Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge
A Series of Modern Surveys in Mathematics
Editorial Board M...

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Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge

A Series of Modern Surveys in Mathematics

Editorial Board 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 Managing Editor R. Remmert, Münster

Volume 50

Sergey Neshveyev Erling Størmer

Dynamical Entropy in Operator Algebras

123

Sergey Neshveyev Erling Størmer Department of Mathematics University of Oslo P. B. 1053 Blindern 0316 Oslo, Norway e-mail: [email protected] [email protected]

Library of Congress Control Number: 2006928835

Mathematics Subject Classiﬁcation (2000): 46L55, 28D20

ISSN 0071-1136 ISBN-10 3-540-34670-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-34670-8 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors using a Springer LATEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

46/3100YL - 5 4 3 2 1 0

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 lattice systems, that is rather straightforward, since in those models the mean entropy deﬁnition in the classical case can be extended to C∗ -algebras by replacing partitions by local algebras. But in more general cases the C∗ -algebras generated by ﬁnite dimensional C∗ -algebras can easily be inﬁnite dimensional, so the mean entropy cannot be used as a deﬁnition. In order to deﬁne dynamical entropy for automorphisms of C∗ -algebras one has to rewrite the classical 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 and Narnhofer and Thirring [50] extended the theory to states of C∗ -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 dynamical entropy and show its main properties, and follow this up in Chap. 5 with a deﬁnition, due to Sauvageot and Thouvenot [188], inspired by the classical 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; the ﬁrst contains chapters of general nature, while we in the second part 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. Ocneanu, 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Relative Entropy for Matrix Algebras . . . . . . . . . . . . . . . . . . . . . . 2.2 Von Neumann Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Relative Entropy for General C*-algebras . . . . . . . . . . . . . . . . . . . 2.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 21 26 30

3

Dynamical Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mutual Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Entropy of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Type I Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 48 54 57

4

Maximality of Entropy and Commutativity . . . . . . . . . . . . . . . . 4.1 Maximal Entropy of Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Independent Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Entropic K-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 66 69 74

5

Dynamical Abelian Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Entropy via Stationary Couplings . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Zero Entropy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 86 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

In this chapter we include some background material on entropy in the classical commutative case. We essentially include all the results from the classical theory which will be needed later, proving those which will be used repeatedly and leaving more diﬃcult and rarely used ones without proof. In the last 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 = {X1 , . . . , Xn } be a ﬁnite measurable partition of X. The entropy of ξ is H(ξ) = η(µ(X1 )) + . . . + η(µ(Xn )), where η is the function deﬁned by −t log t, if t > 0, η(t) = 0, if t = 0, the logarithm being considered with base e(1) . We shall write Hµ (ξ) instead 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 1

In the probabilistic literature the logarithm with base 2 is often used.

4

1 Classical Dynamical Systems

number of measurable sets. Then X/ζ is again a Lebesgue space. The algebra L∞ (X/ζ, µ) can be identiﬁed with the von Neumann subalgebra of L∞ (X, µ) consisting of functions measurable with respect to a complete σalgebra Bζ . This way we get one-to-one correspondences between measurable partitions (considered up to a set of measure zero), von Neumann subalgebras of L∞ (X, µ) and complete σ-subalgebras of B. Let Eζ : L∞ (X) → L∞ (X/ζ) be the µ-preserving conditional expectation, that is, Eζ is determined by the condition X Eζ (f )g dµ = X f g dµ for f ∈ L∞ (X) and g ∈ L∞ (X/ζ). The conditional entropy is deﬁned as H(ξ|ζ) =

n k=1

X

η(Eζ (1Xk )(x))dµ(x) = −

n k=1

Xk

log Eζ (1Xk )(x)dµ(x),

where 1Y denotes the characteristic function of a subset Y ⊂ X. We shall often write Hµ (ξ|ζ) instead of H(ξ|ζ). If ζ = {Yl }m l=1 then Eζ (f ) =

m µ(f 1Yl ) 1Yl , µ(Yl ) l=1

where µ(g) =

X

g dµ for g ∈ L1 (X, µ), and therefore µ(Xk ∩ Yl ) µ(Yl ). η H(ξ|ζ) = µ(Yl ) k,l

In particular, if ν = {X} is the trivial partition then H(ξ|ν) = H(ξ). For measurable partitions ξ and ζ we write ξ ≺ ζ if L∞ (X/ξ) ⊂ L∞ (X/ζ). If ξ = {Xk }nk=1 and ζ = {Yl }m l=1 this means that up to a set of measure zero each Xk is the union of some of the Yl ’s. If {ξi }i∈I is a collection of measurable partitions then we denote by ∨i∈I ξi the measurable partition (deﬁned up to a set of measure zero) corresponding to the von Neumann algebra generated by L∞ (X/ξi ), i ∈ I. If ξ = {Xk }nk=1 and ζ = {Yl }m l=1 then the partition ξ ∨ ζ consists of the sets Xk ∩ Yl . Proposition 1.1.1. Conditional entropy has the following properties: (i) H(ξ|ζ) ≥ 0, with equality if and only if ξ ≺ ζ; (ii) H(ξ ∨ ϑ|ζ) = H(ξ|ζ) + H(ϑ|ξ ∨ ζ); (iii) H(ξ|ζ) is increasing in ξ and decreasing in ζ. Proof. It is clear that H(ξ|ζ) ≥ 0. Assume H(ξ|ζ) = 0. Let ξ = {Xk }nk=1 . Since the function η takes the value 0 only at 0 and 1, it follows that Eζ (1Xk ) = 1Yk for some Yk ⊂ X. We have µ(Yk ) = µ(Eζ (1Xk )) = µ(Xk ), and

1.1 Measure Entropy

5

µ(Yk ∩ Xk ) = µ(1Yk 1Xk ) = µ(1Yk Eζ (1Xk )) = µ(Yk ). Hence 1Xk = 1Yk ∈ L∞ (X/ζ), so that ξ ≺ ζ. Conversely, if ξ ≺ ζ then Eζ (1Xk ) = 1Xk , so that η(Eζ (1Xk )) = 0 and hence H(ξ|ζ) = 0. Thus (i) is proved. To prove (ii) assume ξ = {Xk }nk=1 . One can easily check that for any f ∈ L∞ (X) we have Eξ∨ζ (f ) =

n Eζ (f 1Xk ) 1Xk . Eζ (1Xk )

k=1

Hence log Eξ∨ζ (f ) =

n

(log Eζ (f 1Xk ) − log Eζ (1Xk ))1Xk ,

k=1

whence for ϑ = {Yl }m l=1 we get 1Yl log Eξ∨ζ (1Yl ) = 1Xk ∩Yl log Eζ (1Xk ∩Yl ) − 1Xk log Eζ (1Xk ). l

k,l

k

Integrating over X we get (ii). To show (iii) ﬁrst note that if ξ ≺ ϑ then ξ ∨ ϑ = ϑ, so it follows from (ii) that H(ξ|ζ) ≤ H(ϑ|ζ). Assume now that ζ ≺ ϑ. Then Eζ = Eζ ◦ Eϑ . Since the function η is concave, by Jensen’s inequality for any measurable Y ⊂ X we get Eζ (η(Eϑ (1Y ))) ≤ η((Eζ ◦ Eϑ )(1Y )) = η(Eζ (1Y )), whence

η(Eϑ (1Y ))dµ ≤

X

η(Eζ (1Y ))dµ. X

This implies that H(ξ|ϑ) ≤ H(ξ|ζ).

From parts (ii) and (iii) in the above proposition we see that if ξ, ϑ, ζ and ε are measurable partitions then H(ξ ∨ ϑ|ζ ∨ ε) ≤ H(ξ|ζ) + H(ϑ|ε).

(1.1)

In particular, entropy is subadditive, i.e., H(ξ ∨ ϑ) ≤ H(ξ) + H(ϑ). Let now T be a measure-preserving invertible transformation of a Lebesgue space (X, µ). In such a case we say that (X, µ, T ) is a dynamical system. It will sometimes be convenient to assume that T is a homeomorphism of a compact metric space. More precisely, if (X, µ, T ) is a dynamical system then by a topological model for it we mean a homeomorphism S of a compact metric space Y together with an S-invariant probability measure ν such that (X, µ, T ) and (Y, ν, S) are isomorphic. Topological models always exist: take

6

1 Classical Dynamical Systems

the spectrum of any weakly operator dense invariant separable C∗ -subalgebra of L∞ (X, µ) for Y . Let (X, µ, T ) be a dynamical system. For a ﬁnite measurable partition ξ = {X1 , . . . , Xm } set 1 H(ξ ∨ T ξ ∨ . . . ∨ T n−1 ξ), n→∞ n

h(ξ; T ) = lim

where T ξ = {T X1 , . . . , T Xm }. Note that the limit exists by virtue of subadditivity of entropy and the following standard result. Lemma 1.1.2. Let {an }∞ n=1 be a sequence of real numbers such that an+m ≤ an + am for any n, m ∈ N. Then the limit of the sequence {an /n}n exists and coincides with the inﬁmum. Proof. Fix m ∈ N. Any n ∈ N can be written as n = km + l for some k and l, 0 ≤ l < m. By assumption an ≤ kam + al . It follows that

an am ≤ . m n→∞ n Since this is true for every m, we get the result. lim sup

In fact with a bit more work one can show that monotonically decreasing. Indeed, we have

k ∞ {H(∨n−1 k=0 T ξ)/n}n=1

is

n−1 k k H(∨n−1 k=0 T ξ) = H(ξ) + H(∨k=1 T ξ|ξ) k −1 = H(ξ) + H(∨n−2 ξ) k=0 T ξ|T n−1

= ... =

l H(ξ| ∨−1 l=−k T ξ).

(1.2)

k=0 l Since the sequence {H(ξ| ∨−1 l=−k T ξ)}k is monotonically decreasing by Propok sition 1.1.1(iii), the sequence {H(∨n−1 k=0 T ξ)/n}n is also monotonically decreasing. Furthermore, by the martingale convergence theorem for any f ∈ L1 (X, µ) we have E∨−1 T k ξ (f ) → E∨−1 T k ξ (f ) a.e. So denoting the partik=−n

k − we get tion ∨−1 k=−∞ T ξ by ξ

k=−∞

k − h(ξ; T ) = lim H(ξ| ∨−1 k=−n T ξ) = H(ξ|ξ ). n→∞

(1.3)

The entropy of T is now deﬁned as h(T ) = sup h(ξ; T ), ξ

where the supremum is taken over all ﬁnite measurable partitions of (X, µ). We shall write hµ (ξ; T ) and hµ (T ) instead of h(ξ; T ) and h(T ) when we want to stress that we deal with measure µ. We need a more practical way of computing entropy than taking supremum over all ﬁnite partitions.

1.1 Measure Entropy

7

Lemma 1.1.3. Let ξ and ζ be ﬁnite measurable partitions. Denote by ζ ± the partition ∨k∈Z T k ξ. Then h(ξ; T ) ≤ h(ζ; T ) + H(ξ|ζ ± ). Proof. We have n−1 k n−1 k n−1 k n−1 k k H(∨n−1 k=0 T ξ) ≤ H(∨k=0 T (ξ ∨ ζ)) = H(∨k=0 T ζ) + H(∨k=0 T ξ| ∨k=0 T ζ)

and by (1.1) n−1 k k H(∨n−1 k=0 T ξ| ∨k=0 T ζ) ≤

n−1

H(T k ξ|T k ζ) = nH(ξ|ζ).

k=0

Therefore h(ξ; T ) ≤ h(ζ; T ) + H(ξ|ζ). Since also

h(∨nk=−n T k ζ; T )

= h(ζ; T ) for any n, we then get

h(ξ; T ) ≤ h(ζ; T ) + H(ξ| ∨nk=−n T k ζ). Letting n → ∞ we obtain the result.

±

A measurable partition ξ is called generating if ξ is the partition of X into points. From the above lemma we then get the following result. Theorem 1.1.4. (Kolmogorov-Sinai) Let ξ be a ﬁnite generating partition for a dynamical system (X, µ, T ). Then h(T ) = h(ξ; T ).

The following example was motivating for the whole theory. Example 1.1.5. Let ν be a probability measure on {1, . . . , n}, ν({k}) = λk . Let X = {1, . . . , n}Z , µ the product measure on X and T the shift to the right. The system (X, µ, T ) is called the Bernoulli shift with weights λ1 , . . . , λn . Consider the partition ξ = {X1 , . . . , Xn }, where Xk is the cylindrical set consisting of x = {xn }n∈Z ∈ X such that x0 = k. Then ξ is generating and we get h(T ) = h(ξ; T ) = k η(λk ). The following two theorems are fundamental results on Bernoulli shifts. Theorem 1.1.6. (Ornstein) Bernoulli shifts with the same entropy are isomorphic.

Recall that a system (X, µ, T ) is called ergodic if any T -invariant measurable subset of X has measure either zero or one. If ξ is a T -invariant measurable partition then the system (X/ξ, µ, T ) is called a factor system of (X, µ, T ).

8

1 Classical Dynamical Systems

Theorem 1.1.7. (Sinai) Let (X, µ, T ) be an ergodic dynamical system with entropy h(T ) > 0. Then for any ﬁnite h, 0 < h ≤ h(T ), there exists a factor system which is isomorphic to the Bernoulli shift with entropy h.

Let us now list the basic properties of entropy. Theorem 1.1.8. For any dynamical system (X, µ, T ) we have: (i) h(T n ) = |n|h(T ) for n ∈ Z; (ii) if (Y, ν, S) is a factor system then h(T ) ≥ h(S); (iii) if Y ⊂ X is T -invariant then h(T ) = µ(Y )h(T |Y ) + µ(Y c )h(T |Y c ), where we consider the measures µ(Y )−1 µ|Y and µ(Y c )−1 µ|Y c on Y and Y c = X\Y , respectively; (iv) if (Y, ν, S) is another dynamical system then h(T × S) = h(T ) + h(S).

We shall next show properties of H(ξ) which will be useful later. Proposition 1.1.9. For ﬁnite measurable partitions we have: (i) if ξ = {X1 , . . . , Xn } then H(ξ) ≤ log n, and equality holds if and only if µ(Xk ) = 1/n for all k; (ii) H(ξ ∨ ζ ∨ ϑ) + H(ζ) ≤ H(ξ ∨ ζ) + H(ζ ∨ ϑ); (iii) H(ξ ∨ ζ) ≤ H(ξ) + H(ζ), and equality holds if and only if ξ and ζ are independent, that is, µ(f g) = µ(f )µ(g) for f ∈ L∞ (X/ξ) and g ∈ L∞ (X/ζ); n n n (iv) k=1 H(ξk ) − H(∨nk=1 ξk ) ≤ k=1 H(ζk ) − H(∨k=1 ζk ) if ξk ≺ ζk for k = 1, . . . , n. Proof. Since η is strictly concave,

η(µ(Xk )) ≤ nη

µ(Xk )

k

k

n

= log n,

moreover, equality holds if and only if all µ(Xk )’s are equal, that is, µ(Xk ) = 1/n. Thus (i) is proved. We have by Proposition 1.1.1(ii) H(ξ ∨ ζ ∨ ϑ) − H(ξ ∨ ζ) = H(ϑ|ξ ∨ ζ) and H(ζ ∨ ϑ) − H(ζ) = H(ϑ|ζ). Since H(ϑ|ζ) ≥ H(ϑ|ξ ∨ ζ), this proves (ii). Since H(ξ ∨ ζ) = H(ξ) + H(ζ|ξ), to prove (iii) we have to show that H(ζ|ξ) = H(ζ) if and only if ξ and ζ are independent. Let ζ = {X1 , . . . , Xn }. Since η is strictly concave, H(ζ|ξ) = η(Eξ (1Xk ))dµ ≤ η Eξ (1Xk )dµ = η(µ(Xk )), k

X

k

X

k

1.2 Topological Entropy

9

and equality holds if and only if Eξ (1Xk ) is constant a.e. for all k, that is, Eξ (1Xk ) = µ(Xk ). But the latter condition means exactly that µ(1Xk g) = µ(Xk )µ(g) for any g ∈ L∞ (X/ξ), that is, ξ and ζ are independent. We have by (1.1) H(∨k ζk | ∨k ξk ) ≤ H(ζk |ξk ). k

On the other hand, assuming ξk ≺ ζk , we have H(∨k ζk | ∨k ξk ) = H(∨k ζk ) − H(∨k ξk ) and H(ζk |ξk ) = H(ζk ) − H(ξk ). This gives (iv).

It is useful to note that subadditivity of entropy is equivalent to the following number inequality. Lemma 1.1.10. Let {λi1 ...in }i1 ∈I1 ,...,in ∈In be a ﬁnite set of nonnegative num (k) bers such that i1 ,...,in λi1 ...in = 1. Set λik = i1 ,...,ik−1 ,ik+1 ,...,in λi1 ...in . Then n (k) η(λi1 ...in ) ≤ η(λik ) k=1 ik ∈Ik

i1 ,...,in

Proof. Apply the subadditivity inequality H(∨k ξk ) ≤ k H(ξk ) to the partitions ξk of I1 × . . . × In consisting of the sets I1 × . . . × Ik−1 × {ik } × Ik+1 × In , k = 1, . . . , n, and the measure µ deﬁned by µ({(i1 , . . . , in )}) = λi1 ...in .

1.2 Topological Entropy Let T be a homeomorphism of a compact separable space X. We call (X, T ) a topological dynamical system. Let U = {Ui }i be an open cover of X. Denote by N (U) the minimal number of elements of U needed to cover X. If U = {Ui }i and V = {Vj }j are two open covers, we denote by U ∨ V the open cover formed by the sets Ui ∩ Vj . We obviously have N (U ∨ V) ≤ N (U)N (V). Deﬁne 1 h(U; T ) = lim log N (U ∨ T U ∨ . . . ∨ T n−1 U). n→∞ n The quantity htop (T ) = sup h(U; T ), U

where the supremum is taken over all open covers of X, is called the topological entropy of T . An equivalent deﬁnition is given in terms of separating and spanning sets. Let d be a compatible metric on X. Take ε > 0. A subset F of X is called (n, ε)-spanning with respect to T if for every x ∈ X there exists y ∈ F such that

10

1 Classical Dynamical Systems

max

0≤k≤n−1

d(T k x, T k y) < ε,

and it is called (n, ε)-separated if for any x, y ∈ F , x = y, we have max

0≤k≤n−1

d(T k x, T k y) ≥ ε.

Let spn (ε) be the minimal cardinality of an (n, ε)-spanning set, and srn (ε) the maximal cardinality of an (n, ε)-separated set. If U is an open cover with Lebesgue number ε (that is, any open ball of radius ε is contained in one of the elements of U), and V an open cover with diameter less than ε (that is, the diameter of every element in V is less than ε), then one can check that n−1 k k N (∨n−1 k=0 T U) ≤ spn (ε) ≤ srn (ε) ≤ N (∨k=0 T V).

It follows that 1 1 log spn (ε) = lim lim sup log spn (ε) ε↓0 n→∞ n n 1 1 = lim lim inf log srn (ε) = lim lim sup log srn (ε). n→∞ ε↓0 ε↓0 n→∞ n n

htop (T ) = lim lim inf ε↓0 n→∞

This gives an intuitive interpretation of topological entropy as a measure of the asymptotic distortion of the iterates of T along orbits. The main properties of topological entropy are as follows. Theorem 1.2.1. For every topological dynamical system (X, T ) we have: (i) htop (T n ) = |n|htop (T ) for n ∈ Z; (ii) if (X, T ) → (Y, S) is a factor map then htop (T ) ≥ htop (S); (iii) if Y ⊂ X is a closed T -invariant subset then htop (T ) ≥ htop (T |Y ), and if in addition Y is open then htop (T ) = max{htop (T |Y ), htop (T |X\Y )}; (iv) if (Y, S) is another topological dynamical system then htop (T × S) = htop (T ) + htop (S); (v) if U is an open cover such that the diameter of ∨nk=−n T k U with respect to some compatible metric tends to zero as n → ∞ then htop (T ) = h(U; T ).

The relation between measure and topological entropy is given by the variational principle. Theorem 1.2.2. For every topological dynamical system (X, T ) we have htop (T ) = sup hµ (T ), µ

where the supremum is taken over all T -invariant probability measures.

1.3 Entropy via Partitions of Unity

11

A topological dynamical system (X, T ) is called uniquely ergodic if there exists a unique T -invariant probability measure µ. Then by the variational principle hµ (T ) = htop (T ). In Sect. 1.1 we remarked that any dynamical system has a topological model. The following theorem gives a much more precise result for ergodic systems. Theorem 1.2.3. (Jewett-Krieger) Every ergodic dynamical system has a uniquely ergodic topological model.

Finishing our discussion of connections between measure entropy and topological entropy we shall give an equivalent deﬁnition of measure entropy in terms of spanning sets. Let (X, T ) be a topological dynamical system, and µ a T -invariant probability measure, which for simplicity we assume to be ergodic. A subset F of X is called (n, ε, δ)-spanning for (T, µ) if the set of x ∈ X such that there exists y ∈ F with max d(T k x, T k y) < ε 0≤k≤n−1

has measure greater than δ. Denote by spn (ε; δ) the minimal cardinality of an (n, ε, δ)-spanning set. Then for any δ ∈ (0, 1) hµ (T ) = lim lim inf ε↓0 n→∞

1 log spn (ε; δ). n

The proof can be obtained from the following theorem. Theorem 1.2.4. (Shannon-McMillan-Breiman) Let (X, µ, T ) be an ergodic dynamical system, ξ a ﬁnite measurable partition of X. For x ∈ X and n ∈ N k denote by In (x) the element of the partition ∨n−1 k=0 T ξ containing x. Then the convergence log µ(In (x)) − → h(ξ; T ) n holds in mean and almost everywhere.

1.3 Entropy via Partitions of Unity We shall in the present section rewrite the deﬁnition of H(∨nk=1 ξk ) in a form which is more suitable for generalization to the noncommutative case. Since the algebra generated by noncommuting ﬁnite dimensional algebras can be inﬁnite dimensional, we shall consider H(∨nk=1 ξk ) as a function of n partitions. Consider the case n = 1, so we are dealing with one partition ξ = {Xk }k . Then 1 = k 1Xk is a partition of unity in L∞ (X/ξ), and

12

1 Classical Dynamical Systems

H(ξ) =

η(µ(1Xk )).

k

This expression makes perfect sense for ﬁnite dimensional noncommutative algebras. However, in the noncommutative setting it is diﬃcult to form new partitions of unity consisting of projections from given noncommuting ones. Hence a natural question is how we can express H(ξ) using arbitrary ﬁnite decompositions 1 = i fi , fi ∈ L∞ (X/ξ), fi ≥ 0. The quantity i η(µ(fi )) can be arbitrarily large. So we have to compensate for the fact that the fi ’s are not necessarily projections. The function η(fi ) is then a natural ingredient of a correction term as it vanishes exactly when fi is a projection. Lemma 1.3.1. We have H(ξ) = sup

(η(µ(fi )) − µ(η(fi ))),

i

where the supremum is taken over all ﬁnite families {fi }i of positive functions in L∞ (X/ξ) with sum 1. Proof. We just have to check that the supremum is not larger thanH(ξ). Let n ∞ ξ = {X } and f ∈ L (X/ξ), 0 ≤ f ≤ 1, k i i i fi = 1. Then fi = k λik 1Xk k=1 with i λik = 1. Using that η(x + y) ≤ η(x) + η(y) for any positive numbers x and y we compute: (η(µ(fi )) − µ(η(fi ))) = η(µ(fi )) − µ(η(λik 1Xk )) i

i

≤

i,k

(η(µ(λik 1Xk )) − µ(η(λik 1Xk )))

i,k

=

(η(λik µ(Xk )) − η(λik )µ(Xk ))

i,k

=

η(µ(Xk )),

k

which proves the lemma.

Given a partition of unity 1 = i fi in L∞ (X) we can form a partition of unity in L∞ (X/ξ) by applying the conditional expectation Eξ . Thus H(ξ) is the supremum of (η(µ(fi )) − µ(η(Eξ (fi )))) i

over all partitions of unity 1 =

i

fi in L∞ (X).

Note now that

f (log f − log µ(f ))dµ.

η(µ(f )) − µ(η(f )) = X

1.3 Entropy via Partitions of Unity

13

For arbitrary positive functions f and g in L1 (X, µ) the quantity S(f, g) = f (log f − log g)dµ X

is called the relative entropy of the measures f dµ and g dµ. (k)

Next we turn to the case of n partitions ξ1 , . . . , ξn . Let ξk = {Xjk }jk ∈Jk . Then H(∨k ξk ) = η(µ(1X (1) . . . 1X (n) )). j1

j1 ,...,jn

jn

In other words, we consider the partitions of unity consisting of projections 1X (k) and form a new partition consisting of products of these projk

jections. This is not easy to generalize to the noncommutative case. So instead of trying to form a new partition of unity we consider all possible ﬁnite partitions 1 = i1 ,...,in fi1 ...in and think of them as coming from n parti (k) (k) tions 1 = ik fik , k = 1, . . . , n, wherefik is the sum of the fi1 ...in ’s over i1 , . . . , ik−1 , ik+1 , . . . , in . We then take i1 ,...,in η(µ(fi1 ...in )) as the ﬁrst approximation of the entropy and deﬁne the correction term as the sum of the correction terms for the above n partitions of unity. We thus arrive at the following expression. Proposition 1.3.2. Let ξ1 , . . . , ξn be ﬁnite measurable partitions of X. Then ⎧ ⎫ n ⎨ ⎬ (k) η(µ(fi1 ...in )) − µ(η(Eξk (fik ))) , H(∨nk=1 ξk ) = sup ⎩ ⎭ i1 ,...,in

k=1 ik

where the supremum is taken over all ﬁnite families {fi1 ...in }i1 ,...,in of positive functions in L∞ (X) with sum 1. Proof. Let ξ = ∨nk=1 ξk . Let 1 = i1 ,...,in fi1 ...in be a partition of unity. Since µ ◦ Eξ = µ, by Lemma 1.3.1 H(ξ) ≥ η(µ(fi1 ...in )) − µ(η(Eξ (fi1 ...in ))). i1 ,...,in

i1 ,...,in

Apply Lemma 1.1.10 to the numbers Eξ (fi1 ...in )(x), x ∈ X, and then integrate over X. We then see that the above expression is not smaller than i1 ,...,in

η(µ(fi1 ...in )) −

n

(k)

µ(η(Eξ (fik ))).

k=1 ik

As was already used in the proof of Proposition 1.1.1(iii), since ξk ≺ ξ and therefore Eξk = Eξk ◦ Eξ , it follows from Jensen’s inequality that µ(η(Eξ (f ))) ≤ µ(η(Eξk (f ))) for any f ≥ 0. Hence the above is expression is not smaller than

14

1 Classical Dynamical Systems

i1 ,...,in

η(µ(fi1 ...in )) −

n

(k)

µ(η(Eξk (fik ))).

k=1 ik

Thus the left side of the formula in the proposition majorizes the right side. (k) For the opposite inequality assume ξk = {Xik }ik and let fi1 ...in = 1X (1) ∩...∩X (n) . i1

(k)

in

(k)

Then fik = 1X (k) , so that η(Eξk (fik )) = 0. Thus the right side with these ik

fi1 ...in ’s equals H(ξ).

1.4 Notes As our basic reference on entropy we refer the reader to the book by Glasner [72], which contains all the theorems we stated without proofs. For more on Lebesgue spaces, introduced by Rohlin in [181], one can consult [1] or [184]. Entropy was introduced by Kolmogorov [111], who was inspired by Shannon’s information theory [195] and motivated by the isomorphism problem for Bernoulli shifts. The deﬁnition was improved by Sinai [199]. The isomorphism theorem for Bernoulli shifts with the same entropy was proved by Ornstein [150]. He developed a powerful theory of ﬁnitely determined processes giving several dynamical characterizations of Bernoulli shifts. A more constructive proof was given by Keane and Smorodinsky [104], see also [53] for an exposition of their proof. Theorem 1.1.7, due to Sinai, was proved earlier [200], and then it stated that any Bernoulli shift with entropy h ≤ h(T ) can be obtained as a factor system. The notion of topological entropy in terms of open coverings was introduced by Adler, Konheim and McAndrew [2]. The deﬁnition in terms of spanning and separated sets was given by Bowen [29]. The variational principle, Theorem 1.2.2, was proved by Goodwyn [82] and Dinaburg [55]. A stronger version of the variational principle has been obtained by Blanchard, Glasner and Host [26], [72, Chapter 17]. Theorem 1.2.3 is due to Jewett [96] and Krieger [113]. The deﬁnition of measure entropy in terms of spanning sets was proposed by Katok [99], see also [166] for a proof of equivalence of that deﬁnition to the usual one. Theorem 1.2.4 was obtained by Shannon [195], McMillan [121] and Breiman [31] who established convergence in measure, in mean and ﬁnally almost everywhere, respectively. The formula for entropy in the last section, obtained by Connes and Størmer [51], will be the basis for the deﬁnition of entropy in the noncommutative setting in Chap. 3.

2 Relative Entropy

Relative entropy of measures has appeared naturally in the expression for entropy of partitions given in Proposition 1.3.2. In the present chapter we shall deﬁne relative entropy for states on C∗ -algebras and show its properties needed in the sequel. We shall mainly consider ﬁnite dimensional C∗ -algebras.

2.1 Relative Entropy for Matrix Algebras Let A be a ﬁnite dimensional C∗ -algebra. Denote by TrA the canonical trace on A, so TrA (p) = 1 for every minimal projection p ∈ A. The algebra A is a direct sum of full matrix algebras, and the restriction of TrA to each summand is the usual matricial trace. We shall write Tr instead of TrA if no confusion can arise. Each positive linear functional ϕ on A is of the form ϕ = Tr(· Qϕ ) for a unique positive element Qϕ ∈ A called the density operator. Deﬁnition 2.1.1. If ϕ and ψ are positive linear functionals on A, then the relative entropy of ϕ and ψ is Tr(Qϕ (log Qϕ − log Qψ )), if ϕ ≤ λψ for some λ > 0, S(ϕ, ψ) = +∞, otherwise. Since A is ﬁnite dimensional, ϕ ≤ λψ for some λ > 0 if and only if supp ϕ ≤ supp ψ, that is, ϕ(a) = 0 for a positive a ∈ A as soon as ψ(a) = 0. Two identities are immediate from the deﬁnition: S(λ1 ϕ, λ2 ψ) = λ1 S(ϕ, ψ) + λ1 ϕ(1) log S(ϕ, ψ) =

λ1 if λ1 , λ2 > 0, λ2

d Tr(Qtϕ Q1−t ψ )|t=1 if ϕ ≤ λψ. dt

We next list the main properties of relative entropy.

(2.1) (2.2)

16

2 Relative Entropy

Theorem 2.1.2. We have: (i) S(ϕ, ψ) ≥ 0 for states ϕ and ψ, and equality holds if and only if ϕ = ψ; (ii) S(ϕ, ψ) is decreasing in ψ, i.e., S(ϕ, ψ1 ) ≥ S(ϕ, ψ2 ) if ψ1 ≤ ψ2 ; (iii) S(ϕ, ψ) is a lower semicontinuous function of (ϕ, ψ); moreover, it is continuous on the sets {(ϕ, ψ) | ϕ ≤ λψ}, λ > 0; (iv) S(ϕ, ψ) is a convex function of (ϕ, ψ); (v) if α: B → A is a unital Schwarz map, then S(ϕ ◦ α, ψ ◦ α) ≤ S(ϕ, ψ); (vi) if ϕ and ψ are states on A, B is a C∗ -subalgebra of A, and E: A → B is a ψ-preserving conditional expectation, then S(ϕ, ψ) = S(ϕ|B , ψ|B ) + S(ϕ, ϕ ◦ E); n (vii)if ϕ = i=1 ϕi , then i S(ϕi , ψ) = S(ϕ, ψ) + i S(ϕi , ϕ). Recall that if A and B are C∗ -algebras, a linear map α: B → A is said to be a Schwarz map if α(b∗ b) ≥ α(b)∗ α(b) for each b ∈ B. Then clearly α is positive. The map α is completely positive if α ⊗ id: B ⊗ Matn (C) → A ⊗ Matn (C) is positive for all natural numbers n, where Matn (C) denotes the complex n × n matrices. See App. A, where we have collected some basic facts on such maps. In particular, unital completely positive maps are Schwarz maps by A.2. Recall also that if B ⊂ A, then a conditional expectation of A onto B is a unital positive map E: A → B such that E(bac) = bE(a)c for all b, c ∈ B and a ∈ A. Then E is completely positive, see A.13. If A is ﬁnite dimensional, there always exists a unique TrA -preserving conditional expectation EB of A onto B. However, for an arbitrary state ψ on A a ψ-preserving conditional expectation may not exist. Parts (iv) and (v) of Theorem 2.1.2 require nontrivial estimates. In both cases the proof will be based on (2.2) and the following operator inequality. Proposition 2.1.3. Let x1 and x2 be bounded positive operators on a Hilbert space K, and yi , zi , i = 1, 2, bounded positive operators on a Hilbert space H such that x1 x2 = x2 x1 , y1 y2 = y2 y1 , z1 z2 = z2 z1 . Suppose v: K → H is a bounded operator such that xi ≥ v ∗ (yi + zi )v, i = 1, 2. Then for t ∈ [0, 1].

xt1 x1−t ≥ v ∗ (y1t y21−t + z1t z21−t )v 2

2.1 Relative Entropy for Matrix Algebras

17

Proof. If we can show the proposition for xi replaced by xi + ε1 for ε > 0 then by continuity the conclusion of the proposition follows when ε → 0. We may thus assume x1 and x2 are invertible. We ﬁrst consider the case t = 1/2. Let 1/2 1/2

1/2 1/2

1/2 1/2

x = x1 x2 , y = y1 y2 , z = z1 z2 . Put c = v ∗ (y + z)v, and let ζ, ξ ∈ K. Then by using the Cauchy-Schwarz inequality twice together with the assumption of the proposition we have |(cξ, ζ)| = |((y + z)vξ, vζ)| 1/2

1/2

1/2

1/2

≤ |(y1 vξ, y2 vζ)| + |(z1 vξ, z2 vζ)| ≤ (y1 vξ, vξ)1/2 (y2 vζ, vζ)1/2 + (z1 vξ, vξ)1/2 (z2 vζ, vζ)1/2 ≤ ((y1 + z1 )vξ, vξ)1/2 ((y2 + z2 )vζ, vζ)1/2 ≤ (x1 ξ, ξ)1/2 (x2 ζ, ζ)1/2 . −1/2

Replacing ξ by x1

−1/2

ξ and ζ by x2 −1/2

|(x2 −1/2

−1/2

cx1

ζ in the above inequality, we get

ξ, ζ)| ≤ ξ ζ.

−1/2

−1/2

−1/2

Hence x2 cx1 ≤ 1. In particular, the spectrum of x2 cx1 is contained in the unit disc. Recall that the nonzero parts of the spectra of the operators ab and ba coincide (indeed, if ab − 1 is invertible, then ba − 1 is invertible and (ba − 1)−1 = b(ab − 1)−1 a − 1). Since x1 and x2 commute, we thus have −1/4 −1/4 −1/4 −1/4 x2 cx1 x2 )\{0} −1/2 −1/2 Sp(x2 cx1 )\{0}.

Sp(x−1/2 cx−1/2 )\{0} = Sp(x1 =

Hence x−1/2 cx−1/2 ≤ 1. Thus c ≤ x, so that 1/2 1/2

v ∗ (y1 y2

1/2 1/2

1/2 1/2

+ z1 z2 )v ≤ x1 x2 .

Assuming that the proposition is true for exponents s and t, apply the t 1−t t 1−t s 1−s , y1s y21−s , z1s z21−s instead of above inequality to xt1 x1−t 2 , y1 y2 , z1 z2 , x1 x2 x1 , y1 , z1 , x2 , y2 , z2 . Then we see that the proposition holds for the exponent (s + t)/2. Hence it holds for all dyadic numbers and by continuity, for every t ∈ [0, 1].

Proof of Theorem 2.1.2. Note that for any numbers x, y ≥ 0 we have x(log x − log y) ≥ x − y,

(2.3)

and equality holds only if x = y, which is just another way of writing the inequality log t ≤ t − 1 for t = y/x. Now let ϕ and ψ be states, and Qϕ = λ p and Q = ψ i i i j µj qj the spectral decompositions. It follows that

18

2 Relative Entropy

λi (log λi − log µj )Tr(pi qj ) ≥ (λi − µj )Tr(pi qj ). Summing over i and j and assuming that ϕ(1) = ψ(1) = 1 we get S(ϕ, ψ) ≥ 0. Moreover, if equality holds then (λi − µj )pi qj = 0, so that summing over i and j again, we get Qϕ = Qψ . Thus (i) is proved. Part (ii) is a consequence of operator monotonicity of log, see B.4. To prove (iii) assume ϕn → ϕ, ψn → ψ. If ψ is faithful, then ψn is also faithful for suﬃciently large n, and the equality S(ϕ, ψ) = −Tr(η(Qϕ )) − ϕ(log Qψ ), where η(t) = −t log t, shows that S(ϕn , ψn ) → S(ϕ, ψ). In general, by (ii) we have for ε > 0 S(ϕn , ψn ) ≥ S(ϕn , ψn + εTr) → S(ϕ, ψ + εTr), and since S(ϕ, ψ + εTr) = −Tr(η(Qϕ )) − ϕ(log(Qψ + ε1)) S(ϕ, ψ) as ε → 0, we conclude that lim inf S(ϕn , ψn ) ≥ S(ϕ, ψ). n

If in addition ϕn ≤ λψn , then 0 ≤ S(ϕn , ψn ) − S(ϕn , ψn + εTr) = ϕn (log(Qψn + ε1) − log Qψn ) ≤ λψn (log(Qψn + ε1) − log Qψn ) → λψ(log(Qψ + ε1) − log Qψ ), and since the latter expression tends to zero as ε → 0, we see that S(ϕn , ψn ) → S(ϕ, ψ). To prove (iv), by (2.2) it suﬃces to show (for x = 1) that for x ∈ A and t ∈ [0, 1] Tr(x∗ at xb1−t ) is a concave function of (a, b) ∈ A+ × A+ .

(2.4)

For this consider the Hilbert space H = L2 (A, Tr), that is, the space A with inner product (Λ(y), Λ(z)) = Tr(z ∗ y), where we denote by Λ(y) the element y ∈ A viewed as a vector in H. Assume a = λa1 + (1 − λ)a2 and b = λb1 + (1 − λ)b2 . Deﬁne operators on H by x1 Λ(y) = Λ(ay), y1 Λ(y) = Λ(λa1 y), z1 Λ(y) = Λ((1 − λ)a2 y), x2 Λ(y) = Λ(yb), y2 Λ(y) = Λ(λyb1 ), z2 Λ(y) = Λ((1 − λ)yb2 ). Then x1 x2 = x2 x1 , y1 y2 = y2 y2 , z1 z2 = z2 z1 . We have yi + zi = xi , i = 1, 2, so by Proposition 2.1.3 applied to v = 1 xt1 x1−t ≥ y1t y21−t + z1t z21−t . 2

2.1 Relative Entropy for Matrix Algebras

19

Hence for x ∈ A Tr(x∗ at xb1−t ) = (Λ(xb1−t ), Λ(at x)) = (xt1 x1−t 2 Λ(x), Λ(x)) ≥ ((y1t y21−t + z1t z21−t )Λ(x), Λ(x)) 1−t ∗ t = λTr(x∗ at1 xb1−t 1 ) + (1 − λ)Tr(x a2 xb2 ), and (2.4) is proved. Similarly, by virtue of (2.2) in order to prove (v) it suﬃces to show (for x = 1) that for x ∈ B and t ∈ [0, 1] 1−t ∗ t TrB (x∗ Qtϕ◦α xQ1−t ψ◦α ) ≥ TrA (α(x) Qϕ α(x)Qψ ).

(2.5)

Deﬁne a linear operator v: L2 (B, TrB ) → L2 (A, TrA ) by vΛ(y) = Λ(α(y)), y ∈ B. For y ∈ B, z ∈ A deﬁne x1 Λ(y) = Λ(Qϕ◦α y), y1 Λ(z) = Λ(Qϕ z), x2 Λ(y) = Λ(yQψ◦α ), y2 Λ(z) = Λ(zQψ ), We have

(x1 Λ(y), Λ(y)) = TrB (y ∗ Qϕ◦α y) = ϕ(α(yy ∗ )), (v ∗ y1 vΛ(y), Λ(y)) = TrA (α(y)∗ Qϕ α(y)) = ϕ(α(y)α(y)∗ ).

Since α is a Schwarz map, it follows that x1 ≥ v ∗ y1 v. Similarly we have x2 ≥ v ∗ y2 v. Hence by Proposition 2.1.3, with z1 = z2 = 0, ∗ t 1−t (xt1 x1−t 2 Λ(x), Λ(x)) ≥ (v y1 y2 vΛ(x), Λ(x)),

which is exactly (2.5). To prove (vi), ﬁrst of all note that both sides of the equality are simultaneously either ﬁnite or inﬁnite. Indeed, if supp ϕ ≤ supp ψ, then obviously supp ϕ|B ≤ supp ψ|B . Furthermore, if 0 ≤ a ≤ supp ψ then by A.13 the support s(E(a)) of E(a) is not smaller than s(a), so ϕ(E(a)) > 0 if ϕ(a) > 0, and thus supp ϕ ≤ supp ϕ ◦ E. Conversely, suppose supp ϕ|B ≤ supp ψ|B and supp ϕ ≤ supp ϕ ◦ E. Then if ϕ(a) > 0 for some a ≥ 0, we have ϕ(E(a)) > 0, hence ψ(a) = ψ(E(a)) > 0, so that supp ϕ ≤ supp ψ. Thus we may assume that ϕ ≤ λψ, ϕ ≤ λϕ ◦ E. Let F : A → B be any faithful conditional expectation, and ω a faithful state on A. Then using (iii) we may replace E by Eε = (1 − ε)E + εF and ψ by ((1 − ε)ψ + εω) ◦ Eε , and thus assume that ψ and E are faithful. Replacing further ϕ by (1 − ε)ϕ + εψ, we may assume that ϕ is faithful as well. We claim that

20

2 Relative Entropy −t t Qtϕ◦E Q−t ψ = Qϕ|B Qψ|B .

Then, applying ϕ to both sides and taking the derivative at t = 0, we get S(ϕ|B , ψ|B ) on the right side and ϕ(log Qϕ◦E − log Qψ ) = S(ϕ, ψ) − S(ϕ, ϕ ◦ E) on the left side, proving (vi). It remains to prove the claim. In other words, we must prove that −t t Qtϕ Q−t ψ = Qϕ|B Qψ|B

(2.6)

if ϕ = ϕ ◦ E and ψ = ψ ◦ E. This is a well-known general property of Connes’ Radon-Nikodym cocycle, but an elementary proof of our particular case is possible. We shall reduce the proof of (2.6) to the case when B is abelian. The identity t −t t −t −1 Qtϕ Q−t ψ = Qϕ Qω (Qψ Qω ) for a state ω shows that to prove (2.6) for arbitrary ϕ and ψ, it suﬃces to consider some ﬁxed ψ. Thus we can assume that ψ|B is a trace. Choose a maximal abelian subalgebra C of B containing Qϕ|B . Since Qψ|B is in the center of B, we also have Qψ|B ∈ C. The algebra C is generated by minimal projections p1 , . . . , pm in B. Then EC (b) = i pi bpi is the unique conditional expectation of B onto C. It is ϕ|B - and ψ|B -preserving. Thus to prove (2.6) it is enough to prove the analogous statement for EC ◦ E: A → C and EC : B → C. In other words, we may assume that B = C is abelian. The conditional expectation E: A → B is then of the form E(a) = i . Let Qi be the density i ϕi (a)pi , where ϕi is a state with supp ϕi = p operator of ϕi . Then Qϕ = i ϕ(pi )Qi and Qψ = i ψ(pi )Qi , so that Qtϕ Q−t ψ =

ϕ(pi )t p = Qtϕ|B Q−t ψ|B , t i ψ(p ) i i

and the proof of (vi) is complete. Part (vii) is immediate as Qϕ = i Qϕi .

Remark 2.1.4. (i) Concerning Theorem 2.1.2(i), a more careful analysis shows that S(ϕ, ψ) ≥

1 ϕ − ψ2 2

for any states ϕ and ψ. (ii) If ϕ ≤ ψ are positive linear functionals, then S(ϕ, ψ) ≤ 0, and equality holds if and only if the support p of ϕ commutes with Qψ and Qϕ = pQψ . Indeed, S(ϕ, ψ) ≤ S(ϕ, ϕ) = 0 by Theorem 2.1.2(ii). Assume S(ϕ, ψ) = 0. As

2.2 Von Neumann Entropy

21

−1 −1 Qψ ≥ Qϕ , we have by B.3. So using the integral ϕ + t) ∞ (Qψ +−1t) ≤ (Q−1 formula log a = 0 ((1+t) −(a+t) ) dt as in B.4, we get from S(ϕ, ψ) = 0 that p((Qϕ + t)−1 − (Qψ + t)−1 )p = 0

for any t > 0. In other words, p commutes with (Qψ + t)−1 and p(Qψ + t)−1 = p(Qϕ + t)−1 , so pQψ = pQϕ = Qϕ .

2.2 Von Neumann Entropy A natural generalization of the entropy of a partition is the following notion. Deﬁnition 2.2.1. The entropy, or von Neumann entropy, of a positive linear functional ϕ on a ﬁnite dimensional C∗ -algebra A is S(ϕ) = Tr(η(Qϕ )) = −S(ϕ, Tr), where η(t) = −t log t. Before we proceed to the study of this notion, recall that the centralizer of a positive linear functional ϕ is the set Aϕ of all x ∈ A such that ϕ(xy) = ϕ(yx) for any y ∈ A. This is just the algebra of elements commuting with Qϕ . By the rank of A we mean the dimension of a maximal abelian subalgebra of A. Then rank A = TrA (1). Theorem 2.2.2. We have: (i) 0 ≤ S(ϕ) ≤ log rank A for any state ϕ; moreover, S(ϕ) = 0 if and only if ϕ is pure, and S(ϕ) = log rank A if and only if ϕ is the normalized canonical trace; (ii) the entropy is continuous, concave, and S(ϕ + ψ) ≤ S(ϕ) + S(ψ); in particular, if ϕ, ψ are states and 0 < λ < 1, then S(λϕ + (1 − λ)ψ) ≤ λS(ϕ) + (1 − λ)S(ψ) + η(λ) + η(1 − λ); (iii) if ϕ = i λi ϕi is a convex combination of states, then λi S(ϕi , ϕ), S(ϕ) ≥ i

and equality holds if and only if ϕi is pure for every λi = 0; (iv) if ϕ is a state on A and ψ is a state on B, then S(ϕ ⊗ ψ) = S(ϕ) + S(ψ);

22

2 Relative Entropy

(v) if ϕ is a positive linear functional on A ⊗ B ⊗ C, then S(ϕ) + S(ϕ|B ) ≤ S(ϕ|A⊗B ) + S(ϕ|B⊗C ); (vi) if ϕ is a state on A ⊗ B, then |S(ϕ) − S(ϕ|B )| ≤ S(ϕ|A ); (vii) if B is a maximal abelian subalgebra of A, then S(ϕ|B ) ≥ S(ϕ), and equality holds if and only if B is in the centralizer Aϕ of ϕ. Most of the properties of von Neumann entropy are simple consequences of properties of relative entropy. In order to prove (vi) we shall need the following result. Lemma 2.2.3. We have: (i) if ϕ is a pure state on the algebra Matn (C) ⊗ Matm (C) ∼ = Matnm (C), then S(ϕ|Matn (C) ) = S(ϕ|Matm (C) ); (ii) every state on Matn (C) ⊗ 1 ⊂ Matn (C) ⊗ Matn (C) extends to a pure state on Matn (C) ⊗ Matn (C). Proof. If ϕ is a pure state on Matn (C) ⊗ Matm (C), then the density operator Qϕ is a projection of rank one, so it is the projection onto the line spanned by a unit vector i,j λij ei ⊗ ej ∈ n2 ⊗ m 2 . Thus Qϕ =

¯ kl eik ⊗ ejl , λij λ

i,j,k,l

where {eik }i,k are the matrix units. The density matrices of ϕ|Matn (C) and ϕ|Matm (C) are (id ⊗ Trm )(Qϕ ) and (Tr n ⊗ id)(Qϕ ), respectively, where Trn is the usual trace on Matn (C). Set T = i,j λij eij ∈ Matn×m (C). Then (Trn ⊗ id)(Qϕ ) = (T ∗ T ) , the transpose of the matrix T ∗ T , and (id⊗Trm )(Qϕ ) = T T ∗ . Since the nonzero eigenvalues of T ∗ T and T T ∗ are the same (counting with multiplicities), we get S(ϕ|Matn (C) ) = S(ϕ|Matm (C) ), and (i) is proved. If ϕ is a state non Matn (C), we may assume that its density matrix is diagonal, Qϕ = i=1 λi eii . Then 1/2 1/2 λi λj eij ⊗ eij P = i,j

is a rank one projection in Matn (C) ⊗ Matn (C) such that Qϕ = (id ⊗ Trn )(P ). Thus the pure state with density matrix P has the property that its restriction to Matn (C) ⊗ 1 coincides with ϕ.

2.2 Von Neumann Entropy

23

Proof of Theorem 2.2.2. If ϕ isa state, and {λi }i are the eigenvalues of Qϕ , then i λi = 1 and S(ϕ) = i η(λi ). We see that S(ϕ) = 0 if and only if λi = 1 for some i and λj = 0 for j = i, that is, ϕ is a pure state. The rest of (i) follows from the corresponding properties of entropy of partitions, Proposition 1.1.9(i). As S(ϕ) = −S(ϕ, Tr), the entropy is concave by Theorem 2.1.2(iv). Since η is continuous, the entropy is also continuous. To prove the ﬁrst inequality in (ii) we use operator monotonicity of log, B.4, so S(ϕ + ψ) = −ϕ(log(Qϕ + Qψ )) − ψ(log(Qϕ + Qψ )) ≤ −ϕ(log Qϕ ) − ψ(log Qψ ) = S(ϕ) + S(ψ). This implies the second inequality in (ii), since S(λϕ) = λS(ϕ) + η(λ), if ϕ is a state. To prove (iii) note that λi S(ϕi , ϕ) = λi ϕi (log Qϕi − log Qϕ ) = S(ϕ) − λi S(ϕi ). i

i

i

Since S(ϕi ) ≥ 0, and S(ϕi ) = 0 if and only if ϕi is pure, we get (iii). Part (iv) is immediate. To prove (v) note that S(ϕ|A⊗B ) − S(ϕ|B ) = −S(ϕ|A⊗B , TrA ⊗ ϕ|B ) and similarly S(ϕ) − S(ϕ|B⊗C ) = −S(ϕ, TrA ⊗ ϕ|B⊗C ). Since S(ϕ, TrA ⊗ϕ|B⊗C ) ≥ S(ϕ|A⊗B , TrA ⊗ϕ|B ) by Theorem 2.1.2(v) (applied to the inclusion mapping A ⊗ B → A ⊗ B ⊗ C), we get (v). Part (v) implies in particular that if ϕ is a state on A ⊗ B, then S(ϕ) ≤ S(ϕ|A ) + S(ϕ|B ). So to prove (vi) it is enough to show that S(ϕ|B ) ≤ S(ϕ) + S(ϕ|A ).

(2.7)

Note that if D is a ﬁnite dimensional C∗ -algebra, then there exists an embedding D → Matn (C) with n = rank D such that TrD = Trn |D , where Trn is the usual trace on Matn (C). This allows us to assume without loss of generality that A and B are full matrix algebras. Let C ∼ = A ⊗ B. By Lemma 2.2.3(ii)

24

2 Relative Entropy

there exists a pure state ϕ˜ on A⊗B⊗C extending ϕ. Then S(ϕ| ˜ B⊗C ) = S(ϕ| ˜ A) by Lemma 2.2.3(i). Hence (2.7) follows from (v) applied to the state ϕ. ˜ To prove (vii) note that as B is maximal abelian in A, we have TrA |B = TrB . In particular, if EB : A → B is the trace preserving conditional expectation, then Qϕ◦EB = Qϕ|B . Hence S(ϕ|B ) − S(ϕ) = ϕ(log Qϕ − log Qϕ|B ) = S(ϕ, ϕ ◦ EB ). Of course, this also follows from Theorem 2.1.2(vi). Hence S(ϕ|B ) ≥ S(ϕ). If Qϕ ∈ B, then Qϕ|B = Qϕ as TrA |B = TrB , so S(ϕ|B ) = S(ϕ). Conversely, if S(ϕ|B ) = S(ϕ), then S(ϕ, ϕ◦EB ) = 0. By Theorem 2.1.2(i) we get ϕ = ϕ◦EB , so Qϕ = Qϕ◦EB = Qϕ|B ∈ B, that is, B is in the centralizer of ϕ.

Part (vii) of the above theorem implies that if a subalgebra B of A contains a maximal abelian subalgebra of A, then S(ϕ|B ) ≥ S(ϕ). The following lemma gives a useful estimate of the diﬀerence. Lemma 2.2.4. Let e1 , . . . , en be projections in A with sum 1, ϕ a state on A. Let B = ⊕ni=1 Bi , where Bi = ei Aei . Then η(ϕ(ei )). S(ϕ|B ) − S(ϕ) ≤

i

Proof. Consider a decomposition ϕ = k λk ϕk of ϕ into a convex combination of pure states. Then by monotonicity of relative entropy, Theorem 2.1.2(v), and Theorem 2.2.2(iii) we have S(ϕ) = λk S(ϕk , ϕ) ≥ λk S(ϕk |B , ϕ|B ) k

= S(ϕ|B ) −

k

λk S(ϕk |B ) = S(ϕ|B ) −

i

k

λk S(ϕk |Bi ).

k

The restriction of ϕk to Bi = ei Aei is a scalar multiple of a pure state, whence S(ϕk |Bi ) = η(ϕk (ei )). By concavity of η we then get λk S(ϕk |Bi ) = λk η(ϕk (ei )) ≤ η λk ϕk (ei ) = η(ϕ(ei )), k

k

k

which completes the proof of the lemma.

In Chap. 9 we shall use the following form of positivity of relative entropy, called the thermodynamic inequality. Proposition 2.2.5. If ϕ is a state on a ﬁnite dimensional C∗ -algebra A and H a self-adjoint element in A, then we have S(ϕ) + ϕ(H) ≤ log Tr(eH ), and equality holds if and only if Qϕ = Tr(eH )−1 eH .

2.2 Von Neumann Entropy

25

Proof. Let ψ be the state with density operator Tr(eH )−1 eH . Then S(ϕ, ψ) = −S(ϕ) − ϕ(log Qψ ) = −S(ϕ) − ϕ(H) + log Tr(eH ). So the result follows from Theorem 2.1.2(i).

Finishing our discussion of von Neumann entropy we prove the following estimate. Proposition 2.2.6. If ϕ and ψ are states on a ﬁnite dimensional C∗ -algebra A, d = rank A, and ϕ − ψ ≤ ε ≤ 1/e, then |S(ϕ) − S(ψ)| ≤ ε log d + η(ε). For this we need an elementary inequality 1 . (2.8) 2 To show this note that as the function f (x) = η(x+(s−t))−η(x) is decreasing, we have for x taking the increasing values 0, t, 1 − (s − t) |η(s) − η(t)| ≤ η(s − t) when s, t ∈ [0, 1], 0 ≤ s − t ≤

η(s − t) ≥ η(s) − η(t) ≥ −η(1 − (s − t)). If 1/e ≤ s − t ≤ 1/2, then η(1 − (s − t)) ≤ η(s − t) as η is decreasing on [1/e, 1], so (2.8) is proved in this case. On the other hand, if s − t ≤ 1/e, then using that η(x) + x is increasing on [0, 1], we get η(s) − η(t) ≥ t − s. But t − s ≥ −η(s − t) whenever 0 ≤ s − t ≤ 1/e, so (2.8) follows in this case too. We also need the following well-known inequality. Recall that the trace norm is deﬁned by x1 = Tr(|x|). Then if a and b are positive trace-class operators, λ1 ≥ λ2 ≥ . . . and µ1 ≥ µ2 ≥ . . . the eigenvalues of a and b, respectively, we have a − b1 ≥ |λi − µi |. (2.9) i

Indeed, let x and y be positive operators such that xy = 0 and x − y = a − b. Then a − b1 = Tr(x) + Tr(y). Let c = b + x = a + y, and let ν1 ≥ ν2 ≥ . . . be the eigenvalues of c. Then νi ≥ λi , µi , hence |λi − µi | ≤ 2νi − λi − µi . Thus |λi − µi | ≤ (2νi − λi − µi ) = Tr(2c − a − b) = Tr(x) + Tr(y) = a − b1 , i

i

and (2.9) is proved. Proof of Proposition 2.2.6. Let λ1 ≥ . . . ≥ λd and µ1 ≥ . . . ≥ µ d be the eigenvalues of Qϕ and Qψ , respectively. Put δi = |λi − µi |, δ = i δi . By (2.8) |S(ϕ) − S(ψ)| = (η(λi ) − η(µi )) ≤ |η(λi ) − η(µi )| ≤ η(δi ). i i i Then i η(δi ) = δ i η(δi /δ) − δ log δ ≤ δ log d + η(δ). By (2.9) we have δ ≤ ϕ − ψ, so δ ≤ ε ≤ 1/e and η(δ) ≤ η(ε), proving the proposition.

26

2 Relative Entropy

2.3 Relative Entropy for General C*-algebras A few times, notably in Chap. 5, we shall need relative entropy to be deﬁned for arbitrary C∗ -algebras. The shortest way of introducing it is the following variational expression. Let ϕ and ψ be positive linear functional on a unital C∗ -algebra A. Then ∞ dt , (ϕ(y(t)∗ y(t)) + t−1 ψ(x(t)x(t)∗ )) S(ϕ, ψ) = sup sup ϕ(1) log n − t n∈N x 1/n (2.10) where the second supremum is taken over all step functions x: (1/n, ∞) → A with ﬁnite range, and where y(t) = 1 − x(t). It suﬃces to consider functions x with values in a ﬁxed weakly dense subspace of A containing the unit. It is not at all obvious that in the ﬁnite dimensional case the above expression gives the same value of relative entropy as before. Since all our computations will be done by a reduction to the ﬁnite dimensional case anyway, instead of developing the whole theory we conﬁne ourselves to listing the main properties of relative entropy, see Notes at the end of the chapter for references. Theorem 2.3.1. For positive linear functionals on unital C∗ -algebras we have: λ1 if λ1 , λ2 > 0; (i) S(λ1 ϕ, λ2 ψ) = λ1 S(ϕ, ψ) + λ1 ϕ(1) log λ2 (ii) S(ϕ, ψ) ≥ 0 for states ϕ and ψ, and equality holds if and only if ϕ = ψ; (iii) S(ϕ, ψ) is decreasing in ψ, i.e., S(ϕ, ψ1 ) ≥ S(ϕ, ψ2 ) if ψ1 ≤ ψ2 ; (iv) S(ϕ, ψ) is a weakly∗ lower semicontinuous function of (ϕ, ψ); (v) S(ϕ, ψ) is a convex function of (ϕ, ψ); (vi) if α: B → A is a unital Schwarz mapping, then S(ϕ ◦ α, ψ ◦ α) ≤ S(ϕ, ψ); (vii) if M is a von Neumann algebra, ϕ and ψ are normal states on M , N a von Neumann subalgebra of M , and E: M → N a ψ-preserving faithful normal conditional expectation, then S(ϕ, ψ) = S(ϕ|N , ψ|N ) + S(ϕ, ϕ ◦ E); (viii) if ϕ = i=1 ϕi , then i S(ϕi , ψ) = S(ϕ, ψ) + i S(ϕi , ϕ); (ix) if ψ is a state on A, and ψ¯ its normal extension to πψ (A) , where πψ : A → B(Hψ ) is the GNS-representation, then ¯ ¯ S(ϕ, ¯ ψ), if ϕ = ϕ¯ ◦ πψ for a normal linear functional ϕ, S(ϕ, ψ) = +∞, otherwise; n

(x) if τ is a ﬁnite trace on A, a and b positive elements in A such that a ≤ λb for some λ > 0, then S(τ (· a), τ (· b)) = τ (a(log a − log b)).

2.3 Relative Entropy for General C*-algebras

27

We ﬁnish the chapter with several corollaries of Theorem 2.3.1 (or of Theorem 2.1.2 in the ﬁnite dimensional case). Corollary 2.3.2. If {ϕi }i∈I and {ψi }i∈I are ﬁnite sets of positive linear functionals then ϕi , ψi ≤ S(ϕi , ψi ). S i

i

i

Proof. This follows from the scaling (i) and convexity (v) properties in Theorem 2.3.1.

Corollary 2.3.3. Let A be a C∗ -algebra and E: A → B be a conditional expectation onto a subalgebra B, ϕ and ψ positive linear functionals on A such that ϕ ◦ E = ϕ and ψ ◦ E = ψ. Then S(ϕ|B , ψ|B ) = S(ϕ, ψ). Proof. This is obtained by applying the monotonicity property of relative entropy, Theorem 2.3.1(vi), to E: A → B and to the inclusion map B → A.

Corollary 2.3.4. Let B be a C∗ -subalgebra of a C∗ -algebra A, ψ a state on A, πψ the GNS-representation deﬁned by ψ. Assume πψ (B) is weakly operator dense in πψ (A). Then S(ϕ, ψ) = S(ϕ|B , ψ|B ) for any positive linear functional ϕ ≤ ψ. Proof. We can identify πψ |B with the GNS-representation deﬁned by ψ|B . Then the result follows from Theorem 2.3.1(ix).

Corollary 2.3.5. Let ϕ and ψ be positive linear functionals on a C∗ -algebra A, {γi : Ai → A}i and {θi : A → Ai }i two nets of unital completely positive maps such that ϕ ◦ γi ◦ θi → ϕ and ψ ◦ γi ◦ θi → ψ weakly∗ . Then S(ϕ ◦ γi , ψ ◦ γi ) → S(ϕ, ψ). Proof. By lower semicontinuity, Theorem 2.3.1(iv), S(ϕ, ψ) ≤ lim inf S(ϕ ◦ γi ◦ θi , ψ ◦ γi ◦ θi ). i

On the other hand, by monotonicity, Theorem 2.3.1(vi), S(ϕ ◦ γi ◦ θi , ψ ◦ γi ◦ θi ) ≤ S(ϕ ◦ γi , ψ ◦ γi ) ≤ S(ϕ, ψ). Hence S(ϕ ◦ γi , ψ ◦ γi ) → S(ϕ, ψ).

A related but easier result is the following. Corollary 2.3.6. If {Ai }i is an increasing net of unital C∗ -subalgebras of a C∗ -algebra A such that ∪i Ai is norm-dense, then S(ϕ|Ai , ψ|Ai ) S(ϕ, ψ) for any ϕ and ψ.

28

2 Relative Entropy

Proof. This follows immediately from the variational expression (2.10), since it suﬃces to consider step functions with values in ∪i Ai .

Recall that a unital C∗ -algebra is called nuclear if there exist two nets {γi : Ai → A}i and {θi : A → Ai }i of unital completely positive maps with ﬁnite dimensional C∗ -algebras Ai such that (γi ◦ θi )(a) − a → 0 for every a ∈ A. Then by Corollary 2.3.5 it follows that S(ϕ, ψ) = limi S(ϕ ◦ γi , ψ ◦ γi ) for any ϕ and ψ. This equality could be used as a deﬁnition of relative entropy for nuclear algebras, which would in fact be suﬃcient for us in most cases. Corollary 2.3.7. Let A be a separable C∗ -algebra, (X, µ) a Lebesgue space, X x → ϕx and X x → ψx measurable maps into the state space of A, the state space being considered with the weak∗ topology. Consider the states ϕ and ψ on A ⊗ L∞ (X, µ) deﬁned by ϕ(a ⊗ f ) = ϕx (a)f (x)dµ(x), ψ(a ⊗ f ) = ψx (a)f (x)dµ(x). X

X

S(ϕx , ψx )dµ(x).

Then S(ϕ, ψ) = X

Proof. Note that if A is ﬁnite dimensional, then TrA ⊗ µ is a ﬁnite trace on A ⊗ L∞ (X, µ), and the result can be deduced from Theorem 2.3.1(x). Using Corollary 2.3.5 one can extend the result to nuclear C∗ -algebras. In general we can argue as follows. Elements of the algebraic tensor product A L∞ (X, µ) can be considered as A-valued functions on X. Hence any step function (1/n, ∞) → A L∞ (X, µ) deﬁnes a family of step functions (1/n, ∞) → A indexed by elements of X. By the variational expression (2.10) this implies the inequality ≤. On the other hand, if ξ is a ﬁnite measurable partition of X then S(ϕ, ψ) ≥ S(ϕ|A⊗L∞ (X/ξ,µ) , ψ|A⊗L∞ (X/ξ,µ) ) S(ϕ|A⊗C1Z ) , ψ|A⊗C1Z ) = Z∈ξ

=

S Z

Z∈ξ

=

S

= X

Z

µ(Z)S µ(Z)

Z∈ξ

ψy dµ(y)

ϕy dµ(y), −1

ϕy dµ(y), µ(Z)

−1

Z

−1

µ(Z(x))

ψy dµ(y)

Z −1

ϕy dµ(y), µ(Z(x)) Z(x)

ψy dµ(y) dµ(x),

Z(x)

where Z(x) is the element of ξ containing x. Choose an increasing sequence {ξn }n of ﬁnite measurable partitions such that ∨n ξn is the partition into points. For x ∈ X denote by Zn (x) the element of ξn containing x. By the martingale convergence theorem for any f ∈ L1 (X, µ) we have

2.3 Relative Entropy for General C*-algebras

29

µ(Zn (x))−1 Zn (x) f (y)dµ(y) → f (x) for a.e. x ∈ X. It follows that −1

µ(Zn (x))

−1

Zn (x)

ϕy dµ(y) → ϕx and µ(Zn (x))

Zn (x)

ψy dµ(y) → ψx

weakly∗ for a.e. x ∈ X. Hence the above inequality together with weak∗ lower semicontinuity of relative entropy implies the inequality ≥ in the corollary.

Corollary 2.3.8. Let {ϕij }i∈I,j∈J be a ﬁnite linear functionals set of positive on a C∗ -algebra. Put ϕ = i,j ϕij , ϕIi = j ϕij and ϕJj = i ϕij . Then

S(ϕij , ϕ) ≥

i,j

S(ϕIi , ϕ) +

i

S(ϕJj , ϕ).

j

Proof. Using Theorem 2.3.1(viii) and Corollary 2.3.2 we compute J J S(ϕij , ϕ) − S(ϕj , ϕ) = S(ϕij , ϕ) − S(ϕj , ϕ) i,j

j

j

=

i,j

≥

i

S(ϕij , ϕJj ) ⎞ ⎛ S⎝ ϕij , ϕJj ⎠

i

=

j

j

S(ϕIi , ϕ),

i

which proves the corollary.

