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May 21, 2011 at 17:09 comment added Simon Thomas Rats! I was going to say that!
May 21, 2011 at 17:06 comment added Alain Valette @ Andreas: Correct, the assumption in the OP ("every unitary irrep is finite dimension") implies that the group is type I, so by Thoma's theorem the group is virtually abelian. Your sentence "You could ask...", ending with a (in)famous open question, is a bit tough (:-).
May 21, 2011 at 16:26 history answered Andreas Thom CC BY-SA 3.0