Suppose I have a Hilbert space with a direct sum structure into "superselection sectors", i.e. $\mathcal{H} = \oplus_\alpha \mathcal{H}_\alpha$, where $\alpha$ labels irreps of some group $G$. I want know if there's some sort of decomposition of the space of endomorphisms $\textrm{End}(\mathcal{H})$ into something like the group homomorphisms given by the representation on the total Hilbert space $\pi(g): G \rightarrow GL(\mathcal{H})$ and homomorphisms between superselection sectors $\phi: \mathcal{H}_\alpha \rightarrow \mathcal{H}_\beta$. I'm pretty unfamiliar with even the basics of representation theory so sorry if this is a basic question!
A related question: since I'm dealing with unitary representations, are the endomorphisms automatically automorphisms? I'd guess so just because the group homomorphisms are unitary and therefore invertible, but that may be wrong...
