An analytic algebra without analytic structure in the spectrum
This paper deals with the following example due to J. Wermer: There is a compact subset Y of \({\mathbb{C}}^ 2\) such that \(X\setminus Y\) contains no analytic variety of positive dimension, where X stands for the polynomially convex hull of Y [\textit{J. Wermer}, Ark. Mat. 20, 129-135 (1982; Zbl 0491.32013)]. Here the author proves that, nonetheless, the identity principle holds for the algebra \({\mathcal P}(X)\) of continuous complex-valued functions on X which are uniform limits of polynomials. The proof is quite delicate and uses techniques of Banach algebras and complex analysis. The author also includes an example showing that for an arbitrary Y the conclusion above may fail.
scientific article; zbMATH DE number 4111044