Note: A "representation" for me is a continuous linear action of a topological group on a complex topological vector space. A "subrepresentation" is a closed action-invariant linear subspace.

Question: Is there an irreducible representation of $U(1)$ which is not one dimensional? An example on a Banach space would be particularly nice.

Effort: Such representation cannot be a representation on a Hilbert space. Maybe we can take $1\leq p\leq\infty,p\neq 2$, consider the action on $L^p(S^1)$ and find some irreducible subrepresentation $V$ (the topology of our chosen $V$ must be a topology which cannot be induced from an inner product, otherwise this will not work).

  • $\begingroup$ Maybe relevant: math.stackexchange.com/questions/354926/… ? Maybe there is an analogous result for your general form of representation? $\endgroup$
    – user357980
    Jun 13, 2017 at 9:13
  • $\begingroup$ @user357980: The argument stated in your linked question seems to use Schur's lemma (what is referred to as a "previous exercise" there). I am not aware of a version of Schur's lemma outside the Hilbert space world (but that would be very interesting!). $\endgroup$
    – Jo Lasker
    Jun 13, 2017 at 9:22
  • $\begingroup$ @user357980: Actually, it seems that there are versions of Schur's lemma for Banach spaces, see "A primer on spectral theory / Aupetit", Thm 4.2.2 (so we probably need to look outside of the Banach space world as well). $\endgroup$
    – Jo Lasker
    Jun 13, 2017 at 9:29


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