On algebras of Banach algebra-valued bounded continuous functions.(English)Zbl 1356.46033

Let $$X$$ be a completely regular Hausdorff space and let $$A$$ be a complex commutative Banach algebra with unit $$e$$. Let $$C_b(X,A)$$ denote the space of bounded $$A$$-valued continuous functions on $$X$$, equipped with the uniform norm. Assume that $$A$$ is also completely symmetric and has continuous involution $$*$$. For $$f \in C_b(X,A)$$, the authors show that the statement that the invertibility of the Gelfand transform $$\widetilde{f}$$ implies that of $$f$$ is equivalent to the statement that $$\|\widetilde{f}\|<1$$ implies that $$e-f$$ is invertible.

MSC:

 46E40 Spaces of vector- and operator-valued functions 46J10 Banach algebras of continuous functions, function algebras
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References:

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