The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).


Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the denseness of the invertible group in Banach algebras

Authors: T. W. Dawson and J. F. Feinstein
Journal: Proc. Amer. Math. Soc. 131 (2003), 2831-2839
MSC (2000): Primary 46J10, 46H05
Published electronically: April 7, 2003
MathSciNet review: 1974340
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We examine the condition that a complex Banach algebra $A$ 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 [Enhancements On Off] (What's this?)

  • 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 $C^*$-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 $K$-Theory of $C^*$-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 $C^*$-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

J. F. Feinstein
Affiliation: Division of Pure Mathematics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

Received by editor(s): April 4, 2002
Published electronically: April 7, 2003
Additional Notes: The first author thanks the EPSRC for providing support for this research
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society