The spectrally passive, gauge-invariant, quasi-free stases ω on the C*-algebra of anticommutation relations with respect to a one-parameter quasi-free action τ are described. If the one-particle Hamiltonian H is discrete, the precise condition on the one-particle density Q of ω is combinatorial, but if the Connes spectrum of τ is non-zero, it implies that Q = (I+eβ (H+T))–1 for some β ≥ 0 and some operator T of bounded trace norm, apart from some degenerate possibilities. If H has both discrete and continuous parts, these results can be combined with those of de Cannière for the purely continuous case.
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Charles J.K. Batty, Passive Quasi-free States of the CAR Algebra with Discrete Hamiltonians. Publ. Res. Inst. Math. Sci. 22 (1986), no. 3, pp. 487–506DOI 10.2977/PRIMS/1195177846