Property (x) in Theorem 2.3.1 could also be used as a deﬁnition of entropy for C∗ -algebras with ﬁnite trace. To prove the main properties of entropy such as convexity and monotonicity one could then use the same technique as in the ﬁnite dimensional case. In particular, the following three corollaries do not require the full strength of Theorem 2.3.1 and can be deduced from Proposition 2.1.3. The ﬁrst corollary is the noncommutative extension of Lemma 1.1.10. In Chap. 4 we shall prove a stronger result. Corollary 2.3.9. Let {xi1 ...in }i1 ,...,in be aﬁnite set of positive elements in a C∗ -algebra with a ﬁnite trace τ . Assume i1 ,...,in xi1 ...in = 1. For each k put (k) xik = i1 ,...,ik−1 ,ik+1 ,...,in xi1 ...in . Then i1 ,...,in

τ (η(xi1 ...in )) ≤

k

ik

(k)

τ (η(xik )).

30

2 Relative Entropy

Proof. For n = 2 the result follows by applying Corollary 2.3.8 to ϕi1 i2 = τ (· xi1 i2 ) and using Theorem 2.3.1(x). The general case is proved by induction.

Corollary 2.3.10. Let τ be a ﬁnite trace on a C∗ -algebra A, a and b positive elements in A such that a ≤ λb for some λ > 0. Then τ (a(log a − log b)) ≥ τ (a) − τ (b). Proof. By monotonicity, Theorem 2.3.1(vi), S(τ (· a), τ (· b)) ≥ S(τ (· a)|C1 , τ (· b)|C1 ) = τ (a)(log τ (a) − log τ (b)). On the other hand, τ (a)(log τ (a) − log τ (b)) ≥ τ (a) − τ (b) by (2.3).

In a similar way we get the following Peierls-Bogoliubov inequality. Corollary 2.3.11. Let τ be a ﬁnite trace on a C∗ -algebra A, a and b selfadjoint elements in A. Then log τ (ea+b ) − log τ (eb ) ≥

τ (aeb ) . τ (eb )

In particular, | log τ (ea+b ) − log τ (eb )| ≤ a. Proof. Applying S(ϕ, ψ) ≥ S(ϕ|C1 , ψ|C1 ) = ϕ(1)(log ϕ(1) − log ψ(1)) to ϕ = τ (· eb ) and ψ = τ (· ea+b ) we get the ﬁrst inequality. It follows that log τ (ea+b ) − log τ (eb ) ≥ −a. Applying this to −a and a + b instead of a and b we get the second inequality.

2.4 Notes Our main sources during the preparation of this chapter were the books by Simon [198] and Ohya and Petz [147]. Chapter 5 of the latter book contains in particular proofs of all the properties listed in Theorem 2.3.1. Relative entropy of measures was introduced by Kullback and Leibler [114]. The deﬁnition for normal states on ﬁnite von Neumann algebras was given by Umegaki [218]. One should be aware that in the literature several different conventions are used. Sometimes one uses the opposite order of the

2.4 Notes

31

arguments or the opposite sign. The statement (2.4) is the famous Lieb concavity. It was conjectured by Wigner and Yanase [235] on the basis of remarks of Dyson and proved by Lieb [116]. As we saw, it implies one of the main properties of relative entropy, joint convexity. The monotonicity property, Theorem 2.1.2(v) and Theorem 2.3.1(vi), was proved by Uhlmann [217]. Our deduction of both properties from Proposition 2.1.3 essentially follows Uhlmann’s paper using simpliﬁcations made by Simon [198]. The conditional expectation property, Theorem 2.1.2(vi) and Theorem 2.3.1(vii), is due to Hiai, Ohya and Tsukada [89] in a weaker form and to Petz [158], [159] in general. Although in our exposition entropy of states appears after entropy of partitions, it was introduced by von Neumann [145] more than twenty years earlier than the latter [195]. Of course, both notions have their origin in works of Boltzmann and Gibbs on thermodynamics. Among the properties of von Neumann entropy the one which attracted most attention was strong subadditivity, Theorem 2.2.2(v). It was ﬁrst proved by Lieb and Ruskai [117] using the Lieb concavity. The triangle inequality, Theorem 2.2.2(vi), was established by Araki and Lieb [11]. Proposition 2.2.6 is due to Fannes [66]. The notion of relative entropy was extended to normal states on arbitrary von Neumann algebras by Araki [9], [10]. If ϕ and ψ are normal states on a von Neumann algebra, ϕ ≤ λψ, then, [158], S(ϕ, ψ) = i

d ϕ((Dϕ: Dψ)t )|t=0 , dt

where (Dϕ: Dψ)t is Connes’ Radon-Nikodym cocycle. This is an extension of (2.2). A deﬁnition for states on arbitrary ∗-algebras was given by Uhlmann [217]. Later Kosaki [112], using ideas of Pusz and Woronowicz [176], obtained the variational expression (2.10). The inequality in Corollary 2.3.8, sometimes called the marginal inequality, was ﬁrst used by Connes [49], and the inequality in Corollary 2.3.9 by Connes and Størmer [51]. Finally note that in view of Theorem 2.2.2(iii) one can deﬁne the entropy of a state ϕ on a C∗ -algebra as the supremum of i λi S(ϕi , ϕ) over all convex decompositions ϕ = i λi ϕi . It turns out that if S(ϕ) < ∞, then πϕ (A) is discrete, that is, it is a direct sum of algebras B(H), see [147, Chapter 6].

3 Dynamical Entropy

As we already mentioned in Chap. 1, one of the main problems in deﬁning noncommutative entropy is that two ﬁnite dimensional subalgebras of a noncommutative algebra can generate an inﬁnite dimensional subalgebra. To overcome this diﬃculty we consider the entropy of the join of n partitions as a function of the partitions and try to ﬁnd its noncommutative analogue for each n ∈ N. In the present chapter we shall deﬁne such an analogue and study the dynamical invariant arising from it.

3.1 Mutual Entropy In Chap. 1 we succeeded in writing a formula for H(ξ1 ∨. . .∨ξn ) which does not use joins of the partitions ξi , see Proposition 1.3.2. This formula makes sense for ﬁnite von Neumann algebras. Namely, let N be a von Neumann algebra and τ a faithful normal trace on N . Let N1 , . . . , Nn be ﬁnite dimensional subalgebras of N , and let ENk : N → Nk denote the τ -preserving conditional expectation. Then the mutual entropy of N1 , . . . , Nn ⊂ N with respect to τ is ⎧ ⎫ n ⎨ ⎬ (k) η(τ (xi1 ...in )) − τ η ENk (xik ) , Hτ (N1 , . . . , Nn ) = sup ⎩ ⎭ i1 ,...,in

k=1 ik

where η(t) = −t log t, and the supremum is taken over all ﬁnite partitions of unity xi1 ...in , xi1 ...in ≥ 0, 1= i1 ∈I1 ,...,in ∈In (k) xik

= i1 ,...,ik−1 ,ik+1 ,...,in xi1 ...in . in N , where In the case of one algebra B ⊂ N , the deﬁnition says that Hτ (B) is the supremum of

34

3 Dynamical Entropy

η(τ (xi )) −

i

τ (η(EB (xi ))) = −

i

τ (xi )τ (η(EB (τ (xi )−1 xi ))).

(3.1)

i

We shall see soon that this supremum is just S(τ |B ). This deﬁnition does not make sense if τ is not a trace, since in general a state preserving conditional expectation N → Nk may not exist. However, by Theorem 2.3.1(x) we know that if x ∈ Nk then τ (η(x)) = −S(τ (·x)|Nk , τ |Nk ). Recalling that the map x → τ (·x) establishes a one-to-one correspondence between positive elements x ∈ N , x ≤ 1, and positive linear functionals ϕ ≤ τ on N , we can thus rewrite the deﬁnition of the mutual entropy as ⎧ ⎫ ⎨ ⎬ (k) Hτ (N1 , . . . , Nn ) = sup η(ϕi1 ...in (1)) + S(ϕik |Nk , τ |Nk ) , ⎩ ⎭ i1 ,...,in

ik

k

where the supremum is taken over all ﬁnite decompositions of τ into a sum (k) of positive linear functionals, and ϕik = i1 ,...,ik−1 ,ik+1 ,...,in ϕi1 ...ik . This formula already makes sense for arbitrary states. We shall now extend the deﬁnition to channels. Let A be a unital C∗ algebra. By a channel in A we mean a unital completely positive map γ: B → A, where B is a ﬁnite dimensional C∗ -algebra. The reason to consider channels is that in general a C∗ -algebra may have too few ﬁnite dimensional subalgebras. Deﬁnition 3.1.1. Let A be a unital C∗ -algebra, γk : Ak → A, 1 ≤ k ≤ n, a collection of channels. The mutual entropy Hϕ (γ1 , . . . , γn ) of γ1 , . . . , γn with respect to a state ϕ on A is the supremum of (k) Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) = η(ϕi1 ...in (1)) + S(ϕik ◦ γk , ϕ ◦ γk ) i1 ,...,in

k

ik

(k)

over all ﬁnite decompositions ϕ = i1 ,...,in ϕi1 ...in , where ϕik is obtained by taking the sum of the elements ϕi1 ...in with the k-th index equal to ik . We call Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) the mutual entropy of γ1 , . . . , γn with respect to the decomposition ϕ = i1 ,...,in ϕi1 ...in . Note that since S(λψ, ϕ) = λS(ψ, ϕ) − ψ(1)η(λ), it can also be written as (k) (k) (k) η(ϕi1 ...in (1)) − η(ϕik (1)) + ϕik (1)S(ϕˆik ◦ γk , ϕ ◦ γk ), i1 ,...,in

k

ik

k

ik

(3.2) ˆ where for a positive linear functional ψ we denote by ψ the state ψ(1)−1 ψ. The last expression can equivalently be written as (k) (k) η(ϕi1 ...in (1)) − η(ϕik (1)) + Hϕ (γk ; {ϕik }). (3.3) i1 ,...,in

k

ik

k

ik

3.1 Mutual Entropy

35

In the case when the Ai ’s are subalgebras of A and the γi ’s are the inclusion maps, we write Hϕ (A1 , . . . , An ) instead of Hϕ (γ1 , . . . , γn ). To get a better understanding of the deﬁnition, consider the case of one channel γ: B → A. Then Hϕ (γ) is the supremum of the quantities η(ϕi (1)) + S(ϕi ◦ γ, ϕ ◦ γ). (3.4) i

i

For a state ψ on B, call a decomposition ψ = i ψi orthogonal if the density matrices of the ψi ’s are mutually orthogonal. Equivalently, there exists a partition of unity 1 = i pi consisting of projections in the centralizer of ψ such that ψi = ψ(·pi ). By Remark 2.1.4(ii) this is also the same as requiring S(ψi , ψ) = 0 for all i. Thus the ﬁrst summand in (3.4) is a classical entropy term, while the second one is a correction term which compensates for nonorthogonality of the decomposition ϕ ◦ γ = i ϕi ◦ γ. The quantity S(ϕi ◦ γ, ϕ ◦ γ) = S(ϕi ◦ γ) − S(ϕ ◦ γ) − i

i

=

i

η(ϕi (1)) +

ϕi (1)S(ϕˆi ◦ γ) − S(ϕ ◦ γ) (3.5)

i

is also called the entropy defect of the decomposition ϕ ◦ γ = i ϕi ◦ γ. In the case of several channels we similarly have the classical term i1 ,...,in η(ϕi1 ...in (1)) plus the sum of the correction terms corresponding to (k) the decompositions ϕ ◦ γk = ik ϕik ◦ γk , 1 ≤ k ≤ n. If A is a von Neumann algebra and ϕ a faithful normal state, then instead of decompositions of ϕ we can talk about partitions of unity in A. To show this we ﬁrst need to recall a few facts from modular theory, see e.g. [30, Chapter 2.5] or [205] for details. We may assume A ⊂ B(H) and ϕ is a vector state, ϕ(a) = (aξϕ , ξϕ ), where ξϕ is a cyclic and separating vector in H. The anti-linear map Aξϕ aξϕ → a∗ ξϕ 1/2

is closable. Denote by Sϕ its closure. Let Sϕ = Jϕ ∆ϕ be the polar decomposition, where Jϕ is an anti-linear isometry and ∆ϕ a positive operator, which is in general unbounded. Then Jϕ2 = 1, ∆ϕ is nonsingular and Jϕ ξϕ = ∆ϕ ξϕ = ξϕ . The operators Jϕ and ∆ϕ are called the modular conjugation and the modular operator deﬁned by ϕ. The formula −it σtϕ (a) = ∆it ϕ a∆ϕ , a ∈ A,

deﬁnes a one-parameter automorphism group of A called the modular group of ϕ. The modular group measures how far the state ϕ is from a tracial state. Namely, the centralizer Aϕ of ϕ, that is, the set of all elements a ∈ A such

36

3 Dynamical Entropy

that ϕ(ab) = ϕ(ba) for any b ∈ A, coincides with the ﬁxed point algebra of the modular group. If B is a von Neumann subalgebra of A, then a normal ϕ-preserving conditional expectation A → B exists if and only if σtϕ (B) = B for any t ∈ R. The most important property of Jϕ is that Jϕ AJϕ = A , the commutant of A in B(H). It follows that any positive linear functional ψ ≤ ϕ can be written in the form ψ = (·Jϕ xJϕ ξϕ , ξϕ ) = (·Jϕ xξϕ , ξϕ ) for a unique element x ∈ A, 0 ≤ x ≤ 1. Since 1/2 −1/2 ξϕ , Jϕ xξϕ = Jϕ Sϕ xξϕ = ∆1/2 ϕ xξϕ = ∆ϕ x∆ϕ ϕ (x)). Thus any by slightly abusing notation we can write ψ = ϕ(· σ−i/2 partition of unity 1 = j xj in A gives rise to a decomposition ϕ = ϕ ϕ(· σ (x )), and vice versa. We shall write Hϕ (γ1 , . . . , γn ; {xj1 ...jn }) inj j −i/2 ϕ (xj1 ...jn ))}). stead of Hϕ (γ1 , . . . , γn ; {ϕ(· σ−i/2

Yet another equivalent way of describing decompositions of states is as follows. Let C be a ﬁnite dimensional abelian C∗ -algebra, C1 , . . . , Cn ⊂ C subalgebras, µ a state on C, and P : A → C a unital positive map such that ϕ = µ ◦ P . Then we say that (C, µ, {Ck }nk=1 , P ) is an abelianmodel for (A, ϕ, {γk }nk=1 ). An abelian model deﬁnes a decomposition ϕ = ϕi1 ...in by (1)

(n)

ϕi1 ...in (a) = µ(P (a)pi1 . . . pin ),

(3.6)

(k)

where {pik }ik is the set of atoms of Ck . We call Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) the entropy of the abelian model. By deﬁnition (k) Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) = S(µ|∨k Ck ) + S(µ((P ◦ γk )(·)pik ), ϕ ◦ γk ). k

ik

(3.7) Conversely, starting with a decomposition of ϕ one can easily construct an abelian model deﬁning this decomposition: take C = C(I1 × . . . × In ), Ck = C(Ik ), then (3.6) determines µ and P . Note that if C is a ﬁnite dimensional abelian C∗ -algebra, C1 , . . . , Cn are subalgebras of C, and µ is a state on C, then (C, µ, {Ck }nk=1 , idC ) is an abelian model for (C, µ, {Ck }nk=1 ). The entropy of this model is S(µ|∨k Ck ). The setting of abelian models makes it explicit that in our deﬁnition of mutual entropy we ﬁrst project a noncommutative C∗ -algebra into a commutative one, and then compute the entropy of the images of channels under this projection. More precisely, we have the following. Lemma 3.1.2. The mutual entropy Hϕ (γ1 , . . . , γn ) is the supremum of the entropies Hµ (P ◦ γ1 , . . . , P ◦ γn ) over all unital completely positive maps P : A → C of A into abelian C∗ -algebras and over all states µ such that µ ◦ P = ϕ.

3.1 Mutual Entropy

37

Proof. The inequality Hϕ (γ1 , . . . , γn ) ≥ Hµ (P ◦ γ1 , . . . , P ◦ γn ) will follow from Proposition 3.1.3(ii) below. On the other hand, if (C, µ, {Ck }nk=1 , P ) is an abelian model for (A, ϕ, {γk }nk=1 ), then (C, µ, {Ck }nk=1 , idC ) is an abelian model for (C, µ, {P ◦ γk }nk=1 ). By deﬁnition, see (3.7), the entropies of these two abelian models coincide, so that the entropy of the abelian model (C, µ, {Ck }nk=1 , P ) for (A, ϕ, {γk }nk=1 ) is not larger than Hµ (P ◦γ1 , . . . , P ◦γn ). Thus we get the result.

The main properties of mutual entropy are the following. Proposition 3.1.3. We have: (i) if θk : Bk → Ak , 1 ≤ k ≤ n, are unital completely positive maps, then Hϕ (γ1 ◦ θ1 , . . . , γn ◦ θn ) ≤ Hϕ (γ1 , . . . , γn ); (ii) if θ: A → B is a unital completely positive map and ψ a state on B, then Hψ (θ ◦ γ1 , . . . , θ ◦ γn ) ≤ Hψ◦θ (γ1 , . . . , γn ); in particular, Hψ (θ ◦γ1 , . . . , θ ◦γn ) = Hψ◦θ (γ1 , . . . , γn ) if θ is an isomorphism; (iii) if k < n, then Hϕ (γ1 , . . . , γk ) ≤ Hϕ (γ1 , . . . , γn ), and Hϕ (γ1 , . . . , γn ) ≤ Hϕ (γ1 , . . . , γk ) + Hϕ (γk+1 , . . . , γn ); (iv) Hϕ (γ1 , . . . , γn ) depends only on the set {γ1 , . . . , γn }; in other words, it is invariant under permutations of γ1 , . . . , γn , and if γn−1 = γn , then Hϕ (γ1 , . . . , γn ) = Hϕ (γ1 , . . . , γn−1 ). Proof. Part (i) follows from monotonicity of relative entropy, Theorem 2.1.2(v). Part (ii) is immediate as any decomposition of the state ψ deﬁnes a decomposition of ψ ◦ θ. The inequality Hϕ (γ1 , . . . , γk ) ≤ Hϕ (γ1 , . . . , γn ) follows from the fact that any decomposition ϕ = ϕi1 ...ik can be considered as a decomposition ϕ = ϕi1 ...in with single-point indexsets Ik+1 , . . . , In . Given adecomposition ϕ = ϕi1 ...in , we can construct two decompositions ϕ = ϕi1 ...ik and ϕ = ϕik+1 ...in , which are more transparently described in terms of abelian models: given an abelian model (C, µ, {Cj }nj=1 , P ) for (A, ϕ, {γj }nj=1 ) we get the abelian models (C, µ, {Cj }kj=1 , P ) and (C, µ, {Cj }nj=k+1 , P ) for (A, ϕ, {γj }kj=1 ) and (A, ϕ, {γj }nj=k+1 ), respectively. Then the subadditivity inequality in part (iii) is a consequence of the classical subadditivity S(µ|∨nj=1 Cj ) ≤ S(µ|∨kj=1 Cj ) + S(µ|∨nj=k+1 Cj ). It remains to prove part (iv). The invariance under permutations is obvious. Assume now that γn−1 = γn . By (iii) we already know that

38

3 Dynamical Entropy

Hϕ (γ1 , . . . , γn−1 ) ≤ Hϕ (γ1 , . . . , γn ). To make the proof of the opposite inequality more transparent, consider ﬁrst the case n = 2. Then given any decomposition ϕ = i1 ,i2 ϕi1 i2 we have Hϕ (γ1 ; {ϕi1 i2 }) ≥ Hϕ (γ1 , γ1 ; {ϕi1 i2 }) by Corollary 2.3.8, since the classical terms on both sides of the above inequality coincide. Consider now the general case. Any = ϕi1 ...in can be decomposition ϕ considered as a decomposition ϕ = ϕi1 ...in−1 with Ik = Ik for k < n − 1 = In−1 ×In . In other words, any abelian model (C, µ, {Ck }nk=1 , P ) for and In−1 n−1 (A, ϕ, {γk }nk=1 ) deﬁnes an abelian model (C, µ, {Ck }n−1 k=1 , P ) for (A, ϕ, {γk }k=1 ) with Ck = Ck for k < n − 1 and Cn−1 = Cn−1 ∨ Cn . Comparing the entropies of these abelian models, we see that the classical terms as well as the ﬁrst n−2 correction terms coincide. Thus Corollary 2.3.8 applied to I = In−1 and J = In shows that the entropy of (C, µ, {Ck }n−1 k=1 , P ) is not smaller than the entropy of (C, µ, {Ck }nk=1 , P ). It follows that Hϕ (γ1 , . . . , γn−1 ) ≥ Hϕ (γ1 , . . . , γn ).

So far we have not given any estimates for Hϕ . Lemma 3.1.4. For any channel γ: B → A, we have 0 ≤ Hϕ (γ) ≤ S(ϕ ◦ γ). If, moreover, A is a von Neumann algebra, ϕ a faithful normal state and the image of γ does not consist only of scalars, then Hϕ (γ) > 0. Together with Proposition 3.1.3(iii) this implies that Hϕ (γk ) ≤ S(ϕ ◦ γk ), 0 ≤ Hϕ (γ1 , . . . , γn ) ≤ k

k

and if Im γk ⊂ B ⊂ A for 1 ≤ k ≤ n, then by Proposition 3.1.3(i),(iv) Hϕ (γ1 , . . . , γn ) ≤ Hϕ (B, . . . , B ) = Hϕ (B) ≤ S(ϕ|B ). n

Proof of Lemma 3.1.4. The entropy Hϕ (γ) is the supremum of the quantities (see (3.2)) ϕi (1)S(ϕˆi ◦ γ, ϕ ◦ γ) = S(ϕ ◦ γ) − ϕi (1)S(ϕˆi ◦ γ). i

i

The left hand side of the above equality shows that Hϕ (γ) ≥ 0, while the right hand side implies Hϕ (γ) ≤ S(ϕ ◦ γ). If A is a von Neumann algebra and ϕ is normal and faithful, we may assume that A ⊂ B(H) and ϕ is given vector ξ. by a cyclic and separating Then any partition of unity 1 = x in the commutant A of A deﬁnes i i a decomposition ϕ = ϕ with ϕ = (· x ξ, ξ). If H (γ; {ϕ }) = 0, then i i ϕ i i i S(ϕˆi ◦ γ, ϕ ◦ γ) = 0 for all i with ϕi (1) = 0. By Theorem 2.1.2(i) it follows that ϕ ◦ γ = ϕˆi ◦ γ. In other words, if Hϕ (γ) = 0 and a ∈ Im γ, then

3.1 Mutual Entropy

ϕ(a) =

39

(axξ, ξ) (xξ, ξ)

for any x ∈ A , 0 ≤ x ≤ 1, x = 0. Thus ((a − ϕ(a)1)xξ, ξ) = 0. Since ξ is cyclic for A , this shows that a = ϕ(a)1.

As opposed to the commutative case, Hϕ (B) for B ⊂ A cannot be determined just by the restriction of ϕ to B. For example, if the state ϕ on A is pure, then Hϕ (B) = 0, since there are no nontrivial decompositions of ϕ into a convex combination of states. On the other hand, as we shall see in a moment, Hϕ|B (B) = S(ϕ|B ), which is positive unless ϕ|B is pure. We have however the following simple but useful result. Lemma 3.1.5. If the images of the channels γk : Ak → A, 1 ≤ k ≤ n, are contained in a C∗ -subalgebra B ⊂ A, and there exists a ϕ-preserving conditional expectation E: A → B, then Hϕ|B (γ1 , . . . , γn ) = Hϕ (γ1 , . . . , γn ). Proof. Clearly Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) = Hϕ|B (γ1 , . . . , γn ; {ϕi1 ...in |B }). Since any decomposition of ϕ|B can be extended to a decomposition of ϕ by composing positive functionals on B with E, we get the result.

In general it is rather diﬃcult to compute Hϕ , or at least to ﬁnd a way of producing optimal decompositions. The following proposition gives one of the few examples when an explicit computation is possible. Proposition 3.1.6. Let A1 , . . . , An ⊂ B ⊂ A. Suppose there exist mutually commuting abelian subalgebras Ck ⊂ Ak , 1 ≤ k ≤ n, such that ∨k Ck is maximal abelian in the centralizer of ϕ|B . Suppose also that there exists a ϕ-preserving conditional expectation E: A → B. Then Hϕ (A1 , . . . , An ) = Hϕ (B) = S(ϕ|B ). Proof. Let F : B → C = ∨k Ck be a ϕ-preserving conditional expectation. Then (C, ϕ|C , {Ck }k , F ◦ E) is an abelian model for (A, ϕ, {Ck }k ). Since F ◦ E is the identity map on Ck ⊂ C, the entropy of this model is the same as the entropy of the model (C, ϕ|C , {Ck }k , idC ) for (C, ϕ|C , {Ck }k ). The latter entropy is S(ϕ|C ). Thus Hϕ (C1 , . . . , Cn ) ≥ S(ϕ|C ) = S(ϕ|B ), where the equality follows from Theorem 2.2.2(vii). On the other hand, we have Hϕ (C1 , . . . , Cn ) ≤ Hϕ (A1 , . . . , An ) ≤ Hϕ (B) ≤ S(ϕ|B ). Hence the above inequalities are in fact equalities.

In particular, in the abelian case we recover the classical quantities H(∨k ξk ), which also follows from Proposition 1.3.2. Further properties of mutual entropy are given in the following proposition.

40

3 Dynamical Entropy

Proposition 3.1.7. We have: (i) if πϕ : A → B(Hϕ ) is the GNS-representation and ϕ¯ is the normal extension of ϕ to πϕ (A) , then Hϕ (γ1 , . . . , γn ) = Hϕ¯ (πϕ ◦ γ1 , . . . , πϕ ◦ γn ); (ii) if ψ is another state on A and 0 ≤ λ ≤ 1, then Hλϕ+(1−λ)ψ (γ1 , . . . , γn ) ≥ λHϕ (γ1 , . . . , γn ) + (1 − λ)Hψ (γ1 , . . . , γn ) − (n − 1)(η(λ) + η(1 − λ)); (iii) if ψ is a state on a C∗ -algebra B, 0 ≤ λ ≤ 1 and θk : Bk → B, 1 ≤ k ≤ n, are channels, then on A ⊕ B Hλϕ⊕(1−λ)ψ (γ1 ⊕ θ1 , . . . , γn ⊕ θn ) = λHϕ (γ1 , . . . , γn ) + (1 − λ)Hψ (θ1 , . . . , θn ) + η(λ) + η(1 − λ); (iv) under the assumptions of part (iii), Hϕ⊗ψ (γ1 ⊗ θ1 , . . . , γn ⊗ θn ) ≥ Hϕ (γ1 , . . . , γn ) + Hψ (θ1 , . . . , θn ) on A ⊗ B, and equality holds if B is abelian and the θk ’s are injective homomorphisms; (v) for channels γk : Ak ⊗ Bk → A, 1 ≤ k ≤ n, Hϕ (γ1 , . . . , γn ) ≤ Hϕ (γ1 |A1 , . . . , γn |An ) + 2 S(ϕ ◦ γk |Bk ). k

As we shall see in Chaps. 11 and 12, the inequality in (iv) can be strict. Proof of Proposition 3.1.7. Part (i) follows from the fact that for any ψ ≤ ϕ there exists a unique normal positive linear functional ψ¯ on πϕ (A) such that ψ = ψ¯ ◦ πϕ , so that any decomposition of ϕ comes from a decomposition of ϕ. ¯ To prove (ii), given abelian models (C, µ, {Ck }k , P ) for (A, ϕ, {γk }k ) and (D, ν, {Dk }k , Q) for (A, ψ, {γk }k ), we can form the abelian model (C ⊕ D, λµ ⊕ (1 − λ)ν, {Ck ⊕ Dk }k , P ⊕ Q) for (A, λϕ + (1 − λ)ψ, {γk }k ). In the simplest case n= 1 this means that starting with decompositions ϕ = i ϕi and ψ = j ψj we consider the decomposition λϕ + (1 − λ)ψ = λϕi + (1 − λ)ψj . i

The classical entropy term is then η(λϕi (1)) + η((1 − λ)ψj (1)) i

j

j

3.1 Mutual Entropy

=λ

η(ϕi (1)) + (1 − λ)

i

41

η(ψj (1)) + η(λ) + η(1 − λ).

j

On the other hand, the correction term is by the ﬁrst equality in (3.5) S(λϕ ◦ γ + (1 − λ)ψ ◦ γ) − i S(λϕi ◦ γ) − j S((1 − λ)ψj ◦ γ) = S(λϕ◦γ +(1−λ)ψ ◦γ)−

λS(ϕi ◦γ)−

i

(1−λ)S(ψj ◦γ)−η(λ)−η(1−λ).

j

Recalling that by Theorem 2.2.2(ii) von Neumann entropy is concave, we see that by adding the above expressions for the classical entropy term and the correction term, part (ii) is true for n = 1. For an arbitrary n we similarly get one summand η(λ) + η(1 − λ) from the classical term and n summands −η(λ) − η(1 − λ) from the correction terms. Thus (ii) is proved. To prove (iii) we repeat the computation in (ii), but now for the channels γk ⊕ θk : Ak ⊕ Bk → A ⊕ B, 1 ≤ k ≤ n, and the states ϕ ⊕ 0 and 0 ⊕ ψ. This time, for n = 1, the correction term is S(λϕ ◦ γ ⊕ (1 − λ)ψ ◦ θ) − i S(λϕi ◦ γ) − j S((1 − λ)ψj ◦ θ) = λS(ϕ ◦ γ) + (1 − λ)S(ψ ◦ θ) −

λS(ϕi ◦ γ) −

i

(1 − λ)S(ψj ◦ θ).

j

Thus we do not get n summands −η(λ) − η(1 − λ), which proves the inequality ≥ in (iii). To provethe opposite inequality, starting with a decomposition λϕ ⊕ (1 − λ)ψ = ω , we get decompositions ϕ = ϕi1 ...in and i ...i 1 n ψ = ψi1 ...in such that ωi1 ...in = λϕi1 ...in ⊕ (1 − λ)ψi1 ...in . The correction terms for the ﬁrst decomposition are just convex combinations of the correction terms for the other two decompositions. To estimate the classical term we use the inequality S(λµ + (1 − λ)ν) ≤ λS(µ) + (1 − λ)S(ν) + η(λ) + η(1 − λ) from Theorem 2.2.2(ii). This proves the inequality ≤ in (iii). ϕi1 ...in and ψ = Turning to (iv), note that any decompositions ϕ = ψj1 ...jn give rise to a decomposition ϕi1 ...in ⊗ ψj1 ...jn . ϕ⊗ψ = (i1 ,j1 ),...,(in ,jn )

This proves the inequality in (iv). Assume now that B is abelian, Bk ⊂ B and θk : Bk → B is the inclusion map, 1 ≤ k ≤ n. Since there exists a ψ-preserving conditional expectation B → ∨k Bk , by Lemma 3.1.5 we may without loss of generality assume that B = ∨k Bk . Let {pj }rj=1 be the atoms of B, {χj }rj=1 the corresponding characters. The algebra A ⊗ B can be identiﬁed with ⊕rj=1 A. Under this identiﬁcation ϕ ⊗ ψ = ⊕j ψ(pj )ϕ, and

42

3 Dynamical Entropy

γk (a) ⊗ θk (b) = (χ1 (b)γk (a), . . . , χr (b)γk (a)). Thus the channel γk ⊗ θk factorizes through the channel ⊕j γk : ⊕j Ak → ⊕j A. By part (iii) (or rather by its obvious generalization to direct sums of r algebras) and Proposition 3.1.3(i), we get Hϕ⊗ψ (γ1 ⊗ θ1 , . . . , γn ⊗ θn ) ≤ H⊕j ψ(pj )ϕ (⊕j γ1 , . . . , ⊕j γn ) η(ψ(pj )) ψ(pj )Hϕ (γ1 , . . . , γn ) + = j

j

= Hϕ (γ1 , . . . , γn ) + S(ψ), which completes the proof of (iv), as S(ψ) = Hψ (B1 , . . . , Bn ) by Proposition 3.1.6. It remains to prove (v). So assume we have channels γk : Ak ⊗ Bk → A, 1 ≤ k ≤ n. Any abelian model for (A, ϕ, {γk }k ) can be considered as an abelian model for (A, ϕ, {γk |Ak }k ). The classical terms for the entropies of these models are the same. So we have to compare the correction terms. In view of the second equality in (3.5) we have to prove that if ω = i ωi is a decomposition of a state on Ak ⊗ Bk , then ωi (1)S(ˆ ωi ) ≤ S(ω|Ak ) − ωi (1)S(ˆ ωi |Ak ) + 2S(ω|Bk ). S(ω) − i

i

It suﬃces to show that each of the quantities ωi (1)(S(ˆ ωi |Ak ) − S(ˆ ωi )) S(ω) − S(ω|Ak ) and i

does not exceed S(ω|Bk ). By Theorem 2.2.2(vi) we have |S(ω) − S(ω|Ak )| ≤ S(ω|Bk ) and |S(ˆ ωi ) − S(ˆ ωi |Ak )| ≤ S(ˆ ωi |Bk ). It remains to note that ωi (1)S(ˆ ωi |Bk ) ≤ S(ω|Bk ) i

by concavity of von Neumann entropy, Theorem 2.2.2(ii).

Our next goal is to establish continuity of the entropy Hϕ (γ1 , . . . , γn ) as a function of γ1 , . . . , γn . For this we need to show that there exist models of a ﬁxed size. The idea is quite simple. Given a decomposition ϕ = ϕi1 ...in , we deﬁne a coarser decomposition by collecting indices i, k ∈ Ij such that the (j) (j) states ϕˆi and ϕˆk are close to each other. We shall formulate this procedure in a more reﬁned form, which will be useful later. We shall ﬁrst describe one more way of encoding decompositions of states. If we have a ﬁnite decomposition ϕ = ϕi1 ...in , we can deﬁne a state λ on A ⊗ C(X), where X = I1 × . . . × In , by

3.1 Mutual Entropy

43

λ(a ⊗ 1{(i1 ,...,in )} ) = ϕi1 ...in (a), where 1Z denotes the characteristic function of the set Z. If ξk is the partition of X with atoms I1 × . . . × {ik } × . . . × In , then the original decomposition can be written as λ(· ⊗ 1X1 ∩...∩Xn ). (3.8) ϕ= X1 ∈ξ1 ,...,Xn ∈ξn

More generally, let (X, µ) be a probability space. Deﬁnition 3.1.8. A coupling of (A, ϕ) with (X, µ) is a state λ on the tensor product A ⊗ L∞ (X, µ) such that λ|A = ϕ and λ|L∞ (X,µ) = µ. If X = S(A) is the state space of A, and µ is a regular Borel measure on X, then a coupling is called canonical if λ(a ⊗ f ) = ψ(a)f (ψ) dµ(ψ). X

Thus a canonical coupling is determined by a probability measure µ on S(A) with barycenter ϕ, that is, ϕ = ψ dµ(ψ). Note that for any a ∈ A the functional λ(a ⊗ ·) is normal on L∞ (X, µ), since if a is positive then it is dominated by a scalar multiple of µ. It follows that if X is a compact space and µ is a regular Borel measure on X, then every coupling λ of (A, ϕ) with (X, µ) is determined by its restriction to A ⊗ C(X). Moreover, any state λ on A⊗C(X) such that λ|A = ϕ and λ|C(X) = µ extends uniquely to a coupling of (A, ϕ) with (X, µ). Now if ξ1 , . . . , ξn are ﬁnite measurable partitions of (X, µ), we can deﬁne a decomposition of ϕ by (3.8). For any channels γ1 , . . . , γn we denote their mutual entropy with respect to this decomposition by Hλ (γ1 , . . . , γn ; ξ1 , . . . , ξn ). Thus by deﬁnition Hλ (γ1 , . . . , γn ; ξ1 , . . . , ξn ) = Hµ (∨k ξk ) + S(λ(γk (·) ⊗ 1Z ), ϕ ◦ γk ). k Z∈ξk

(3.9) By virtue of (3.2), the right hand side can be written as Hµ (∨k ξk )− Hµ (ξk )+ µ(Z)S(µ(Z)−1 λ(γk (·)⊗1Z ), ϕ◦γk ). (3.10) k

k Z∈ξk

Putting ϕZ = µ(Z)−1 λ(· ⊗ 1Z ), we can also write this expression as ⎛ ⎞ ⎝S(ϕ ◦ γk ) − Hµ (ξk ) + µ(Z)S(ϕZ ◦ γk )⎠ . (3.11) Hµ (∨k ξk ) − k

k

Note that (3.3) now becomes

Z∈ξk

44

3 Dynamical Entropy

Hλ (γ1 , . . . , γn ; ξ1 , . . . , ξn ) = Hµ (∨k ξk ) −

Hµ (ξk ) +

k

Hλ (γk ; ξk ). (3.12)

k

coupling in an obvious Any decomposition ϕ = i ϕi deﬁnes a canonical way: take the measure on X = S(A) to be ϕ (1)δ ˆi is the i ϕ ˆi , where δϕ i δ-measure concentrated at the point ϕˆi . To put it diﬀerently, given a coupling λ of (A, ϕ) with (Y, µ), and a ﬁnite measurable partition ξ of Y , we can construct a map f : Y → X which sends an atom Z of ξ to the state ϕZ = µ(Z)−1 λ(·⊗1Z ). Consider the image µ = f∗ (µ) of the measure µ, and put λ = λ◦(idA ⊗f ∗ ), where f ∗ : L∞ (X, µ ) → L∞ (Y, µ) is the map induced by f . Then λ is the canonical coupling deﬁned by µ . In ∗ deed, for any bounded Borel function g on X we have f (g) = Z∈ξ g(ϕZ )1Z , whence λ (a ⊗ g) = g(ϕZ )λ(a ⊗ 1Z ). Z

On the other hand, g(ψ)ψ(a) dµ (ψ) = g(ψZ )ψZ (a)µ(Z) = g(ψZ )λ(a ⊗ 1Z ). X

Z

Z

Thus the coupling λ is canonical. Note further that if ζ is a ﬁnite Borel partition of X, then the decompositions of ϕ deﬁned by the pairs (λ, f −1 (ζ)) and (λ , ζ) are the same, so that Hλ (γ; f −1 (ζ)) = Hλ (γ; ζ) for any channel γ: B → A. In particular, if ζ is such that f −1 (ζ) = ξ, then Hλ (γ; ξ) = Hλ (γ; ζ). But in fact any suﬃciently ﬁne ζ gives Hλ (γ; ζ) close to Hλ (γ; ξ). Lemma 3.1.9. Let notation be as above. For ε > 0 choose δ > 0 such that |S(ψ1 ) − S(ψ2 )| < ε for states ψ1 and ψ2 on B as soon as ψ1 − ψ2 < δ. Assume ζ is a ﬁnite Borel partition of S(A) such that ϕ1 ◦ γ − ϕ2 ◦ γ < δ as soon as ϕ1 and ϕ2 are in the same atom of ζ. Then |Hλ (γ; ξ) − Hλ (γ; ζ)| < ε. Proof. For W ∈ ζ, set ϕW = µ (W )−1 λ (· ⊗ 1W ). Then ϕW = µ (W )−1

Z∈ξ:Z⊂f −1 (W )

λ(· ⊗ 1Z ) =

Z∈ξ:Z⊂f −1 (W )

µ(Z) ϕZ . µ(f −1 (W ))

By assumption, the states ϕZ with Z ⊂ f −1 (W ) are δ-close to each other when composed with γ. It follows that (ϕW − ϕZ ) ◦ γ < δ, and thus |S(ϕW ◦ γ) − S(ϕZ ◦ γ)| < ε for Z ∈ ξ, Z ⊂ f −1 (W ). Since

3.1 Mutual Entropy

S(ϕ ◦ γ) − Hλ (γ; ζ) =

45

µ(f −1 (W ))S(ϕW ◦ γ)

W

by (3.11), and similarly S(ϕ ◦ γ) − Hλ (γ; ξ) =

µ(Z)S(ϕZ ◦ γ) =

Z

µ(Z)S(ϕZ ◦ γ),

W Z⊂f −1 (W )

we see that |Hλ (γ; ξ) − Hλ (γ; ζ)| < ε.

Note that a partition ζ satisfying the conditions of the previous lemma can be chosen such that the number |ζ| of elements of ζ is not bigger than the number of balls of diameter δ which are needed to cover the state space of B. Thus, what we have proved, is that if we want to compute Hϕ (γ) up to ε, it suﬃces to consider canonical couplings and one Borel partition of S(A) of size depending only on dim B and ε. More generally, if λ is a coupling of (A, ϕ) with (Y, µ), and ξ1 , . . . , ξn are ﬁnite measurable partitions of Y , we can deﬁne a measure µ on X n , where X = S(A), and a coupling λ of (A, ϕ) with (X n , µ ) as follows. Let fk : Y → X be the map deﬁned by λ and ξk , so fk maps an atom Z of ξk to the state µ(Z)−1 λ(·⊗1Z ) ∈ X. Then the map f = (f1 , . . . , fn ): Y → X n is measurable, and we set µ = f∗ (µ) and λ = λ ◦ (id ⊗ f ∗ ). The properties of the coupling λ are summarized in the following proposition. Proposition 3.1.10. Under the above assumptions, ﬁx ε > 0 and choose δ > 0 such that |S(ψ1 ) − S(ψ2 )| < ε for states ψ1 and ψ2 on Ak as soon as ψ1 − ψ2 < δ, 1 ≤ k ≤ n. For each k let ζk be a ﬁnite Borel partition of X such that ϕ1 ◦ γk − ϕ2 ◦ γk < δ as soon as ϕ1 and ϕ2 are in the same atom of ζk . Then (i) fk−1 (ζk ) ≺ ξk ; (ii) the coupling λk = λ ◦ (id ⊗ pr∗k ) = λ ◦ (id ⊗ fk∗ ) of (A, ϕ) with (X, µk ), where prk : X n → X is the projection onto the k-th factor and µk = (prk )∗ (µ ) = (fk )∗ (µ), is canonical and |Hλ (γk ; ξk ) − Hλk (γk ; ζk )| < ε; −1 (iii) Hλ (γ1 , . . . , γn ; ξ1 , . . . , ξn ) < Hλ (γ1 , . . . , γn ; pr−1 1 (ζ1 ), . . . , prn (ζn )) + nε.

Proof. Part (i) is obvious as by deﬁnition fk maps the atoms of ξk to points. Part (ii) follows from Lemma 3.1.9 and the discussion preceding it. −1 To prove (iii) note that since Hλ (γ1 , . . . , γn ; pr−1 1 (ζ1 ), . . . , prn (ζn )) is −1 −1 −1 nothing but Hλ (γ1 , . . . , γn ; f1 (ζ1 ), . . . , fn (ζn )), and Hλ (γk ; fk (ζk )) equals Hλk (γk ; ζk ), in view of (ii) and equality (3.12), it suﬃces to show that Hµ (∨k ξk ) −

k

Hµ (ξk ) ≤ Hµ (∨k fk−1 (ζk )) −

Hµ (fk−1 (ζk )).

k

Since fk−1 (ζk ) ≺ ξk , this is true by Proposition 1.1.9(iv).

46

3 Dynamical Entropy

The previous proposition in its full generality will be used in subsequent chapters. What is important for us at the moment, is that it implies existence of models of a ﬁxed size. Namely, given ε > 0 and d ∈ N there exists r > 0 such that for any n ∈ N and any channels γk : Ak → A, 1 ≤ k ≤ n, such that dim Ak ≤ d, there exist a coupling λ of (A, ϕ) with a probability space (Y, µ), and ﬁnite measurable partitions ζ1 , . . . , ζn such that |ζk | ≤ r, 1 ≤ k ≤ n, and Hϕ (γ1 , . . . , γn ) < Hλ (γ1 , . . . , γn ; ζ1 , . . . , ζn ) + nε. Explicitly, ﬁx ε < ε. Then take r to be the number of balls of diameter δ which are needed to cover the state space of a C∗ -algebra B with dim B ≤ d, where δ is such that |S(ψ1 ) − S(ψ2 )| < ε for states ψ1 and ψ2 on B as soon as ψ1 − ψ2 < δ. Take a coupling Λ and partitions ξ1 , . . . , ξn such that HΛ (γ1 , . . . , γn ; ξ1 , . . . , ξn ) is close to Hϕ (γ1 , . . . , γn ) up to ε−ε . Then applying the procedure described above we get λ and ζ1 , . . . , ζn . For channels γ, γ : B → A, put γ − γ ϕ =

sup

γ(x) − γ (x)ϕ ,

x∈B,x≤1

where aϕ = ϕ(a∗ a)1/2 . Proposition 3.1.11. For every ε > 0 and d ≥ 1 there exists δ > 0 such that for any C∗ -algebra A with a state ϕ, n ∈ N and channels γk , γk : Ak → A such that dim Ak ≤ d and γk − γk ϕ < δ, 1 ≤ k ≤ n, we have |Hϕ (γ1 , . . . , γn ) − Hϕ (γ1 , . . . , γn )| < nε. Proof. By Proposition 3.1.10 and the discussion following it we can ﬁnd r ∈ N depending only on ε and d, and a decomposition ϕ = i1 ,...,in ∈I ϕi1 ...in with |I| ≤ r such that Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) > Hϕ (γ1 , . . . , γn ) −

nε . 2

To estimate the diﬀerence between the mutual entropies of γ1 , . . . , γn and γ1 , . . . , γn with respect to the decomposition ϕ = i1 ,...,in ϕi1 ...in we have to estimate the diﬀerences between the correction terms. Thus, what we need to prove is the following: there exists δ > 0such that if γ, γ : B → A are channels, dim B ≤ d, γ − γ ϕ < δ and ϕ = i∈I ϕi with |I| ≤ r, then ε S(ϕi ◦ γ, ϕ ◦ γ) − S(ϕi ◦ γ , ϕ ◦ γ ) < . (3.13) 2 i i We have S(ϕi ◦ γ, ϕ ◦ γ) − S(ϕi ◦ γ , ϕ ◦ γ ) i

i

3.1 Mutual Entropy

= S(ϕ ◦ γ) − S(ϕ ◦ γ ) −

47

ϕi (1)(S(ϕˆi ◦ γ) − S(ϕˆi ◦ γ )).

i

Choose δ0 > 0 such that |S(ψ1 ) − S(ψ2 )| < ε/6 for states ψ1 and ψ2 on B as soon as ψ1 − ψ2 < δ0 and dim B ≤ d. If δ < δ0 , then as |ϕ(x)| ≤ xϕ , we have ϕ ◦ γ − ϕ ◦ γ ≤ γ − γ ϕ < δ0 , so that

ε . 6 Set I0 = {i ∈ I | ϕi (1) > (δ/δ0 )2 }. Then for i ∈ I0 we have |S(ϕ ◦ γ) − S(ϕ ◦ γ )|

0 there exists a ﬁnite index set I depending only on ε and the dimensions of the Ak ’s such that for any state ϕ there is a decomposition ϕ = i1 ,...,in ∈I ϕi1 ...in with Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) > Hϕ (γ1 , . . . , γn ) − nε. Now if {ϕj }j∈J is a net converging to ϕ we can choose decompositions of the ϕj ’s as above, and passing to a subnet may assume that these decompositions converge to a decomposition of ϕ. Since the map (ω, ψ) → S(ω◦γk , ψ◦γk ) is continuous on the set {ω ≤ ψ} by Theorem 2.1.2(iii), it follows that Hϕ (γ1 , . . . , γn ) ≥ lim sup Hϕj (γ1 , . . . , γn ) − nε. j

Since ε > 0 was arbitrary, we get the result.

48

3 Dynamical Entropy

3.2 Entropy of Dynamical Systems We are now ready to deﬁne entropy of dynamical systems. By a C∗ -dynamical system we mean a triple (A, ϕ, α), where A is a unital C∗ -algebra, ϕ a state on A, α a ϕ-preserving automorphism of A. If A is a von Neumann algebra and the state is normal, we call (A, ϕ, α) a W∗ -dynamical system. Deﬁnition 3.2.1. The entropy of α with respect to a channel γ: B → A is 1 Hϕ (γ, α ◦ γ, . . . , αn−1 ◦ γ). n→∞ n

hϕ (γ; α) = lim

The dynamical entropy hϕ (α) is the supremum of hϕ (γ; α) for all possible γ’s. Note that if we denote Hϕ (γ, α ◦ γ, . . . , αn−1 ◦ γ) by an , then an+m ≤ an + am by properties (ii) and (iii) in Proposition 3.1.3. Hence the limit in the above deﬁnition exists by Lemma 1.1.2. Note also that the deﬁnition makes sense for any unital completely positive ϕ-preserving map α. Theorem 3.2.2. Let (A, ϕ, α) be a C∗ -dynamical system. Then (i) if β: A → B is an isomorphism, then hϕ◦β −1 (β ◦ α ◦ β −1 ) = hϕ (α); (ii) if πϕ : A → B(H) is the GNS-representation and ϕ¯ and α ¯ are the normal state and the automorphism of πϕ (A) deﬁned by ϕ and α, respectively, then hϕ¯ (¯ α) = hϕ (α); (iii) hϕ (α−1 ) = hϕ (α), and hϕ (αn ) ≤ |n|hϕ (α) for any n ∈ Z; (iv) for any C∗ -dynamical system (B, ψ, β), hλϕ⊕(1−λ)ψ (α ⊕ β) = λhϕ (α) + (1 − λ)hψ (β) and hϕ⊗ψ (α ⊗ β) ≥ hϕ (α) + hψ (β); (v) if B is an α-invariant subalgebra of A and there exists a ϕ-preserving conditional expectation A → B, then hϕ|B (α|B ) ≤ hϕ (α). Proof. Part (i) follows immediately from the deﬁnitions. ¯ ) = hϕ (γ; α) for any channel Proposition 3.1.7(i) shows that hϕ¯ (πϕ ◦ γ; α γ: B → A. By Proposition 3.1.11 we also know that if two channels γ, γ : B → πϕ (A) are close in the · ϕ¯ -seminorm, then hϕ¯ (γ; α ¯ ) and hϕ¯ (γ ; α ¯ ) are close. Thus to prove (ii) we have to show that for any channel γ: B → πϕ (A) there exists a channel γ˜ : B → A such that γ − πϕ ◦ γ˜ ϕ¯ is arbitrarily small. This is possible by A.10 and A.11, and (ii) is proved. By Proposition 3.1.3(ii) we have Hϕ (γ, α ◦ γ, . . . , αn ◦ γ) = Hϕ (α−n ◦ γ, α−n+1 ◦ γ, . . . , γ).

3.2 Entropy of Dynamical Systems

49

It follows that hϕ (α−1 ) = hϕ (α). If n ≥ 1, we have Hϕ (γ, αn ◦ γ, . . . , α(m−1)n ◦ γ) ≤ Hϕ (γ, α ◦ γ, . . . , αmn−1 ◦ γ), whence hϕ (γ; αn ) ≤ nhϕ (γ; α). Thus (iii) is proved. Part (iv) follows from Proposition 3.1.7(iii)-(iv), while (v) is a consequence of Lemma 3.1.5.

To proceed further we need to restrict ourselves to systems satisfying the following approximation property. We say that a net of channels {γi : Ai → A}i is ϕ-approximating if there exists a net {θi : A → Ai }i of unital completely positive maps such that (γi ◦ θi )(x) − xϕ → 0 for any x ∈ A. If γ: B → A is an arbitrary channel, then γi ◦ θi ◦ γ − γϕ → 0. Hence by Proposition 3.1.11 lim hϕ (γi ◦ θi ◦ γ; α) = hϕ (γ; α). i

On the other hand, hϕ (γi ◦ θi ◦ γ; α) ≤ hϕ (γi ; α) ≤ hϕ (α) by Proposition 3.1.3(i). We thus get the following weak analogue of the Kolmogorov-Sinai theorem. Theorem 3.2.3. If {γi : Ai → A}i is a ϕ-approximating net for a C∗ -dynamical system (A, ϕ, α), then hϕ (α) = lim hϕ (γi ; α).

i

Approximating nets exist for nuclear C∗ -algebras and injective von Neumann algebras, and in fact existence of a ϕ-approximating net is close to πϕ (A) being injective. If {Ai }i is an increasing net of ﬁnite dimensional C∗ -subalgebras of a C∗ -algebra A with ∪i Ai norm-dense in A, then the net {Ai → A}i is ϕ-approximating (take arbitrary conditional expectations A → Ai for θi ’s). If A is a von Neumann algebra, ϕ is normal and ∪i Ai is dense in A in the strong operator topology, then the net {Ai → A}i is again ϕ-approximating, but this is not so simple, see e.g. [91]. Note that we do not really need this fact, since what is important for Theorem 3.2.3, is that any channel into A can be approximated by channels into ∪i Ai , which is easy by A.10. We also have the following related approximation result. Proposition 3.2.4. Let (A, ϕ, α) be a C∗ -dynamical system, {Ai }i∈I an increasing net of α-invariant C∗ -subalgebras of A such that ∪i πϕ (Ai ) is strongly operator dense in πϕ (A). Then hϕ (α) ≤ lim inf i hϕ|Ai (α|Ai ). Proof. By A.10 and A.11 any channel into A can be approximated in the · ϕ -seminorm by channels into ∪i Ai . But if γ: B → Ai is a channel, then hϕ (γ; α) ≤ hϕ|Ai (γ; α|Ai ) by Theorem 3.1.3(ii) applied to the inclusion map θ: Ai → A.

We now show how approximation properties translate into properties of dynamical entropy.

50

3 Dynamical Entropy

Theorem 3.2.5. Let (A, ϕ, α) be a C∗ -dynamical system having a ϕ-approximating net. Then (i) hϕ (αn ) = |n|hϕ (α) for any n ∈ Z; moreover, if {αt }t∈R is a ϕ-preserving one-parameter automorphism group of A such that α1 = α, then hϕ (αt ) = |t|hϕ (α) for any t ∈ R; (ii) if ϕ = λω + (1 − λ)ψ, where ω and ψ are α-invariant states, then hϕ (α) ≥ λhω (α) + (1 − λ)hψ (α); (iii) if (B, ψ, β) is an abelian C∗ -dynamical system, then hϕ⊗ψ (α ⊗ β) = hϕ (α) + hψ (β); (iv) if ψ is a state on a C∗ -algebra B, and there exists a ψ-approximating net, then hϕ⊗ψ (α ⊗ idB ) = hϕ (α). In Sect. 3.3 we shall extend (iii) to type I algebras. Proof of Theorem 3.2.5. We already know from Theorem 3.2.2(iii) that hϕ (αn ) ≤ |n|hϕ (α). To prove the opposite inequality we may by the same theorem assume n ≥ 1. Let {γi : Ai → A}i be a ϕ-approximating net, and θi : A → Ai , i ∈ I, be maps as in the deﬁnition of such a net. Fix a channel γ: B → A and δ > 0, and choose i ∈ I such that γi ◦ θi ◦ αk ◦ γ − αk ◦ γϕ < δ for 0 ≤ k ≤ n − 1. Then αln ◦ γi ◦ θi ◦ αk ◦ γ − αln+k ◦ γϕ < δ for any l ≥ 0 and 0 ≤ k ≤ n − 1. It follows by Propositions 3.1.11 and 3.1.3(i),(iv) that Hϕ (γ, α ◦ γ, . . . , αnm−1 ◦ γ) ≤ Hϕ ({αln ◦ γi ◦ θi ◦ αk ◦ γ}0≤l≤m−1,0≤k≤n−1 ) + nmε ≤ Hϕ (γi , αn ◦ γi , . . . , αn(m−1) ◦ γi ) + nmε, where ε = ε(δ, dim B) → 0 as δ → 0. Thus hϕ (γ; α) ≤ n−1 hϕ (γi ; αn ) + ε, so that hϕ (αn ) ≥ nhϕ (α). Assume now we have a ﬂow {αt }t with α1 = α. To prove that t−1 hϕ (αt ) is independent of t > 0, it suﬃces to show that t2 hϕ (αt1 ) ≤ t1 hϕ (αt2 ) for any t1 , t2 > 0. Replacing {αt }t by {αt2 t }t , we just have to show that hϕ (αt ) ≤ thϕ (α1 ) for any t > 0. The proof is similar to the argument above. Since the set {αs (a) | 0 ≤ s ≤ 1} is compact for any a ∈ A, given a channel γ: B → A and δ > 0 we can ﬁnd i ∈ I such that γi ◦ θi ◦ αs ◦ γ − αs ◦ γϕ < δ for 0 ≤ s ≤ 1. [lt]

Then α1 ◦ γi ◦ θi ◦ αlt−[lt] ◦ γ − αtl ◦ γϕ < δ for any l ≥ 0, where [lt] denotes the integer part of lt. Hence

3.2 Entropy of Dynamical Systems [(n−1)t]

Hϕ (γ, αt ◦ γ, . . . , αtn−1 ◦ γ) ≤ Hϕ (γi , α1 ◦ γi , . . . , α1

51

◦ γi ) + nε,

so that in the limit hϕ (γ; αt ) ≤ thϕ (γi ; α1 ) + ε. Thus (i) is proved. By Proposition 3.1.7(ii) we have hϕ (α) ≥ λhω (α) + (1 − λ)hψ (α) − η(λ) − η(1 − λ). We can apply this inequality to αn . By virtue of (i), dividing by n and letting n → ∞, we get (ii). Let (B, ψ, β) be an abelian C∗ -dynamical system. By Proposition 3.2.2(ii), in proving (iii) we may assume that B is a von Neumann algebra. Then B is in particular an inductive limit of ﬁnite dimensional abelian C∗ -algebras, so there exists a ψ-approximating net {γk }k consisting of injective homomorphisms. Then {γi ⊗ γk }(i,k) is a (ϕ ⊗ ψ)-approximating net, and (iii) follows from Proposition 3.1.7(iv). To prove (iv), note that hϕ⊗ψ (α ⊗ idB ) ≥ hϕ (α) by Theorem 3.2.2(v). To prove the opposite inequality consider ﬁrst the case when B is ﬁnite dimensional. If {γi }i is a ϕ-approximating net, then the net {γi ⊗ idB }i is (ϕ ⊗ ψ)-approximating. By Proposition 3.1.7(v) we have hϕ⊗ψ (γi ⊗ idB ; α ⊗ idB ) ≤ hϕ (γi ; α) + 2S(ψ), so that hϕ⊗ψ (α ⊗ idB ) ≤ hϕ (α) + 2S(ψ). As in the proof of (ii), applying this inequality to αn we conclude that hϕ⊗ψ (α ⊗ idB ) ≤ hϕ (α). Consider now the general case. Let {γk : Bk → B}k be a ψ-approximating net. Then {γi ⊗γk }(i,k) is a (ϕ⊗ψ)-approximating net. By Proposition 3.1.3(ii) and ﬁnite dimensionality of the Bk ’s we have hϕ⊗ψ (γi ⊗ γk ; α ⊗ idB ) ≤ hϕ⊗(ψ◦γk ) (γi ⊗ idBk ; α ⊗ idBk ) ≤ hϕ (α). Hence hϕ⊗ψ (α ⊗ idB ) ≤ hϕ (α).

Similarly to the classical case, the general theory we have developed so far was ﬁrst applied to compute entropy of noncommutative Bernoulli shifts. We next discuss these systems. Example 3.2.6. (i) Noncommutative Bernoulli shifts. Let B be a ﬁnite dimensional C∗ algebra, ψ a state on B, (M, ϕ) = (B, ψ)⊗Z the inﬁnite W∗ -tensor product (that is, if A = B ⊗Z is the inﬁnite C∗ -tensor product, then M is the weak operator closure of A in the GNS-representation corresponding to the state ψ ⊗Z ), and α the shift to the right on M . Then hϕ (α) = S(ψ). To see this, let πn : B → M be the canonical homomorphism onto the n-th factor, so that α(πn (b)) = πn+1 (b). Let B[n,m] be the algebra generated by πk (B) for n ≤ k ≤ m. If D is a maximal abelian subalgebra in the centralizer of ψ, then the algebras πk (D), n ≤ k ≤ m, mutually commute and generate a

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3 Dynamical Entropy

maximal abelian subalgebra in the centralizer of the restriction of ϕ to B[n,m] . By Proposition 3.1.6 it follows that for any n, m ∈ N we have Hϕ (B[−n,n] , . . . , αm (B[−n,n] )) = S(ϕ|B[−n,n+m] ) = (m + 2n + 1)S(ψ). Thus hϕ (B[−n,n] ; α) = S(ψ), and hϕ (α) = S(ψ) by Theorem 3.2.3. We remark that the same result is true if B is inﬁnite dimensional and we use the deﬁnition of entropy of a state mentioned in Notes to Chap. 2. (ii) Bernoulli shifts on the hyperﬁnite II1 -factor. Consider the same system as above, but assume that B ∼ = Matn (C) and ψ is faithful. Let N = Mϕ be the centralizer of ϕ, so β = α|N is an automorphism and τ = ϕ|N is a faithful normal β-invariant trace on N . Then hτ (β) = S(ψ). To see this we can repeat the above argument with B[n,m] replaced by N ∩ B[n,m] . Another possibility is to argue as follows. Denote by C the von Neumann subalgebra of M generated by πn (D), n ∈ Z. There exist ϕ-preserving conditional expectations M → N and N → C. Hence hτ |C (β|C ) ≤ hτ (β) ≤ hϕ (α). But (C, τ |C , β|C ) is the classical Bernoulli shift with entropy S(ψ). Since also hϕ (α) = S(ψ), we see that hτ (β) = S(ψ). Observe next that N is the hyperﬁnite II1 -factor. The hyperﬁniteness is clear. We shall give two proofs of the factoriality. The ﬁrst one is shorter, but the second one can be applied in more general situations. The group S∞ of ﬁnite permutations of Z acts naturally on M . Denote this action by α: S∞ → Aut(M ). If x, y ∈ ∪n B[−n,n] , then ϕ(αg (x)y) = ϕ(x)ϕ(y) for g outside a ﬁnite subset of S∞ . It follows that for any x, y ∈ M we have ϕ(αg (x)y) → ϕ(x)ϕ(y) as g goes to inﬁnity, meaning that g eventually leaves every ﬁnite subset of S∞ . In particular, the ﬁxed point algebra M S∞ is trivial. On the other hand, since any automorphism of a full matrix algebra is inner, for each g ∈ S∞ there exists ug ∈ ∪n B[−n,n] such that αg = Ad ug . Since ϕ is αg -invariant, we have ug ∈ Mϕ = N . Hence if z is in the center of N , then z ∈ M S∞ . Therefore z is a scalar. For the second proof note that M , being an inﬁnite tensor product of factors with respect to a product-state, is a factor. Thus it suﬃces to check that the center of N is contained in the center of M . We shall prove that the relative commutant N ∩ M is contained in the center of M . Let z ∈ N ∩ M , x ∈ B[−m,m] , x = 0. We want to prove that xz = zx. Since σtϕ |B[−m,m] has pure point spectrum, without loss of generality we may assume that σtϕ (x) = λit x for t ∈ R and some λ > 0. Then σtϕ (αn (x∗ )x) = αn (x∗ )x, so αn (x∗ )x ∈ N . Hence αn (x∗ )xz = zαn (x∗ )x, and thus αn (xx∗ )xz = αn (x)zαn (x∗ )x. We claim that αn (xx∗ ) → ϕ(xx∗ )1 in the weak operator topology and [αn (x), z] → 0 in the strong operator topology as n → ∞. Thus letting n → ∞ in the equality above we get ϕ(xx∗ )xz on the left hand side and ϕ(xx∗ )zx on the right hand side, whence xz = zx.

3.2 Entropy of Dynamical Systems

53

It remains to prove the claims. If y ∈ ∪k B[−k,k] , then ϕ(αn (xx∗ )y) = ϕ(xx∗ )ϕ(y) if n is suﬃciently large. Thus any weak operator limit point a of the sequence {αn (xx∗ )}n has the property ϕ(ay) = ϕ(xx∗ )ϕ(y). Hence a = ϕ(xx∗ )1, and the ﬁrst claim is proved. To prove the second claim, note 1/2 that αn (x)ξϕ = Jϕ ∆ϕ αn (x∗ )ξϕ = λ−1/2 Jϕ αn (x∗ )ξϕ , where we use the same notation as on page 35. Then the equality [αn (x), y]ξϕ = αn (x)yξϕ − yαn (x)ξϕ = αn (x)yξϕ − λ−1/2 Jϕ αn (x∗ )Jϕ yξϕ shows that to prove that [αn (x), y] → 0 strongly for any y ∈ M , it suﬃces to consider y lying in a strongly dense subspace. But if y ∈ ∪k B[−k,k] then [αn (x), y] = 0 for all n suﬃciently large. This completes our second proof of factoriality of N . It follows from the discussion prior to Lemma 3.1.5 that the entropy of a system can be smaller than the entropy of a subsystem. Bernoulli shifts provide a simple example. If in Example 3.2.6 we took an arbitrary maximal abelian subalgebra D of B, then we would get an abelian system (C, ϕ|C , α|C ) with entropy S(ψ|D ), which is strictly larger than S(ψ) unless D is in the centralizer of ψ. Nevertheless if we have a conditional expectation onto a subalgebra commuting with an automorphism, then any state on the subalgebra extends to a state on the algebra such that the entropy of the system becomes at least as large as the entropy of the subsystem. The following result is a simple but useful extension of this fact. Proposition 3.2.7. Let α be an automorphism of a unital C∗ -algebra A, B ⊂ A an α-invariant unital C∗ -subalgebra, ψ an α-invariant state on B. Assume there exists a ψ-approximating net. Then there exists an α-invariant state ϕ on A such that ϕ|B = ψ and hϕ (α) ≥ hψ (α|B ). Proof. Let {γi : Bi → B}i be a ψ-approximating net, and θi : B → Bi , i ∈ I, be maps as in the deﬁnition of such a net. By Arveson’s extension theorem, A.8, we can extend each θi to a unital completely positive map θ¯i : A → Bi . Consider the GNS-representation π: B → B(H) corresponding to ψ, and let ψ¯ and β be the normal state and the automorphism of π(B) deﬁned by ψ and α|B , respectively. Let Φ: A → π(B) be any pointwise weak operator limit point of {π ◦γi ◦ θ¯i }i . Then Φ is a unital completely positive map, and Φ(b)ξψ = π(b)ξψ for any b∈ B. Replacing further Φ by a pointwise weak operator limit point n−1 of {n−1 k=0 β k ◦ Φ ◦ α−k }, we may also assume that β ◦ Φ = Φ ◦ α. Then ¯ ϕ = ψ ◦ Φ is an α-invariant state extending ψ. Moreover, if ω is a positive linear functional on B such that ω ≤ ψ, then ω extends to a normal positive functional ω ¯ on π(B) , and ω ¯ ◦ Φ extends ω. Hence any decomposition of ψ extends to a decomposition of ϕ, so that hϕ (α) ≥ hψ (α|B ).

