An optimal linear approximation for a class of nonlinear operators between uniform algebras
Let \(X,Y\) be compact Hausdorff spaces and let \(A,B\) be uniform algebras on \(X,Y\), respectively, and let \(\partial A, \partial B\) be the respective Šilov boundaries. This interesting paper deals with the structure of maps \(T: A \rightarrow B\) such that \(T(1) = 1\), \(T\) maps the set of invertible elements of \(A\) onto the set of invertible elements in \(B\) and satisfies certain norm inequalities. Results of this nature for spectrally multiplicative maps when the uniform algebra is considerd on its maximal ideal spaces were obtained by \textit{L. Molnár} [Proc. Am. Math. Soc. 130, No. 1, 111--120 (2002; Zbl 0983.47024)], \textit{N. V. Rao} and \textit{A. K. Roy} [Proc. Edinb. Math. Soc., II. Ser. 48, No. 1, 219--229 (2005; Zbl 1074.46033)]. The authors show that the conditions on \(T\) imply that the Šilov boundaries are homeomorphic by a homeomorphism \(\tau\) and obtain pointwise inequalities on the Šilov boundary comparing \(T(f)\) and \(f \circ \tau\). An interesting step in the proof is to get pointwise estimates for maps defined only on the set of invertible elements that preserve constants (Theorem 2.4).
scientific article; zbMATH DE number 6918459
scientific article