We introduce a Banach Lie group of unitary operators subject to a natural trace condition. We compute the homotopy groups of , describe its cohomology and construct an -central extension. We show that the central extension determines a non-trivial gerbe on the action Lie groupoid , where denotes the Hilbert space of self-adjoint Hilbert–Schmidt operators. With an eye towards constructing elements in twisted K-theory, we prove the existence of a cubic Dirac operator in a suitable completion of the quantum Weil algebra , which is subsequently extended to a projective family of self-adjoint operators on . While the kernel of is infinite-dimensional, we show that there is still a notion of finite reducibility at every point, which suggests a generalized definition of twisted K-theory for action Lie groupoids.
Cite this article
Pedram Hekmati, Jouko Mickelsson, Projective families of Dirac operators on a Banach Lie groupoid. J. Noncommut. Geom. 10 (2016), no. 1, pp. 1–28DOI 10.4171/JNCG/227