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On the solubility of transcendental equations in commutative C*-algebras. (English) Zbl 1197.46028
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.
46J10Banach algebras of continuous functions, function algebras
46T25Holomorphic maps in nonlinear functional analysis
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