On Bremermann's conjecture for the Shilov boundary of pseudoconvex Riemann domains (Q1065973)

Let \(\Omega\) be a relatively compact subdomain of a Riemann domain spread over \({\mathbb{C}}^ n\). Let A be a uniform subalgebra of A(\({\bar \Omega}\))\(=C({\bar \Omega})\cap {\mathcal O}(\Omega)\), where \({\mathcal O}(X)\) denotes the algebra of all functions holomorphic in some neighborhood of X. In this setting the author proves two results: (1) For \(\Omega\) with (\({\bar \Omega}\))\({}^ 0=\Omega\) and for subalgebras A containing the component functions of the spread map, if \({\bar \Omega}\) is A-convex, then \(\Omega\) is a Stein domain spread over \({\mathbb{C}}^ n.\) (2) For \(\Omega\) with \(C^{(2)}\)-boundaries, if \({\bar \Omega}\) is A- convex and \(A\supset {\mathcal O}({\bar \Omega})\), then for any uniform subalgebra A' such that \(A\subset A'\subset A({\bar \Omega})\), the Shilov boundary of A' is the closure of the set of strictly pseudoconvex boundary points of \(\Omega\). These partially generalize results of the reviewer [''Algebras of holomorphic functions'' (1972; Zbl 0253.46115) and of \textit{M. Hakim} and \textit{N. Sibony} [C. R. Acad. Sci., Paris Sér. A 281, 959-962 (1975; Zbl 0324.46058)].