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Let $\Omega\subset\mathbb{C}^n$ be a bounded domain. Consider the Banach algebra $A(\Omega):=\mathcal{C}(\overline{\Omega})\cap\mathcal{O}(\Omega).$ Let $\partial_S\Omega$ denote the Bergmann Shilov boundary of $\overline{\Omega}$ with respect to $A(\Omega)$. Now from the very definition of "Strong boundary points" and "Peak points" we know that these are always contained in $\partial_S\Omega$. I have two questions:

  1. Does there exist a large class of domains in $\mathbb{C}^n$ such that $\partial_S\Omega$ contains points different from strong boundary points and peak points.

  2. Is there a classification of domains where $\partial_S\Omega$ coincides with set of peak points or strong boundary points.

P.S.:References related to the above will also be helpful.

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