A commuting family of conditional expectations for Fermion C*-algebras and applications to equilibrium statistical mechnics.

Abstract: For a C*-algebra of a lattice system with a finite number of Fermions and spins on each lattice site, conditional expectations with respect to an even product state is introduced and the corresponding standard potential for any given even *-derivation of strictly local operators is defined, where the product property of the reference state is for mutually non-commutative subalgebras, with the tracial state and the Fermion Fock vacuum state as examples. The potentials of a given *-derivation relative to different product states are necessarily different but they are shown to give the same set of equilibrium state, where one can use for the characterization of equilibrium states either the variational principle (for translation invariant states) or the local thermodynamical stability or the Gibbs condition. Equivalence between different characterization of equilibrium states with the KMS condition for dynamics is shown under the minimal assumptions on dynamics.