Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-04-25T19:19:52.066Z Has data issue: false hasContentIssue false
  • English
  • Français

Primitive ideals with bounded approximate units in L1-algebras of exponential lie groups

Part of: Abstract harmonic analysis Lie groups

Published online by Cambridge University Press:  09 April 2009

Mohammed El Bachir Bekka
Mohammed El Bachir Bekka
Affiliation:
Fachbereich Mathematik/Informatik, der Universität-Gesamthochschule Paderborn, Warburgerstr. 100, D-4790 Paderborn, Federal Republic of Germany
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.

Let G be an exponential Lie group. We study primitive ideals (i.e. kernels of irreducible *-representations of L1(G)), with bounded approximate units (b.a.u.). We prove a result relating the existence of b.a.u. in certain primitive ideals with the geometry of the corresponding Kirillov orbits. This yields for a solvable group of class 2, a characterization of the primitive ideals with b.a.u.

MSC classification

Secondary: 43A20: $L^1$-algebras on groups, semigroups, etc. 22E30: Analysis on real and complex Lie groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 3 , December 1986 , pp. 411 - 420
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Bekka, M. B., ‘On bounded approximate units in ideals of group algebras’, Math. Ann. 266 (1984), 391–396.CrossRefGoogle Scholar
[2]Bernat, P., ‘Sur les représentations unitaires des groupes de Lie résolubles’, Ann. École Norm. Sup. 82 (1965), 37–99.CrossRefGoogle Scholar
[3]Bernat, P. and Conze, N., Représentations des groupes de Lie résolubles (Dunod, Paris, 1972).Google Scholar
[4]Dixmier, J., Les C*-algèbres et leurs représentations (Gauthier-Villars, Paris, 1969).Google Scholar
[5]Diximier, J., ‘Sur les représentations des groupes de Lie nilpotents III’, Canad. J. Math. 10 (1958), 321–348.CrossRefGoogle Scholar
[6]Doran, R. S. and Wichmann, J., Approximate identities and factorization in Banach modules (Lecture Notes in Mathematics 768, Springer-Verlag, Berlin, 1979).CrossRefGoogle Scholar
[7]Duflo, M., ‘Caractères des groupes et des algèbres de Lie résolubles’, Ann. Ecole Norm. Sup. 3, Sér. 4 (1970), 23–74.CrossRefGoogle Scholar
[8]Grélaud, G., ‘Désintégration des représentations induites d'un groupe de Lie résoluble exponentiel’, C. R. Acad. Sci. Paris 277, Série A (1973), 327–330.Google Scholar
[9]Kirillov, A. A., ‘Unitary representations of nilpotent Lie groups’, Uspehi Mat. Nauk 17 (1962), 57–110.Google Scholar
[10]Liu, T. S., van Rooij, A. and Wang, J. K., ‘Projections and approximate identities for ideals in group algebras’, Trans. Amer. Math. Soc. 175 (1973), 469–482.CrossRefGoogle Scholar
[11]Ludwig, J., ‘On primary ideals in the group algebra of a nilpotent Lie group’, Math. Ann. 262 (1983), 287–304.CrossRefGoogle Scholar
[12]Penney, R. C., ‘Canonical objects in the Kirillov theory of nilpotent Lie groups’, Proc. Amer. Soc. 66 (1977), 175–178.CrossRefGoogle Scholar
[13]Pukanszky, L., Lecons sur les représentations des groupes (Dunod, Paris, 1966).Google Scholar
[14]Pukanszky, L., ‘On Kirillov's character formula’, J. Reine Angew. Math. 311/312 (1979), 408–440.Google Scholar
[15]Pukanszky, L., ‘On the unitary representations of exponential groups’, J. Funct. Anal. 2 (1968), 73–113.CrossRefGoogle Scholar
[16]Reiter, H., L1-Algebras and Segal Algebras (Lecture Notes in Mathematics 231, Springer-Verlag, Berlin, 1971).CrossRefGoogle Scholar
[17]Schreiber, B. M., ‘On the coset ring and strong Ditkin sets’, Pacific J. Math. 32 (1970), 805–812.CrossRefGoogle Scholar