Relative modular operator in semifinite von Neumann algebras and its use

Abstract

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 Rényi's relative entropy known in finite dimension.

Quantum $f$-divergences via Nussbaum-Szkoła Distributions with applications to Petz-Rényi and von Neumann Relative Entropy

“…Recently, there have been attempts [17] to prove a similar formula as in (3.37) when dim K = ∞. In this section, we provide a counter example to show that (3.37) will not hold in general when K is infinite dimensional.…”

“…A specific example of such a state is obtained by taking ρ = i r i |u i u i | where {u i } is an orthonormal basis and In this section we discuss a counter example to Theorem 20 of [17] thus disproving that result. Consider ρ and σ as in Example 3.20.…”

“…In this section, we provide a quick exposition to the definition of the relative modular ∆ ρ,σ . The references for this exposition are [5,17]. Let (B 2 (K), •|• 2 ) denote the Hilbert space of Hilbert-Schmidt operators on K. Then define the closable antilinear operator S on the dense domain…”