Polynomial approximation on three-dimensional real-analytic submanifolds of 𝐂ⁿ

Anderson, John; Izzo, Alexander; Wermer, John

doi:10.1090/S0002-9939-01-05911-1

Polynomial approximation on three-dimensional real-analytic submanifolds of $\mathbf {C}^n$
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by John T. Anderson, Alexander J. Izzo and John Wermer PDF

Proc. Amer. Math. Soc. 129 (2001), 2395-2402 Request permission

Abstract:

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

Herbert Alexander and John Wermer, Several complex variables and Banach algebras, 3rd ed., Graduate Texts in Mathematics, vol. 35, Springer-Verlag, New York, 1998. MR 1482798
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)
Richard F. Basener, On rationally convex hulls, Trans. Amer. Math. Soc. 182 (1973), 353–381. MR 379899, DOI 10.1090/S0002-9947-1973-0379899-1
Andrew Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0246125
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Michael Freeman, Some conditions for uniform approximation on a manifold, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 42–60. MR 0193538
L. Hörmander and J. Wermer, Uniform approximation on compact sets in $C^{n}$, Math. Scand. 23 (1968), 5–21 (1969). MR 254275, DOI 10.7146/math.scand.a-10893
Alexander J. Izzo, Failure of polynomial approximation on polynomially convex subsets of the sphere, Bull. London Math. Soc. 28 (1996), no. 4, 393–397. MR 1384828, DOI 10.1112/blms/28.4.393
Ricardo Nirenberg and R. O. Wells Jr., Approximation theorems on differentiable submanifolds of a complex manifold, Trans. Amer. Math. Soc. 142 (1969), 15–35. MR 245834, DOI 10.1090/S0002-9947-1969-0245834-9
A. G. O’Farrell, K. J. Preskenis, and D. Walsh, Holomorphic approximation in Lipschitz norms, Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983) Contemp. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 187–194. MR 769507, DOI 10.1090/conm/032/769507
Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0423083
J. Wermer, Polynomially convex disks, Math. Ann. 158 (1965), 6–10. MR 174968, DOI 10.1007/BF01370392