Isomorphisms of algebras of symmetric functions on spaces $\ell_p$

Abstract

The work is devoted to the study of algebras of entire symmetric functions on some Banach spaces of sequences. A function on a vector space is called symmetric with respect to some fixed group $G$ of operators acting on this space, or $G$-symmetric, if it is invariant under the action of elements of the group $G$ on its argument. For different vector spaces there exist some natural groups of symmetries. In the case of vector spaces of sequences the most natural are groups of operators permuting coordinates of sequences. Such groups of operators are generated by some groups of bijections on the set $\mathbb{N}$ of positive integers. The most commonly used for this purpose is the group $\mathcal{S}$ of all bijections on $\mathbb{N}.$ We consider entire functions and polynomials that are symmetric with respect to the group of operators, generated by $\mathcal{S},$ on the complex Banach space $\ell_p(\mathbb{C}^n)$ of all absolutely summable in a power $p\in [1,+\infty)$ sequences of $n$-dimensional complex vectors. We construct some natural isomorphism between the space $\ell_p(\mathbb{C}^n)$ and its partial case -- the classical Banach space $\ell_p.$ Also we construct the group of operators on $\ell_p$ that is consistent with the isomorphism and the above-mentioned group of operators on $\ell_p(\mathbb{C}^n).$ This group is generated by the subgroup of $\mathcal{S}$, elements of which permute elements of $\mathbb{N}$ ``by blocks''. We obtain the isomorphism between Frechet algebras of complex-valued entire functions of bounded type on $\ell_p$ and $\ell_p(\mathbb{C}^n)$ that are symmetric with respect to the above-mentioned respective groups of operators. The respective subalgebras of continuous symmetric polynomials on these spaces are also isomorphic.