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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.