Dual spaces of groups with precompact conjugacy classes

Liukkonen, John R.

doi:10.1090/S0002-9947-1973-0318390-5

Dual spaces of groups with precompact conjugacy classes
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by John R. Liukkonen

Trans. Amer. Math. Soc. 180 (1973), 85-108

DOI: https://doi.org/10.1090/S0002-9947-1973-0318390-5

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Abstract:

We show that a second countable locally compact type I group with a compact invariant neighborhood of the identity is CCR, and has a Hausdorff dual if and only if its conjugacy classes are precompact. We obtain sharper results if the group is almost connected or has a fundamental system of invariant neighborhoods of the identity. Along the way we show that for a locally compact abelian group $A$ and a group $B$ of topological group automorphisms of $A, A$ has small $B$ invariant neighborhoods at 1 if and only if $\hat A$ has precompact orbits under the dual actions of $B$.

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On existence of compact open normal subgroups of

-dimensional groups

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