Field algebras in quantum theory with indefinite metric. IV

Construction of basic elements of modular theory for $WJ^*$-algebras (weakly closed unital $J$-involutive operator algebras) in Pontryagin spaces is completed. Tomita's fundamental theorem is proved without the restrictions imposed in the part III. It is found that for a $WJ^*$-algebra satisfying the bicommutant theorem separating vectors are cyclic for its commutant. KMS boundary condition and the uniqueness of the modular group implementing this condition are proved.