Presence or absence of analytic structure in maximal ideal spaces (Q330860)

This paper presents several interesting extensions of Wermer's maximality theorem to several complex variables. Here are just two sample results: 1) If \(\Omega\subset \mathbb C^n\) is a bounded open set with \(C^1\)-smooth boundary and polynomially convex closure, then \([z,h_1,\dots, h_n]_{b\Omega} =C(b\Omega,\mathbb C)\) implies that \([z,h_1,\dots,h_n]_{\overline\Omega}=C(\overline\Omega,\mathbb C)\) whenever the functions \(h_j\in C(\overline\Omega,\mathbb C)\) are pluriharmonic in \(\Omega\). 2) There exists a real-valued smooth function \(f\) on \(\mathbb T^2\subset \mathbb C^2\) such that the polynomially convex hull of the graph \(\{(z, f(z)): z\in \mathbb T^2\}\subset \mathbb C^3\) of \(f\) is non-trivial but contains no analytic disk.