A related result will be proved in Chap. 5.

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3 Dynamical Entropy

3.3 Type I Algebras In this section we consider type I algebras. Not surprisingly the theory of dynamical entropy in this case reduces completely to abelian systems. Recall that a type I von Neumann algebra is a direct sum of algebras of ¯ the form A⊗B(H) with A abelian. If the algebra is σ-ﬁnite, that is, every set of mutually orthogonal projections is at most countable, then one can equivalently say that the algebra is a direct integral of algebras B(H), see App. C. Theorem 3.3.1. Let (M, ϕ, α) be a W∗ -dynamical system with M a type I von Neumann algebra. Let Z = Z(M ) be the center of M . Then hϕ (α) = hϕ|Z (α|Z ). More generally, if (B, ψ, β) is a C∗ -dynamical system having a ψ-approximating net, then hψ⊗ϕ (β ⊗ α) = hψ (β) + hϕ|Z (α|Z ). Proof. Since the modular group acts trivially on the center, there exists a ϕ-preserving conditional expectation M → Z, and hence the inequality ≥ follows from Theorem 3.2.2(iv),(v). To prove the opposite inequality consider ﬁrst the case when M ∼ = Z ⊗ Matn (C). The proof in this case is similar to that of Theorem 3.2.5(iv). Let {γi }i be a ψ-approximating net and {γk }k be a ϕ|Z -approximating net. Then the net {γi ⊗ γk ⊗ idMatn (C) }(i,k) is (ψ ⊗ ϕ)approximating. By Proposition 3.1.7(v) we have hψ⊗ϕ (γi ⊗ γk ⊗ idMatn (C) ; β ⊗ α) ≤ hψ⊗(ϕ|Z) (γi ⊗ γk ; β ⊗ α|Z ) + 2 log n, so that hψ⊗ϕ (β ⊗ α) ≤ hψ⊗(ϕ|Z) (β ⊗ α|Z ) + 2 log n. Applying this inequality to β m and αm we conclude that hψ⊗ϕ (β ⊗ α) ≤ hψ⊗(ϕ|Z) (β ⊗ α|Z ) = hψ (β) + hϕ|Z (α|Z ), where the equality follows from Theorem 3.2.5(iii). Consider next the case when M is ﬁnite, so M is a (possibly inﬁnite) direct sum of algebras Zn ⊗Matn (C) with Zn abelian. The required inequality would follow from the previous case if we could use an inﬁnite analogue of Theorem 3.2.2(iv). In other words, we have to check that if (N, ϕ, γ) is a W∗ dynamical system and zn , n ∈ N, are γ-invariant central projections in N such that n zn = 1, then hϕ (γ) = λn hϕn (γ|N zn ), n

where λn = ϕ(zn ) and ϕn = λ−1 n ϕ|N zn . Set Mn = N (z1 + . . . + zn ) + C(1 − z1 − . . . − zn ).

3.3 Type I Algebras

55

Then ∪n Mn is strongly operator dense in N . Since there exist ϕ-preserving conditional expectations N → Mn , by Theorem 3.2.2(v) and Proposition 3.2.4 we get hϕ|Mn (γ|Mn ) → hϕ (γ). Since hϕ|Mn (γ|Mn ) =

n

λk hϕk (γ|N zk )

k=1

by Theorem 3.2.2(iv), we get the result. Consider now the general case. Replacing M by pM p, where p is the support of ϕ, we may assume that ϕ is faithful, in particular M is σﬁnite. We claim that there exists an increasing sequence {pn }∞ n=1 of ﬁnite α-invariant projections which belongs to the centralizer Mϕ of ϕ and is such that 1 − pn ϕ → 0. Indeed, identify Z with L∞ (X, µ), so that ϕ|Z is deﬁned by µ. Then, see C.3 for the notation, we have a direct integral decomposition ⊕ ⊕ M= B(Hx )dµ(x), ϕ = ϕx dµ(x). X

X

Let Qx be the density operator of the state ϕx , so ϕx = Tr(· Qx ). Let pn (x) be the spectral projection of Qx corresponding to the interval [1/n, 1]. Then pn (x) is a ﬁnite projection in B(Hx ) belonging to the centralizer of ϕx , and 1 − pn (x)ϕx → 0. Hence

⊕

pn =

pn (x)dµ(x), n ∈ N,

X

are the required projections. Set Mn = pn M pn + C(1 − pn ). Then a → pn apn + ϕ(a(1 − pn ))(1 − pn ) is a ϕ-preserving conditional expectation onto Mn . Denote it by En . Since 1 − pn ϕ → 0, we have a − En (a)ϕ → 0 for any a ∈ M . It follows that hψ⊗ϕ (β ⊗ α) = lim hψ⊗(ϕ|Mn ) (β ⊗ α|Mn ). n

The algebra Mn is ﬁnite and of type I. Denote by Zn its center. By the previous case hψ⊗(ϕ|Mn ) (β ⊗ α|Mn ) = hψ (β) + hϕ|Zn (α|Zn ). To ﬁnish the proof, it thus suﬃces to check that hϕ|Zn (α|Zn ) ≤ hϕ|Z (α|Z ) for any n. If pn = 1, there is nothing to prove. Otherwise let zn ∈ Z be the central support of pn . Then Zn = Zpn ⊕ C(1 − pn ) is isomorphic to Zzn ⊕ C. Under this isomorphism the state ϕ|Zn becomes ϕ(pn )ψn ⊕ ϕ(1 − pn ), where ψn = ϕ(pn )−1 ϕ(· pn )|Zzn . Hence hϕ|Zn (α|Zn ) = ϕ(pn )hψn (α|Zzn ). On the other hand, consider the state ϕn = ϕ(zn )−1 ϕ|Zzn . Since Zzn + C(1 − zn ) is an α-invariant subalgebra of Z, we have hϕ|Z (α|Z ) ≥ ϕ(zn )hϕn (α|Zzn ). Hence we just have to prove that

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3 Dynamical Entropy

hψn (α|Zzn ) ≤

ϕ(zn ) hϕ (α|Zzn ). ϕ(pn ) n

Since ψn ≤ ϕ(pn )−1 ϕ(zn )ϕn , and the function η(t) = −t log t is monotone for small t, this follows easily from the classical deﬁnition of entropy.

Recall that a C∗ -algebra is of type I if its second dual is a type I von Neumann algebra. It follows from Theorem 3.3.1 that inner automorphisms of type I C∗ -algebras have zero entropy. It turns out this is a characteristic property of the type. Theorem 3.3.2. Let A be a unital C∗ -algebra. Then A is of type I if and only if for every unitary u ∈ A and (Ad u)-invariant state ϕ on A we have hϕ (Ad u) = 0. Proof. One implication follows from Theorem 3.3.1. So assuming that A is not of type I we have to show that there exists an inner automorphism with nonzero entropy. By Glimm’s theorem, see e.g. [156, Corollary 6.7.4], there exists a unital C∗ -subalgebra B of A and an ideal I ⊂ B such that B/I is a UHF-algebra. Assume there exists a unitary v ∈ B/I such that hϕ (Ad v) > 0 for a state ϕ on B/I. Since the unitary group of an AF-algebra is connected, the unitary v lifts to a unitary u in B by [24, Proposition 3.4.5]. Let ψ be the state on B obtained by composing ϕ with the quotient map B → B/I. Then hψ (Ad u|B ) = hϕ (Ad v) by Theorem 3.2.2(ii). By Proposition 3.2.7 we can extend the state ψ to a state ϕ on A such that hϕ (Ad u) ≥ hψ (Ad u|B ) > 0. Thus it remains to show that UHF-algebras have inner automorphisms with positive entropy. This will be done in the next lemma.

Lemma 3.3.3. Let R be the hyperﬁnite II1 -factor, u ∈ R a unitary with nonatomic spectral measure, A ⊂ R a strongly operator dense UHF-subalgebra. Then there exists a unitary v ∈ R such that vuv ∗ ∈ A. In particular, there exist inner automorphisms of A having positive entropy with respect to the tracial state. Proof. Let τ be the tracial state on R. We are going to use the fact that if p and q are projections in R such that p − q2 = p − qτ is small and τ (p) = τ (q), then there exists a partial isometry w such that w∗ w = p, ww∗ = q and w −p2 is small, see e.g. [214, Lemma XIV.2.1]. Here is a sketch of the proof. Let qp = w1 |qp| be the polar decomposition. Then p − w1 2 is small. Let w2 be any partial isometry such that w2∗ w2 = p − w1∗ w1 and w2 w2∗ = q − w1 w1∗ . Put w = w1 + w2 . Let F ⊂ [0, 1] be the set of values of τ on projections in A. Since A is UHF, if s, t ∈ F and s+t ≤ 1, then s+t ∈ F. Choose an increasing sequence {ξn }∞ n=1 of ﬁnite partitions of T into intervals such that if Y ∈ ξn then the length of Y is at most 2−n , and if pY is the spectral projection of u corresponding to Y then τ (pY ) ∈ F.

3.4 Notes

57

Fix ε > 0. The projections pY , Y ∈ ξ1 , can be approximated arbitrarily close in the L2 -norm by mutually orthogonal projections qY , Y ∈ ξ1 , in A such that τ (pY ) = τ (qY ). It follows that there exists a unitary w1 ∈ R such that w1 − 12 < ε/2 and qY = w1∗ pY w1 ∈ A for Y ∈ ξ1 . If Z ∈ ξ2 and Z ⊂ Y for some Y ∈ ξ1 , we can approximate w1 pZ w1∗ by projections in qY AqY ⊂ A and repeat the above argument. By continuing this process we obtain a sequence of unitaries wn ∈ R, n ∈ N, such that wn − 12 < 2−n ε and vn pY vn∗ ∈ A for Y ∈ ξn , where vn = wn . . . w1 , and if Z ⊂ Y for some Z ∈ ξn+1 and Y ∈ ξn then vn+1 pZ vn+1 ≤ vn pY vn∗ . The sequence {vn }∞ n=1 converges in the strong operator topology to a unitary v ∈ R. Moreover, 1 − v2 < ε. We claim that vuv ∗ ∈ A. To show this choose a point tY ∈ Y ⊂ T for each Y ∈ ξn , n ∈ N, and set un = Y ∈ξn tY pY . Then u − un ≤ 2−n . Since ∗ |tZ − tY | ≤ 2−m and vn pZ vn∗ ≤ vm pY vm for Z ⊂ Y if Z ∈ ξn and Y ∈ ξm , we also have ∗ vn un vn∗ − vm um vm ≤ 2−m , n ≥ m, ∗ whence vn uvn∗ − vm um vm ≤ 2−n + 2−m , so that ∗ ∗ ≤ lim inf vn un vn∗ − vm um vm ≤ 2−m . vuv ∗ − vm um vm n→∞

∗ vm um vm

∈ A, we see that vuv ∗ ∈ A. Since Since hτ |A (Ad vuv ∗ |A ) = hτ (Ad vuv ∗ ) = hτ (Ad u), to prove the second part of the lemma it suﬃces to show that there exists a unitary u ∈ R with nonatomic spectral measure such that hτ (Ad u) > 0. Take any ergodic dynamical system (X, µ, T ) with positive entropy. Then L∞ (X, µ) Z is the hyperﬁnite II1 -factor, see [214, Chapters XIII and XIV]. For the canonical unitary u in the crossed product we have hτ (Ad u) ≥ hτ (Ad u|L∞ (X,µ) ) = hµ (T ) > 0 (the ﬁrst inequality is actually equality, as will be shown in Chap. 8). This completes the proof of the lemma and thus also of Theorem 3.3.2.

It is worth mentioning that there are shorter but less elementary ways of ﬁnishing the proof of Theorem 3.3.2. For example, one could use a result of Voiculescu [222] stating that the crossed product of Mat2 (C)⊗Z by the shift automorphism embeds into the UHF-algebra ⊗n∈N Matn (C).

3.4 Notes Dynamical entropy for automorphisms preserving a tracial state was introduced by Connes and Størmer [51]. Later Connes [49] emphasized the role of relative entropy and showed how the deﬁnition can be extended to arbitrary W∗ -dynamical system. The formula for Hϕ (B) was also independently obtained by Narnhofer and Thirring [129]. The deﬁnition of dynamical entropy was extended to C∗ -dynamical systems by Connes, Narnhofer and Thirring [50], who introduced the concept of an abelian model and gave a detailed treatment of the foundations of the theory. The material in

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3 Dynamical Entropy

Sects. 3.1 and 3.2 is almost entirely based on their paper. A few properties of entropy observed later include the superadditivity inequality in Proposition 3.1.7(iv) [212], with equality in the case when one of the factors is abelian [141], the apparently new inequality in Proposition 3.1.7(v), the formula for the entropy of a ﬂow [130], Theorem 3.2.5(iii) [141] and Theorem 3.2.5(iv) [79], and Proposition 3.2.7, which is essentially from [34]. The concept of a coupling was introduced by Sauvageot and Thouvenot [188]. It will play a central role in Chap. 5, where we shall give an equivalent deﬁnition of dynamical entropy. The notion of an approximating net is borrowed from the Ohya and Petz book [147]. It allows us to simultaneously discuss nuclear C∗ -algebras and injective von Neumann algebras. Theorem 3.3.1 is due to the authors [141], and Theorem 3.3.2 to Brown [34]. As we already mentioned, the problem of ﬁnding optimal decompositions of a state is quite complicated, see [18], [19] for some results in the ﬁnite dimensional case. Though mutual and dynamical entropies share most properties of their abelian counterparts, there are some properties which are still missing. E.g. if {γi }i∈I is a ﬁnite collection of channels, and I = J ∪ K, it is to the best of our knowledge unknown whether the strong subadditivity inequality Hϕ ({γi }i∈I ) + Hϕ ({γi }i∈J∩K ) ≤ Hϕ ({γi }i∈J ) + Hϕ ({γi }i∈K ) holds. It is also unknown whether the entropy hϕ (α) is an aﬃne function of the state, though this is stated in [50]. There is one aspect of the theory which may be viewed as a drawback, namely, that it is in a sense too abelian, the entropy being deﬁned by abelian models. We shall see later that the entropy tends to be the smaller the less abelian the system is. Similarly to abelian systems the notion of entropy can be extended to actions of arbitrary discrete amenable groups. Namely, let α: G → Aut(A) be an action of a countable amenable group G on A by ϕ-preserving automorphisms. Then for any channel γ: B → A the function m deﬁned on ﬁnite subsets of G by m(F ) = Hϕ ({αg ◦ γ}g∈F ) for F = ∅, m(∅) = 0, is positive, nondecreasing, left-invariant and subadditive, i.e., m(F1 ∪ F2 ) ≤ m(F1 ) + m(F2 ). By [120, Theorem 6.1] it follows that if {Fn }n is a sequence of ﬁnite sets such that |Fn g ∩ Fn | → 1 as n → ∞, |Fn | then the sequence {m(Fn )/|Fn |}n converges, and its limit does not depend on the choice of {Fn }n . Denote this limit by hϕ (γ; αG ). Then we can deﬁne

3.4 Notes

59

hϕ (αG ) by taking supremum over all channels γ into A. Note that if we knew in addition that m is strongly subadditive, by [124, Proposition 3.1.9 and Remark 3.1.7] we would have hϕ (γ; αG ) =

inf

F ⊂G,F =∅

Hϕ ({αg ◦ γ}g∈F ) . |F |

So far the entropy for actions of groups on noncommutative algebras has been considered for very few models, such as shifts on UHF-algebras [92], [102] and Bogoliubov actions [155], [21], [149], [78]. Although the Connes-Størmer-Narnhofer-Thirring deﬁnition of entropy is arguably the most successful extension of the classical theory, it is not the only one available. The ﬁrst deﬁnition of entropy for noncommutative systems is due to Emch [59]. It is as follows. Let (M, ϕ, α) be a W∗ -dynamical system with a faithful normal state ϕ. Then the entropy is deﬁned (compare with (1.3)) as the supremum of the quantities lim ϕ(η(En (pi ))), n→∞

i∈I

where the pi ’s are projections in the centralizer Mϕ of ϕ such that i pi = 1, and En is the ϕ-preserving conditional expectation onto the von Neumann algebra generated by α−k (pi ), i ∈ I, k = 1, . . . , n. Very little is known about this entropy. Note also that it depends only on the restriction of the automorphism to the centralizer of the state, which can be quite small. We shall see later examples of systems with trivial centralizer and inﬁnite dynamical entropy. A diﬀerent approach was suggested by Lindblad [119] and developed by Alicki and Fannes [4]. Let (A, ϕ, α) be a C∗ -dynamical system, and A0 ⊂ A an α-invariant unital ∗-subalgebra. By an operational partition of unity in A0 one means a ﬁnite subset X = {xk }nk=1 of A0 such that k x∗k xk = 1. Such a partition deﬁnes a completely positive map θX : Matn (C) → A by θX (eij ) = x∗i xj . Denote by H[ϕ, X] the entropy of the state ϕ ◦ θX . Note that the density matrix of ϕ◦θX is (ϕ(x∗j xi ))i,j . If Y = {yl }m l=1 is another operational partition of unity, we can deﬁne a new partition X ◦ Y consisting of the elements xk yl . Then put h[ϕ, α, X] = lim sup n→∞

1 H[ϕ, αn−1 (X) ◦ αn−2 (X) ◦ . . . ◦ X]. n

Finally deﬁne the entropy h[ϕ, α, A0 ] of the C∗ -dynamical system (A, ϕ, α) with respect to A0 as the supremum of h[ϕ, α, X] over all operational partitions of unity in A0 . Computability of h[ϕ, α, A0 ] depends very much on the choice of A0 , and in this respect most choices of A0 give too large subalgebras. As a result one can hardly use this entropy as an invariant of a dynamical system. On the other hand, in most examples there is a natural choice of A0 .

60

3 Dynamical Entropy

E.g. for models of quantum statistical mechanics this is the algebra of local observables. We refer the reader to the Alicki and Fannes book [5] for interesting examples. There is also a completely diﬀerent class of entropies based on the idea of approximation of ﬁnite sets. We shall discuss this topic in Chap. 6.

4 Maximality of Entropy and Commutativity

Since our deﬁnition of entropy was based on abelian models, one may expect that the entropy is the larger the more amount of commutativity is present in the system. In this chapter we shall prove several results in this direction. We shall mainly consider tracial states, where proofs are simpler and available results are stronger.

4.1 Maximal Entropy of Subalgebras If N1 , . . . , Nm are ﬁnite dimensional subalgebras of a C∗ -algebra then in general the C∗ -algebra N = ∨k Nk they generate is inﬁnite dimensional. But if N is ﬁnite dimensional, a natural question is, under which conditions on the relative positions of the Nk ’s the entropy of N coincides with the mutual entropy of N1 , . . . , Nm . It follows from Proposition 3.1.6 that if τ is a tracial state and N1 , . . . , Nm contain pairwise commuting abelian subalgebras A1 , . . . , An , respectively, such that ∨k Ak is maximal abelian in N then Hτ (N1 , . . . , Nm ) = Hτ (N ). In the present section we show the converse to this result. Theorem 4.1.1. Let N be a ﬁnite dimensional C∗ -algebra with a faithful tracial state τ , N1 , . . . , Nm ⊂ N . Suppose Hτ (N1 , . . . , Nm ) = Hτ (N ). Then there exist mutually commuting abelian subalgebras Ak ⊂ Nk such that ∨k Ak is maximal abelian in N . The key point in the proof of this theorem is the following reﬁnement of the subadditivity inequality from Corollary 2.3.9. Proposition 4.1.2. Let 1 = i1 ,...,im xi1 ...im be a ﬁnite partition of unity in a C∗ -algebra with a tracial state τ . Then

62

4 Maximality of Entropy and Commutativity

k

(k)

τ (η(xik )) −

ik

τ (η(xi1 ...im )) ≥

i1 ,...,im

1 (k) (l) [(xik )1/2 , (xil )1/2 ]22 , max 2 k=l i ,i k

l

where x2 = τ (x∗ x)1/2 . We shall postpone the proof of this inequality. Note that if the elements of the partition lie in a ﬁnite dimensional C∗ -algebra N , then it can be written as (k) (k) η(τ (xik )) − Hτ (N ; {xik }) k

ik

⎛

⎞

−⎝

η(τ (xi1 ...im )) − Hτ (N ; {xi1 ...im })⎠

i1 ,...,im

≥

1 (k) (l) [(xik )1/2 , (xil )1/2 ]22 . max 2 k=l i ,i k

(4.1)

l

Proof of Theorem 4.1.1. Let X = S(N ) be the state space of N , {ξ (n) }n a sequence of ﬁnite Borel partitions of X with diameter tending to zero as n → ∞. By Proposition 3.1.10 we can ﬁnd a sequence {λ(n) }n of states on N ⊗ C(X m ) such that λ(n) |N = τ and (n)

(i) the coupling λk = λ(n) ◦ (id ⊗ pr∗k ) is canonical, where prk : X m → X is the projection onto the k-th factor, 1 ≤ k ≤ m; (n) (n) ), . . . , pr−1 )) → Hτ (N ) as n → ∞. (ii) Hλ(n) (N1 , . . . , Nm ; pr−1 m (ξ 1 (ξ Choosing a subsequence, we may assume that {λ(n) }n converges to a state λ. Let µ(n) (resp. µ) be the measure on X m deﬁned by λ(n) |C(X m ) (resp. (n) (n) by λ|C(X m ) ), µk = (prk )∗ (µ(n) ), µk = (prk )∗ (µ), and λk = λ(n) ◦ (id ⊗ pr∗k ). By subadditivity of the function ξ → Hµ(n) (ξ) − Hλ(n) (N ; ξ) = S(λ(n) (· ⊗ 1Z ), τ ), Z∈ξ (n) ), we have see Corollary 2.3.8, applied to ξ = ∨k pr−1 k (ξ (n) (n) )) − Hλ(n) (N ; ∨k pr−1 )) Hµ(n) (∨k pr−1 k (ξ k (ξ

≤

k

(Hµ(n) (ξ (n) ) − Hλ(n) (N ; ξ (n) )). k

k

Using (3.12), monotonicity of relative entropy, Theorem 2.1.2(v), and the above inequality we then get (n) (n) ), . . . , pr−1 )) Hλ(n) (N1 , . . . , Nm ; pr−1 m (ξ 1 (ξ

4.1 Maximal Entropy of Subalgebras (n) = Hµ(n) (∨k pr−1 )) − k (ξ

Hµ(n) (ξ (n) ) + k

k

(n) )) − ≤ Hµ(n) (∨k pr−1 k (ξ

k

Hµ(n) (ξ (n) ) + k

k

k

63

Hλ(n) (Nk ; ξ (n) ) k

Hλ(n) (N ; ξ (n) ) k

(n) ≤ Hλ(n) (N ; ∨k pr−1 )) k (ξ ≤ Hτ (N ).

It follows that the above inequalities become close to equalities as n → ∞. In other words, Hλ(n) (N ; ξ (n) ) − Hλ(n) (Nk ; ξ (n) ) → 0, 1 ≤ k ≤ m, k

and

(4.2)

k

(Hµ(n) (ξ (n) ) − Hλ(n) (N ; ξ (n) )) k

k

k

(n) (n) −(Hµ(n) (∨k pr−1 )) − Hλ(n) (N ; ∨k pr−1 )) → 0. k (ξ k (ξ

(4.3)

sa Let aϕ ∈ N+ be the density operator of a state ϕ on N with respect to τ , (n) so that ϕ = τ (· aϕ ). Since the coupling λk is canonical, the decomposition (n) (n) of τ deﬁned by λk and ξ (n) consists of the functionals Z ϕ dµk (ϕ), Z ∈ ξ (n) . (n) In other words, the partition of unity in N deﬁned by λk and ξ (n) is (n) aϕ dµk (ϕ). 1= Z∈ξ (n)

Z

Hence, see (3.1), Hλ(n) (Nk ; ξ (n) ) is equal to k

−

(n) µk (Z)τ

(n) (n) −1 η ENk µk (Z) aϕ dµk (ϕ) . Z

Z∈ξ (n)

Since aϕ depends continuously on ϕ, and the diameters of the atoms of ξ (n) (n) tend to zero, this expression becomes close to − X τ (η(ENk (aϕ )))dµk (ϕ) as n → ∞, so that Hλ(n) (Nk ; ξ (n) ) → − τ (η(ENk (aϕ )))dµk (ϕ). (4.4) k

X

Similarly, Hλ(n) (N ; ξ (n) ) → − k

τ (η(aϕ ))dµk (ϕ). X

Thus (4.2) implies that τ (η(ENk (aϕ )))dµk (ϕ) = τ (η(aϕ ))dµk (ϕ). X

X

64

4 Maximality of Entropy and Commutativity

Since η(ENk (aϕ )) ≥ ENk (η(aϕ )) by strict operator concavity of η, see B.2 and B.4 , we conclude that η(ENk (aϕ )) = ENk (η(aϕ )) for ϕ ∈ supp µk . By B.5 this implies that aϕ ∈ Nk for ϕ ∈ supp µk . Denote by Bk the C∗ -subalgebra of Nk generated by aϕ for ϕ ∈ supp µk . (k) (n) Then (4.3) and inequality (4.1) applied to {xik } = { Z aϕ dµk (ϕ)} imply that for k = l 1/2 1/2 2 (n) (n) aϕ dµk (ϕ) , aψ dµl (ψ) → 0. Z W (n) Z,W ∈ξ

2

(n) (n) Since µk (Z)−1 Z aϕ dµk (ϕ) becomes arbitrarily close to aψ for ψ ∈ Z, as n → ∞ the above expression gets close to 1/2 2 (n) (n) [a1/2 ϕ , aψ ]2 dµk (ϕ)dµl (ψ), X×X

so that in the limit we obtain 1/2 2 [a1/2 ϕ , aψ ]2 dµk (ϕ)dµl (ψ) = 0. X×X

Thus aϕ aψ = aψ aϕ for ϕ ∈ supp µk and ψ ∈ supp µl . Therefore the algebras B1 , . . . , Bm mutually commute. Analogously to (4.4) we have Hλ(n) (Bk ; ξ (n) ) → − τ (η(EBk (aϕ )))dµk (ϕ). k

X

But aϕ = ENk (aϕ ) = EBk (aϕ ) for ϕ ∈ supp µk , so that Hλ(n) (Nk ; ξ (n) ) − Hλ(n) (Bk ; ξ (n) ) → 0. k

k

(n) (n) It follows that Hλ(n) (B1 , . . . , Bm ; pr−1 ), . . . , pr−1 )) becomes close to m (ξ 1 (ξ (n) −1 (n) Hλ(n) (N1 , . . . , Nm ; pr−1 (ξ ), . . . , pr (ξ )) as n → ∞. Since the latter exm 1

pression converges to Hτ (N ), it follows that Hτ (B1 , . . . , Bm ) = Hτ (N ). Let Ak be a maximal abelian subalgebra of Bk . Since B1 , . . . , Bm mutually commute, by Proposition 3.1.6 Hτ (B1 , . . . , Bm ) = S(τ |∨k Ak ), hence Hτ (N ) = S(τ |∨k Ak ). Let A be a maximal abelian subalgebra of N containing ∨k Ak . Then S(τ ) = S(τ |A ) by Theorem 2.2.2(vii). Thus S(τ |∨k Ak ) =

4.1 Maximal Entropy of Subalgebras

65

S(τ |A ). Since the atoms of ∨k Ak are sums of the atoms of A, and η(x + y) < η(x)+η(y) for any numbers x, y > 0, this is possible only when ∨k Ak = A.

In order to prove Proposition 4.1.2 we shall ﬁrst strengthen the GoldenThompson inequality Tr(ea eb ) ≥ Tr(ea+b ). Lemma 4.1.3. For self-adjoint elements a and b in a C∗ -algebra with a tracial state τ we have 1 τ (ea eb ) − τ (ea+b ) ≥ [ea/2 , eb/2 ]22 . 2 Proof. By the Trotter-Lie formula, see e.g. [177, Theorem VIII.29], (ea/2 eb/2 )2 → ea+b in norm as n → ∞. n

n

n

Note that since |ea/2 eb/2 |2 = eb/2 ea/2 n

n

n

|ea/2 eb/2 |2 = (eb/2 ea/2 n

n

n

n

n−1

eb/2 )2 n

n−1

n−1

n

eb/2 , we have n

= eb/2 (ea/2

n−1

eb/2

n−1

)2

n−1

e−b/2 . n

Using the generalized H¨older inequality x1 . . . xp 1 ≤ x1 p . . . xp p , see B.6, we then get τ ((ea/2 eb/2 )2 ) ≤ (ea/2 eb/2 )2 1 n n n ≤ ea/2 eb/2 22n n n n = τ (|ea/2 eb/2 |2 ) n

n

n

n

= τ ((ea/2

n

n−1

n

eb/2

n−1

)2

n−1

).

Thus τ ((ea/2 eb/2 )2 ) τ (ea+b ). In particular, τ (ea+b ) ≤ τ ((ea/2 eb/2 )2 ). It remains to note that n

n

n

τ (ea eb ) − τ ((ea/2 eb/2 )2 ) =

1 a/2 b/2 2 [e , e ]2 . 2

Proof of Proposition 4.1.2. Consider ﬁrst the case m = 2. So we have a partition of unity 1 = i1 ∈I1 ,i2 ∈I2 xi1 i2 . Replacing xi1 i2 by (1 − ε)xi1 i2 +

ε 1, |I1 ||I2 |

we may assume that the xi1 i2 ’s are invertible. Then by Corollary 2.3.10 and Lemma 4.1.3 we have (1) (2) τ (η(xi1 )) + τ (η(xi2 )) − τ (η(xi1 i2 )) i1

i2

=

i1 ,i2 (1)

τ (xi1 i2 (log xi1 i2 − log elog xi1

i1 ,i2

≥

i1 ,i2

(1)

(τ (xi1 i2 ) − τ (elog xi1

(2) 2

+log xi

(2) 2

+log xi

))

))

66

4 Maximality of Entropy and Commutativity

= 1− =

(1)

τ (elog xi1

(2) 2

+log xi

)

i1 ,i2 (1)

(1) (2)

(τ (xi1 xi2 ) − τ (elog xi1

(2) 2

+log xi

))

i1 ,i2

≥

1 (1) (2) [(xi1 )1/2 , (xi2 )1/2 ]22 . 2 i ,i 1

2

Consider now the case m ≥ 3. It is clear that it suﬃces to prove the inequality for k = 1 and l = 2. Put xi1 ...im and zi3 ...im = xi1 ...im , y i1 i2 = i3 ,...,im (1)

(1)

(2)

i1 ,i2

(2)

(n)

(n)

so that yi1 = xi1 , yi2 = xi2 and zin = xin for n = 3, . . . , m. Then τ (η(xi1 ...im )) ≤ τ (η(yi1 i2 )) + τ (η(zi3 ...im )) i1 ,...,im

i1 ,i2

≤

τ (η(yi1 i2 )) +

i3 ,...,im m

(n)

τ (η(xin )),

n=3 in

i1 ,i2

whence m n=1 in

(n)

τ (η(xin ))−

τ (η(xi1 ...im )) ≥

i1 ,...,im

2

(n)

τ (η(yin ))−

n=1 in

Therefore the result follows from the case m = 2.

τ (η(yi1 i2 )).

i1 ,i2

4.2 Independent Algebras Let us now consider the problem of maximizing Hτ (N1 , . . . , Nm ) without assuming that N1 , . . . , Nm generate a ﬁnite dimensional algebra. By subadditivity we have Hτ (N1 , . . . , Nm ) ≤ Hτ (N1 ) + . . . + Hτ (Nm ). In the commutative case, by Proposition 1.1.9(iii), the equality holds exactly when N1 , . . . , Nm are mutually independent. For a C∗ -algebra A and a faithful tracial state τ on A let us say that C∗ -subalgebras A1 , . . . , Am of A are τ independent if they mutually commute and τ (a1 . . . am ) = τ (a1 ) . . . τ (am ) for ai ∈ Ai . Let N1 , . . . , Nm be ﬁnite dimensional C∗ -subalgebras of A. Assume there exist maximal abelian subalgebras Ai ⊂ Ni such that A1 , . . . , Am are τ -independent. Since

4.2 Independent Algebras

Hτ (A1 , . . . , Am ) ≤ Hτ (N1 , . . . , Nm ) ≤

Hτ (Nk ) =

k

and Hτ (A1 , . . . , Am ) = Hτ (∨k Ak ) =

67

Hτ (Ak )

k

Hτ (Ak ),

k

we conclude that Hτ (N1 , . . . , Nm ) = Hτ (N1 ) + . . . + Hτ (Nm ). One may expect that conversely, if the above equality holds, then there exist τ -independent maximal abelian subalgebras Ak ⊂ Nk . This is indeed the case by Theorem 4.1.1 and Proposition 1.1.9(iii) if we in addition assume that N1 , . . . , Nm ⊂ N and Hτ (N1 , . . . , Nm ) = Hτ (N ). We do not know whether this is true in general. We have, however, the following weaker result. Theorem 4.2.1. Let A1 , . . . , Am be ﬁnite dimensional abelian subalgebras of a von Neumann algebra N with a faithful normal trace τ . Suppose Hτ (A1 , . . . , Am ) = Hτ (A1 ) + . . . + Hτ (Am ). Then the algebras A1 , . . . , Am are τ -independent. Note that as soon as the mutual commutativity is established, the independence becomes a purely classical statement, Proposition 1.1.9(iii). We shall use that for abelian algebras any state has a unique pure state decomposition, and any optimal model arises from this decomposition. Lemma 4.2.2. For any d ∈ N and ε > 0, there exists δ > 0 such that if A is an abelian C∗ -algebra of dimension d with pure states ϕ1 , . . . , ϕd , and τ = i∈I λi ψi is a ﬁnite convex combination of states with

λi S(ψi , τ ) > S(τ ) − δ,

i

then there exists a partition I = I(0) . . . I(d) of the index set I such that λi < ε, and ψi − ϕk < ε for i ∈ I(k) and k = 1, . . . , d. i∈I(0)

Proof. Since the von Neumann entropy is a continuous function which is zero only on pure states, we can chose δ0 depending on d and ε such that a state ψ on A is ε-close to a pure state as soon as S(ψ) < δ0 . If λi S(ψi ) = S(τ ) − λi S(ψi , τ ) < δ, i

i

then putting I(0) = {i ∈ I | S(ψi ) ≥ δ0 }, we get

68

4 Maximality of Entropy and Commutativity

λi

0. The system is called uniformly mixing if for any measurable A ⊂ X and any ﬁnite measurable partition ζ we have lim sup{|µ(A ∩ B) − µ(A)µ(B)| : B is (∨∞ k=n ζ)-measurable} = 0.

n→∞

One can then check that H(ξ| ∨∞ k=n ζ) → H(ξ) for any ﬁnite measurable partition ξ, which in view of (1.3) implies h(ξ; T n ) → H(ξ). The latter property is in fact equivalent to uniform mixing. It has a direct generalization to noncommutative systems. Deﬁnition 4.3.1. A W∗ -dynamical system (M, ϕ, α) with faithful state ϕ and M = C is called an entropic K-system if for any channel γ: A → M we have hϕ (γ; αn ) → Hϕ (γ) as n → ∞. Note that since hϕ (γ; αn ) ≤ nhϕ (γ; α), Lemma 3.1.4 implies that for any γ with Im γ = C1 we have hϕ (γ; α) > 0. In particular, hϕ (α) > 0. The simplest examples of K-systems in the classical case are Bernoulli shifts. If ξ is the standard generating partition of a Bernoulli shift then the partition ζ = ∨0k=−∞ T k ξ has the following properties: ζ ≺ T ζ, the algebra ∪n∈Z L∞ (X/T n ζ) is weakly operator dense in L∞ (X), and ∩n∈Z L∞ (X/T n ζ) = C1. It turns out that existence of a partition ζ with such properties is yet another equivalent form of the K-property. This is not the case for noncommutative systems. Nevertheless we have the following result. Theorem 4.3.2. Let (M, ϕ, α) be a W∗ -dynamical system with faithful ϕ and M = C. Suppose there exists a von Neumann subalgebra M0 of M such that (i) M0 ⊂ α(M0 ); (ii) ∪n∈N (α−n (M0 ) ∩ αn (M0 )) is weakly operator dense in M ; (iii) ∩n∈Z αn (M0 ) = C1. Then (M, ϕ, α) is an entropic K-system. For the proof we need two lemmas. Lemma 4.3.3. Let M be a von Neumann algebra, {Mn }∞ n=1 a decreasing sequence of von Neumann subalgebras, ϕ a normal faithful state on M . Assume ∩n Mn = C1. Then for any a ∈ M we have |ϕ(ab) − ϕ(a)ϕ(b)| → 0 as n → ∞, bϕ + b∗ ϕ b∈Mn \{0} sup

where xϕ = ϕ(x∗ x)1/2 .

70

4 Maximality of Entropy and Commutativity

Proof. We may assume that M ⊂ B(H), and ϕ is the state deﬁned by a cyclic and separating vector ξ ∈ H, ϕ(x) = (xξ, ξ). Assume the conclusion of the lemma is false. Then there exist ε > 0, a ∈ M and a sequence {bn }n such that bn ∈ Mn , bn ϕ + b∗n ϕ = 1 and |ϕ(abn ) − ϕ(a)ϕ(bn )| ≥ ε. Replacing {Mn }n by a subsequence we may assume that the bounded sequences {bn ξ}n and {b∗n ξ}n are weakly convergent. Denote by ζ and ϑ their weak limits. Then |(aζ, ξ) − (aξ, ξ)(ζ, ξ)| ≥ ε. (4.5) On the other hand, if x ∈ Mm then for n ≥ m we have (xbn ξ, ξ) = (xξ, b∗n ξ), whence (xζ, ξ) = (xξ, ϑ). Since ∩m Mm = C1, the algebra ∪m Mm is weakly operator dense in B(H). It follows that the above identity holds for all x ∈ B(H). Then, denoting the projection onto Cξ by p, for any x ∈ B(H) we get

(pζ, xξ) = (x∗ pζ, ξ) = (x∗ pξ, ϑ) = (x∗ ξ, ϑ) = (x∗ ζ, ξ) = (ζ, xξ). Hence pζ = ζ, so that ζ ∈ Cξ. But this contradicts (4.5).

Lemma 4.3.4. For any m ∈ N and ε > 0 there exists δ > 0 satisfying the ∗ ∗ following property. m If A is a C -algebra, B ⊂ A a C -subalgebra, ϕ a state on A, and 1 = i=1 xi a partition of unity in B ∩ A such that |ϕ(xi y) − ϕ(xi )ϕ(y)| ≤ δ||y|| for any y ∈ B, i = 1, . . . , m, then

η(ϕ(xi yj )) >

i,j

i

η(ϕ(xi )) +

η(ϕ(yj )) − ε

j

for any ﬁnite partition of unity 1 = j yj in B. m m Proof. Let δ > 0 be such that | i=1 η(λi ) − i=1 η(µi )| < ε/2 for probability distributions (λ1 , . . . , λm ) and (µ1 , . . . , µm ) as soon as m

|λi − µi | ≤ 2δε−1 m log m.

i=1

For every element y of the form y = j ±yj we have −1 ≤ y ≤ 1, and consequently |ϕ(xi y) − ϕ(xi )ϕ(y)| ≤ δ. Hence |ϕ(xi yj ) − ϕ(xi )ϕ(yj )| ≤ δ, j

and thus

4.3 Entropic K-systems

71

|ϕ(xi yj ) − ϕ(xi )ϕ(yj )| ≤ mδ.

i,j

Consider the set J consisting of the indices j such that |ϕ(xi yj ) − ϕ(xi )ϕ(yj )| ≤ 2ϕ(yj )ε−1 mδ log m. i

Since mδ ≥

j ∈J /

we have

j ∈J /

j ∈J /

|ϕ(xi yj ) − ϕ(xi )ϕ(yj )| > 2ε−1 mδ log m

ϕ(yj ),

j ∈J /

i

ϕ(yj ) < ε/2 log m. Hence

ϕ(x y ) ε i j − ϕ(yj ) η(ϕ(xi )) ≤ ϕ(yj ) log m < , η ϕ(yj ) 2 i i j ∈J /

since neither sum over i is larger than log m. On the other hand, by our choice of δ, for every j ∈ J we have ε ϕ(x y ) i j − η η(ϕ(xi )) < . 2 ϕ(y ) j i i It follows that j

ϕ(xi yj ) − ϕ(yj ) η η(ϕ(xi )) > −ε, ϕ(yj ) i i

which is exactly what we need.

Proof of Theorem m 4.3.2. Let γ: A → M be a channel. There exists a partition of unity 1 = i=1 xi in M such that Hϕ (γ; {xi }) is arbitrarily close to Hϕ (γ). Since ∪N (α−N (M0 ) ∩ αN (M0 )) is strongly operator dense in M , we may assume that the xi ’s belong to α−N (M0 ) ∩ αN (M0 ) for some N ∈ N (note that one can think of a partition of unity as a unital completely positive map Cm → M , so the claim follows from A.10). Then αn (xi ) commutes with xj for n ≥ 2N . For n ≥ 2N and k ∈ N we want to estimate Hϕ (γ, αn ◦ γ, . . . , α(k−1)n ◦ γ; {xi1 αn (xi2 ) . . . α(k−1)n (xik )}) = η(ϕ(xi1 αn (xi2 ) . . . α(k−1)n (xik ))) − k η(ϕ(xi )) + kHϕ (γ; {xi }), i1 ,...,ik

i

see (3.3). To prove the theorem it suﬃces to show that for any ε > 0 the above expression is (kε)-close to kHϕ (γ; {xi }) when n is suﬃciently large and k is arbitrary.

72

4 Maximality of Entropy and Commutativity

Choose δ according to Lemma 4.3.4. Since the algebras αl (M0 ) have trivial intersection, by Lemma 4.3.3 there exists n0 ≤ −N such that |ϕ(xi y) − ϕ(xi )ϕ(y)| ≤ δ||y|| for y ∈ αn0 (M0 ), i = 1, . . . , n. Then by Lemma 4.3.4 for any l ∈ Z and any partition of unity 1 = j yj in αn0 +ln (M0 ) we get η(ϕ(yj αln (xi ))) > η(ϕ(xi )) + η(ϕ(yj )) − ε. i,j

i

j

Since xi1 αn (xi2 ) . . . α(l−1)n (xil ) ∈ αN −n+ln (M0 ) ⊂ αn0 +ln (M0 ) if n ≥ N −n0 , applying recursively the above inequality we obtain η(ϕ(xi )) − (k − 1)ε, η(ϕ(xi1 αn (xi2 ) . . . α(k−1)n (xik ))) > k i1 ,...,ik

i

which completes the proof of the theorem.

The simplest examples of entropic K-systems are noncommutative Bernoulli shifts. We shall see more interesting examples in Chaps. 11 and 13. It is worth noticing that the condition (ii) in the above theorem can not be weakened to the requirement that ∪n αn (M0 ) is weakly operator dense in M . In fact, as we shall see in Chap. 12, we can then even get a system with zero entropy. Our next goal is to show that an entropic K-system is necessarily asymptotically abelian, at least when the state is tracial. Theorem 4.3.5. Let (N, τ, α) be an entropic K-system, where τ is a faithful normal trace. Then the system is strongly asymptotically abelian and strongly mixing. In other words, for any x, y ∈ N we have [αn (x), y] → 0 in the strong operator topology and τ (αn (x)y) → τ (x)τ (y) as n → ∞. Proof of Theorem 4.3.5. Let A be a ﬁnite dimensional abelian subalgebra of N . Since Hτ (A, αn (A), . . . , α(2m−1)n (A)) ≤ mHτ (A, αn (A)), so that 2hτ (A; αn ) ≤ Hτ (A, αn (A)), it follows from our assumptions that Hτ (A, αn (A)) converges to 2Hτ (A). We thus need an approximate version of Theorem 4.2.1. A standard trick in such cases is to use ultraproducts. So let ω be a free ultraﬁlter on N. Consider the ultraproduct N ω . By deﬁnition this is the quotient of the algebra ∞ (N, N ) of bounded sequences in N by the ideal Iω consisting of sequences (xn )n such that xn 2 → 0 as n → ω. This is a von Neumann algebra with a faithful normal trace τω deﬁned by

4.3 Entropic K-systems

73

τω ((xn )n ) = lim τ (xn ). n→ω

We identify N with the subalgebra of N ω consisting of constant sequences. Consider the automorphism θ of N ω deﬁned by θ((xn )n ) = (αn (xn ))n . We claim that Hτω (A, θ(A)) = 2Hτ (A). Indeed, by Proposition 3.1.10 and the discussion following it, for any ε > 0 there exists a ﬁnite index set I such that for each n there exists a partition of unity 1 = i,j∈I x(n)ij in N such that Hτ (A, αn (A); {x(n)ij }) is ε-close to Hτ (A, αn (A)) (in fact, if we consider only suﬃciently large n, then by the proof of Theorem 4.2.1 we may even take I consisting of d + 1 elements, where d is the dimension of A). Then yij = (x(n)ij )n , i, j ∈ I, form a partition of unity in N ω such that Hτω (A, θ(A); {yij }) = lim Hτ (A, αn (A); {x(n)ij }) ≥ 2Hτ (A) − ε. n→ω

Thus we can apply Theorem 4.2.1 and conclude that A and θ(A) are τω independent. To conclude that N and θ(N ) are τω -independent we need the following lemma. Lemma 4.3.6. Let M be a von Neumann algebra with a faithful normal trace τ , N ⊂ M , and θ a τ -preserving automorphism of M . Assume that for any commuting projections p and q in N we have θ(p)q = qθ(p) and τ (θ(p)q) = τ (p)τ (q). Then the algebras N and θ(N ) are τ -independent. Proof. We shall ﬁrst check that τ (xθ(y)) = τ (x)τ (y) for x, y ∈ N . Since every self-adjoint operator is approximated by a linear combination of its spectral projections, we have θ(a)b = bθ(a) and τ (θ(a)b) = τ (a)τ (b) for any commuting self-adjoint elements a, b ∈ N . Let A ⊂ N be a maximal abelian von Neumann subalgebra of N containing a. Since θ(a) commutes with any element in N commuting with a, the same is true for EN (θ(a)). Hence EN (θ(a)) lies in A. On the other hand, τ (EN (θ(a))b) = τ (θ(a)b) = τ (a)τ (b). for every b ∈ A. It follows that EN (θ(a)) is a scalar, so that τ (xθ(a)) = τ (x)τ (a) for every x ∈ N .

74

4 Maximality of Entropy and Commutativity

To prove that N and θ(N ) commute, it is enough to check that θ(p) and q commute for any projections p and q in N . Since [p + q, θ(p + q)] = 0, [p, θ(p)] = 0 and [q, θ(q)] = 0, we get [p, θ(q)] = [θ(p), q]. Using this identity we compute τ (θ(p)qθ(p)q) = τ (θ(p)qθ(p)) + τ (θ(p)q[θ(p), q]) = τ (θ(p)q) + τ (θ(p)q[p, θ(q)]) = τ (θ(p)q) + τ (θ(p)qpθ(q)) − τ (θ(p)qθ(q)p) = τ (θ(p)q) + τ (θ(q)θ(p)qp) − τ (θ(p)θ(q)qp) = τ (p)τ (q) + τ (qp)τ (qp) − τ (pq)τ (qp) = τ (p)τ (q), whence [θ(p), q]22 = τ (qθ(p)q) − τ (θ(p)qθ(p)q) − τ (qθ(p)qθ(p)) + τ (θ(p)qθ(p)) = 0. Thus θ(p)q = qθ(p).

Returning to the proof of the theorem, we conclude that N and θ(N ) are τω -independent. That is, for any x, y ∈ N we get [αn (x), y]2 → 0 and τ (αn (x)y) → τ (x)τ (y) as n → ω. Since ω was an arbitrary free ultraﬁlter, we conclude that these convergences hold as n → ∞.

4.4 Notes Theorem 4.1.1 was proved by Haagerup and Størmer [83]. Theorem 4.2.1 is contained in the work of Benatti and Narnhofer [15]. Curiously enough it has apparently never been formulated explicitly. K-systems (called also Kolmogorov systems, or systems regular in the sense of Kolmogorov) were introduced by Kolmogorov [111], who called them quasiregular. The main deﬁnition is the last one we mentioned. So (X, µ, T ) is a K-system if there exists a measurable partition ζ such that ζ ≺ T ζ, ∨n T n ζ is the partition of (X, µ) into points, and the partition ∧n T n ζ corresponding to ∩n L∞ (X/T n ζ) is trivial. Equivalent forms of the K-property were obtained by Rohlin and Sinai [183]. For some time the only known examples of Ksystems were Bernoulli shifts. It was Ornstein who gave the ﬁrst example of a K-system which is not a Bernoulli shift [151]. Ornstein and Shields constructed an uncountable family of pairwise nonisomorphic K-systems with

4.4 Notes

75

the same entropy [152]. A simple example of a non-Bernoullian K-system was given by Kalikow [98]. Katok found examples of smooth non-Bernoullian Ksystems [100]. The ﬁrst appearance of K-systems in the quantum setting is the work of Emch [59], [60], [61]. One says that a W∗ -dynamical system (M, ϕ, α) is an algebraic K-system, the name suggested by Narnhofer and Thirring [132], if there exists a W∗ -subalgebra M0 ⊂ M such that M0 ⊂ α(M0 ), ∪n αn (M0 ) is dense in M , and ∩n αn (M0 ) = C1. As was already mentioned, in Chap. 12 we shall see examples of algebraic K-systems with zero entropy. The notion of entropic K-system was proposed by Narnhofer and Thirring [131]. A stronger result than that formulated in Theorem 4.3.2 was stated by Narnhofer [126]. Explicitly Theorem 4.3.2 appeared in the work of Golodets and Neshveyev [76]. Lemma 4.3.3 is due to Powers [170]. Theorem 4.3.5 was proved by Benatti and Narnhofer [15]. An attempt to capture the K-property by imposing certain hyperbolicity conditions was made by Emch, Narnhofer, Sewell and Thirring [62]. See [135], [136] for results in this direction.

5 Dynamical Abelian Models

In Chap. 3 we used abelian models to deﬁne mutual entropy of channels into A. It is then natural to consider dynamical abelian models, that is, abelian dynamical systems (X, µ, T ) together with equivariant maps A → L∞ (X, µ). In this chapter we shall prove that there always exist enough such models to compute the entropy.

5.1 Entropy via Stationary Couplings As in Chap. 3 we prefer to work with states on A ⊗ L∞ (X, µ) rather than with maps A → L∞ (X, µ). Then by a stationary coupling of (A, ϕ, α) with (X, µ, T ) we mean an (α ⊗ β)-invariant state λ on A ⊗ L∞ (X, µ), where β(f ) = f ◦ T −1 , such that λ|A = ϕ and λ|L∞ (X,µ) = µ. Given a stationary coupling λ, a channel γ into A and a ﬁnite measurable partition ξ of X, we have (see (3.12)) Hλ (γ, α ◦ γ, . . . , αn−1 ◦ γ; ξ, T ξ, . . . , T n−1 ξ) k = Hµ (∨n−1 k=0 T ξ) − nHµ (ξ) + nHλ (γ; ξ).

(5.1)

This motivates the following deﬁnition of entropy. Deﬁnition 5.1.1. The Sauvageot-Thouvenot entropy of a C∗ -dynamical system (A, ϕ, α) is hST ϕ (α) = sup{hµ (ξ; T ) − Hµ (ξ) + Hλ (A; ξ)}, where the supremum is taken over all stationary couplings λ of (A, ϕ, α) with abelian dynamical systems (X, µ, T ) and over all ﬁnite measurable partitions ξ of X. To make sense of the expression

78

5 Dynamical Abelian Models

Hλ (A; ξ) = Hµ (ξ) +

S(λ(· ⊗ 1Z ), ϕ)

Z∈ξ

we need relative entropy to be deﬁned for arbitrary C∗ -algebras. However, essentially all we have to know in order to deal with, say, nuclear C∗ -algebras is that if {γi }i is a ϕ-approximating net, then Hλ (γi ; ξ) Hλ (A; ξ) by Corollary 2.3.5. Let us ﬁrst observe that in some cases just existence of nontrivial couplings, that is, couplings diﬀerent from product-states, implies positivity of entropy. Proposition 5.1.2. If a C∗ -dynamical system (A, ϕ, α) has a nontrivial stationary coupling with a classical Bernoulli shift then hϕ (α) > 0. Proof. Let λ be a nontrivial stationary coupling of (A, ϕ, α) with a Bernoulli shift (X, µ, T ). Let ξ be the standard generating partition for (X, µ, T ). Since λ is assumed to be nontrivial, there exist a self-adjoint element a ∈ A and an element Y ∈ ∨nk=−n T k ξ such that λ(a ⊗ 1Y ) = ϕ(a)µ(Y ). Since hϕ (αm ) ≤ mhϕ (α) for any m ∈ N, it suﬃces to show that hϕ (α2n+1 ) > 0. Hence replacing α by α2n+1 , ξ by ∨nk=−n T k ξ and T by T 2n+1 we may assume n = 0. Since hµ (ξ; T ) = Hµ (ξ), by virtue of (5.1) and (3.10) for any channel γ into A we get hϕ (γ; α) ≥ Hλ (γ; ξ) = µ(Z)S(µ(Z)−1 λ(γ(·) ⊗ 1Z ), ϕ ◦ γ). Z∈ξ

If a ∈ Im γ, then by assumption µ(Y )−1 λ(γ(·) ⊗ 1Y ) = ϕ ◦ γ, and by Theorem 2.1.2(i) we conclude that hϕ (γ; α) > 0.

In the next section we shall prove a partial converse to the above proposition. The quantity Hµ (ξ) − Hλ (A; ξ), which is the entropy defect of the decomposition of ϕ deﬁned by λ and ξ, should be thought of as an analogue of the conditional entropy of L∞ (X/ξ) with respect to A in (A ⊗ L∞ (X, µ), λ). We shall now show that all the main properties of the classical conditional entropy are satisﬁed, which allows us to deal eﬃciently with our new deﬁnition of entropy. For γ: B → A denote by Hλ,γ (ξ) the entropy defect of the abelian model deﬁned by λ and ξ, so by (3.9) Hλ,γ (ξ) = Hµ (ξ) − Hλ (γ; ξ) = − S(λ(γ(·) ⊗ 1Z ), ϕ ◦ γ). (5.2) Z∈ξ

This is well-deﬁned for an arbitrary unital completely positive mapping γ: B → A, not only for ﬁnite dimensional B’s. More generally, for any measurable (not necessarily ﬁnite) partition ζ of Y , denote by iζ the embedding L∞ (Y /ζ) → L∞ (Y ), and put

5.1 Entropy via Stationary Couplings

Hλ,γ (ξ|ζ) = −

79

S(λ(γ(·) ⊗ iζ (·)1Z ), λ ◦ (γ ⊗ iζ )).

Z∈ξ

Equivalently, using the diagonal embedding Y → Y × Y , for any coupling λ of (A, ϕ) with (Y, µ) we can deﬁne a coupling of (A ⊗ L∞ (Y, µ), λ) with (Y, µ), which we again denote by λ, so that a ⊗ f ⊗ g → λ(a ⊗ f g). Then Hλ,γ (ξ|ζ) = Hλ,γ⊗iζ (ξ). If ζ is ﬁnite then Hλ,γ (ξ|ζ) = − S(λ(γ(·) ⊗ 1Z∩W ), λ(γ(·) ⊗ 1W )). Z∈ξ,W ∈ζ

Proposition 5.1.3. For any unital completely positive map γ: B → A we have: (i) Hλ (γ; ξ) is increasing in γ and ξ, that is, Hλ (γ ◦ ρ; ξ) ≤ Hλ (γ; ϑ) for any unital completely positive mapping ρ into B and any ﬁnite partition ϑ ξ; (ii) 0 ≤ Hλ (γ; ξ) ≤ Hµ (ξ); (iii) the function λ → Hλ (γ; ξ) is convex; (iv) Hλ,γ (ξ|ζ) is increasing in ξ and decreasing in γ and ζ; moreover, if {ζn }n is an increasing sequence of measurable partitions with ∨n ζn = ζ, then Hλ,γ (ξ|ζn ) Hλ,γ (ξ|ζ); (v) 0 ≤ Hλ,γ (ξ|ζ) ≤ Hµ (ξ|ζ); (vi) Hλ,γ (ξ ∨ ϑ|ζ) = Hλ,γ (ξ|ζ) + Hλ,γ (ϑ|ξ ∨ ζ); in particular, Hλ,γ (ξ ∨ ϑ) ≤ Hλ,γ (ξ) + Hλ,γ (ϑ). Proof. Essentially all these statements appeared in one or another form in Chap. 3, but we shall spell out the proofs for convenience. The fact that Hλ (γ; ξ) is increasing in γ follows from monotonicity of relative entropy, Theorem 2.3.1(vi). That it is increasing in ξ can be deduced either from convexity of relative entropy or from (v) and (vi) as follows. If ϑ ξ, then Hλ,γ (ϑ) = Hλ,γ (ξ) + Hλ,γ (ϑ|ξ) ≤ Hλ,γ (ξ) + Hµ (ϑ|ξ) = Hλ,γ (ξ) + Hµ (ϑ) − Hµ (ξ), so Hλ (γ; ξ) ≤ Hλ (γ; ϑ) by (5.2). The ﬁrst inequality in (ii) follows from S(ψ, ω) ≤ 0 for ψ ≤ ω, Theorem 2.3.1(iii). On the other hand, S(ψ, ω) ≥ 0 if ψ and ω are states, Theorem 2.3.1(ii). Hence −S(λ(γ(·)⊗1Z ), ϕ◦γ) = −S(µ(Z)−1 λ(γ(·)⊗1Z ), ϕ◦γ)+η(µ(Z)) ≤ η(µ(Z)), which gives the second inequality in (ii). Part (iii) follows from (3.10) and convexity of relative entropy, Theorem 2.3.1(v).

80

5 Dynamical Abelian Models

Since Hλ,γ (ξ|ζ) = Hλ,γ⊗iζ (ξ), it follows from (i) that Hλ,γ (ξ|ζ) = Hµ (ξ) − Hλ (γ ⊗ iζ ; ξ) is decreasing in γ and ζ. Since Hλ,γ (ϑ|ξ ∨ ζ) ≥ 0 by (ii), it will follow from (vi) that Hλ,γ (ξ|ζ) is increasing in ξ. The convergence statement in (iv) follows from Corollaries 2.3.6 and 2.3.4. We have already mentioned that positivity of Hλ,γ follows from (ii). Since Hλ,γ (ξ|ζ) is decreasing in γ, to prove the second inequality in (v) we may assume that B = C. Then Hλ,γ (ξ|ζ) = − S(µ(·1Z )|L∞ (X/ζ) , µ|L∞ (X/ζ) ). Z∈ξ

Since µ(·1Z ) = µ(·Eζ (1Z )) on L∞ (X/ζ), where Eζ is the µ-preserving conditional expectation onto L∞ (X/ζ), by Theorem 2.3.1(x) we have S(µ(·1Z )|L∞ (X/ζ) , µ|L∞ (X/ζ) ) = − η(Eζ (1Z ))dµ. X

Hence Hλ,γ (ξ|ζ) = Hµ (ξ|ζ). To prove (vi) ﬁrst notice that Hλ,γ (ϑ|ξ ∨ ζ) = Hλ,γ⊗iζ (ϑ|ξ), or in other words, Hλ,γ⊗iξ∨ζ (ϑ) = Hλ,γ⊗iξ ⊗iζ (ϑ). This follows from Corollary 2.3.4, since the homomorphism L∞ (X/ξ) ⊗ L∞ (X/ζ) → L∞ (X/ξ ∨ ζ), f ⊗ g → f g, has dense image in the strong operator topology. It follows that in proving (vi) we may replace γ by γ ⊗ iζ and thus assume that ζ is trivial. By Theorem 2.3.1(viii) we then have S(λ(γ(·) ⊗ 1Z∩W ), ϕ ◦ γ) Z∈ϑ

= S(λ(γ(·) ⊗ 1W ), ϕ ◦ γ) +

S(λ(γ(·) ⊗ 1Z∩W ), λ(γ(·) ⊗ 1W ))

Z∈ϑ

for W ∈ ξ. Hence Hλ,γ (ξ ∨ ϑ) = Hλ,γ (ξ) + Hλ,γ (ϑ|ξ). The following formula for

hST ϕ (α)

is reminiscent of (1.3).

Proposition 5.1.4. For any C∗ -dynamical system (A, ϕ, α) we have − − hST ϕ (α) = sup{Hµ (ξ|ξ ) − Hλ,A (ξ|ξ )},

where the supremum is taken over all stationary couplings λ of (A, ϕ, α) with abelian dynamical systems (Y, µ, T ) and over all ﬁnite measurable partitions ξ. Proof. As Hλ,A (ξ|ξ − ) ≤ Hλ,A (ξ) = Hµ (ξ) − Hλ (A; ξ) and Hµ (ξ|ξ − ) = hµ (ξ; T ), the entropy hST ϕ (α) is not bigger than the supremum.

5.1 Entropy via Stationary Couplings

81

To prove the opposite inequality, ﬁx a stationary coupling λ. First of all note that using Proposition 5.1.3(iv),(vi) the same argument as in the proof of (1.3) shows that Hλ,A (ξ|ξ − ) = lim

n→∞

1 k Hλ,A (∨n−1 k=0 T ξ), n

and hence Hµ (ξ|ξ − ) − Hλ,A (ξ|ξ − ) = lim

n→∞

1 k Hλ (A; ∨n−1 k=0 T ξ). n

(5.3)

Fix ε > 0 and choose n ∈ N such that 1 k − − Hλ (A; ∨n−1 k=0 T ξ) > Hµ (ξ|ξ ) − Hλ,A (ξ|ξ ) − ε n and

1 k Hµ (∨n−1 k=0 T ξ) < hµ (ξ; T ) + ε. n Consider the space Y = Y × (Z/nZ) with the transformation T (x, k) = (T x, k + 1) and the measure µ given by the product of µ with the counting measure. By taking the tensor product of λ with the counting measure, we get a stationary coupling λ of (A, ϕ, α) with (Y , µ , T ). Let ξ be the partition of Y consisting of the elements Y × {1, . . . , n − 1} and Z × {0}, where Z runs k through the elements of ∨n−1 k=0 T ξ. It is easy to check that 1 1 1 n−1 k +η 1 − and hµ (ξ ; T ) = hµ (ξ; T ), Hµ (ξ ) = Hµ (∨k=0 T ξ) + η n n n so that

1 1 −η 1− > −2ε n n if n is suﬃciently large. On the other hand, if ϕ = i ϕi is the decomposition k deﬁned by λ and ∨n−1 k=0 T ξ, then 1 1 ϕ ϕi + 1 − ϕ= n n i hµ (ξ ; T ) − Hµ (ξ ) > −ε − η

is the decomposition deﬁned by λ and ξ . Recalling that we denote by ϕˆi the state ϕi (1)−1 ϕi , by virtue of (3.10) we get ϕi (1) 1 S(ϕ, ϕ) S(ϕˆi , ϕ) + 1 − Hλ (A; ξ ) = n n i = It follows that

1 k − − Hλ (A; ∨n−1 k=0 T ξ) > Hµ (ξ|ξ ) − Hλ,A (ξ|ξ ) − ε. n

82

5 Dynamical Abelian Models

hµ (ξ ; T ) − Hµ (ξ ) + Hλ (A; ξ ) > Hµ (ξ|ξ − ) − Hλ,A (ξ|ξ − ) − 3ε,

and the proof is complete. We can now formulate the main result of the chapter.

∗ Theorem 5.1.5. We have hST ϕ (α) ≥ hϕ (α) for any C -dynamical system (A, ϕ, α). If the system has a ϕ-approximating net, then hST ϕ (α) = hϕ (α).

Proof. By (5.1) we have hϕ (γ; α) ≥ hµ (ξ; T ) − Hµ (ξ) + Hλ (γ; ξ). If {γi }i is a ϕ-approximating net, then Hλ (γi ; ξ) Hλ (A; ξ) by Corollary 2.3.5. Hence hST ϕ (α) ≤ hϕ (α). To prove the opposite inequality in general we shall ﬁnd a channel γ: B → A, a sequence {λn }n of nonstationary couplings of (A, ϕ, α) with (Y, T, µn ), where Y is a compact metric space, and a partition ξ of Y consisting of clopen sets such that 1 Hλ (γ, α ◦ γ, . . . , αn−1 ◦ γ; ξ, T ξ, . . . , T n−1 ξ) > hϕ (α) − ε. n n Then to geta stationary coupling it is natural to take a weak∗ limit point of the n−1 states n−1 k=0 λn ◦(αk ⊗β k ), n ∈ N, on A⊗C(Y ). However, relative entropy is convex, which makes it impossible in general to estimate the entropy of the coupling we get. The key observation allowing to overcome this problem is that although Hλ (γ; ξ) is a convex function of λ, it is almost aﬃne on canonical couplings, see Deﬁnition 3.1.8. Lemma 5.1.6. Let γ: B → A be a channel. For ε > 0 choose δ > 0 such that |S(ψ1 ) − S(ψ2 )| < ε whenever ψ1 and ψ2 are states on B satisfying ψ1 − ψ2 < δ. Let ζ be a ﬁnite Borel partition of the state space X of A for which ω1 ◦ γ − ω2 ◦ γ < δ when ω1 and ω2 belong to the same element of ζ.Then if λi is a canonical coupling of (A, ϕ) with (X, µi ), i ∈ I, and λ = i αi λi is a convex combination, we have αi Hλi (γ; ζ) < ε. Hλ (γ; ζ) − i

Proof. For any canonical coupling Λ we have by (3.11) HΛ (γ; ζ) = S(ϕ ◦ γ) − ν(Z)S ν(Z)−1 ω ◦ γ dν(ω) , Z∈ζ

Z

where ν = Λ|C(X) . Let µ = i αi µi , and set ϕZ = µ(Z)−1 Z ω dµ(ω). Then by the choice of ζ we have ω ◦ γ − ϕZ ◦ γ < δ for any ω ∈ Z. Hence

5.1 Entropy via Stationary Couplings

83

HΛ (γ; ζ) − S(ϕ ◦ γ) + ν(Z)S(ϕZ ◦ γ) < ε. Z∈ζ In other words, if we put f (Λ) = S(ϕ ◦ γ) − Z∈ζ ν(Z)S(ϕZ ◦ γ), then f is an aﬃne function on the space of canonical couplings such that f (λ) = Hλ (γ, ζ) and |HΛ (γ, ζ) − f (Λ)| < ε for any Λ. This gives the result.

