Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-29T21:54:48.971Z Has data issue: false hasContentIssue false
  • English
  • Français

Compact open sets in duals and projections in L1-algebras of certain semi-direct product groups

Published online by Cambridge University Press:  24 October 2008

Karlheinz Gröchenig ,
Eberhard Kaniuth  and
Keith F. Taylor
Karlheinz Gröchenig
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269, U.S.A.
Eberhard Kaniuth
Affiliation:
Fachbereich Mathematik/Informatik, Universität Paderborn, D-W-4790 Paderborn, Germany
Keith F. Taylor
Affiliation:
Department of Mathematics, University of Saskatchewan, Saskatoon, Saskatchewan S7N OWO, Canada
Get access Rights & Permissions [Opens in a new window]

Extract

The main purpose of this paper is to study projections, that is, self-adjoint idempotents, in L1-algebras of semi-direct products G = ℝ ⋉ ℝd, d ≥ 2. We establish necessary and sufficient conditions for the existence of non-zero projections in terms of the action of ℝ on ℝd. In the cases where such projections exist, we describe minimal ones in detail.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 3 , May 1992 , pp. 545 - 556
Copyright
Copyright © Cambridge Philosophical Society 1992

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]Barnes, B. A.. The role of minimal idempotents in the representation theory of locally compact groups. Proc. Edinburgh Math. Soc. (2) 23 (1980), 229–238.CrossRefGoogle Scholar
[2]Bourbaki, N.. Eléments de Mathématique, Livre VI: Intégration. Chapitre 6 (Hermann, 1959).Google Scholar
[3]Dixmier, J.. Les C*-algèbres et leurs Représentations (Gauthier-Villars, 1964).Google Scholar
[4]Eymard, P. and Terp, M.. La transformation de Fourier et son inverse sur le groupe de ax+b d'un corps local. In Analyse Harmonique sur les Groupes de Lie. II, Lecture Notes in Math. vol. 739 (Springer-Verlag, 1979), pp. 207–248.CrossRefGoogle Scholar
[5]Federer, H. and Morse, A. P.. Some properties of measurable functions. Bull. Amer. Math. Soc. 49 (1943), 270–277.CrossRefGoogle Scholar
[6]Fell, J. M. G.. Weak containment and induced representations II. Trans. Amer. Math. Soc. 110 (1964), 424–447.Google Scholar
[7]Fell, J. M. G. and Doran, R. S.. Representations of *-algebras, Locally Compact Groups, and Banach *-algebraic Bundles, vol. 2 (Academic Press, 1988).Google Scholar
[8]Gaal, S. A.. Linear Analysis and Representation Theory (Springer-Verlag, 1973).CrossRefGoogle Scholar
[9]Kaniuth, E. and Taylor, K. F.. Projections in C*-algebras of nilpotent groups. Manuscripta Math. 65 (1989), 93–111.CrossRefGoogle Scholar
[10]Leptin, H.. On onesided harmonic analysis in non commutative locally compact groups. J. Reine Angew. Math. 306 (1979), 122–153.Google Scholar
[11]Lipsman, R. L.. Solvability of invariant second-order differential operators on metabelian groups. Michigan Math. J. 35 (1988), 132–139.CrossRefGoogle Scholar
[12]Rieffel, M. A.. Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner. Adv. in Math. Suppl. Stud. 4 (1979), 43–82.Google Scholar
[13]Rosenberg, J.. The C*-algebras of some real and p-adic solvable groups. Pacific J. Math. 65 (1976), 175–192.CrossRefGoogle Scholar