Minimum index for subfactors and entropy. II (Q1176088)

The author investigates in detail the indices and entropies in von Neumann algebras. Given von Neumann algebras \(M\supset N\) and a normal state \(\varphi\) on \(M\), the entropy \(H_ \varphi(M\mid N)\) of \(M\) relative to \(N\) and \(\varphi\) was introduced by \textit{A. Connes} in ``Entropie de Kolmogoroff-Sinai et mécanique statistique quantique'', C. R. Acad. Sci., Paris, Sér. I 301, 1-6 (1985; Zbl 0591.46058). In his earlier works, the author defined the entropies \(H_ E(M\mid N)\) and \(K_ E(M\mid N)\) for a given conditional expectation \(E\) from \(M\) onto \(N\); he also defined the minimum index \([M:N]_ 0\). In this paper, he continues to study the relationships among \(H_ E(M\mid N)\), \(K_ E(M\mid N)\) and \([M:N]_ 0\).