Returning to the proof of Theorem 5.1.5, ﬁx h < hϕ (α) and ε > 0. Choose a channel γ: B → A such that hϕ (γ; α) > h. Choose also a ﬁnite Borel partition ζ of X = S(A) as in Lemma 5.1.6. +∞ Put Z = −∞ X, and deﬁne a homeomorphism T of Z by ˆ (ωn−1 ) = ωn−1 ◦ α−1 for ω = (ωn )n ∈ Z. (T ω)n = α Let β be the corresponding automorphism of C(Z), so β(f ) = f ◦ T −1 . Let prk : Z → X be the projection onto the k-th factor. Set ξ = pr−1 0 (ζ). k Then T k ξ = pr−1 (ˆ α (ζ)). By Proposition 3.1.10 applied to the channels αk ◦γ k k and the partitions α ˆ (ζ) of X, 0 ≤ k ≤ n − 1, there exists a coupling λn of (A, ϕ) with (Z, µn ) such that 1 Hλ (γ, α ◦ γ, . . . , αn−1 ◦ γ; ξ, T ξ, . . . , T n−1 ξ) > h − ε n n n−1

(more precisely, we ﬁrst get a state on A ⊗ C( k=0 X), and then extend it arbitrarily to A ⊗ C(Z)). In other words, by (5.1) and (5.2), n−1 1 k Hµn (∨n−1 Hλn ,αk ◦γ (T k ξ) > h − ε. k=0 T ξ) − n k=0

By a computation similar to (1.2) this can be written as n−1 1 j k (Hµn (T k ξ| ∨k−1 j=0 T ξ) − Hλn ,αk ◦γ (T ξ)) > h − ε. n k=0

For any ﬁxed p ∈ N the contribution of the summands with k < p becomes k−1 j j negligible as n → ∞. On the other hand, if k ≥ p then ∨k−1 k−p T ξ ≺ ∨j=0 T ξ, k−1 j j T ξ) ≤ Hµn (T k ξ| ∨k−1 so that Hµn (T k ξ| ∨j=0 k−p T ξ). Hence n−1 1 j k (Hµn (T k ξ| ∨k−1 k−p T ξ) − Hλn ,αk ◦γ (T ξ)) > h − 2ε n k=0

for any ﬁxed p and any n suﬃciently large. By Proposition 5.1.3(vi), j −Hλn ,αk ◦γ (T k ξ) ≤ −Hλn ,αk ◦γ (∨kk−p T j ξ) + Hλn ,αk ◦γ (∨k−1 k−p T ξ), k−1 j j j k so that using Hµn (T k ξ| ∨k−1 k−p T ξ) = Hµn (∨k−p T ξ) − Hµn (∨k−p T ξ) we get

84

5 Dynamical Abelian Models n−1 1 j (Hλn (αk ◦ γ; ∨kk−p T j ξ) − Hλn (αk ◦ γ; ∨k−1 k−p T ξ)) > h − 2ε n

(5.4)

k=0

for any ﬁxed p and any n suﬃciently large. Our goal is to replace ∨kk−p T j ξ by T k ξ in the ﬁrst summand to be able to use Lemma 5.1.6. Set an,p =

n−1 1 Hλn (αk ◦ γ; ∨kk−p T j ξ). n k=0

We have 0 ≤ an,p ≤ Hϕ (γ). Thus there exists an inﬁnite subset I ⊂ N such that the sequence {an,p }n∈I converges as I n → ∞ for any p ∈ N. Denote its limit by ap . By Proposition 5.1.3(i) we have an,p ≥ an,r if p ≥ r, so ap ≥ ar . Let a∞ be the limit of {ap }p as p → ∞. Fix r ∈ N such that ar > a∞ − ε. As ap ≤ a∞ , for any ﬁxed p and any n ∈ I suﬃciently large we have an,r > an,p − ε. Thus (5.4) implies that n−1 1 k−1 j (Hλn (αk ◦ γ; ∨kk−r T j ξ) − Hλn (αk ◦ γ; ∨k−p T ξ)) > h − 3ε n

(5.5)

k=0

for any ﬁxed p and any n ∈ I suﬃciently large. Now apply Proposition 3.1.10 to the coupling λn , the channels αk ◦ γ, the partitions ∨kk−r T j ξ of Z and the partitions α ˆ k (ζ) of X, 0 ≤ k ≤ n − 1. Thus we get a Borel map f : Z → Z such that for 0 ≤ k ≤ n − 1 we have (i) f −1 (T k ξ) ≺ ∨kk−r T j ξ; (ii) if we set λn = λn ◦ (id ⊗ f ∗ ), then |Hλn (αk ◦ γ; ∨kk−r T j ξ) − Hλn (αk ◦ γ; T k ξ)| < ε; (iii) the coupling λn ◦ (id ⊗ pr∗k ) is canonical. (Again, to be more precise, we have maps fk : Z → X for 0 ≤ k ≤ n − 1, but then we take arbitrary Borel maps Z → X for all other k’s.) k−1 j j By virtue of (i) we have f −1 (∨k−1 k−p+r T ξ) ≺ ∨k−p T ξ if k ≥ p − r > 0 and k ≤ n − 1. By Proposition 5.1.3(i) we thus have j k −1 j Hλn (αk ◦ γ; ∨k−1 (∨k−1 k−p T ξ) ≥ Hλn (α ◦ γ; f k−p+r T ξ)) j = Hλn (αk ◦ γ; ∨k−1 k−p+r T ξ)

if 0 < p − r ≤ k ≤ n − 1, so that n−1 n−1 1 1 j j Hλn (αk ◦ γ; ∨k−1 T ξ) > Hλn (αk ◦ γ; ∨k−1 k−p+r T ξ) − ε k−p n n k=0

k=0

for p > r and n suﬃciently large, since the contribution of the summands with k < p − r becomes negligible as n → ∞. Thus property (ii) of the map f and inequality (5.5) imply

5.1 Entropy via Stationary Couplings

85

n−1 1 k−1 (Hλn (αk ◦ γ; T k ξ) − Hλn (αk ◦ γ; ∨k−p+r T j ξ)) > h − 5ε, n k=0

or equivalently, n−1 1 j (Hλn ◦(αk ⊗β k ) (γ; ξ) − Hλn ◦(αk ⊗β k ) (γ; ∨−1 −p+r T ξ)) > h − 5ε n k=0

for any ﬁxed p > r and any n ∈ I suﬃciently large. Set λn =

n−1 1 λn ◦ (αk ⊗ β k ). n k=0

ˆ ∗ )−k , The couplings λn ◦ (αk ⊗ β k ) ◦ (id ⊗ pr∗0 ) = λn ◦ (id ⊗ pr∗k ) ◦ (α−1 ⊗ α ∗ where α ˆ (g) = g ◦ α ˆ for g ∈ C(X), are canonical for 0 ≤ k ≤ n − 1 by property (iii) of f and the fact that the automorphism α−1 ⊗ α ˆ ∗ of A ⊗ C(X) −1 preserves the set of canonical couplings. Since ξ = pr0 (ζ), by Lemma 5.1.6 and Proposition 5.1.3(iii) we get j Hλn (γ; ξ) − Hλn (γ; ∨−1 −p+r T ξ) > h − 6ε.

Now replace Z by the spectrum Y of a separable β-invariant C∗ -algebra consisting of bounded Borel functions on Z and containing the characteristic functions of the elements of ξ. Denote the homeomorphism of Y deﬁned by T and the restriction of λn to A ⊗ C(Y ) by the same symbols. We can think of ξ as a partition of Y consisting of clopen sets. Let λ be a weak∗ limit point of {λn }n∈I . Since the von Neumann entropy is continuous, we have j Hλ (γ; ξ) − Hλ (γ; ∨−1 −k T ξ) ≥ h − 6ε

for any k ∈ N. Hence j Hλ (γ; ∨0−k T j ξ) − Hλ (γ; ∨−1 −k T ξ) ≥ h − 6ε,

equivalently, with µ = λ|C(Y ) , −1 j j 0 j Hµ (∨0−k T j ξ) − Hµ (ξ| ∨−1 −k T ξ) − Hλ,γ (∨−k T ξ) + Hλ,γ (∨−k T ξ) ≥ h − 6ε,

which by Proposition 5.1.3(vi) can be written as −1 j j Hµ (ξ| ∨−1 −k T ξ) − Hλ,γ (ξ| ∨−k T ξ) ≥ h − 6ε.

Letting k → ∞, by Proposition 5.1.3(iv) we get Hµ (ξ|ξ − ) − Hλ,γ (ξ|ξ − ) ≥ h − 6ε. Since Hλ,γ (ξ|ξ − ) ≥ Hλ,A (ξ|ξ − ) by Proposition 5.1.3(iv), using Proposition 5.1.4 we conclude that hST ϕ (α) ≥ h − 6ε, which completes the proof of the theorem.

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5 Dynamical Abelian Models

In Proposition 3.2.7 we proved that a state on a nuclear subalgebra of a C∗ -algebra can be extended to a state on the algebra in an entropy-increasing way. As our ﬁrst application of existence of abelian models we shall now prove another similar result. Proposition 5.1.7. Let α be an automorphism of a nuclear C∗ -algebra A, B ⊂ A an α-invariant C∗ -subalgebra, ψ an α-invariant state on B. Then for every h < hψ (α|B ) there exists an α-invariant state ϕ on A such that ϕ|B = ψ and hϕ (α) > h. Proof. Let λ be a stationary coupling of (B, ψ, α|B ) with (X, µ, T ), and ξ a measurable partition of (X, µ) such that hµ (ξ; T ) − Hµ (ξ) + Hλ (B; ξ) > h. Extend λ to an (α ⊗ β)-invariant state Λ on A ⊗ L∞ (X, µ), where β is the automorphism of L∞ (X, µ) deﬁned by T , and put ϕ = Λ|A . By Proposition 5.1.3(i) we have HΛ (A; ξ) ≥ Hλ (B; ξ), hence hϕ (α) = hST

ϕ (α) > h. If (X/ξ, µ, T ) is a factor system of (X, µ, T ) such that the map X → X/ξ is ﬁnite-to-one almost everywhere, then it is not diﬃcult to check that both systems have the same entropy. The following is a noncommutative analogue of this result. Proposition 5.1.8. Let (M, ϕ, α) be a W∗ -dynamical system having a ϕapproximating net, N ⊂ M an α-invariant von Neumann subalgebra, E: M → N a ϕ-preserving faithful normal conditional expectation commuting with α. Assume there exists a constant c > 0 such that E(a) ≥ ca for any positive a ∈ M . Then hϕ (α) = hϕ|N (α|N ). Proof. Using the conditional expectation E, any stationary coupling of the system (N, ϕ|N , α|N ) with an abelian system can be extended to a coupling of (M, ϕ, α) with the same system. This implies hϕ (α) ≥ hϕ|N (α|N ). Conversely, let λ be a stationary coupling of (M, ϕ, α) with (X, µ, T ), and ξ a measurable partition of (X, µ). For any normal state ψ on M we have ψ ◦ E ≥ cψ. By Theorem 2.3.1(vii) we then get S(ψ, ϕ) − S(ψ|N , ϕ|N ) = S(ψ, ψ ◦ E) ≤ S(ψ, cψ) = − log c. It follows that Hλ (M ; ξ) − Hλ (N ; ξ) ≤ − log c, and hence hϕ (α) ≤ hϕ|N (α|N ) − log c. Applying this to αn , dividing by n and letting n → ∞ we get hϕ (α) ≤ hϕ|N (α|N ).

5.2 Zero Entropy Systems Let (A, ϕ, α) be a C∗ -dynamical system. Consider the space of the GNSrepresentation deﬁned by ϕ. On this space the automorphism α is implemented by the unitary U deﬁned by U aξϕ = α(a)ξϕ for a ∈ A. We say that

5.2 Zero Entropy Systems

87

the system has singular spectrum if the spectral measure of U is singular with respect to the Lebesgue measure on T. If the space is nonseparable, we mean by this that the spectral measure of the restriction of U to any separable invariant subspace is singular. Equivalently, for any vector ξ the measure νξ on T such that (U n ξ, ξ) = T z n dνξ (z) for n ∈ Z is singular. Theorem 5.2.1. Let (A, ϕ, α) be a C∗ -dynamical system with singular spectrum. Then hϕ (α) = hST ϕ (α) = 0. Proof. We shall ﬁrst reduce the proof to the case when A separable. Choose an increasing net {Ai }i of separable α-invariant C∗ -subalgebras with dense union. Let λ be a stationary coupling of (A, ϕ, α) with an abelian system (X, µ, T ), and ξ a ﬁnite measurable partition of (X, µ). By Corollary 2.3.6 it follows that Hλ (A; ξ) = lim Hλ (Ai ; ξ). i

≤ So it is enough to show that the enHence tropy of α|Ai is zero for every i. Since the systems (Ai , ϕ|Ai , α|Ai ) still have singular spectrum, we see that is suﬃces to prove the theorem for separable C∗ -algebras. By Proposition 5.1.4 we have to show that hST ϕ (α)

lim inf i hST ϕ|Ai (α|Ai ).

Hµ (ξ|ξ − ) = Hλ,A (ξ|ξ − )

(5.6)

for every stationary coupling λ with (X, µ, T ) and every ﬁnite measurable partition ξ. We shall do this in three steps, ﬁrst assuming that (X, µ, T ) is a Bernoulli shift, then that it is ergodic, and ﬁnally in general. So assume that (X, µ, T ) is a Bernoulli shift. We claim that then λ is trivial, λ = ϕ ⊗ µ. Let β be the automorphism of L∞ (X, µ) deﬁned by T , β(f ) = f ◦ T −1 for f ∈ L∞ (X, µ). Denote by V the canonical unitary on L2 (X, µ) implementing β, V f = f ◦ T −1 for f ∈ L2 (X, dµ). Let ζ be the standard generating partition for (X, µ, T ). Then, as we already remarked in Sect. 4.3, ζ − ≺ T ζ − , the algebra ∪n∈Z L∞ (X/T n ζ − ) is weakly operator dense in L∞ (X, µ), and ∩n∈Z L∞ (X/T n ζ − ) = C1. It follows that if we set L = L2 (X/T ζ − , dµ) L2 (X/ζ − , dµ) then L2 (X, dµ) C = ⊕ V n L. n∈Z

Thus on the orthogonal complement of the constants the unitary V has homogeneous Lebesgue spectrum, see C.7. Now let H be the space of the GNSrepresentation deﬁned by λ, and v ∈ H the corresponding cyclic vector. Let W be the unitary on H implementing α ⊗ β. According to C.7 the space H decomposes into a direct sum Hs ⊕ Ha such that the restriction of W to Hs has singular spectral measure, and the restriction of W to Ha has absolutely continuous spectral measure. The restriction of W to Av can be identiﬁed with U , while the restriction of W to L∞ (X, µ)v can be identiﬁed with V . It

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follows that av ∈ Hs for any a ∈ A, and f v ∈ Ha for any f ∈ L∞ (X, µ) with µ(f ) = 0. Hence av and f v are orthogonal, that is, λ(a ⊗ f¯) = 0. For any f ∈ L∞ (X, µ) we then have λ(a ⊗ f ) = ϕ(a)µ(f ) + λ(a ⊗ (f − µ(f )1)) = ϕ(a)µ(f ), and our claim is proved. Assume now that (X, µ, T ) is ergodic. Replacing (X, µ, T ) by the factor system (X/ξ ± , µ, T ), where ξ ± = ∨n∈Z T n ξ, we may assume that ξ is a generating partition. By Sinai’s Theorem 1.1.7 there exists a Bernoullian factor system with the same entropy. In other words, there exists a ﬁnite measurable partition ζ such that the partitions T n ζ, n ∈ Z, are mutually independent, and hµ (T ) = Hµ (ζ). Since ξ is generating and, as we already observed in the proof of Proposition 5.1.4, Hλ,A (ϑ|ϑ− ) = lim

n→∞

1 k Hλ,A (∨n−1 k=0 T ϑ) n

for any ﬁnite ϑ, the same proof as that of Lemma 1.1.3 shows that Hλ,A (ζ|ζ − ) ≤ Hλ,A (ξ|ξ − ) + Hλ,A (ζ|ξ ± ) = Hλ,A (ξ|ξ − ). On the other hand, by the case of Bernoullian systems considered above, we have Hλ,A (ζ|ζ − ) = Hµ (ζ|ζ − ) = hµ (T ) = Hµ (ξ|ξ − ). Since we always have Hλ,A (ξ|ξ − ) ≤ Hµ (ξ|ξ − ) by Proposition 5.1.3(iv), we conclude that (5.6) holds. Consider the general case. As before, let β be the automorphism of L∞ (X, µ) deﬁned by T , β(f ) = f ◦ T −1 for f ∈ L∞ (X, µ). Let ϑ be the measurable partition such that L∞ (X/ϑ) = L∞ (X)β , the ﬁxed point algebra of β. Then, see C.1-C.4, the space H of the GNS-representation deﬁned by λ ⊕ decomposes into a direct integral X/ϑ Hy dµ(y), elements of A⊗L∞ (X, µ) are decomposable, and a normal extension Λ of λ to the commutant of L∞ (X/ϑ) ⊕ decomposes into a direct integral Λ = X/ϑ Λy dµ(y). Without loss of generality we may assume that X and X/ϑ are compact metric spaces, X → X/ϑ is continuous, and T is a homeomorphism. Then ⊕ by C.5 the representation of A ⊗ C(X) decomposes into a direct integral X/ϑ πy dµ(y) of representations. Set λy = Λy ◦ πy , ϕy = λy |A and µy = λy |C(X) . The state λy is (α ⊗ β)invariant for a.e. y ∈ X/ϑ, so that λy is a stationary coupling of (A, ϕy , α) with (X, µy , T ). ∗ n a ∈ A and ν (resp. νy ) is the nmeasure on T such that ϕ(a α (a)) = If n ∗ n z dν(z) (resp. ϕy (a α (a)) = T z dνy (z)), then ν = X/ϑ νy dµ(y). Since T ν is singular by assumption, it follows that νy is singular for a.e. y. Thus the system (A, ϕy , α) has singular spectrum for a.e. y. Next note that (X, µy , T ) is ergodic for a.e. y. We shall give a sketch of the argument, see e.g. [72, Theorem 3.42] for details. By Birkhoﬀ’s ergodic

5.2 Zero Entropy Systems

89

n−1 −1

−k theorem, for any f ∈ C(X), the sequence {n x)}∞ n=1 converges k=0 f (T ∞ ∞ to E(f )(x) for a.e. x ∈ X, where E: L (X) → L (X/ϑ) is the µ-preserving conditional expectation. Since the operator E(f ) is diagonal, it follows that for n−1 f ∈ C(X) and a.e. y ∈ X/ϑ the sequence {n−1 k=0 πy (β k (f ))}n converges to a scalar operator in the strong operator topology. But if for a ﬁxed y this convergence holds for a countable dense subset of functions f ∈ C(X), then (X, µy , T ) is ergodic. We can thus apply the previous case to the couplings λy and conclude that

Hµy (ξ|ξ − ) = Hλy ,A (ξ|ξ − ) for any ﬁnite Borel partition ξ of X and a.e. y ∈ X/ϑ. To prove that (5.6) holds, it is then enough to show that Hµ (ξ|ξ − ) = Hµy (ξ|ξ − )dµ(y) and Hλ,A (ξ|ξ − ) = Hλy ,A (ξ|ξ − )dµ(y). X/ϑ

X/ϑ

We claim that

Hλy ,A (ξ|ξ − )dµ(y) = Hλ,A (ξ|ξ − ∨ ϑ).

(5.7)

X/ϑ

Since Hλy ,A (ξ| ∨nk=1 T −k ξ) Hλy ,A (ξ|ξ − ), and similarly Hλ,A (ξ|ξ − ∨ ϑ) is the limit of Hλ,A (ξ|ϑ ∨ ∨nk=1 T −k ξ), it suﬃces to check that Hλy ,A (ξ|ζ)dµ(y) = Hλ,A (ξ|ζ ∨ ϑ) X/ϑ

for any ﬁnite Borel partition ζ. By Proposition 5.1.3(vi) it then suﬃces to check that Hλy ,A (ζ)dµ(y) = Hλ,A (ζ|ϑ) X/ϑ

for any ζ. By (5.2) for this, in turn, it is enough to check that for any Borel subset Z ⊂ X we have S(λy (· ⊗ ·1Z ), λy )dµ(y) = S(λ(· ⊗ ·1Z ), λ). (5.8) X/ϑ

˜ and λ ˜ Z on A ⊗ C(X) ⊗ C(X/ϑ) deﬁned by Consider the states λ ˜ ˜ Z (a ⊗ f ⊗ g) = λ(a ˜ ⊗ f 1Z ⊗ g). λ(a ⊗ f ⊗ g) = λy (a ⊗ f )g(y)dµ(y) and λ X/ϑ

˜ Z , λ), ˜ Then by Corollary 2.3.7 the left hand side of (5.8) coincides with S(λ ˜ ˜ while the right hand side is S(λZ |A⊗C(X) , λ|A⊗C(X) ). Since the homomorphism A ⊗ C(X) ⊗ C(X/ϑ) → A ⊗ C(X), a ⊗ f ⊗ g → a ⊗ f g, which is also a

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5 Dynamical Abelian Models

conditional expectation, preserves both states, both sides indeed coincide by Theorem 2.3.1(ix) or Corollary 2.3.3. Thus (5.7) is established. Any ﬁnite measurable partition ζ ≺ ϑ of (X, µ) is T -invariant. By Proposition 5.1.3(vi) it follows that Hλ,A (ξ ∨ ζ|ξ − ∨ ζ − ) = Hλ,A (ξ ∨ ζ|ξ − ∨ ζ) = Hλ,A (ξ|ξ − ∨ ζ). On the other hand, 1 k Hλ,A (∨n−1 k=0 T ξ) n 1 k − − ≤ lim Hλ,A (∨n−1 k=0 T (ξ ∨ ζ)) = Hλ,A (ξ ∨ ζ|ξ ∨ ζ ). n→∞ n

Hλ,A (ξ|ξ − ) = lim

n→∞

Consequently

Hλ,A (ξ|ξ − ) = Hλ,A (ξ|ξ − ∨ ζ).

Choosing an increasing sequence of ﬁnite measurable partitions ζn , n ∈ N, such that ∨n ζn = ϑ, we conclude that Hλ,A (ξ|ξ − ) = Hλ,A (ξ|ξ − ∨ ϑ). Together with (5.7) this shows that Hλ,A (ξ|ξ − ) = X/ϑ Hλy ,A (ξ|ξ − )dµ(y). A similar argument gives Hµ (ξ|ξ − ) = X/ϑ Hµy (ξ|ξ − )dµ(y), which completes the proof of the theorem.

Proposition 5.1.2 implies that if a system has a nontrivial stationary coupling with a classical Bernoulli shift, then hST ϕ (α) > 0. The proof of the above theorem shows that the converse would be true if we could show that if the system does not have nontrivial couplings with Bernoulli shifts, and ϕ = X ϕx dµ(x), then the same is true for (A, ϕx , α) for almost every x. This is obviously true if we do not have nontrivial integral decompositions of ϕ into α-invariant states. Thus we get the following. Theorem 5.2.2. Let (A, ϕ, α) be a C∗ -dynamical system with separable A. Assume ϕ is ergodic, that is, ϕ is an extremal α-invariant state. Then hST ϕ (α) > 0 if and only if there exists a nontrivial stationary coupling with a classical Bernoulli shift.

We shall now introduce a class of systems which do not have nontrivial couplings with abelian systems. Deﬁnition 5.2.3. A system (A, α) consisting of a unital C∗ -algebra A and an automorphism α is called asymptotically highly anticommutative if there exists a set {xi }i∈I of elements of A which together with the unit spans a dense subspace of A and satisﬁes the following property. For any i ∈ I, ε > 0 and n ∈ N there exist m1 , . . . , mn ∈ N such that αmk (x∗i )αml (xi )+αml (xi )αmk (x∗i ) < ε for all k, l ≤ n with k = l.

5.3 Notes

91

Proposition 5.2.4. If (A, α) is asymptotically highly anticommutative then there exists a unique α-invariant state on A. Proof. Let ϕ be an α-invariant state, and {xi }i∈I a set as in the deﬁnition of an asymptotically highly anticommutative system. Fix i ∈ I. Let ε > 0 and mk m1 , . . . , mn ∈ N be such that αmk (x∗i )αml (xi ) + αml (x (x∗i ) < ε for i )α n −1 mk all k, l ≤ n with k = l. Consider the element x = n (xi ). Then k=1 α ∗ ∗ x x + xx equals n 1 mk ∗ ml 1 mk ∗ ∗ α (x x + x x ) + (α (xi )α (xi ) + αml (xi )αmk (x∗i )), i i i i n2 n2 k=l

k=1

whence x∗ x + xx∗ ≤

2xi 2 (n − 1)ε + . n n

Since |ϕ(xi )| = |ϕ(x)| ≤

1 1 (ϕ(x∗ x)1/2 + ϕ(xx∗ )1/2 ) ≤ √ ϕ(x∗ x + xx∗ )1/2 , 2 2

it follows that ϕ(xi ) = 0. Since {1} ∪ {xi }i∈I spans a dense subspace of A, the condition ϕ(xi ) = 0 completely determines the state.

The above proof implies the following. Corollary 5.2.5. If (A, ϕ, α) is an asymptotically highly anticommutative C∗ dynamical system, then any stationary coupling with an abelian dynamical system is trivial. In particular, hϕ (α) = hST ϕ (α) = 0. Proof. Let λ be a stationary coupling of (A, ϕ, α) with (X, µ, T ). Then the same estimate as in the proof of the previous proposition applied to xi ⊗ f instead of xi shows that λ(xi ⊗f ) = 0 = ϕ(xi )µ(f ). Together with the equality λ(1 ⊗ f ) = µ(f ) this implies λ = ϕ ⊗ µ.

We shall see examples of asymptotically highly anticommutative systems in Chaps. 11 and 12. In Chap. 14 we shall consider a system which is not asymptotically highly anticommutative, but which still has only trivial stationary couplings with abelian systems.

5.3 Notes The notion of a stationary coupling was introduced by Sauvageot and Thouvenot [188], who obtained all the main results of the present chapter. In the classical case stationary couplings are called joinings. They were introduced by Furstenberg [68]. See [72], [184] for an exposition of ergodic theory from the theory of joinings point of view.

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Propositions 5.1.7 and 5.1.8 are due to the authors [140], [141]. Theorem 5.2.1, proved by Sauvageot and Thouvenot [188], is an extension to the noncommutative case of a result of Rohlin and Sinai [183]. In the classical case the result states a bit more: if a system has positive entropy then the spectrum has a countably multiple Lebesgue component. Asymptotically highly anticommutative systems were introduced by Narnhofer, Størmer and Thirring [128], who proved Proposition 5.2.4 and Corollary 5.2.5. The name was coined later by Narnhofer and Thirring [134].

6 Topological Entropy

In this chapter we shall study the extension of topological entropy to C∗ algebras. The main problem is to ﬁnd the right analogue of the cardinality N (U) of a minimal subcover for an open cover U. If X is a totally disconnected compact space and U a clopen partition of X, then N (U) is the rank of the abelian algebra spanned by the characteristic functions of the elements of U. In the general case instead of the algebra generated by the characteristic functions of the elements of U we shall consider ﬁnite collections of functions which are almost constant on the elements of U. It turns out that instead of N (U) one can then take the ranks of completely positive approximations of these collections.

6.1 Rank of a Completely Positive Approximation Let A be a unital C∗ -algebra, Ω a ﬁnite subset of A. Deﬁnition 6.1.1. For δ > 0 denote by rcpA (Ω, δ) the minimal number n ∈ N for which there exist a unital injective homomorphism π: A → B for some C∗ -algebra B and unital completely positive maps θ: A → Matn (C) and γ: Matn (C) → B such that (γ ◦ θ)(a) − π(a) < δ for any a ∈ Ω. Put rcpA (Ω, δ) = +∞ if no such approximation exists. We shall write rcp(Ω, δ) instead of rcpA (Ω, δ) when no confusion can arise. If D is a ﬁnite dimensional C∗ -algebra, then D can be imbedded into Matn (C), where n is the rank of D. A unital completely positive map D → B can be extended to a unital completely positive map Matn (C) → B by composing it with the trace preserving conditional expectation Matn (C) → D. So we could deﬁne rcpA (Ω, δ) as the minimal number n ∈ N for which there exist an injective homomorphism π: A → B for some C∗ -algebra B and unital completely positive maps θ: A → D and γ: D → B such that n = rank D and (γ ◦ θ)(a) − π(a) < δ for any a ∈ Ω.

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6 Topological Entropy

Observe next that there is no need to consider all possible embeddings of A. Lemma 6.1.2. In computing rcpA (Ω, δ) it is enough to consider one faithful representation π: A → B(H). It also suﬃces to take a nuclear embedding π: A → B, if one exists. Recall that a homomorphism π: A → B is called nuclear if there exist two nets {θi : A → Matni (C)}i∈I and {γi : Matni (C) → B}i∈I of unital completely positive maps such that (γi ◦ θi )(a) − π(a) → 0 for any a ∈ A. The algebras admitting nuclear embeddings are said to be nuclearly embeddable, or exact. If rcp(Ω, δ) is ﬁnite for any Ω ⊂ A and any δ > 0, then Lemma 6.1.2 implies that any faithful representation π: A → B(H) is nuclear, so that A is exact. If A is nuclear, then the identity map A → A is nuclear, so we can take B = A. At least one reason to consider the class of exact C∗ -algebras is that subalgebras of nuclear algebras are not necessarily nuclear, but always exact. The class of exact algebras is quite large. It is obviously closed under taking subalgebras and minimal tensor products. In Chap. 8 we shall show that it is closed under taking crossed products by actions of amenable locally compact groups, while in Chap. 14 we shall prove that reduced free products of exact algebras are exact. Quotients of exact algebras are also exact. This is nontrivial, and will not be used in the sequel. Proof of Lemma 6.1.2. Denote by rcp(π, Ω, δ) the quantity deﬁned exactly as rcp(Ω, δ) but using a ﬁxed embedding π: A → B. Let π: A → B(H) be a faithful representation. Given an arbitrary embedding π0 : A → B, by Arveson’s extension theorem A.8 there exists a unital completely positive map Φ: B → B(H) such that Φ ◦ π0 = π. Given a unital completely positive map γ: Matn (C) → B we can now consider Φ◦γ: Matn (C) → B(H). Since (Φ◦γ ◦θ)(a)−π(a) = Φ((γ ◦θ)(a)−π0 (a)), we get rcp(π, Ω, δ) ≤ rcp(π0 , Ω, δ). Similarly, assume π: A → B is a nuclear embedding. Let {θi : A → Matni (C)}i∈I and {γi : Matni (C) → B}i∈I be nets of unital completely positive maps such that (γi ◦ θi )(a) − π(a) → 0 for any a ∈ A. Let π0 : A → B0 be another embedding, and let θ0 : A → Matn (C) and γ0 : Matn (C) → B0 be unital completely positive maps such that (γ0 ◦θ0 )(a)−π0 (a) < δ for a ∈ Ω. Again by Arveson’s extension theorem there exist unital completely positive maps θ¯i : B0 → Matni (C), i ∈ I, such that θ¯i ◦ π0 = θi . Consider the maps γi ◦ θ¯i ◦ γ0 : Matn (C) → B. We have (γi ◦ θ¯i ◦ γ0 ◦ θ0 )(a) − π(a) ≤ (γi ◦ θ¯i ◦ γ0 ◦ θ0 )(a) − (γi ◦ θ¯i ◦ π0 )(a) + (γi ◦ θ¯i ◦ π0 )(a) − π(a). ≤ (γ0 ◦ θ0 )(a) − π0 (a) + (γi ◦ θi )(a) − π(a). Since (γi ◦ θi )(a) − π(a) → 0, we see that for suﬃciently large i we get (γi ◦θ¯i ◦γ0 ◦θ0 )(a)−π(a) < δ for any a ∈ Ω. Thus rcp(π, Ω, δ) ≤ rcp(π0 , Ω, δ).

6.1 Rank of a Completely Positive Approximation

95

In order to obtain a better understanding of the deﬁnition consider the abelian case. Lemma 6.1.3. Let X be a compact space, Ω ⊂ C(X) a ﬁnite subset, and δ > 0. Let U be a ﬁnite open cover of X such that |f (x) − f (y)| < δ for every f ∈ Ω and x, y ∈ X lying in the same element of U. Then rcp(Ω, δ) ≤ N (U). Proof. Let V be a subcover of U with minimal number N = N (U) of elements. Let V = {V1 , . . . , VN }, and for each i choose xi ∈ Vi . Denote the subset of X consisting of the xi ’s by Y . Let further {χi }N i=1 be a partition of unity subordinate to V, that is, supp χi ⊂ Vi and i χi = 1. We set B = C(Y ), and deﬁne θ: C(X) → B, θ(f ) = f |Y , and γ: B → C(X), γ(g) =

N

g(xi )χi .

i=1

If f ∈ Ω then for any x ∈ X |f (x) − ((γ ◦ θ)(f ))(x)| ≤

N

χi (x)|f (x) − f (xi )|

i=1

=

χi (x)|f (x) − f (xi )| < δ.

i : x∈Vi

Hence rcp(Ω, δ) ≤ rank B = N (U).

We next list some properties of rcp. Lemma 6.1.4. For every unital C∗ -algebra A we have: (i) rcp(Ω, δ) = rcp(Ω ∪ Ω ∗ ∪ {1}, δ); (ii) if Ω ⊂ Ω0 then rcp(Ω, δ) ≤ rcp(Ω0 , δ); (iii) if Ω ⊂δ0 Ω0 , meaning that for any a ∈ Ω there is a0 ∈ Ω0 such that a − a0 < δ0 , then rcp(Ω, δ + 2δ0 ) ≤ rcp(Ω0 , δ); (iv) if each a ∈ Ω is a linear combination of at most k elements of Ω0 with coeﬃcients of modulus not greater than l, then rcp(Ω, klδ) ≤ rcp(Ω0 , δ); (v) if Ω consists of unitary operators and contains the unit, then rcp(Ω 2 , 4δ) ≤ rcp(Ω, δ) ≤ rcp(Ω 2 , δ), where Ω 2 = {uv | u, v ∈ Ω}; (vi) if α is an automorphism of A, then rcp(α(Ω), δ) = rcp(Ω, δ); (vii)if β: A → C is an injective unital homomorphism, then rcpC (β(Ω), δ) = rcpA (Ω, δ). Proof. Parts (i)-(iv) are straightforward. For example, part (iii) follows from the inequality (γ ◦ θ)(a) − π(a) ≤ (γ ◦ θ)(a0 ) − π(a0 ) + (γ ◦ θ)(a − a0 ) + π(a − a0 ).

96

6 Topological Entropy

To prove (v) note that the second inequality is trivial as Ω ⊂ Ω 2 . To prove the other inequality suppose A ⊂ B, and let θ: A → Matn (C) and γ: Matn (C) → B be unital completely positive maps such that (γ ◦ θ)(a) − a < δ for a ∈ Ω. Put Φ = γ ◦ θ. Then for u ∈ Ω Φ(u)Φ(u)∗ − 1 ≤ (Φ(u) − u)Φ(u)∗ + u(Φ(u)∗ − u∗ ) < 2δ, and similarly Φ(u)Φ(v) − uv < 2δ and Φ(v)∗ Φ(v) − 1 < 2δ when furthermore v ∈ Ω. Thus by A.5 we have Φ(uv) − uv < Φ(uv) − Φ(u)Φ(v) + 2δ ≤ Φ(uu∗ ) − Φ(u)Φ(u)∗ 1/2 Φ(v ∗ v) − Φ(v)∗ Φ(v)1/2 + 2δ = 1 − Φ(u)Φ(u)∗ 1/2 1 − Φ(v)∗ Φ(v)1/2 + 2δ < (2δ)1/2 (2δ)1/2 + 2δ = 4δ. It follows that rcp(Ω 2 , 4δ) ≤ rcp(Ω, δ). Part (vi) is a particular case of (vii). To prove (vii), by Lemma 6.1.2 it suﬃces to take a faithful representation π: C → B(H) and check that rcp(π ◦ β, Ω, δ) = rcp(π, β(Ω), δ). This follows again from Arveson’s extension theorem, which implies that every unital completely positive map A → Matn (C) extends to C.

6.2 Topological Entropy We are now in position to deﬁne topological entropy. Deﬁnition 6.2.1. Let A be a unital C∗ -algebra, and α an automorphism of A. Let 1 log rcpA (Ω ∪ α(Ω) ∪ . . . ∪ αn−1 (Ω), δ), n ht(Ω; α) = sup ht(Ω, δ; α),

ht(Ω, δ; α) = lim sup n→∞

δ>0

ht(α) = sup ht(Ω; α), Ω

where the last supremum is taken over all ﬁnite subsets Ω of A. The quantity ht(α) is called the topological entropy of α. Note that if δ1 < δ2 , then rcp(Ω, δ1 ) ≥ rcp(Ω, δ2 ). So the supremum in the deﬁnition of ht(Ω; α) can be replaced by the limit as δ → 0. We know that rcp(Ω, δ) is ﬁnite for any δ > 0 and any ﬁnite Ω ⊂ A if and only if A is exact. Thus, even though our deﬁnition of topological entropy makes sense for arbitrary C∗ -algebras, it is inﬁnity for any automorphism of a nonexact algebra.

6.2 Topological Entropy

97

Theorem 6.2.2. Let α be an automorphism of an exact C∗ -algebra A. Then (i) if β: A → C is an isomorphism then ht(α) = ht(β ◦ α ◦ β −1 ); (ii) ht(αn ) = |n|ht(α) for every n ∈ Z; if {αt }t∈R is a one-parameter automorphism group of A such that α1 = α, then ht(αt ) = |t|ht(α) for t ∈ R; (iii) if B is an α-invariant unital subalgebra of A, then ht(α) ≥ ht(α|B ); (iv) if β is an automorphism of a unital exact C∗ -algebra B, then ht(α ⊗ β) ≤ ht(α) + ht(β); in particular, ht(α ⊗ idB ) = ht(α); (v) under the assumptions of (iv), ht(α ⊕ β) = max{ht(α), ht(β)}. Proof. Parts (i) and (iii) are obvious. By Lemma 6.1.4(vi) we have m−1 −k k (Ω), δ), rcp(∪m−1 k=0 α (Ω), δ) = rcp(∪k=0 α

which shows ht(α) = ht(α−1 ). Thus to prove the ﬁrst part of (ii) we may n(m−1) k nk α (Ω), so that assume n > 0. We have ∪m−1 k=0 α (Ω) ⊂ ∪k=0 n(m−1) k

nk rcp(∪m−1 k=0 α (Ω), δ) ≤ rcp(∪k=0

α (Ω), δ),

which implies ht(Ω, δ; αn ) ≤ n ht(Ω, δ; α), and thus ht(αn ) ≤ n ht(α). For the reverse inequality note that if Ω ⊂ A, then Ω0 = ∪n−1 k=0 (Ω) satisﬁes m−1 ! k=0

[m/n]

αk (Ω) ⊂

!

αnk (Ω0 ).

k=0

Thus we have n ht(Ω, δ; α) ≤ ht(Ω0 , δ; αn ). Since Ω is an arbitrary ﬁnite subset of A, it follows that n ht(α) ≤ ht(αn ). Let now {αt }t∈R be a one-parameter automorphism group of A such that α1 = α. As in the proof of Theorem 3.2.5(i), to prove that t−1 ht(αt ) is independent of t > 0, it suﬃces to check that ht(αt ) ≤ t ht(α). Given a ﬁnite subset Ω ⊂ A and δ > 0 ﬁnd a ﬁnite Ω0 ⊂ A such that αs (Ω) ⊂δ Ω0 for 0 ≤ s ≤ 1. Then by Lemma 6.1.4(iii) [(n−1)t] k

k rcp(∪n−1 k=0 αt (Ω), 3δ) ≤ rcp(∪k=0

α (Ω0 ), δ),

so that ht(Ω, 3δ; αt ) ≤ t ht(Ω0 , δ; α) ≤ t ht(α). Hence ht(αt ) ≤ t ht(α). To prove the inequality in (iv) note that by Lemma 6.1.4(iii),(iv) in order to compute the entropy it suﬃces to consider ﬁnite subsets of A ⊗ B of the form Ω = Ω1 ⊗ Ω2 , where Ω1 and Ω2 are ﬁnite subsets of the unit balls of A and B, respectively. Since Matk (C) ⊗ Matm (C) ∼ = Matkm (C), it is then clear that

98

6 Topological Entropy

n−1 j n−1 j j rcp(∪n−1 j=0 (α ⊗ β) (Ω1 ⊗ Ω2 ), 2δ) ≤ rcp(∪j=0 α (Ω1 ), δ) rcp(∪j=0 β (Ω2 ), δ).

This in turn gives ht(Ω1 ⊗ Ω2 , 2δ; α ⊗ β) ≤ ht(Ω1 , δ; α) + ht(Ω2 , δ; β), and hence we get the desired conclusion. In particular, ht(α ⊗ idB ) ≤ ht(α) if B is exact, since ht(idB ) = 0 in this case. Since A ⊂ A ⊗ B, we also have the opposite inequality. It remains to prove (v). By considering direct sums of unital completely positive maps it is clear that if Ω1 ⊂ A and Ω2 ⊂ B, then rcp(Ω1 ⊕ Ω2 , δ) ≤ rcpA (Ω1 , δ) + rcpB (Ω2 , δ). Since log(s + t) ≤ log 2 + max{log s, log t} for any positive numbers s, t, we conclude that ht(Ω1 ⊕ Ω2 , δ; α ⊕ β) ≤ max{ht(Ω1 , δ; α), ht(Ω2 , δ; β)}. If A ⊂ B(H) and B ⊂ B(K), then A ⊕ B ⊂ B(H ⊕ K). Now given θ: A ⊕ B → Matn (C) and γ: Matn (C) → B(H ⊕ K), ﬁx a state ϕ on A and ˜ A → Matn (C) deﬁned by θ(a) ˜ = θ(a⊕ϕ(a)1), and the composition consider θ: γ˜ : Matn (C) → B(H) of γ with the compression map B(H ⊕ K) → B(H). It ˜ δ), where Ω ˜ consists of the elements follows that rcpA (Ω, δ) ≤ rcpA⊕B (Ω, a ⊕ ϕ(a)1, a ∈ Ω. Thus ht(α) ≤ ht(α ⊕ β), and analogously ht(β) ≤ ht(α ⊕ β).

Recall that by Theorem 3.2.2(iv) the inequality for the dynamical entropy of a tensor product automorphism goes the opposite way. We next show two Kolmogorov-Sinai type results. We need the following simple lemma. Lemma 6.2.3. Let Ω ⊂ A be ﬁnite, k ∈ N, α ∈ Aut(A), and Ωk = ∪kj=−k αj (Ω). Then ht(Ωk , δ; α) = ht(Ω, δ; α). j −k (∪n+2k−1 αj (Ω)), by Lemma 6.1.4(vi) we have Proof. Since ∪n−1 j=0 α (Ωk ) = α j=0 n+2k−1 j j rcp(∪n−1 α (Ω), δ). j=0 α (Ωk ), δ) = rcp(∪j=0

Taking the logarithm, dividing by n and letting n → ∞, we get the result.

Theorem 6.2.4. Let A be a C∗ -algebra and α ∈ Aut(A). Suppose {Ωλ }λ∈Λ is an increasing net of ﬁnite subsets of A such that the linear span of ∪λ∈Λ,n∈Z αn (Ωλ ) is dense in A. Then ht(α) = lim ht(Ωλ ; α). λ

Proof. By Lemma 6.1.4(iii),(iv) in order to compute the entropy it suﬃces to consider ﬁnite subsets of ∪n∈Z ∪λ∈Λ αn (Ωλ ). Given such a subset Ω we can ﬁnd λ0 and n ∈ N such that Ω ⊂ ∪nk=−n αk (Ωλ0 ). Then by the previous lemma and Lemma 6.1.4(ii) ht(Ω, δ; α) ≤ ht(∪nk=−n αk (Ωλ0 ), δ; α) = ht(Ωλ0 , δ; α),

6.2 Topological Entropy

so that ht(Ω; α) ≤ ht(Ωλ ; α) for any λ λ0 .

99

If we consider sets consisting of unitary elements, we get a much more satisfactory result. Lemma 6.2.5. Assume Ω consists of unitary operators and contains the unit. Put Ω n = {u1 . . . un | ui ∈ Ω}. Then ht(Ω; α) = ht(Ω n ; α). Proof. By Lemma 6.1.4(v) we have ht(Ω; α) = ht(Ω 2 ; α). Thus by iteration k k ht(Ω; α) = ht(Ω 2 ; α) for all k ∈ N. Since Ω ⊂ Ω n ⊂ Ω 2 for suﬃciently large k the lemma follows from Lemma 6.1.4(ii).

Theorem 6.2.6. Let A be a unital C∗ -algebra and α ∈ Aut(A). Suppose {Ωλ }λ∈Λ is an increasing net of ﬁnite sets of unitary operators in A such that the ∗-algebra generated by ∪λ∈Λ,n∈Z αn (Ωλ ) is dense in A. Then ht(α) = lim ht(Ωλ ; α). λ

Proof. By Lemma 6.1.4(i) we may assume 1 ∈ Ωλ and Ωλ∗ = Ωλ for any λ ∈ Λ. Set Ωλ,n = (∪nk=−n αk (Ωλ ))n . Deﬁne an order on the set {(λ, n) | λ ∈ Λ, n ∈ N} by letting (λ1 , n1 ) (λ2 , n2 ) if λ1 λ2 and n1 ≥ n2 . Then {Ωλ,n }λ,n is an increasing net, and ∪λ,n Ωλ,n spans a dense subspace of A. Hence ht(α) = lim ht(Ωλ,n ; α). (λ,n)

Since ht(Ωλ,n ; α) = ht(Ωλ ; α) by Lemmas 6.2.3 and 6.2.5, the proof of the theorem is complete.

In the abelian case the entropy of a homeomorphism with respect to an invariant probability measure is majorized by the topological entropy of the homeomorphism. We now show a noncommutative analogue of this result. Proposition 6.2.7. Let (A, ϕ, α) be a C∗ -dynamical system with A exact. Then hϕ (α) ≤ ht(α). Proof. Assume A ⊂ B(H). Let N be a ﬁnite dimensional C∗ -algebra and γ: N → A be a channel. Fix δ > 0 and choose a ﬁnite subset ω of N such that its convex hull contains the unit ball. Put Ω = γ(ω). Let n ∈ N, and θ: A → Matm (C) and ρ: Matm (C) → B(H) be unital completely positive maps such that (ρ ◦ θ)(a) − a < δ for a ∈ αk (Ω), 0 ≤ k ≤ n − 1. Then ρ ◦ θ ◦ αk ◦ γ − αk ◦ γ < δ for 0 ≤ k ≤ n − 1. Let ϕ = i1 ,...,in ϕi1 ...in be a ﬁnite decomposition. Extend each ϕi1 ...in to a positive linear functional ψi1 ...in on B(H), and set ψ = i1 ,...,in ψi1 ...in . We obviously have

100

6 Topological Entropy

Hϕ (γ, . . . , αn−1 ◦ γ; {ϕi1 ...in }) = Hψ (γ, . . . , αn−1 ◦ γ; {ψi1 ...in }). On the other hand, by Proposition 3.1.11 |Hψ (γ, . . . , αn−1 ◦ γ) − Hψ (ρ ◦ θ ◦ γ, . . . , ρ ◦ θ ◦ αn−1 ◦ γ)| < nε for some ε depending only on δ and dim N . Since Hψ (ρ ◦ θ ◦ γ, . . . , ρ ◦ θ ◦ αn−1 ◦ γ) ≤ S(ψ ◦ ρ) ≤ log m by Proposition 3.1.3(i),(iv) and Lemma 3.1.4, we thus get Hϕ (γ, . . . , αn−1 ◦ γ; {ϕi1 ...in }) < log m + nε. It follows that hϕ (γ; α) ≤ ht(Ω, δ; α) + ε ≤ ht(α) + ε. Since ε = ε(δ, dim N ) can be made arbitrarily small, we obtain the desired inequality.

Remark 6.2.8. For the entropy hST introduced in Chap. 5, which always majorizes the dynamical entropy and coincides with it in the nuclear case by Theorem 5.1.5, we also have hST ϕ (α) ≤ ht(α). Indeed, assume A ⊂ B(H). We may also assume that α extends to an automorphism α ¯ of B(H) (represent A α Z faithfully and take α ¯ = Ad u, where u is the canonical unitary in the crossed product). Let λ be a stationary coupling of (A, ϕ, α) with (X, µ, T ). ¯ on B(H)⊗C(X), where β(f ) = f ◦T −1 Extend λ to an (¯ α⊗β)-invariant state λ ¯ for f ∈ C(X). Set ϕ¯ = λ|B(H) . By deﬁnition of hST we have to prove that S(λ(· ⊗ 1Y ), ϕ) ≤ ht(α) hµ (ξ; T ) + Y ∈ξ

for any ﬁnite measurable partition ξ of X. Fix ε > 0. Since relative entropy is weakly∗ lower semicontinuous by Theorem 2.3.1(iv), there exist δ > 0 and a ﬁnite subset Ω ⊂ A such that for Y ∈ ξ S(λ(· ⊗ 1Y ), ϕ) < S(ω, ψ) +

ε |ξ|

as soon as |ω(a) − λ(a ⊗ 1Y )| < δ and |ψ(a) − ϕ(a)| < δ for a ∈ Ω. Let n ∈ N, and θ: A → Matm (C) and γ: Matm (C) → B(H) be unital completely positive maps such that (γ ◦ θ)(a) − a < δ for a ∈ αk (Ω), 0 ≤ k ≤ n − 1. Then using monotonicity of relative entropy, Theorem 2.3.1(vi), we get (see (3.9)) k Hλ¯ (γ, . . . , γ; ξ, T ξ, . . . , T n−1 ξ) − Hµ (∨n−1 k=0 T ξ)

6.2 Topological Entropy

=

n−1

101

¯ S(λ(γ(·) ⊗ 1T k Y ), ϕ¯ ◦ γ)

k=0 Y ∈ξ

=

n−1

¯ α−k ◦ γ)(·) ⊗ 1Y ), ϕ¯ ◦ γ) S(λ((¯

k=0 Y ∈ξ

≥

n−1

¯ α−k ◦ γ ◦ θ ◦ αk )(·) ⊗ 1Y ), ϕ¯ ◦ γ ◦ θ ◦ αk ) S(λ((¯

k=0 Y ∈ξ

>n

S(λ(· ⊗ 1Y ), ϕ) − nε.

Y ∈ξ

On the other hand, Hλ¯ (γ, . . . , γ; ξ, . . . , T n−1 ξ) ≤ Hϕ¯ (γ) ≤ S(ϕ¯ ◦ γ) ≤ log m. It follows that n−1 k k log rcp(∪n−1 k=0 α (Ω), δ) > Hµ (∨k=0 T ξ) + n

S(λ(· ⊗ 1Y ), ϕ) − nε.

Y ∈ξ

Dividing by n and letting n → ∞ we get the desired conclusion, as ε can be made arbitrarily small. We can now prove that our deﬁnition of topological entropy reduces to the classical one in the abelian case. Proposition 6.2.9. Let T : X → X be a homeomorphism of a compact metric space X, and let α be the automorphism of C(X) induced by T via α(f ) = f ◦ T −1 . Then ht(α) = htop (T ). Proof. Let Ω be a ﬁnite subset of C(X), δ > 0, and let U be an open cover of X such that |f (x) − f (y)| < δ for any f ∈ Ω and any x, y ∈ X lying in the j same element of U. Let n ∈ N. Then if f ∈ ∪n−1 j=0 α (Ω) and x, y belong to the n−1 j same element of Un = ∨j=0 T U we have |f (x) − f (y)| < δ. By Lemma 6.1.3 we get j rcp(∪n−1 j=0 α (Ω), δ) ≤ N (Un ), whence ht(Ω, δ; α) ≤ h(U; T ). It follows that ht(α) ≤ htop (T ). The opposite inequality is more involved. By Proposition 6.2.7, we have ht(α) ≥ hϕ (α) for every α-invariant state ϕ on C(X). In other words, ht(α) ≥ hµ (T ) for every T -invariant probability measure µ. Thus by the variational principle, Theorem 1.2.2, we have htop (T ) = sup hµ (T ) ≤ ht(α), µ

and the proof is complete.

We consider next two simple examples of computations of topological entropy.

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6 Topological Entropy

Example 6.2.10. (i) Let α be an automorphism of A = K(H) + C1 ⊂ B(H), where K(H) is the algebra of compact operators on a Hilbert space H. Then ht(α) = 0. It suﬃces to prove that ht(Ω; α) = 0 for any ﬁnite Ω ⊂ pB(H)p, where k p is a ﬁnite rank projection. Let pn = ∨n−1 k=0 α (p). Then pn is a ﬁnite rank n−1 k projection such that ∪k=0 α (Ω) ⊂ pn B(H)pn . So if we put Bn = pn B(H)pn + C(1 − pn ), and take θ: A → Bn to be a conditional expectation (e.g. θ(a) = pn apn +ϕ(a)(1−pn ), where ϕ is a state with support ≤ 1−pn ) and γ: Bn → A to be the embedding map, then for any δ > 0 we get k rcp(∪n−1 k=0 α (Ω), δ) ≤ rank Bn = Tr(pn ) + 1 ≤ nTr(p) + 1.

Hence ht(Ω, δ; α) = 0. (ii) Let B be a ﬁnite dimensional C∗ -algebra, A = B ⊗Z the inﬁnite C∗ -tensor product, α the shift automorphism of A. Then ht(α) = log rank B. The inequality ≤ follows easily from rank B ⊗n = (rank B)n . On the other hand, if τ0 is the normalized canonical trace on B and τ = τ0⊗Z , then by Example 3.2.6(i) we have hτ (α) = S(τ0 ) = log rank B. Since hτ (α) ≤ ht(α), this gives the opposite inequality. We ﬁnish the chapter with some technical remarks. Remark 6.2.11. (i) Even when dealing with unital algebras it is sometimes convenient to consider nonunital maps. Deﬁne rcp (Ω, δ) exactly as rcp(Ω, δ) but by requiring θ and γ to be contractive completely positive maps instead of unital ones (whether we require π: A → B to be unital or not is not important as we can always replace B by π(1)Bπ(1)), and then deﬁne ht (α). We obviously have rcp (Ω, δ) ≤ rcp(Ω, δ). But if 1 ∈ Ω, so x = (γ ◦ θ)(1) is close to 1, we can consider the completely positive map γ˜ = x−1/2 γ(·)x−1/2 , which is close to γ. Then by A.12 we can modify θ and γ˜ to unital completely positive maps keeping γ˜ ◦ θ unchanged. This shows that if 1 ∈ Ω, then ht (Ω; α) = ht(Ω; α), so that ht (α) = ht(α). (ii) Let A be a nonunital C∗ -algebra, α an automorphism of A. One way to deﬁne the topological entropy of α is to consider the algebra A∼ obtained by adjoining unit to A, extend α to an automorphism α ˜ of A∼ , and deﬁne ht(α) = ht(˜ α). Another possibility is to deﬁne ht(α) using rcp from the previous remark, that is, by considering contractive completely positive maps. The result will be the same. Indeed, by A.4 any contractive completely positive maps A → Matn (C) and Matn (C) → B extend to unital completely positive maps A∼ → Matn (C)∼ ∼ = Matn (C) ⊕ C and Matn (C)∼ → B ∼ . It is then easy to check that all the main properties of the topological entropy remain true for nonunital C∗ -algebras. (iii) Lemma 6.1.4(vii) shows that rcpA (Ω, δ) depends only on the position

6.3 Notes

103

of Ω in the C∗ -algebra generated by Ω. Using Arveson’s extension theorem for operator systems we can conclude that it actually depends only on the position of Ω in the operator system spanned by Ω, Ω ∗ and 1. More precisely, recall that an operator system is a closed self-adjoint subspace X of a unital C∗ -algebra containing the unit of the algebra. The notion of a completely positive map makes sense for operator systems. Then a unital completely positive map θ: X → Y is called a complete order embedding if it is injective and the inverse map θ(X) → X is completely positive. Equivalently, θ is completely isometric, that is, the map θ ⊗ id: X ⊗ Matn (C) → Y ⊗ Matn (C) is isometric for all n ∈ N. Let Ω be a ﬁnite subset of an operator system X. Deﬁne rcpX (Ω, δ) by considering approximate factorizations of complete order embeddings of X into C∗ -algebras through ﬁnite dimensional C∗ -algebras. Then if θ: X → A is a complete order embedding, we have rcpA (θ(Ω), δ) = rcpX (Ω, δ). In particular, we get the following strengthening of Theorem 6.2.2(iii). If θ: B → A is a complete order embedding of B into A, and α ◦ θ = θ ◦ β, then ht(β) ≤ ht(α).

6.3 Notes The notion of topological entropy presented here was introduced by Voiculescu [227] for nuclear C∗ -algebras and extended by Brown to exact algebras [32] (we refer the reader to the book by Wassermann [232] for more information on exact C∗ -algebras). Most of the results in this chapter are contained in those papers. An important exception is property (v) in Lemma 6.1.4 implying Theorem 6.2.6, which was observed by Ozawa [153]. As we proved in Chap. 3, inner automorphisms of type I algebras have zero dynamical entropy. This fact and Example 6.2.10(i) suggest that the topological entropy of inner automorphisms of type I C∗ -algebras is zero. This is, however, known only in a few cases, e.g. for homogeneous C∗ -algebras [33]. As we noted in Remark 6.2.11(i), instead of unital completely positive maps we can deal with contractive completely positive maps. Pop and Smith [167] showed that we can equally well consider completely contractive maps, that is, maps θ: A → B such that θ ⊗ id: A ⊗ Matn (C) → B ⊗ Matn (C) is contractive for any n ∈ N. The reason is brieﬂy that if we have a factorization of an algebra through a matrix algebra via completely contractive maps such that their composition almost preserves the unit, then we can construct a contractive completely positive factorization through the same matrix algebra [204]. It is, however, unknown whether it is possible instead to use completely bounded maps with a universal bound on the norms. Another open problem is how topological entropy behaves in quotients. In other words, if α is an automorphism of A leaving a two-sided ideal I ⊂ A invariant, and α ¯ is the automorphism of B = A/I deﬁned by α, the question is

104

6 Topological Entropy

whether the inequality ht(¯ α) ≤ ht(α) holds. If there exists a unital completely positive splitting σ: B → A such that α ◦ σ = σ ◦ α ¯ , then this is true by Remark 6.2.11(iii). It is shown in [36] that if we in addition assume that there exists an approximate unit {ei }i in I such that α(ei ) = ei for any i, then moreover ht(α) = max{ht(¯ α), ht(α|I )}. As in the case of dynamical entropy, the deﬁnition of topological entropy used in this chapter is not the only one possible. Similarly to the LindbladAlicki-Fannes entropy discussed in Notes to Chap. 3, there is an approach based on partitions of unity due to Hudetz [93], [94], [14]. Let us say that a ﬁnite set X = {xi }i of operators is an operator cover if i x∗i xi is invertible. ¯ (X) the cardinality of a minimal operator subcover of X. Given Denote by N two operator covers X = {xi }i and Y = {yj } we can form a cover X ∨ Y consisting of the elements yj xi (in Chap. 3 we denoted this operation by Y ◦ X). One of the unpleasant features of this notion is that if we consider ¯ (X (n) ) can grow exponentially fast as X (n) = X ∨ . . . ∨ X (n times) then N n → ∞. Still it is possible to manufacture a notion of entropy using it, see the above mentioned papers for details. Thomsen [215] proposed a diﬀerent operation of joining operator covers. He requires operator covers to be in addition self-adjoint sets and calls them partitions. Given n partitions X1 , . . . , Xn he deﬁnes X1 ∨ . . . ∨ Xn as the partition consisting of the elements xσ(1) . . . xσ(n) , where xi ∈ Xi and σ ∈ Sn . As was the case for the Lindblad-Alicki-Fannes entropy, in either approach one cannot consider all possible operator covers or partitions in a C∗ -algebra, since this leads to an essentially uncomputable invariant. So Hudetz introduces a notion of appropriately approximated C∗ -algebras, while Thomsen works with local C∗ -algebras. There are a number of dynamical invariants whose deﬁnitions follow the same pattern as that of topological entropy, the main diﬀerence being the way a ﬁnite set Ω is approximated and what kind of a rank-type function is used, see e.g. [44], [110], [109]. We would like to mention one of them, which was introduced by Voiculescu [227]. Let (A, ϕ, α) be a C∗ -dynamical system. For a ﬁnite Ω ⊂ A and δ > 0 denote by rcpϕ (Ω, δ) the minimal n ∈ N for which there exist unital completely positive maps θ: A → Matn (C) and γ: Matn (C) → πϕ (A) such that ϕ ◦ γ ◦ θ = ϕ and (γ ◦ θ)(a) − πϕ (a)ϕ < δ for any a ∈ Ω, where xϕ = ϕ(x∗ x)1/2 . Then we deﬁne the completely positive approximation entropy hcpaϕ (α) just as we deﬁned the topological entropy ht(α). Most of the results on ht(α) can be proved in roughly the same way for hcpaϕ (α). In particular, if (B, ψ, β) is another C∗ -dynamical system, then hcpaϕ⊗ψ (α ⊗ β) ≤ hcpaϕ (α) + hcpaψ (β). Since we have the opposite inequality for the entropy hϕ (α), we get the equality hϕ⊗ψ (α ⊗ β) = hϕ (α) + hψ (β)

6.3 Notes

105

whenever hcpaϕ (α) = hϕ (α) and hcpaψ (β) = hψ (β). If (X, T ) is a topological dynamical system, µ a T -invariant probability measure on X, α(f ) = f ◦ T −1 and ϕ(f ) = f dµ for f ∈ C(X), then hµ (T ) ≤ hcpaϕ (α) ≤ htop (T ). If the system is ergodic, then hµ (T ) = hcpaϕ (α). This can be deduced either from the above inequalities and the Jewett-Krieger theorem, Theorem 1.2.3, or using the Shannon-McMillan-Breiman theorem, Theorem 1.2.4. Without the ergodicity assumption the equality hµ (T ) = hcpaϕ (α) is not always true. To see this, observe that if (A, ϕ, α) = (A1 ⊕A2 , λϕ1 ⊕(1−λ)ϕ2 , α1 ⊕α2 ), 0 < λ < 1, then hcpaϕ (α) = maxi hcpaϕi (αi ), while hϕ (α) = λhϕ1 (α1 )+(1−λ)hϕ2 (α2 ). To get a similar invariant which coincides with hϕ (α) for arbitrary abelian systems, we should have used exp(S(ϕ ◦ γ)) in place of n in the deﬁnition of rcpϕ . Voiculescu’s motivation for introducing completely positive approximation entropy hcpaϕ (α) was that the equality hϕ (α) = hcpaϕ (α), when it holds, should be thought of as a weak form of the Shannon-McMillan-Breiman theorem. For a result in this direction see [142]. The idea of using some sort of approximation of ﬁnite sets of an algebra to deﬁne entropy was apparently ﬁrst used in the work of Connes and Størmer [52]. Their deﬁnition is as follows. Let M ⊂ B(H) be a von Neumann algebra, ξ a cyclic and separating tracial vector, and τ = ωξ the trace on M . For δ > 0 and a ﬁnite set Ω of unitaries in M deﬁne hτδ (Ω) = inf{S(ϕ) | ϕ ∈ B(H)∗ , ϕ|M = τ, S(ϕ, ϕ ◦ Ad u) < δ for all u ∈ Ω}. Then deﬁne the entropy of a τ -preserving automorphism α of M by the same procedure as before: take lim sup as n → ∞ of the quantities 1 τ h (Ω ∪ α(Ω) ∪ . . . ∪ αn−1 (Ω)), n δ and then supremum over δ > 0 and ﬁnite sets of unitaries Ω ⊂ M . The only fact which is known about this entropy is that it coincides with the Kolmogorov-Sinai entropy for abelian M . Voiculescu himself introduced another type of approximation entropy, called the perturbation-theoretic entropy [223], [224], [225]. To deﬁne − it recall that the Macaev ideal C∞ (H) is the space of compact operators T on H such that ∞ sn (T ) T − < ∞, ∞ = n n=1 where s1 (T ) ≥ s2 (T ) ≥ . . . are the eigenvalues of |T |. For a ﬁnite set Ω = {x1 , . . . , xn } of operators on H put − (Ω) = lim inf max [a, xi ]− k∞ ∞, a

1≤i≤n

106

6 Topological Entropy

where the lim inf is taken over the net of ﬁnite rank operators a, 0 ≤ a ≤ 1, with its natural order. Now if (A, ϕ, α) is a C∗ -dynamical system, and U is the canonical unitary implementing α in the GNS-representation, we deﬁne − HP (U, πϕ (A)) = sup k∞ (Ω ∪ {U }),

where the supremum is taken over all ﬁnite subsets Ω of πϕ (A) generating a ﬁnite dimensional C∗ -algebra. It is shown in [223] that if (X, µ, T ) is an ergodic dynamical system, and UT is the unitary on L2 (X, µ) implementing T , then 1 h(T ) ≤ HP (UT , L∞ (X, µ)) ≤ 6h(T ). 2 It is also known that for noncommutative systems this entropy is quite different from the dynamical entropy considered in Chap. 3. E.g. the dynamical entropy of a free shift to be considered in Chap. 14 is zero, while the perturbation-theoretic entropy is inﬁnite [223]. It is interesting to note [224] that the perturbation-theoretic entropy allows us to deﬁne the entropy of an action of a ﬁnitely generated group G with a ﬁxed set S of generators on a von Neumann algebra M . We do not have to assume that G is amenable or that the action preserves a state. Namely, if we consider M in its standard form, M ⊂ B(H), then any automorphism of M is implemented by a canonical unitary. If us , s ∈ S, are the unitaries corresponding to the generators of the group, we can then deﬁne HP (S, M ) = HP ({us }s , M ). See [148] for an estimate of this entropy for the action of a free group on its boundary, which may suggest that perturbation-theoretic entropy is related to entropy of pseudogroups [71], [229].

7 Dynamics on the State Space

Throughout this chapter we ﬁx a unital separable C∗ -algebra A and an automorphism α of A. Denote by X = S(A) the state space of A, and by T the homeomorphism of X deﬁned by T ϕ = ϕ ◦ α−1 . It would be too simple (and actually quite disappointing) if the entropy of α could be computed by looking at (X, T ). Nevertheless it is a natural problem to investigate relations between entropic properties of (A, α) and (X, T ). We shall see that either both systems have positive entropy or both have zero entropy.

