Algebras of symmetric and block-symmetric functions on spaces of Lebesgue measurable functions

In this work, we investigate algebras of symmetric and block-symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions for which the pth power of the absolute value is Lebesgue integrable, where p∈[1,+∞), and Lebesgue measurable essentially bounded functions on $[0,1]$. We show that spectra of Fréchet algebras of block-symmetric entire functions of bounded type on these spaces consist only of point-evaluation functionals. Also we construct algebraic bases of algebras of continuous block-symmetric polynomials on these spaces. We generalize the above-mentioned results to a wide class of algebras of symmetric entire functions.