I asked this question on math stack exchange but figured I might get a quicker answer here. I included a bit more information at the end. Thank you for your help!
As I understand it, for the Lie group of positive real numbers under multiplication, the exp map from its Lie algebra as a vector space of all real numbers to this group is a global diffeomorphism.
I am tempted to think that the same conclusion would hold for the infinite-dimensional Lie group (in the IHL or Frechet sense, if appropriate) of positive real-valued functions on a (compact) manifold under multiplication whose Lie algebra is the vector space of all real-valued functions. Assume proper Sobolev spaces are used, just for the peace of mind.
According to Khesin and Wendt's "Geometry of infinite-dimensional groups", this is a particular example of what they call "current group/algebra". In the general case, the exp map from the current algebra to its current group is not even surjective. However, is it surjective -- and hopefully even a global diffeomorphism -- for this particular example?
While it is a yes or no question, a little further explanation will certainly be appreciated.
