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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On rationally convex hulls

Author: Richard F. Basener
Journal: Trans. Amer. Math. Soc. 182 (1973), 353-381
MSC: Primary 32E20; Secondary 32E30, 46J10
MathSciNet review: 0379899
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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

$\displaystyle {h_r}({X_S}) = \{ (z,w) \in S \times S\vert{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}{\vert _{\partial {\mathbf{\Delta }}}} = 0,{u_S}{\vert _{\partial S\backslash \partial {\mathbf{\Delta }}}} = 1$ and $ {u_S}{\vert _{\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}\vert z \in T,\vert w\vert = \sqrt {1 - \vert z{\vert^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}$.

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Keywords: Rationally convex hull, analytic structure, peak point
Article copyright: © Copyright 1973 American Mathematical Society