Dynamical entropy in operator algebras

293 Pages·2006·1.892 MB·English

by  Sergey Neshveyev, Erling Størmer

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 quan-
tum 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 fi-
nite 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 def-
inition 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 mea-
sure 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 mea-
sure 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.