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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Weak containment and weak Frobenius reciprocity


Author: Elliot C. Gootman
Journal: Proc. Amer. Math. Soc. 54 (1976), 417-422
MSC: Primary 22D10
MathSciNet review: 0435286
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Abstract: We study weak containment relations between unitary representations of a group $ G$ and a closed normal subgroup $ K$ by exploiting a property of $ G$-ergodic quasi-invariant measures on the primitive ideal space of $ K$. By this means, we prove that every irreducible representation of $ G$ is weakly contained in a representation induced from an irreducible representation of $ K$ if the quotient group $ G/K$ is amenable; and that the pair $ (G,K)$ satisfies a weak Frobenius reciprocity property if and only if $ G/K$ is amenable and $ G$ acts minimally on the primitive ideal space of $ K$. If $ G/K$ is compact, $ G$ acts minimally if and only if the primitive ideal space of $ K$ is $ {T_1}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0435286-7
PII: S 0002-9939(1976)0435286-7
Keywords: Unitary representation, induced representation, weak containment, weak Frobenius reciprocity, primitive ideal space, kernel of a representation, amenable
Article copyright: © Copyright 1976 American Mathematical Society