Topology on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_∞$ - Vasylyshyn, T.V. (Carpathian Mathematical Publications, 2017)
It is known that the so-called elementary symmetric polynomials $R_n(x) = \int_{[0,1]}(x(t))^n\,dt$ form an algebraic basis in the algebra of all symmetric continuous polynomials on the complex Banach space $L_∞,$ which is dense in the Fréchet algebra $H_{bs}(L_∞)$ of all entire symmetric functions of bounded type on $L_∞.$ Consequently, every continuous homomorphism $φ: H_{bs}(L_∞) → \mathbb{C}$ is uniquely determined by the sequence $\{φ(R_n)\}_{n=1}^∞.$ By the continuity of the homomorphism $φ,$ the sequence $\{\sqrt[n]{|φ(R_n)|}\}_{n=1}^∞$ is bounded. On the other hand, for every sequence $\{ξ_n\}_{n=1}^∞ \subset \mathbb{C},$ such that the sequence $\{\sqrt[n]{|ξ_n|}\}_{n=1}^∞$ is bounded, there exists $x_ξ \in L_∞$ such that $R_n(x_ξ) = ξ_n$ for every $n \in \mathbb{N}.$ Therefore, for the point-evaluation functional $δ_{x_ξ}$ we have $δ_{x_ξ}(R_n) = ξ_n$ for every $n \in \mathbb{N}.$ Thus, every continuous complex-valued homomorphism of $H_{bs}(L_∞)$ is a point-evaluation functional at some point of $L_∞.$ Note that such a point is not unique. We can consider an equivalence relation on $L_∞,$ defined by $x\sim y \Leftrightarrow δ_x = δ_y.$ The spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_∞)$ is one-to-one with the quotient set $L_∞/_\sim.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, it is naturally to identify $M_{bs}$ with the set of all sequences $\{ξ_n\}_{n=1}^∞ \subset \mathbb{C}$ such that the sequence $\{\sqrt[n]{|ξ_n|}\}_{n=1}^∞$ is bounded. We show that the quotient topology is Hausdorffand that $M_{bs}$ with the operation of coordinate-wise addition of sequences forms an abelian topological group.