7.1 Measure Entropy We ﬁrst consider measure-theoretic entropy. Theorem 7.1.1. Let notation be as above and let ϕ be an α-invariant state on A. Assume hµ (T ) = 0 for any T -invariant probability measure µ on X with barycenter ϕ. Then hϕ (α) = 0. In particular, if htop (T ) = 0 then hϕ (α) = 0 for every α-invariant state ϕ. This is a consequence of the inequality hϕ (α) ≤ hST ϕ (α) from Theorem 5.1.5 and the following result of independent interest, which in particular implies that hST ϕ (α) is not larger than the supremum of hµ (T ) over all T -invariant probability measures µ on X with barycenter ϕ. Proposition 7.1.2. We have − − hST ϕ (α) = sup{Hµ (ξ|ξ ) − Hλ,A (ξ|ξ )},

where the supremum is taken over all stationary canonical couplings λ of (A, ϕ, α) with (X, µ, T ) and over all ﬁnite measurable partitions ξ of (X, µ). For the proof we need the following Pinsker formula, which is a reﬁnement of the inequality from Lemma 1.1.3.

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7 Dynamics on the State Space

Lemma 7.1.3. If (Y, ν, S) is a dynamical system, then for any two ﬁnite measurable partitions ξ and ζ of (Y, ν) we have k h(ξ ∨ ζ; S) = h(ζ; S) + H(ξ|ξ − ∨ ζ ± ), where ζ ± = ∨∞ −∞ S ζ.

Proof. Observe ﬁrst that by a computation similar to (1.3) we have H(ξ|ξ − ∨ ζ ± ) = lim

n→∞

1 k ± H(∨n−1 k=0 S ξ|ζ ). n

(7.1)

n−1 k k Denote the partitions ∨n−1 k=0 S ξ and ∨k=0 S ζ by ξn and ζn , respectively. By Lemma 1.1.3,

h(ξn ∨ ζn ; S n ) ≤ h(ζn ; S n ) + H(ξn |ζ ± ). Since h(ξn ∨ ζn ; S n ) = nh(ξ ∨ ζ; S) and h(ζn ; S n ) = nh(ζ; S), dividing the above inequality by n, letting n → ∞ and using (7.1) we get the inequality ≤ in the lemma. On the other hand, H(ξn ∨ ζn ) = H(ζn ) + H(ξn |ζn ) ≥ H(ζn ) + H(ξn |ζ ± ). Dividing by n, letting n → ∞ and using (7.1) again, we get the inequality ≥ in the lemma.

Proof of Proposition 7.1.2. Let Λ be a stationary coupling of (A, ϕ, α) with (Y, ν, S). We can desintegrate the state Λ over (Y, ν). That is, there exists a measurable map Y y → Λy ∈ X such that Λy (a)f (y) dν(y) for a ∈ A, f ∈ L∞ (Y, ν). Λ(a ⊗ f ) = Y

Existence of such a map can be deduced from C.3-C.5, by ﬁrst decomposing the space of the GNS-representation deﬁned by Λ into a direct integral of Hilbert spaces over (Y, ν), and then decomposing the representation of A and a normal extension of Λ to the algebra of decomposable operators into corresponding direct integrals (we used a similar argument in the proof of Theorem 5.2.1). Let µ be the image of ν under the map y → Λy . Since Λ is invariant, we have ΛSy = Λy ◦ α−1 for a.e. y ∈ Y . Since ν is S-invariant, it follows that µ is T -invariant. Hence µ deﬁnes a stationary canonical coupling λ, λ(a ⊗ f ) = ψ(a)f (ψ)dµ(ψ) = Λy (a)f (Λy ) dν(y) for a ∈ A, f ∈ C(X). X ∞

Y

Identify L (X, µ) with a subalgebra of L∞ (Y, ν) using the map y → Λy . Let εX be the measurable partition of Y such that L∞ (X) = L∞ (Y /εX ). Then λ is just the restriction of Λ to A ⊗ L∞ (Y /εX ). By Proposition 5.1.4 it suﬃces to show that λ is as good as Λ for computing the entropy, that is, the supremum

7.1 Measure Entropy

109

sup{Hν (ξ|ξ − ) − HΛ,A (ξ|ξ − )} over ﬁnite partitions ξ of Y will not change if we only consider ξ ≺ εX . Let ξ be a ﬁnite partition of Y . Since by (5.3) Hν (ξ|ξ − ) − HΛ,A (ξ|ξ − ) = lim

n→∞

1 k HΛ (A; ∨n−1 k=0 S ξ), n

by virtue of Proposition 5.1.3(i), we see that the expression Hν (ξ|ξ − ) − HΛ,A (ξ|ξ − ) is increasing in ξ, so that Hν (ξ|ξ − ) − HΛ,A (ξ|ξ − ) ≤ Hν (ξ ∨ ϑ|ξ − ∨ ϑ− ) − HΛ,A (ξ ∨ ϑ|ξ − ∨ ϑ− ) for any ﬁnite partition ϑ ≺ εX . By the Pinsker formula, Lemma 7.1.3, Hν (ξ ∨ ϑ|ξ − ∨ ϑ− ) = Hν (ϑ|ϑ− ) + Hν (ξ|ξ − ∨ ϑ± ), k where ϑ± = ∨∞ −∞ S ϑ. Using Proposition 5.1.3(iv),(vi) we similarly get

HΛ,A (ξ ∨ ϑ|ξ − ∨ ϑ− ) = HΛ,A (ϑ|ϑ− ) + HΛ,A (ξ|ξ − ∨ ϑ± ). Therefore Hν (ξ|ξ − ) − HΛ,A (ξ|ξ − ) ≤ Hν (ϑ|ϑ− ) − HΛ,A (ϑ|ϑ− ) +Hν (ξ|ξ − ∨ ϑ± ) − HΛ,A (ξ|ξ − ∨ ϑ± ). Thus it suﬃces to show that Hν (ξ|ξ − ∨ ϑ± ) − HΛ,A (ξ|ξ − ∨ ϑ± ) can be made arbitrarily small for some ϑ ≺ εX . If {ϑn }n is an increasing sequence of ﬁnite partitions such that ∨n ϑn = εX , then − − ± − Hν (ξ|ξ − ∨ ϑ± n ) Hν (ξ|ξ ∨ εX ) and HΛ,A (ξ|ξ ∨ ϑn ) HΛ,A (ξ|ξ ∨ εX )

by Proposition 5.1.3(iv). Hence it is enough to prove that Hν (ξ|ζ) = HΛ,A (ξ|ζ) for any ζ εX . For this, in turn, it is enough to check that S(ν(·1Z )|L∞ (Y /ζ) , ν|L∞ (Y /ζ) ) = S(Λ(· ⊗ ·1Z )|A⊗L∞ (Y /ζ) , Λ|A⊗L∞ (Y /ζ) ) for any measurable Z ⊂ Y . This is indeed the case by Corollary 2.3.3, since the map E: A ⊗ L∞ (Y /ζ) → L∞ (Y /ζ), E(a ⊗ f )(y) = Λy (a)f (y), is a conditional expectation which preserves any functional of the form a ⊗ f → Λ(a ⊗ f 1Z ) = Λy (a)f (y) dν(y), Z ⊂ Y, Z

on A ⊗ L∞ (Y /ζ).

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7 Dynamics on the State Space

7.2 Topological Entropy It is natural to conjecture that if hST ϕ (α) = 0 for any α-invariant state ϕ then htop (T ) = 0. We do not know whether this is true. We have however the following result. Theorem 7.2.1. If ht(α) = 0 then htop (T ) = 0. The proof is based on the following combinatorial statement. Proposition 7.2.2. For given ε > 0 and a > 0, there exist n0 ∈ N and b > 0 such that if n ≥ n0 and Φ: Matm (C)∗ → n∞ is a linear contractive map such that the image of the unit ball contains at least ean real elements that are ε-separated, then m ≥ ebn . Recall that we say that a subset of a metric space is ε-separated if the distance between any two diﬀerent elements of the set is at least ε. We shall postpone the proof of Proposition 7.2.2 and ﬁrst use it to deduce Theorem 7.2.1. Proof of Theorem 7.2.1. Assume A ⊂ B(H). Choose a relatively compact subset Ω0 of the unit ball of A which consists of self-adjoint elements and spans a dense subspace of A (e.g. take a sequence which converges to zero and spans a dense subspace of A). Then the weak∗ topology on X is deﬁned by the metric d(ϕ, ψ) = sup |ϕ(y) − ψ(y)|. y∈Ω0

Assume htop (T ) > a0 > 0. Recalling the deﬁnition of the topological entropy in terms of separated sets from Sect. 1.2, this means that for any suﬃciently small ε > 0 and for all n large enough there exist at least ea0 n points of X that are (n, 4ε)-separated with respect to T −1 , that is, they are 4ε-separated in the metric dn (ϕ, ψ) =

max

0≤k≤n−1

d(T −k ϕ, T −k ψ) =

max

0≤k≤n−1

d(ϕ ◦ αk , ψ ◦ αk ).

Choose a ﬁnite subset Ω ⊂ Ω0 such that d(ϕ, ψ) < max |ϕ(y) − ψ(y)| + ε for ϕ, ψ ∈ X. y∈Ω

Given n ∈ N and a unital completely positive map σ: A → Matm (C), deﬁne the map k Φ: Matm (C)∗ → n|Ω| ∞ , Φ(f ) = {(f ◦ σ ◦ α )(y)}0≤k≤n−1,y∈Ω ,

where we ﬁx a certain order on the set {0, . . . , n − 1} × Ω. Assume there exists a unital completely positive map γ: Matm (C) → B(H) such that (γ ◦ σ ◦ αk )(y) − αk (y) < ε for 0 ≤ k ≤ n − 1, y ∈ Ω.

7.2 Topological Entropy

111

Extend each state ϕ on A to a state ϕ˜ on B(H). Then |(ϕ˜ ◦ γ ◦ σ ◦ αk )(y) − (ϕ ◦ αk )(y)| < ε. If two states ϕ and ψ are (n, 4ε)-separated with respect to T −1 , there exists k, 0 ≤ k ≤ n − 1, such that d(ϕ ◦ αk , ψ ◦ αk ) ≥ 4ε. Hence there exists y ∈ Ω such that |(ϕ ◦ αk )(y) − (ψ ◦ αk )(y)| > 3ε, whence |(ϕ˜ ◦ γ ◦ σ ◦ αk )(y) − (ψ˜ ◦ γ ◦ σ ◦ αk )(y)| > ε, ˜ so that Φ(ϕ◦γ) ˜ and Φ(ψ◦γ) are ε-separated. Thus the image of the unit ball of Matm (C)∗ under Φ contains at least ea0 n real elements that are ε-separated. Hence by Proposition 7.2.2 applied to a = a0 /|Ω| we can ﬁnd b > 0 such that m ≥ ebn if n is suﬃciently large. In other words, rcp(Ω ∪ α(Ω) ∪ . . . αn−1 (Ω), ε) ≥ ebn , so that ht(α) ≥ b > 0.

In order to prove Proposition 7.2.2 we shall compare the type 2 constants of the dual spaces. Recall that for a Banach space X the type 2 constant T2 (X ) is deﬁned as the inﬁmum of the positive numbers T such that for any N ∈ N and x1 , . . . , xN ∈ X we have N 2 N 1 2 ε x ≤ T xk 2 . (7.2) k k 2N ε1 ,...,εN =±1

k=1

k=1

Lemma 7.2.3. We have: √ (i) T2 (n1 ) ≥ n; √ (ii) there is C > 0 such that T2 (Matn (C)) ≤ C log n (for n ≥ 2). Proof. To prove the ﬁrst inequality it suﬃces to take N = n and the canonical basis of n1 for xk ’s. To prove the second inequality consider the Schatten p-class Sp (n), that is, the space Matn (C) with the norm xp = Tr(|x|p )1/p . We claim that there exists a constant C such that √ (7.3) T2 (S2m (n)) ≤ C m for any m, n ∈ N. Assuming this and using that x ≤ x2m ≤ n1/2m x, we get from (7.2) √ T2 (Matn (C)) ≤ C n1/2m m. Applying this to m = [log n], the integer part of log n, we obtain the second inequality of the lemma.

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7 Dynamics on the State Space

It remains to prove (7.3). We shall prove that T2 (S2m (n)) ≤

(2m)! 2m m!

1/2m ,

√ which implies (7.3) by the Stirling formula m! ∼ (m/e)m 2πm. Consider the measure with weights ( 12 , 12 ) on the two-point set {−1, 1}. Consider then the space Y = {−1, 1}N with the product measure µ, and let rk : Y → {±1} be the projection onto the k-th coordinate. Then, by H¨ older’s inequality, 1 2N

ε1 ,...,εN =±1

N 2 εk xk

2m

k=1

2 N = rk (t)xk dµ(t) Y k=1 2m ⎞1/m ⎛ 2m N rk (t)xk dµ(t)⎠ . ≤⎝ Y 2m

k=1

∗ m Since x2m 2m = Tr((x x) ), the m-th power of the right hand side is equal to N k1 ,...,k2m =1

Y

rk1 (t) . . . rk2m (t) dµ(t)Tr(x∗k1 xk2 . . . x∗k2m−1 xk2m ).

Since the random variables r1 , . . . , rN are independent, and one depending on whether l is odd or even, the integral rk1 (t) . . . rk2m (t) dµ(t)

Y

rkl dµ is zero or

Y

is zero unless each k, 1 ≤ k ≤ N , appears in the sequence k 1 , . . . , k2m an even number of times, and then the integral is 1. Denoting by the summation over the indices satisfying the latter property, we thus have to show that m (2m)! 2 ∗ ∗ xk 2m . (7.4) Tr(xk1 xk2 . . . xk2m−1 xk2m ) ≤ m 2 m! k

k1 ,...,k2m

Apply the generalized Horn inequality Tr(|y1 . . . yp |) ≤

n

sj (y1 ) . . . sj (yp )

j=1

(see e.g. [198, Theorem 1.15] for p = 2, the same proof works in general), where sj (yk ) is the j-th s-number of yk ∈ Matn (C), i.e., s1 (yk ) ≥ . . . ≥ sn (yk ) are the eigenvalues of |yk |. Using that sj (xk ) = sj (x∗k ), we then see that the left hand side of (7.4) is not larger than

7.2 Topological Entropy

j

k1 ,...,k2m

113

sj (xk1 ) . . . sj (xk2m ).

Since the number of multi-indices (k1 , . . . , k2m ) containing each k, 1 ≤ k ≤ N , 2mk times is equal to (2m)!((2m1 )! . . . (2mN )!)−1 , the above expression equals

j

m1 ,...,mN m1 +...+mN =m

(2m)! sj (x1 )2m1 . . . sj (xN )2mN . (2m1 )! . . . (2mN )!

By virtue of the inequality (2mk )! ≥ 2mk mk !, the last expression is not larger than (2m)! m! sj (x1 )2m1 . . . sj (xN )2mN m 2 m! j m ! . . . m ! 1 N m1 ,...,mN m1 +...+mN =m

m (2m)! = m sj (xk )2 2 m! j k 1/m m (2m)! 2m ≤ m sj (xk ) 2 m! j k

(by the triangle inequality for N m) m (2m)! xk 22m , = m 2 m! k

which establishes (7.4) and thus completes the proof of the lemma.

We shall also need the following form of the combinatorial Sauer-Shelah lemma. For a set I we denote by 2I the set of all subsets of I. If J ⊂ I and A ⊂ 2I , we denote by A(J) the subset of 2J consisting of all sets of the form H ∩ J, H ∈ A. Lemma 7.2.4. For every c > 0 there exist d > 0 and n0 ∈ N such that if I is a ﬁnite set, |I| = n ≥ n0 , and A ⊂ 2I satisﬁes |A| ≥ ecn , then there exists J ⊂ I such that |J| ≥ dn and A(J) = 2J . Proof. We shall prove the following more precise result. If m ≤ n is such that |A| >

m−1 k=0

n , k

then there exists J such that |J| = m and A(J) = 2J . To show that this is indeed enough, it suﬃces to check that there exists d > 0 such that the than ecn for suﬃciently large n and m = "above # sum " n # is " msmaller # " [dn] # + 1. Since n m n k ≤ m k for k ≤ m, the above sum is smaller than 2 m , and by the

114

7 Dynamics on the State Space

Stirling formula we get 2dn+1

n [dn]+1

≤ 2c(d)n for suﬃciently large n, with

c(d) → 0 as d → 0. So we just have to choose d such that c(d) ≤ c. The proof of the above statement is by induction on n. Since the result is obvious for m = 1 and m = n, assume 1 < m < n. Fix any point p ∈ I, and set Ip = I\{p}, Ap = A(Ip ). m−1 " n−1 # If |Ap | > k=0 , the result is true by induction. So assume k |Ap | ≤

m−1 k=0

n−1 k

.

Deﬁne B = {H ⊂ Ip | H ∈ A, H ∪ {p} ∈ A}. Then every element in B has two preimages in A under the map A H → H ∩ Ip ∈ Ap , while every element in Ap \B has one preimage. Therefore |A| = |Ap \B| + 2|B| = |Ap | + |B|. Using " n # " n−1 # n−1 that k − k = k−1 for k ≥ 1, we then get |B| = |A| − |Ap | >

m−1 k=0

m−2 m−1 n − 1 n n − 1 = . − k k k k=0

k=0

By induction there exists Jp ⊂ Ip such that |Jp | = m−1 and B(Jp ) = 2Jp . Put J = Jp ∪ {p}. Then by deﬁnition of B, for any H ⊂ Jp there exists H ⊂ Ip such that H ∩ Jp = H and both H and H ∪ {p} belong to A. This shows that A(J) = 2J .

This lemma allows us to replace the assumption on the number of εseparated elements in Proposition 7.2.2 by the assumption that the image of the unit ball under the map Φ in the proposition contains a ball of a ﬁxed radius. Lemma 7.2.5. For given ε > 0 and a > 0, there exist n0 ∈ N, d > 0 and δ > 0 such that if n ≥ n0 and Ω is a symmetric and convex subset of the unit ball of n∞ (R) containing at least ean elements that are ε-separated, then there exists a subset J ⊂ {1, . . . , n} such that |J| ≥ dn and the image of Ω under the projection πJ : n∞ (R) → ∞ (J; R) contains the ball of radius δ. Proof. Let Y ⊂ Ω be an ε-separated set, |Y | ≥ ean . Fix N ∈ N to be speciﬁed later, and for each k, 0 ≤ k < N , consider the set Uk ⊂ R deﬁned by Uk =

M !

[mε + kδ0 , mε + (k + 1)δ0 ),

m=−M

where M = 1 + [1/ε] and δ0 = ε/N . Let Yk be the subset of Y consisting of elements y = (y1 , . . . , yn ) such that the set Jk (y) = {j | yj ∈ Uk } contains no more than n/N elements. Since the sets Uk are disjoint and cover the

7.2 Topological Entropy

115

interval [−1, 1], the sets Jk (y), 0 ≤ k ≤ N − 1, are disjoint, and their union −1 is {1, . . . , n}. Hence |Jk (y)| ≤ n/N for some k. Therefore ∪N k=0 Yk = Y . Fix k such that |Yk | ≥ |Y |/N . For all m, |m| ≤ M , and I ⊂ {1, . . . , n} with |I| > n(N − 1)/N consider Am,I ⊂ 2I deﬁned as follows. A subset H ⊂ I is in Am,I if there exists y ∈ Yk such that yj ≥ mε + (k + 1)δ0 , if j ∈ H, yj < mε + kδ0 , if j ∈ I\H, yj ∈ [mε + kδ0 , mε + (k + 1)δ0 ) ⊂ Uk , if j ∈ {1, . . . , n}\I. For any m and y ∈ Yk there exists exactly one set I such that y deﬁnes an element of Am,I . In other words, for each m we have a map from Yk into the disjoint union I Am,I . If two points y, z ∈ Yk have the same images under these maps, that is, they deﬁne the same element of 2Im for each m (for some uniquely deﬁned Im ), then they cannot be ε-separated, and hence y = z. It follows that $ |Yk | ≤ |Am,I | . m

I

1)/N . Equivalently, Let cn (N ) be the number of I’s such that |I| > n(N − cn (N ) is the number of J’s with |J| ≤ n/N , so cn (N ) = j≤n/N ( nj ). Then

M $ m=−M

|Am,I |

≤

2M +1 cn (N ) max |Am,I | .

I

m,I

Since |Yk | ≥ |Y |/N , we get −1

max |Am,I | ≥ cn (N ) m,I

|Y | N

1/(2M +1)

≥ cn (N )−1 N −1/(2M +1) ean/(2M +1) .

(7.5) We have already remarked in the proof of Lemma 7.2.4 that cn (N ) does not grow faster than ec(N )n as n → ∞, with c(N ) → 0 as N → ∞. Thus we can choose N and c > 0 such that the right hand side of (7.5) is not smaller than ecn for suﬃciently large n. Fix m and I, |I| > n(N − 1)/N , such that |Am,I | ≥ ecn . By Lemma 7.2.4 there exists J ⊂ I such that |J| ≥ dn and Am,I (J) = 2J . In other words, for any H ⊂ J there is y ∈ Y satisfying yj ≥ mε + (k + 1)δ0 , if j ∈ H, yj < mε + kδ0 , if j ∈ J\H. Since the set πJ (Ω) is convex, this implies (e.g. by the Hahn-Banach theorem) that it contains all elements y ∈ ∞ (J; R) such that mε + kδ0 ≤ yj ≤ mε + (k + 1)δ0 , j ∈ J.

116

7 Dynamics on the State Space

In other words, it contains a translation of the ball of radius δ0 /2 in ∞ (J; R), namely, Bδ0 /2 (y) = y + Bδ0 /2 (0) with yj = mε + kδ0 + δ0 /2 for j ∈ J. Since πJ (Ω) is also symmetric, it follows that it contains 12 (y + Bδ0 /2 (0) − y) = Bδ0 /4 (0), the ball of radius δ = δ0 /4.

Proof of Proposition 7.2.2. By Lemma 7.2.5 we can ﬁnd J ⊂ {1, . . . , n} such that |J| ≥ dn and the image of the unit ball of Matm (C)∗ under the map πJ ◦ Φ contains the ball of radius δ in ∞ (J; R). Decomposing elements of ∞ (J) into real and imaginary parts, we conclude that the image of the unit ball contains the ball of radius δ0 = δ/2 in ∞ (J). Then the dual map S = (πJ ◦ Φ)∗ : 1 (J) → Matm (C) has the property δ0 y1 ≤ Sy ≤ y1 . Hence the type 2 constants satisfy T2 (Matm (C)) ≥ δ0 T2 (1 (J)). By Lemma 7.2.3 this gives % % √ C log m ≥ δ0 |J| ≥ δ0 dn, so that m ≥ e(δ0 /C)

2

dn

.

Although we have proved that ht(α) is positive as soon as htop (T ) is positive, our proof does not provide any estimates for ht(α). This is not a drawback of the proof, as in fact htop (T ) can take only two values, zero and inﬁnity. Proposition 7.2.6. We have either htop (T ) = 0 or htop (T ) = ∞. Proof. As in the proof of Theorem 7.2.1, let the metric on X be given by d(ϕ, ψ) = sup |ϕ(y) − ψ(y)|. y∈Ω0

If htop (T ) > a > 0, there exists ε > 0 such that for all n suﬃciently large −1 there exists ean points that are (n, ε)-separated with respect to T . Fix r ∈ N and choose ε > 0 and λ1 , . . . , λr > 0 such that j λj = 1 and ελi − 2

r

λj > ε for i = 1, . . . , r − 1.

j=i+1

Let En be an (n, ε)-separated set. Consider the set En consisting of states of the form r λj ϕj , where ϕj ∈ En for j = 1, . . . , r. j=1

We claim that |En | = |En |r and the set En is (n, ε )-separated. This would imply that htop (T ) ≥ ra, so that htop (T ) = ∞, as r was arbitrary.

7.2 Topological Entropy

117

Let (ϕ1 , . . . , ϕr ) and (ψ1 , . . . , ψr ) be diﬀerent r-tuples in Enr , and let i be the minimal number such that ϕi = ψi . As ϕi and ψi are (n, ε)-separated, there exist y ∈ Ω0 and k, 0 ≤ k ≤ n − 1, such that |(ϕi ◦ αk )(y) − (ψi ◦ αk )(y)| ≥ ε. Then, since ϕj = ψj for j < i, the distance between T −k ( j λj ϕj ) and T −k ( j λj ψj ) is not smaller than # " k k λj (ϕj ◦ α )(y) − (ψj ◦ α )(y) j r

≥ λi |(ϕi ◦ αk )(y) − (ψi ◦ αk )(y)| −

λj (|(ϕj ◦ αk )(y)| + |(ψj ◦ αk )(y)|)

j=i+1

≥ ελi − 2

r

λj > ε ,

j=i+1

and the proof is complete.

Remark 7.2.7. The above proof actually shows that the topological entropy of any aﬃne homeomorphism of a convex compact subset of a locally convex space is either zero or inﬁnity. We ﬁnish this chapter with the following suﬃcient condition for positivity of entropy, which is implicit in the proof of Theorem 7.2.1. Recall ﬁrst that a subset I of N is said to have positive upper density if lim sup n→∞

|I ∩ {1, . . . , n}| > 0. n

Proposition 7.2.8. Suppose there exist a self-adjoint element a ∈ A and a subset I = {n1 < n2 < . . .} ⊂ N of positive upper density such that the map Γ : 1 → A deﬁned by Γ (ek ) = αnk (a), k ∈ N, where {ek }k denotes the canonical basis of 1 , is an isomorphism of 1 onto a closed subspace of A. Then ht(α) > 0. Proof. Assume A ⊂ B(H). We may also assume that a ≤ 1, so that Γ is a contraction. Let δ > 0 be such that Γ (x) ≥ δx1 for x ∈ 1 . For each n ∈ N set In = I ∩ {1, . . . , n}. Denote by Γn the restric|I | tion of Γ to 1 n . If unital completely positive maps σ: A → Matm (C) and γ: Matm (C) → B(H) are such that (γ ◦ σ ◦ αk )(a) − αk (a) < δ/2 for k = 1, . . . , n, then Γn − γ ◦ σ ◦ Γn < δ/2. It follows that (γ ◦ σ ◦ Γn )(x) ≥

δ |I | x1 for x ∈ 1 n . 2

118

7 Dynamics on the State Space

Since γ is a contraction, the same inequality holds for the contraction |I | σ ◦ Γn : 1 n → Matm (C). The same argument as in the proof of Proposition 7.2.2 then shows that C

% δ% log m ≥ |In |, 2

2

so m ≥ e(δ/2C) |In | . Since |In | ≥ dn for certain suﬃciently large n’s and d > 0 by assumption, we conclude that ht({a}, δ/2; α) ≥

δ 2C

2 d > 0,

and the proof is complete. {an }∞ n=1

∗

is a sequence of self-adjoint elements in the unit ball of a C If algebra A, then in order to check that the map 1 → A, en → an , is an isomorphism onto a closed subspace it suﬃces to ﬁnd ε > 0 such that for any n ∈ N and ε1 , . . . , εn = ±1 there exists a state ϕε1 ...εn satisfying εk ϕε1 ...εn (ak ) ≥ ε, k = 1, . . . , n. Indeed, if λ1 , . . . , λn ∈ R and εk is the sign of λk , then ϕε1 ...εn (λ1 a1 + . . . + λn an ) ≥ ε(|λ1 | + . . . + |λn |), so that λ1 a1 + . . . + λn an ≥ ε(λ1 , . . . , λn )1 . By the proof of the above proposition it then follows that there exist δ > 0 and d > 0 such that rcp({a1 , . . . , an }, δ) ≥ edn for all suﬃciently large n ∈ N. Note however that in this case the same conclusion can be made without relying on the estimates of the type 2 constants from Lemma 7.2.3. Indeed, consider the channels γk : C2 → A deﬁned by γk (p1 ) =

1 1 (1 + ak ), γk (p2 ) = (1 − ak ), 2 2

where p1 = (1, 0), p2 = (0, 1). For n ∈ N choose ϕε1 ...εn as above and put ϕ=

1 2n

ϕε1 ...εn .

ε1 ,...,εn =±1

Then Hϕ (γ1 , . . . , γn ; {2−n ϕε1 ...εn }) =

n k=1

By the choice of the ϕε1 ...εn ’s,

Hϕ (γk ; {2−n ϕ(k) εk }).

7.3 Notes (k)

2−n+1 (ϕ1 ◦ γk )(p1 ) ≥

119

1 1 (k) (1 + ε) and 2−n+1 (ϕ−1 ◦ γk )(p1 ) ≤ (1 − ε). 2 2

Note that if µ1 and µ2 are two states on C2 such that µ1 (p1 ) ≥ (1 + ε)/2 and µ2 (p1 ) ≤ (1 − ε)/2, and µ = (µ1 + µ2 )/2, then µ − µi ≥ |µ(p1 ) − µi (p1 )| =

1 ε |µ1 (p1 ) − µ2 (p1 )| ≥ . 2 2

Hence there exists ε > 0 depending only on ε such that 1 1 S(µ1 , µ) + S(µ2 , µ) ≥ ε . 2 2 It follows that Hϕ (γ1 , . . . , γn ) ≥ nε . Then by an argument similar to that in the proof of Proposition 6.2.7, for any d < ε we can ﬁnd δ > 0 such that rcp({a1 , . . . , an }, δ) ≥ edn for any n ∈ N. Example 7.2.9. Let u be the shift to the right on 2 (Z). Denote by α the automorphism Ad u of B(H). Let a ∈ B(H) be a self-adjoint operator such that for any n ∈ N and ε1 , . . . , εn = ±1 there exists m ∈ Z such that (aem−k , em−k ) = (αk (a)em , em ) = εk for k = 1, . . . , n, where {ek }k∈Z is the canonical basis in 2 (Z). Then for any α-invariant C∗ subalgebra A of B(H) containing a we have ht(α|A ) > 0.

7.3 Notes Proposition 7.1.2, of which Theorem 7.1.1 is an obvious consequence, was proved by Sauvageot [190]. The Pinsker formula, Lemma 7.1.3, appeared in [164]. In the abelian case Theorem 7.2.1 was proved by Glasner and Weiss [73]. They gave two diﬀerent proofs of the result, one measure-theoretic, and one using geometry of the Banach spaces 1 and ∞ . The second proof was extended to noncommutative systems by Kerr [107]. The estimate of the type 2 constants of matrix algebras in Lemma 7.2.3 is attributed in [107] to TomczakJaegermann, who obtained the estimate of the type 2 constants of the Schatten ideals in [216]. Inequality (7.3) can be extended to noninteger m ≥ 1 by interpolation [216]. Lemma 7.2.4, or more precisely the statement established in the course of its proof, is a famous result of Perles and Sauer [187], ˇ Shelah [196] and Vapnik and Cervonenkis [219]. Lemma 7.2.5 is apparently well-known to specialists on geometric Banach space theory. We follow Glasner and Weiss [73] in its proof. Proposition 7.2.6 was proved by Sigmund [197]

120

7 Dynamics on the State Space

in the abelian case. As was observed in [107], the same proof works for noncommutative systems. Proposition 7.2.8 is due to Kerr and Li [108]. In the abelian case the converse statement is also true [108], which extends a result of Glasner and Weiss [73]. Example 7.2.9 is from [107]. In the abelian case the converse to Theorem 7.2.1 clearly holds, since the spectrum of an abelian algebra can be identiﬁed with the pure state space. To put it diﬀerently, if A is a C∗ -algebra and X = S(A) is its state space, then we can deﬁne an injective unital completely positive map θ: A → C(X) by θ(a)(ϕ) = ϕ(a). If α is an automorphism of A, T ϕ = ϕ ◦ α−1 , and α ˆ is the automorphism of C(X) deﬁned by T , α ˆ (f ) = f ◦ T −1 , then α ˆ ◦ θ = θ ◦ α. If A is abelian then θ is a complete order embedding, so by Remark 6.2.11(iii) we have ht(α) ≤ ht(ˆ α) = htop (T ). For general A the map θ is far from being a complete order embedding. In fact, the inverse map θ−1 : θ(A) → A is not even completely bounded unless A is subhomogeneous, that is, a subalgebra of C(Y ) ⊗ Matn (C) for a compact set Y and n ∈ N [95], [203]. Kerr and Li [108] introduced the contractive approximation entropy hc(α), where they use isometric embeddings instead of injective homomorphisms or complete order embeddings, and contractive factorizations through abelian ﬁnite dimensional C∗ -algebras instead of contractive completely positive factorizations through matrix algebras. One of the advantages of this entropy is that the converse to Theorem 7.2.1 holds [108], as one can see using the ˜ where X ˜ is the unit ball in A∗ , and the isometric embedding of A into C(X), ˜ deﬁned by α is fact that the topological entropy of the homeomorphism of X zero as soon as htop (T ) = 0 [107]. The price one pays is that for noncommutative systems contractive approximation entropy behaves very diﬀerently from the Voiculescu-Brown topological entropy. E.g. if α is the shift on Matm (C)⊗Z then, as we know, ht(α) = log m, while hc(α) = +∞ [108]. It remains an open question whether the converse to Theorem 7.2.1 is true for exact or nuclear algebras. In this respect it would be interesting to investigate automorphisms of noncommutative tori, see Chap. 11, where we have positive topological entropy, but generically only one invariant state with zero dynamical entropy.

8 Crossed Products

In this chapter we shall prove that canonical extensions of automorphisms to crossed products have the same entropy as the original systems. This is interesting and nontrivial already for crossed products by Z, but we shall work with larger classes of groups and with twisted crossed products as this does not require signiﬁcant additional eﬀorts.

8.1 Crossed Products by Discrete Amenable Groups Let G be a discrete group, and β: G → Aut(A) an action of G on a unital C∗ -algebra A. Let also ω be a T-valued 2-cocycle on G, so ω(h, k)ω(gh, k)ω(g, hk)ω(g, h) = 1 for g, h, k ∈ G. Note that if e ∈ G is the unit element then the cocycle identity implies that ω(g, e) = ω(e, e) = ω(e, k) for any g, k ∈ G (put h = k = e and g = h = e), and ω(g, g −1 ) = ω(g −1 , g) for any g ∈ G (put h = g −1 and k = g). Consider the full crossed product Aβ,ω G, which is the universal unital C∗ -algebra generated by a copy of A and by unitaries ug , g ∈ G, such that ug a = βg (a)ug and ug uh = ω(g, h)ugh . Note that ue = ω(e, e)1 and u∗g = ω(g, g −1 )ω(e, e)ug−1 for any g ∈ G. Assume A ⊂ B(H). Then we can deﬁne a representation π of A β,ω G on H ⊗ 2 (G) as follows: π(a)(ξ ⊗ δh ) = βh−1 (a)ξ ⊗ δh , π(ug )(ξ ⊗ δh ) = ω(g, h)ξ ⊗ δgh ,

(8.1)

where δg ∈ 2 (G) is deﬁned by δg (h) = 0 for h = g and δg (g) = 1. The reduced crossed product A r,β,ω G is by deﬁnition the quotient of the full crossed product by the kernel of π, which is independent of the embedding of A into B(H). Recall that if G is amenable then the full and reduced crossed

122

8 Crossed Products

product coincide, which we shall actually prove as a byproduct. If ω ≡ 1 we write A β G and A r,β G for the crossed products. If G = Z then the action of G on A is given by one automorphism β of A, and the crossed product A β Z is generated by A and a unitary u such that ua = β(a)u. We say that u ∈ A β Z is the canonical unitary implementing β. Let α be an automorphism of A commuting with βg for every g ∈ G. Then α extends to an automorphism α ¯ of A r,β,ω G such that α(u ¯ g ) = ug . Using the canonical conditional expectation E: A r,β,ω G → A deﬁned by E(aug ) = 0 for g = e, we can also extend any state ϕ on A to a state ϕ¯ on the crossed product. If ϕ is α-invariant, then ϕ¯ is α ¯ -invariant. Theorem 8.1.1. Let G be a discrete amenable group, ω ∈ Z 2 (G; T) a 2cocycle, β: G → Aut(A) an action of G on A, and α an automorphism of A commuting with β. Then for the automorphism α ¯ of A β,ω G we have (i) ht(¯ α) = ht(α); (ii) if ϕ is an α- and β-invariant state on A such that there exists a ϕapproximating net, then hϕ¯ (¯ α) = hϕ (α). If A ⊂ B(H) is a von Neumann algebra, we can also deﬁne the von Neumann algebra crossed product of A by G, which is by deﬁnition the von Neumann algebra generated by A r,β,ω G in B(H ⊗ 2 (G)). Since the restriction of an automorphism of a von Neumann algebra to a dense C∗ -subalgebra gives the same dynamical entropy, part (ii) is of course also true for W∗ -dynamical systems and von Neumann algebra crossed products. The most interesting particular case of the above theorem is formulated in the following corollary. Corollary 8.1.2. Let α be an automorphism of a unital C∗ -algebra A. Let u ∈ A α Z be the canonical unitary implementing α. Then (i) ht(Ad u) = ht(α); (ii) if ϕ is an α-invariant state on A such that there exists a ϕ-approximating net, then hϕ¯ (Ad u) = hϕ (α).

To prove Theorem 8.1.1 we shall approximate the system (A r,β,ω G, α ¯) by ampliﬁcations of the system (A, α). Let f be a positive ﬁnitely supported function on G such that f 1 = h f (h) = 1. Denote by F the support of f . We shall construct two unital completely positive maps Φf : A r,β,ω G → A ⊗ B(2 (F )) and Ψf : A ⊗ B(2 (F )) → A r,β,ω G. To deﬁne Φf assume A ⊂ B(H), identify 2 (F ) with a subspace of 2 (G), and let PF : H ⊗ 2 (G) → H ⊗ 2 (F ) be the orthogonal projection. Then consider the representation π of A r,β,ω G on H ⊗ 2 (G) deﬁned above, and set

8.1 Crossed Products by Discrete Amenable Groups

123

Φf (x) = PF π(x)PF for x ∈ A r,β,ω G. A priori we have Φf (x) ∈ B(H) ⊗ B(2 (F )), but the equality ω(g, h)(βh−1 g−1 (a)ξ ⊗ δgh ), if gh ∈ F , PF π(aug )(ξ ⊗ δh ) = 0, otherwise, see (8.1), shows that Φf (x) ∈ A ⊗ B(2 (F )). More precisely, we have Φf (aug ) = ω(g, h)βh−1 g−1 (a) ⊗ egh,h ,

(8.2)

h∈F ∩g −1 F

where {es,t }s,t∈F is the set of matrix units in B(2 (F )) corresponding to the basis {δh }h∈F . In particular, βh−1 (a) ⊗ eh,h . Φf (a) = h∈F

To deﬁne Ψf note that for any elements xs ∈ A r,β,ω G, s ∈ F , we have a completely positive map A ⊗ B(2 (F )) → A r,β,ω G, a ⊗ es,t → xs ax∗t . This is proved similarly to A.9. Namely, identify A r,β,ω G with (A r,β,ω G) ⊗ Ceh,h ⊂ A ⊗ B(2 (F )) for some ﬁxed h ∈ F , and write the map as A ⊗ B(2 (F )) x → V xV ∗ ∈ (A r,β,ω G) ⊗ Ceh,h , where V = s∈F xs ⊗ eh,s . Apply this construction to xs = f (s)1/2 us , s ∈ F , to get Ψf . Thus Ψf (a ⊗ es,t ) = f (s)1/2 us a(f (t)1/2 ut )∗ = f (s)1/2 f (t)1/2 βs (a)us u∗t . Since h f (h) = 1, the map Ψf is unital.

(8.3)

Proposition 8.1.3. For every positive ﬁnitely supported function f on G such that f 1 = 1, the unital completely positive maps Φf and Ψf have the following properties: ¯ = (α ⊗ id) ◦ Φf and Ψf ◦ (α ⊗ id) = α ¯ ◦ Ψf ; (i) Φf ◦ α (ii) if ϕ is a G-invariant state, then (ϕ ⊗ ϕf ) ◦ Φf = ϕ¯ and ϕ¯ ◦ Ψf = ϕ ⊗ ϕf , where ϕf is the state on B(2 (F )) given by ϕf (es,t ) = δs,t f (s); (iii) (Ψf ◦ Φf )(aug ) = ( h∈G f (gh)1/2 f (h)1/2 )aug , so that 1/2

(Ψf ◦ Φf )(aug ) − aug ≤ g f − f 1 a, where g f (h) = f (g −1 h).

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8 Crossed Products

Proof. Parts (i) and (ii) are immediate consequences of (8.2) and (8.3). To check (iii) we compute ⎛ ⎞ (Ψf ◦ Φf )(aug ) = Ψf ⎝ ω(g, h)βh−1 g−1 (a) ⊗ egh,h ⎠ h∈F ∩g −1 F

=

f (gh)1/2 f (h)1/2 ω(g, h)augh u∗h

h∈F ∩g −1 F

=

1/2

f (gh)

1/2

f (h)

aug .

h∈G

Then the inequality in (iii) follows from f (gh)1/2 f (h)1/2 − 1 = |(f 1/2 , g f 1/2 ) − (f 1/2 , f 1/2 )| h∈G

≤ g f 1/2 − f 1/2 2 1/2

≤ g f − f 1 , where the last inequality holds, as (x−y)2 ≤ |x2 −y 2 | for any positive numbers x and y.

Since the group G is amenable, there exists a net {fi }i of positive ﬁnitely supported functions on G such that fi 1 = 1 and g fi − fi 1 → 0 for any g ∈ G. Then (Ψfi ◦ Φfi )(x) → x in norm for any x ∈ A r,β,ω G. ¯f from the full Note that for full crossed products we can deﬁne a map Φ crossed into A⊗B(2 (F )) by composing Φf with the quotient map Aβ,ω G → Ar,β,ω G, and then deﬁne a map Ψ¯f into the full crossed product by the same ¯f → id pointwise. Since by deﬁnition formula as for Ψf . Then we get Ψ¯fi ◦ Φ i ¯ Φf factorizes through the reduced crossed product, it follows that full and reduced crossed products by amenable groups coincide. To complete the proof of the theorem we need the following approximation result similar to Proposition 3.2.4. Lemma 8.1.4. Let α be an automorphism of a C∗ -algebra B, {Bi }i a net of C∗ -algebras together with automorphisms αi ∈ Aut(Bi ), and {Φi : B → Bi }i and {Ψi : Bi → B}i two nets of equivariant (that is, Φi ◦ α = αi ◦ Φi and α ◦ Ψi = Ψi ◦ αi ) unital completely positive maps such that (Ψi ◦ Φi )(b) → b in norm for any b ∈ B. Then (i) ht(α) ≤ lim inf i ht(αi ); (ii) if ϕ is an α-invariant state on B such that ϕ ◦ Ψi ◦ Φi = ϕ for any i, then hϕ (α) ≤ lim inf i hϕ◦Ψi (αi ). Proof. Let B ⊂ B(H), Bi ⊂ B(Hi ). By Arveson’s extension theorem, A.8, we can extend Ψi : Bi → B to a unital completely positive map Ψ¯i : B(Hi ) →

8.2 Generalizations

125

B(H). Then given unital completely positive maps γ: Bi → Matn (C) and θ: Matn (C) → B(Hi ) we can consider γ ◦ Φi : B → Matn (C) and Ψ¯i ◦ θ: Matn (C) → B(H), and estimate (Ψ¯i ◦ θ ◦ γ ◦ Φi )(b) − b ≤ Ψ¯i ((θ ◦ γ ◦ Φi )(b) − Φi (b)) + (Ψi ◦ Φi )(b) − b ≤ (θ ◦ γ)(Φi (b)) − Φi (b) + (Ψi ◦ Φi )(b) − b. It follows that if Ω ⊂ B is a ﬁnite subset, then ht(Ω, 2ε; α) ≤ ht(Φi (Ω), ε; αi ) ≤ ht(αi ) as soon as (Ψi ◦ Φi )(b) − b < ε for b ∈ Ω. Thus (i) is proved. Similarly, if γ: A → B is a channel, then by Proposition 3.1.3(ii) hϕ (Ψi ◦ Φi ◦ γ; α) ≤ hϕ◦Ψi (Φi ◦ γ; αi ) ≤ hϕ◦Ψi (αi ). On the other hand, by Proposition 3.1.11 hϕ (γ; α) ≤ hϕ (Ψi ◦ Φi ◦ γ; α) + ε if Ψi ◦ Φi ◦ γ − γ is suﬃciently small. This proves (ii).

Proof of Theorem 8.1.1. If {fi }i is a net of positive ﬁnitely supported functions on G such that fi 1 = 1 and g fi − fi 1 → 0 for any g ∈ G, then by Proposition 8.1.3, the discussion following it and Lemma 8.1.4(i) we have ht(¯ α) ≤ lim inf ht(α ⊗ idB(2 (Fi )) ), i

where Fi is the support of fi . By Theorem 6.2.2(iv), ht(α ⊗ id) = ht(α). Thus we get ht(¯ α) ≤ ht(α). On the other hand, the opposite inequality holds by Theorem 6.2.2(iii). The case of dynamical entropy is similar. The inequality hϕ¯ (¯ α) ≥ hϕ (α) holds by Theorem 3.2.2(v). In order to get the opposite inequality we just have to recall that by Theorem 3.2.5(iv), if there exists a ϕ-approximating net, then hϕ⊗ψ (α ⊗ id) = hϕ (α) for any state ψ on a ﬁnite dimensional C∗ algebra.

8.2 Generalizations In this section we shall discuss several generalizations of Theorem 8.1.1. First of all, what are the minimal assumptions on G for the equality ht(¯ α) = ht(α) to be true? Since by our deﬁnition of topological entropy an algebra is exact if and only if the entropy of the identity automorphism is

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8 Crossed Products

zero, for ω ≡ 1 the equality implies that A r,β G is exact as soon as A is exact. But then the reduced group C∗ -algebra Cr∗ (G) ⊂ A r,β G is also exact. Groups with this property are called exact. We shall use an equivalent definition of exactness saying that there exists an amenable action of G on a compact space. That is, there are an action γ: G → Aut(B) of G on an abelian unital C∗ -algebra B (in fact, we can always take the action by translations on ∞ (G))and a net {fi }i of positive ﬁnitely supported functions G → B such that h fi (h) = 1 and lim |fi (gh) − γg (fi (h))| = 0 for any g ∈ G. i h∈G

Note that if B = C1, then we get the usual deﬁnition of amenability of G. The class of exact groups is however much bigger than the class of amenable groups. For example, free groups are exact (in Chap. 14 we shall prove a more general result). Theorem 8.2.1. Let G be a discrete exact group, ω ∈ Z 2 (G; T) a 2-cocycle, β: G → Aut(A) an action of G on A, and α an automorphism of A commuting with β. Then for the automorphism α ¯ of A r,β,ω G we have ht(¯ α) = ht(α). Proof. The proof is a slight modiﬁcation of that of Theorem 8.1.1, so we will be sketchy. Assume ﬁrst that there exists a unital C∗ -subalgebra B of the center of A such that the action β|B of G on B is amenable and α| B is trivial. Given a positive ﬁnitely supported function f : G → B such that h f (h) = 1 we can deﬁne unital completely positive maps Φf and Ψf by formulas (8.2) and (8.3), so, with F = supp f , ω(g, h)βh−1 g−1 (a) ⊗ egh,h , Φf (aug ) = h∈F ∩g −1 F

Ψf (a ⊗ es,t ) = f (s)1/2 us a(f (t)1/2 ut )∗ = f (s)1/2 βst−1 (f (t)1/2 )βs (a)us u∗t . Then (Ψf ◦ Φf )(aug ) =

f (gh)1/2 βg (f (h)1/2 )aug ,

h

and as f (gh)1/2 βg (f (h)1/2 ) − 1 = (f (gh)1/2 − βg (f (h))1/2 )βg (f (h))1/2 h h 1/2 ≤ |f (gh) − βg (f (h))| , h

we see that the maps Φf and Ψf have properties similar to those in Proposition 8.1.3(i),(iii). Hence ht(¯ α) = ht(α).

8.2 Generalizations

127

In the general case, given an amenable action γ of G on a unital abelian C∗ -algebra B, we can conclude that the extension of the automorphism id ⊗ α of B ⊗A to (B ⊗A)r,γ⊗β,ω G has entropy not greater than ht(id⊗α) = ht(α). As A r,β,ω G is a subalgebra of (B ⊗ A) r,γ⊗β,ω G, we get ht(¯ α) ≤ ht(α), whereas the opposite inequality always holds.

Note that the previous argument does not allow us to prove an analogue of Theorem 8.1.1(ii) for exact groups. The problem is that if a group acts amenably and preserves a state, then it is amenable. Thus if G is exact but nonamenable, there is no G-invariant state on B. In particular, there is no canonical extension of a state on A to a state on (B ⊗ A) r,γ⊗β G. Moreover, there exists no conditional expectation (B ⊗ A) r,γ⊗β G → A r,β G, since otherwise the restriction to B of the composition of ϕ¯ with such a conditional expectation would be a G-invariant state. So even if we manage to get an estimate of the dynamical entropy of the extension of idB ⊗ α to the crossed product with respect to some invariant state, it will not give us an immediate estimate of the entropy of α. ¯ We next turn to nondiscrete groups. Then the crossed products are nonunital, but the notion of topological entropy still makes sense by Remark 6.2.11(ii). Recall the deﬁnition of a crossed product. Let G be a locally compact group, ω: G × G → T a continuous 2-cocycle, β: G → Aut(A) a strongly continuous action of G on a C∗ -algebra A. Fix a left-invariant Haar measure 1 on G. Consider the Banach ∗-algebra Lω (G, A) consisting of functions f : G → A such that G f (g)dg < ∞, with product (f1 ∗ω f2 )(g) = ω(h, h−1 g)f1 (h)βh (f2 (h−1 g))dh G

and involution f ∗ (g) = ω(g, g −1 )ω(e, e)∆(g −1 )βg (f (g −1 )∗ ), where ∆ is the modular function of G. Then the full crossed product algebra A β,ω G is the enveloping C∗ -algebra of L1ω (G, A). The algebra A is in general a subalgebra not of the crossed product, but of its multiplier algebra. Namely, for a ∈ A and f ∈ L1ω (G, A) we deﬁne (af )(g) = af (g) and (f a)(g) = f (g)βg (a). Similarly we have unitary elements ug , g ∈ G, in the multiplier algebra deﬁned by (ug f )(h) = ω(g, g −1 h)βg (f (g −1 h)) and (f ug )(h) = ω(hg −1 , g)∆(g −1 )f (hg −1 ). If A ⊂ B(H) then similarly to (8.1) deﬁne a representation π of A β,ω G on L2 (G, H) ∼ = H ⊗ L2 (G). Its extension to the multiplier algebra is given by (π(a)ξ)(h) = βh−1 (a)ξ(h), (π(ug )ξ)(h) = ω(g, g −1 h)ξ(g −1 h), so that for f ∈ L1ω (G, A) we have

(8.4)

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8 Crossed Products

π(f ) =

π(f (g))π(ug )dg. G

Then A r,β,ω G is the quotient of the full crossed product by the kernel of this representation. Again, this kernel is independent of the choice of the embedding of A into B(H), and if G is amenable then the kernel is trivial. Finally, if α is an automorphism of A commuting with the automorphisms βg , g ∈ G, we can deﬁne an automorphism α ¯ of A r,β,ω G such that (¯ α(f ))(g) = α(f (g)) for f ∈ L1ω (G, A). Theorem 8.2.2. Let G be a locally compact amenable group, ω ∈ Z 2 (G; T) a continuous 2-cocycle, β: G → Aut(A) a strongly continuous action of G on a C∗ -algebra A, and α an automorphism of A commuting with β. Then for the automorphism α ¯ of A β,ω G we have ht(¯ α) = ht(α). Proof. The proof is again similar to that of Theorem 8.1.1, but the details become a bit more involved. This time the maps Φf and Ψf will be deﬁned for positive compactly supported functions f on G, so that Φf : A r,β,ω G → A ⊗ K(L2 (F )) and Ψf : A ⊗ K(L2 (F )) → A r,β,ω G, where F = supp f , and K(L2 (F )) denotes the algebra of compact operators on L2 (F ). To deﬁne these maps represent A faithfully on a Hilbert H, and consider the representation of the crossed product on H0 = L2 (G, H) as above. Let PF : H0 → L2 (F, H) be the orthogonal projection, and set Φf (x) = PF π(x)PF . Then, using (8.4), similarly to (8.2) we have (Φf (aug )ξ)(h) = ω(g, g −1 h)βh−1 (a)ξ(g −1 h) for ξ ∈ L2 (F, H).

(8.5)

To deﬁne Ψf we shall consider A⊗K(L2 (F )) acting on H0 ⊗L2 (F ) = L2 (F, H0 ) rather than on H ⊗L2 (F ), and deﬁne a map B(L2 (F, H0 )) → B(H0 ). Consider the operator Vf : H0 → L2 (F, H0 ), (Vf ξ)(s) = f (s)1/2 π(us )∗ ξ. The adjoint operator is given by ∗ Vf ζ = f (s)1/2 π(us )ζ(s) ds. F

In particular, Vf∗ Vf is just the scalar operator f 1 , so that if f 1 = 1 we get a unital completely positive map Ψ˜f : B(L2 (F, H0 )) → B(H0 ), Ψ˜f (x) = Vf∗ xVf . To see that Ψ˜f (π(A) ⊗ K(L2 (F ))) ⊂ π(A r,β,ω G), consider an integral operator T on L2 (F ) with continuous kernel k ∈ C(F × F ), in other words

8.2 Generalizations

129

(T ξ)(s) =

k(s, t)ξ(t) dt. F

Then a simple computation yields (compare with (8.3)) ˜ Ψf (π(a)⊗T ) = f (s)1/2 f (t)1/2 k(s, t)π(βs (a)us u∗t )ds dt ∈ π(Ar,β,ω G). F ×F

Since integral operators are dense in the algebra of compact operators, we conclude that Ψ˜f (π(A) ⊗ K(L2 (F ))) ⊂ π(A r,β,ω G). Hence there exists a contractive completely positive map Ψf : A ⊗ K(L2 (F )) → A r,β,ω G such that (π ◦ Ψf )(a ⊗ T ) = Ψ˜f (π(a) ⊗ T ) for a ∈ A and T ∈ K(L2 (F )). Consider the representation ρ of A⊗K(L2 (F )) on L2 (F, H0 ) = H0 ⊗L2 (F ) deﬁned by ρ(a ⊗ T ) = π(a) ⊗ T . Then, denoting the extension of ρ to the multiplier algebra by the same letter, (8.5) can be written as (ρ(Φf (aug ))ζ)(s) = ω(g, g −1 s)π(βs−1 (a))ζ(g −1 s) for ζ ∈ L2 (F, H0 ), s ∈ F. For ξ ∈ H0 we then compute π((Ψf ◦ Φf )(aug ))ξ = Vf∗ ρ(Φf (aug ))Vf ξ = f (s)1/2 π(us )(ρ(Φf (aug ))Vf ξ)(s) ds F f (s)1/2 ω(g, g −1 s)π(us )π(βs−1 (a))(Vf ξ)(g −1 s) ds = F f (s)1/2 f (g −1 s)1/2 ω(g, g −1 s)π(us βs−1 (a)u∗g−1 s )ξ ds = F f (s)1/2 f (g −1 s)1/2 π(aug )ξ ds, = F

so that exactly as in Proposition 8.1.3(iii) 1/2 −1 1/2 f (s) f (g s) ds aug . (Ψf ◦ Φf )(aug ) = F

By an analogue of Lemma 8.1.4(i) for contractive completely positive maps instead of unital ones, this allows us to conclude that ht(¯ α) ≤ ht(α). If G is nondiscrete, A is not a subalgebra of the crossed product, so we need an additional argument for ht(α) ≤ ht(¯ α). This has nothing to do with amenability and is done as follows. It suﬃces to construct two nets {ϕi : A → A r,β,ω G}i and {ψi : A r,β,ω G → A}i of equivariant completely positive contractive maps such that (ψi ◦ ϕi )(a) − a → 0 for any a ∈ A. Choose a net {fi }i ⊂ L1 (G) of functions such that fi 1 = 1, and the support of fi is eventually contained in an arbitrarily small neighbourhood of the unit element e ∈ G. Put xi = G fi (g)ug dg. Then set ϕi (a) = xi ax∗i . For ζ ∈ L2 (G), ζ2 = 1, consider the Fubini map

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8 Crossed Products

ψζ = id ⊗ ωζ : B(H ⊗ L2 (G)) → B(H), where ωζ is the state on B(L2 (G)) given by ωζ (T ) = (T ζ, ζ). We shall see in a moment that ψζ (π(A r,β,ω G)) ⊂ A. We claim that if βh−1 (a) − a < ε for h ∈ supp ζ, then (ψζ ◦ π ◦ ϕi )(a) − a < ε for suﬃciently large i. Thus, replacing {ϕi }i by a subnet, an appropriate net {ψi }i can be chosen from the maps ψζ ◦ π, ζ ∈ L2 (G). To prove our claim consider the C∗ -subalgebra Cr∗ (G, ω) = C1 r,ω G of the multiplier algebra of Ar,β,ω G. Consider the representation π ¯ of Cr∗ (G, ω) 2 −1 −1 on L (G) given by (¯ π (ug )ϑ)(h) = ω(g, g h)ϑ(g h). Thus π(x) = 1 ⊗ π ¯ (x) for x ∈ Cr∗ (G, ω). The equality ψζ (π(aug )) = ω(g, g −1 h)ζ(g −1 h)ζ(h)βh−1 (a)dh shows that if βh−1 (a) − a < ε for h ∈ supp ζ, then π (ug ))a < ε, ψζ (π(aug )) − ωζ (¯ π (xi x∗i ))a < ε. Hence ψζ (π(axi x∗i )) − a < ε so that ψζ (π(axi x∗i )) − ωζ (¯ for i large enough, since π ¯ (xi x∗i )ζ → ζ. On the other hand, fi (g)(βg (a) − a)ug dg → 0, [xi , a] = G

axi x∗i

so becomes arbitrarily close to xi ax∗i = ϕi (a). This proves our claim and thus completes the proof of the theorem.

Note that similarly to the case of a discrete group the maps Φf and Ψf can be deﬁned for full crossed products. This is clearly true for Φf . To construct Ψf it suﬃces to take a faithful representation of the full crossed product on a Hilbert space H0 and use the same formulas as in the proof of the theorem. Thus, exactly as for discrete groups, we may conclude that if G is amenable then the full and reduced crossed products coincide. Moreover, then the identity map of A β,ω G approximately factorizes through the algebras A ⊗ K(H), where K(H) is the algebra of compact operators on H. So if in addition A is nuclear then A β,ω G is also nuclear, and if A is exact then by an argument similar to the proof of Lemma 6.1.2 the algebra A β,ω G is exact.

8.3 Notes The question whether for an ergodic dynamical system (X, µ, T ) the dynamical entropy of the inner automorphism of L∞ (X, µ)Z deﬁned by the canonical unitary implementing T coincides with hµ (T ), was raised by Størmer [208].

8.3 Notes

131

Voiculescu [227] answered this question positively by constructing the maps Φf and Ψf from Sect. 8.1. These maps were also independently introduced by Sinclair and Smith [201] for diﬀerent purposes. The observation made in the proof of Theorem 8.2.2 that a similar construction works for crossed products by arbitrary locally compact groups, seems to be new. Remark that the composition of the maps Ψf and Φf is a map of the form aug → ϕ(g)aug , where ϕ is a ﬁnitely supported positive deﬁnite function. Such maps were considered already by Zeller-Meier [237]. But the fact that they have nice factorizations through tensor products of the algebra and matrix algebras had apparently not been observed before Voiculescu [227] and Sinclair and Smith [201]. The result of Voiculescu was independently extended to general C∗ dynamical systems by Brown [32] in the case of topological entropy and by Golodets and Neshveyev [79] in the case of dynamical entropy. A similar result is true for other types of dynamical invariants, see e.g. [35], [110]. Essentially, one has to check that such invariants (i) have the approximation property stated in Lemma 8.1.4, (ii) do not change if we tensor an automorphism with the trivial automorphism of a full matrix algebra. For equivalence of various notions of exactness for discrete groups we refer the reader to the work of Anantharaman-Delaroche [6]. Theorem 8.2.1 was obtained by Choda [47] and Germain [70]. Theorem 8.2.2, in the case of the trivial cocycle, is due to Smith and Pop [167], whose proof is based on completely contractive factorizations of crossed products and the Takesaki-Takai duality. Smith and Pop prove, in fact, a slightly stronger result. Namely, assume that α does not commute with β, but there exists a continuous automorphism γ of G such that α ◦ βg = βγ(g) ◦ α. Assume also that the cocycle ω is γ-invariant. Then there exists an automorphism α ¯ of A r,β,ω G such that α ¯ (aug ) = α(a)uγ(g) . Then, assuming that the group {γ n }n∈Z is relatively compact in Aut(G), we have ht(¯ α) = ht(α). In order to show this one just has to use γ-invariant functions in the constructions of the maps involved, which is possible by the relative compactness assumption, and instead of the automorphism α ⊗id of A⊗K(L2 (F )) use the automorphism α ⊗Ad U , where U is the unitary on L2 (G) deﬁned by γ, so (U ξ)(h) = ξ(γ −1 (h)). Without the assumption that {γ n }n∈Z is relatively compact it is natural to ask whether the inequality ht(¯ α) ≤ ht(α) + ht(αγ ) holds, where αγ is the automorphism of Cr∗ (G, ω) deﬁned by γ. Instead of ht(αγ ) one can try to use approximation entropies of automorphisms of discrete groups, e.g. the ones introduced by Choda [45] and Brown and Germain [37]. Very little is known about this problem, see [79], [45], [75] for some examples and partial results. In Chap. 11 we shall see that the inequality above can be strict already for discrete abelian groups and ω ≡ 1. The results of the present chapter show that the entropy of inner automorphisms is a computable invariant. Moreover, this invariant tells something

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about the position of the algebra generated by the unitary deﬁning the automorphism. E.g. it is shown in [207] and [141] that if hϕ (Ad u) > 0 for a unitary u in a hyperﬁnite factor, then u is not contained in any Cartan subalgebra. See [33], [143], [144] for some other results in this direction. It thus seems natural to pose the following question. Given a von Neumann algebra M , a normal state ϕ on M , and a von Neumann subalgebra N of the centralizer Mϕ of the state ϕ, is it possible to deﬁne a computable entropic invariant of the action of the unitary group of N on M ?

9 Variational Principle

The variational principle formulated in Chap. 1 states that for topological dynamical systems the topological entropy is the supremum of the measure entropies over all invariant measures. More generally, given a continuous function f one introduces the notion of pressure and proves that it can be obtained as the supremum of the quantities hµ (T ) + f dµ. The measures where the supremum is attained are called equilibrium, as they indeed are equilibrium states in gas lattice models. In this form, using mean entropy and a formulation of equilibrium in terms of the KMS condition, the variational principle was extended to quantum spin lattice system. The notion of mean entropy depends on the choice of a sequence of ﬁnite subsystems and so does not make sense in general. It is natural to replace it by dynamical entropy, and our goal is to prove a variational principle in this setting for certain asymptotically abelian systems. In particular, we shall prove that under certain assumptions the topological entropy is the supremum of the dynamical entropies, and if the supremum is attained at some state, then this state must be tracial.

9.1 Pressure In order to deﬁne pressure we follow the setup of Chap. 6 of the deﬁnition of topological entropy. Let A be a unital C∗ -algebra, α an automorphism of A, H ∈ Asa a selfadjoint element. For δ > 0 and a ﬁnite subset Ω ⊂ A denote by P (H, Ω; δ) the inﬁmum of the quantities log TrB (e−θ(H) ), where B is a ﬁnite dimensional C∗ -algebra and θ: A → B is a unital completely positive map such that there exist a unital injective homomorphism π: A → C for some C∗ -algebra C and a unital completely positive map γ: B → C with (γ ◦ θ)(a) − π(a) < δ for any a ∈ Ω. Put P (H, Ω; δ) = +∞ if no such B and θ exist. As before, TrB denotes the canonical trace on B, i.e., the trace which takes the value 1 on each minimal projection.

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By the argument in the proof of Lemma 6.1.2 it suﬃces to consider one ﬁxed faithful representation π: A → B(H). If A is nuclear we may also take C = A and π the identity map. Note also that if H = 0 then P (H, Ω; δ) = log rcp(Ω, δ). Now put ⎞ ⎛ n−1 n−1 ! 1 ⎝ j Pα (H, Ω; δ) = lim sup P α (H), αj (Ω); δ ⎠ , n→∞ n j=0 j=0 Pα (H, Ω) = sup Pα (H, Ω; δ). δ>0

Deﬁnition 9.1.1. The pressure of α at H is Pα (H) = sup Pα (H, Ω), Ω

where the supremum is taken over all ﬁnite subsets of Ω ⊂ A. Note that it suﬃces to consider ﬁnite subsets Ω of a set spanning a dense subspace of A. It is clear also that if H = 0 then the pressure coincides with the topological entropy ht(α). Proposition 9.1.2. For any α-invariant state ϕ of A we have Pα (H) ≥ hϕ (α) − ϕ(H). Proof. Assume A ⊂ B(H). Let N be a ﬁnite dimensional C∗ -algebra and γ: N → A be a channel. Fix δ > 0. Choose a ﬁnite subset Ω0 of N such that its convex hull contains the unit ball. Put Ω = {H} ∪ γ(Ω0 ). Let B be a ﬁnite dimensional C∗ -algebra, and θ: A → B and ψ: B → B(H) be unital completely positive maps such that (ψ ◦ θ)(a) − a < δ for a ∈ αk (Ω), 0 ≤ k ≤ n − 1. Then ψ ◦ θ ◦ αk ◦ γ − αk ◦ γ < δ for 0 ≤ k ≤ n − 1. Let ϕ = i1 ,...,in ϕi1 ...in be a ﬁnite decomposition. Extend each ϕi1 ...in to a positive linear functional ϕ˜i1 ...in on B(H), and set ϕ˜ = i1 ,...,in ϕ˜i1 ...in . We obviously have Hϕ (γ, . . . , αn−1 ◦ γ; {ϕi1 ...in }) = Hϕ˜ (γ, . . . , αn−1 ◦ γ; {ϕ˜i1 ...in }). On the other hand, by Proposition 3.1.11 |Hϕ˜ (γ, . . . , αn−1 ◦ γ) − Hϕ˜ (ψ ◦ θ ◦ γ, . . . , ψ ◦ θ ◦ αn−1 ◦ γ)| < nε for some ε depending only on δ and dim N . Since Hϕ˜ (ψ ◦ θ ◦ γ, . . . , ψ ◦ θ ◦ αn−1 ◦ γ) ≤ S(ϕ˜ ◦ ψ) by Proposition 3.1.3(i),(iv) and Lemma 3.1.4, we get

9.1 Pressure

135

Hϕ (γ, . . . , αn−1 ◦ γ; {ϕi1 ...in }) ≤ S(ϕ˜ ◦ ψ) + nε. By Proposition 2.2.5 we have log TrB (e−K ) ≥ S(ω) − ω(K) for any state ω and any self-adjoint K. Therefore n−1 n−1 j S(ϕ˜ ◦ ψ) ≤ log TrB e−θ( j=0 α (H)) + ϕ˜ (ψ ◦ θ) αj (H) j=0 n−1 n−1 j αj (H) + nδ ≤ log TrB e−θ( j=0 α (H)) + ϕ j=0 j −θ ( n−1 α (H) ) + nϕ(H) + nδ. j=0 = log TrB e

P P P

Thus

⎞ ⎛ n−1 n−1 ! 1 1 αj (H), αj (Ω); δ ⎠ + ϕ(H) + δ + ε. Hϕ ({αj ◦ γ}0≤j≤n−1 ) ≤ P ⎝ n n j=0 j=0

It follows that hϕ (γ; α) ≤ Pα (H, Ω) + ϕ(H), hence hϕ (α) ≤ Pα (H) + ϕ(H).

