A \(C^*\)-dynamical entropy and applications to canonical endomorphisms
type: Item
instance of: scholarly article
title: A \(C^*\)-dynamical entropy and applications to canonical endomorphisms
zbMATH Open document ID: 0962.46048
author: Marie Choda
published in: Journal of Functional Analysis
publication date: 2000-11-13T00:00:00Z
review text: 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 \…
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L37
zbMATH DE Number: 1484768
zbMATH Keywords: \( C^* \)-dynamical entropy
zbMATH Keywords: endomorphism
zbMATH Keywords: index for subfactors
zbMATH Keywords: standard inclusion
zbMATH Keywords: graded algebra
zbMATH Keywords: Voiculescu's topological entropy
zbMATH Keywords: Connes-Narnhofer-Thirring entropy
zbMATH Keywords: mean entropy
zbMATH Keywords: AF \(C^*\)-dynamical systems
zbMATH Keywords: graded standard AF endomorphism
zbMATH Keywords: Cuntz \(C^*\)-algebra
zbMATH Keywords: graded standard AFD-inclusion
zbMATH Keywords: Jones index for subfactors
zbMATH Keywords: index for conditional expectation
reviewed by: Andrey V. Bulinski
MaRDI profile type: Publication
full work available at URL: jfan.2000.3564
OpenAlex ID: W2059240658
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DOI: 10.1006/JFAN.2000.3564
description: scientific article; zbMATH DE number 1484768