On the solubility of transcendental equations in commutative C*-algebras

TL;DR

This paper investigates conditions under which transcendental equations in commutative C*-algebras, specifically in spaces of continuous functions, have solutions, extending known algebraic closure results to more general settings.

Contribution

It generalizes the algebraic closure of $C(X)$ to transcendental equations with power series coefficients, under certain restrictions, in commutative C*-algebras.

Findings

Solutions exist for certain transcendental equations in $C(X)$

Extension of algebraic closure results to transcendental cases

Conditions on power series coefficients ensure solvability

Abstract

It is known that $C(X)$ is algebraically closed if $X$ is a locally connected, hereditarily unicoherent compact Hausdorff space. For such spaces, we prove that if $F:C(X) \to C(X)$ is given by an everywhere convergent power series with coefficients in $C(X)$ and satisfies certain restrictions, then it has a root in $C(X)$. Our results generalizes the monic algebraic case.