Localization for uniform algebras generated by real-analytic functions
Authors:
John T. Anderson and Alexander J. Izzo
Journal:
Proc. Amer. Math. Soc. 145 (2017), 4919-4930
MSC (2010):
Primary 46J10, 46J15; Secondary 32A38, 32A65
DOI:
https://doi.org/10.1090/proc/13640
Published electronically:
June 22, 2017
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: It is shown that if 
is a uniform algebra generated by real-analytic functions on a suitable compact subset 
of a real-analytic variety such that the maximal ideal space of 
is 
and every continuous function on 
is locally a uniform limit of functions in 
, then 
. This gives an affirmative answer to a special case of a question from the Proceedings of the Symposium on Function Algebras held at Tulane University in 1965.
- [1] H. Alexander, Polynomial approximation and analytic structure, Duke Math. J. 38 (1971), 123–135. MR 0283244
- [2] John T. Anderson and Alexander J. Izzo, A peak point theorem for uniform algebras generated by smooth functions on two-manifolds, Bull. London Math. Soc. 33 (2001), no. 2, 187–195. MR 1815422, https://doi.org/10.1112/blms/33.2.187
- [3] John T. Anderson, Alexander J. Izzo, and John Wermer, Polynomial approximation on three-dimensional real-analytic submanifolds of 𝐶ⁿ, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2395–2402. MR 1823924, https://doi.org/10.1090/S0002-9939-01-05911-1
- [4] John T. Anderson, Alexander J. Izzo, and John Wermer, Polynomial approximation on real-analytic varieties in 𝐂ⁿ, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1495–1500. MR 2053357, https://doi.org/10.1090/S0002-9939-03-07263-0
- [5] John T. Anderson and Alexander J. Izzo, Peak point theorems for uniform algebras on smooth manifolds, Math. Z. 261 (2009), no. 1, 65–71. MR 2452637, https://doi.org/10.1007/s00209-008-0313-x
- [6] John T. Anderson and Alexander J. Izzo, A peak point theorem for uniform algebras on real-analytic varieties, Math. Ann. 364 (2016), no. 1-2, 657–665. MR 3451401, https://doi.org/10.1007/s00208-015-1224-x
- [7] Function algebras, Proceedings of an International Symposium on Function Algebras held at Tulane University, vol. 1965, Scott, Foresman and Co., Chicago, Ill., 1966. MR 0193471
- [8] Brian James Cole, ONE-POINT PARTS AND THE PEAK POINT CONJECTURE, ProQuest LLC, Ann Arbor, MI, 1968. Thesis (Ph.D.)–Yale University. MR 2617861
- [9] 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
- [10] Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR 0410387
- [11] Alexander J. Izzo, Localization for uniform algebras generated by smooth functions on two-manifolds, Bull. Lond. Math. Soc. 42 (2010), no. 4, 652–660. MR 2669686, https://doi.org/10.1112/blms/bdq024
- [12] Alexander J. Izzo, Uniform approximation on manifolds, Ann. of Math. (2) 174 (2011), no. 1, 55–73. MR 2811594, https://doi.org/10.4007/annals.2011.174.1.2
- [13] Alexander J. Izzo, The peak point conjecture and uniform algebras invariant under group actions, Function spaces in modern analysis, Contemp. Math., vol. 547, Amer. Math. Soc., Providence, RI, 2011, pp. 135–146. MR 2856487, https://doi.org/10.1090/conm/547/10814
- [14] Alexander J. Izzo, Nonlocal uniform algebras on three-manifolds, Pacific J. Math. 259 (2012), no. 1, 109–116. MR 2988485, https://doi.org/10.2140/pjm.2012.259.109
- [15] Eva Kallin, A nonlocal function algebra, Proc. Nat. Acad. Sci. U.S.A. 49 (1963), 821–824. MR 0152907
- [16] Raghavan Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR 0217337
- [17] John Rainwater, A remark on regular Banach algebras, Proc. Amer. MAth. Soc. 18 (1967), 255–256. MR 0208413, https://doi.org/10.1090/S0002-9939-1967-0208413-9
- [18] Edgar Lee Stout, Holomorphic approximation on compact, holomorphically convex, real-analytic varieties, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2302–2308. MR 2213703, https://doi.org/10.1090/S0002-9939-06-08250-5
- [19] J. Wermer, Approximation on a disk, Math. Ann. 155 (1964), 331–333. MR 0165386, https://doi.org/10.1007/BF01354865
- [20] J. Wermer, Polynomially convex disks, Math. Ann. 158 (1965), 6–10. MR 0174968, https://doi.org/10.1007/BF01370392
- [21] Donald R. Wilken, Approximate normality and function algebras on the interval and the circle, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 98–111. MR 0196525
, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2395-2402. 
, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1495-1500. Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46J10, 46J15, 32A38, 32A65
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Additional Information
John T. Anderson
Affiliation:
Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, Massachusetts 01610
Email:
janderso@holycross.edu
Alexander J. Izzo
Affiliation:
Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403
Email:
aizzo@bgsu.edu
DOI:
https://doi.org/10.1090/proc/13640
Received by editor(s):
May 30, 2016
Received by editor(s) in revised form:
December 24, 2016
Published electronically:
June 22, 2017
Communicated by:
Franc Forstneric
Article copyright:
© Copyright 2017
American Mathematical Society
