Zerofree extension of continuous functions on compact Hausdorff space
Let \(X\) be a compact Hausdorff space and \(C_R(X)\) denote the algebra of all real-valued functions on \(X\) with the sup norm. The author's main result is that, for any \(g\in C_R(X)\) with zero set \(Z\) and any \(n\)-tuple \((f_1, \dots, f_n)\in C^n_R(X)\) without common zeros in \(Z\), the following are equivalent: (i) the restriction \((f_1,\dots,f_n)|_Z\) has an extension \((F_1, \dots, F_n)\in C^n_R(X)\) without common zeros in \(X\); (ii) there exists \((h_1, \dots, h_n)\in C^n_R(X)\) such that \((f_1+h_1g, \dots, f_n+h_ng)\in C^n_R(X)\) has no common zeros in \(X\). He then proves an inclusion relation for the homotopy class of \((f_1, \dots, f_n)|_Z\) and shows how his methods lead to a general result for commutative Banach algebras with unit. The latter results were proved earlier by \textit{G. Corach} and \textit{F. D. Suárez} [ibid. 21, 1-8 (1985; Zbl 0606.46033)] in the complex case using different methods.
scientific article; zbMATH DE number 1307863