Excuse the awkwardly long title.
Let $G$ be a simply connected semi-simple complex lie group. Then I know that every smooth finite dimensional projective representation of $G$, $G\to PGL(V)$, can be lifted to an honest representation $G\to SL(V)$.
However I would like to know whether this $\textit{always}$ holds: if $G\to PGL(V)$ is a projective representation of $G$ of any dimension (finite or infinite, and no further requirements on the map), can we always lift this to an honest representation $G\to GL(V)$?
I understand this question is somehow related to group cohomology but the precise relationship is eluding me. Any help is appreciated!