For the group $LG$ of smooth loops into a simple compact 1-connected Lie group $G$ there is a well-known universal central extension. My qustion is basically whether this extension also exists for the group $L_c G$ of continuous loops in $G$ (note that the standard argument for its non-existence does not apply, since I want an extension of topological groups and not of Lie groups).
The usual argument for its non-existence is that one shows that the Lie algebra $L_c \mathfrak{g}$ of continuous loops has no central extensions and then lifts this argument to the group level. However, if one asks for an extension of $L_c G$ as a topological group, then this argument does not apply. Moreover, the bundle underlying the universal extension of $LG$ also exists for $L_c G$, since this is simply the line bundle $P\to L_c G$ whose first Chern class is a generator in $H^2(LG,\mathbb{Z})\cong H^2(L_c G,\mathbb{C})\cong \mathbb{Z}$ (note that the inclusion $LG \to L_c G$ is a dense homotopy equivalence).
Now there are two conditions for this bundle to admit a compatible group structure. One (the triviality of $\pi_1^* P\otimes \mu^* P\otimes \pi_2^* P$) is homotopy invariant and thus also satisfied for $L\to L_c G$. This tells us that $L$ has a compatible $A_\infty$-structure. However, there is a second obstruction that forces the $A_\infty$-structure to really being a topological group structure. There are some analytic properties of $L_c G$ that would ensure the latter obstruction to vanish but so far I found none of them to hold (and I really expect them to fail in general).
Another possibility to get ones hands on this problem is to consider the (topological) basic gerbe over $G$ and then transgressing it to a line bundle over $L_c G$ (along with all the structure morphisms defining the multiplicative structure), but I don't know whether this also works for the continuous loops.
