Topological Entropy for Appropriately Approximated C -algebras
Topological Entropy for Appropriately Approximated C -algebras
复制标题
适当近似C代数的拓扑熵
DOI：
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
A. Mossi
中科院分区：
文献类型：
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作者：
H. Treichel;S. Golunski;A. F. Camargo;T. Scapini;T. A. Modkovski;B. Venturin;E. Bordin;Vanusa Rossetto;A. Mossi
The \classical" topological entropy is one of the main numerical invariants in topological dynamics on compact spaces. Here, the author's recent development of a non{commutative generalization of topological entropy, in the natural setting of general C {algebras as the non{commutative counterpart of continuous function algebras on compact spaces, is presented in a slightly modiied and improved form. This includes both a survey of earlier results with some important corrections, and also new general results in response to (and inspired by) a more recent counter{ proposal for a non{commutative topological entropy by K. Thom-sen. Finally, some partially new examples for the calculation of the deened topological entropy are shown. The rather self{evident physical interpretation in the framework of (operator{algebraic) quantum statistical mechanics and of \chaotic" quantum dynami-cal systems is brieey touched upon. Supported by Fonds zur FF orderung der wissenschaftlichen Forschung in Osterreich as Erwin Schrr odinger Fellow (J0852-Phy). I TOPOLOGICAL ENTROPY FOR AA C-ALGEBRAS 1 I. Introduction The notion of topological entropy had been introduced in topolog-ical dynamics by Adler, Konheim and McAndrew 1] in 1965, rst by purely formal analogy with the Kolmogorov{Sinai (KS) entropy of measure{theoretic ergodic theory that had been created by the two named mathematicians about ten years earlier. Since then, however, on the one hand the topological entropy has become one of the main numerical invariants in topological dynamics, and until quite recently still, it has been more and more successfully applied to deterministically chaotic classical physical systems, see in particular 2]. On the other hand, the KS entropy (among other measure{theoretic entropy{like quantities) has been more and more successfully generalized to \quantum ergodic theory" in the framework of the operator{ algebraic approach to quantum statistical mechanics (cf. for example 3, 4]). We can already now refer the reader to at least two recent books on these quantum generalizations of measure{theoretic (dynam-ical) entropy: From the more mathematical point of view, the book by Petz and Ohya 5] is a complete introduction to the subject of quantum entropy theory, whereas the book by Benatti 6] concentrates on the quantum ergodic theory aspects, more from the point of view of mathematical physics. In particular, the work by Connes 7] and Connes, Narnhofer and Thirring 8] (CNT for short, and also subsequent work) reviewed in both books has been a breakthrough in the non{commutative generalization of the classical measure{theoretic KS entropy; see also 9] for a short overview …
The \classical" topological entropy is one of the main numerical invariants in topological dynamics on compact spaces. Here, the author's recent development of a non{commutative generalization of topological entropy, in the natural setting of general C {algebras as the non{commutative counterpart of continuous function algebras on compact spaces, is presented in a slightly modiied and improved form. This includes both a survey of earlier results with some important corrections, and also new general results in response to (and inspired by) a more recent counter{ proposal for a non{commutative topological entropy by K. Thom-sen. Finally, some partially new examples for the calculation of the deened topological entropy are shown. The rather self{evident physical interpretation in the framework of (operator{algebraic) quantum statistical mechanics and of \chaotic" quantum dynami-cal systems is brieey touched upon. Supported by Fonds zur FF orderung der wissenschaftlichen Forschung in Osterreich as Erwin Schrr odinger Fellow (J0852-Phy). I TOPOLOGICAL ENTROPY FOR AA C-ALGEBRAS 1 I. Introduction The notion of topological entropy had been introduced in topolog-ical dynamics by Adler, Konheim and McAndrew 1] in 1965, rst by purely formal analogy with the Kolmogorov{Sinai (KS) entropy of measure{theoretic ergodic theory that had been created by the two named mathematicians about ten years earlier. Since then, however, on the one hand the topological entropy has become one of the main numerical invariants in topological dynamics, and until quite recently still, it has been more and more successfully applied to deterministically chaotic classical physical systems, see in particular 2]. On the other hand, the KS entropy (among other measure{theoretic entropy{like quantities) has been more and more successfully generalized to \quantum ergodic theory" in the framework of the operator{ algebraic approach to quantum statistical mechanics (cf. for example 3, 4]). We can already now refer the reader to at least two recent books on these quantum generalizations of measure{theoretic (dynam-ical) entropy: From the more mathematical point of view, the book by Petz and Ohya 5] is a complete introduction to the subject of quantum entropy theory, whereas the book by Benatti 6] concentrates on the quantum ergodic theory aspects, more from the point of view of mathematical physics. In particular, the work by Connes 7] and Connes, Narnhofer and Thirring 8] (CNT for short, and also subsequent work) reviewed in both books has been a breakthrough in the non{commutative generalization of the classical measure{theoretic KS entropy; see also 9] for a short overview …