A normal uniform algebra that fails to be strongly regular at a peak point (Q6629510)

The author shows that there exists a normal uniform algebra \(A\), on a compact metrizable space, that fails to be strongly regular at a peak point (Theorem 1.2). The algebra \(A\) is \(R(K)\) for a certain compact set \(K\) in the complex plane.\N\NRecall that for a compact planar set \(K\) the symbol \(R(K)\) denotes the uniform closure on \(K\) of the holomorphic rational functions with poles off \(K\).\N\NA uniform algebra \(A\) on \(X\) (that is, a closed subalgebra of \(C(X)\)) is said to be:\N\begin{itemize}\N\item \textit{regular on \(X\)} if for each closed set \(K_{0}\) of \(X\) and each point \(x\in X\setminus K_{0}\), there exists a function \(f\) in \(A\) such that \(f(x_{0})=1\) and \(f=0\) on \(K_{0}\),\N\item \textit{normal on \(X\)} if for each pair of disjoint closed sets \(K_{0}\) and \(K_{1}\) of \(X\), there exists a function \(f\) in \(A\) such that \(f=1\) on \(K_{1}\) and \(f=0\) on \(K_{0}\).\N\end{itemize}\NLet\N\begin{align*}\NM_{x}&=\{f\in A\colon f(x)=0\},\\\NJ_{x}&=\{f\in A\colon f^{-1}(0) \text{\:contains a neighborhood of \(x\) in \(X\)}\}.\N\end{align*}\NThe uniform algebra \(A\) is \textit{strongly regular at \(x\)} if \(\bar{J}_{x}=M_{x}\), and \(A\) is \textit{strongly regular} if \(A\) is strongly regular at every point of \(X\). It is known that every strongly regular uniform algebra is normal.\N\NThe main result of the paper is Theorem 1.4, which says: For each \(r>0\), there exists a sequence of open discs \(\{D_{k}\}_{k=1}^{\infty}\) such that the sum of the radii of the discs \(D_{k}\) is \(<r\), the point \(1\) is in \(K=\bar{\mathbb{D}}\setminus \bigcup_{k=1}^{\infty}D_{k}\), and the following conditions hold:\N\begin{itemize}\N\item[(i)] \(R(K)\) is normal.\N\item[(ii)] \(R(K)\) is strongly regular at every point of \(K\setminus \{1\}\).\N\item[(iii)] \(R(K)\) is not strongly regular at the point \(1\).\N\end{itemize}\NA modification of the proof of Theorem 1.4 (which is Theorem 1.5) shows that a normal uniform algebra can fail to be strongly regular at an uncountable set of peak points. The uniform algebras in Theorem 1.4 and Theorem 1.5 are strongly regular at every nonpeak point. These theorems also show that \(R(K)\) has no non-zero bounded point derivations.