Isolated point theorems for uniform algebras on two- and three-manifolds

Author:
Swarup N. Ghosh

Journal:
Proc. Amer. Math. Soc. **144** (2016), 3921-3933

MSC (2010):
Primary 32E30, 46J10

DOI:
https://doi.org/10.1090/proc/13036

Published electronically:
March 30, 2016

MathSciNet review:
3513549

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1957, Andrew Gleason conjectured that if is a uniform algebra on its maximal ideal space and every point of is a one-point Gleason part for , then must contain all continuous functions on . However, in 1968, Brian Cole produced a counterexample to disprove Gleason's conjecture. In this paper, we establish that Gleason's conjecture still holds for two important classes of uniform algebras considered by John Anderson, Alexander Izzo and John Wermer in connection with the peak point conjecture. In fact, we prove stronger results by weakening the hypothesis of Gleason's conjecture for those two classes of uniform algebras.

**[1]**H. Alexander,*Polynomial approximation and analytic structure*, Duke Math. J.**38**(1971), 123-135. MR**0283244 (44 #477)****[2]**H. Alexander,*Structure of certain polynomial hulls*, Michigan Math. J.**24**(1977), no. 1, 7-12. MR**0463500 (57 #3449)****[3]**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 (98g:32002)****[4]**John T. Anderson,*Finitely generated algebras and algebras of solutions to partial differential equations*, Pacific J. Math.**133**(1988), no. 1, 1-12. MR**936353 (89d:46055)****[5]**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 (2002j:32035)**, https://doi.org/10.1112/blms/33.2.187**[6]**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 (2009m:46076)**, https://doi.org/10.1007/s00209-008-0313-x- [7]
J. T. Anderson and A. J. Izzo,
*A peak point theorem for uniform algebras on real-analytic varieties*, Math. Ann. (to appear) **[8]**John T. Anderson, Alexander J. Izzo, and John Wermer,*Polynomial approximation on three-dimensional real-analytic submanifolds of*, Proc. Amer. Math. Soc.**129**(2001), no. 8, 2395-2402. MR**1823924 (2002d:32021)**, https://doi.org/10.1090/S0002-9939-01-05911-1**[9]**Richard F. Basener,*On rationally convex hulls*, Trans. Amer. Math. Soc.**182**(1973), 353-381. MR**0379899 (52 #803)****[10]**H. S. Bear,*Complex function algebras*, Trans. Amer. Math. Soc.**90**(1959), 383-393. MR**0107164 (21 #5889)****[11]**Errett Bishop,*A minimal boundary for function algebras*, Pacific J. Math.**9**(1959), 629-642. MR**0109305 (22 #191)**- [12]
A. Boggess,
*CR Manifolds and the Tangential Cauchy-Riemann Complex*, CRC Press, Inc., 1991. **[13]**Andrew Browder,*Introduction to function algebras*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0246125 (39 #7431)****[14]**Brian James Cole,*ONE-POINT PARTS AND THE PEAK POINT CONJECTURE*, ProQuest LLC, Ann Arbor, MI, 1968. Thesis (Ph.D.)-Yale University. MR**2617861****[15]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325 (41 #1976)****[16]**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 (33 #1758)**- [17]
T. W. Gamelin,
*Uniform Algebras*, 2nd ed., Chelsea Publishing Company, New York, NY, 1984. **[18]**John Garnett,*A topological characterization of Gleason parts*, Pacific J. Math.**20**(1967), 59-63. MR**0205107 (34 #4942)****[19]**Swarup N. Ghosh,*Isolated point theorems for uniform algebras on manifolds*, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)-Bowling Green State University. MR**3321927**- [20]
A. Gleason,
*Function algebras*, Seminar on Analytic Functions, vol. II, Institute for Advanced Study, Princeton (1957), 213-226. **[21]**Victor Guillemin and Alan Pollack,*Differential topology*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR**0348781 (50 #1276)**- [22]
J. R. Munkres,
*Topology*, 2nd ed., Prentice-Hall of India Private Limited, New Delhi, 2005. **[23]**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 (86c:32015)**, https://doi.org/10.1090/conm/032/769507**[24]**Arthur Sard,*The measure of the critical values of differentiable maps*, Bull. Amer. Math. Soc.**48**(1942), 883-890. MR**0007523 (4,153c)****[25]**Nessim Sibony,*Multi-dimensional analytic structure in the spectrum of a uniform algebra*, Spaces of analytic functions (Sem. Functional Anal. and Function Theory, Kristiansand, 1975) Springer, Berlin, 1976, pp. 139-165. Lecture Notes in Math., Vol. 512. MR**0632106 (58 #30277)****[26]**S. J. Sidney,*Properties of the sequence of closed powers of a maximal ideal in a sup-norm algebra*, Trans. Amer. Math. Soc.**131**(1968), 128-148. MR**0222651 (36 #5701)**- [27]
E. L. Stout,
*The Theory of Uniform Algebras*, Bogden & Quigley, New York, 1971.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
32E30,
46J10

Retrieve articles in all journals with MSC (2010): 32E30, 46J10

Additional Information

**Swarup N. Ghosh**

Affiliation:
Department of Mathematics, Southwestern Oklahoma State University, Weatherford, Oklahoma 73096

Email:
swarup.ghosh@swosu.edu

DOI:
https://doi.org/10.1090/proc/13036

Received by editor(s):
June 14, 2015

Received by editor(s) in revised form:
June 22, 2015, and November 12, 2015

Published electronically:
March 30, 2016

Communicated by:
Pamela B. Gorkin

Article copyright:
© Copyright 2016
American Mathematical Society