Within the setting of infinite-dimensional self-dual CAR C* algebras describing fermions in the Zd lattice, we depart from the well-known Araki–Evans σ(P1,P2)Z2 index for quasi-free fermion states and rewrite it in terms of states rather than in terms of basis projections. Furthermore, we reformulate results that relate equivalences of Fock representations to the index parity into results that relate equivalences of Gel’fand–Naimark–Segal representations and the associated index parity.