Noncoherent uniform algebras in $\mathbb C^n$

TL;DR

This paper provides simple proofs that certain uniform algebras of holomorphic functions in several complex variables are noncoherent, extending known results and establishing new cases using classical theorems.

Contribution

It offers concise, non-technical proofs of noncoherence for algebras like $A(ar{ D}^n)$ and $A( B_n)$, and introduces new noncoherence results for $A(K)$ on various compact sets.

Findings

Proved noncoherence of $A(ar{ D}^n)$ and $A( B_n)$

Established noncoherence of $A(K)$ for certain compact sets

Showed no uniformly closed subalgebra between polynomials and continuous functions is coherent under specific conditions

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.