On rationally convex hulls

Abstract:

For a compact set $X \subseteq {{\mathbf {C}}^n}$, let ${h_r}(X)$ denote the rationally convex hull of X; let ${\mathbf {\Delta }}$ denote the closed unit disk in C; and, following Wermer, for a compact set S such that $\partial {\mathbf {\Delta }} \subseteq S \subseteq {\mathbf {\Delta }}$ let ${X_S} = S \times S \cap \partial {{\mathbf {\Delta }}^2}$. It is shown that \[ {h_r}({X_S}) = \{ (z,w) \in S \times S|{u_S}{(z)^ + }{u_S}(w) \leq 1\} \] where ${u_S}$ is a function on S which, in the case when S is smoothly bounded, is specified by requiring ${u_S}{|_{\partial {\mathbf {\Delta }}}} = 0,{u_S}{|_{\partial S\backslash \partial {\mathbf {\Delta }}}} = 1$ and ${u_S}{|_{\operatorname {int} S}}$ harmonic. In particular this provides a precise description of ${h_r}(X)$ for certain sets $X \subseteq {{\mathbf {C}}^2}$ with the property that ${h_r}(X) \ne X$, but ${h_r}(X)$ does not contain analytic structure (as Wermer demonstrated, there are S for which $X = {X_S}$ has these properties). Furthermore, it follows that whenever ${h_r}({X_S}) \ne {X_S}$ then there is a Gleason part of ${h_r}({X_S})$ for the algebra $R({X_S})$ with positive four-dimensional measure. In fact, the Gleason part of any point $(z,w) \in {h_r}({X_S}) \cap \operatorname {int} {{\mathbf {\Delta }}^2}$ such that ${u_S}(z) + {u_S}(w) < 1$ has positive four-dimensional measure. A similar idea is then used to construct a compact rationally convex set $Y \subseteq {{\mathbf {C}}^2}$ such that each point of Y is a peak point for $R(Y)$ even though $R(Y) \ne C(Y)$; namely, $Y = {\tilde X_T} = \{ (z,w) \in {{\mathbf {C}}^2}|z \in T,|w| = \sqrt {1 - |z{|^2}} \}$ where T is any compact subset of $\operatorname {int} {\mathbf {\Delta }}$ having the property that $R(T) \ne C(T)$ even though there are no nontrivial Jensen measures for $R(T)$. This example is more concrete than the original example of such a uniform algebra which was discovered by Cole. It is possible to show, for instance, that $R({\tilde X_T})$ is not even in general locally dense in $C({\tilde X_T})$, a possibility which had been suggested by Stuart Sidney. Finally, smooth examples (3-spheres in ${{\mathbf {C}}^6}$) with the same pathological properties are obtained from ${X_S}$ and ${\tilde X_T}$.