Maximality of entropy in finite von Neumann algebras (Q1127748)

Entropy for trace preserving automorphisms of finite von Neumann algebras was introduced in [\textit{A. Connes}, \textit{E. Størmer}, Acta Math. 134, 289-306 (1975; Zbl 0326.46032)]. Let \(H(N_1,\ldots, N_k)\) denote the entropy function on finite dimensional von Neumann subalgebras of a finite von Neumann algebra. The main result of the paper is the following theorem: Let \(R\) be a finite von Neumann algebra with a faithful normal trace \(\tau\) with \(\tau(1)=1\). Let \(N_1, \ldots, N_k\) be finite dimensional von Neumann subalgebras of \(R\) and let \(N=\bigvee_{l=1}^k N_l\). Then the following two conditions are equivalent: (i) \(H(N_1,\ldots,N_k)=H(N)\). (ii) There exists a maximal abelian subalgebra \(A\) in \(N\) such that \(A=\bigvee_{l=1}^k(A\cap N_l)\). In particular, if the above conditions are satisfied, then \(N\) is finite-dimensional and \(\text{rank} N\leq\prod_{l=1}^k \text{rank} N_l\).