This is related to Theo's question about the abelianizations of finite dimensionsal Lie groups.

I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-dimensional Hilbert space and GL(H) represent the bounded two-sided invertible operators on H. Is there a nice description of the commutator subgroup G (the group generated be elements of the form ABA^{-1}B^{-1}) and the abelianization GL(H)/G?