# Projective families of Dirac operators on a Banach Lie groupoid

### Pedram Hekmati

University of Adelaide, Australia### Jouko Mickelsson

University of Helsinki, Finland

## Abstract

We introduce a Banach Lie group $G$ of unitary operators subject to a natural trace condition. We compute the homotopy groups of $G$, describe its cohomology and construct an $S^1$-central extension. We show that the central extension determines a non-trivial gerbe on the action Lie groupoid $G \ltimes \frak k$, where $\frak k$ 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 $\mathbb D$ in a suitable completion of the quantum Weil algebra $\mathcal{U}(\frak{g}) \otimes Cl(\frak{k})$, which is subsequently extended to a projective family of self-adjoint operators $\mathbb D_A$ on $G \ltimes \frak k$. While the kernel of $\mathbb D_A$ 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. 1–28

DOI 10.4171/JNCG/227