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?

irreducibleunitary 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