Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Polynomial approximation on three-dimensional real-analytic submanifolds of $\mathbf{C}^n$

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
Published electronically: January 18, 2001
MathSciNet review: 1823924
Full-text PDF

Abstract | References | Similar Articles | Additional Information


It was once conjectured that if $A$ is a uniform algebra on its maximal ideal space $X$ and if each point of $X$ is a peak point for $A$, then $A = C(X)$. 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.

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

  • 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 $\mathbf{ C}^n$, 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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32E30, 46J10

Retrieve articles in all journals with MSC (2000): 32E30, 46J10

Additional Information

John T. Anderson
Affiliation: Department of Mathematics, College of the Holy Cross, Worcester, Massachusetts 01610

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

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

Received by editor(s): December 28, 1999
Published electronically: January 18, 2001
Communicated by: Steven R. Bell
Article copyright: © Copyright 2001 American Mathematical Society