Geodesic
Spectral properties of modular operators and the asymptotic ratio set
Izvestiya. Mathematics , Tome 9 (1975) no. 3, pp. 599-619.

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In the paper we study the asymptotic ratio set (an invariant for von Neumann algebras introduced by H. Araki and E. J. Woods) with the aid of spectral properties of modular operators of the von Neumann algebra. We give an equivalent description of this set in terms of modular operators and indicate a constructive method for its evaluation. Bibliography: 19 items. 
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V. Ya. Golodets. Spectral properties of modular operators and the asymptotic ratio set. Izvestiya. Mathematics , Tome 9 (1975) no. 3, pp. 599-619. http://geodesic.mathdoc.fr/item/IM2_1975_9_3_a7/

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