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.
Deliyannis, Platon C. Group Representations and Cardinal Algebras. Canadian journal of mathematics, Tome 22 (1970) no. 4, pp. 759-772. doi: 10.4153/CJM-1970-086-4
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author = {Deliyannis, Platon C.},
title = {Group {Representations} and {Cardinal} {Algebras}},
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pages = {759--772},
year = {1970},
volume = {22},
number = {4},
doi = {10.4153/CJM-1970-086-4},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-086-4/}
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[3] 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
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Mackey, G. W., The theory of group representations, Mimeographed notes, University of Chicago, Chicago, Illinois, 1955. Google Scholar
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Tarski, A., Cardinal algebras (Oxford Univ. Press, New York, 1949). Google Scholar
