Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-08T18:49:11.622Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonstandard Ideals in Radical Convolution Algebras on a Half-Line

Published online by Cambridge University Press:  20 November 2018

H. G. Dales  and
J. P. McClure
H. G. Dales
Affiliation:
University of Leeds, Leeds, England
J. P. McClure
Affiliation:
University of Manitoba, Winnipeg, Manitoba
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This note is about the interplay between two classes of radical Banach algebras, and we begin by describing the algebras in question.

A weight sequence is a positive sequence w = (wn) defined on Z+ (the non-negative integers) and satisfying w0 = 1 and wm+n ≦ wmwn for all m and n in Z+. For such a sequence w, the Banach space

is a Banach algebra with respect to the convolution product, defined by

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 2 , 01 April 1987 , pp. 309 - 321
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Allan, G. R., Ideals of rapidly growing functions, Proceedings International Symposium on Functional Analysis and its Applications, Ibadan, Nigeria (1977).Google Scholar
2. Bachar, J. M. et al., Editors, Radical Banach algebras and automatic continuity. Lecture notes in Mathematics 975 (Springer-Verlag, Berlin and New York, 1983).CrossRefGoogle Scholar
3. Bade, W. G. and Dales, H. G., Norms and ideals in radical convolution algebras, J. Functional Analysis 41 (1981), 77–109.Google Scholar
4. Bade, W. G., Dales, H. G. and Laursen, K. B., Multipliers of radical Banach algebras of power series, Mem. American Math. Soc. 303 (Providence, R.I., 1984).Google Scholar
5. Domar, Y., Cyclic elements under translation in weighted L1 spaces on R + , Ark. Mat. 19 (1981), 137–144.Google Scholar
6. Domar, Y., A solution of the translation-invariant subspace problem for weighted Lp on R, R +, or Z , in [2], 214–226.Google Scholar
7. Gelfand, I., Raikov, D. and Sĭlov, , Commutative normed rings (Chelsea, New York, 1964).Google Scholar
8. Grabiner, S., A formal power series operational calculus for quasinilpotent operators II, J. Math. Anal. Appl. 43 (1973), 170–192.Google Scholar
9. Grabiner, S., Weighted shifts and Banach algebras of power series, American J. Math. 97 (1975), 16–42.Google Scholar
10. Grabiner, S. and Thomas, M. P., Non-unicellular strictly cyclic quasinilpotent shifts on Banach spaces, J. Operator Theory 13 (1985), 163–170.Google Scholar
11. Hille, E. and Phillips, R. S., Functional analysis and semi-groups, American Math. Soc. Colloquium Publications 31 (Providence, R. F, 1957).Google Scholar
12. Nikolskiĭ, N. K., Selected problems of weighted approximation and spectral analysis, Proc. Steklov Inst. Math. 720 (1974) (American Math. Soc. Translation, 1976).Google Scholar
13. Shields, A. L., Weighted shift operators and analytic function theory, in Topics in operator theory, American Math. Soc. Mathematical Surveys 13 (Providence, R. I., 1974).Google Scholar
14. Thomas, M. P., Closed ideals and biorthogonal systems in radical Banach algebras of power series, Proc. Edinburgh Math. Soc. 25 (1982), 245–257.Google Scholar
15. Thomas, M. P., Closed ideals of l1{ωn} when {ωn} is star-shaped, Pacific J. Math. 105 (1983), 237–255.Google Scholar
16. Thomas, M. P., Approximation in the radical algebra l1{ωn} when {ωn} is star-shaped, in [2], 258–272.Google Scholar
17. Thomas, M. P., A nonstandard ideal of a radical Banach algebra of power series, Acta Math. 152 (1984), 199–217.Google Scholar
18. Thomas, M. P., Quasinilpotent strictly cyclic unilateral weighted shift operators on lp which are not unicellular, Proc. London Math. Soc. (3) 51 (1985), 127–145.Google Scholar