The basic bundle gerbe on unitary groups

We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n) and Up(H), the Banach Lie group of unitaries differing from the identity by an element of a Schatten ideal. In all these cases we give an explicit connection and curving on the basic bundle gerbe and calculate the real Dixmier-Douady class. Extensive use is made of the holomorphic functional calculus for operators on a Hilbert space.