On centralizers of generalized uniform subgroups of locally compact groups

Sit, Kwan-Yuk Law

doi:10.1090/S0002-9947-1975-0354923-2

On centralizers of generalized uniform subgroups of locally compact groups
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Trans. Amer. Math. Soc. 201 (1975), 133-146 Request permission

Abstract:

Let $G$ be a locally compact group and $H$ a closed subgroup of $G$ such that the homogeneous space $G/H$ admits a finite invariant measure. Let ${Z_G}(H)$ be the centralizer of $H$ in $G$. It is shown that if $G$ is connected then ${Z_G}(H)$ modulo its center is compact. If $G$ is only assumed to be locally connected it is shown that the commutator subgroup of ${Z_G}(H)$ has compact closure. Consequences of these results are found for special classes of groups, such as Lie groups. An example of a totally disconnected group $G$ is given to show that the results for ${Z_G}(H)$ need not hold if $G$ is not connected or locally connected.

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On bounded elements and centralizers of generalized uniform subgroups of locally compact groups

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