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

## Keywords:

symmetric function, block-symmetric function, analytic function on Banach space, space of Lebesgue measurable functions, spectrum of algebra, algebraic basis### Abstract

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 $p$th power of the absolute value is Lebesgue integrable, where $p\in[1,+\infty),$ 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.

*Carpathian Math. Publ.*

**2024**,

*16*, 174-189.