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For which properties $(P)$ is the following statement known to be true?

In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the limit of a convergent sequence of $(P)$-elements has $(P)$.

For example, 'unimodular' (conjugating by the element preserves the Haar measure) is such a property.

I'm also interested in the analogous question just for the class of totally disconnected locally compact groups (e.g. 'uniscalar' is such a property for t.d.l.c. groups, but I don't think it has been defined for general locally compact groups).

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    $\begingroup$ I think it follows from a result of Trofimov that the topological FC-center, namely those $g$ whose conjugacy class has a compact closure, is closed if $G$ is in addition compactly generated; this is not true for general locally compact groups. $\endgroup$
    – YCor
    Jul 7, 2014 at 11:03
  • $\begingroup$ btw there's a natural extension of "uniscalar" to general locally compact (LC) groups, namely a LC-group $G$ is uniscalar if given its unit component $G_0$ and $W$ the maximal compact normal subgroup in $G_0$, then $G/G_0$ is uniscalar, and the action by conjugation of $G$ on the Lie algebra of $G_0/W$ is distal (=all eigenvalues have modulus 1). $\endgroup$
    – YCor
    Sep 6, 2014 at 16:10

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