Comparison of dynamical entropies for the noncommutative shifts
Let \(M\) be a von Neumann algebra, \(\Theta\) an automorphism acting on \(M\), \(A_0\) subalgebra satisfying \(\Theta A_0\subset A_0\), and \(\omega\) a \(\Theta\)-invariant state over \(M\). The authors show the definitions of two dynamical entropies \(h^I (\omega, \Theta, A_0)\) and \(h^{II} (\Theta)\). They treat here an example of \(A_0\) given by \(w_I= e_{i1} e_{i2} \dots e_{in}\) with \(I= \{i_1< i_2< \dots< i_n\}\). Here the operators \(\{e_i\); \(i\in \mathbb{Z}\}\) satisfy \(e_i e_j= e_j e_i (-1)^{g (|i-j|)}\), \(g: \mathbb{N}\to \{0,1\}\), and the shift acts on \(A_0\) as \(\Theta e_i= e_{i+1}\). Next they think that \(\text{II}_1\) factor \({\mathfrak L} ({\mathfrak F}_\infty)\) from the free group \({\mathfrak F}_\infty\) is the example constructed by the similar way. They give the following results: For the first example, \(h^I (\omega, \Theta, A_0)= \ln 2\) independently of \(g\), \(h^{II} (\Theta)= \ln 2\) for \(g\equiv 0\), and \(h^{II} (\Theta)\leq \ln 2\) for \(g\neq 0\). For the second example, \(h^I (\omega, \Theta, A_0)= \infty\) and \(h^{II} (\Theta^k) =0\), \(k\geq 1\) hold.
scientific article; zbMATH DE number 807835