Doesn't seem exactly a research level question to me, so posting here instead of MO.
I seem to know a proof of the (counter-intuitive, in my opinion) fact that a countable abelian group may not have an infinite-dimensional irreducible complex representation. (Although I was very reluctant to believe it at first.)
1) Can someone knowledgeable confirm that this is indeed true?
2) How does this agree with this passage (and, in general, the existence of counterexamples to the invariant subspace problem)? (I.e., is it true that some adjusted notion of a "representation" or "subrepresentation" is being used?)
Update. Since the link doesn't work for some people, here's a screenshot from said book (Harmonic Analysis: Proceedings of the International Symposium, held at the Centre Universitaire of Luxembourg, September 7-11, 1987). Hope I'm within the law here =)
In particular, I'm referring to the paragraph starting with "on the other hand, ..."