Polynomial approximation on three-dimensional real-analytic submanifolds of

Authors:
John T. Anderson, Alexander J. Izzo and John Wermer

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2395-2402

MSC (2000):
Primary 32E30; Secondary 46J10

DOI:
https://doi.org/10.1090/S0002-9939-01-05911-1

Published electronically:
January 18, 2001

MathSciNet review:
1823924

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Abstract | References | Similar Articles | Additional Information

It was once conjectured that if is a uniform algebra on its maximal ideal space and if each point of is a peak point for , then . This peak point conjecture was disproved by Brian Cole in 1968. However, it was recently shown by Anderson and Izzo that the peak point conjecture does hold for uniform algebras generated by smooth functions on smooth two-manifolds with boundary. Although the corresponding assertion for smooth three-manifolds is false, we establish a peak point theorem for real-analytic three-manifolds with boundary.

**1.**H. Alexander and J. Wermer,*Several Complex Variables and Banach Algebras*, Third edition, Springer, 1998. MR**98g:32002****2.**J. T. Anderson and A. J. Izzo,*A Peak Point Theorem for Uniform Algebras Generated by Smooth Functions On a Two-Manifold*, Bull. London Math. Soc. (to appear)**3.**R. F. Basener,*On Rationally Convex Hulls*, Trans. Amer. Math. Soc.**182**(1973), pp. 353-381. MR**52:803****4.**A. Browder,*Introduction to Function Algebras*, Benjamin, New York, 1969. MR**39:7431****5.**H. Federer,*Geometric Measure Theory*, Springer, 1969. MR**41:1976****6.**M. Freeman,*Some Conditions for Uniform Approximation on a Manifold*, in: Function Algebras, F. Birtel (ed.), Scott, Foresman and Co., Chicago, 1966, pp. 42-60. MR**33:1758****7.**L. Hörmander and J. Wermer,*Uniform Approximation on Compact Subsets in*, Math. Scand.**23**(1968), pp. 5-21. MR**40:7484****8.**A. J. Izzo,*Failure of Polynomial Approximation on Polynomially Convex Subsets of the Sphere*, Bull. London Math. Soc.**28**(1996), pp. 393-397. MR**98d:32017****9.**R. Nirenberg and R. O. Wells,*Approximation Theorems on Differentiable Submanifolds of a Complex Manifold*, Trans. Amer. Math. Soc.**142**(1969), pp. 15-35. MR**39:7140****10.**A. J. O'Farrell, K. J. Preskenis, and D. Walsh,*Holomorphic Approximation in Lipschitz Norms*, in Proceedings of the Conference on Banach Algebras and Several Complex Variables, Contemporary Math. v. 32, American Mathematical Society, 1983. MR**86c:32015****11.**E.L. Stout,*The Theory of Uniform Algebras*, Bogden and Quigley, 1971. MR**54:11066****12.**J. Wermer,*Polynomially Convex Disks*, Math. Ann.**158**(1965), pp. 6-10. MR**30:5158**

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Additional Information

**John T. Anderson**

Affiliation:
Department of Mathematics, College of the Holy Cross, Worcester, Massachusetts 01610

Email:
anderson@math.holycross.edu

**Alexander J. Izzo**

Affiliation:
Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403

Address at time of publication:
Department of Mathematics, Texas A & M University, College Station, Texas 77843

Email:
aizzo@math.bgsu.edu, aizzo@math.tamu.edu

**John Wermer**

Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island 02912

Email:
wermer@math.brown.edu

DOI:
https://doi.org/10.1090/S0002-9939-01-05911-1

Received by editor(s):
December 28, 1999

Published electronically:
January 18, 2001

Communicated by:
Steven R. Bell

Article copyright:
© Copyright 2001
American Mathematical Society