# 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.