Groups with finite dimensional irreducible representations

Moore, Calvin C.

doi:10.1090/S0002-9947-1972-0302817-8

Groups with finite dimensional irreducible representations
HTML articles powered by AMS MathViewer

by Calvin C. Moore PDF

Trans. Amer. Math. Soc. 166 (1972), 401-410 Request permission

Abstract:

It will be shown that a locally compact group has a finite bound for the dimensions of its irreducible unitary representations if and only if it has a closed abelian subgroup of finite index. It will further be shown that a locally compact group has all of its irreducible representations of finite dimension if and only if it is a projective limit of Lie groups with the same property, and finally that a Lie group has this property if and only if it has a closed subgroup H of finite index such that H modulo its center is compact.

References

Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62 (1966), 199. MR 207910
Jacques Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (Algèbres de von Neumann), Cahiers Scientifiques, Fasc. XXV, Gauthier-Villars, Paris, 1957 (French). MR 0094722
Hans Freudenthal, Topologische Gruppen mit genügend vielen fastperiodischen Funktionen, Ann. of Math. (2) 37 (1936), no. 1, 57–77 (German). MR 1503269, DOI 10.2307/1968687
Siegfried Grosser and Martin Moskowitz, On central topological groups, Trans. Amer. Math. Soc. 127 (1967), 317–340. MR 209394, DOI 10.1090/S0002-9947-1967-0209394-9
Siegfried Grosser and Martin Moskowitz, Representation theory of central topological groups, Trans. Amer. Math. Soc. 129 (1967), 361–390. MR 229753, DOI 10.1090/S0002-9947-1967-0229753-8
James Glimm, Families of induced representations, Pacific J. Math. 12 (1962), 885–911. MR 146297, DOI 10.2140/pjm.1962.12.885
I. M. Isaacs and D. S. Passman, Groups with representations of bounded degree, Canadian J. Math. 16 (1964), 299–309. MR 167539, DOI 10.4153/CJM-1964-029-5
Nathan Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, 190 Hope Street, Providence, R.I., 1956. MR 0081264, DOI 10.1090/coll/037
Irving Kaplansky, Group algebras in the large, Tohoku Math. J. (2) 3 (1951), 249–256. MR 48712, DOI 10.2748/tmj/1178245477
Irving Kaplansky, Groups with representations of bounded degree, Canad. J. Math. 1 (1949), 105–112. MR 28317, DOI 10.4153/cjm-1949-011-9
George W. Mackey, Infinite-dimensional group representations, Bull. Amer. Math. Soc. 69 (1963), 628–686. MR 153784, DOI 10.1090/S0002-9904-1963-10973-8
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
George W. Mackey, Induced representations of locally compact groups. II. The Frobenius reciprocity theorem, Ann. of Math. (2) 58 (1953), 193–221. MR 56611, DOI 10.2307/1969786
George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI 10.1007/BF02392428
Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
Calvin C. Moore, Extensions and low dimensional cohomology theory of locally compact groups. I, II, Trans. Amer. Math. Soc. 113 (1964), 40–63; ibid. 113 (1964), 64–86. MR 171880, DOI 10.1090/S0002-9947-1964-0171880-5

Extensions and low dimensional cohomology of locally compact groups

113

30

Calvin C. Moore, Decomposition of unitary representations defined by discrete subgroups of nilpotent groups, Ann. of Math. (2) 82 (1965), 146–182. MR 181701, DOI 10.2307/1970567
Calvin C. Moore, On the Frobenius reciprocity theorem for locally compact groups, Pacific J. Math. 12 (1962), 359–365. MR 141737, DOI 10.2140/pjm.1962.12.359
Lewis C. Robertson, A note on the structure of Moore groups, Bull. Amer. Math. Soc. 75 (1969), 594–599. MR 245721, DOI 10.1090/S0002-9904-1969-12252-4
Elmar Thoma, Über unitäre Darstellungen abzählbarer, diskreter Gruppen, Math. Ann. 153 (1964), 111–138 (German). MR 160118, DOI 10.1007/BF01361180
Elmar Thoma, Eine Charakterisierung diskreter Gruppen vom Typ I, Invent. Math. 6 (1968), 190–196 (German). MR 248288, DOI 10.1007/BF01404824

L’intégration dans les groupes topologiques et ses applications

The theory of groups and quantum mechanics