Is "the set of all unitary representations of a locally compact group $G$" a set? If not, what about the set of equivalence classes of unitary representations?
Note that the latter is different from the unitary dual since I do not restrict to irreducible representations.
Thanks in advance!
Given a Hilbert space $H$, there is a proper class of Hilbert spaces that are isomorphic to $H$. This means that the collection of all unitary representations of a locally compact group $G$ is not a set: given any representation $\theta : G \to U(H)$, you have a a representation $\theta' : G \to U(H')$ for any $H'$ isomorphic to $H$.
The equivalence classes aren't a set, either. But you can form a set with one representative from each equivalence class (by restricting your attention to Hilbert spaces with $|G|$ as a basis).