Field algebras in quantum theory with indefinite metric. IV

The construction of the fundamentals of Tomita-Takesaki modular theory for weakly closed unital J-involutive operator algebras (WJ* algebras) in Pontryagin spaces is completed. Tomita's fundamental theorem is proved without the restrictions imposed in Part III. It is established that the separating vectors of an algebra that satisfies the bicommutant theorem are cyclic for its commutant. A KMS boundary condition and uniqueness of the modular group as group realizing this condition are proved.