I know not much is known about the left-regular representation of non-compact groups, but I was wondering the following: if we have a subquotient $$R\hookleftarrow M\twoheadrightarrow \mathbb{C}(\chi)$$ of $R$ the left regular representation of an infinite discrete group $G$ (or even locally compact, more generally), is it the case that the subquotient above is unique? i.e. that the multiplicity one for characters from the Peter-Weyl theorem holds in this generality? If not, is there a counterexample, and are there specific circumstances where it might be known?
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