Noncoherent uniform algebras in $\mathbb C^n$

Abstract

Let $\mathbf D=\bar{\mathbb D}$ be the closed unit disk in $\mathbb C$ and $\mathbf B_n=\bar{\mathbb B_n}$ 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 $A(\bar{\mathbb D}^n)$ 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 $A(\bar\Omega)$, where $\Omega$ 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.