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 ₀;₁ be the set of all bijections: 0;1 0;1 such that both and ^-1 are measurable and preserve Lebesgue measure, i. e. ( (E) ) = (^-1 (E) ) = (E) for every Lebesgue measurable set E 0;1, where is Lebesgue measure. A function f on the Cartesian product L₏_₁ L ₏_₂ L₏_₍, where p₁, p₂, , pₙ 1;+), is called symmetric if f ( (x₁;x₂;;xₙ) ) =f ( (x₁;x₂;;xₙ) ) for every [₀;₁ and (x₁;x₂;;xₙ) L₏_₁ L ₏_₂ L₏_₍. We construct an algebraic basis of the algebra of all complex-valued continuous symmetric polynomials on L₏_₁ L ₏_₂ L₏_₍. Also we construct some isomorphisms of Fréchet algebras of complex-valued entire symmetric functions of bounded type on L₏_₁ L ₏_₂ L₏_₍.