Cohomology and Extensions
For a Hilbert space, the projective unitary group is the quotient of the unitary group by its center , the circle group
This is naturally a topological group. For of finite-dimension is also naturally a Lie group.
For an infinite-dimensional separable Hilbert space -principal bundles over a topological space are classidied by third integral cohomology of
By prop. 1 we have that the classifying space of itself is an Eilenberg-MacLane space
This is the classifying space for degree-3 integral cohomology (see Eilenberg-MacLane spectrum for more on this).
Every circle 2-bundle/bundle gerbe on is equivalent to the lifting gerbe of some -principal bundle to a -bundle, and the equivalence classes of these structures correspond uniquely.
The twisted bundles of a given bundle gerbe are given by the twisted cohomology relative to the morphism that is part of the long fiber sequence
Since the topological space underlying is contractible, on the underlying topological spaces this is
This means that the morphism that sends -bundles to the twist that they induce is an isomorphism.
(Somebody should force me to say this in more detail).
For more on this see also twisted K-theory.
Action on Fredholm operators
Let be an infinite-dimensional separable Hilbert space.
Since by the above and since there is a canonical action of line bundles on complex vector bundles, hence on the topological K-theory of a manifold , there must also be a natural action of of on the space of Fredholm operators (on connected components).
This is given by letting a projective unitary act by conjugation on a Fredholm operator: .
On the cohomology of its classifying space:
- Xing Gu, On the Cohomology of the Classifying Spaces of Projective Unitary Groups (arXiv:1612.00506)