Noncoherent uniform algebras in $\mathbb {C}^n$

Raymond Mortini¹

¹ Université de Lorraine Institut Élie Cartan de Lorraine, UMR 7502 F-57045 Metz, France

Studia Mathematica, Tome 234 (2016) no. 1, pp. 83-95

Résumé

Let $\mathbf D=\overline {\mathbb {D}}$ be the closed unit disk in $\mathbb C$ and $\mathbf {B}_n=\overline {\mathbb {B}}_n$ the closed unit ball in $\mathbb C^n$. For a compact subset $K$ in $\mathbb C^n$ with nonempty interior, let $A(K)$ be the uniform algebra of all complex-valued continuous functions on $K$ that are holomorphic in the interior of $K$. We give short and non-technical proofs of the known facts that $A(\overline {\mathbb D}^n)$ and $A(\mathbf B_n)$ are noncoherent rings. Using, additionally, Earl’s interpolation theorem in the unit disk and the existence of peak functions, we also establish with the same method the new result that $A(K)$ is not coherent. As special cases we obtain Hickel’s theorems on the noncoherence of $A(\overline \varOmega )$, where $\varOmega $ runs through a certain class of pseudoconvex domains in $\mathbb C^n$, results that were obtained with deep and complicated methods. Finally, using a refinement of the interpolation theorem we show that no uniformly closed subalgebra $A$ of $C(K)$ with $P(K)\subseteq A\subseteq C(K)$ is coherent provided the polynomial convex hull of $K$ has no isolated points.

Détail

DOI : 10.4064/sm8517-5-2016

Keywords: mathbf overline mathbb closed unit disk mathbb mathbf overline mathbb closed unit ball mathbb compact subset mathbb nonempty interior uniform algebra complex valued continuous functions holomorphic interior nbsp short non technical proofs known facts overline mathbb mathbf noncoherent rings using additionally earl interpolation theorem unit disk existence peak functions establish method result coherent special cases obtain hickel theorems noncoherence overline varomega where varomega runs through certain class pseudoconvex domains mathbb results obtained deep complicated methods finally using refinement interpolation theorem uniformly closed subalgebra subseteq subseteq coherent provided polynomial convex hull has isolated points