Remark 9.1.3. We have chosen the minus sign e−θ(H) in the deﬁnition of P (H, Ω; δ) because of its use in physical applications [30], rather than the plus sign used in ergodic theory [231]. If A is abelian, A = C(X), and Pαcl (H) denotes the pressure as deﬁned in [231], see also Notes at the end of the chapter, then Pα (H) = Pαcl (−H). The inequality ≤ can be proved similarly to the proof of Lemma 6.1.3. The converse inequality follows from Proposition 9.1.2 and the classical variational principle. We list some properties of the function H → Pα (H) on Asa . Proposition 9.1.4. We have: (i) if H ≤ K then Pα (H) ≥ Pα (K); (ii) Pα (H + c1) = Pα (H) − c for c ∈ R; (iii) Pα (H) is either inﬁnite for all H or is ﬁnite valued; (iv) if Pα is ﬁnite valued then |Pα (H) − Pα (K)| ≤ ||H − K||; k−1 (v) for k ∈ N, Pαk ( j=0 αj (H)) = kPα (H); (vi) Pα (H + α(K) − K) = Pα (H). Proof. Let θ: A → D be a unital completely positive map. If H ≤ K, then by the Peierls-Bogoliubov inequality, Corollary 2.3.11, n−1 j n−1 j log TrB e−θ( j=0 α (H)) ≥ log TrB e−θ( j=0 α (K)) .

P

Thus (i) follows. Since

P

136

9 Variational Principle

P

log TrB e−θ(

n−1 j=0

αj (H+c1))

P

= log TrB e−θ(

n−1 j=0

αj (H))

− nc,

we get (ii). By (i) and (ii) we have Pα (H) ≥ Pα (||H||) = Pα (0) − ||H|| = ht(α) − ||H||, and similarly Pα (H) ≤ ht(α) + ||H||. This shows (iii). By the Peierls-Bogoliubov inequality, Corollary 2.3.11, 1 n−1 n−1 log TrB e−θ( j=0 αj (H)) − 1 log TrB e−θ( j=0 αj (K)) ≤ ||H − K||, n n

P

P

and (iv) follows. Let Ω be a ﬁnite subset of A, and k ∈ N. Given n ∈ N choose m ∈ N such k−1 j that mk ≤ n < (m + 1)k. Set Hk = j=0 αj (H) and Ωk = ∪k−1 j=0 α (Ω). Then, using the Peierls-Bogoliubov inequality once again, n−1 j m−1 jk log TrB e−θ( j=0 α (H)) ≥ log TrB e−θ( j=0 α (Hk )) − k||H||.

P

Similarly

P

P

log TrB e−θ(

n−1 j=0

αj (H))

P

≤ log TrB e−θ(

m j=0

αjk (Hk ))

+ k||H||.

n−1 j jk m jk Since ∪m−1 j=0 α (Ωk ) ⊂ ∪j=0 α (Ω) ⊂ ∪j=0 α (Ωk ), it follows that ⎛ ⎞ k−1 k−1 ! 1 Pα (H, Ω; δ) = Pαk ⎝ αj (H), αj (Ω); δ ⎠ , k j=0 j=0

k−1 and hence Pα (H) = k1 Pαk ( j=0 αj (H)), which proves (v). k−1 Set Hk = j=0 αj (H) and Hk =

k−1

αj (H + α(K) − K) = Hk + αk (K) − K.

j=0

Then by (iv) and (v) we have |Pα (H) − Pα (H + α(K) − K)| = Thus (vi) follows.

2||K|| 1 |Pαk (Hk ) − Pαk (Hk )| ≤ . k k

It follows from Proposition 9.1.2 that if ϕ is an α-invariant state then −ϕ(H) ≤ Pα (H) for any H ∈ Asa . It is interesting that the converse is also true.

9.1 Pressure

137

Proposition 9.1.5. Suppose ht(α) < ∞. Let ϕ be a self-adjoint linear functional on A. Then ϕ is an α-invariant state if and only if −ϕ(H) ≤ Pα (H) for all H ∈ Asa . Proof. As we already remarked, in one direction the result follows from the inequality −ϕ(H) ≤ Pα (H) − hϕ (α) for α-invariant states, Proposition 9.1.2. Conversely, if −ϕ(H) ≤ Pα (H) for all H ∈ Asa then by Proposition 9.1.4(vi) 1 1 1 −ϕ(α(H) − H) = − ϕ(α(nH) − nH) ≤ Pα (α(nH) − nH) = Pα (0). n n n Since Pα (0) = ht(α) < ∞, letting n → ∞ we get ϕ(H) ≤ ϕ(α(H)). Applying this also to −H we see that ϕ is α-invariant. Furthermore, by Proposition 9.1.4(i),(ii) 1 1 1 −ϕ(H) = − ϕ(nH) ≤ Pα (nH) ≤ ht(α) + ||H|| −→ ||H||, n→∞ n n n so that ||ϕ|| ≤ 1. For c ∈ R we have −cϕ(1) ≤ Pα (c1) = ht(α) − c.

Hence ϕ(1) = 1, and ϕ is a state.

Deﬁnition 9.1.6. We say that an α-invariant state ϕ is an equilibrium state at H if Pα (H) = hϕ (α) − ϕ(H). If ϕ is an equilibrium state then, by Proposition 9.1.2, hϕ (α) − ϕ(H) = sup(hψ (α) − ψ(H)), ψ

where the sup is taken over all α-invariant states. Even though we cannot prove in general that the pressure is a convex function, this is the case in our main examples. The notion of equilibrium state is then closely related to diﬀerentiability of the pressure. Recall that if F is a real convex continuous function on a real Banach space X, then a linear functional f on X is called a subgradient, or tangent functional, of F at x ∈ X if F (x + y) − F (x) ≥ f (y) for any y ∈ X. We shall usually identify self-adjoint linear functionals on A with real linear functionals on Asa . Proposition 9.1.7. Suppose ht(α) < ∞ and the pressure is a convex function on Asa . Then (i) if ϕ is an equilibrium state at H then −ϕ is a subgradient for the pressure at H; (ii) if −ϕ is a subgradient for the pressure at H then ϕ is an α-invariant state.

138

9 Variational Principle

Proof. Assume ϕ is an equilibrium state at H. Let K ∈ Asa . Then by Proposition 9.1.2 Pα (H + K) − Pα (H) ≥ (hϕ (α) − ϕ(H + K)) − (hϕ (α) − ϕ(H)) = −ϕ(K), so −ϕ is a subgradient. Assume now that ϕ is a subgradient at H. If K ∈ Asa then by Proposition 9.1.4(vi) −ϕ(α(K) − K) ≤ Pα (H + α(K) − K) − Pα (H) = 0. Applying this also to −K we see that ϕ is α-invariant. Now note that ||ϕ|| ≤ 1 by Proposition 9.1.4(iv). By Proposition 9.1.4(ii) we have also −c ≥ −cϕ(1) for any c ∈ R. Hence ϕ(1) = 1, and ϕ is a state.

9.2 The variational principle We shall prove the variational principle for the following class of systems. Deﬁnition 9.2.1. A system (A, α) consisting of a unital C∗ -algebra A and an automorphism α of A is called asymptotically abelian with locality if there is a dense α-invariant ∗-subalgebra A of A such that for each pair a, b ∈ A the C∗ -algebra generated by a and b is ﬁnite dimensional, and for some p = p(a, b) ∈ N we have [αj (a), b] = 0 whenever |j| ≥ p. We call elements of A for local operators and ﬁnite dimensional C∗ subalgebras of A for local algebras. We may assume that 1 ∈ A. Since each ﬁnite dimensional C∗ -algebra is singly generated, an easy induction argument shows that the C∗ -algebra generated by a ﬁnite set of local operators is ﬁnite dimensional. In particular, if A is separable then A is an AF-algebra. Note also that for each local algebra N there is p ∈ N such that [αj (a), b] = 0 for all a, b ∈ N whenever |j| ≥ p. Theorem 9.2.2. Let (A, α) be asymptotically abelian with locality, and H ∈ Asa . Then Pα (H) = sup{hϕ (α) − ϕ(H)}, ϕ

where the supremum is taken over all α-invariant states of A. In particular, the topological entropy satisﬁes ht(α) = sup hϕ (α). ϕ

Consider ﬁrst the case when there exists a ﬁnite dimensional C∗ -subalgebra N of A such that H ∈ N , αj (N ) commutes with N for j = 0, and the C∗ -algebra generated by αj (N ), j ∈ Z, coincides with A.

9.2 The variational principle

139

Lemma 9.2.3. Under the above assumptions there exists an α-invariant state ϕ such that n−1 j 1 Pα (H) = hϕ (α) − ϕ(H) = lim log Tr∨n−1 αj (N ) e− j=0 α (H) . j=0 n→∞ n

P

Proof. Of course, this is essentially an abelian situation, so the result can be deduced from the classical variational principle. But we shall give a selfcontained argument. First note that if A1 and A2 are commuting ﬁnite dimensional C∗ -algebras, and ai ∈ Ai , ai ≥ 0, i = 1, 2, then TrA1 ∨A2 (a1 a2 ) ≤ TrA1 (a1 )TrA2 (a2 ), since if pi is a minimal projection in Ai , i = 1, 2, then p1 p2 is either zero or minimal in A1 ∨ A2 . Hence the sequence & ' n−1 j log Tr∨n−1 αj (N ) e− j=0 α (H)

P

j=0

n

is subadditive, so by Lemma 1.1.2 the limit in the formulation of the lemma exists and coincides with the inﬁmum. We denote it by P˜α (H). It is clear that P˜α (H) ≥ Pα (H). Consider M = N ⊗Z , the inﬁnite C∗ -tensor product of N with itself. Denote the n-th factor of M by Nn . Let β be the shift to the right on M , and π: M → A the homomorphism which intertwines β with α, and identiﬁes N0 with N . Set I = Ker π. For each n ∈ N let Mn = N0 ⊗ . . . ⊗ Nn−1 , In = I ∩ Mn , πn = π|Mn . j Consider the state fn on πn (Mn ) = ∨n−1 j=0 α (N ) with density operator

Tr∨n−1 αj (N ) e−

P

n−1 j=0

αj (H)

−1

e−

P

n−1 j=0

αj (H)

.

j=0

Then fn ◦ πn is a state on Mn , and identifying M with Mn⊗Z we denote the corresponding product-state on M by ψn . The state ψn is β n -invariant. Set n−1 ϕn = n−1 j=0 ψn ◦ β j . Then ϕn is β-invariant. Using concavity of entropy, Theorem 3.2.5(ii), and that hψn (β n ) = S(fn ◦ πn ) by Example 3.2.6(i), we obtain n−1 1 1 1 1 hψn ◦β j (β n ) = hψn (β n ) = S(fn ◦ πn ) hϕn (β n ) ≥ 2 n n j=0 n n ⎛ ⎞ n−1 1 n−1 j 1 1 αj (H)⎠ = S(fn ) = log Tr∨n−1 αj (N ) e− j=0 α (H) + fn ⎝ j=0 n n n j=0 ⎛ ⎞ n−1 1 αj (H)⎠ ≥ P˜α (H) + fn ⎝ n j=0

hϕn (β) =

P

140

9 Variational Principle

⎛ ⎞ n−1 1 β j (H)⎠ = P˜α (H) + ϕn (H). = P˜α (H) + ψn ⎝ n j=0 Let ϕ˜ be any weak∗ limit point of the sequence {ϕn }n . Then ϕ˜ is β-invariant. Let D be a maximal abelian subalgebra of N0 containing H. Then D is in the centralizer of the state ϕn . Hence, using Proposition 3.1.6, by the same reasoning as in Example 3.2.6(i) we get 1 Hϕn (D, β(D), . . . , β k−1 (D)). k∈N k

hϕn (β) = hϕn (D; β) = inf

By Proposition 3.1.12 the map ψ → hψ (D; β) is weak∗ upper semicontinuous. Hence hϕ˜ (β) ≥ P˜α (H) + ϕ(H). ˜ Now observe that ϕ˜ is zero on I. Indeed, if x ∈ In then β j (x) ∈ Im for j = 0, . . . , m − n and m ≥ n, whence |ϕm (x)| ≤

m−1 1 n−1 |(ψm ◦ β j )(x)| ≤ ||x||, m j=m−n+1 m

so ϕ(x) ˜ = 0. Thus ϕ˜ deﬁnes a state ϕ on A. We have hϕ (α) = hϕ˜ (β) ≥ P˜α (H) + ϕ(H), where the ﬁrst equality follows from Theorem 3.2.2(ii). Since by Proposition 9.1.2, hϕ (α) − ϕ(H) ≤ Pα (H) ≤ P˜α (H), the proof of the lemma is complete.

We shall reduce the general case to the case considered above by replacing α by its powers. For this suppose that N is a local subalgebra of A, and H ∈ N . Choose p ≥ 1 such that αj (N ) commutes with N whenever |j| ≥ p. k−p j j For k ≥ p set Mk = ∨k−p j=0 α (N ), Hk = j=0 α (H). Then Hk ∈ Mk , and αjk (Mk ) commutes with Mk for j = 0. Lemma 9.2.4. For every ﬁnite subset Ω of N we have Pα (H, Ω) ≤ lim inf k→∞

1 (Hk ). P k jk k α |∨j∈Z α (Mk )

Proof. The idea of the proof is to reduce to the situation of Lemma 9.2.3 by showing that the contribution of the indices in the intervals [jk − p + 1, jk − 1], j ∈ N, becomes negligible for large k. Fix δ > 0. Choose m0 ∈ N such that 2(p − 1)||a|| < δ for a ∈ Ω. m0

9.2 The variational principle

141

Take k ≥ m0 + p. Let n ∈ N. Then (m − 1)k ≤ n < mk for some m ∈ N. Set jk i m B0 = ∨m j=0 α (Mk ) = ∨i∈X α (N ), where X = ∪j=0 [jk, (j + 1)k − p]. Put B = B0 ⊕ . . . ⊕ B0 . m0

Choose a conditional expectation E: A → B0 , and deﬁne unital completely positive maps ψ: B → A and θ: A → B by ψ(b1 , . . . , bm0 ) =

m0 1 α−i+1 (bi ), m0 i=1

θ(a) = (E(a), (E ◦ α)(a), . . . , (E ◦ αm0 −1 )(a)). / B0 }. Then Let a ∈ A, and put S = {i | 0 ≤ i ≤ m0 − 1, αi (a) ∈ (ψ ◦ θ)(a) − a =

1 −i ((α ◦ E ◦ αi )(a) − a). m0 i∈S

Assume now that a = αl (b) for some b ∈ Ω and l, 0 ≤ l ≤ n − 1. Then l < mk, and hence [l, l + m0 − 1] is contained in Y = [0, (m + 1)k − p]. The set X is a union of m + 1 intervals of length k − p ≥ m0 , and it is obtained from Y by removing m open intervals of length p. Hence [l, l + m0 − 1] can intersect at most one of the latter intervals. It follows that S contains at most p − 1 elements. Hence ||(ψ ◦ θ)(a) − a|| ≤

2(p − 1)||a|| < δ. m0

Consequently ⎛ ⎞ n−1 n−1 ! P⎝ αj (H), αj (Ω); δ ⎠ ≤ log TrB e−θ(

P

j=0

n−1 j=0

αj (H))

.

j=0

Denoting by #I the number of integer points in a set I ⊂ R, note that for Xi = [i, i + n − 1], 0 ≤ i ≤ m0 − 1, we have #(Xi X) ≤ #(Y \Xi ) + #(Y \X) = ((m + 1)k − p + 1 − n) + m(p − 1) ≤ (m + 1)k − p + 1 − (m − 1)k + m(p − 1) < mp + 2k. Since αj (H) ∈ B0 for j ∈ X, we then get ⎛ ⎛ ⎞ ⎞ n−1 m j jk j j (E ◦ αi ) ⎝ ⎠ ⎝ ⎠ α (H) α (H) − α (Hk ) = E α (H) − j=0 j=0 j∈Xi j∈X

142

9 Variational Principle

⎛ ⎞ j j ⎝ ⎠ α (H) = E α (H) − ≤ (mp + 2k)||H||. j∈X\Xi j∈Xi \X By the Peierls-Bogoliubov inequality, Corollary 2.3.11, we obtain n−1 j i m jk TrB0 e−(E◦α )( j=0 α (H)) ≤ e(mp+2k)||H|| TrB0 e− j=0 α (Hk ) ,

P

so

P

TrB e−θ(

n−1 j=0

P

αj (H))

≤ m0 e(mp+2k)||H|| TrB0 e−

P

m j=0

αjk (Hk )

.

Taking the logarithm, dividing by n and letting n → ∞, we get 1 p||H|| 1 + lim log Tr∨m−1 αjk (Mk ) e− j=0 k k m→∞ m p||H|| 1 + Pαk |∨j∈Z αjk (Mk ) (Hk ), = k k

Pα (H, Ω; δ) ≤

P

m−1 j=0

αjk (Hk )

where the last equality follows from Lemma 9.2.3. Since this is true for any k ≥ m0 + p, we get the result.

Proof of Theorem 9.2.2. The inequality ≥ has been proved in Proposition 9.1.2. Since the pressure is continuous by Proposition 9.1.4(iv), to prove the converse inequality it suﬃces to consider local H. Then by Lemma 9.2.4 we have only to show that if H is contained in a local algebra N then with k−p Hk = j=0 αj (H) as in Lemma 9.2.4 sup(hϕ (α) − ϕ(H)) ≥ lim inf k→∞

ϕ

1 (Hk ). P k jk k α |∨j∈Z α (Mk )

By Lemma 9.2.3, for each k ∈ N there exists an αk -invariant state ψk on ∨j∈Z αjk (Mk ) such that hψk (αk |∨j∈Z αjk (Mk ) ) − ψk (Hk ) = Pαk |∨j∈Z αjk (Mk ) (Hk ). By Proposition 3.2.7 we can extend ψk to an αk -invariant state ϕ˜k on A such that hϕ˜k (αk ) ≥ hψk (αk |∨j∈Z αjk (Mk ) ). Set ϕk =

k−1 1 ϕ˜k ◦ αj . Then, as in the proof of Lemma 9.2.3, k j=0

hϕk (α) ≥ Since

1 1 hϕ˜k (αk ) ≥ hψk (αk |∨j∈Z αjk (Mk ) ). k k

9.2 The variational principle

ϕk (H) =

143

k−1 1 1 p−1 p−1 1 ϕ˜k (αj (H)) ≤ ϕ˜k (Hk ) + ||H|| = ψk (Hk ) + ||H||, k j=0 k k k k

we get 1 1 (p − 1)||H|| hψ (αk |∨j∈Z αjk (Mk ) ) − ψk (Hk ) − k k k k (p − 1)||H|| 1 , = Pαk |∨j∈Z αjk (Mk ) (Hk ) − k k

hϕk (α) − ϕk (H) ≥

and the proof is complete.

Corollary 9.2.5. If (A, α) is asymptotically abelian with locality then the pressure Pα (H) is a convex function of H. Proof. Indeed, it is the supremum of the aﬃne functions H → hϕ (α) − ϕ(H).

Corollary 9.2.6. If (A1 , α1 ) and (A2 , α2 ) are asymptotically abelian with locality then ht(α1 ⊗ α2 ) = ht(α1 ) + ht(α2 ). Proof. If ϕi is an αi -invariant state, i = 1, 2, then by Theorem 3.2.2(iv), Proposition 6.2.7 and Theorem 6.2.2(iv), used in that order, hϕ1 (α1 ) + hϕ2 (α2 ) ≤ hϕ1 ⊗ϕ2 (α1 ⊗ α2 ) ≤ ht(α1 ⊗ α2 ) ≤ ht(α1 ) + ht(α2 ). Taking the supremum over ϕi we get the conclusion.

In the next section we shall need an explicit formula for the pressure, which is a consequence of our proof of the variational principle. Corollary 9.2.7. Assume A is separable. Let N be a local algebra. Then there exist a sequence {An }n of local algebras containing N and three sequences {pn }n , {mn }n , {kn }n of positive integers such that (i) αp (An ) commutes with An whenever |p| ≥ pn ; pn (ii) → 0 as n → ∞; kn j 1 log Tr∨j∈In αj (An ) e− j∈In α (H) for all H ∈ N sa , (iii) Pα (H) = lim n→∞ kn mn mn −1 [jkn , (j + 1)kn − pn ] ∩ Z. where In = ∪j=0

P

Proof. Let {An }n be an increasing sequence of local algebras containing N such that ∪n An is dense in A, Ωn a ﬁnite subset of An such that the linear span of Ωn is An . Let {pn }n be a sequence satisfying condition (i). By Lemma 9.2.4 ⎛ ⎞ k−p n 1 Pα (H, Ωn ) ≤ lim inf Pαk |∨j∈Z αjk (An,k ) ⎝ αj (H)⎠ for H ∈ N sa , k→∞ k j=0

144

9 Variational Principle

n j where An,k = ∨k−p j=0 α (An ). On the other hand, by Theorem 9.2.2 and its proof ⎛ ⎞ k−p n 1 Pα (H) ≥ lim sup Pαk |∨j∈Z αjk (An,k ) ⎝ αj (H)⎠ for H ∈ N sa , n ∈ N. k→∞ k j=0

Choose a countable dense subset X of N sa . Since Pα (H, Ωn ) Pα (H) for any H ∈ X, we can ﬁnd a sequence {kn }n such that condition (ii) is satisﬁed and ⎛ ⎞ kn −pn 1 αj (H)⎠ for H ∈ X. Pα (H) = lim Pαkn |∨j∈Z αjkn (An,kn ) ⎝ n→∞ kn j=0 Since by Lemma 9.2.3

⎛

Pαkn |∨j∈Z αjkn (An,kn ) ⎝

kn −pn

⎞ αj (H)⎠

j=0

1 − log Tr∨j∈In,m αj (An ) e m→∞ m

= lim

P

j∈In,m

αj (H)

,

where In,m = ∪m−1 j=0 [jkn , (j + 1)kn − pn ], we can choose a sequence {mn }n such that condition (iii) is satisﬁed for all H ∈ X. But then it is satisﬁed for all H ∈ N sa by Proposition 9.1.4(iv) and the Peierls-Bogoliubov inequality, Corollary 2.3.11.

9.3 KMS-states Let A be a ﬁnite dimensional C∗ -algebra, H ∈ Asa a self-adjoint element. By the thermodynamic inequality, Proposition 2.2.5, for any state ϕ on A we have S(ϕ) − ϕ(H) ≤ log Tr(e−H ), and equality holds if and only if Qϕ = Tr(e−H )−1 e−H , in which case ϕ is called the Gibbs state. Consider the one-parameter automorphism group σ of A deﬁned by σt (a) = eitH ae−itH . Then one can check that if A is a full matrix algebra then the Gibbs state is the unique state with the property ϕ(ab) = ϕ(bσi (a)). In this section we shall show that equilibrium states have a similar property. Let A be a C∗ -algebra, σ a strongly continuous one-parameter automorphism group of A. Recall that an element a ∈ A is called an entire analytic element for σ if the map R t → σt (a) ∈ A extends to an entire analytic function. Let β ∈ R. Then a state ϕ on A is called a σ-KMSβ -state if

9.3 KMS-states

145

ϕ(ab) = ϕ(bσiβ (a)) for any entire analytic element a ∈ A and any b ∈ A. If β = 0 then ϕ is automatically σ-invariant by [30, Proposition 5.3.3]. Furthermore, by [30, Corollary 5.3.9 and Theorem 5.3.10] the normal state on πϕ (A) deﬁned by ϕ is faithful, and its modular group is given by σtϕ (πϕ (a)) = πϕ (σ−βt (a)). In particular, if ϕ is faithful and β = 0 then there exists at most one oneparameter group σ for which ϕ is a KMSβ -state. We shall use the following form of the KMS-condition which does not involve analytic continuation of σ and hence is better suited for passing to limits. By [30, Proposition 5.3.12] a state ϕ is a σ-KMSβ if and only if ˆ (9.1) f (t)ϕ(aσt (b))dt = fˆ(t + iβ)ϕ(σt (b)a)dt R

R

for all a, b ∈ A and f ∈ D, where D is the space of compactly supported C ∞ -functions, and fˆ(t) = (2π)−1/2 R f (x)e−ixt dx is the Fourier transform of f . Return now to our setting of asymptotically abelian systems with locality. Let H be a local self-adjoint element. Our ﬁrst goal is to show that there is a natural dynamics associated with H. For a subset I ⊂ Z and a local operator a put δH,I (a) = [αj (H), a]. j∈I

Then δH,I is a derivation on the algebra of local operators. Put σtH,I = exp(itδH,I ). We shall write δH for δH,Z , and σtH for σtH,Z . It will follow from the next lemma that σ H,I is a well-deﬁned strongly continuous one-parameter automorphism group of A. Lemma 9.3.1. Let r ∈ N, and let cn be the number of n-tuples (k1 , . . . , kn ) ∈ Zn such that min |kj − ki | ≤ r for j = 1, . . . , n, (9.2) 0≤i<j

where k0 = 0. Put c0 = 1. Then the series convergence.

∞ cn n z has inﬁnite radius of n! n=0

Proof. For m ∈ N let Kn (m) denote the set of n-tuples (k1 , . . . , kn ) such that condition (9.2) is satisﬁed, and furthermore max ki = min ki + m − 1,

0≤i≤n

0≤i≤n

146

9 Variational Principle

that is, the minimal interval containing k0 , . . . , kn has length m−1. Let cn (m) denote the number of elements inKn (m) for n ≥ 1, and put c0 (1) = 1 and ∞ c0 (m) = 0 for m ≥ 2. Then cn = m=1 cn (m). We assert that the following recurrence relation holds: cn+1 (m) = mcn (m) + 2

r

cn (m − k),

k=1

with the convention that cn (l) = 0 for l ≤ 0. To show this let (k1 , . . . , kn ) ∈ ∪m−1 k=0 Kn (m − k). Let [a, b] be the minimal interval containing k0 , . . . , kn . We want to see how many choices there are for kn+1 such that (k1 , . . . , kn+1 ) ∈ Kn+1 (m). If (k1 , . . . , kn ) ∈ Kn (m) then we can take any integer in [a, b] for kn+1 , so there are m choices. If (k1 , . . . , kn ) ∈ Kn (m − k) with 1 ≤ k ≤ r, there are two choices, kn+1 = a − k and kn+1 = b + k. And if k > r, no kn+1 with the required properties exists. This computation proves the recurrence formula. By virtue of (9.2) we have cn (m) = 0 for m > nr + 1. Thus by the recurrence formula max cn+1 (m) ≤ (nr + 1 + 2r) max cn (m), m

m

and consequently max cn+1 (m) ≤ m

n $

(1 + r(k + 2)).

k=0

Hence the series f (x, y) =

∞ ∞ cn (m) n m x y n! n=0 m=1

deﬁnes an analytic function in a neighborhood of (0, 0) ∈ C2 . We have, using the recurrence formula, ∞ ∞ ∂f cn+1 (m) n m = x y ∂x n! n=0 m=1

=

∞ ∞

m

n=0 m=1 ∞ ∞

∞ ∞ r cn (m − k) n m cn (m) n m x y +2 x y n! n! n=0 m=1 k=1

cn (m) n m = m yk . x y + 2f (x, y) n! n=0 m=1 r

k=1

On the other hand, y

∞ ∞ ∂f cn (m) n m m = x y . ∂y n! n=0 m=1

9.3 KMS-states

147

We thus get a partial diﬀerential equation ∂f ∂f yk . −y = 2f (x, y) ∂x ∂y r

k=1

Working in a neighbourhood of a point (0, y0 ) with y0 = 0 consider the change of variables x = u + v, y = ev−u . Then

∂x ∂y = 1 and = −y, and the above equation can be written as ∂u ∂u ∂ log f ∂ yk = −2 . ∂u ∂u k r

k=1

Hence log f = g(v) − 2

r yk k=1

k

=g

x + log y 2

−2

r yk

k

k=1

for a function g. To ﬁnd g let x = 0. Since f (0, y) = y, we get g

log y 2

= log y + 2

r yk

k

k=1

that is, g(v) = 2v + 2

r e2kv k=1

k

,

.

We thus have log f (x, y) = x + log y + 2

r y k ekx

k

k=1

whence

f (x, y) = y exp x + 2

r yk k=1

k

−2

r yk k=1

k

,

(ekx − 1) .

We see that f is analytic on C2 . In particular, the series has inﬁnite radius of convergence.

∞ cn n z = f (z, 1) n! n=0

Proposition 9.3.2. Let (A, α) be asymptotically abelian with locality, H ∈ Asa a local self-adjoint element. Then the series σzH,I (a) =

∞ (iz)n n δ (a) n! H,I n=0

148

9 Variational Principle

converges absolutely in norm for any z ∈ C and any local operator a. Moreover, if N is a local algebra containing H, Ω ⊂ C a compact subset and ε > 0, then (i) there exists a ﬁnite set I0 ⊂ Z such that σzH (a) − σzH,I (a) ≤ εa for any I ⊃ I0 , z ∈ Ω and a ∈ N ; (ii) there is δ > 0 such that if K ∈ N sa with H − K < δ then σzH (a) − σzK (a) ≤ εa for all z ∈ Ω and a ∈ N . Proof. Let N be a local algebra containing H. Let r ∈ N be so large that αn (N ) commutes with N for |n| > r. For a ∈ N , n (a) = [αkn (H), [. . . [αk1 (H), a] . . .]]. δH k1 ,...,kn ∈Z

The commutator in the sum can be nonzero only when the n-tuple (k1 , . . . , kn ) satisﬁes condition (9.2), hence with cn as in Lemma 9.3.1 we have n δH (a) ≤ cn 2n Hn a. ∞ n It follows from that lemma that the series n=0 (z n /n!)δH (a) converges in norm for any z ∈ C. n If I ⊂ Z then in the formula for δH,I (a) we sum over a smaller set, so convergence holds in this case too. Moreover, for any m ∈ N, if I contains n n Z ∩ [−mr, mr] then δH,I (a) = δH (a) for n ≤ m, whence

σzH (a) − σzH,I (a) ≤ 2

cn 2n |z|n Hn a. n! n>m

This implies part (i). This also implies that to show (ii) it suﬃces to prove a similar statement for σ H,I and σ K,I for someﬁnite I. But then it is immediate, since σzH,I (a) = eizHI ae−izHI , where HI = j∈I αj (H).

We can now formulate the main result of the present section. Theorem 9.3.3. Let (A, α) be asymptotically abelian with locality. Assume ht(α) < ∞. If H is a local self-adjoint operator in A and ϕ is an equilibrium state at H then ϕ is a σ H -KMS1 -state. In particular, if ht(α) = hϕ (α) then ϕ is a trace. By Corollary 9.2.5 the pressure Pα is a convex function of H, so that Proposition 9.1.7 shows that Theorem 9.3.3 will follow from the following more general result.

9.3 KMS-states

149

Theorem 9.3.4. If −ϕ is a subgradient of Pα at H then ϕ is a σ H -KMS1 state. The proof is based on Corollary 9.2.7 which, as we shall see, implies that for generic H a subgradient of the pressure at H can be obtained as a limit of Gibbs states on local algebras. Note that if B is a ﬁnite dimensional C∗ -algebra, H0 ∈ B sa , and ϕ is the state with density operator Qϕ = TrB (e−H0 )−1 TrB (e−H0 ) then for any H ∈ B sa we have d log TrB (e−H0 −tH )|t=0 = −TrB (e−H0 )−1 TrB (He−H0 ) = −ϕ(H), dt so that −ϕ is the gradient of the function H → log TrB (e−H ) at H0 . Note also that this function is convex, being the supremum of the aﬃne functions H → S(ψ) − ψ(H) by the thermodynamic inequality, Proposition 2.2.5. Lemma 9.3.5. Let N be a local algebra, H ∈ N sa , −ϕ ∈ N ∗ a subgradient of (Pα )|N sa at H. Let E: A → N be a conditional expectation. Then for every function f in the space D of C ∞ -functions with compact support and a, b ∈ N we have fˆ(t)ϕ(aE(σtH (b)))dt − fˆ(t + i)ϕ(E(σtH (b))a)dt R

≤ ||a||

R

R

(|fˆ(t)| + |fˆ(t + i)|)||σtH (b) − E(σtH (b))||dt.

Proof. We may assume that A is separable. Indeed, for every K ∈ Asa there exists a sequence {Ωn }∞ n=1 of ﬁnite subsets of A such that Pα (H, Ωn ) Pα (H). Then for any α-invariant C∗ -subalgebra A0 containing K and the Ωn ’s we have Pα (K) = Pα|A0 (K). It follows that there exists a separable α-invariant C∗ -subalgebra A0 containing N such that (A0 , α|A0 ) is asymptotically abelian with locality and Pα|A0 (K) = Pα (K) for a dense family of elements K in N sa . But then the equality Pα|A0 (K) = Pα (K) holds for all K ∈ N sa by continuity. Therefore we may replace A by A0 and thus assume that A is separable. First consider the case when (Pα )|N sa has a unique subgradient at H. With the notation of Corollary 9.2.7 consider the state fn on ∨j∈In αj (An ) with density operator

Tr∨j∈In αj (An ) e−

P

j∈In

αj (H)

−1

e−

P

Then deﬁne a positive linear functional ϕn on N by ϕn (x) =

1 fn (αj (x)). kn mn j∈In

j∈In

αj (H)

.

150

9 Variational Principle

Note that ||ϕn || = ϕn (1) ≤ 1. As we remarked before the formulation of the lemma, the functional −fn is the gradient of the convex function x → log Tr∨j∈In αj (An ) (e−x ) on (∨j∈In αj (An ))sa at the point j∈In αj (H). Hence −ϕn is the gradient of the convex function j 1 log Tr∨j∈In αj (An ) e− j∈In α (x) N sa x → kn mn

P

at H. It follows that any limit point of the sequence {−ϕn }n is a subgradient of (Pα )|N sa at H. Since the latter is unique by assumption, ϕn → ϕ as n → ∞. Since fn is a σ H,In -KMS1 -state, by (9.1) for every j ∈ In we have

R

fˆ(t)fn (αj (a)σtH,In (αj (b)))dt =

R

fˆ(t + i)fn (σtH,In (αj (b))αj (a))dt. (9.3)

Note that σtH,In (αj (b)) = αj (σtH,In −j (b)). Fix q ∈ N, and put n −1 In,q = ∪m j=0 [jkn + q, (j + 1)kn − pn − q] ∩ Z,

which is nonempty for suﬃciently large n. Then In − j contains [−q, q] ∩ Z for j ∈ In,q . By Proposition 9.3.2(i), if q is large enough then σtH,In −j (b) is arbitrarily close to σtH (b) for every j ∈ In,q and t in a ﬁxed compact subset of R. But then σtH,In (αj (b)) − αj (E(σtH (b))) is arbitrarily close to αj (σtH (b) − E(σtH (b))). It follows that for all suﬃciently large n ∈ N 1 H,I fˆ(t) fn αj (a)σt n (αj (b)) − αj (a)αj (E(σtH (b))) dt kn mn R j∈In,q ≤ ||a||

R

|fˆ(t)| ||σtH (b) − E(σtH (b))||dt + ε(q),

where ε(q) → 0 as q → ∞. Since |In,q |/|In | → 1 as n → ∞, letting n → ∞ we may replace averaging over the set In,q by averaging over In , and then obtain 1 H,In j j H ˆ ˆ (α (b)) dt − f (t)ϕ(aE(σt (b)))dt f (t)fn α (a)σt kn mn R j∈In R ≤ ||a||

R

|fˆ(t)| ||σtH (b) − E(σtH (b))||dt + εn ,

→ ∞. Since an analogous estimate holds for the integral with εn → 0 as n ˆ(t + i)ϕ(E(σ H (b))a)dt, we obtain the conclusion of the lemma by virtue f t R of (9.3). To ﬁnish the proof we need a few facts about diﬀerentiability of convex functions, see e.g. [179, Section 25]. Let F be a convex function on Rn . For

9.4 Notes

151

any x ∈ Rn the function F is diﬀerentiable at x if and only if it has a unique subgradient at x, and then this subgradient coincides with the gradient of F at x. The points where F is diﬀerentiable form a dense Gδ -set of full measure. Finally, for any x ∈ Rn the set of subgradients of F at x coincides with the closed convex hull of subgradients which are limits of sequences of the form {fn (xn )}n such that xn → x, F is diﬀerentiable at xn , and fn (xn ) is the gradient of F at xn . Returning to the proof of the lemma, we conclude that ϕ lies in the closed convex hull of linear functionals ϕ˜ for which there exist sequences {Hn }n ⊂ N sa and {ϕn }n ⊂ N ∗ such that Hn → H, ϕn → ϕ, ˜ and −ϕn is the unique subgradient of (Pα )|N sa at Hn . Since for ϕn the lemma is already proved (for Hn instead of H), using Proposition 9.3.2(ii) we conclude that the conclusion of the lemma is true for ϕ. ˜ But then it is true for any functional in the closed convex hull of the ϕ’s. ˜

Proof of Theorem 9.3.4. If −ϕ is a subgradient of Pα at H then −ϕ|N is a subgradient of (Pα )|N sa at H for any local algebra N containing H. Thus applying Lemma 9.3.5 to an increasing sequence of local algebras we conclude that the equality H ˆ f (t)ϕ(aσt (b))dt = fˆ(t + i)ϕ(σtH (b)a)dt R

R

holds for all f ∈ D and all local a, b, hence for all a, b ∈ A. But this is one of the equivalent forms of the KMS-condition.

Remark 9.3.6. Under the assumptions of Theorem 9.3.3, any weak∗ limit point of a sequence on which the supremum in the variational principle is attained is a subgradient of the pressure, hence a σ H -KMS1 state. In order to deﬁne the dynamics σ H we need some assumptions on α, such as asymptotic abelianness with locality. On the other hand, the question when hϕ (α) = ht(α) < ∞ forces ϕ to be tracial makes sense in general. In Chap. 13 we shall give an example of an asymptotically abelian system (A, α) with zero topological entropy and many nontracial invariant states. In Chaps. 11 and 12 we shall see examples of an automorphism α with a unique invariant state ϕ such that 0 = hϕ (α) < ht(α).

9.4 Notes Let B be a ﬁnite dimensional C∗ -algebra, A = B ⊗Z the inﬁnite C∗ -tensor product, α the shift to the right on A. For a subset Λ ⊂ Z denote by AΛ the C∗ -subalgebra of A generated by the factors corresponding to the elements of Λ. Given an α-invariant state ϕ on A, the quantity s(ϕ) = lim

n→∞

1 S(ϕ|A[0,n−1] ) n

152

9 Variational Principle

is called the mean entropy of ϕ. We always have hϕ (α) ≤ s(ϕ). By an interaction potential one understands a map Φ which associates a self-adjoint element Φ(X) ∈ AX to every X ⊂ Z. We assume that Φ is translationally invariant, Φ(n + X) = Φ(X), and that the series Φ(X) |X|

H=

X0

is absolutely convergent. For a ﬁnite subset Λ ⊂ Z deﬁne the local Hamiltonian by H(Λ) = Φ(X). X⊂Λ

The pressure (at inverse temperature β = 1) is deﬁned by P (Φ) = lim

n→∞

1 log TrA[0,n−1] (e−H([0,n−1]) ). n

Then the variational principle of Ruelle [185] (classical lattices, B is abelian) and Robinson [180] (quantum lattices, B is a full matrix algebra) states that P (Φ) = sup{s(ϕ) − ϕ(H)}. ϕ

Moreover, as was proved by Lanford and Robinson [115], see also [30] for a detailed exposition, under suitable conditions on the potential, the limit σt (x) = lim (Ad eitH([−n,n]) )(x) n→∞

deﬁnes a one-parameter automorphism group of A, and if the supremum above is attained at a state ϕ, then ϕ is a σ-KMS1 -state. Note that if Φ has ﬁnite range in the sense that there exist only ﬁnitely many sets X 0 such that Φ(X) = 0, then the above σ coincides with the dynamics σ H deﬁned in Sect. 9.3 by a result of Araki [7], who also proved that local operators are σ H -analytic, Proposition 9.3.2 and Lemma 9.3.1. In the classical case there is a more general result subsuming also the variational principle for topological entropy. Let (X, T ) be a topological dynamical system, and f ∈ C(X). For an open cover U of X put f (x) q(T, f, U) = inf , inf e V

V ∈V

x∈V

where the ﬁrst inﬁmum is taken over all ﬁnite subcovers V of U, and then denoting f + f ◦ T + . . . + f ◦ T n−1 by Sn f deﬁne P (T, f ) = sup lim sup U

n→∞

1 −k U). log q(T, Sn f, ∨n−1 k=0 T n

9.4 Notes

153

Then P (T, f ) = supµ {hµ (T ) + X f dµ}. This was proved by Ruelle [186] under additional assumptions, and by Walters [230] in general, see [231] for details. It is therefore only natural to try to generalize the variational principle for lattice systems to C∗ -dynamical systems using dynamical entropy instead of mean entropy. The ﬁrst attempt in this direction was made by Narnhofer [125]. She studied the question whether a state ϕ on a lattice system such that hϕ (α)−ϕ(H) = sup{hψ (α)−ψ(H)} satisﬁes the KMS-condition. Moriya [123] approached this problem diﬀerently by proving that sup{hψ (α) − ψ(H)} = sup{s(ψ) − ψ(H)}. ψ

ψ

This implies in particular that the pressure P (Φ) deﬁned above coincides with the pressure Pα (H). The pressure Pα (H) for nuclear C∗ -algebras was introduced by the authors [140], who obtained the main results of this chapter. But of course most of the arguments are already found in the above mentioned papers. The notion of pressure was extended to exact C∗ -algebras by Kerr and Pinzari [110]. For further results and relevant examples of systems see [105], [110], [213], [77], [3]. There are a number of nonasymptotically abelian systems where the topological entropy is the supremum of the dynamical entropies. E.g. Choda [43] showed that both the topological entropy and the dynamical entropy with respect to a certain state of the canonical endomorphism of the Cuntz algebra On are equal to log n; see [27], [165], [110] for generalizations. Nevertheless in all known examples there always exists an asymptotically abelian system around which completely determines the entropy. Although, as we mentioned above, for lattice systems we can use both dynamical entropy and mean entropy in the variational principle, it is unknown whether the equality hϕ (α) = s(ϕ) always holds. In order to prove it one probably needs a better understanding not only of dynamical entropy, but also of mean entropy. E.g. if ϕ is pure, we clearly have hϕ (α) = 0, but it is still an open problem whether s(ϕ) = 0. The equality hϕ (α) = s(ϕ) holds for states derived from quasi-free states (see Chap. 13), Markov states [160], [154] (which under the assumption of faithfulness can be described as the states such that σtϕ (A0 ) ⊂ A[−1,1] [137], [76]), and of course the states coming from diagonal maximal abelian subalgebras (that is, obtained by composing a state on D = C ⊗Z , where C is a maximal abelian subalgebra of B, with the trace preserving conditional expectation A → D). It is stated in [50] that if Φ is a ﬁnite range interaction then hϕ (α) = s(ϕ) for the state ϕ such that s(ϕ) − βϕ(H) = sup{s(ψ) − βψ(H)} for suﬃciently small β > 0 (such ϕ is known to be unique [8]). However, we were unable to follow the proof. A related open problem is to ﬁnd conditions on automorphisms of C∗ -algebras under which the map ϕ → hϕ (α) is weakly∗ upper semicontinuous.

Part II

Special Topics

10 Relative Entropy and Subfactors

Conditional entropy is, as we saw in Chap. 1, an important concept in the classical theory. In the noncommutative case it was introduced under the name of relative entropy as a tool to take care of approximation of entropy. We shall in the present chapter develop the theory of relative entropy and show its relationship to subfactors of II1 -factors. Then we shall show a formula analogous to the classical formula h(T ) = H(ξ|ξ − ). Finally we shall give applications to the canonical shift on the tower of relative commutants deﬁned by an inclusion of II1 -factors and for shifts on the Jones projections.

10.1 Relative Entropy Let M be a von Neumann algebra, ϕ a normal state on M , P and Q von Neumann subalgebras of M . Deﬁnition 10.1.1. The relative entropy of P and Q with respect to ϕ is Hϕ (P |Q) = sup (S(ϕi |P , ϕ|P ) − S(ϕi |Q , ϕ|Q )), i

where the supremum is taken over all ﬁnite decompositions ϕ = into a sum of positive linear functionals.

i

ϕi of ϕ

Note that if ϕ = τ is a trace, which is the case we are mainly interested in, any ψ ≤ τ has the form ψ = τ (·x). Hence by Theorem 2.3.1(x) the relative entropy can be written as Hτ (P |Q) = sup (τ (η(EQ (xi ))) − τ (η(EP (xi )))), (10.1) i

where the supremum is taken over all ﬁnite partitions of unity 1 = i xi in M , and EP and EQ are the τ -preserving conditional expectations on P and Q, respectively. One often suppresses τ in the notation for relative entropy. The main properties of relative entropy are as follows.

158

10 Relative Entropy and Subfactors

Theorem 10.1.2. We have: (i) Hϕ (P |Q) ≥ 0, and if ϕ is a faithful tracial state then Hϕ (P |Q) = 0 if and only if P ⊂ Q; (ii) Hϕ (P |R) ≤ Hϕ (P |Q) + Hϕ (Q|R); (iii) Hϕ (P |Q) is increasing in P and decreasing in Q; (iv) if Q ⊂ P and there is a ϕ-preserving faithful normal conditional expectation E: P → Q, then Hϕ (P |Q) = sup S(ϕi |P , ϕi ◦ E), i

where the supremum is taken over all ﬁnite decompositions ϕ =

i

ϕi .

Proof. Taking the trivial decomposition ϕ = ϕ we see that Hϕ (P |Q) ≥ 0. If P ⊂ Q then Hϕ (P |Q) = 0 by monotonicity of relative entropy S, Theorem 2.3.1(vi). Let now ϕ = τ be a faithful tracial state. If P is not a subset of Q, there exists a projection e ∈ P such that e ∈ / Q. Then by (10.1) Hτ (P |Q) ≥ τ (η(EQ (e))) + τ (η(EQ (1 − e))). The element EQ (e) is not a projection (this follows e.g. by A.13, as if EQ (e) is a projection then e ≤ s(EQ (e)) = EQ (e), where s(a) is the support of a self-adjoint element a, and hence e = EQ (e) by faithfulness of EQ ). Therefore τ (η(EQ (e))) > 0 and similarly τ (η(EQ (1 − e))) > 0. This completes the proof of (i). Part (ii) is immediate from the deﬁnition of Hϕ (P |Q). Part (iii) follows from monotonicity of relative entropy S, while part (iv) follows from Theorem 2.3.1(vii).

Let us show next that relative entropy coincides with conditional entropy in the abelian case. So let M = L∞ (X, µ), ξ and ζ be measurable partitions of X such that ξ is ﬁnite, P = L∞ (X/ξ) and Q = L∞ (X/ζ). We want to show that Hτ (P |Q) = Hµ (ξ|ζ). If p1 , . . . , pn are the atoms of P , by (10.1) we have Hτ (P |Q) ≥ τ (η(EQ (pi ))). i

The latter expression is exactly Hµ (ξ|ζ). Thus Hτ (P |Q) ≥ Hµ (ξ|ζ). It suﬃces to prove the opposite inequality for ﬁnite ζ. Indeed, if {ζn }n is an increasing sequence of ﬁnite measurable partitions such that ∨n ζn = ζ then Hτ (P |L∞ (X/ζn )) ≥ Hτ (P |Q) ≥ Hµ (ξ|ζ), and Hµ (ξ|ζn ) Hµ (ξ|ζ) by the martingale convergence theorem. Thus if Hτ (P |L∞ (X/ζn )) = Hµ (ξ|ζn ), we conclude that Hτ (P |Q) = Hµ (ξ|ζ).

10.1 Relative Entropy

159

So assume ζ is ﬁnite. Then Hτ (P |Q) ≤ Hτ (P ∨ Q|Q) and Hµ (ξ|ζ) = Hµ (ξ ∨ ζ|ζ). It follows that to prove the inequality Hτ (P |Q) ≤ Hµ (ξ|ζ) we may assume that ζ ≺ ξ, so Q ⊂ P . In this case it is enough to consider partitions of unity in P . Then part (iv) of Theorem 10.1.2 and convexity of relative entropy of positive functionals, Corollary 2.3.2, show that it suﬃces to consider partitions 1 = i xi such that each xi is a scalar multiple of a minimal projection in P . Since S(λψ, λϕ) = λS(ψ, ϕ), we then see that this is the same as to consider one partition 1 = i pi , where p1 , . . . , pn are the minimal projections in P . But then clearly Hτ (P |Q) = Hµ (ξ|ζ). Next we want to show that in some cases the computation of relative entropy can be reduced to the ﬁnite dimensional case. For this we need the following notion. If P , Q, R are von Neumann subalgebras of a von Neumann algebra M , and EP : M → P , EQ : M → Q, ER : M → R are conditional expectations, then we say that P ∪ R

⊂ M ∪ ⊂ Q

is a commuting square if ER = EQ ◦EP = EP ◦EQ . Equivalently, R = P ∩Q and EQ (P ) ⊂ R, EP (Q) ⊂ R. Remark that if the conditional expectations preserve a faithful normal state ϕ, then is suﬃces to check that, say, ER = EQ ◦EP . Indeed, if M ⊂ B(H) and the state ϕ is deﬁned by a cyclic and separating vector ξ ∈ H, then ER (x)ξ = eR xξ for x ∈ M , where eR ∈ B(H) is the projection onto Rξ. Then ER = EQ ◦ EP implies that eR = eQ eP . But then eR = eP eQ , and so ER = EP ◦ E Q . Proposition 10.1.3. Let M be a von Neumann algebra with a faithful normal state ϕ, N a von Neumann subalgebra. Suppose {Mn }n and {Nn }n are increasing sequences of von Neumann subalgebras of M such that Nn ⊂ Mn , M = (∪n Mn ) and N = (∪n Nn ) . Suppose there exist ϕ-preserving conditional expectations EMn : M → Mn and ENn : M → Nn such that Mn ∪ Nn

⊂ ⊂

Mn+1 ∪ Nn+1

is a commuting square for every n. Then Hϕ (M |N ) = limn Hϕ (Mn |Nn ). Proof. If ψ ≤ ϕ then S(ψ|Mn , ϕ|Mn ) S(ψ, ϕ) by Corollary 2.3.5, and a similar convergence holds for Nn and N . This implies that Hϕ (M |N ) ≤ lim inf Hϕ (Mn |Nn ). n

160

10 Relative Entropy and Subfactors

Since ENn+1 ◦ EMn = ENn for all n, we have ENn+k+1 ◦ EMn = ENn+k+1 ◦ EMn+k ◦ EMn = ENn+k ◦ EMn , so by induction ENn+k ◦EMn = ENn for all k ∈ N. Next note that {ENn }n converges in the pointwise strong operator topology to a ϕ-preserving conditional expectation EN : M → N . It follows that for any n Mn ∪ Nn

⊂ M ∪ ⊂ N

is a commuting square. In other words, EN |Mn = ENn |Mn . Hence for any positive linear functional ψ we have S(ψ, ψ ◦ EN ) ≥ S(ψ|Mn , ψ ◦ EN |Mn ) = S(ψ|Mn , ψ ◦ ENn |Mn ). By Theorem 10.1.2(iv) it follows that Hϕ (M |N ) ≥ Hϕ (Mn |Nn ), which completes the proof.

From now onwards we shall only consider tracial states, and our next goal is to compute the relative entropy Hτ (M |N ) when N ⊂ M are ﬁnite dimensional (and τ is a trace on M ). Let Z(M ) and Z(N ) be the centers of M and N , respectively, m

n

l=1

k=1

M = ⊕ Ml , N = ⊕ N k , where Ml ∼ = Matml (C) and Nk ∼ = Matnk (C). Let wl and zk be the central projections in M and N such that Ml = M wl and Nk = N zk . Let akl be the multiplicity of Nk wl in Ml . Thus if tl denotes the trace of a minimal projection in Ml and sk that of a minimal projection in Nk , nk akl tl and τ (zk ) = nk sk = nk akl tl . τ (wl ) = ml tl = k

l

Theorem 10.1.4. With the above notation the relative entropy Hτ (M |N ) is (2Hτ (M ) − Hτ (Z(M ))) − (2Hτ (N ) − Hτ (Z(N ))) + nk akl tl log ckl , k,l

where ckl = min

( nk ,1 . akl

In particular, if M is abelian we have Hτ (M |N ) = Hτ (M ) − Hτ (N ), which we already know, and if M = Matm (C) and N = Matn (C) then

10.1 Relative Entropy

Hτ (M |N ) = 2 log m − 2 log n + log min

n2 ,1 m

(

161

& m' = min log m, 2 log . n

A straightforward computation of the formula in the theorem shows that it is equivalent to the formula ml tl log tl + ml tl log ml + nk sk log sk Hτ (M |N ) = − l

−

l

nk sk log nk +

k

k

nk akl tl log ckl .

(10.2)

k,l

Note that if τ = i λi ϕi is a decomposition of τ into a convex combination of states, then S(λi ϕi , τ ) = S(τ ) − λi S(ϕi ) − η(λi ), i

i

i

S(λi ϕi |N , τ |N ). It follows that λi (S(ϕi |N ) − S(ϕi )) , (10.3) Hτ (M |N ) = sup Hτ (M ) − Hτ (N ) +

and we have a similar expression for

i

i

where the supremum is taken over all decompositions τ = i λi ϕi of τ into a convex combination of states. This is the expression we shall use in the proof of the theorem. Note also that instead of ﬁnite convex decompositions we can use integral decompositions of τ . Proof of Theorem 10.1.4. We ﬁrst show that the left side is majorized by the right side in equation (10.2). Consider an integral decomposition τ = ϕ dµ(x). By (10.3) we have to estimate X x Hτ (M ) − Hτ (N ) + (S(ϕx |N ) − S(ϕx ))dµ(x). (10.4) X

By Theorem 2.2.2(ii) we have S(ϕx |N zk ) ≤ S(ϕx |N ) = S(ϕx |N zk wl ). k

k,l

For ﬁxed k and l consider the state ψ = ϕx (zk wl )−1 ϕx on zk M wl zk . By Theorem 2.2.2(i) we have S(ψ|N zk wl ) − S(ψ) ≤ S(ψ|N zk wl ) ≤ log nk . On the other hand, since zk M wl zk ∼ = N zk wl ⊗Matakl (C), by Theorem 2.2.2(vi) we have S(ψ|N zk wl ) − S(ψ) ≤ log akl .

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10 Relative Entropy and Subfactors

Using S(λψ) = λS(ψ) + η(λ) with λ = ϕx (zk wl ), we therefore get S(ϕx |N zk wl ) − S(ϕx |zk M wl zk ) ≤ ϕx (zk wl ) log min{nk , akl }. Finally, from Lemma 2.2.4 applied to the state ϕx (wl )−1 ϕx on M wl , we obtain ϕx (zk wl ) . S(ϕx |zk M wl zk ) − S(ϕx |M wl ) ≤ ϕx (wl ) η ϕx (wl ) k

k

Thus, since l S(ϕx |M wl ) = S(ϕx ), the integral in (10.4) is estimated from above by ϕx (zk wl ) dµ(x). ϕx (zk wl ) log(min{nk , akl })dµ(x) + ϕx (wl )η ϕx (wl ) X X k,l

k,l

Since X ϕx dµ(x) = τ and τ (zk wl ) = nk akl tl , the ﬁrst summand in the above expression is equal to nk akl tl log min{nk , akl }. k,l

On the other hand, using concavity of η and that τ (wl ) = ml tl , we can estimate the second summand by ϕx (wl ) ϕx (zk wl ) ϕx (zk wl ) ϕx (wl ) dµ(x) ≤ ml tl η ml tl η dµ(x) ϕx (wl ) ml tl ϕx (wl ) X m l tl X nk akl tl = ml tl η ml tl nk akl = −tl nk akl log . ml Thus we estimate (10.4) by nk akl Hτ (M ) − Hτ (N ) + nk akl tl log(min{akl , nk }) − nk akl tl log ml k,l

k,l

= Hτ (M ) − Hτ (N ) + −

nk akl tl log akl

k,l

nk akl tl (log akl + log nk − log ml )

= Hτ (M ) − Hτ (N ) +

nk akl tl log ckl +

k,l

k,l

−

nk akl tl log ckl

k,l

nk sk log nk +

k

where we used that sk =

ml tl log ml ,

l

l

akl tl and ml =

k

nk akl . Since

10.1 Relative Entropy

Hτ (M ) = −

ml tl log tl and Hτ (N ) =

l

163

nk sk log sk ,

k

we have shown that the left side of equation (10.2) is majorized by the right side. In order to prove the opposite inequality choose a pure state ϕl of M wl such that nk akl ϕl (zk wl ) = , ml ϕl |N zk wl is a trace if nk < akl , ϕl |(N T M )zk wl is a trace if nk ≥ akl . nk akl l Explicitly ϕl can be constructed as follows. Identify m 2 with ⊕k (2 ⊗ 2 ), and let {ei }i∈N denote the standard basis in 2 . Put

ξl =

) k

nk akl min{nk , akl }ml

1/2

min{nk ,akl }

ei ⊗ ei .

i=1

Then let ϕl = ωξl be the vector state deﬁned by ξl . Let Kl denote the subalgebra of ⊕k zk M wl zk ⊂ M wl consisting of elements ⊕k xk such that xk ∈ N zk wl if nk < akl , and xk ∈ (N ∩ M )zk wl if nk ≥ akl . Then ϕl is a pure state on M wl such that its restriction to Kl coincides with the restriction of the unique tracial state τl on M wl to Kl . Furthermore, the map El : M wl → Kl deﬁned by El (x) = (Ad u)(x)dµl (u), U (Kl ∩M wl )

where µl is the normalized Haar measure on the unitary group U (Kl ∩ M wl ) of Kl ∩ M wl , is a τl -preserving conditional expectation. It follows that ml tl ϕl ◦ Ad u dµl (u). τ= l

U (Kl ∩M wl )

Thus by (10.3) Hτ (M |N ) ≥ Hτ (M ) − Hτ (N ) +

ml tl S(ϕl |N zk wl ).

(10.5)

k,l

The restricition of ϕl to zk M wl zk is a scalar multiple of a pure state. Hence by Lemma 2.2.3(i) ml ml ϕl |N zk wl = S ϕl |(N ∩M )zk wl = log min{nk , akl }, S nk akl nk akl whence

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10 Relative Entropy and Subfactors

S(ϕl |N zk wl ) =

nk akl nk akl nk akl (log ckl + log akl ) − log . ml ml ml

Thus the right side of (10.5) equals Hτ (M ) − Hτ (N ) +

nk akl tl (log ckl + log akl ) −

k,l

k,l

nk akl tl log

nk akl , ml

which is exactly what we need.

10.2 Index of Subfactors Our goal in this section is to show that relative entropy is related to index of subfactors. We shall brieﬂy recall some basic facts of the latter theory referring the reader to [214, Chapter XIX] for more details. Let M be a II1 -factor with trace τ . If M acts on a Hilbert space H then H can be embedded into L2 (M ) ⊗ K for a suﬃciently large Hilbert space K. Let ¯ p ∈ (M ⊗ C1) = M ⊗B(K) be the projection onto H. Then the dimension of H relative to M is deﬁned by dimM (H) = (τ ⊗ Tr)(p), where τ is the unique tracial state on M ⊂ B(L2 (M )) and Tr is the canonical trace on B(K). The dimension can take any value in (0, +∞], and if H is countably generated in the sense that it is a direct sum of at most countably many cyclic subspaces for M , then dimM (H) is a complete invariant of the unitary equivalence class of the representation M → B(H). The dimension dimM (H) is ﬁnite if and only if the commutant M of M in B(H) is a II1 -factor. In the latter case dimM (H) =

τ ([M ξ]) , τ ([M ξ])

(10.6)

where ξ ∈ H is any nonzero vector, τ is the unique tracial state on M ⊂ B(H), and [M ξ] denotes the projection onto the subspace M ξ. If N is a subfactor of M , the index of N in M is [M : N ] = dimN (L2 (M )). If M acts on H then dimN (H) = [M : N ] dimM (H). It follows that if N ⊂ M ⊂ L then [L: N ] = [L: M ][M : N ]. It follows also that if [M : N ] < ∞ and N is represented on H then this representation extends to a representation of M on H. Indeed, decomposing H into a direct sum of cyclic subspaces we may assume that dimN (H) ≤ 1. Choose a representation of M on a Hilbert space H such that dimM (H ) = [M : N ]−1 dimN (H). Then dimN (H ) = dimN (H).

10.2 Index of Subfactors

165

Hence the representations of N on H and H are equivalent. Since the representation of N on H extends to a representation of M , the same is true for H. Using Proposition 10.1.3 and Theorem 10.1.4 it is easy to check that if R is the hyperﬁnite II1 -factor then Hτ (R ⊗ Matn (C)|R ⊗ 1) = 2 log n. On the other hand, [R ⊗ Matn (C): R ⊗ 1] = n2 . So we see that in this case the relative entropy is the logarithm of the index. This is a particular case of a far more general result. For simplicity of the presentation we stick to the irreducible case, that is, when N ∩ M = C1. Theorem 10.2.1. Let N ⊂ M be II1 -factors such that [M : N ] < ∞ and N ∩ M = C1. Then Hτ (M |N ) = log[M : N ]. To prove this theorem we need two properties of ﬁnite index subfactors. The ﬁrst one is the Pimsner-Popa inequality stating that EN (x) ≥ [M : N ]−1 x for any x ∈ M+ , where EN : M → N is the trace preserving conditional expectation, see [214, Theorem XIX.4.14]. Proposition 10.2.2. If N ⊂ M is a ﬁnite index subfactor, then Hτ (M |N ) ≤ log[M : N ]. Proof. Let λ = [M : N ]−1 . Then by the Pimsner-Popa inequality ϕ ◦ EN ≥ λϕ for any positive linear functional ϕ on M . Since the relative entropy S is decreasing in the second variable, Theorem 2.3.1(iii), we then have S(ϕ, ϕ ◦ EN ) ≤ S(ϕ, λϕ) = −ϕ(1) log λ. By Theorem 10.1.2(iv) it follows that Hτ (M |N ) ≤ − log λ = log[M : N ].

The second property of subfactors which we need, is existence of special projections. Lemma 10.2.3. If N ⊂ M is a ﬁnite index subfactor then there exists a projection q ∈ M such that EN (q) = [M : N ]−1 1. Moreover, any other projection with this property is of the form vqv ∗ , where v is a unitary in N . Proof. See [214, Theorem XIX.4.12]. The existence will also essentially be shown in Lemma 10.4.1 below.

Proof of Theorem 10.2.1. Let λ = [M : N ]−1 and q be the projection from Lemma 10.2.3. The idea of the proof is to consider the state ϕ = λ−1 τ (· q)

166

10 Relative Entropy and Subfactors

and the decomposition τ = U (N ) ϕ ◦ Ad u du, which formally holds because of EN ∩M (q) = λ1. For II1 -factors the above decomposition does not make sense, but it is still possible to get an approximate version of it. Let x = q − λ1. Then τ (x) = 0. Let Kx = conv{vxv ∗ : v ∈ U (N )}, where the closure is in the weak operator topology. Then Kx is a convex wcompact set in M , and τ (y) = 0 for all y ∈ Kx . The weak operator closure of a bounded convex set coincides with the closure with respect to the L2 -norm. Thus there exists a unique y0 ∈ Kx such that y0 2 = inf{y2 : y ∈ Kx }. But vy0 v ∗ ∈ Kx for all v ∈ U (N ), and vy0 v ∗ 2 = y0 2 , so that vy0 v ∗ = y0 , i.e., y0 ∈ N ∩ M . Since N ∩ M = C1, it follows that y0 is a scalar, so y0 = 0. Hence for any ﬁxed ε > 0 there are unitaries v1 , . . . , vn ∈ N such that n 1 ∗ vi qvi − λ1 < ε2 λ. n i=1

2

Let y = (λn)−1 i vi qvi∗ . Then 0 ≤ y ≤ λ−1 1 and y − 12 ≤ ε2 . Let p be the spectral projection of y corresponding to the interval [0, 1 + ε]. Set xi = ((1 + ε)λn)−1 vi qvi∗ ∧ p. Then i xi ≤ ((1 + ε)λn)−1 i pvi qvi∗ p = (1 + ε)−1 yp ≤ 1. Hence Hτ (M |N ) ≥ τ (η(EN (xi ))−η(xi )) = ((1+ε)λn)−1 τ (η(EN (vi qvi∗ ∧p))). i

i

Using operator monotonicity of log as in the proof of Theorem 2.2.2(ii), we get τ (η(e + f )) ≤ τ (η(e)) + τ (η(f )) for any positive e and f . Hence τ (η(EN (vi qvi∗ ∧ p))) ≥ τ (η(EN (vi qvi∗ ))) − τ (η(EN (vi qvi∗ − vi qvi∗ ∧ p))) ≥ η(λ) − η(τ (vi qvi∗ − vi qvi∗ ∧ p)), where in the second inequality we used that EN (q) = λ1 and that η ◦ τ ≥ τ ◦ η by concavity of η. Thus Hτ (M |N ) ≥ −(1+ε)−1 log λ−((1+ε)λn)−1 η(τ (vi qvi∗ −vi qvi∗ ∧p)). (10.7) i

Now recall that τ (e ∨ f ) − τ (f ) = τ (e) − τ (e ∧ f ) for any projections e and f . Hence

10.3 Generators and Relative Entropy

167

τ (vi qvi∗ − vi qvi∗ ∧ p) ≤ 1 − τ (p). Since y(1 − p) ≥ (1 + ε)(1 − p), we have y − 122 ≥ τ ((y − 1)2 (1 − p)) ≥ ε2 τ (1 − p), so that 1 − τ (p) ≤ ε−2 y − 122 ≤ ε2 . Thus τ (vi qvi∗ − vi qvi∗ ∧ p) ≤ ε2 . Since η is increasing on [0, e−1 ], for ε small enough we obtain η(τ (vi qvi∗ − vi qvi∗ ∧ p)) ≤ η(ε2 ). From (10.7) we then get Hτ (M |N ) ≥ −(1 + ε)−1 log λ − ((1 + ε)λ)−1 η(ε2 ). Letting ε → 0 we conclude that Hτ (M |N ) ≥ − log λ = log[M : N ]. The opposite inequality follows from Proposition 10.2.2.

