## Localization for uniform algebras generated by real-analytic functions

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- by John T. Anderson and Alexander J. Izzo PDF
- Proc. Amer. Math. Soc.
**145**(2017), 4919-4930 Request permission

## Abstract:

It is shown that if $A$ is a uniform algebra generated by real-analytic functions on a suitable compact subset $K$ of a real-analytic variety such that the maximal ideal space of $A$ is $K$ and every continuous function on $K$ is locally a uniform limit of functions in $A$, then $A=C(K)$. 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.## References

- H. Alexander,
*Polynomial approximation and analytic structure*, Duke Math. J.**38**(1971), 123β135. MR**283244** - 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.Β 2, 187β195. MR**1815422**, DOI 10.1112/blms/33.2.187 - John T. Anderson, Alexander J. Izzo, and John Wermer,
*Polynomial approximation on three-dimensional real-analytic submanifolds of $\textbf {C}^n$*, Proc. Amer. Math. Soc.**129**(2001), no.Β 8, 2395β2402. MR**1823924**, DOI 10.1090/S0002-9939-01-05911-1 - John T. Anderson, Alexander J. Izzo, and John Wermer,
*Polynomial approximation on real-analytic varieties in $\mathbf C^n$*, Proc. Amer. Math. Soc.**132**(2004), no.Β 5, 1495β1500. MR**2053357**, DOI 10.1090/S0002-9939-03-07263-0 - John T. Anderson and Alexander J. Izzo,
*Peak point theorems for uniform algebras on smooth manifolds*, Math. Z.**261**(2009), no.Β 1, 65β71. MR**2452637**, DOI 10.1007/s00209-008-0313-x - John T. Anderson and Alexander J. Izzo,
*A peak point theorem for uniform algebras on real-analytic varieties*, Math. Ann.**364**(2016), no.Β 1-2, 657β665. MR**3451401**, DOI 10.1007/s00208-015-1224-x - Frank T. Birtel (ed.),
*Function algebras*, Scott, Foresman & Co., Chicago, Ill., 1966. MR**0193471** - Brian James Cole,
*ONE-POINT PARTS AND THE PEAK POINT CONJECTURE*, ProQuest LLC, Ann Arbor, MI, 1968. Thesis (Ph.D.)βYale University. MR**2617861** - 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.Β 42β60. MR**0193538** - Theodore W. Gamelin,
*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR**0410387** - Alexander J. Izzo,
*Localization for uniform algebras generated by smooth functions on two-manifolds*, Bull. Lond. Math. Soc.**42**(2010), no.Β 4, 652β660. MR**2669686**, DOI 10.1112/blms/bdq024 - Alexander J. Izzo,
*Uniform approximation on manifolds*, Ann. of Math. (2)**174**(2011), no.Β 1, 55β73. MR**2811594**, DOI 10.4007/annals.2011.174.1.2 - 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.Β 135β146. MR**2856487**, DOI 10.1090/conm/547/10814 - Alexander J. Izzo,
*Nonlocal uniform algebras on three-manifolds*, Pacific J. Math.**259**(2012), no.Β 1, 109β116. MR**2988485**, DOI 10.2140/pjm.2012.259.109 - Eva Kallin,
*A nonlocal function algebra*, Proc. Nat. Acad. Sci. U.S.A.**49**(1963), 821β824. MR**152907**, DOI 10.1073/pnas.49.6.821 - Raghavan Narasimhan,
*Introduction to the theory of analytic spaces*, Lecture Notes in Mathematics, No. 25, Springer-Verlag, Berlin-New York, 1966. MR**0217337** - John Rainwater,
*A remark on regular Banach algebras*, Proc. Amer. MAth. Soc.**18**(1967), 255β256. MR**0208413**, DOI 10.1090/S0002-9939-1967-0208413-9 - Edgar Lee Stout,
*Holomorphic approximation on compact, holomorphically convex, real-analytic varieties*, Proc. Amer. Math. Soc.**134**(2006), no.Β 8, 2302β2308. MR**2213703**, DOI 10.1090/S0002-9939-06-08250-5 - J. Wermer,
*Approximation on a disk*, Math. Ann.**155**(1964), 331β333. MR**165386**, DOI 10.1007/BF01354865 - J. Wermer,
*Polynomially convex disks*, Math. Ann.**158**(1965), 6β10. MR**174968**, DOI 10.1007/BF01370392 - 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.Β 98β111. MR**0196525**

## Additional Information

**John T. Anderson**- Affiliation: Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, Massachusetts 01610
- MR Author ID: 251416
- Email: janderso@holycross.edu
**Alexander J. Izzo**- Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403
- MR Author ID: 307587
- Email: aizzo@bgsu.edu
- 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
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3692006