A \(C^*\)-dynamical entropy and applications to canonical endomorphisms (Q1572905)

A modification of Voiculescu's topological entropy \( ht(\alpha) \) is proposed as a \( C^* \)-dynamical entropy \( ht_\phi(\alpha) \) of an automorphism \( \alpha \) of an unital (nuclear) \( C^* \)-algebra \( A \), with respect to an \( \alpha \)-invariant state \( \phi \) of \( A \). The general relation \( h_\phi(\alpha)\leq ht_\phi(\alpha)\leq ht(\alpha) \) gives a bound for the Connes-Narnhofer-Thirring entropy \( h_\phi(\alpha) \). Relations with the mean entropy \( s(\phi) \) of \( \phi \) on AF \( C^* \)-algebra \( A \) are established, when \( s(\phi) \) exists; e.g., for AF \( C^* \)-dynamical systems in [\textit{F. Hiai} and \textit{D. Petz} Rev. Mod. Phys. 8, No. 6, 819-859 (1996; Zbl 0948.81023)] \( h_\phi(\alpha)=ht_\phi(\alpha)=s(\phi) \). The paper is mainly devoted to refined applications of these entropies in a wider context of \( \phi \)-preserving unital endomorphisms \( \rho \) of \( A \) having a positive left inverse \( \psi \). The (globally stable under \( \rho \)) graded \( C^*\)-algebra \( O_\rho\subset A \) and its grade 0-part \( O_\rho^{(o)} \) [see \textit{S. Doplicher} and \textit{R. Longo}, Invent. Math. 75, 493-536 (1984; Zbl 0539.46043)], are revised and employed to construct a \( \rho \)-invariant state \( \Phi \) of \({\mathcal O}_\rho^{(o)} \), induced by \( \psi \), and extend it to \({\mathcal O}_\rho \). Given that \( A'\cap A=\mathbb C\mathbf{1}\) and \( \dim(\rho^n(A)'\cap A)<\infty \), \( \forall n\in\mathbb N \), \( \rho \) is a standard AF endomorphism when \({\mathcal O}_\rho^{(o)}=A \). If \( va=\rho(a)v \), \( \forall a\in A \), for some isometry \( v\in A \) and \({\mathcal O}_\rho=A \), \( \rho \) is called a graded standard AF endomorphism [the suitable counterparts \( \rho \) over von Neumann factors are termed approximately finite dimensional (AFD), c.f. \textit{R. Longo} and \textit{J. E. Roberts}, \(K\)-theory 11, No. 2, 103-159 (1997; Zbl 0874.18005)]. Under some conditions the entropy of graded standard AF \( \rho \) is determined by \( \rho\mid{\mathcal O}_\rho^{(o)} \) and coincides with \( s(\phi) \). In particular, this is the case for the Cuntz's canonical inner endomorphism \( \Phi \) of the Cuntz \( C^* \)-algebra \({\mathcal O}_n \) (\( 2\leq n\leq\infty \)). Another important example is the Longo canonical endomorphism \( \gamma \) for ``graded standard AFD-inclusion'' \( N\subset M \) of infinite factors with finite index [see \textit{R. Longo} and \textit{J. E. Roberts}, loc. cit.]. Furthermore, if \( N'\cap M=\mathbb C\mathbf{1}\), the entropy can be expressed in terms of the Jones index for subfactors and the index for conditional expectation \( E_\gamma:M\to\gamma(M) \).