Remark 10.2.4. What we actually used in the proof is not the irreducibility N ∩ M = C1, but that EN ∩M (q) is a scalar. Subfactors satisfying the latter condition are called extremal, and in fact the equality Hτ (M |N ) = log[M : N ] holds if and only if N ⊂ M is extremal [162, Corollary 4.5]. Note that if [M : N ] < 4 then automatically N ∩ M = C1, see [214, Corollary XIX.2.10]. Recall also that the possible values of the index is the set {4 cos2 π/n | n ≥ 3} ∪ [4, +∞], see [214, Theorem XIX.2.22]. Finally note that with some extra work the above theorem can be extended to arbitrary subfactors [162, Theorem 4.5]. In that case Hτ (M |N ) = ∞ whenever N ∩ M has a diﬀuse part. If N ∩ M is atomic, and {fk }k are minimal projections in N ∩ M with sum 1, then Hτ (M |N ) = 2 η(τ (fk )) + τ (fk ) log[fk M fk : fk N fk ]. k

k

10.3 Generators and Relative Entropy In this section we shall prove an analogue of the formula h(T ) = H(ξ|ξ − ), see (1.3) and Theorem 1.1.4. Before we state the main result we introduce some notation. Let (M, τ, α) be a W∗ -dynamical system, where τ is a faithful normal trace. Let {An }∞ n=1 be an increasing sequence of ﬁnite dimensional C∗ -subalgebras of M . We say that {An }∞ n=1 is a generating sequence for α if

168

10 Relative Entropy and Subfactors

(i) α(An ) ⊂ An+1 , n ∈ N; (ii) the von Neumann subalgebra generated by αk (An ), k ∈ Z, n ∈ N, coincides with M ; 1 (iii) hτ (α) = lim Hτ (An ). n→∞ n We say {An }n satisﬁes the commuting square condition if An+2 An+1 ⊂ ∪ ∪ α(An ) ⊂ α(An+1 ) is a commuting square with respect to the τ -preserving conditional expectak tions for every n ∈ N. Remark that if M is abelian and An = ∨n−1 k=0 α (A1 ), then this condition is satisﬁed if and only if A1 corresponds to the generating partition of a Markov process. Write An in the form An = ⊕ Mkn , l∈Kn

∼ where = Matmnk (C). Let (ankl )k∈Kn−1 ,l∈Kn be the inclusion matrix for α(An−1 ) ⊂ An . Denote by Z(An ) the center of An . Mkn

Theorem 10.3.1. Let (M, τ, α) be a W∗ -dynamical system, where τ is a faithful normal trace and M is of type II1 . Suppose {An }∞ n=1 is a generating sequence for α satisfying the commuting square condition. Assume also that ankl n−1 < ∞. n,k,l mk

hτ (α) < ∞ and sup

1 Hτ (Z(An )) exists, and with R = (∪n An ) we have n→∞ n

Then the limit lim

hτ (α) =

1 1 1 Hτ (R|α(R)) + lim Hτ (Z(An )). n→∞ 2 2 n

If we combine this theorem with Theorem 10.2.1, we immediately obtain the following corollary. Corollary 10.3.2. If we in the above theorem add the assumptions that α(R) ∩ R = C1 and limn n1 Hτ (Z(An )) = 0, then hτ (α) = 12 log[R: α(R)].

To prove the theorem we need some preparation. Lemma 10.3.3. Fix r > 0. Let fn be the central projection in An such that An fn = ⊕mnk ≤r Mkn . Then {fn }n is a decreasing sequence converging strongly to zero.

10.3 Generators and Relative Entropy

169

Proof. First note that R = (∪n An ) is of type II1 . Indeed, let z be the central projection in R corresponding to the sum of the type Ik components of R with k ≤ r, and assume z = 0. Since α(R) ⊂ R, and type II and type Ik algebras can not be embedded into a type Il algebra with l < k, we have z ≤ α(z). As τ is α-invariant and faithful, it follows that z = α(z). The maximal number of mutually equivalent orthogonal projections in Rz is not larger than r. By the deﬁnition of a generating sequence, the sequence {α−n (R)}∞ n=1 is increasing with union dense in M . Since z = α−n (z) is central in each α−n (R), the projection z is central in M , and the maximal number of mutually equivalent orthogonal projections in M z is not larger than r. In other words, M z is a sum of type Ik algebras with k ≤ r. But this contradicts our assumption on M . Since a type Ik algebra can not be embedded into a type Il algebra with l < k, it is clear that {fn }n is a decreasing sequence. Let f be its strong limit. Since fn is central in An , the projection f is central in R. Then Rf = (∪n An f ) , which by the same argument as above contradicts the fact that R is of type II1 unless f = 0.

Denote by tnk the trace of a minimal projection in Mkn ⊂ An . Since Hτ (An−1 ) = Hτ (α(An−1 )) and Hτ (Z(An−1 )) = Hτ (Z(α(An−1 ))), by Theorem 10.1.4 we have Hτ (An |α(An−1 )) = (2Hτ (An ) − Hτ (Z(An ))) − (2Hτ (An−1 ) − Hτ (Z(An−1 ))) mn−1 ankl tnl log cnkl , (10.8) + k k,l

/ankl , 1}. where cnkl = min{mn−1 k Lemma 10.3.4. Suppose C = sup N

Then

N 1 n−1 n n mk akl tl log ankl < ∞. N n=2 k,l

N 1 n−1 n n mk akl tl log cnkl = 0. N →∞ N n=2

lim

k,l

Proof. Since by assumption of the theorem there is c > 0 such that c ≤ cnkl ≤ 1 for all k, l and n, it suﬃces to prove that lim N

N 1 N n=2

mn−1 ankl tnl = 0. k

n−1 k,l:an kl >mk

Fix r > 0. Let fn be as in Lemma 10.3.3. Then N 1 N n=2

k,l:mn−1 ≤r k

mn−1 ankl tnl = k

N 1 N n=2

k:mn−1 ≤r k

mn−1 tn−1 = k k

N 1 τ (fn−1 ), N n=2

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10 Relative Entropy and Subfactors

and the latter expression tends to zero by Lemma 10.3.3. On the other hand, N 1 N n=2

mn−1 ankl tnl k

−1

≤ (log r)

n−1 k,l:an >r kl >mk

N 1 n−1 n n mk akl tl log ankl N n=2 k,l

−1

≤ C(log r)

.

Since we can take an arbitrary large r, this proves the lemma.

Proof of Theorem 10.3.1.Let us check that the assumption of the previous n−1 n n n n a = m and , we have lemma is satisﬁed. Since k mn−1 kl l l akl tl = tl k

mnl mn−1 k k,l = mnl tnl log mnl − mn−1 tn−1 log mn−1 k k k

mn−1 ankl tnl log ankl ≤ k

k,l

mn−1 ankl tnl log k

l

k

= Hτ (An ) − Hτ (Z(An )) − Hτ (An−1 ) + Hτ (Z(An−1 )). Hence, with A0 = C1, N 1 1 1 1 n−1 n n mk akl tl log ankl ≤ Hτ (AN ) − Hτ (Z(AN )) ≤ Hτ (AN ). N n=1 N N N k,l

Since the sequence { N1 Hτ (AN )}N is bounded by assumption of the theorem, it follows of Lemma 10.3.4 is satisﬁed. Hence if we put that then assumption n n Cn = k,l mn−1 a t log c , we get kl l kl k N 1 Cn → 0. N n=1

By equation (10.8) we have N N 2 1 1 1 Hτ (An |α(An−1 )) = Hτ (AN ) − Hτ (Z(AN )) + Cn . N n=1 N N N n=1

Since {An }n satisﬁes the commuting square condition, by Proposition 10.1.3 Hτ (An |α(An−1 )) → Hτ (R|α(R)). Since N1 Hτ (AN ) → hτ (α) by assumption, it follows that limN exists and 1 Hτ (R|α(R)) = 2hτ (α) − lim Hτ (Z(AN )). N N Thus 1 1 1 hτ (α) = Hτ (R|α(R)) + lim Hτ (Z(AN )). 2 2 N N

1 N Hτ (Z(AN ))

10.4 The Canonical Shift

171

10.4 The Canonical Shift In the present section we shall apply the results of the preceding sections to an interesting automorphism, called the canonical shift, deﬁned on the tower of relative commutants associated with an inclusion N ⊂ M of II1 -factors of ﬁnite index. In order to introduce it we need a few more facts from index theory, again see [214, Chapter XIX] for more details. Let τ be the trace on M , ξ ∈ L2 (M ) the cyclic vector deﬁning τ . Denote by eN the projection onto N ξ, and by M1 = !M, eN " the von Neumann subalgebra of B(L2 (M )) generated by M and eN . Since eN xeN = EN (x)eN = eN EN (x) for x ∈ M , we have {eN } ∩ M = N . In other words, N is generated by M and eN . So if we denote by J the modular conjugation deﬁned by τ , Jxξ = x∗ ξ, so that JM J = M and JeN J = eN , then JM1 J = N . It follows that M1 is a II1 -factor and dimM1 (L2 (M )) = dimN (L2 (M )) =

1 1 = , 2 dimN (L (M )) [M : N ]

where the second equality follows from (10.6). Hence [M1 : M ] = [M : N ]. Furthermore, by (10.6) the unique tracial state τ on N has the property τ (eN ) = [M : N ]−1 . The unique tracial state on M1 , which we again denote by τ , is such that τ (x) = τ (Jx∗ J). Hence τ (eN ) = [M : N ]−1 . Since N is a factor, τ (ab) = τ (a)τ (b) for a ∈ N and b ∈ N ∩ M1 . Hence, for x ∈ M , τ (xEM (eN )) = τ (xeN ) = τ (EN (x)eN ) = τ (EN (x))τ (eN ) = [M : N ]−1 τ (x), so we also have EM (eN ) = [M : N ]−1 1, where EM : M1 → M is the trace preserving conditional expectation. Thus starting with a ﬁnite index inclusion N ⊂ M we construct a II1 factor M1 and a projection eN ∈ M1 such that [M1 : M ] = [M : N ], M1 = !M, eN ", N = {eN } ∩ M and EM (eN ) = [M : N ]−1 1. This is called the basic construction. Iterating it we get the Jones tower of II1 -factors M−1 = N ⊂ M0 = M ⊂ M1 ⊂ . . . ⊂ Mk ⊂ . . . and projections ek ∈ Mk+1 such that Mk+1 = !Mk , ek ", k ≥ 0, e0 = eN . There is also a downward construction. So there exists a projection e−1 and a subfactor N−1 ⊂ N such that N−1 ⊂ N ⊂ !N, e−1 " = M is the basic construction. Since EN (e−1 ) = [M : N ]−1 1, by Lemma 10.2.3 the projection e−1 is deﬁned up to conjugation by a unitary in N . Since N−1 = {e−1 } ∩ N , the subfactor N−1 is also deﬁned up to conjugation by a unitary in N . Denoting N−1 by M−2 and iterating the downward construction we get a tunnel . . . ⊂ M−k ⊂ . . . ⊂ M−1 ⊂ M0

172

10 Relative Entropy and Subfactors

and projections e−k ∈ M−k+1 , k ≥ 1. Using that ek xek = ek EMk−1 (x) = EMk−1 (x)ek for x ∈ Mk and EMk (ek ) = [M : N ]−1 1 one then checks that the projections ek , k ∈ Z, satisfy the relations ek ek±1 ek = [M : N ]−1 ek , ek ej = ej ek if |k − j| ≥ 2. Since [Mk+1 : Mk ] = [M : N ] < ∞, by the discussion at the beginning of Sect. 10.2 we can inductively construct a representation of ∪k Mk on L2 (M ) extending the representation of M1 . This representation is not unique, but as soon as we ﬁx a tunnel there exists a canonical one. Lemma 10.4.1. With the above notation there exists a unique extension of 2 the representation of M1 on L2 (M ) to a representation of ∪∞ k=1 Mk on L (M ) such that ek = Je−k J for k ≥ 1. In this representation JMk J = M−k for every k ∈ Z. Proof. Since Mk+1 = !Mk , ek ", the condition ek = Je−k J completely determines the representation. We therefore only have to prove its existence. We shall do this by constructing the required representation of Mk by induction on k. For k = 1 we have a representation by deﬁnition, and M1 = JM−1 J. So assume the representation is well-deﬁned for some k, and Mj = JM−j J for j = 1, . . . , k, where we identify Mj with its image under the representation. Since [Mk+1 : Mk ] < ∞, the identity map Mk → Mk ⊂ B(L2 (M )) extends to a representation π: Mk+1 → B(L2 (M )). Let f = Jπ(ek )J. Denote [M : N ]−1 by λ. We assert that f ∈ M−k+1 and EM−k (f ) = λ1. Indeed, since ek ∈ Mk−1 ∩ Mk+1 , we have π(ek ) ∈ Mk−1 , so that f ∈ JMk−1 J = M−k+1 . Then EM−k (f )e−k+1 = e−k+1 f e−k+1 = Jek−1 JJπ(ek )JJek−1 J = Jπ(ek−1 ek ek−1 )J = λJek−1 J = λe−k+1 , so that EM−k (f ) = λ1. But then by Lemma 10.2.3 there exists a unitary u ∈ M−k such that e−k = uf u∗ = J(JuJπ(ek )Ju∗ J)J. Since JuJ ∈ JM−k J = Mk , we thus see that x → JuJπ(x)Ju∗ J is a representation of Mk+1 which coincides with π on Mk and maps ek onto Je−k J. Therefore it is the required representation of Mk+1 . Since M−k−1 = {e−k } ∩ M−k , we have J, Je−k J" = !Mk , ek " = Mk+1 , J = !JM−k JM−k−1

which completes the proof of the induction step.

J = Mn , it follows that M−n ⊂ M0 ⊂ Mn is the basic conSince JM−n struction for each n ∈ N. Hence, more generally, Mk−n ⊂ Mk ⊂ Mk+n is the basic construction for any k ∈ Z and n ∈ N. Consider now the AF-algebra A∞ obtained by taking the inductive limit of the algebras Mk ∩ Ml as k → −∞ and l → +∞ (note that Mk ∩ Ml is ﬁnite dimensional by [214, Corollary XIX.2.9]). By the above lemma we get

10.4 The Canonical Shift

173

a mirroring γ0 , the involutive anti-automorphism of A∞ deﬁned by γ0 (x) = Jx∗ J for x ∈ ∪k,l (Mk ∩ Ml ). It has the properties γ0 (Mk ∩ Ml ) = M−l ∩ M−k and γ0 (ek ) = e−k . On the other hand, we can consider . . . ⊂ Mk ⊂ Mk+1 ⊂ . . . as the tower and the tunnel associated with the inclusion M0 ⊂ M1 , and get an anti-automorphism γ1 of A∞ such that γ1 (Mk ∩ Ml ) = M−l+2 ∩ M−k+2 and γ1 (ek ) = e−k+2 . The canonical shift for the inclusion N ⊂ M is the automorphism Γ = γ1 ◦ γ0 of A∞ . It has the properties Γ (Mk ∩ Ml ) = Mk+2 ∩ Ml+2 and Γ (ek ) = ek+2 .

Since each algebra Mk has a unique trace, the C∗ -algebra A∞ has a canonical tracial state, which we continue to denote by τ . Our next goal is to show that Γ is τ -preserving. By [214, Proposition XIX.4.19] in the ﬁnite depth case, which we shall consider below, the trace τ is the unique tracial state on A∞ . Hence in that case it is γ0 - and γ1 -invariant. However, this is not true in general. To prove that nevertheless the trace is Γ -invariant we need to compare the constructed representations of ∪k Mk on L2 (M0 ) and L2 (M1 ). To simplify the notation we consider ∪k Mk as a subalgebra of B(L2 (M1 )), and denote by π the representation of ∪k Mk on L2 (M0 ). Lemma 10.4.2. If we identify L2 (M0 ) with e1 L2 (M1 ), then the representation π satisﬁes π(x) = λ−k e1 . . . ek xek+1 ek . . . e1 for x ∈ Mk , k ≥ 1, where λ = [M : N ]−1 . Proof. Using the relations ei ei−1 ei = λei , i = 1, . . . , k, we get ek+1 ek . . . e1 e1 . . . ek ek+1 = λk ek+1 . Since ek+1 x = xek+1 for x ∈ Mk , it follows that the expression in the formulation of the lemma deﬁnes a homomorphism. For x ∈ M0 we have π(x) = xe1 . Since Mk is generated by M0 and e0 , . . . , ek−1 , all that remains to show is that the equality in the formulation holds for x = e0 , . . . , ek−1 . In other words, π(ek−1 ) = ek+1 e1 for k ≥ 2, and π(e0 ) = λ−1 e1 e0 e2 e1 . Denote by J1 and J0 the modular conjugations on L2 (M1 ) and L2 (M0 ), respectively. Since J0 = J1 |e1 L2 (M1 ) , for k ≥ 2 we have π(ek−1 ) = J0 π(e−k+1 )J0 = J1 e−k+1 e1 J1 = ek+1 e1 , proving the ﬁrst identity. To prove the second note that if ξ1 is the cyclic trace vector in L2 (M1 ), then

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10 Relative Entropy and Subfactors

(λ−1 e1 e0 e2 e1 ξ1 , ξ1 ) = λ−1 (e2 ξ1 , e0 ξ1 ) = λ−1 (J1 e0 J1 ξ1 , e0 ξ1 ) = λ−1 (e0 ξ1 , e0 ξ1 ) = λ−1 τ (e0 ) = 1. Since λ−1 e1 e0 e2 e1 is a projection, we conclude that ξ1 = λ−1 e1 e0 e2 e1 ξ1 . Since e1 e0 e2 e1 ∈ M−1 , it follows that the projection [M−1 ξ1 ] is majorized −1 by λ e1 e0 e2 e1 . On the other hand, since M1 is generated by M0 and e0 , we have e0 M1 e0 = e0 M−1 . Hence [M−1 ξ1 ] = [M−1 e1 e2 e0 e1 ξ1 ] = [e1 e2 e0 M−1 ξ1 ] = [e1 e2 e0 M1 e0 ξ1 ]. Since e2 ξ1 = J1 e0 J1 ξ1 = e0 ξ1 and e2 ∈ M1 , the above projection equals [e1 e2 e0 M1 e2 ξ1 ] = [e1 e2 e0 M1 ξ1 ] = [e1 e2 e0 L2 (M1 )] ≥ λ−1 e1 e0 e2 e1 . By deﬁnition of e0 = eN we get π(e0 ) = [M−1 ξ1 ] = λ−1 e1 e0 e2 e1 , completing the proof of the lemma.

Proposition 10.4.3. The canonical shift Γ is trace preserving. Proof. We use the same notation as in the proof of the previous lemma. Since x = γ0 (γ1 (Γ (x))), π ◦ γ0 = J0 π(·)∗ J0 and J0 = J1 |e1 L2 (M1 ) , for any x ∈ A∞ we have π(x) = J1 π(γ1 (Γ (x∗ )))J1 . ∩ M−k . If x ∈ Mk ∩ Ml with k ≤ −1 and l ≥ 1, then γ1 (Γ (x)) = γ0 (x) ∈ M−l Using Lemma 10.4.2 and that J1 ei J1 = e−i+2 , we conclude that the above expression equals

λk J1 e1 . . . e−k J1 Γ (x)J1 e−k+1 e−k . . . e1 J1 = λk e1 e0 . . . ek+2 Γ (x)ek+1 . . . e0 e1 . Thus λk e1 e0 . . . ek+2 Γ (x)ek+1 . . . e0 e1 = π(x) = λ−l e1 . . . el xel+1 . . . e1 .

(10.9)

Applying the trace to the left side and using that ei ei+1 ei = λei and Γ (x) ∈ Mk+2 ∩ Ml+2 we get λk τ (Γ (x)ek+1 . . . e0 e1 e0 . . . ek+1 ) = τ (Γ (x)ek+1 ) = λτ (Γ (x)). Similarly applying τ to the right side of (10.9) we get λτ (x), so that τ (Γ (x)) = τ (x).

Remark 10.4.4. If x ∈ Mk ∩ Ml then multiplying (10.9) by ek+1 . . . e0 from the left and by e0 . . . ek+1 from the right, we conclude that Γ (x) is an element in Mk+2 ∩ Ml+2 such that Γ (x)ek+1 = λk−l ek+1 . . . el xel+1 . . . ek+1 .

10.4 The Canonical Shift

175

Since Mk+2 ⊂ B(L2 (M1 )) is a factor and ek+1 ∈ Mk+2 , the homomorphism Mk+2 a → aek+1 is faithful. Hence Γ (x) is uniquely determined by the above identity. In particular, the restriction of Γ to ∪n≥0 (N ∩ Mn ) ⊂ A∞ is independent of the choice of the tunnel. Often it is this restriction which is called the canonical shift. It follows also that if for some n ∈ Z we consider

. . . ⊂ Mk ⊂ Mk+1 ⊂ . . . as the tower and the tunnel for Mn ⊂ Mn+1 and deﬁne the corresponding mirrorings γn and γn+1 , then Γ = γn+1 ◦ γn . To compute the entropy of Γ we shall use Theorem 10.3.1. We have to check that its assumptions are satisﬁed. To show that {M ∩ M2n }n is a generating sequence for Γ we prove the next general proposition. Proposition 10.4.5. Let (P, τ, σ) be a W∗ -dynamical system, where τ is a normal tracial state. Let {Pn }n be an increasing sequence of ﬁnite dimensional C∗ -subalgebras such that σ(Pn ) ⊂ Pn+1 and P is generated by σ k (Pn ), k ∈ Z, n ∈ N. Assume there exists a sequence {kn }n of natural numbers such that kn → 1 as n → ∞; (i) n (ii) the algebras σ mkn (Pn ), m ∈ N, are mutually τ -independent. Then {Pn }n is a generating sequence for σ. Recall that we say that A1 , . . . , An are τ -independent if they mutually commute, and τ (a1 . . . an ) = τ (a1 ) . . . τ (an ) for ai ∈ Ai . Proof of Proposition 10.4.5. Since σ k (Pl ) ⊂ σ −[n/2] (Pn ) for −[n/2] ≤ k ≤ [n/2] − l, the algebra ∪n σ −[n/2] (Pn ) is dense in P . Hence, by the KolmogorovSinai type theorem, Theorem 3.2.3, hτ (σ) = lim hτ (Pn ; σ). n→∞

Since σ i (Pk ) ⊂ Pk+i , for any n ∈ N we have Hτ (Pk , σ(Pk ), . . . , σ n (Pk )) ≤ Hτ (Pn+k ), whence hτ (Pk ; σ) ≤ lim inf n→∞

1 Hτ (Pn ). n

On the other hand, since the algebras σ mkn (Pn ), m ∈ Z, are independent, we have

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10 Relative Entropy and Subfactors

hτ (Pn ; σ kn ) = Hτ (Pn ). Hence hτ (σ) = Since

1 1 1 hτ (σ kn ) ≥ hτ (Pn ; σ kn ) = Hτ (Pn ). kn kn kn

kn → 1, it follows that n hτ (σ) ≥ lim sup n→∞

1 Hτ (Pn ). n

1 Thus the limit lim Hτ (Pn ) exists and equals hτ (σ), that is, {Pn }n is a genn n erating sequence.

The proposition implies that {M ∩ M2n }n is a generating sequence for the canonical shift Γ . Indeed, since Γ mn (M ∩ M2n ) = M2mn ∩ M2(m+1)n , we mn see that Γ (M ∩ M2n ) commutes with M ∩ M2n for any m ≥ 1. Moreover, since M2n is a factor, we have τ (xy) = τ (x)τ (y) for any x ∈ M2n and y ∈ ∪k (M2n ∩M2n+k ). Thus the algebras Γ mn (M ∩M2n ), m ∈ N, are independent, and we can apply the previous proposition with kn = n. Next note that the von Neumann algebra generated by ∪n (M ∩ M2n ) in the GNS-representation of A∞ deﬁned by τ has type II1 , since already the von Neumann algebra generated by the projections en , n ≥ 1, is a II1 -factor by [214, Theorem XIX.3.1]. The generating sequence {M ∩ M2n }n satisﬁes the commuting square condition, that is, M ∩ M2n ⊂ M ∩ M2n+2 ∪ ∪ M2 ∩ M2n ⊂ M2 ∩ M2n+2 is a commuting square for any n ∈ N. Indeed, the trace preserving conditional expectation M ∩ M2n+2 → M ∩ M2n is the restriction of the conditional expectation EM2n : M2n+2 → M2n . But then we obviously have EM2n (M2 ∩ M2n+2 ) ⊂ M2 ∩ M2n . Next we check that the entropy is ﬁnite. Proposition 10.4.6. We have hτ (Γ ) ≤ ht(Γ ) ≤ log[M : N ]. Proof. If Ω is a ﬁnite subset of M ∩Mk , then Γ n (Ω) ⊂ M ∩M2n+k . It follows that 1 ht(Ω; Γ ) ≤ lim sup log rank M ∩ M2n . n→∞ n By [214, Proposition XIX.2.8] if p1 , . . . , pm are projections in M ∩ M2n with sum 1, then

10.4 The Canonical Shift

[pi M2n pi : M pi ] Hence

i

i

τ (pi )

177

= [M2n : M ].

τ (pi )−1 ≤ [M2n : M ], so by the Cauchy-Schwarz inequality m=

m

τ (pi )1/2 τ (pi )−1/2 ≤ [M2n : M ]1/2 .

i=1

Therefore rank M ∩ M2n ≤ [M2n : M ]1/2 = [M : N ]n , and the proof of the proposition is complete.

Finally, the multiplicities for the inclusion M ∩ Mn−1 ⊂ M ∩ Mn are bounded by [M : N ] (in fact, by [M : N ]1/2 ). To see this let p be a minimal projection in M ∩Mn−1 . Then q = pen is a minimal projection in M ∩Mn+1 . Indeed, since en Mn+1 en = Mn−1 en , we have en (M ∩Mn+1 )en = (M en ) ∩en Mn+1 en = (M en ) ∩Mn−1 en = (M ∩Mn−1 )en , so that q(M ∩ Mn+1 )q = p(M ∩ Mn−1 )pen = Cq. Furthermore, if f is a projection in M ∩ Mn and f ≤ p, then qf q = en f en = EMn−1 (f )en = 0. Therefore if p majorizes a minimal projection f ∈ M ∩ Mn then the central support of f in M ∩ Mn+1 majorizes q. In other words, if (aij )i,j and (bjk )j,k are the inclusion matrices for M ∩ Mn−1 ⊂ M ∩ Mn and M ∩ Mn ⊂ M ∩ Mn+1 , respectively, i0 corresponds p and k0 corresponds to q, then bjk0 = 0 as soon as ai0 j = 0. If {zk }k is the set of minimal central projections in M ∩ Mn+1 , then denoting by sk the trace of a minimal projection majorized by zk , we get ai0 j bjk sk ≥ ai0 j bjk0 τ (q) = [M : N ]−1 ai0 j bjk0 τ (p). τ (p) = j,k

It follows that ai0 j ≤ [M : N ] (in fact, it is known that ai0 j = bjk0 , so that ai0 j ≤ [M : N ]1/2 ). Since . . . ⊂ M2k−2 ⊂ M2k ⊂ . . . can be considered as the tower and the tunnel associated with M−2n ⊂ M−2n+2 , we conclude that the multiplicities for the inclusion M−2n ∩ M−2 ⊂ M−2n ∩ M are bounded by [M : N ]2 . Since γ0 (M2 ∩ M2n ) = M−2n ∩ M−2 and γ0 (M ∩ M2n ) = M−2n ∩ M , it follows that the multiplicities for the inclusion Γ (M ∩ M2n−2 ) ⊂ M ∩ M2n are also bounded by [M : N ]2 . We have thus checked that all the assumptions of Theorem 10.3.1 are satisﬁed, so we get our ﬁrst and most general result on the entropy of Γ . Theorem 10.4.7. Let Γ be the canonical shift for the inclusion N ⊂ M of II1 -factors of ﬁnite index. Let R = πτ (∪∞ n=1 (M ∩ M2n )) . Then 1 1 1 Hτ (M ∩ M2n ) = Hτ (R|Γ (R)) + lim Hτ (Z(M ∩ M2n )). n→∞ 2n n→∞ n 2

hτ (Γ ) = lim

178

10 Relative Entropy and Subfactors

Combining this theorem and Proposition 10.4.6 we obtain the following estimates. Corollary 10.4.8. We have 1 Hτ (R|Γ (R)) ≤ hτ (Γ ) ≤ ht(Γ ) ≤ log[M : N ]. 2

In particular, if Hτ (R|Γ (R)) = 2 log[M : N ], then both the topological entropy and the dynamical entropy of Γ with respect to τ are equal to log[M : N ]. By [169, Theorem 5.3.1] the condition Hτ (R|Γ (R)) = 2 log[M : N ] is one of several equivalent characterizations of extremality of N ⊂ M and strong amenability of the standard invariant of the inclusion. The simplest case when this condition is satisﬁed, is that of ﬁnite depth inclusions. A ﬁnite index inclusion N ⊂ M of II1 -factors is said to have ﬁnite depth if sup dim Z(M ∩ Mn ) < ∞. n

Then, by the discussion prior to [214, Proposition XIX.4.19], there exists n0 such that if G denotes the matrix of the inclusion M ∩ M2n0 ⊂ M ∩ M2n0 +1 , then the inclusions M ∩ M2n ⊂ M ∩ M2n+1 and M ∩ M2n+1 ⊂ M ∩ M2n+2 are given by the matrices G and Gt , respectively, for any n ≥ n0 . Moreover, the matrix GGt is primitive, and if s denotes the vector formed by the values of the trace on minimal projections in M ∩ M2n0 , then GGt s = [M : N ]s. We shall now show that this allows us to compute the entropy without relying on the deep results of Popa [169]. Recall that a matrix B ∈ Matr (R) with nonnegative coeﬃcients is called primitive if there exists m ∈ N such that B m has only positive entries. By the Perron-Frobenius theorem, see e.g. [193, Section 1.1], the spectral radius β of B is an eigenvalue of B, called the Perron-Frobenius eigenvalue, and the following conditions are satisﬁed: (i) there is an eigenvector ξ of B with eigenvalue β whose coordinates are all positive; similarly, there is an eigenvector ζ of B ∗ with eigenvalue β and positive coordinates; (ii) if ζ is normalized such that (ξ, ζ) = 1 then β −n B n → P as n → +∞, where P = (·, ζ)ξ is the projection onto Rξ along ζ ⊥ . These properties imply ∩n∈N B n (Rr+ ) = R+ ξ. Indeed, assume ϑn ∈ Rr+ , n ≥ 0, are such that ϑ0 = β −n B n ϑn . Since ζ has positive coordinates, there exists ε > 0 such that P ϑ = (ϑ, ζ)ξ ≥ εϑ for ϑ ∈ Rr+ . Then if ε < ε and n is such that β −n B n − P ≤ ε , we get β −n B n ϑ ≥ (ε − ε )ϑ. It follows that the sequence {ϑn }n is bounded. But then the vectors β −n B n ϑn become arbitrarily close to P ϑn as n → ∞. Hence ϑ0 ∈ R+ ξ. It follows that ξ is the unique (up to a scalar factor) eigenvector of B with nonnegative coordinates. In particular, if B = GGt corresponds to a ﬁnite depth inclusion N ⊂ M , then the Perron-Frobenius eigenvalue is [M : N ].

10.4 The Canonical Shift

179

Proposition 10.4.9. Let A = lim An be an AF-algebra such that for every n −→ the inclusion An ⊂ An+1 is deﬁned by a ﬁxed primitive matrix B. Then there exists a unique tracial state τ on A, and lim

n→∞

1 Hτ (An ) = log β, n

where β is the Perron-Frobenius eigenvalue of B. Proof. Let An = ⊕rk=1 Matmnk (C). A tracial state τ on A is deﬁned by a sequence of vectors sn ∈ Rr+ , n ∈ N, such that k s1k m1k = 1 and sn = Bsn+1 . Let of B normalized by the condition ξ be1 the Perron-Frobenius eigenvector n −n+1 ξ m = 1. Then we can take s = β ξ and obtain a tracial state on A. k k k Conversely, given a tracial state τ on A, we have sn ∈ ∩m B m (Rr+ ). Hence by the discussion before the formulation of the proposition the vector sn is a scalar multiple of ξ for any n, and the scalar is completely determined by the condition τ (1) = 1. Then we have Hτ (An ) = −

r

mnk snk log snk = −

k=1

r

mnk snk log(β −n+1 ξk )

k=1

= (n − 1) log β −

r

mnk snk log ξk .

k=1

Since

k

mnk snk = 1, dividing by n and letting n → ∞ we get the result.

Applying this proposition to An = M ∩ M2n for a ﬁnite depth inclusion, we then get 1 lim Hτ (M ∩ M2n ) = log[M : N ]. n→∞ n We also obviously have lim

n→∞

1 Hτ (Z(M ∩ M2n )) = 0. n

Thus combining Proposition 10.4.6 and Theorem 10.4.7 we obtain the following result. Theorem 10.4.10. Let N ⊂ M be an inclusion of II1 -factors of ﬁnite index and ﬁnite depth. Let R = πτ (∪∞ n=1 (M ∩ Mn )) . Then 1 Hτ (R|Γ (R)) = hτ (Γ ) = ht(Γ ) = log[M : N ]. 2

Remark that by [214, Corollary XIX.4.9] any inclusion of index < 4 has ﬁnite depth.

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10 Relative Entropy and Subfactors

10.5 Shifts on Temperley-Lieb Algebras π Let λ be a real number such that λ−1 ∈ {4 cos2 n+1 | n ≥ 3} ∪ [4, ∞). Consider ∗ the universal C -algebra A generated by a sequence of projections {ek }k∈Z such that ek ek±1 ek = λek , ek ej = ej ek if |k − j| ≥ 2.

As we saw in the previous section, a representation of this algebra arises naturally from an inclusion of II1 -factors with index λ−1 . We also saw that there exists a trace τ on A such that τ (wek ) = λτ (w)

(10.10)

for any k ∈ Z and any w in the algebra generated by projections ej with j < k. For n ≥ 2 denote by An the C∗ -algebra generated by e1 , . . . , en−1 , and set A0 = A1 = C1. An easy induction argument shows that An = An−1 + An−1 en−1 An−1 , n ≥ 2. In particular, the trace is completely determined by (10.10). The dimension and the trace vectors mn and sn of the algebras An , n ∈ N, are described as follows, see § 3 of Chapter XIX in [214] for a proof. If λ−1 ≥ 4 then n n [n/2] , snk = λk Pn−2k (λ), An ∼ − = ⊕ Matmnk (C), mnk = k k−1 k=0 where {Pn }∞ n=0 is the sequence of polynomials such that P0 = P1 = 1, Pn+1 (t) = Pn (t) − tPn−1 (t), n ≥ 1. π with n ≥ 3 then for any k ≥ 0 the On the other hand, if λ−1 = 4 cos2 n+1 embeddings An+2k−1 → An+2k+1 are given by a ﬁxed primitive matrix with the Perron-Frobenius eigenvalue λ−1 . Denote by θλ the automorphism of A deﬁned by θλ (ek ) = ek+1 .

Theorem 10.5.1. We have: 1 (i) hτ (θλ ) = − log λ, if λ−1 ≤ 4; 2 (ii) hτ (θλ ) = η(α) + η(1 − α), where α =

1+

√

1 − 4λ , if λ−1 ≥ 4. 2

Proof. The algebras θλnm (An ), m ∈ Z, are mutually τ -independent. By Proposition 10.4.5 it follows that {An }n is a generating sequence, that is, hτ (θλ ) = lim

n→∞

If λ−1 < 4, then by Proposition 10.4.9

1 Hτ (An ). n

10.5 Shifts on Temperley-Lieb Algebras

181

1 Hτ (A2n ) = log λ. n→∞ n lim

Hence hτ (θλ ) = − 12 log λ. It remains to consider the case λ−1 ≥ 4. Assume ﬁrst that λ−1 > 4. Set √ √ 1 + 1 − 4λ 1 − 1 − 4λ α= and β = 1 − α = . 2 2 Then using that αβ = λ we check that Pn (λ) =

αn+1 − β n+1 , n ≥ 0. α−β

k k 2n−2k (1 − (β/α)2n−2k+1 )(1 − β/α)−1 , we see Since s2n k = λ P2n−2k (λ) = λ α that the diﬀerence k 2n−2k log s2n ) = log s2n k − log(λ α k − ((2n − k) log α + k log β)

is uniformly bounded. Since Hτ (A2n ) = −

n

2n m2n k sk

log s2n k

and

k=0

n

2n m2n k sk = 1,

k=0

the limit of Hτ (A2n )/2n coincides with the limit of n α 2n k m2n . k sk β 2n k=0 (10.11) = (αβ)k (α2n−2k+1 − β 2n−2k+1 )/(α − β), we have

1 2n 2n mk sk ((2n − k) log α + k log β) = − log α + log − 2n n

k=0

Since s2n k n

2n m2n k sk

k=0

n n k α 2n 2n−k k k β 2n k 2n−k k mk α β mk α β = − . 2n α−β 2n α − β 2n k=0

As α > β, m2n k ≤ n

" 2n # k

k=0

and 4αβ = 4λ < 1, k ≤ (αβ)n 2n n

k 2n−k m2n k α β

k=0

k=0

2n k

≤ (4αβ)n → 0.

On the other hand, n 2n k=0

k

α

n 2n − 1 k α2n−k β k β = 2n k−1 k=1 n−1 2n − 1 α2n−1−k β k → β, =β k

2n−k k

k=0

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10 Relative Entropy and Subfactors

since

2n−1 " 2n−1 # k=n

k

α2n−1−k β k ≤ α−1 (αβ)n

2n−1 " 2n−1 # k=n

k

→ 0. Similarly

n n 2n k 2n k−1 α2n−k β k α2n−k β k = lim n→∞ k−1 k−1 2n n→∞ 2n k=0 k=1 n−1 β 2n k β2 α2n−k β k = lim = . n→∞ α k 2n α lim

k=0

Therefore, since m2n k = lim

n

n→∞

" 2n # k

2n m2n k sk

k=0

−

2n k−1

,

n k α 2n−k k k m2n β = lim k α 2n α − β n→∞ 2n k=0 2 β α β− = β, = α−β α

so that from (10.11) we get that α hτ (θλ ) = − log α + log β = η(α) + η(β). β Finally consider the case λ−1 = 4. Then Pn (λ) =

n+1 , 2n

−2n whence s2n (2n − 2k + 1), and k =2

1 1 2n 2n 2n log(2n − 2k + 1) m2n mk sk log s2n Hτ (A2n ) = − . k sk k = log 2 − 2n 2n 2n n

n

k=0

k=0

Letting n → ∞ we get hτ (θ1/4 ) = log 2.

Denote by M the von Neumann subalgebra of πτ (A) generated by en , n ≤ −1, and by R the von Neumann subalgebra generated by en , n ≥ 1. By [214, Theorem XIX.3.1] both M and R are the hyperﬁnite II1 -factors, N = θλ−1 (M ) ⊂ M is a subfactor of index λ−1 , and N ⊂ M ⊂ θλ (M ) = !M, e0 " is the basic construction. It follows that the automorphism θλ2 of πτ (A) is (the extension to the weak operator closure of) the canonical shift associated with the inclusion N ⊂ M . It is clear that An ⊂ M ∩ Mn for any n ≥ 0. It can be shown that if −1 λ ≤ 4 then An = M ∩ Mn . For λ−1 < 4 the inclusion N ⊂ M has ﬁnite depth, and from Theorem 10.4.10 we again obtain hτ (θλ ) =

1 1 1 hτ (θλ2 ) = log[M : N ] = − log λ. 2 2 2

10.6 Notes

183

On the other hand, if λ−1 > 4 then it can be shown that already M ∩M1 is diﬀerent from A1 = C1. It is not diﬃcult to check that the generating sequence {An }n satisﬁes the commuting square condition. Hence by Theorem 10.3.1 hτ (θλ ) =

1 Hτ (R|θλ (R)). 2

Therefore Hτ (R|θλ (R)) = 2(η(α) + η(β)). This equality, as well as the entropy hτ (θλ ), can also be obtained from the embedding A → Mat2 (C)⊗Z , % k+1 k+1 k k ek → (1 − λ)ek11 ⊗ ek+1 λ(1 − λ)(ek12 ⊗ ek+1 22 + λe22 ⊗ e11 + 21 + e21 ⊗ e12 ), where {ekij }2i,j=1 are the matrix units in the k-th factor Mat2 (C). It can be shown, see [162], that in the GNS-representation of Mat2 (C)⊗Z corresponding to the product-state α 0 ψ = ϕ⊗Z , Qϕ = , 0 β the weak operator closure of A coincides with the centralizer of the state. Thus θλ is one of the Bernoulli shifts on the hyperﬁnite factor considered in Example 3.2.6(ii). Moreover, the von Neumann algebra generated by the projections en , n ≥ 1, coincides with the centralizer of ψ on πψ (Mat2 (C)⊗N ) . In particular, there exists a projection p ∈ R which is mapped onto e111 under the above embedding, and then τ (p) = α, p ∈ θλ (R) ∩ R, pRp = θλ (R)p, (1 − p)R(1 − p) = θλ (R)(1 − p). It should be remarked that the projections en , n ≤ −1, do not generate the centralizer of ψ on πψ (⊗n≤0 Mat2 (C)) .

10.6 Notes The notion of relative entropy for subalgebras of a von Neumann algebra with a normal tracial state was introduced by Connes and Størmer [51], who used it as a tool to prove continuity of mutual entropy. The deﬁnition was extended to arbitrary states by Connes [49]. To see that relative entropy is relevant for estimates of mutual entropy, observe that for ﬁnite dimensional C∗ -subalgebras P1 , . . . , Pn and Q1 , . . . , Qn of M we have Hϕ (P1 , . . . , Pn ) ≤ Hϕ (Q1 , . . . , Qn ) +

n

Hϕ (Pk |Qk ).

k=1

It was Pimsner and Popa [162] who undertook a serious study of this notion and discovered its connection with Jones’ index of II1 -factors. Sects. 10.1 and 10.2 are almost entirely based on their paper. For some examples of

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10 Relative Entropy and Subfactors

computations of relative entropy see [22], [103], [233]. The results of Pimsner and Popa were extended to type III factors by Hiai [86]. Theorem 10.3.1 was proved by Størmer [210] extending results of Choda [40] and Hiai [88]. See [81] for further discussion of generating sequences. The canonical shift was introduced by Pimsner and Popa [162] and Ocneanu [146]. One usually deﬁnes the canonical shift a bit diﬀerently. As we remarked after the proof of Lemma 10.4.1, for any n ∈ N the algebras M−1 ⊂ Mn ⊂ M2n+1 form a basic construction. Pimsner and Popa [163] found an explicit formula for the corresponding Jones projection in M2n+1 . In other words, they found an explicit representation of M2n+1 on L2 (Mn ) extending the representation of Mn+1 . Using this representation we can deﬁne an antiautomorphism of M−1 ∩M2n+1 , which for the moment we denote by γn . Then one checks that γn+2 ◦γn+1 and γn+1 ◦γn agree on M−1 ⊂ Mn ⊂ M2n+1 , ∩M so there exists a well-deﬁned endomorphism of ∪n (M−1 2n+1 ). To see that we get the same endomorphism as the one deﬁned in the present chapter, by Remark 10.4.4 it suﬃces to check that γn = γn on M−1 ∩ M2n+1 , or even 2 better, the representation of M2n+1 on L (Mn ) alluded to above coincides with the representation deﬁned as in Lemma 10.4.1. Since these representations are determined by the images of en+1 , . . . , e2n , and γn (en+k ) = en−k by construction, we just have to check that γn (en+k ) = en−k for k = 1, . . . , n. This is indeed the case, see e.g. [54]. Alternatively one can use the identity in Remark 10.4.4 and an explicit formula for the canonical shift, see e.g. [23]. The entropy of the shifts on the Temperley-Lieb algebras, Theorem 10.5.1, was computed by Pimsner and Popa [162] for all λ = 1/4. For the case λ < 1/4 instead of the elementary but tedious computations presented above they proved that the shifts are isomorphic to the Bernoulli shifts on the hyperﬁnite factor considered in Example 3.2.6(ii); see [191] for a more general result. The missing case λ = 1/4 was treated by Choda [40] and Yin [236]. The latter paper also contains the computation of the entropy for λ < 1/4, which we used. For arbitrary subfactors the study of entropic properties of the canonical shift was initiated by Choda [41]. In particular, she obtained the inequalities from Corollary 10.4.8 and computed the entropy in the case of a ﬁnite depth subfactor, Theorem 10.4.10. The most general result, Theorem 10.4.7, was proved by Hiai [88]. For similar results in the type III case see [48], [87]. An argument similar to the proof of Proposition 10.4.5 appeared already in the paper of Connes and Størmer [51] and had been used by several authors until it was formalized by Choda [40], [41]. Proposition 10.4.9 was discovered by so many people that it should probably be considered a folklore. It was successfully applied by Choda [40], [41] to a number of models.

11 Systems of Algebraic Origin

Given an automorphism T of a discrete abelian group preserving a 2-cocycle, we deﬁne an automorphism αT of the corresponding twisted group C∗ -algebra. If the cocycle is trivial, we get an abelian dynamical system deﬁned by the ˆ Dynamical properties of such automorphisms are dual automorphism Tˆ of G. well understood. In this chapter we shall study entropic properties of αT for general cocycles.

11.1 Twisted Group C*-algebras For a discrete abelian group G and a 2-cocycle ω ∈ Z 2 (G; T) consider the twisted group C∗ -algebra C ∗ (G, ω). By deﬁnition it is the universal C∗ -algebra generated by unitaries ug , g ∈ G, such that ug uh = ω(g, h)ug+h . We have considered a more general class of algebras in Chap. 8. In particular, we know that C ∗ (G, ω) is nuclear and the regular representation C ∗ (G, ω) → B(2 (G)), ug δh = ω(g, h)δg+h , is faithful, where δg ∈ 2 (G) is deﬁned by δg (h) = 0 for h = g and δg (g) = 1. Recall also that ω(g, 0) = ω(0, g) = ω(0, 0) for any g ∈ G. The vector δ0 deﬁnes the canonical trace τ on C ∗ (G, ω) such that τ (ug ) = 0 for g = 0. Since δ0 is cyclic and the regular representation is faithful, the trace τ is faithful. For any 2-cocycle ω the cocycle ρω (g, h) = ω(g, h)ω(h, g) is a skewsymmetric bicharacter, that is, ρω (g, g) = 1 and ρω (g1 + g2 , h) = ρω (g1 , h)ρω (g2 , h), ρω (h, g1 + g2 ) = ρω (h, g1 )ρω (h, g2 ). Indeed, using that ug uh = ρω (g, h)uh ug in C ∗ (G, ω) we get

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11 Systems of Algebraic Origin

ug1 +g2 uh = ρω (g1 + g2 , h)uh ug1 +g2 , and on the other hand ug1 +g2 uh = ω(g1 , g2 )ug1 ug2 uh = ω(g1 , g2 )ρω (g1 , h)ρω (g2 , h)uh ug1 ug2 = ρω (g1 , h)ρω (g2 , h)uh ug1 +g2 , so that ρω (g1 + g2 , h) = ρω (g1 , h)ρω (g2 , h). Note also that any bicharacter is a 2-cocycle. It is now easy to describe when the twisted group C∗ -algebra is simple. Theorem 11.1.1. For any ω ∈ Z 2 (G; T) the following conditions are equivalent: (i) the C∗ -algebra C ∗ (G, ω) is simple; (ii) τ is the unique tracial state on C ∗ (G, ω); (iii) the skew-symmetric bicharacter ρω is nondegenerate. Recall that nondegeneracy of ρω means by deﬁnition that for any h = 0 there exists g such that ρω (g, h) = 1. In other words, the subgroup of the dual ˆ formed by the characters ρω (g, ·), g ∈ G, has trivial annihilator in G, group G ˆ so that it is dense in G. Proof of Theorem 11.1.1. Since ug uh u∗g = ρω (g, h)uh , if ρω is nondegenerate then ϕ(uh ) = 0 for every tracial state ϕ. Thus (iii)⇒(ii). If h = 0 is in the kernel of ρω , so that ρω (g, h) = 1 for any g ∈ G, then uh belongs to the center of C ∗ (G, ω). Hence C ∗ (G, ω) is nonsimple. Moreover, then πτ (C ∗ (G, ω)) has nontrivial center and consequently possesses a normal trace diﬀerent from τ . Thus (i)⇒(iii) and (ii)⇒(iii). ˆ on C ∗ (G, ω), It remains to prove (iii)⇒(i). Consider the dual action α of G so ˆ h ∈ G. αχ (uh ) = !χ, h"uh for χ ∈ G, Since ρω is nondegenerate, any character of G can be approximated in the topology of pointwise convergence by characters ρω (g, ·). Since (Ad ug )(uh ) = ρω (g, h)uh , we conclude that the automorphism αχ is approximately inner for ˆ in the sense that it can be approximated by inner automorphisms any χ ∈ G in the pointwise norm topology. Hence any ideal in C ∗ (G, ω) is αχ -invariant. Since τ (·)1 = αχ (·)dχ, ˆ G

it follows that if a ≥ 0 lies in an ideal I, then either 1 ∈ I or τ (a) = 0, and hence a = 0, as τ is faithful. Therefore C ∗ (G, ω) is simple.

In particular, if G is countable and ρω is nondegenerate, then the weak operator closure W ∗ (G, ω) = πτ (C ∗ (G, ω)) of C ∗ (G, ω) is a II1 -factor. This factor is hyperﬁnite since C ∗ (G, ω) is nuclear.

11.2 Estimates of Topological Entropy

187

If ρω is degenerate, the argument in the proof of Theorem 11.1.1 essentially gives a description of the ideal structure of C ∗ (G, ω). The following statement will be suﬃcient for our purposes. Lemma 11.1.2. Let H be the kernel of ρω , that is, H consists of the elements h ∈ G such that ρω (g, h) = 1 for every g ∈ G. Then the center of C ∗ (G, ω) is generated by the elements uh , h ∈ H, so it is isomorphic to C ∗ (H, ω|H ). Note that it follows from Lemma 11.1.3 below that C ∗ (H, ω|H ) is isomorˆ but the isomorphism is noncanonical. phic to C(H), Proof of Lemma 11.1.2. It is clear that C ∗ (H, ω|H ) is contained in the center. On the other hand, the map E deﬁned by E(x) = αχ (x)dχ G/H

is a conditional expectation onto C ∗ (H, ω|H ). Since (Ad ug )(uh ) = ρω (g, h)uh , the autoand the characters ρω (g, ·), g ∈ G, form a dense subgroup of G/H, morphism αχ is approximately inner for any χ ∈ G/H. Hence E(x) = x for any element x in the center, so the center is contained in C ∗ (H, ω|H ).

Finally remark that the map ω → ρω deﬁnes an embedding of H 2 (G; T) into the group of skew-symmetric bicharacters of G. Lemma 11.1.3. If ρω (g, h) = 1 for every g, h ∈ G, then ω is a coboundary, that is, there exists a function G g → zg ∈ T such that ω(g, h) = zg zh z¯g+h . Proof. By assumption the C∗ -algebra C ∗ (G, ω) is abelian. Choose a character f on C ∗ (G, ω). Then applying it to the identity ug uh = ω(g, h)ug+h we see that ω is the coboundary of G g → f (ug ).

Notice that in the notation of the lemma the elements vg = z¯g ug satisfy ˆ the relations vg vh = vg+h , so that C ∗ (G, ω) ∼ = C ∗ (G) ∼ = C(G).

11.2 Estimates of Topological Entropy Let T be an automorphism of a discrete countable abelian group G. If a cocycle ω ∈ Z 2 (G; T) is T -invariant, then T deﬁnes an automorphism αT of C ∗ (G, ω) such that αT (ug ) = uT g . We shall write αT,ω for αT and ug,ω for ug when we want to stress that we consider an automorphism of C ∗ (G, ω). ˆ so that !Tˆχ, g" = !χ, T −1 g". Consider the dual automorphism Tˆ of G, ˆ Then Tˆ is a homeomorphism of the compact space G. Theorem 11.2.1. For the automorphism αT of C ∗ (G, ω) we have 1 htop (Tˆ) ≤ ht(αT ) ≤ htop (Tˆ). 2

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11 Systems of Algebraic Origin

ˆ deﬁned by Tˆ, so that If ω ≡ 1 then αT is the automorphism of C(G) ht(αT ) = htop (Tˆ). Thus the upper bound in the theorem is optimal. In Example 11.2.7 below we shall see that the lower bound is also optimal. Proof of the lower bound in Theorem 11.2.1. Let ω ¯ be the cocycle deﬁned by ω ¯ (g, h) = ω(g, h). The elements ug,ω ⊗ug,¯ω , g ∈ G, generate a C∗ -subalgebra D ˆ Under this isoof C ∗ (G, ω)⊗C ∗ (G, ω ¯ ), which is isomorphic to C ∗ (G) ∼ = C(G). ˆ morphism the automorphism αT,ω ⊗αT,¯ω becomes the automorphism of C(G) deﬁned by Tˆ. Using that the topological entropy is subadditive and monotone we then get ht(αT,ω ) + ht(αT,¯ω ) ≥ ht(αT,ω ⊗ αT,¯ω ) ≥ ht((αT,ω ⊗ αT,¯ω )|D ) = htop (Tˆ). Thus we just have to prove that ht(αT,ω ) = ht(αT,¯ω ). ¯ ) can Assume ﬁrst that ω(g, h) = ω(h, g) for any g, h ∈ G. Then C ∗ (G, ω be identiﬁed with the opposite algebra C ∗ (G, ω)op , since for any g, h ∈ G we have uh,¯ω ug,¯ω = ω ¯ (h, g)ug+h,¯ω = ω(g, h)ug+h,¯ω . Hence it suﬃces to prove that if α is an automorphism of a C∗ -algebra and we consider it as an auop tomorphism αop of the opposite algebra, then ht(α) ). By A.1 a n= ht(α ∗ map γ: A → B is completely positive if and only if i,j bi γ(a∗i aj )bj ≥ 0 for any n and a1 , . . . , an ∈ A, b1 , . . . , bn ∈ B. It follows that any completely positive map A → B remains completely positive being considered as a map Aop → B op . Consequently, if Ω is a ﬁnite subset of a C∗ -algebra A, then rcpA (Ω, δ) = rcpAop (Ω, δ). Hence ht(α) = ht(αop ). In the general case consider the cocycle ω op (g, h) = ω(h, g), so that ∗ C (G, ω op ) can be identiﬁed with C ∗ (G, ω)op . Then it is enough to show that ht(αT,ωop ) = ht(αT,¯ω ). Since ρωop = ρω¯ , by Lemma 11.1.3 the cocycles ω op and ω ¯ are cohomologous, so that there exists a function G g → zg ∈ T such that ω ¯ (g, h) = ω op (g, h)zg zh z¯g+h . ¯ ) → C ∗ (G, ω op ) deﬁned by ug,¯ω → We thus get an isomorphism γ: C ∗ (G, ω zg ug,ωop . Even though this isomorphism is not equivariant, the elements αT,¯ω (ug,¯ω ) and (γ −1 ◦αT,ωop ◦γ)(ug,¯ω ) diﬀer only by a scalar factor of modulus one. It follows that for any ﬁnite set Ω consisting of elements ug,¯ω we have ht(Ω, δ; αT,¯ω ) = ht(Ω, δ; γ −1 ◦ αT,ωop ◦ γ). Hence ht(αT,¯ω ) = ht(γ −1 ◦ αT,ωop ◦ γ) = ht(αT,ωop ), and the proof of the lower bound for ht(αT ) is complete.

To obtain the upper bound we need a way to compute htop (Tˆ) by looking at the action of T on G. For a ﬁnite set F ⊂ G let h(F ; T ) = lim

n→∞

1 log |F + T (F ) + . . . + T n−1 (F )|. n

Then deﬁne h(T ) = supF ⊂G h(F ; T ).

11.2 Estimates of Topological Entropy

189

Lemma 11.2.2. We have ht(αT ) ≤ h(T ). Proof. To estimate ht(αT ) we shall use the maps Φf and Ψf from Chap. 8. Let X be a ﬁnite subset of G. Fix δ > 0. Since G is amenable, there exists a ﬁnitely supported positive function f on G such that g f − f 1 < δ 2 for g ∈ X, where g f (h) = f (h − g), and f 1 = 1. Let F be the support of f . For n ∈ N consider the function f (n) = f ∗ (f ◦ T −1 ) ∗ . . . ∗ (f ◦ T −n+1 ). Notice that for any h ∈ G and 0 ≤ k ≤ n − 1 we can write h (f

(n)

) = f ∗ . . . ∗ (f ◦ T −k+1 ) ∗ h (f ◦ T −k ) ∗ (f ◦ T −k−1 ) ∗ . . . ∗ (f ◦ T −n+1 ).

Then for g ∈ X and 0 ≤ k ≤ n − 1 we have T k g (f (n) ) − f (n) 1 ≤ f 1 . . . T k g (f ◦ T −k ) − f ◦ T −k 1 . . . f ◦ T −n+1 1 = (g f ) ◦ T −k − f ◦ T −k 1 = g f − f 1 < δ 2 . Let Ω = {ug | g ∈ X}. Using the maps Φf (n) and Ψf (n) from Chap. 8 we then get by Proposition 8.1.3(iii) that rcp(Ω ∪ αT (Ω) ∪ . . . ∪ αTn−1 (Ω), δ) ≤ |supp f (n) |. Since the support of f (n) is contained in F + T (F ) + . . . + T n−1 (F ), we thus obtain ht(Ω, δ; αT ) ≤ h(F ; T ). Hence ht(αT ) ≤ h(T ).

To ﬁnish the proof of Theorem 11.2.1 it is now suﬃcient to prove the following. Theorem 11.2.3. We have h(T ) = htop (Tˆ). For this we need yet another formula for the entropy of automorphisms of compact groups. Theorem 11.2.4. Let S be a continuous automorphism of a compact separable group Γ . Then htop (S) = hµ (S) = sup lim sup − V

n→∞

1 log µ(V ∩ S −1 (V ) ∩ . . . ∩ S −n+1 (V )), n

where µ is the Haar measure on Γ and the supremum is taken over all neighbourhoods V of the unit element e ∈ Γ . Proof. Since hµ (S) ≤ htop (S), it suﬃces to prove that htop (S) ≤ sup lim sup − V

n→∞

1 log µ(V ∩S −1 (V )∩. . .∩S −n+1 (V )) ≤ hµ (S). (11.1) n

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11 Systems of Algebraic Origin

Choose a left Γ -invariant metric d on Γ . For a ﬁxed ε > 0 let V = Bε/2 (e), the open ball of radius ε/2 with center at e. If x and y are (n, ε)-separated with respect to S, then the sets x(V ∩S −1 (V )∩. . .∩S −n+1 (V )) and y(V ∩S −1 (V )∩ . . . ∩ S −n+1 (V )) are disjoint. Indeed, otherwise for each k = 0, . . . , n − 1 we could ﬁnd a, b ∈ V such that xS −k (a) = yS −k (b), or equivalently, S k (x)a = S k (y)b. But then d(S k (x), S k (y)) < ε, so that x and y are not (n, ε)-separated. Hence if srn (ε) is the maximal number of (n, ε)-separated points, then srn (ε)µ(V ∩ S −1 (V ) ∩ . . . ∩ S −n+1 (V )) ≤ 1, which implies the ﬁrst inequality in (11.1). To prove the second inequality ﬁx a neighbourhood V of e. Let ε > 0 be such that Bε (e) ⊂ V . Choose a measurable partition ξ = {X1 , . . . , Xm } such that the diameter of each atom is smaller than ε. If x ∈ Xi0 ∩ S −1 (Xi1 ) ∩ . . . ∩ S −n+1 (Xin−1 ), then Xik ⊂ Bε (S k (x)) ⊂ S k (x)V for k = 0, . . . , n − 1, so that Xi0 ∩ S −1 (Xi1 ) ∩ . . . ∩ S −n+1 (Xin−1 ) ⊂ x(V ∩ S −1 (V ) ∩ . . . ∩ S −n+1 (V )) and hence µ(Xi0 ∩ S −1 (Xi1 ) ∩ . . . ∩ S −n+1 (Xin−1 )) ≤ µ(V ∩ S −1 (V ) ∩ . . . ∩ S −n+1 (V )). Taking the logarithm, multiplying by µ(Xi0 ∩ S −1 (Xi1 ) ∩ . . . ∩ S −n+1 (Xin−1 )) and summing over all multi-indices (i0 , . . . , in−1 ) we get Hµ (ξ ∨ S −1 ξ ∨ . . . ∨ S −n+1 ξ) ≥ − log µ(V ∩ S −1 (V ) ∩ . . . ∩ S −n+1 (V )). Dividing by n and letting n → ∞ we get the second inequality in (11.1).

Note that instead of the supremum over all neighbourhoods we can take the limit along a basis of neighbourhoods. Before we return to the computation of the entropy of αT , we shall illustrate this theorem by a computation of the entropy of toral automorphisms. Example 11.2.5. Let T ∈ GLn (Z). Then T deﬁnes an automorphism ST of the n-dimensional torus Tn ∼ = Rn /Zn . Then log |λi |, htop (ST ) = i:|λi |>1

where Sp T = {λ1 , . . . , λn } (counting with multiplicities). To see this consider the quotient map π: Rn → Tn . Let U be a relatively compact neighbourhood of 0 ∈ Rn such that π is injective on U ∪ T (U ). Let V = π(U ). If x ∈ U is such that π(T (x)) = ST (π(x)) ∈ V , then T (x) ∈ U . It follows that V ∩ ST−1 (V ) ∩ . . . ∩ ST−n+1 (V ) = π(U ∩ T −1 (U ) ∩ . . . ∩ T −n+1 (U )). Hence

11.2 Estimates of Topological Entropy

htop (ST ) = lim lim sup − U

n→∞

191

1 log µ(U ∩ T −1 (U ) ∩ . . . ∩ T −n+1 (U )), n

where µ is now the Lebesgue measure on Rn . Since for any two relatively compact neighbourhoods U1 , U2 of 0 ∈ Rn we have c−1 U1 ⊂ U2 ⊂ cU1 for some c > 0, the lim sup above is actually the same for all U ’s. Replacing Rn by Cn , which simply doubles the lim sup, we may assume that T is an operator on a complex vector space. By considering the normal Jordan form of T , we can reduce the computation to the case when T ∈ GLm (C) has only one eigenvalue λ. If we write T in the normal Jordan form then the matrix ST S −1 can be made arbitrarily close to the matrix λ1 by a suitable choice of a diagonal matrix S. In other words, for any ε > 0, we can introduce a norm on Cm such that T ±1 − λ±1 1 < ε. Take the unit ball in this norm for U . Then T −k (U ) is contained in the ball of radius (|λ|−1 + ε)k and contains the ball of radius (|λ| + ε)−k . So if |λ| + ε ≤ 1 then U ⊂ T −1 (U ) and the lim sup is zero, and if |λ| + ε ≥ 1 then it lies between −2m log(|λ|−1 + ε) and 2m log(|λ| + ε). It follows that the lim sup is zero when |λ| < 1, and it is 2m log |λ| when |λ| ≥ 1. We can now prove one half of Theorem 11.2.3. Lemma 11.2.6. We have h(T ) ≤ htop (Tˆ). Proof. It is clear that h(F ; T ) is nondecreasing in F . Thus in proving that h(F ; T ) ≤ htop (Tˆ) we may assume that F = −F and 0 ∈ F . For m ∈ N set F (m) = F + . . . + F . m

By replacing, if necessary, F by a larger set we may also assume that |F (m) | → 1 as m → ∞. |F (m+1) |

(11.2)

Indeed, consider the group generated by F . It is a ﬁnite product G1 × . . . × Gn of cyclic groups. We may then assume that F is of the form F1 × . . . × Fn . Thus it suﬃces to prove the claim for cyclic groups. But then it is obvious: if the group is ﬁnite, replace F by the group itself, and if the group is isomorphic to Z, take a suﬃciently large symmetric interval. We assert now that for any n, m ∈ N we have |F + T (F ) + . . . + T n−1 (F )| |F (m+1) |2n , (1F (m) ∗ 1F (m) ∗ 1T (F (m) ) ∗ 1T (F (m) ) ∗ . . . ∗ 1T n−1 (F (m) ) ∗ 1T n−1 (F (m) ) )(0) (11.3) where 1X denotes the characteristic function of a set X ⊂ G. Indeed, denote by N the denominator in the above expression. It is exactly the number of ≤

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11 Systems of Algebraic Origin

elements (g1 , . . . , g2n ) ∈ F (m) × F (m) × . . . × T n−1 (F (m) ) × T n−1 (F (m) ) such that g1 + . . . + g2n = 0. For every such (g1 , . . . , g2n ) and an element h = h1 + . . . + hn ∈ F + T (F ) + . . . + T n−1 (F ) consider the element (h1 + g1 , g2 , . . . , hn + g2n−1 , g2n ) of F (m+1) ×F (m+1) ×. . .×T n−1 (F (m+1) )×T n−1 (F (m+1) ). This way we get N diﬀerent elements representing h, which proves (11.3). Fix m and consider the function f = |F (m) |−2 1F (m) ∗ 1F (m) on G. Then taking the logarithm of the both sides of (11.3) and dividing by n we get 1 log |F + T (F ) + . . . + T n−1 (F )| n |F (m+1) | 1 . (11.4) log(f ∗ (f ◦ T −1 ) ∗ . . . ∗ (f ◦ T −n+1 ))(0) + 2 log n |F (m) | Denote by ϕ the Fourier transform of f , so ϕ(χ) = g !χ, g"f (g). Then, (m) ˆ such that = −F (m) , ϕ is a positive continuous function on G since F ϕ(e) = g f (g) = 1. Fix ε > 0 and put ≤−

ˆ | ϕ(χ) > 1 − ε}. V = {χ ∈ G ˆ and Then V is an open neighbourhood of the unit e ∈ G, (f ∗ (f ◦ T −1 ) ∗ . . . ∗ (f ◦ T −n+1 ))(0) =

ˆ G

ϕ(χ)(ϕ ◦ Tˆ−1 )(χ) . . . (ϕ ◦ Tˆ−n+1 )(χ)dµ(χ) ≥ (1 − ε)n µ

n−1 *

k ˆ T (V ) .

k=0

This and (11.4) then imply that 1 log |F + T (F ) + . . . + T n−1 (F )| n n−1 * 1 |F (m+1) | k Tˆ (V ) − log(1 − ε) + 2 log , ≤ − log µ n |F (m) | k=0 whence by Theorem 11.2.4 |F (m+1) | . h(F ; T ) ≤ htop (Tˆ) − log(1 − ε) + 2 log |F (m) | In view of (11.2) and since ε could be taken arbitrarily small, we get h(F ; T ) ≤

htop (Tˆ). By virtue of Lemma 11.2.2 the previous lemma completes the proof of Theorem 11.2.1.

