Entropy of endomorphisms and relative entropy in finite von Neumann algebras (Q1971929)

If \(T\) is a measure-preserving nonsingular transformation on a probability space and \(\mathcal P\) is a generator then the entropy of \(T\) is given by the relative entropy \(H(T)=H(\bigvee_0^\infty T^{-i}\mathcal P \mid \bigvee_1^\infty T^{-i}\mathcal P)\). The author proves an analogue of this result for automorphisms of finite von Neumann algebras. Suppose \(R\) is a finite von Neumann algebra with a faithful normal tracial state \(\tau\) and \(\alpha\) is a \(\tau\)-invariant endomorphism of \(R\). The entropy of \(\alpha\), \(H(\alpha)\), was defined by the author and \textit{A. Connes} in [Acta Math. 134, 289-306 (1975; Zbl 0326.46032)] via finite dimensional subalgebras of \(R\) and relative entropy of a subalgebra \(P\) of \(R\), \(H(R\mid P)\) is defined in terms of \(\tau\)-invariant conditional expectations (the infinite dimensional case was developed by \textit{M. Pimsner} and \textit{S. Popa} in [Ann. Sci. Éc. Norm. Supér, IV. Sér. 19, 57-106 (1986; Zbl 0646.46057)]). Call \((A_n)_{n\in {\mathbb N}}\) a generating sequence for \(\alpha\) satisfying the commuting square condition if it is an increasing sequence of finite dimensional von Neumann subalgebras of \(R\) satisfying \(R=\overline{\bigcup_n A_n}\), \(\alpha(A_n)\subset A_{n+1}\), \(H(\alpha)=\lim_{n\to\infty} 1/n H(A_n)\), and \(E_{\alpha(A_n)}=E_{\alpha(A_{n+1})}\circ E_{A_{n+1}}\) for all \(n\in {\mathbb N}\), where \(E_P\) is the \(\tau\)-invariant conditional expectation of \(R\) onto \(P\). The main result of the paper is that if \(H(\alpha)<\infty\) and \((A_n)_{n\in {\mathbb N}}\) a generating sequence for \(\alpha\) satisfying the commuting square condition then \(\lim_{n\to\infty}1/n H(Z(A_n))\) exists and \(H(\alpha)=1/2H(R\mid \alpha(R))+ 1/2\lim_{n\to\infty}1/n H(Z(A_n))\) where \(Z(P)\) denotes the centre of \(P\). Furthermore, if \(R\) is of type \(I\) then \(H(\alpha)=H(R\mid \alpha(R))\).