Topological isomorphism of a countably generated algebra of entire functions on $\ell_\infty$ and the algebra of symmetric entire functions on $L_{\infty}^2[0,1]$

Abstract

In the paper, we establish some general results on countably generated algebras of entire functions of bounded type on a complex Banach space. Specifically, we study the Fréchet subalgebra $H_{b\mathbf{P}}(X)$ of the algebra of all entire functions of bounded type $H_b(X),$ generated by a countable set $\mathbf{P}$ of continuous algebraically independent complex-valued homogeneous polynomials on a complex Banach space $X$ such that some finite number of elements of the set $\mathbf{P}$ can share the same degree of homogeneity. We investigate the form of elements of this subalgebra. Furthermore, we show that every linear multiplicative functional, acting from $H_{b\mathbf{P}}(X)$ to $\mathbb{C},$ is completely determined by its values on the elements of $\mathbf{P}.$ We also establish an upper estimate for the value of such a functional on an arbitrary $n$-homogeneous polynomial in $H_{b\mathbf{P}}(X).$ We apply these results to some specific algebras.

Let $L_{\infty}[0,1]$ be the complex Banach space of all complex-valued Lebesgue measurable essentially bounded functions on $[0,1].$ Let $L_{\infty}^{2}[0,1]$ be the Cartesian square of $L_{\infty}[0,1].$ We consider the Fréchet algebra $H_{bs}(L_{\infty}^{2}[0,1])$ of all entire symmetric functions of bounded type on $L_{\infty}^{2}[0,1].$ In the paper, we construct a countably generated Fréchet subalgebra of the Fréchet algebra $H_b(\ell_\infty),$ which is topologically isomorphic to $H_{bs}(L_{\infty}^{2}[0,1]),$ where $\ell_{\infty}$ is the complex Banach space of all bounded sequences of complex numbers. Namely, let \[\mathcal{I}=\big(I_{11}, I_{12}, I_{21}, I_{22}, I_{23}, \ldots, I_{n1}, I_{n2}, \ldots, I_{n,n+1}, \ldots\big),\] where
$I_{11}(x) = x_1$, $I_{12}(x) = x_2$, $I_{21}(x) = x_3^2$, $I_{22}(x) = x_4^2$, $I_{23}(x) = x_5^2$, $I_{31}(x) = x_6^3$, $I_{32}(x) = x_7^3$, $I_{33}(x) = x_8^3$, $I_{34}(x) = x_9^3, \ldots $ for $x = (x_1, x_2, \ldots ) \in \ell_{\infty}$. We denote by $H_{b\mathcal{I}}(\ell_{\infty})$ the Fréchet subalgebra of the algebra $H_b(\ell_\infty),$ generated by the sequence of polynomials $\mathcal{I}.$ We construct a topological isomorphism between the algebras $H_{b\mathbb{\mathcal{I}}}(\ell_{\infty})$ and $H_{bs}(L_{\infty}^{2}[0,1]).$

Results of the paper can be used for investigations of the algebras of symmetric analytic functions on Banach spaces.