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 \R, C\ of all K-valued functions on [0;1, the p-th 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^ (K) L₏_₍^ (K), where p₁, , pₙ 1;+), is called symmetric if f ( (x₁;;xₙ) ) =f ( (x₁;;xₙ) ) for every [₀;₁ and (x₁;;xₙ) L₏_₁^ (K) L₏_₍^ (K). We describe spectra of Fréchet algebras of entire symmetric functions of bounded type on L₏_₁^ (K) L₏_₍^ (K). Also we construct some isomorphisms of these algebras.