A counterexample to a theorem of Bremermann on Shilov boundaries

Abstract

We give a counterexample to the following theorem of Bremermann on Shilov boundaries: if $D$ is a bounded domain in ${\mathbb{C}}^n$ having a univalent envelope of holomorphy, say $\tilde{D}$, then the Shilov boundary of $D$ with respect to the algebra $\mathcal{A}(D)$ coincides with the corresponding one for $\tilde{D}$.