Relative modular operator in semifinite von Neumann algebras and its use

Andrzej {\L}uczak; Hanna Pods\k{e}dkowska; Rafa{\l} Wieczorek

Relative modular operator in semifinite von Neumann algebras and its use

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arxiv 1912.09633 v2 pith:SAD3FDHD submitted 2019-12-20 math.OA math-phmath.FAmath.MPmath.QAquant-ph

Relative modular operator in semifinite von Neumann algebras and its use

Andrzej {\L}uczak , Hanna Pods\k{e}dkowska , Rafa{\l} Wieczorek This is my paper

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We present some results concerning the relative modular operator in semifinite von Neumann algebras. These results allow one to prove some basic formula for trace, to obtain equivalence between Araki's relative entropy and Umegaki's information as well as to derive some formulae for quasi-entropies, and R\'enyi's relative entropy known in finite dimension.

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Integral representations of $f$-divergences for general von Neumann algebras
math.OA 2026-07 accept novelty 7.0

The f_0-divergence defined via Jordan decomposition integrals coincides with Araki's relative entropy on arbitrary von Neumann algebras, extending Frenkel's finite-dimensional formula.