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Notions of boundaries for function spaces

Author: Richard F. Basener
Journal: Proc. Amer. Math. Soc. 130 (2002), 2397-2402
MSC (2000): Primary 32Fxx, 46J10
Published electronically: February 12, 2002
MathSciNet review: 1897465
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Abstract: Sibony and the author independently defined a higher order generalization of the usual Shilov boundary of a function algebra which yielded extensions of results about analytic structure from one dimension to several dimensions. Tonev later obtained an alternative characterization of this generalized Shilov boundary by looking at closed subsets of the spectrum whose image under the spectral mapping contains the topological boundary of the joint spectrum. In this note we define two related notions of what it means to be a higher order/higher dimensional boundary for a space of functions without requiring that the boundary be a closed set. We look at the relationships between these two boundaries, and in the process we obtain an alternative proof of Tonev's result. We look at some examples, and we show how the same concepts apply to convex sets and linear functions.

References [Enhancements On Off] (What's this?)

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Additional Information

Richard F. Basener
Affiliation: 24 Namoth Road, Wappingers Falls, New York 12590

Received by editor(s): March 23, 2001
Published electronically: February 12, 2002
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2002 American Mathematical Society