I'm working on a problem with unitary representations. Basically the irreducible representations of a group $G$ have been shown to be of the form of $\chi(z) \otimes \beta$ where $\chi(z)$ maps the elements of the group to a complex number $z$ and $\beta$ is a irreducible representation of $G$. I've shown that this is unitary if and only if the $\beta(g) = U$ have the property that $\langle Uv | Uw \rangle_A = c\langle v | w \rangle_A$ for some Hermitian inner product $\langle \cdot | \cdot \rangle_A$ which I have taken to calling near unitarizable.
It would be very helpful if near unitarizable implied unitarizable (ie. $\langle Uv | Uw \rangle_B = \langle v | w \rangle_B$ for some inner product). I have been trying to find either a proof or a counter example with no luck. If you have any imput on either it would be very helpful.