Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T08:56:24.611Z Has data issue: false hasContentIssue false
  • English
  • Français

Factorization and unbounded approximate identities in Banach algebras

Published online by Cambridge University Press:  24 October 2008

P. G. Dixon
P. G. Dixon
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH
Get access Rights & Permissions [Opens in a new window]

Extract

Cohen's Factorization Theorem says, in its basic form, that if A is a Banach algebra with a bounded left approximate identity, then every element x ∈ A may be written as a product x = ay for some a, y ∈ A. Such is the beauty and importance of this result that much interest attaches to the question of whether the hypothesis of a bounded left approximate identity can be weakened, or whether a converse result exists. This paper contributes to the study of that question.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 3 , May 1990 , pp. 557 - 571
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Altman, M.. Contractors, approximate identities and factorization in Banach algebras. Pacific J. Math. 48 (1973), 323–334.CrossRefGoogle Scholar
[2]Bonsall, F. F. and Duncan, J.. Complete Normed Algebras (Springer-Verlag, 1973).CrossRefGoogle Scholar
[3]Browder, A.. Introduction to Function Algebras (Benjamin, 1969).Google Scholar
[4]Cohen, P. J.. Factorization in group algebras. Duke Math. J. 26 (1959), 199–205.CrossRefGoogle Scholar
[5]Dixon, P. G.. Spectra of approximate identities in Banach algebras. Math. Proc. Cambridge Philos. Soc. 86 (1979), 271–278.CrossRefGoogle Scholar
[6]Dixon, P. G.. Generalized open mapping theorems for bilinear maps, with an application to operator algebras. Proc. Amer. Math. Soc. 104 (1988), 106–110.CrossRefGoogle Scholar
[7]Doran, R. S. and Wichmann, J.. Approximate Identities and Factorization in Banach Modules. Lecture Notes in Math. vol. 768 (Springer-Verlag, 1979).CrossRefGoogle Scholar
[8]Feichtinger, H. G. and Leinert, M.. Individual factorization in Banach modules. Colloq. Math. 51(1987), 107–117.CrossRefGoogle Scholar
[9]Gamelin, T. W.. Uniform Algebras (Prentice-Hall, 1969).Google Scholar
[10]Hallstrom, A. P.. On bounded point derivations and analytic capacity J. Funct. Anal. 4 (1969), 153–165.CrossRefGoogle Scholar
[11]Leinert, M.. A commutative Banach algebra which factorizes but has no approximate units. Proc. Amer. Math. Soc. 55 (1976), 345–346.CrossRefGoogle Scholar
[12]Loy, R. J.. Multilinear mappings and Banach algebras. J. London Math. Soc. (2) 14 (1976), 423–429.CrossRefGoogle Scholar
[13]Loy, R. J.. Notes on analytic spaces. (Mimeographed notes.)Google Scholar
[14]Mel'nikov, M. S.. Estimate of the Cauchy integral along an analytic curve. Mat. Sb. 71 (113) (1966), 503–514Google Scholar
translated in Amer. Math. Soc. Transl. (2) 80(1968), 243–255.Google Scholar
[15]Ouzomgi, S.. Factorization in topological algebras. Ph.D. Thesis, University of Leeds (1985).Google Scholar
[16]Rickart, C. E.. Natural Function Algebras (Springer-Verlag, 1979).CrossRefGoogle Scholar
[17]Stout, E. L.. The Theory of Uniform Algebras (Bogden and Quigley, 1971).Google Scholar
[18]Willis, G. A.. Examples of factorization without bounded approximate units. (Research Report no. 12, 1989. Mathematical Sciences Research Centre, Australian National University, Canberra.)Google Scholar
[19]Zalcman, L.. Analytic Capacity and Rational Approximation. Lecture Notes in Math. vol. 50 (Springer-Verlag, 1968).CrossRefGoogle Scholar