The unitary group in the strong topology and a construction of Dixmier-Douady

By a theorem of Dixmier-Douady the unitary group of an infinite-dimensional separable Hilbert space $H$ in the strong operator topology is contractible. The Dixmier-Douady proof is based on an explicit construction of families of subspaces and operators in $H$ with rather special properties. Unfortunately, this proof leaves hidden the geometric meaning of the theorem. The first goal of this note is to give a direct geometric proof of this theorem. The second goal is to provide a geometic analogue of Dixmier-Douady construction.