On algebras of Banach algebra-valued bounded continuous functions (Q309455)

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.