙ∘σ))=f((x₁;…;xₙ)) for every σ∈ Ξ[0;1] and (x₁;…;xₙ)∈ L_p₁(K) × … × L_pₙ(K). We describe spectra of Fréchet algebras of entire symmetric functions of bounded type on L_p₁(K) × … × L_pₙ(K). Also we construct some isomorphisms of these algebras.