# Metric on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$

• T.V. Vasylyshyn Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://doi.org/10.15330/cmp.9.2.198-201
Keywords: symmetric function, spectrum of the algebra
Published online: 2018-01-02

### Abstract

It is known that every complex-valued homomorphism of the Fréchet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on the complex Banach space $L_\infty$ is a point-evaluation functional $\delta_x$ (defined by $\delta_x(f) = f(x)$ for $f \in H_{bs}(L_\infty)$) at some point $x \in L_\infty.$ Therefore, the spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim,$ where an equivalence relation "$\sim$'' on $L_\infty$ is defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, $M_{bs}$ has a natural representation as a set of sequences which endowed with the coordinate-wise addition and the quotient topology forms an Abelian topological group. We show that the topology on $M_{bs}$ is metrizable and it is induced by the metric $d(\xi, \eta) = \sup_{n\in\mathbb{N}}\sqrt[n]{|\xi_n-\eta_n|},$ where $\xi = \{\xi_n\}_{n=1}^\infty,\eta = \{\eta_n\}_{n=1}^\infty \in M_{bs}.$

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Vasylyshyn T. Metric on the Spectrum of the Algebra of Entire Symmetric Functions of Bounded Type on the Complex $L_\infty$. Carpathian Math. Publ. 2018, 9 (2), 198-201.