SPECTRUM OF ALGEBRA OF BLOCK-SYMMETRIC ANALYTIC FUNCTIONS ON SPACE l1
DOI:
https://doi.org/10.31471/1993-9981-2024-1(52)-98-101Keywords:
block-symmetric polynomials, block-symmetric analytic functions, spectrum, character.Abstract
The work is dedicated to the study of the algebra of block-symmetric polynomials and analytic functions of infinitely many variables on the product of Banach spaces. The aim of this research was to investigate the spectrum (the set of characters) of the algebra of block-symmetric analytic functions of bounded type on a given space. During the study, a connection was established between the spectrum of algebras of analytic functions of bounded type and analytic functions of exponential type. Additionally, an example was provided to demonstrate that the spectrum of the algebra of symmetric analytic functions of bounded type does not coincide with the set of equivalence classes of point evaluation functionals. As a result, it was possible to partially describe the spectrum of the algebra of block-symmetric analytic functions of bounded type on the space. Since there is a one-to-one correspondence between maximal ideals and complex homomorphisms (characters) of a Banach algebra, established via the Gelfand transform, we can interpret the elements of the original algebra as functions on the space of maximal ideals. Thus, the spectrum of a functional topological algebra is a natural domain for its elements. Therefore, describing the spectrum is the first important task that arises when studying a specific commutative Banach algebra. This article is therefore of significant importance in the study of algebras of block-symmetric analytic functions of bounded type on products of Banach spaces.
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