Algebras of symmetric analytic functions on Cartesian powers of Lebesgue integrable in a power $p\in [1,+\infty)$ functions

The work is devoted to the study of Fréchet algebras symmetric (invariant under composition every components its argument with any measure preserving bijection domain argument) analytic functions on Cartesian powers complex Banach spaces Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued segment $[0,1]$ and semi-axis. We show that algebra all entire bounded type $n$th space $L_p[0,1]$ isomorphic $\mathbb C^m,$ where $m$ cardinality algebraic basis continuous polynomials this power. analogical result for $L_p[0,+\infty)$ semi-axis $[0,+\infty)$ proved.