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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Topology and the duals of certain locally compact groups

Author: I. Schochetman
Journal: Trans. Amer. Math. Soc. 150 (1970), 477-489
MSC: Primary 22.60
MathSciNet review: 0265513
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Abstract: We consider some topological questions concerning the dual space of a (separable) extension $ G$ of a type I, regularly embedded subgroup $ N$. The dual $ \hat G$ is known to have a fibre-like structure. The fibres are in bijective correspondence with certain subsets of dual spaces of associated stability subgroups. These subsets in turn are in bijective correspondence with certain projective dual spaces. Under varying hypotheses, we give sufficient conditions for these bijections to be homeomorphisms, we determine the support of the induced representation $ U^L$ (for $ L \in \hat N$) and we give necessary and sufficient conditions for a union of fibres in $ \hat G$ to be closed.

In a much more general context we study the Hausdorff and CCR separation properties of the dual of an extension. We then completely describe the dual space topology of the above extension $ G$ in an interesting case.

The preceding results are then applied to the case where $ N$ is abelian and $ G/N$ is compact.

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Keywords: Locally compact group, dual space, group extension, regularly embedded, stability subgroup, projective representation, projective dual space, induced representation, amenable group, CCR representation, weak containment, hull-kernel topology, support of a representation, double coset, transformation group, orbit space
Article copyright: © Copyright 1970 American Mathematical Society

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