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Transactions of the American Mathematical Society

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



Dual spaces of groups with precompact conjugacy classes

Author: John R. Liukkonen
Journal: Trans. Amer. Math. Soc. 180 (1973), 85-108
MSC: Primary 22D05
MathSciNet review: 0318390
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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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Keywords: <!– MATH ${[FC]^ - }$ –> <IMG WIDTH="62" HEIGHT="43" ALIGN="MIDDLE" BORDER="0" SRC="images/img3.gif" ALT="${[FC]^ - }$"> group, <IMG WIDTH="45" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$[IN]$"> group, <IMG WIDTH="59" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$[SIN]$"> group, <IMG WIDTH="32" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img15.gif" ALT="$[Z]$"> group, Fell dual space, Hausdorff dual, type I, CCR
Article copyright: © Copyright 1973 American Mathematical Society