Local automorphisms of the unitary group and the general linear group on   a Hilbert space
  We prove that every 2-local automorphism of the unitary group or the general linear group on a complex infinite-dimensional separable Hilbert space is an automorphism. Thus these types of transformations are completely determined by their local actions on the two-points subsets of the groups in question. 
Lajos Molnar and Peter Semrl