Topology and the duals of certain locally compact groups

Schochetman, I.

doi:10.1090/S0002-9947-1970-0265513-X

Topology and the duals of certain locally compact groups
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Trans. Amer. Math. Soc. 150 (1970), 477-489 Request permission

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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