Field algebras in quantum theory with indefinite metric. III. Spectrum of modular operator and Tomita's fundamental theorem

Formulation of modular theory for weakly closed $J$-involutive algebras of bounded operators in Pontryagin spaces is continued. Spectrum of the modular operator of such an algebra is investigated in detail. The existence of strongly continuous $J$-unitary group $\Delta^{it}$, $t\in \mathbb{R}$, is established and Tomita's fundamental theorem is proved under the assumption that the spectrum of $\Delta$ belongs to the right half-plane.