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Group Representations and Cardinal Algebras

Published online by Cambridge University Press:  20 November 2018

Platon C. Deliyannis
Platon C. Deliyannis*
Affiliation:
Illinois Institute of Technology, Chicago, Illinois
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Our aim in this work is to show that the global theory of group representations (as presented in [4] for example) is actually and naturally a part of the theory of cardinal algebras. In this sense it is analogous to the work of Kaplansky [2] and Loomis [3] on operator rings. In his paper Loomis proposed an abstract scheme for representation theory; it appears, however, that the idea of abstracting the equivalence class of a representation is more suitable. It is the set of all these classes that forms the cardinal algebra which we study.

The connection between cardinal algebras and operator theory has been made explicit by Fillmore [1], where he worked out a dimension theory for a class of cardinal algebras satisfying certain conditions. In the next section we shall discuss the relation between the two systems; we could say that the main difference lies in our introducing such axioms as to make possible the study of type III cases.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 759 - 772
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Fillmore, P. A., The dimension theory of certain cardinal algebras, Trans. Amer. Math. Soc. 117 (1965), 21–36.Google Scholar
2. Kaplansky, I., Rings of operators, Mimeographed notes, University of Chicago, Chicago, Illinois, 1955.Google Scholar
3. Loomis, L. H., The lattice-theoretic background of the dimension theory of operator algebras, Mem. Amer. Math. Soc. no. 18 (1955), 36 pp.Google Scholar
4. Mackey, G. W., The theory of group representations, Mimeographed notes, University of Chicago, Chicago, Illinois, 1955.Google Scholar
5. Tarski, A., Cardinal algebras (Oxford Univ. Press, New York, 1949).Google Scholar