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I have been thinking about existence of faithful representation of locally compact groups. This representation exists for example for compact lie groups. But I am curious to know if one can say some more general statement like:

Every locally compact group G admits a faithful unitary representation (not necessarily finite dimensional.)?

I guess answer should be yes, if one look at the regular representation of G. But I could not verify details. Do you think that statement is true?

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It's an easy exercise that $G$ acts faithfully on $L^2(G)$. –  Martin Brandenburg Nov 3 '10 at 23:41
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To add just a little detail to Martin's comment, if you accept the existence of left invariant Haar measure then it is a trivial fact that left translations preserve the $L^2$ (and in fact $L^p$) norms of continuous functions with compact support with respect to this measure, and since these are dense in $L^2$ you are done. –  Dick Palais Nov 3 '10 at 23:48
    
Ah, I misread (part of) the question as asking if there were sufficiently many irreducible unitary representations to separate points -- which is a different question! There, the answer is "yes" - this is the Gel'fand-Raikov theorem. –  Yemon Choi Nov 4 '10 at 3:02

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