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Ergebnisse der Mathematik Volume 50 und ihrer Grenzgebiete 3.Folge A Series of Modern Surveys in Mathematics EditorialBoard M.Gromov,Bures-sur-Yvette J.Jost,Leipzig J.Kollár,Princeton G.Laumon,Orsay H.W.Lenstra,Jr.,Leiden J.Tits,Paris D.B.Zagier,Bonn/Paris G.M.Ziegler,Berlin ManagingEditor R.Remmert,Münster Sergey Neshveyev Erling Størmer Dynamical Entropy in Operator Algebras 123 SergeyNeshveyev ErlingStørmer DepartmentofMathematics UniversityofOslo P.B.1053Blindern 0316Oslo,Norway e-mail:[email protected] [email protected] LibraryofCongressControlNumber:2006928835 MathematicsSubjectClassification(2000):46L55,28D20 ISSN0071-1136 ISBN-10 3-540-34670-8 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-34670-8 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerial isconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyright LawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtained fromSpringer.ViolationsareliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com ©Springer-VerlagBerlinHeidelberg2006 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantpro- tectivelawsandregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsusingaSpringerLATEXmacropackage Production:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Coverdesign:ErichKirchner,Heidelberg Printedonacid-freepaper 46/3100YL-543210 Preface When the algebraic formalism of quantum statistical mechanics and quantum field theory gained momentum in the 1960’s, it started a very fruitful interplay between mathematical physics and operator algebras. The study of automorphisms and their invariant states became a blooming discipline, and a subject of noncommutative ergodic theory evolved. With the great success of entropy in classical (abelian) ergodic theory it was natural to extend that theory to operator algebras. In some cases, like that of quantum spin lat- tice systems, that is rather straightforward, since in those models the mean entropy definition in the classical case can be extended to C∗-algebras by re- placingpartitionsbylocalalgebras.ButinmoregeneralcasestheC∗-algebras generatedbyfinitedimensionalC∗-algebrascaneasilybeinfinitedimensional, sothemeanentropycannotbeusedasadefinition.Inordertodefinedynam- icalentropyforautomorphismsofC∗-algebrasonehastorewritetheclassical definition in a form independent of the join of partitions and use that as the basis for a definition. This was done by Connes and Størmer in 1975 for finite von Neumann algebras, and a useful definition was accomplished, giving in particular the expected entropy for noncommutative Bernoulli shifts [51]. The theory evolved slowly; it took 10 years before Connes extended the definition to normal states of von Neumann algebras [49], and a little later he andNarnhoferandThirring[50]extendedthetheorytostatesofC∗-algebras. Several other attempts have been made to define dynamical entropy for C∗- algebras, see Notes to Chaps. 3 and 6, the most useful of which being those of Voiculescu [227]. His idea was to consider finite dimensional C∗-algebras which approximately contain finite sets of operators instead of the algebras they generate. In particular he obtained a definition of topological entropy which is an extension of topological entropy in the classical case. In the present book we shall develop the basic theory for the dynamical and topological entropies alluded to above. Then we shall discuss the special situations which have attracted most attention. We start with a chapter on the classical case, mainly for motivation and background. Then we develop in Chap. 2 the necessary theory of relative entropy for states, which is in- VI Preface dispensable for noncommutative entropy. In Chap. 3 we give the definition of dynamicalentropyandshowitsmainproperties,andfollowthisupinChap.5 withadefinition,duetoSauvageotandThouvenot[188],inspiredbytheclas- sical concept of joinings. Topological entropy is treated in Chap. 6. The rest of the contents of the book depends heavily on the above chapters, while the other chapters are more loosely connected. The book is divided into two parts;thefirstcontainschaptersofgeneralnature,whileweinthesecondpart consider dynamical systems in more special settings. At this stage it should also be remarked that parts of the theory have been treated in the books by Benatti [13], Ohya and Petz [147] and the survey article by Størmer [211]. We are grateful to our colleagues N. Brown, M. Choda, D. Kerr, A. Oc- neanu, and Y. Ueda for useful comments concerning the preparations of the manuscript. S. Neshveyev, E. Størmer Contents Part I General Theory 1 Classical Dynamical Systems .............................. 3 1.1 Measure Entropy ........................................ 3 1.2 Topological Entropy ..................................... 9 1.3 Entropy via Partitions of Unity ........................... 11 1.4 Notes .................................................. 14 2 Relative Entropy .......................................... 15 2.1 Relative Entropy for Matrix Algebras ...................... 15 2.2 Von Neumann Entropy................................... 21 2.3 Relative Entropy for General C*-algebras................... 26 2.4 Notes .................................................. 30 3 Dynamical Entropy........................................ 33 3.1 Mutual Entropy......................................... 33 3.2 Entropy of Dynamical Systems............................ 48 3.3 Type I Algebras......................................... 