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    QUANTUM STATISTICAL MECHANICS AND FELLER SEMIGROUP

    We review the mathematical formalism of the equilibrium quantum statistical mechanics of lattice models with an infinite degree of freedom. The equivalence of the KMS boundary condition, the Gibbs-Araki condition and the variational principle is established for a class of long range interactions. Our technical tool is one-parameter semigroups of completely positive maps on UHF C*-algebras, which we call Feller semigroups. Uniqueness of Gibbs and ground sates is re-examined from the viewpoint of unique ergodicity of Feller semigroups.