Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-18T00:15:50.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Topologically nilpotent Banach algebras and factorisation

Published online by Cambridge University Press:  14 November 2011

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

Synopsis

A Banach algebra A is said to be topologically nilpotent if sup {‖x1x2…xn‖1/n: xi ∈ A, ‖xi‖ ≦ 1 (1 ≦ i ≦ n)} tends to zero as n → ∞. A Banach algebra A is uniformly topologically nil if sup {‖xn‖ 1/n: x ∈ A, ‖x‖ ≦ 1} tends to zero as n → ∞. These notions are equivalent for commutative algebras and a topological version of the Nagata-Higman Theorem gives a partial result for the non-commutative case. Topologically nilpotent algebras have a strong non-factorisation property and this yields theorems of the type “factorisation implies the existence of arbitrarily slowly decreasing powers”. Extensions of topologically nilpotent algebras by topologically nilpotent algebras are topologically nilpotent.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 3-4 , 1991 , pp. 329 - 341
Copyright
Copyright © Royal Society of Edinburgh 1991

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

1Allan, G. R. and Sinclair, A. M.. Power factorization in Banach algebras with a bounded approximate identity. Studia Math. 56 (1976), 31–38.Google Scholar
2Dixon, P. G.. Factorization and unbounded approximate identities in Banach algebras. Math. Proc. Cambridge Philos. Soc. 107 (1990), 557–571.CrossRefGoogle Scholar
3Doran, R. S. and Wichmann, J.. Approximate identities and facorization in Banach modules. Lecture Notes in Mathematics 768 (Berlin: Springer, 1979).Google Scholar
4Grønbæk, N.. Power factorization in Banach modules over commutative radical Banach algebras. Math. Scand. 50 (1982), 123–134.CrossRefGoogle Scholar
5Grønbaeæk, N.. Weighted discrete convolution algebras. In Radical Banach algebras and automatic continuity: Proceedings of a conference held at California State University, Long Beach, July 17–31, 1981, Lecture Notes in Mathematics 975 295–300 (Berlin: Springer, 1983).CrossRefGoogle Scholar
6Higman, G.. On a conjecture of Nagata. Proc. Cambridge Philos. Soc. 52 (1956), 1–4.CrossRefGoogle Scholar
7Miziolek, J., Müldner, T. and Rek, A.. On the topologically nilpotent algebras. Studia Math. 43 (1972), 41–50.Google Scholar
8Razmyslov, Ju. P.. Trace identities of full matrix algebras over a field characteristic zero. Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723–756; English translation: Math. USSR Izv. 8 (1974), 727–760.Google Scholar
9Rickart, C. E.. General theory of Banach algebras (Princeton: van Nostrand, 1960).Google Scholar
10Willis, G.. Examples of factorization without bounded approximate units. Proc. London Math. Soc. (to appear).Google Scholar