  The fermionic projector state is a distinguished quasi-free state for the
algebra of Dirac fields in a globally hyperbolic spacetime. We construct and
analyze it in the four-dimensional de Sitter spacetime, both in the closed and
in the flat slicing. In the latter case we show that the mass oscillation
properties do not hold due to boundary effects. This is taken into account in a
so-called mass decomposition. The involved fermionic signature operator defines
a fermionic projector state. In the case of a closed slicing, we construct the
fermionic signature operator and show that the ensuing state is maximally
symmetric and of Hadamard form, thus coinciding with the counterpart for
spinors of the Bunch-Davies state.
