A multiple time expectation φ(AB₁(t₁)…B_n(t_n)) in a stationary state φ satisfying the KMS boundary condition is studied. It is found to be holomorphic in a simplicial tube domain 0<Im t₁<Im t₂<…<Im t_n<β, continuous and bounded in the closure and the expectation of cyclic permutation of operators are obtained as its values on various distinguished boundaries of the domain.

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An invariant state satisfying the Kubo-Martin-Schwinger condition is studied. It is shown that the decomposition of a given state into extremal invariant states yields states satisfying the KMS boundary condition if and only if the cyclic representation associated with the given state is η-abelian, and that, if this is the case, the decomposition coincides with the standard central decomposition. The structure of the cyclic representation when it is non η-abelian is analyzed and typical examples are given. One of the examples gives a case where the cyclic representation is G-abelian but not η-abelian.

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Under the condition that a certain hermitian operator has a self-adjoint extension a necessary and sufficient condition that a bilinear Fermion Hamiltonian can be diagonalized by a Bogoliubov transformation is obtained. Under the same assumption, any bilinear Fermion Hamiltonian can be diagonalized in a slightly extended sense by an extended Bogoliubov transformation. The meaning of this diagonalization from the view point of the Clifford C^* algebra is discussed. It is shown that a parallel treatment is possible for a bilinear Boson Hamiltonian (with complications concerning unbounded operators and an indefinite metric) if a spectral theory of pseudo hermitian operator on a Hilbert space of an indefinite metric hold in parallel with that of definite metric Hilbert space.

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