Hostname: page-component-7479d7b7d-767nl Total loading time: 0 Render date: 2024-07-14T10:32:33.688Z Has data issue: false hasContentIssue false
  • English
  • Français

The role of minimal idempotents in the representation theory of locally compact groups

Published online by Cambridge University Press:  20 January 2009

Bruce A. Barnes
Bruce A. Barnes
Affiliation:
University of Oregon Eugene, Oregon 97403, U.S.A.
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.

In the representation theory of finite groups, the minimal idempotents of the group algebra play a central role. In this case the minimal idempotents determine irreducible modules over the group algebra, which in turn are in direct correspondence with the irreducible matrix representations of the group; see Chapter IV of the book of C. Curtis and I. Reiner (2). Many of the same ideas generalise to the situation where the group is compact. In addition, minimal idempotents are involved in some important parts of the theory of Hubert algebras; see M. Rieffel's paper (20).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 23 , Issue 2 , June 1980 , pp. 229 - 238
Copyright
Copyright © Edinburgh Mathematical Society 1980

References

REFERENCES

(1)Barnes, B. A., Locally separable Banach *-algebras: preprint.Google Scholar
(2)Curtis, C. W. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras (Interscience Publishers, John Wiley & Sons, New York, 1966).Google Scholar
(3)Dixmier, J., Les C*-algebres et leurs Representations (Gautier-Villars, Paris, 1969).Google Scholar
(4)Dixmier, J., Points isoles dans le dual d'un groupe localement compact, Bull. Se. Math. 85 (1961), 91–96.Google Scholar
(5)Greenleaf, F. P., Invariant Means on Topological Groups (Van Nostrand, New York, 1969).Google Scholar
(6)Kato, T., Pertubation Theory for Linear Operators (Springer-Verlag, New York, 1966).Google Scholar
(7)Lipsman, R. L., Dual topology for principal and discrete series, Trans. Amer. Math. Soc. 152 (1970), 399–417.CrossRefGoogle Scholar
(8)Milicec, D., Representations of almost connected groups, Proc. Amer. Math. Soc. 47 (1975), 517–518.Google Scholar
(9)Palmer, T. W., Hermitian Banach *-algebras, Bull. Amer. Math. Soc. 78 (1972), 522–524.CrossRefGoogle Scholar
(10)Palmer, T. W., Classes of nonabelian, noncompact, locally compact groups, Rocky Mountain J. of Math. 8 (1978), 683–741.CrossRefGoogle Scholar
(11)Pytlik, T., On the spectral radius of elements in group algebras, Bull. de l'Academie Polonaise des Sci., Serie des sci. math., astron., et phys. 21 (1973), 899–902.Google Scholar
(12)Reiter, J. H., Classical Harmonic Analysis and Locally Compact Groups (Oxford Univ. Press, London, 1968).Google Scholar
(13)Rickart, C. E., Banach Algebras (D. Van Nostrand, Princeton, N.J., 1960).Google Scholar
(14)Sund, T., Isolated points in duals of certain locally compact groups, Math. Ann. 224 (1976), 33–39.CrossRefGoogle Scholar
(15)Wang, P. S., On integrable representations, Math. Z. 147 (1976), 201–203.CrossRefGoogle Scholar
(16)Wang, P. S., On isolated points in the dual spaces of locally compact groups, Math. Ann. 218 (1975), 19–34.CrossRefGoogle Scholar
(17)Warner, G., Harmonic Analysis on Semi-simple Lie Groups, Vol. I (Springer-Verlag, Berlin, 1972).Google Scholar
(18)Fountain, J., Ramsey, R. and Williamson, J. H., Functions of measures on compact groups, Proc. Royal Irish Acad. Sect. A 76 (1976), 231–251.Google Scholar
(19)Ludwig, J., A class of symmetric and a class of Wiener group algebras, J. Functional Analysis 31 (1979), 187–194.CrossRefGoogle Scholar
(20)Rieffel, M., Square integrable representations of Hubert algebras, J. Functional Analysis 3 (1969), 265–300.CrossRefGoogle Scholar