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Groups with Representations of Bounded Degree

Published online by Cambridge University Press:  20 November 2018

Irving Kaplansky
Irving Kaplansky*
Affiliation:
University of Chicago
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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

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