Hostname: page-component-66644f4456-976bt Total loading time: 0 Render date: 2025-02-12T15:55:10.530Z Has data issue: true hasContentIssue false
  • English
  • Français

Groups with Representations of Bounded Degree

Published online by Cambridge University Press:  20 November 2018

Irving Kaplansky
Irving Kaplansky*
Affiliation:
University of Chicago
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.

Let G be a compact group. According to the celebrated theorem of Peter-Weyl there exists a complete set of finite-dimensional irreducible unitary representations of G, the completeness meaning that for any group element other than the identity there is a representation sending it into a matrix other than the unit matrix. If G is commutative, the representations are necessarily one-dimensional. It is an immediate consequence of the Peter-Weyl theorem that the converse also holds: if every representation is one-dimensional, G is commutative. The main theorem in the present paper is a generalization of this result to the case where the representations have bounded degree.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 1 , 01 February 1949 , pp. 105 - 112
Copyright
Copyright © Canadian Mathematical Society 1949

References

[1] Halmos, P. R., “On Automorphisms of Compact Groups,” Bull. Amer. Math, Soc, vol. 49 (1943), 619–624.Google Scholar
[2] Jacobson, N., “The Radical and Semi-simplicity for Arbitrary Rings,” Amer. J. of Math., vol. 67 (1945), 300–320.Google Scholar
[3] Kaplansky, I., “Rings with a Polynomial Identity,” Bull. Amer. Math. Soc, vol. 54 (1948), 575–580.Google Scholar
[4] Segal, I. E., “The Group Algebra of a Locally Compact Group,” Trans. Amer. Math. Soc, vol. 61 (1947), 69–105.Google Scholar
[5] Weil, A., “L'Intégration dans les groupes topologiques et ses applications,” Actualités Scientifiques et Industrielles, no. 869, Paris, 1938.Google Scholar
[6] Chandra, Harish, “On Representations of Lie Algebras,” to appear in the Annals of Mathematics.Google Scholar