Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Subrepresentations of direct integrals and finite volume homogeneous spaces


Author: Elliot C. Gootman
Journal: Proc. Amer. Math. Soc. 88 (1983), 565-568
MSC: Primary 22D30; Secondary 46A35
DOI: https://doi.org/10.1090/S0002-9939-1983-0699435-1
MathSciNet review: 699435
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a result on representations of separable $ {C^*}$-algebras which has application to, and was in fact motivated by, a problem concerning relations between unitary representations of a second countable locally compact group $ G$ and those of a closed subgroup $ K$, when $ G/K$ is of finite volume. The result is that if an irreducible representation $ \pi $ is contained in $ \int_X {{\pi _x}} d\mu (x)$, then $ \pi \subseteq {\pi _x}$ for all $ x$ in a set of positive measure. With $ G$ and $ K$ as above, it follows that for each $ \pi \in \hat G$ there exists $ \sigma \in \hat K$ with $ \pi \subseteq {U^\sigma }$, the induced representation. Frobenius reciprocity type results are derived as further consequences.


References [Enhancements On Off] (What's this?)

  • [1] L. Baggett, A unimodular Lie group having purely atomic Plancherel measure is compact, preprint.
  • [2] J. Dixmier, Les $ {C^*}$-algèbres et leurs représentations, Cahiers Scientifiques, fasc. 29, Gauthier-Villars, Paris, 1964. MR 0171173 (30:1404)
  • [3] E. G. Effros and F. Hahn, Locally compact transformation groups and $ {C^*}$-algebras, Mem. Amer. Math. Soc. No. 75 (1967). MR 0227310 (37:2895)
  • [4] E. C. Gootman, Weak containment and weak Frobenius reciprocity, Proc. Amer. Math. Soc. 54 (1976), 417-422. MR 0435286 (55:8246)
  • [5] -, Induced representations and finite volume homogeneous spaces, J. Funct. Anal. 24 (1977), 223-240. MR 0442144 (56:532)
  • [6] G. W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101-139. MR 0044536 (13:434a)
  • [7] -, The theory of group representations, Mimeographed notes, University of Chicago, 1955.
  • [8] C. C. Moore, On the Frobenius reciprocity theorem for locally compact groups, Pacific J. Math. 12 (1962), 359-365. MR 0141737 (25:5134)
  • [9] R. R. Phelps, Lectures on Choquet's Theorem, Van Nostrand Mathematical Studies, No. 7, Van Nostrand, Princeton, N. J., 1966. MR 0193470 (33:1690)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 22D30, 46A35

Retrieve articles in all journals with MSC: 22D30, 46A35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0699435-1
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society