Preliminaries -- Modular Hilbert algebras -- Generalized Hilbert algebras -- The commutation theorem for modular Hilbert algebras -- Self-adjoint subalgebras of generalized Hilbert algebras -- The spectral algebra -- The modular operator ? -- The resolvent of the modular operator ? -- The one-parameter automorphisms defined by the modular operator ? -- Formulation of the modular Hilbert algebra -- Tensor product and direct sum of modular Hilbert algebras -- The standard representation of von Neumann algebras -- The modular automorphism group and the Kubo-Martin-Schwinger boundary condition -- Semi-finiteness and the modular automorphism group.