# On Shilov boundary and Gelfand spectrum of algebras of generalized analytic functions

## TL;DR

This paper investigates the structure of boundaries and the Gelfand spectrum of algebras of generalized analytic functions on semigroup character spaces, revealing conditions under which these boundaries coincide with the character group.

## Contribution

It characterizes the strong and Shilov boundaries of these algebras as unions of maximal subgroups and computes the Gelfand spectrum when certain ideal conditions are met.

## Key findings

- Boundaries are unions of maximal subgroups of the semicharacter semigroup.
- Boundaries coincide with the character group if no nontrivial simple ideals exist.
- Gelfand spectrum is explicitly calculated in the absence of simple ideals.

## Abstract

Let $S$ be a discrete abelian semigroup with unit and concellations and $\widehat {S}$ the semigroup of semicharacters of $S$. We shaw that the strong boundary and the Shilov boundary of the algebra of generalized analytic functions defined on $\widehat {S}$ are unions of some maximal subgroups of $\widehat {S}$. We shaw also that the both boundaries coincide with the character group of $S$ if $S$ does not contain nontrivial simple ideals. In the last case the Gelfand spectrum of the algebra under consideration is calculated.