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 $\widetilde {D}$, then the Shilov boundary of $D$ with respect to the algebra $\mathcal {A}(D)$, call it $\partial _SD$, coincides with the corresponding one for $\widetilde {D}$, called $\partial _S\widetilde {D}$.