On rationally convex hulls

Author:
Richard F. Basener

Journal:
Trans. Amer. Math. Soc. **182** (1973), 353-381

MSC:
Primary 32E20; Secondary 32E30, 46J10

DOI:
https://doi.org/10.1090/S0002-9947-1973-0379899-1

MathSciNet review:
0379899

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Abstract: For a compact set , let denote the rationally convex hull of *X*; let denote the closed unit disk in **C**; and, following Wermer, for a compact set *S* such that let . It is shown that

*S*which, in the case when

*S*is smoothly bounded, is specified by requiring and harmonic. In particular this provides a precise description of for certain sets with the property that , but does not contain analytic structure (as Wermer demonstrated, there are

*S*for which has these properties). Furthermore, it follows that whenever then there is a Gleason part of for the algebra with positive four-dimensional measure. In fact, the Gleason part of any point such that has positive four-dimensional measure.

A similar idea is then used to construct a compact rationally convex set such that each point of *Y* is a peak point for even though ; namely, where *T* is any compact subset of having the property that even though there are no nontrivial Jensen measures for . 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 is not even in general locally dense in , a possibility which had been suggested by Stuart Sidney.

Finally, smooth examples (3-spheres in ) with the same pathological properties are obtained from and .

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0379899-1

Keywords:
Rationally convex hull,
analytic structure,
peak point

Article copyright:
© Copyright 1973
American Mathematical Society