11.2 Estimates of Topological Entropy

193

Proof of Theorem 11.2.3. By Lemmas 11.2.2 and 11.2.6 we have ht(αT ) ≤ h(T ) ≤ htop (Tˆ). ˆ deﬁned by Tˆ, so that But if ω ≡ 1 then αT is the automorphism of C(G) ht(αT ) = htop (Tˆ). Hence h(T ) = htop (Tˆ).

Examples of nontrivial invariant cocycles can be obtained as follows. Let R = Z[t, t−1 ] be the ring of Laurent polynomials over Z. For g ∈ R deﬁne g˜(t) = g(t−1 ). Let f ∈ Z[t], f = 1, be a polynomial with nonzero constant term and no cyclotomic factors, that is, its roots are not roots of unity. Consider the abelian group G = R/(f˜) ⊕ R/(f ). Let T be the automorphism of G of multiplication by t ∈ R. Then for any character χ of the abelian group R/(f ) we have a pairing (g1 , g2 ) → χ(˜ g1 g2 )

(11.5)

of R/(f˜) with R/(f ), and a T -invariant bicharacter ω on G deﬁned by ˜ 1 g2 ). ω((g1 , g2 ), (h1 , h2 )) = χ(h The corresponding bicharacter ρω is then ˜ 1 g2 − g˜1 h2 ). ρω ((g1 , g2 ), (h1 , h2 )) = χ(h This bicharacter is nondegenerate if and only if the pairing (11.5) is nonde˜2 g˜1 , it suﬃces to check nondegeneracy in the second generate. Since g˜1 g2 = g˜ ), which is dual to the automorvariable. Let S be the automorphism of R/(f phism of multiplication by t. Then the right hand side of (11.5) can be written as χ(g1 (T −1 )g2 ) = !g1 (S)χ, g2 ". Hence (11.5) is nondegenerate if and only if ). By Lemma 11.3.2 the group generated by the S-orbit of χ is dense in R/(f below and our assumption on f , the automorphism S is ergodic, so the orbit of χ is dense for almost every χ. Note that if the leading coeﬃcient and the constant term of f are equal to 1, then G ∼ = Z2n , where n is the degree of f . Then by Example 11.2.5, if λ1 , . . . , λn are the roots of f , we have 1 ˆ log |λj | = 2 log |f (e2πis )|ds, htop (T ) = 2 j:|λj |>1

0

1 where the second equality follows from the fact that 0 log |λ − e2πis |ds is zero 1 if |λ| ≤ 1, and it is log |λ| if |λ| ≥ 1. The number m(f ) = 0 log |f (e2πis )|ds is called the logarithmic Mahler measure of f , and the formula htop (S) = m(f ) holds in fact for arbitrary f . It follows from Theorem 11.2.1 that

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11 Systems of Algebraic Origin

m(f ) ≤ ht(αT ) ≤ 2m(f ). Note also that if we denote R/(f˜) by G1 and R/(f ) by G2 , then the ˆ 1 . Using this homomorphism pairing (11.5) deﬁnes a homomorphism G2 → G ˆ 1 . Then C ∗ (G, ω) ∼ we can deﬁne the action by right translations of G2 on G = ∗ ˆ ˆ C(G1 ) G2 . In particular C(G1 ) is a subalgebra of C (G, ω), which gives an alternative way of obtaining the lower bound for ht(αT ). Example 11.2.7. Let f = 2. Then G = R/(2) ⊕ R/(2) ∼ = ⊕ (Z/2Z ⊕ Z/2Z), n∈Z

T is the shift to the right. We have htop (Tˆ) = 2 log 2, so that log 2 ≤ ht(αT ) ≤ 2 log 2. A character on ⊕n∈Z Z/2Z is just a sequence χ = (χn )n of numbers χn = ±1. Consider the simplest nontrivial character, namely, χ0 = −1 and χn = 1 for n = 0. In other words, if we consider χ as a character on R = Z[t, t−1 ] then χ(1) = −1 and χ(tn ) = 1 for n ∈ Z\{0}. Then the pairing (11.5) is $ ⊕ (Z/2Z ⊕ Z/2Z) (g1 , g2 ) → (−1)g1 (n)g2 (n) . n∈Z

n∈Z

It follows that not only the bicharacter ρω is nondegenerate, but its restriction to the subgroup n G(n) = ⊕ (Z/2Z ⊕ Z/2Z) ⊂ G i=1

is nondegenerate for any n ∈ N. It follows by Theorem 11.1.1 that the C∗ subalgebra C ∗ (G(n) , ω|G(n) ) of C ∗ (G, ω) is a simple C∗ -algebra of dimension 4n , hence it is isomorphic to Mat2n (C). But then ht(αT ) ≤ lim sup n→∞

1 log rank C ∗ (G(n) , ω|G(n) ) = log 2. n

Thus ht(αT ) = log 2. This example shows that the lower bound in Theorem 11.2.1 is optimal.

11.3 K-systems Keeping the notation of the previous section, it is known that if the dual automorphism Tˆ is ergodic then Tˆ, considered as a transformation preserving the Haar measure, is a Bernoulli shift, see Notes at the end of the chapter. Although we cannot prove such a result for the automorphism αT of C ∗ (G, ω), we shall show that if the cocycle ω is in a sense asymptotically trivial then the powers of αT can be approximated by shifts on UHF-algebras. As we shall see,

11.3 K-systems

195

this implies that we get an entropic K-system, in particular hτ (αT ) > 0. Notice that since by Theorem 11.2.4 the topological entropy of Tˆ coincides with the ˆ Theorem 11.2.1 measure entropy with respect to the Haar measure µ on G, implies that hτ (αT ) ≤ hµ (Tˆ) = htop (Tˆ). However, since the dynamical entropy is superadditive we can not obtain a lower bound for hτ (αT ) analogous to that in Theorem 11.2.1. We start with the following classical characterization of ergodicity. Extend αT to W ∗ (G, ω) = πτ (C ∗ (G, ω)) and continue to denote the extended automorphism by αT . Lemma 11.3.1. The following conditions are equivalent: (i) the automorphism αT of W ∗ (G, ω) is ergodic; (ii) the automorphism T of G is aperiodic, that is, the orbit of any nonzero element is inﬁnite. Proof. If T is not aperiodic, thenthere exist n ∈ N and g =

0 such that n−1 T n g = g. Then the element x = k=0 αTk (ug ) of W ∗ (G, ω) is αT -invariant and not a scalar, since the vector xδ0 =

n−1

ω(0, 0)δT k g ∈ 2 (G)

k=0

is nonzero and orthogonal to δ0 . Thus αT is not ergodic. Conversely, assume that T is aperiodic. The automorphism αT is implemented by the unitary U on 2 (G) deﬁned by U δg = δT g . To prove ergodicity of αT it suﬃces to show that if U ξ = ξ then ξ is a scalar multiple of the trace vector δ0 . This is indeed the case, since if ξ = g ag δg then aT n g = ag for every n ∈ Z and g ∈ G, which is possible only when ag = 0 for every g = 0.

In particular, Tˆ is ergodic if and only if T is aperiodic. As we already mentioned, then Tˆ is a Bernoulli shift. At the level of G the Bernoullicity manifests itself in the following property of T . Proposition 11.3.2. Let G be a discrete abelian group, T an aperiodic endomorphism of G, Y a ﬁnite subset of G containing 0 ∈ G. Then there exists m ∈ N such that whenever y1 , . . . , yn ∈ Y and m1 , . . . , mn ∈ N satisfy mk+1 − mk ≥ m, k = 1, . . . , n − 1, then n k=1

only if y1 = . . . = yn = 0.

T mk yk = 0

(11.6)

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11 Systems of Algebraic Origin

Proof. First consider the case when G is ﬁnitely generated. Then the periodic part of G is ﬁnite. Since T acts on it aperiodically, the periodic part is trivial, so G ∼ = ZN for some N ∈ N. Then T is deﬁned by a nondegenerate matrix with integral coeﬃcients, which we denote by the same letter T . It is clear that the aperiodicity is equivalent to the fact that T has no roots of unity as eigenvalues. Let P be the spectral projection of T corresponding to the part of the spectrum lying outside the closed unit disc. Thus V0 = Ker P ⊂ CN is the sum of the root spaces of T corresponding to the eigenvalues lying in the closed unit disc, and V1 = P (CN ) is the sum of the root spaces corresponding to the eigenvalues lying outside the closed unit disc. We claim that ZN ∩ V0 = 0. Indeed, if W is the linear span of ZN ∩ V0 ∼ = Zl , then T |W is represented by an l by l matrix with integral coeﬃcients, all of whose eigenvalues lie in the unit disc, hence (if l = 0) on the unit circle, since the determinant is a nonzero integer. But then the eigenvalues are roots of unity by the Dirichlet theorem, see e.g. [178, 10.1.B], which contradicts our assumptions. It follows that W = 0. It follows that (11.6) is equivalent to n

T mk P yk = 0.

(11.7)

k=1

Since the eigenvalues of T |V1 have absolute values greater than 1, writing T |V1 in the normal Jordan form we conclude that there exist δ > 0 and C > 0 such that T −k |V1 ≤ C(1 − δ)k for all k ∈ N. Choose C1 > 0 such that C1−1 ≤ P y ≤ C1 for y ∈ Y \{0}. Now choose m ∈ N such that ∞ CC1 (1 − δ)k < C1−1 . k=m

Then if mk+1 − mk ≥ m, we have n−1 n−1 CC1 (1 − δ)mn −mk < C1−1 . T mk −mn P yk ≤ k=1

k=1

So if (11.7) holds then P yn < C1−1 , and consequently yn = 0 and n−1 mk P yk = 0. Thus we recursively get yn = yn−1 = . . . = y1 = 0. k=1 T Consider now the general case and prove the proposition by induction on |Y |. Let H0 be the group generated by T k (Y ), k = 0, 1, . . . , ∞. Set Hk = T k (H0 ), H∞ = ∩k Hk , Y = Y ∩ H∞ . Suppose Y =Y . Since Y is ﬁnite, there exists l ∈ N such that Y = Y ∩Hl . n We claim that if k=1 T mk yk ∈ H∞ and mk+1 −mk ≥ l, then y1 , . . . , yn ∈ Y . If g ∈ H∞ ⊂ T (H0 ), then g = T h for a unique element h, since Ker T = 0. As g ∈ T k (H0 ), we conclude that h ∈ T k−1 (H0 ) for any k ∈ N, so that

11.3 K-systems

197

h ∈ H∞ . In other words, T deﬁnes an automorphism of H∞ . It follows that n n mk −m1 mk −m1 T y ∈ H . Hence y ∈ H + T yk , so y1 ∈ Y ∩Hl = k ∞ 1 ∞ k=1 k=2 n Y ⊂ H∞ and therefore k=2 T mk yk ∈ H∞ . Continuing this process we get that y1 , . . . , yn ∈ Y . In particular, if (11.6) holds with mk+1 − mk ≥ l, then y1 , . . . , yn ∈ Y . Since |Y | < |Y |, we may then apply the inductive assumption. On the other hand, if Y = Y , then Y ⊂ H1 . Hence there exists l ∈ N such that if H is the group generated by Y, T (Y ), . . . , T l (Y ), then Y ⊂ T (H), so that H ⊂ T (H). Then H is a ﬁnitely generated group, T −1 is a aperiodic endomorphism of H, and (11.6) can be written as well-deﬁned n −1 mn −mk (T ) yk = 0. Thus we can apply the case of a ﬁnitely generated k=1 group to the endomorphism T −1 of H, the elements yn , . . . , y1 ∈ Y and the numbers 0, mn − mn−1 , . . . , mn − m1 .

Using this proposition we shall show that under an additional assumption on the cocycle the powers of αT can be approximated by shifts on UHFalgebras. Theorem 11.3.3. Let T be an aperiodic automorphism of a discrete abelian group G, ω ∈ Z 2 (G; T) a T -invariant cocycle such that ∞

|1 − ρω (T n g, h)| < ∞ for any g, h ∈ G.

n=1

Then for any ﬁnite subset Ω ⊂ C ∗ (G, ω) and ε > 0 there exist a full matrix algebra B and n0 ∈ N such that for any n ≥ n0 there exist trace preserving unital completely positive maps Φ: C ∗ (G, ω) → A = B ⊗Z and Ψ : A → C ∗ (G, ω) such that (i) α ◦ Φ = Φ ◦ αTn and αTn ◦ Ψ = Ψ ◦ α, where α is the shift to the right on A; (ii) Φ(Ω) ⊂ε B0 and (Ψ ◦ Φ)(a) − a < ε for any a ∈ Ω, where B0 ∼ = B is the 0-th factor of A. Before we turn to the proof, let us show that this theorem implies a strong form of positivity of entropy. Recall that according to Deﬁnition 4.3.1 we say that a W∗ -dynamical system (M, ϕ, α) with M = C1 and faithful ϕ is an entropic K-system if hϕ (γ; αn ) → Hϕ (γ) as n → ∞ for any channel γ. Corollary 11.3.4. Under the assumptions of Theorem 11.3.3 the system (W ∗ (G, ω), τ, αT ) is an entropic K-system (assuming G = 0). In particular, hτ (αT ) > 0. Proof. Let γ: C → W ∗ (G, ω) be a channel. By deﬁnition H τ (γ) is the supremum of Hτ (γ; {xi }) over all ﬁnite partitions of unity 1 = i xi in W ∗ (G, ω). Any partition of unity in W ∗ (G, ω) can be approximated in the strong operator topology by a partition of unity in C ∗ (G, ω) (this follows from A.10, since a partition of unity can be considered as a unital completely positive map from

198

11 Systems of Algebraic Origin

an abelian algebra into W ∗ (G, ω)). Thus it suﬃces to consider partitions of unity in C ∗ (G, ω). Fix a partition Ω = {xi }i such that Hτ (γ; {xi }) is δ-close to Hτ (γ) for some δ > 0. Then ﬁx ε > 0 and choose n0 and B according to Theorem 11.3.3. Let n ≥ n0 and let Φ and Ψ be the corresponding maps. Let E: A → B0 be a conditional expectation. Put bi = E(Φ(xi )), and then for k ∈ N let yi1 ...ik = Ψ (bi1 α(bi2 ) . . . αk−1 (bik )). Thus for each k we get a partition of unity in C ∗ (G, ω). Since Ψ is assumed to be trace preserving, and the unique trace on A is the product trace, we have τ (yi1 ...ik ) = τ (yi1 ) . . . τ (yik ). (l)

(l−1)n

Since also yil = Ψ (αl−1 (bil )) = αT see (3.3), we obtain (k−1)n

Hτ (γ, αTn ◦ γ, . . . , αT

(yil ) for l = 1, . . . , k, by deﬁnition,

◦ γ; {yi1 ...ik }) = kHτ (γ; {yi }).

Hence hτ (γ; αTn ) ≥ Hτ (γ; {yi }). Since bi − Φ(xi ) ≤ ε by assumption on Φ, we also have yi − xi ≤ 2ε. Thus we could choose ε such that Hτ (γ; {yi }) is δ-close to Hτ (γ; {xi }). Then hτ (γ; αTn ) is (2δ)-close to Hτ (γ) for any n ≥ n0 . Hence hτ (γ; αTn ) → Hτ (γ) as n → ∞.

Remark that by Theorem 4.3.5 the convergence ρω (T n g, h) → 1 as n → ∞ is a necessary condition for (W ∗ (G, ω), τ, αT ) to be a K-system. Proof of Theorem 11.3.3. Without loss of generality we may assume that Ω consists of the elements ug for g in a ﬁnite subset of G. By Proposition 8.1.3(iii) we can ﬁnd a ﬁnite subset X ⊂ G such that the maps ΦX : : C ∗ (G, ω) → B(2 (X)) and ΨX : B(2 (X)) → C ∗ (G, ω) from Chap. 8 corresponding to the function f = 1X satisfy (ΨX ◦ ΦX )(a) − a

0 such that there exists a trigonometric polynomial f satisfying (i) |f (z)| ≤ 1 for any z ∈ T; (ii) |f (z)| ≤ δ for any z ∈ T such that |z + 1| ≥ ε; (iii) | T f (z)dµ(z)| > δ 1/(N −1) . Let f (z) = |l|<m al z l . Our assumption on g implies that g has inﬁnite N −1 order. Hence by Proposition 11.3.2 there exists M0 such that i=1 li T ki g = 0 for any k1 , . . . , kN −1 such that ki+1 − ki ≥ M0 and (l1 , . . . , lN −1 ) = (0, . . . , 0) such that |li | < m. Assuming that M ≥ M0 choose m1 , . . . , mN −1 satisfying (11.13). Then by the same argument as before we get N −1 n−1 N −1 1 $ −1 f (ρω (T k g, T mi g)) → aN = f (z)dµ(z) . 0 n T i=1 k=0

Hence for inﬁnitely many k’s N −1 $

|f (ρω (T k g, T mi g))| > δ,

i=1

so that |1 + ρω (T g, T can be taken for mN . k

mi

g)| < ε for i = 1, . . . , N − 1. Any such k ≥ mN −1 + M

Finishing our discussion of entropic properties of the automorphisms αT , we shall show that the ergodic zero entropy systems are exactly those which are uniquely ergodic.

11.4 Zero Entropy Systems

203

Theorem 11.4.2. Let T be an aperiodic automorphism of G, ω ∈ Z 2 (G; T) a T -invariant cocycle. Then the following conditions are equivalent: (i) hτ (αT ) > 0; (ii) there exists an αT -invariant state diﬀerent from τ . Proof. Let ϕ be an αT -invariant state, ϕ = τ . Using ϕ we shall construct a nontrivial stationary coupling with a Bernoulli shift. Consider the positive ˆ deﬁned by unital map P : C ∗ (G, ω) → C(G) ˆ P (x)(χ) = ϕ(αχ (x)) for x ∈ C ∗ (G, ω), χ ∈ G, where αχ is the dual automorphism, so αχ (ug ) = !χ, g"ug . Since τ (·)1 = ˆ ˆ αχ (·)dµ(χ), where µ is the Haar measure on G, we have µ ◦ P = τ . G ˆ Consider the automorphism αTˆ of C(G) deﬁned by the dual automorphism Tˆ. Then αTˆ (P (a))(χ) = P (a)(Tˆ−1 χ) = ϕ(αTˆ−1 χ (a)) = ϕ((αT−1 ◦ αχ ◦ αT )(a)) = ϕ((αχ ◦ αT )(a)) = P (αT (a))(χ), so that αTˆ ◦ P = P ◦ αT . Since ϕ = τ , there exists g = 0 such that ϕ(ug ) = 0. Then P (ug ) = 0, so that P = τ (·)1. Fix a self-adjoint element a ∈ C ∗ (G, ω) such that P (a) = τ (a)1. Let ε = P (a) − τ (a)1. By Theorem 11.3.3 applied to the trivial cocycle there exist n ∈ N, a full matrix algebra B and a unital completely ˆ → A = B ⊗Z such that α ◦ Φ = Φ ◦ αn , where α is the positive map Φ: C(G) Tˆ shift to the right on A, ψ ◦ Φ = µ, where ψ is the unique tracial state on A, Φ(P (a)) − a0 < ε/4 for some self-adjoint element a0 in the 0-th factor B0 of A, and Φ(P (a)) − τ (a)1 ≥ ε/2. Let C be an abelian C∗ -subalgebra of B0 containing a0 . Put D = C ⊗Z , an let E: A → D be the ψ-preserving conditional expectation. Consider the map Θ = E ◦ Φ ◦ P : C ∗ (G, ω) → D. Then ψ ◦ Θ = τ and α ◦ Θ = Θ ◦ αTn . Moreover, Θ(a) = τ (a)1, since otherwise in view of Φ(P (a)) − a0 < ε/4 we get τ (a)1 − a0 < ε/4, so that Φ(P (a)) − τ (a)1 < ε/2, which is a contradiction. It follows that the formula λ(x ⊗ y) = ψ(Θ(x)y) deﬁnes a nontrivial stationary coupling of (C ∗ (G, ω), τ, αTn ) with the classical Bernoulli shift (D, ψ|D , α|D ). By Proposition 5.1.2 we get hτ (αTn ) > 0, and hence hτ (αT ) > 0. Note that we could make the proof a bit shorter if we used the fact that ˆ µ, Tˆ) is already a Bernoulli shift. the system (G, Conversely, let hτ (αT ) > 0. Then there exists a nontrivial coupling with an abelian system. Thus there exists an abelian C∗ -algebra B, an automorphism β of B, an (αT ⊗ β)-invariant state λ on C ∗ (G, ω) ⊗ B and a nonzero element g ∈ G such that λ(ug ⊗ · ) = 0. Then there exists a unitary v ∈ B such

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11 Systems of Algebraic Origin

that λ(ug ⊗ v) = 0. Multiplying v by a scalar we may also assume that λ(ug ⊗ (v + v ∗ )) = 0. By Proposition 11.3.2 applied to the set {±2g, ±g, 0} there exists n ∈ N such that if N

ai T in g = 0

i=1

for some N ∈ N and a1 , . . . , aN ∈ {−2, −1, 0, 1, 2}, then ai g = 0 for all i. We claim that there exists a unital completely positive map Φ: C ∗ (G, ω) → C ∗ (G, ω) ⊗ B such that Φ ◦ αTn = (αTn ⊗ β n ) ◦ Φ and ⎧ 1 ⎪ ⎨ ug ⊗ v, if 2g = 0, 2 Φ(ug ) = 1 ⎪ ⎩ ug ⊗ (v + v ∗ ), if 2g = 0. 2 Assuming that the claim is proved we then get an αTn -invariant state ψ = λ◦Φ such that ψ(ug ) = 0, so ψ = τ . Consider ϕ=

n−1 1 ψ ◦ αTk . n k=0

Then ϕ is an αT -invariant state. If ϕ = τ , then ψ ≤ nτ and hence ψ extends to a normal αTn -invariant state on W ∗ (G, ω). Since T n is aperiodic, the automorphism αTn of W ∗ (G, ω) is ergodic by Lemma 11.3.1. Hence ψ = τ , which is a contradiction. Thus ϕ = τ . It remains to construct the map Φ. For k, l ∈ Z such that k < l and ε = (εk , . . . , εl ) ∈ {0, 1}l−k+1 set u(ε) = uεTkkn g . . . uεTlln g and v(ε) = β kn (v εk ) . . . β ln (v εl ). Consider the element Tk,l ∈ C ∗ (G, ω) ⊗ C ∗ (G, ω) ⊗ B deﬁned by Tk,l = 2−(l−k+1)/2 u(ε)∗ ⊗ u(ε) ⊗ v(ε), εk ,...,εl =0,1

and deﬁne a completely positive map Φk,l : C ∗ (G, ω) → C ∗ (G, ω) ⊗ B by ∗ Φk,l (a) = (τ ⊗ id ⊗ id)(Tk,l (a ⊗ 1 ⊗ 1)Tk,l ) ∗ −(l−k+1) τ (au(ε )u(ε) )u(ε)u(ε )∗ ⊗ v(ε)v(ε )∗ . =2 ε,ε

l If a = uh then τ (au(ε )u(ε)∗ ) = 0 unless h = i=k (εi − εi )T in g. In the latter case u(ε)u(ε )∗ = zuh for a scalar z ∈ T, so that u(ε )u(ε)∗ = z¯u∗h and thus τ (uh u(ε )u(ε)∗ )u(ε)u(ε )∗ = uh and v(ε)v(ε )∗ =

l $ i=k

β in (v εi −εi ).

11.4 Zero Entropy Systems

Therefore

⎛ Φk,l (uh ) = 2−(l−k+1) uh ⊗ ⎝

l $ ε,ε

205

⎞ β in (v

εi −εi

)⎠ ,

i=k

l where the summation is over all ε and ε such that h = i=k (εi − εi )T in g. By ourchoice of n, if 2g = 0, a representation of an element h in the l form h = i=k ai T in g, ai ∈ {−1, 0, 1}, is unique if it exists. In this case the numbers εi and εi in {0, 1} such that εi − εi = ai are uniquely determined if ai = 0, and we have exactly two choices εi = εi = 0 and εi = εi = 1 for them if ai = 0. Hence l $ −N in ai Φk,l (uh ) = 2 uh ⊗ β (v ) , i=k

where N is the number of i’s such that ai = 0. If 2g = 0, then the numbers ai are determined only up to a sign, and we similarly get ⎛ ⎞ $ Φk,l (uh ) = 2−N uh ⊗ ⎝ β in (v + v −1 )⎠ . i:ai =0

l If no representation of h in the form i=k ai T in g exists, then Φk,l (uh ) = 0. In particular, we see that Φk,l is a unital map, and for any ﬁxed h ∈ G the image Φk,l (uh ) of uh is the same for all suﬃciently small k ∈ Z and suﬃciently large l ∈ Z. Since also (αTn ⊗ β n ) ◦ Φk,l = Φk+1,l+1 ◦ αTn , letting k → −∞ and l → +∞ we get the required map Φ.

Remark 11.4.3. If the assumptions of Theorem 11.3.3 are satisﬁed, invariant states diﬀerent from the trace can be obtained from product-states on UHFalgebras. Such states can be thought of as noncommutative analogues of Riesz products. Explicitly, let g ∈ G, and assume 2g = 0. Consider the map Φ from Theorem 11.3.3 corresponding to the set X = {0, g}. On B(2 (X))⊗Z consider the product state deﬁned by the density matrix 1/2 ω(0, 0)/2 . ω(0, 0)/2 1/2 Let ψ be the composition of this state with the map Φ. Using (11.10) and that ΦX (ug ) = ω(0, 0)eg0 we then get ∞ 1 $ 1 ρω (T kn g, g) . + ψ(ug ) = 2 2 2 k=1

So if n is suﬃciently large, ψ is an αTn -invariant state diﬀerent from the trace.

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11 Systems of Algebraic Origin

Using the method of the proof of the previous theorem the same state can be obtained as the limit of the states ψk,l (a) = 2−(l−k+1) τ (u(ε)au(ε )∗ ). εk ,...,εl =0,1 ε ,...,ε =0,1 k l

In fact, the map Φ from the previous theorem can be obtained as the composition of the map Φ and a modiﬁcation of the map Ψ from Theorem 11.3.3. Although those maps were constructed under additional assumptions on the cocycle, their composition makes sense in general.

11.5 Automorphisms of Noncommutative Tori Let G = Z2 . Consider the Z-valued skew-symmetric form σ deﬁned by σ(g, h) = g1 h2 − g2 h1 . For θ ∈ R set ωθ (g, h) = eπiθσ(g,h) . The algebra Aθ = C ∗ (Z2 , ωθ ) is called the irrational rotation algebra. Since σ is nondegenerate, by Theorem 11.1.1 the algebra Aθ is simple if θ is irrational. Denote by uθ and vθ the unitaries in C ∗ (Z2 , ωθ ) corresponding to the elements (1, 0) and (0, 1), respectively. Then uθ vθ = e2πiθ vθ uθ . One can alternatively deﬁne Aθ as the universal C∗ -algebra generated by unitaries uθ and vθ satisfying the above relation. Any matrix T ∈ SL2 (Z) preserves the form σ and hence deﬁnes an automorphism of Aθ , which we denote by αT,θ . We also denote by ρθ the bicharacter deﬁned by ωθ , and by τθ the canonical trace on Aθ . Let {λ, λ−1 } be the spectrum of T . Assume |λ| > 1. Then it is known that λ is irrational (otherwise λ, being a root of a monic polynomial with integral coeﬃcients, must be an integer, which is impossible as λ + λ−1 = Tr(T ) is an integer). For any n ∈ Z we have Tn =

1 λn λ−n n −1 −n (λ −T )+λ (T −λ)) = (1−λT )+ (λ (T −λ), λ−1 − λ 1 − λ2 λ−1 − λ

as can be checked on the eigenvectors of T . Hence for any g, h ∈ Z2 there exists C > 0 such that for any n ∈ N n σ(T n g, h) − λ (σ(g, h) − λσ(T g, h)) ≤ C|λ|−n . (11.14) 2 1−λ We need the following classical result of Weyl. Lemma 11.5.1. For any real number λ, |λ| > 1, the sequence {eiθλ }∞ n=1 is uniformly distributed on T for almost all θ ∈ R. n

11.5 Automorphisms of Noncommutative Tori

207

Proof. Fix m ∈ Z, m = 0. For n ∈ N put 1 imθλk . e n n

Vn (θ) =

k=1

We have to prove that Vn (θ) → 0 as n → ∞ for almost all θ ∈ R. Let a, b ∈ R, a < b. We have b b n k j 1 2 |Vn (θ)| dθ = 2 eimθ(λ −λ ) dθ n a a k,j=1

=

k j k j 1 eimb(λ −λ ) − eima(λ −λ ) b−a + 2 . n n im(λk − λj )

k=j

Since |λk − λj | ≥ |λ|k+1 − |λ|k for j > k, k=j

n−1 n−1 n 1 1 1 2n ≤ 2n ≤ . ≤ 2 k j k+1 k k+1 k |λ − λ | |λ| − |λ| |λ| − |λ| (|λ| − 1)2 k=1 j=k+1

k=1

b

Thus a |Vn (θ)|2 dθ ≤ C/n for a constant C depending on λ, m, a and b. In particular, b ∞ |Vk2 (θ)|2 dθ < ∞. a k=1

It follows that Vk2 (θ) → 0 as k → ∞ for almost all θ ∈ [a, b]. If this convergence holds for some θ, then for k 2 ≤ n < (k + 1)2 we have 2 (k+1) (k+1)2 l l 1 eimθλ − eimθλ |Vn (θ)| = n l=1

≤

l=n+1

(k + 1) (k + 1)2 − n |V(k+1)2 (θ)| + . n n

Hence Vn (θ) → 0 as n → ∞.

2

We are now in position to apply the results of the previous sections to the automorphisms of noncommutative tori. Theorem 11.5.2. For any T ∈ SL2 (Z) with Sp T = {λ, λ−1 } and |λ| > 1 we have: (i) 12 log |λ| ≤ ht(αT,θ ) ≤ log |λ| for any θ ∈ R; (ii) if θ ∈ Q then hτθ (αT,θ ) = ht(αT,θ ) = log |λ|; (iii) if θ lies in the additive group generated by 1 and λm (1 − λ2 ), m ∈ Z, then (πτθ (Aθ ) , τθ , αT,θ ) is an entropic K-system; in particular, hτθ (αT,θ ) > 0 for any θ ∈ Q(λ) = Qλ + Q; (iv) hτθ (αT,θ ) = 0 for almost all θ ∈ R, and for any such θ the trace τθ is the unique αT,θ -invariant state on Aθ .

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11 Systems of Algebraic Origin

Proof. Part (i) follows from Theorem 11.2.1 and Example 11.2.5. For any n ∈ N we have an embedding An2 θ → Aθ given by un2 θ → unθ , vn2 θ → vθn . In particular, if θ is rational then we have an embedding of C(T2 ) into Aθ , so that the entropy hτθ (αT,θ ) is at least as large as in the classical case, where it is log |λ| by Example 11.2.5. This proves (ii). If θ = λm (1 − λ2 ), then by virtue of (11.14) for any g and h we have |θσ(T n g, h) − λn+m σ(g, h) + λn+m+1 σ(T g, h)| ≤ C|λ|m−n . Since λn+m = Tr(T n+m ) − λ−n−m and λn+m+1 = Tr(T n+m+1 ) − λ−n−m−1 , and the numbers Tr(T n+m ) and Tr(T n+m+1 ) are integers, we see that there exist integers mn such that |θσ(T n g, h) − mn | ≤ C0 |λ|−n for some new constant C0 . Hence |1 − ρθ (T n g, h)| = |1 − e2πiθσ(T

n

g,h)

| ≤ 2πC0 |λ|−n .

∞ Therefore n=1 |1 − ρθ (T n g, h)| < ∞. The same is true if we assume that θ lies in the group H generated by 1 and λm (1−λ2 ), m ∈ Z. By Corollary 11.3.4 we then conclude that (πτθ (Aθ ) , τθ , αT,θ ) is an entropic K-system. Since λ ∈ Qλ2 + Q and 1, λ2 ∈ H, for any θ ∈ Q(λ) we can ﬁnd n ∈ N such that n2 θ ∈ H. Then, since An2 θ embeds into Aθ , hτθ (αT,θ ) ≥ hτn2 θ (αT,n2 θ ) > 0, which completes the proof of (iii). Remark that in fact by Proposition 5.1.8 we have hτθ (αT,θ ) = hτn2 θ (αT,n2 θ ). To prove the ﬁrst statement in (iv) note that if {an }n and {bn }n are two sequences of real numbers such that bn → 0, then {ei(an +bn ) }n is uniformly distributed on T if and only if {eian }n is uniformly distributed. So by (11.14) and Theorem 11.4.1 it suﬃces to check that the sequence {e2πiθλ

n

(1−λ2 )−1 (σ(g,h)−λσ(T g,h)) ∞ }n=1

is uniformly distributed for all g, h = 0 and almost all θ. But this is true by Lemma 11.5.1, since σ(g, h) − λσ(T g, h) = 0. Indeed, since λ is irrational the latter expression is zero only if σ(g, h) = σ(T g, h) = 0. But then both g and T g are scalar multiples of h, which is impossible as T has no rational eigenvalues. The last statement in (iv) follows from Theorem 11.4.2.

Finally we give an example of systems for which the tensor product formula for the entropy fails. Choose any θ such that hτθ (αT,θ ) = 0. Then the same argument as in the proof of the lower bound in Theorem 11.2.1 shows that also hτ−θ (αT,−θ ) = 0, while hτθ ⊗τ−θ (αT,θ ⊗ αT,−θ ) ≥ log |λ|.

11.6 Notes

209

11.6 Notes Theorem 11.1.1 is a result of Slawny [202]. The idea of using the diagonal subalgebra of C ∗ (G, ω) ⊗ C ∗ (G, ω ¯ ) to estimate the entropy of the tensor product automorphism is due to Narnhofer; it was ﬁrst , employed in [128] for binary shifts, which corresponds to the group G = Z Z/2Z, see Chap. 12. As was observed by Kerr and Li [108] in the context of higher dimensional noncommutative tori, it allows one to estimate not only the topological entropy of αT , but also the topological entropy of αχ ◦ αT ˆ We use a similar argument in the proof of the lower bound in for χ ∈ G. Theorem 11.2.1 to show that ht(αT,ω ) depends only on the cohomology class of ω. The upper bound in Theorem 11.2.1 was obtained by Voiculescu [227] for G = Zn , and by Brown and Germain [37] in general. Theorem 11.2.3 is due to Peters [157]. Theorem 11.2.4 was proved by Bowen [29]. The fact that the Haar measure is the measure of maximal entropy for automorphisms of compact groups was established earlier by Berg [20]. The entropy of automorphisms of the two-dimensional torus was computed already by Sinai [199]. For general compact groups it was done by Juzvinskii [97]. For more information on entropic properties of automorphisms (and, more generally, automorphic actions of Zn ) of compact groups we refer the reader to the book by Schmidt [192]. The examples of cocycles at the end of Sect. 11.2 are due to Golodets and Neshveyev [79]. They were used to construct automorphisms α of the hyperﬁnite II1 -factor with a ﬁxed Cartan subalgebra A such that hτ (α) = t and hτ |A (α|A ) = s for any given t and s, 0 ≤ s ≤ t ≤ +∞. In this respect an interesting open problem is to ﬁnd an automorphism of a II1 -factor with entropy larger than the supremum of the entropies of the restrictions of the automorphism to invariant abelian subalgebras. Existence of such an automorphism was announced in [74], but the details have never appeared. The discussion before Example 11.2.7 shows that ,the example can be interpreted as follows. There exist an action of G = Z Z/2Z on X = {0, 1}Z and an automorphism α of C(X) G such that ht(α) = ht(α|C(X) ) = ht(α|C ∗ (G) ) = log 2. In particular, ht(α) < ht(α|C(X) ) + ht(α|C ∗ (G) ). Such an example was promised in Notes to Chap. 8. It was proved by Benatti, Narnhofer and Sewell [17] that for a ﬁxed T ∈ SL2 (Z) the automorphism αT,θ of Aθ can be asymptotically abelian only for a countable number of θ’s. Later Narnhofer observed that the automorphism has indeed a strong form of asymptotic abelianness for certain irrational θ’s in the quadratic ﬁeld generated by the eigenvalues of T , and suggested that we then get an entropic K-system [127]. The proof of the more general result in Sect. 11.3 is an elaboration on the work of Neshveyev [138]. Note that Lemma 11.3.1 is a classical result of Halmos [84], while the argument in the

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11 Systems of Algebraic Origin

proof of Proposition 11.3.2 essentially goes back to Rohlin [182], who proved that ergodic automorphisms of compact abelian group have the K-property. Later, using powerful Ornstein’s machinery, it was proved that any ergodic automorphism of a compact separable group is Bernoullian (here one should allow the Bernoulli shift with inﬁnite entropy). The result was obtained by Katznelson [101] for tori, and by Lind [118] and Miles and Thomas [122] in general. It was Narnhofer who suggested that a suﬃciently chaotic behavior of the cocycle should lead to zero dynamical entropy. Such results were independently obtained by Narnhofer and Thirring [134] for binary shifts and noncommutative tori and by Sauvageot [189] for noncommutative tori. Our exposition in Sect. 11.4 is based on the unpublished work of Sauvageot [189]. Lemma 11.5.1 is a classical result of Weyl [234]. Finally remark that the results of this chapter are valid with minor changes ˆ Some of these automorfor the automorphisms of the form αχ ◦ αT , χ ∈ G. a b ∈ SL2 (Z) phisms may look even more natural than αT . E.g. if T = c d then with an appropriate choice of χ we get the automorphism of Aθ such that uθ → uaθ vθc , vθ → ubθ vθd .

12 Binary Shifts

A rich source of C∗ -dynamical systems is obtained from bitstreams, i.e. sequences of 0’s and 1’s. Given such a sequence we shall construct a C∗ -algebra, which in the interesting cases is the UHF-algebra of type 2∞ , and an automorphism, called a binary shift. Such a system is an example of systems considered in Chap. 11. Diﬀerent bitstreams can give rise to quite diﬀerent C∗ -dynamical systems, and in this chapter we shall study their entropic properties.

12.1 The C*-algebra of a Bitstream With every bitstream {εn }∞ n=1 , where εn ∈ {0, 1}, we associate a subset X of N by X = {n ∈ N | εn = 1}. Conversely each subset X ⊂ N gives rise to a bitstream in this way. Let , Z/2Z. Denote by gi the element 1 in the i-th summand (Z/2Z)i G = Z of G. Deﬁne a bicharacter ω on G by −1, if j − i ∈ X, ω(gi , gj ) = +1, otherwise, so that

⎞ ⎛ gi , gj ⎠ = ω⎝ i∈I

j∈J

$

ω(gi , gj ).

i∈I,j∈J

As in Chap. 11 consider the algebra A(X) = C ∗ (G, ω), which is the universal C∗ -algebra generated by unitaries ug , g ∈ G, such that ug uh = ω(g, h)ug+h . Recall that the canonical trace τ is deﬁned by τ (ug ) = 0 for g = 0. Let si = ugi . Then si is a symmetry, that is, a self-adjoint unitary. We have −ugi +gj , if j − i ∈ X, si sj = ω(gi , gj )ugi +gj = ugi +gj , otherwise.

212

12 Binary Shifts

Thus, if χ denotes the characteristic function of X, we have si sj = (−1)χ(|i−j|) sj si . We could alternatively deﬁne A(X) = C ∗ (G, ω) as the universal C∗ -algebra generated by symmetries sn , n ∈ Z, such that the above relation is satisﬁed. Let G[n,m] = ⊕m i=n (Z/2Z)i ⊂ G. We shall write Gn instead of G[1,n] . Let An = C ∗ (Gn , ω|Gn ) ⊂ C ∗ (G, ω). So An is the C∗ -algebra generated by s1 , . . . , sn . Our goal in this section is to determine the structure of the C∗ algebras A(X) and A+ (X) = ∪∞ n=1 An . From Chap. 11 we know that for this we have to study the bicharacter ω(g, h)ω(h, g), which can be written as (−1)B(g,h) , where B: G × G → Z/2Z is the skew-symmetric bilinear form given by 1, if |j − i| ∈ X, B(gi , gj ) = 0, otherwise. We consider G as a vector space over the ﬁeld F2 = Z/2Z. Then gi , i ∈ Z, form a basis. Let Hn denote the kernel of the restriction of B to Gn , so Hn = {h ∈ Gn | B(h, g) = 0 for every g ∈ Gn }. By Lemma 11.1.2 the center Z(An ) of An coincides with C ∗ (Hn , ω|Hn ). Set cn = dimF2 Hn . Thus Z(An ) is an abelian algebra of dimension 2cn . Note that c1 = 1. Denote by T the shift to the right on G, so T (G[n,m] ) = G[n+1,m+1] . The form B is clearly T -invariant. Lemma 12.1.1. For each n ≥ 1 we have one of the following possibilities: (i) cn+1 = cn + 1 and Hn = Gn ∩ Hn+1 = T −1 (Hn+1 ) ∩ Hn+1 , and if in addition Hn = 0 then Hn+1 = Hn + T (Hn ); (ii) cn+1 = cn − 1 and Hn+1 = Hn ∩ T (Hn ), and if in addition Hn+1 = 0 then Hn = Hn+1 + T −1 (Hn+1 ). Proof. Let us ﬁrst show that if V is a ﬁnite dimensional vector space, W a subspace of V of codimension one, and F a skew-symmetric bilinear form on V , then dim Ker F = dim Ker(F |W ) ± 1. Fix a vector e ∈ V \W . Consider the functional F (·, e) on Ker(F |W ). Assume ﬁrst that this functional is nonzero, so there exists h ∈ Ker(F |W ) such that F (h, e) = 0. Then F (h, g + e) = F (h, e) = 0 for every g ∈ W , which implies that Ker F contains no elements in V \W , so that Ker F ⊂ W . But then Ker F is exactly the kernel of the nonzero functional F (·, e) on Ker(F |W ). So Ker F ⊂ Ker(F |W ) and dim Ker F = dim Ker(F |W ) − 1. Assume now that F (h, e) = 0 for every h ∈ Ker(F |W ). Then Ker(F |W ) ⊂ Ker F . Since F deﬁnes a nondegenerate form on W/Ker(F |W ), there exists f ∈ W such that F (g, e) = F (g, f ) for every g ∈ W . It follows that f − e ∈ Ker F (note that F (e, f − e) = F (e, f ) = F (f, f ) = 0). Thus

12.1 The C*-algebra of a Bitstream

213

the inclusion Ker(F |W ) ⊂ Ker F is proper. In particular, dim Ker F ≥ dim Ker(F |W )+1. On the other hand, we always have W ∩Ker F ⊂ Ker(F |W ), so that dim Ker F − 1 ≤ dim Ker(F |W ). Hence dim Ker F = dim Ker(F |W ) + 1 and W ∩ Ker F = Ker(F |W ). We have thus proved that cn+1 = cn ± 1. Assume cn+1 = cn + 1. It follows from our considerations that this implies Hn = Gn ∩ Hn+1 . Applying this to the inclusion T (Gn ) ⊂ Gn+1 instead of Gn ⊂ Gn+1 we can also conclude that T (Hn ) = T (Gn ) ∩ Hn+1 . Hence Hn ⊂ T −1 (Hn+1 ) ∩ Hn+1 . Since T −1 (Hn+1 ) ∩ Hn+1 ⊂ G[1,n] , the opposite inclusion is obvious. In particular, Hn +T (Hn ) ⊂ Hn+1 . If Hn = 0 then Hn is a proper subspace of Hn + T (Hn ), whence Hn+1 = Hn + T (Hn ). Assume now cn+1 = cn − 1. As we have seen, this implies Hn+1 ⊂ Hn . Applying this to T (Gn ) ⊂ Gn+1 we also conclude Hn+1 ⊂ T (Hn ). Hence Hn+1 ⊂ Hn ∩ T (Hn ). The opposite inclusion is obvious. In particular, Hn+1 + T −1 (Hn+1 ) ⊂ Hn . If Hn+1 = 0 then Hn+1 is a proper subspace of Hn+1 + T −1 (Hn+1 ). Hence Hn = Hn+1 + T −1 (Hn+1 ).

From the previous lemma we get further restrictions on the cn ’s. Lemma 12.1.2. If cn−1 = cn+1 = cn + 1 for some n ≥ 2 then cn = 0. Proof. Since cn = cn−1 − 1, by Lemma 12.1.1(ii) we have Hn = Hn−1 ∩ T (Hn−1 ) ⊂ G[2,n−1] . On the other hand, since cn = cn+1 − 1, by Lemma 12.1.1(i) we have Hn Hn+1 . Since we always have T −1 (Hn+1 ) ∩ Gn ⊂ Hn , we get T −1 (Hn ) ∩ Gn Hn . Since T −1 (Hn ) ⊂ G[1,n−2] ⊂ Gn , we see that T −1 (Hn ) ⊂ Hn . This possible only when Hn = 0.

⊂ ⊂ is

It follows that the sequence {cn }∞ n=1 increases by one to some point, then it decreases by one to zero, and so on. We shall next study the case when the sequence only increases after some n. We say a subset X of N is periodic if the subset −X ∪ X of Z is periodic. In terms of the form B this means that there exists m ≥ 1 such that B(T m g, h) = B(g, h) for g, h ∈ G. Otherwise we say X is nonperiodic. Lemma 12.1.3. The following conditions are equivalent: (i) X is periodic; (ii) there exists n ≥ 0 such that cn+k = k for any k ≥ 0. Here we put for convenience G0 = 0 and c0 = 0.

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12 Binary Shifts

Proof of Lemma 12.1.3. If B(T m g, h) = B(g, h) for g, h ∈ G, then g1 + gm+1 is in the kernel of B, so cn ≥ 1 for all n ≥ m + 1. Hence there exists a largest number n ≤ m such that cn = 0, and then cn+k = k for any k ≥ 1 by Lemmas 12.1.1 and 12.1.2. Conversely, assume (ii) is satisﬁed. Then Hn+1 is one-dimensional, and Hn+k is spanned by Hn+1 , T (Hn+1 ), . . . , T k−1 (Hn+1 ) by Lemma 12.1.1(i). In particular, the kernel H∞ of the restriction of B to G∞ = ∪n Gn is spanned by T k (Hn+1 ), k ≥ 0, so that T (H∞ ) ⊂ H∞ . Hence T induces a linear operator T¯ on G∞ /H∞ . Since Hn+k is k-dimensional and has trivial intersection with Gn (otherwise cn = 0), we have Gn+k = Gn ⊕ Hn+k for any k ∈ N. Hence G∞ = Gn ⊕ H∞ . Thus G∞ /H∞ is a ﬁnite set. It follows that there exist k ≥ 0 and m ≥ 1 such that T¯k+m = T¯k . Hence B(T k+m g, h) = B(T k g, h) for any g, h ∈ G∞ . Since B is T -invariant, we conclude that B(T m g, h) = B(g, h) for all g, h ∈ G, so that X is periodic.

Let us now summarize properties of the sequence {An }n . ∞ Theorem 12.1.4. There exist sequences {cn }∞ n=0 and {dn }n=1 of nonnegatiN ve integers and a strictly increasing sequence {nl }l=1 of even integers, where N ∈ N ∪ {∞}, such that ∼ Mat2dn (C) ⊗ C2cn , and the trace τ takes the value 2−cn −dn on every (i) An = minimal projection in An ; (ii) n = cn + 2dn for any n ≥ 1; (iii) n1 = 0, and if ml = (nl+1 − nl )/2 then k, if 0 ≤ k < ml , cnl +k = 2ml − k, if ml ≤ k ≤ 2ml ,

Z(Anl +k ) ⊂ Z(Anl +k+1 ) for 0 ≤ k ≤ ml − 1 and Z(Anl +k ) ⊃ Z(Anl +k+1 ) for ml ≤ k ≤ 2ml − 1. Furthermore, N < ∞ if and only if X is periodic. Proof. Let Gn be a subspace of Gn such that Gn = Gn ⊕ Hn . Since ug uh = uh ug for g ∈ Gn and h ∈ Hn , we have An ∼ = C ∗ (Gn , ω|Gn ) ⊗ C ∗ (Hn , ω|Hn ), and the canonical trace on An is the tensor product of the canonical traces. By Lemma 12.1.1 the numbers n and cn have the same parity, so Gn has even dimension 2dn = n − cn , which also follows from nondegeneracy of the skew-symmetric form B on Gn . By Lemma 11.1.2 the nondegeneracy implies also that C ∗ (Gn , ω|Gn ) is simple, so it is a full matrix algebra of dimension 22dn . Thus C ∗ (Gn , ω|Gn ) ∼ = Mat2dn (C). The algebra Z(An ) = C ∗ (Hn , ω|Hn ) is an abelian C∗ -algebra of dimension 2cn . Moreover, it is a tensor product of cn twisted group C∗ -algebras of the

12.1 The C*-algebra of a Bitstream

215

group Z/2Z. Since the latter algebras have minimal projections of trace 1/2, we conclude that every minimal projection in Z(An ) has trace 2−cn . We have thus proved (i) and (ii). The rest of the theorem follows from Lemmas 12.1.1–12.1.3.

Corollary 12.1.5. Let X ⊂ N. Then Mat2m (C) ⊗ C({0, 1}N ), A+ (X) ∼ = ⊗∞ n=1 Mat2 (C),

if X is periodic, if X in nonperiodic.

The same is true for A(X). Proof. In the notation of Theorem 12.1.4 if X is nonperiodic then N = ∞, so that A2n ∼ = Mat2n (C) for an inﬁnite number of n’s. Hence A+ (X) is the UHF-algebra of type 2∞ . If X is periodic, then there exists n = 2m such that cn+k = k for all k ≥ 0. Then An ∼ = Mat2m (Z) and An+k ∼ = An ⊗ Z(An+k ). Since Z(An+k ) ⊂ Z(An+k+1 ), we get A+ (X) ∼ = An ⊗ (∪k Z(An+k )) ∼ = Mat2m (C) ⊗ C({0, 1}N ). To see that the same is true for A(X), denote by A˜2n the algebra generated by s−n+1 , . . . , sn , and by A˜2n+1 the algebra generated by s−n , . . . , sn . Then ˜ {A˜n }∞ n=1 is an increasing sequence, and A(X) = ∪n An . It is then not diﬃcult to see that all the properties of the sequence {An }n stated in Theorem 12.1.4 remain true for the sequence {A˜n }n . So the same proof as the one for A+ (X) gives the result for A(X).

For the rest of the section we discuss further properties of the sequence {An }n , which are of independent interest, but are not used later. Note ﬁrst that the proof of Lemma 12.1.3 gives a more precise information about the center of A+ (X) for periodic X than that formulated in the above corollary. Namely, let n = 2m be such that cn+k = k for k ≥ 0. Then in the notation of Lemma 12.1.3 the space H∞ is spanned by T k h, k ≥ 0, where h is the unique nonzero element in Hn+1 . Hence the center Z = C ∗ (H∞ , ω|H∞ ) of A+ (X) is isomorphic to C({0, 1}N ) in such a way that the endomorphism α|Z becomes the usual shift endomorphism of C({0, 1}N ). It follows also that the kernel H of B is spanned by T k h, k ∈ Z. Thus the center Z(X) of A(X) is isomorphic to C({0, 1}Z ) in such a way that the automorphism α|Z(X) becomes the shift automorphism of C({0, 1}Z ). Let us call a word in A every nonempty product w = si1 . . . sik with i1 < . . . < ik . Then we can reformulate the above remark as follows. If X is periodic then there exists a unique word w ∈ A+ (X) such that the center Z+ (X) of A+ (X) is generated by αk (w), k ≥ 0, and then the center Z(X) of A(X) is generated by αk (w), k ∈ Z. If w = si1 . . . sik , then i1 = 1 (otherwise α−1 (w) would be in the center), ik = 2m + 1 for some m ≥ 0, and A+ (X) ∼ = Mat2m (C) ⊗ Z+ (X) and A(X) ∼ = Mat2m (C) ⊗ Z(X).

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12 Binary Shifts

The sequences {cn }n and {dn }n in Theorem 12.1.4 are determined by {nl }l . Any sequence {nl }l can arise this way (assuming that it is strictly increasing, consists of even numbers, and n1 = 0). In other words, the restrictions on {cn }n given by Lemmas 12.1.1 and 12.1.2 are the only ones. To see this note that H1 , . . . , Hn depend only on X∩[1, n−1]. Hence we can construct X inductively. So let {cn }n be the sequence deﬁned by {nl }N l=1 according to part (iii) of Theorem 12.1.4, and assume we have constructed Xn ⊂ {1, . . . , n − 1} such that for the corresponding form Bn we have dim Hk = ck for k ≤ n. We have to set either Xn+1 = Xn or Xn+1 = Xn ∪ {n} such that for the corresponding form Bn+1 we get dim Hn+1 = cn+1 . If cn = 0, then cn+1 = 1 and dim Hn+1 = 1 independently of how we deﬁne Xn+1 . Similarly, if cn > 0 and cn = cn−1 − 1, then cn+1 = cn − 1 and dim Hn+1 = cn − 1 independently of the deﬁnition. So we may assume that for some m < n we have ck = k − m for m ≤ k ≤ n. Let h ∈ Hm+1 be the unique nonzero element. By Lemma 12.1.1 the space Hn is spanned by h, T h, . . . , T n−m−1 h. Since Hm = 0, we have h = g1 + h for some h ∈ G[2,m+1] . Since Bn+1 (g, gn+1 ) = Bn+1 (T −1 g, gn ) = Bn (T −1 g, gn ) for g ∈ G[2,n+1] , we have T h, . . . , T n−m−1 h ∈ Hn+1 independently of how we deﬁne Xn+1 . On the other hand, since Bn+1 (h, gn+1 ) = Bn+1 (g1 , gn+1 ) + Bn (T −1 h , gn ), we see that we can arrange so that we have both h ∈ Hn+1 or h ∈ / Hn+1 . In the ﬁrst case Hn ⊂ Hn+1 , so that dim Hn+1 = cn + 1 by Lemma 12.1.1. In the second case Hn is not contained in Hn+1 , so that dim Hn+1 = cn − 1. So whatever cn+1 is, we can deﬁne Xn+1 such that dim Hn+1 = cn+1 . The above argument shows also that if N = ∞ then for inﬁnitely many n’s it does not matter how we deﬁne Xn+1 from Xn . Thus there exist uncountably many sets X giving the same sequence {nl }N l=1 . In the next section we shall see that an interesting class of systems is obtained when X is ﬁnite. We ﬁnish this section by showing that in this case the sequence {cn }n is periodic. Proposition 12.1.6. Let X ⊂ N be ﬁnite and nonempty. Then there exists m ∈ N such that cn+m = cn for any n ≥ 0. n Proof. Let d be the largest number in X. If g = j=1 ξj gj , ξj ∈ F2 , then n g ∈ Hn if and only if B(g, gi ) = 0 for i = 1, . . . , n, that is, j=1 χ(|j−i|)ξj = 0, where χ is the characteristic function of X. Letting ξ−d+1 = . . . = ξ0 = ξn+1 = . . . = ξn+d = 0 we can write this condition as ξi−d +

d−1 k=−d+1

χ(|k|)ξi+k + ξi+d = 0

(12.1)

12.2 Entropy of Binary Shifts

217

for i = 1, . . . , n. Consider (12.1) for all i ∈ Z as a system of equations. If ξ−d+1 , . . . , ξn+d satisfy (12.1) for i = 1, . . . , n then all other ξj ’s are uniquely determined. Thus there is a bijection between Hn and the set of solutions ξ = (ξi )i∈Z of (12.1) such that ξ−d+1 = . . . = ξ0 = ξn+1 = . . . = ξn+d = 0. ¯ = ∞ Z/2Z. Let T¯ We consider sequences ξ = (ξi )i∈Z as elements of G −∞ ¯ so (T¯ξ)i = ξi−1 . Then equations (12.1) can be be the shift to the right on G, written as p(T¯)ξ = 0, where p(t) ∈ F2 [t] is the polynomial deﬁned by p(t) = t2d +

d−1

χ(|k|)td−k + 1.

k=−d+1

¯ To summarize, there is an isomorphism between Hn and the space of ξ ∈ G ¯ such that p(T )ξ = 0 and ξ−d+1 = . . . = ξ0 = ξn+1 = . . . = ξn+d = 0. The ring F2 [t]/(p(t)) is ﬁnite. The element t is invertible in this ring, so it is of ﬁnite order. In other words, there exists m ∈ N such that tm + 1 lies in the ideal generated by p(t). But then if p(T¯)ξ = 0 we have T¯m ξ = ξ, that is, ξi+m = ξi . Hence for solutions of p(T¯)ξ = 0 instead of the condition ξ−d+1 = . . . = ξ0 = ξn+1 = . . . = ξn+d = 0 we can equally well consider the condition ξ−d+1 = . . . = ξ0 = ξn+m+1 = . . . = ξn+m+d = 0. But the latter condition determines Hn+m . Thus we get an isomorphism ben+m tween Hn+m and Hn . Explicitly, the isomorphism maps j=1 ξj gj ∈ Hn+m n to j=1 ξj gj ∈ Hn .

In particular, if X is ﬁnite and nonempty, the sequence {cn }n is bounded. If d is the largest number in X, then using that B(g, gn+d ) = 1 for any g ∈ Gn \Gn−1 , we see that if cn−1 = 0 then Hn , which is nonzero and contained in Gn \Gn−1 , is not contained in Hn+d . It follows that the sequence 1 = cn , cn+1 , . . . , cn+d cannot be increasing. Hence ck ≤ d for any k ∈ N.

12.2 Entropy of Binary Shifts In the notation of the previous section let X be a subset of N associated with a bitstream, and let A(X) be the C∗ -algebra generated by symmetries sn , n ∈ Z, satisfying the commutation relations si sj = (−1)χ(|i−j|) sj si , where χ is the characteristic function of X. Let also τX be the canonical tracial state on A(X). The shift sn → sn+1 deﬁnes an automorphism σX of A(X) which is called the binary shift corresponding to X. In the present section we shall study entropic properties of such automorphisms. We start with a simple estimate. Proposition 12.2.1. We have 21 log 2 ≤ ht(σX ) ≤ log 2. If X is nonperiodic and ϕ is a σX -invariant state on A(X) then hϕ (σX ) ≤ 12 log 2. Proof. The estimates for the topological entropy follow from Theorem 11.2.1. They will also follow from the argument below and Proposition 12.2.4.

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12 Binary Shifts

As in the previous section denote by An the C∗ -algebra generated by s1 , . . . , sn . Then by the by now standard argument, see e.g. the proof of Proposition 10.4.5, we have hϕ (σX ) ≤ lim inf n

1 1 log rank An and ht(σX ) ≤ lim sup log rank An . n n n

By Theorem 12.1.4 the rank of An is not greater than 2n , and if X is nonperiodic it is equal to 2n/2 for an inﬁnite number of n’s. This gives the required upper estimates for ht(σX ) and hϕ (σX ).

Our next goal is to show that generically a binary shift has zero entropy and diﬀerent bitstreams give rise to nonconjugate systems. , As in the previous section let G = Z Z/2Z and let ωX = ω denote the bicharacter on G deﬁned by X. Let T denote the shift to the right on G, so that σX is the automorphism of A(X) = C ∗ (G, ωX ) deﬁned by T . We shall ∞ identify a subset X of N with the point (1X (i))i≥1 in Y = i=1 {0, 1}, where 1X is the characteristic function of X. Let µ denote the product measure on Y with weights ( 12 , 12 ). Lemma 12.2.2. Let ρX (g, h) = ωX (g, h)ωX (h, g) for g, h ∈ G. Then the sequence {ρX (T n g, h)}∞ n=1 is uniformly distributed for g, h = 0 for µ-almost every X. Recall that a sequence {an }n such that an = ±1 is said to be uniformly n distributed if n−1 m=1 am → 0 as n → ∞. Proof of Lemma 12.2.2. Let f denote the function on Y deﬁned by f (y) = (−1)y1 , y = (y1 , y2 , . . .) ∈ Y. Let S be the shift to the left on Y , S(y1 , y2 , . . .) = (y2 , y3 , . . .). Then if i > j we have ρX (gi , gj ) = (f ◦ S i−j−1 )(X), where as before gi = 1 in the i-th summand (Z/2Z)i . Let g = gi1 + . . . + gik , i1 < . . . < ik , and h = gj1 + . . . + gjl , j1 < . . . < jl , be given elements in G. Replacing g by T n0 g for some large n0 we may assume is > jt for all s, t. Let F denote the function on Y deﬁned by F (X) = ρX (g, h). Then $ (f ◦ S is −jt −1 )(X). F (X) = 1sk,1tl

It follows that ρX (T n g, h) = (F ◦ S n )(X). By Birkhoﬀ’s ergodic theorem, see e.g. [72, Theorem 3.41], for almost all X we have n 1 (F ◦ S m )(X) → F dµ. n m=1 Y

12.2 Entropy of Binary Shifts

219

The function f takes the value ±1 with probability 12 . Hence Y f dµ = 0, and similarly Y f ◦ S ik −j1 −1 dµ = 0. Now F is a product of f ◦ S ik −j1 −1 and a function depending onlyon the ﬁrst ik − j1 − 1 coordinates. Since µ is a product measure, we have Y F dµ = 0, whence {ρX (T n g, h)}∞ n=1 is uniformly distributed.

If we combine the above lemma with Theorems 11.4.1 and 11.4.2 we obtain the following result. Theorem 12.2.3. For almost all subsets X of N the binary shift σX has entropy hτ (σX ) = 0 with respect to the trace τ = τX . For any such X the trace τX is the unique σX -invariant state on A(X).

Though almost every binary shift has zero entropy, we shall see soon that in many cases the entropy is 12 log 2. Similarly to the discussion at the end of Sect. 11.5, we conclude that binary shifts provide many examples of systems for which the tensor product formula for the entropy fails. More precisely, we have the following result. Proposition 12.2.4. If X is nonperiodic then hτ ⊗τ (σX ⊗ σX ) = log 2. Proof. Similarly to the proof of Proposition 12.2.1 we have hτ ⊗τ (σX ⊗ σX ) ≤ lim inf n

1 log rank An ⊗ An = log 2. n

The argument from the proof of Theorem 11.2.1 for the converse inequality now goes as follows. Let D denote the C∗ -subalgebra of A(X) ⊗ A(X) generated by the symmetries sn ⊗ sn , n ∈ Z. Then D is abelian, and the system (D, (τ ⊗ τ )|D , (σX ⊗ σX )|D ) is isomorphic to the Bernoulli shift with weights ( 12 , 12 ), hence it has entropy log 2. Thus hτ ⊗τ (σX ⊗ σX ) ≥ h(τ ⊗τ )|D (σX ⊗ σX |D ) = log 2.

Denote by M (X) the weak operator closure of A(X) in the GNS-representation corresponding to τ = τX . Denote the extensions of τX and σX to M (X) by the same symbols. Note that if X is nonperiodic then by Corollary 12.1.5 the algebra M (X) is the hyperﬁnite II1 -factor. Remark 12.2.5. In analogy with the classical situation if (M, τ, σ) is a W∗ dynamical system we can say that a σ-invariant von Neumann subalgebra Mπ is a Pinsker algebra if it satisﬁes the following conditions: (i) hτ |Mπ (σ|Mπ ) = 0, (ii) if B is a ﬁnite dimensional C∗ -subalgebra of Mπ such that hτ (B; σ) = 0 then B ⊂ Mπ . In the abelian case such an algebra always exists. In the noncommutative case, however, this is false. Indeed, by Theorem 12.2.3 there exists a set X such that hτ (σX ) = 0. Then Hτ (B; σX ) = 0 for all ﬁnite dimensional C∗ -subalgebras B of M (X). Consider now the W∗ -dynamical system

220

12 Binary Shifts

¯ (X), τ ⊗ τ, σX ⊗ σX ). Then hτ ⊗τ (σX ⊗ σX ) = log 2 by Proposi(M (X)⊗M tion 12.2.4. For each ﬁnite dimensional C∗ -subalgebra B of M (X) we have hτ ⊗τ (B ⊗ C; σX ⊗ σX ) = 0. Hence, if the Pinsker algebra for the system ¯ (X), τ ⊗ τ, σX ⊗ σX ) existed, it would contain M (X) ⊗ C, and by (M (X)⊗M ¯ (X), a contrasymmetry C ⊗ M (X). Thus it would coincide with M (X)⊗M diction. Lemma 12.2.6. Let X1 , X2 ⊂ N. Suppose there exist sequences {nk }∞ k=1 ⊂ X1 and {mk }∞ k=1 ⊂ N such that nk , mk → ∞ and Z ∩ [nk − mk , nk + mk ] ⊂ N\X2 . Then the systems (M (X1 ), τX1 , σX1 ) and (M (X2 ), τX2 , σX2 ) are nonconjugate. Proof. Let s1n and s2n be the symmetries corresponding to X1 and X2 , respectively. Then we have nk 1 nk 1 1 σX (sn )sn = −s1n σX (sn ) 1 1 nk 1 (sn ), s1n ]2 = 2. On the other hand, we have for all n and k, so that [σX 1 nk 2 2 nk 2 (si )sj = s2j σX (si ) σX 2 2

whenever |i − j| ≤ mk , from which it follows that nk (x), y]2 = 0 for any x, y ∈ M (X2 ). lim [σX 2

k→∞

Thus σX1 and σX2 are nonconjugate.

Theorem 12.2.7. For almost all pairs (X1 , X2 ) (with respect to the measure µ × µ on Y × Y ) the W∗ -dynamical systems (M (X1 ), τX1 , σX1 ) and (M (X2 ), τX2 , σX2 ) are nonconjugate. Proof. Choose a strictly increasing sequence {mk }k of natural numbers. For each k ∈ N let fk be the function on Y × Y deﬁned by = 1, yl = 0 for 1 ≤ l ≤ 2mk + 1, 1, if ym k +1 fk (y , y ) = 0, otherwise. As before let S be the shift to the left on Y . Then for n ≥ mk + 1 the conditions n ∈ X1 and Z ∩ [n − mk , n + mk ] ⊂ N\X2 mean exactly that fk (S n−mk −1 X1 , S n−mk −1 X2 ) = 1. For almost every pair (X1 , X2 ) Birkhoