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Zbl 1197.46028
Garc\'ia Armas, Mario; Sánchez Fernández, Carlos
(Sanches Fernandes, K.)
On the solubility of transcendental equations in commutative C*-algebras. (English)
[J] Banach J. Math. Anal. 4, No. 2, 45-52, electronic only (2010). ISSN 1735-8787/e

Summary: 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 an entire function in the sense of Lorch, i.e., 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.

MSC 2000:: *46J10 Banach algebras of continuous functions
46T25 Holomorphic maps in nonlinear functional analysis
46-01 Textbooks (functional analysis)

Keywords: Banach algebras of continuous functions; transcendental equations; entire functions

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