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# The right regular representation of a compact right topological group

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http://dx.doi.org/10.4153/CMB-1998-060-2
Canad. Math. Bull. 41(1998), 463-472
Published:1998-12-01
Printed: Dec 1998
• Alan Moran
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## Abstract

We show that for certain compact right topological groups, $\overline{r(G)}$, the strong operator topology closure of the image of the right regular representation of $G$ in ${\cal L}({\cal H})$, where ${\cal H} = \L2$, is a compact topological group and introduce a class of representations, ${\cal R}$, which effectively transfers the representation theory of $\overline{r(G)}$ over to $G$. Amongst the groups for which this holds is the class of equicontinuous groups which have been studied by Ruppert in [10]. We use familiar examples to illustrate these features of the theory and to provide a counter-example. Finally we remark that every equicontinuous group which is at the same time a Borel group is in fact a topological group.

MSC Classifications:  22D99 - None of the above, but in this section
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