Geodesic
Field algebras in quantum theory with indefinite metric.~III. Spectrum of modular operator and Tomita's fundamental theorem
Teoretičeskaâ i matematičeskaâ fizika, Tome 70 (1987) no. 2, pp. 181-191

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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. 
@article{TMF_1987_70_2_a2,
     author = {K. Yu. Dadashyan and S. S. Horuzhy},
     title = {Field algebras in quantum theory with indefinite {metric.~III.} {Spectrum} of modular operator and {Tomita's} fundamental theorem},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {181--191},
     publisher = {mathdoc},
     volume = {70},
     number = {2},
     year = {1987},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1987_70_2_a2/}
}
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K. Yu. Dadashyan; S. S. Horuzhy. Field algebras in quantum theory with indefinite metric.~III. Spectrum of modular operator and Tomita's fundamental theorem. Teoretičeskaâ i matematičeskaâ fizika, Tome 70 (1987) no. 2, pp. 181-191. http://geodesic.mathdoc.fr/item/TMF_1987_70_2_a2/