Dynamical entropy for automorphisms of exact \(C^*\)-algebras.
Connes-Narnhofer-Thirring entropy \(h_\phi(\alpha)\) was defined for an automorphism \(\alpha\) of a \(C^*\)-algebra \(A\) and an \(\alpha\)-invariant state \(\phi\). The topological entropy \(ht(\alpha)\) was defined by Voiculescu for an automorphism \(\alpha\) of a nuclear \(C^*\)-algebra and extended by Brown to exact \(C^*\)-algebras. The author extends Voiculescu's topological entropy to a \(C^*\)-dynamical entropy \(ht_\phi(\alpha)\) for an automorphism \(\alpha\) of a nuclear \(C^*\)-algebra \(A\) with respect to an \(\alpha\)-invariant state \(\phi\). Assume that \(A\) is acting on a Hilbert space \(H\) and that an isometry \(s\) of \(H\), \(\sigma(a)= sas^*\), \((a\in A)\), \(A\rtimes_\sigma N\) is an crossed-product \(C^*\)-algebra. Let \[ \pi_\sigma(a)= \sum_{i< 0} e_{i,i}\otimes a+ \sum_{i\geq 0} e_{i,i}\otimes\sigma^i(a),\quad v= \sum_{i< 0} e_{i+1,i}\otimes I+ \sum_{i\geq 0} e_{i+1,i}\otimes s^i s^{*i}. \] Let \(\xi\in H\) be a unit vector, and \(\overline\xi= 0\oplus\xi\oplus 0\in \bigoplus_{i< 0} H_i\oplus H_0\bigoplus_{i> 0} H_i\). Assume \(\overline\xi\) is cyclic for \(A\rtimes_\sigma N\) for \(\xi\in H\), \(\phi= \omega|_A\), \(\overline\phi= \omega_{\overline\xi}|_{A\rtimes_\sigma N}\). It is proved that: Let \(\gamma\in \Aut(A\rtimes_\sigma N)\) satisfy that \(\gamma(\pi_\sigma(A))= \pi_\sigma(A)\) and \(\gamma(v)= v\). Let \(\gamma_A= \pi^{-1}_\sigma\circ \gamma\circ \pi_\sigma\). If \(A\) is exact, then \[ ht_{\overline\phi}(\gamma)= ht_\phi(\text{id}_A; \gamma_A). \] Moreover, if \(\xi\in H\) is cyclic for \(A\), then \(ht_{\overline\phi}(\gamma)= ht_\phi(\gamma_A)\).
scientific article; zbMATH DE number 1901778
scientific article