I'm trying to solve the following exercise: prove that if $(\rho,V)$ is a unitary representation, then there's a linear isomorphism $\operatorname{End}_\mathbb{C}(\rho)\cong\rho^\ast\otimes\rho$ that induces an isomorphism of $G\times G$ representations.
I guess (it was not defined in the notes I'm reading) that $\operatorname{End}_\mathbb{C}(\rho)$ is the set of endomorphisms $V\to V$.
If the representation is finite-dimensional, the exercise if pretty easy (it basically follows from that fact that $\operatorname{Hom}(V,W)\cong V^\ast\otimes W$). But if I had to guess, I would've guessed it does not hold in the infinite-dimensional case.
The exercise, though, seems to ask about general representations (the context is Peter-Weyl theorem).
So my question is: does the result hold for infinite-dimensional representations? If so, what's the proof idea?
Thanks!
That isomorphism does not work for infinite dimensional V. In that case, the tensor product is isomorphic to the space of linear maps from V to V of finite rank.