Supersymmetric Polynomials on the Space of Absolutely Convergent Series

Abstract

We consider an algebra $H_b^{sup}$ of analytic functions on the Banach space of two-sided absolutely summing sequences which is generated by so-called supersymmetric polynomials. Our purpose is to investigate $H_b^{sup}$ and its spectrum with using methods of infinite dimensional complex analysis and the theory of Fr{\'e}chet algebras. Some algebraic bases of $H_b^{sup}$ are described. Also, we show that the spectrum of the algebra of supersymmetric analytic functions of bounded type contains a metric ring $\mathcal{M}.$ We prove that $\mathcal{M}$ is a complete metric (nonlinear) space and investigate homomorphisms and additive operators on this ring. Some possible applications are discussed.