TITLE: When does $BG \to BA$ loop to a homomorphism?
QUESTION [8 upvotes]: If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) homomorphism $G\to A$? This seems like it can't always be true, and also should be some kind of classical result, but I'm not sure where to look.

REPLY [13 votes]: If $G$ is a compact connected topological group and $A$ is a locally compact abelian topological group, then for any map $f:BG\to BA$ the looped map $\Omega f:\Omega BG\to \Omega BA$ is homotopically equivalent to a homomorphism $\phi:G\to A$. This follows from the main result of 
Scheffer, Wladimiro, Maps between topological groups that are homotopic to homomorphisms, Proc. Am. Math. Soc. 33, 562-567 (1972). ZBL0236.22008.
I'm not sure if this answers your question, however, which seems to be about when $f$ is in the image of the classifying space functor $B:\mathsf{TopGrp}\to\mathsf{Top}$.