Regularity points and Jensen measures for \(R(X)\) (Q2833666)

The authors continue research from \textit{J. F. Feinstein} and \textit{D. W. B. Somerset} [Stud. Math. 141, No. 1, 53--68 (2000; Zbl 0976.46036)], respectively \textit{J. Feinstein} and \textit{R. Mortini} [Math. Z. 271, No. 1--2, 139--155 (2012; Zbl 1256.46032)], on certain types of regularity points in natural Banach function algebras \(A\) on compact Hausdorff spaces \(X\). Recall that \(x\in X\) is a regularity point of type I (or a point of continuity) if the identity map from \(X=M(A)\) with the hull-kernel topology to \(X\) with the original topology is continuous at \(\{x\}\). Moreover, \(x\in X\) is a regularity point of type II (or an \(R\)-point) if the hull (zero-set) of the ideal \(J_x\) of all functions in \(A\) vanishing in a neighborhood of \(x\) coincides with \(\{x\}\). In this nice work, planar compacta \(X\) are unveiled displaying the connections/differences between these two types of regularity points. The two types of regularity points can be different (in both ways) even for natural uniform algebras such as \(R(X)\) for compact plane sets, as is illustrated in the paper. A Swiss cheese \(X\) is constructed containing the compact set of segments NEWLINE\[NEWLINEK=\{z\in \mathbb C: \operatorname{Re}z\in F,\;\operatorname{Im} z\in [-1/2,1/2]\},NEWLINE\]NEWLINE where \(F\) is a Cantor subset of \([-1/2,1/2]\) with positive linear measure, such that every point of \(X\setminus K\) is a regularity point of type I and II for \(R(X)\), that every point in \(K\) is neither a point of regularity type I nor II, and that \(R(X)\) has no nontrivial Jensen measures.