Hostname: page-component-669899f699-2mbcq Total loading time: 0 Render date: 2025-04-26T11:28:08.998Z Has data issue: false hasContentIssue false
  • English
  • Français

The global dimension theorem for weighted convolution algebras

Published online by Cambridge University Press:  20 January 2009

F. Ghahramani  and
Yu. V. Selivanov
F. Ghahramani
Affiliation:
University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2, Email address: [email protected]
Yu. V. Selivanov
Affiliation:
Russian State University of Aviation Technology, Petrovka 27, Moscow 103767, Russia
Rights & Permissions [Opens in a new window]

Abstract

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.

We show that the global dimension, dgA, of every commutative Banach algebra A whose radical is a weighted convolution algebra is strictly greater than one. As an application, we see that in this case H2(A, X) ≠ 0 for some Banach A-bimodule X and thus there exists an unsplittable singular extension of the algebra A.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , Issue 2 , June 1998 , pp. 393 - 406
Copyright
Copyright © Edinburgh Mathematical Society 1998

References

REFERENCES

1.Bade, W. G. and Dales, H. G., Norms and ideals in radical convolution algebras, J. Funct. Anal. 41 (1981), 77–109.CrossRefGoogle Scholar
2.Bade, W. G., Dales, H. G. and Laursen, K. B., Multipliers of radical Banach algebras of power series (Mem. Amer. Math. Soc. 303, 1984).Google Scholar
3.Dales, H. G., Convolution algebras on the real line, in Proceedings of Conference on Radical Banach Algebras and Automatic Continuity (Lecture Notes in Mathematics 975, Springer-Verlag, Berlin, 1983).Google Scholar
4.Dales, H. G. and McClure, J. P., Nonstandard ideals in radical convolution algebras on a half-line, Canad. J. Math. 39 (1987), 309–321.CrossRefGoogle Scholar
5.Despić, M., Ghahramani, F. and Grabiner, S., Weighted convolution algebras without bounded approximate identities, Math. Scand. 76 (1995), 257–272.CrossRefGoogle Scholar
6.Dixon, P. G. and Willis, G. A., Approximate identities in extensions of topologically nilpotent Banach algebras, Proc. Roy. Soc. Edinburgh 122A (1992), 45–52.CrossRefGoogle Scholar
7.Domar, Y., Extensions of the Titchmarsh convolution theorem with applications in the theory of invariant subspaces, Proc. London Math. Soc. 46 (1983), 288–300.CrossRefGoogle Scholar
8.Enflo, P., A counterexample to the approximation problem in Banach spaces, Acta Math. 130(1973), 309–317.CrossRefGoogle Scholar
9.Grothendieck, A., Produits tensoriels topologiques et espaces nucleaires (Mem. Amer. Math. Soc. 16, 1955).Google Scholar
10.Helemskii, A. Ya., On the homological dimension of normed modules over Banach algebras, Mat. Sb. 81(123) (1970), 430–444 (Russian); English transl, in Math. USSR-Sb. 10 (1970), 399–411.Google Scholar
11.Helemskii, A. Ya., Global dimension of a Banach function algebra is different from one, Funktsional. Anal, i Prilozhen. 6 (1972), 95–96 (Russian); English transl. in Functional Anal. Appl. 6 (1972), 166–168.Google Scholar
12.Helemskii, A. Ya., Smallest values assumed by the global homological dimension of Banach function algebras, Trudy Sem. Petrovsk. 3 (1978), 223–242 (Russian); English transl. in Amer. Math. Soc. Transl. 124 (1984), 75–96.Google Scholar
13.Helemskii, A. Ya., The homology of Banach and topological algebras (Moscow Univ. Press, 1986 (Russian); English transl.: Kluwer Academic Press, Dordrecht, 1989).Google Scholar
14.Helemskii, A. Ya., 31 problems of the homology of the algebras of analysis, in Linear and complex analysis problem book 3, Part I (Havin, V. P., Nikolski, N. K., eds., Lecture Notes in Math. 1573, Springer-Verlag, Berlin, 1994), 54–78.Google Scholar
15.Hille, E. and Philips, R. S., Functional Analysis and Semi-groups (Colloquium Publication Series, 31, Amer. Math. Soc., Providence, R.I., 1957).Google Scholar
16.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I, Sequence spaces (Ergebnisse Math. 92, Springer-Verlag, New York, 1977).Google Scholar
17.Selivanov, Yu. V., Computing and estimating the global dimension in certain classes of Banach algebras, Math. Scand. 72 (1993), 85–98.CrossRefGoogle Scholar
18.Selivanov, Yu. V., Projective ideals and estimating the global dimension of commutative Banach algebras, in Third International Conference on Algebra (Krasnoyarsk,1993), 297–298 (Russian).Google Scholar
19.Selivanov, Yu. V., Projective Fréchet modules with the approximation property, Uspekhi Matem. Nauk 50:1 (1995), 209–210 (Russian); English transl, in Russian Math. Surveys. 50:1 (1995), 211–213.Google Scholar
20.Selivanov, Yu. V., Homological characterizations of the approximation property for Banach spaces, Glasgow Math. J. 34 (1992), 229–239.CrossRefGoogle Scholar
21.Wang, J. K., Multipliers of commutative Banach algebras, Pacific J. Math. 11 (1961), 1131–1149.CrossRefGoogle Scholar