In the spacetime of a spherically-symmetric (Schwarzschild) black hole it is possible to construct a quantum state which represents the black hole in thermal equilibrium with its own radiation, the so-called Hartle-Hawking state, and also a quantum state which is empty at radial infinity, the so-called Boulware state. The Hartle-Hawking state is of further physical significance since it satisfies the symmetries of the spacetime and it is regular everywhere. It is known that equivalent states do not exist for bosons in the spacetime of a rotating (Kerr) black hole due to the property of superradiance. Fermions, however, do not exhibit classical superradiance, and so one may wonder whether a Hartle-Hawking-like state and a Boulware-like state can be constructed for fermions in Kerr. In this talk we will present work in progress that constructs candidates for such states for fermions and investigates their properties.