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)



Non-closed sums of closed ideals in Banach algebras

Author: P. G. Dixon
Journal: Proc. Amer. Math. Soc. 128 (2000), 3647-3654
MSC (2000): Primary 46H10; Secondary 46J20, 46J15
Published electronically: June 21, 2000
MathSciNet review: 1778282
Full-text PDF

Abstract | References | Similar Articles | Additional Information


A major difficulty in Banach algebra theory is that sums of closed ideals need not be closed. We survey the known results and present examples showing that they are in most directions the best possible. We also give a new sufficient condition for closure in the uniform algebra setting.

References [Enhancements On Off] (What's this?)

  • 1. F. Combes, Sur les faces d'une C*-algèbre Bull. Sci. Math., 93 (1969), 37-62. MR 42:856
  • 2. A. M. Davie, Quotient algebras of uniform algebras, J. London Math. Soc. (2), 7 (1973), 31-40. MR 48:2779
  • 3. J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964. MR 30:1404
  • 4. P. G. Dixon, Topologically irreducible representations and radicals in Banach algebras, Proc. London Math. Soc. (3), 74 (1997), 174-200. MR 30:46066
  • 5. Robert S. Doran and Josef Wichmann, Approximate Identities and Factorization in Banach Modules, Springer, Lecture Notes in Math., 768, Berlin, 1979. MR 83e:46044
  • 6. E. Kirchberg, On restricted perturbations in inverse images and a description of normalizer algebras in C*-algebras, J. Functional Analysis, 129 (1995), 1-34. MR 95m:46094a
  • 7. W. Rudin, Spaces of type $H^\infty + C$, Ann. Inst. Fourier, Grenoble, 25 (1975), 99-125. MR 57:13692; errata MR 57:7134
  • 8. D. A. Stegenga, Ideals in the disk algebra, J. Functional Analysis, 25 (1977), 335-337. MR 58:2307
  • 9. Simon Wassermann, Exact C*-algebras and related topics, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University, Lecture Note Series, 19, 1994. MR 95b:46081

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46H10, 46J20, 46J15

Retrieve articles in all journals with MSC (2000): 46H10, 46J20, 46J15

Additional Information

P. G. Dixon
Affiliation: Department of Pure Mathematics, University of Sheffield, Sheffield, S3 7RH, England

Received by editor(s): March 1, 1999
Published electronically: June 21, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society