Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-24T05:31:01.660Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak Containment and Induced Representations of Groups

Published online by Cambridge University Press:  20 November 2018

J. M. G. Fell
J. M. G. Fell*
Affiliation:
University of Washington
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 locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 237 - 268
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Dixmier, J., Sur les représentations unitaires des groupes de Lie nilpotents, III, Can. J. Math., 10 (1958), 321–348.Google Scholar
2. Dixmier, J., Sur les représentations unitaires des groupes de Lie nilpotents, VI, Can. J. Math., 12 (1960), 324–352.Google Scholar
3. Fell, J. M. G., The dual spaces of C*-algebras, Trans. Amer. Math. Soc, 94 (1960), 365–403.Google Scholar
4. Fell, J. M. G., C*-algebras with smooth dual, Illinois J. Math., 4 (1960), 221–230.Google Scholar
5. Glimm, J. C., Type I C*-algebras, Ann. Math. 73 (1961), 572–612.Google Scholar
6. Godement, R., Les fonctions de type positif et la théorie des groupes, Trans. Amer. Math. Soc, 63 (1948), 1–84.Google Scholar
7. Mackey, G. W., Induced representations of locally compact groups I, Ann. Math., 55 (1952), 101–139.Google Scholar
8. Mackey, G. W., Induced representations of locally compact groups II. The Frobenius reciprocity theorem, Ann. Math., 58 (1953), 193–220.Google Scholar
9. Mackey, G. W., Borel structure in groups and their duals, Trans. Amer. Math. Soc, 85 (1957), 134–165.Google Scholar
10. Nelson, Edward and Forrest Stinespring, W., Representation of elliptic operators in an enveloping algebra, Amer. J. Math., 81 (1959), 547–559.Google Scholar
11. Takenouchi, O., Sur une classe de fonctions continues de type positif sur un groupe localement compact, Math. J. Okayama Univ., 4 (1955), 143–173.Google Scholar
12. Takenouchi, O., Sur la facteur-représentation d'un groupe de Lie resoluble de type (E), Math. J. Okayama Univ., 7 (1957), 151–161.Google Scholar