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Proceedings of the American Mathematical Society
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
MathSciNet review: 699435
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0699435-1
PII: S 0002-9939(1983)0699435-1
Article copyright: © Copyright 1983 American Mathematical Society