54 3.4 Notes .................................................. 57 4 Maximality of Entropy and Commutativity ................ 61 4.1 Maximal Entropy of Subalgebras .......................... 61 4.2 Independent Algebras.................................... 66 4.3 Entropic K-systems...................................... 69 4.4 Notes .................................................. 74 5 Dynamical Abelian Models ................................ 77 5.1 Entropy via Stationary Couplings ......................... 77 5.2 Zero Entropy Systems.................................... 86 5.3 Notes .................................................. 91 VIII Contents 6 Topological Entropy ....................................... 93 6.1 Rank of a Completely Positive Approximation .............. 93 6.2 Topological Entropy ..................................... 96 6.3 Notes ..................................................103 7 Dynamics on the State Space ..............................107 7.1 Measure Entropy ........................................107 7.2 Topological Entropy .....................................110 7.3 Notes ..................................................119 8 Crossed Products..........................................121 8.1 Crossed Products by Discrete Amenable Groups.............121 8.2 Generalizations..........................................125 8.3 Notes ..................................................130 9 Variational Principle.......................................133 9.1 Pressure................................................133 9.2 The variational principle .................................138 9.3 KMS-states.............................................144 9.4 Notes ..................................................151 Part II Special Topics 10 Relative Entropy and Subfactors...........................157 10.1 Relative Entropy ........................................157 10.2 Index of Subfactors ......................................164 10.3 Generators and Relative Entropy ..........................167 10.4 The Canonical Shift .....................................171 10.5 Shifts on Temperley-Lieb Algebras.........................180 10.6 Notes ..................................................183 11 Systems of Algebraic Origin ...............................185 11.1 Twisted Group C*-algebras...............................185 11.2 Estimates of Topological Entropy..........................187 11.3 K-systems ..............................................194 11.4 Zero Entropy Systems....................................201 11.5 Automorphisms of Noncommutative Tori ...................206 11.6 Notes ..................................................209 12 Binary Shifts ..............................................211 12.1 The C*-algebra of a Bitstream ............................211 12.2 Entropy of Binary Shifts .................................217 12.3 Notes ..................................................224 Contents IX 13 Bogoliubov Automorphisms ...............................227 13.1 Canonical Anticommutation Relations .....................227 13.2 Topological Entropy .....................................229 13.3 Classical Bernoullian Subsystems ..........................233 13.4 Dynamical Entropy ......................................238 13.5 Notes ..................................................248 14 Free Products .............................................251 14.1 Free Products of Algebras and Maps .......................251 14.2 Free Shifts..............................................254 14.3 Free Product Automorphisms .............................257 14.4 Notes ..................................................263 A Completely Positive Maps .................................265 B Operator Inequalities ......................................271 C Direct Integrals............................................275 References.....................................................279 List of Symbols ................................................291 Index..........................................................295 Part I General Theory 1 Classical Dynamical Systems Inthischapterweincludesomebackgroundmaterialonentropyintheclassi- cal commutative case. We essentially include all the results from the classical theory which will be needed later, proving those which will be used repeat- edlyandleavingmoredifficultandrarelyusedoneswithoutproof.Inthelast section we show that the classical definition of entropy can be reformulated to a form which has a natural noncommutative extension and which will be the basis for the general theory later on. 1.1 Measure Entropy Let (X,B,µ) be a Lebesgue space, that is, after removing a subset of measure zero X can be given the topology of a complete separable metric space such that µ is a regular probability measure on X, and B is the algebra of measurable subsets of X. We shall usually suppress B in the notation. Let ξ ={X ,...,X } be a finite measurable partition of X. The entropy of ξ is 1 n H(ξ)=η(µ(X ))+...+η(µ(X )), 1 n where η is the function defined by (cid:1) −tlogt, if t>0, η(t)= 0, if t=0, the logarithm being considered with base e(1). We shall write H (ξ) instead µ of H(ξ) when we want to stress that our measure is µ. More generally, assume we also have a measurable partition ζ = {Yi}i∈I of X. If I is uncountable, this means by definition that up to a set of measure zero the points of the quotient space X/ζ are separated by a countable 1In the probabilistic literature the logarithm with base 2 is often used.