|
On the denseness of the invertible group in Banach algebras
Author(s):
T.
W.
Dawson;
J.
F.
Feinstein
Journal:
Proc. Amer. Math. Soc.
131
(2003),
2831-2839.
MSC (2000):
Primary 46J10, 46H05
Posted:
April 7, 2003
MathSciNet review:
1974340
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We examine the condition that a complex Banach algebra  has dense invertible group. We show that, for commutative algebras, this property is preserved by integral extensions. We also investigate the connections with an old problem in the theory of uniform algebras.
References:
-
- 1.
- Bollobás, B. (1999): `Linear Analysis', 2nd edition, Cambridge: Cambridge University Press. MR 2000g:46001
- 2.
- Bourbaki, N. (1988) `Algebra II, Chapters 4-7', US: Springer-Verlag. MR 91h:00003
- 3.
- Cole, B. J. (1968): `One-Point Parts and the Peak Point Conjecture', Ph.D. Thesis, Yale University.
- 4.
- Cole, B. J. (2002): Private communication.
- 5.
- Corach, G. and Suárez, F. D. (1988): `Thin Spectra and Stable Range Conditions', J. Funct. Anal., 81, 432-442. MR 89k:46061
- 6.
- Dales, H. G. (2001): `Banach Algebras and Automatic Continuity', New York: Oxford University Press, Inc. MR 2002e:46001
- 7.
- Falcón-Rodríguez, C. M. (1988) `Sobre la densidad del grupo de los elementos invertibles de un álgebra uniforme', Revista Ciencias Matemáticas, IX, no. 2, 11-17. MR 91c:46072
- 8.
- Feinstein, J. F. (1992): `A Non-Trivial, Strongly Regular Uniform Algebra', J. Lond. Math. Soc., 45, no. 2, 288-300. MR 93i:46086
- 9.
- Feinstein, J. F. and Somerset, D. W. B. (1999): `Strong Regularity for Uniform Algebras', Contemp. Math., 232, 139-149. MR 2000i:46042
- 10.
- Gelfand, I. M. (1957): `On the Subrings of a Ring of Continuous Functions', Usp. Mat. Nauk, 12, no. 1, 249-251. MR 18:913a
- 11.
- Gelfand, I. M., Kapranov, M. M., and Zelevinsky, A. V. (1994) `Discriminants, Resultants, and Multidimensional Determinants', United States of America: Birkhäuser Boston. MR 95e:14045
- 12.
- Grigoryan, S. A. (1984): `Polynomial Extensions of Commutative Banach Algebras', Russian Math. Surveys, 39, no. 1, 161-162. MR 85j:46086
- 13.
- Hatori, O. and Miura, T.: (1999) `On a Characterization of the Maximal Ideal Spaces of Commutative
 -Algebras in which Every Element is the Square of Another', Proc. Am. Math. Soc., 128, no. 4, 1185-1189. MR 2000k:46072a - 14.
- Helson, H. and Quigley, F. (1957): `Existence of Maximal Ideals in Algebras of Continuous Functions', Proc. Am. Math. Soc., 8, 115-119. MR 18:911e
- 15.
- Henkin, G. M. and Cirka, E. M. (1976): `Boundary Properties of Holomorphic Functions of Several Complex Variables', J. Soviet Math., 5, no. 5, 612-687.
- 16.
- Jacobson, N. (1996) `Basic Algebra I' (2nd ed.) New York: W. H. Freeman and Company.
- 17.
- Karahanjan, M. I. (1979): `Some algebraic characterizations of the algebra of all continuous functions on a locally connected compactum', Math. USSR Sb., 35, 681-696. MR 82b:46065
- 18.
- Leibowitz, G. M. (1970) `Lectures on Complex Function Algebras', United States of America: Scott, Foresman and Company. MR 55:1072
- 19.
- Lindberg, J. A. (1973): `Integral Extensions of Commutative Banach Algebras' Can. J. Math., 25, 673-686. MR 48:9401
- 20.
- Palmer, T. W. (1994) `Banach Algebras and the General Theory of *-Algebras' (vol. 1), Cambridge: CUP. MR 95c:46002
- 21.
- Pears, A. R. (1975) `Dimension Theory of General Spaces', Cambridge: CUP. MR 52:15405
- 22.
- Rieffel M. A. (1983): `Dimension and Stable Rank in
 -Theory of  -Algebras', Proc. Lond. Math. Soc., 46, no. 3, 577-600. MR 84g:46085 - 23.
- Robertson, G. (1976): `On the Density of the Invertible Group in
 -Algebras', Proc. Edinb. Math. Soc., 20, 153-157. MR 54:5845 - 24.
- Stout, E. L. (1973) `The Theory of Uniform Algebras', Tarrytown-on-Hudson, New York: Bogden and Quigley, Inc. MR 54:11066
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
46J10, 46H05
Retrieve articles in all Journals with
MSC (2000):
46J10, 46H05
Additional Information:
T.
W.
Dawson
Affiliation:
Division of Pure Mathematics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom
Email:
pmxtwd@nottingham.ac.uk
J.
F.
Feinstein
Affiliation:
Division of Pure Mathematics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom
Email:
Joel.Feinstein@nottingham.ac.uk
DOI:
10.1090/S0002-9939-03-07058-8
PII:
S 0002-9939(03)07058-8
Received by editor(s):
April 4, 2002
Posted:
April 7, 2003
Additional Notes:
The first author thanks the EPSRC for providing support for this research
Communicated by:
N. Tomczak-Jaegermann
Copyright of article:
Copyright
2003,
American Mathematical Society
|
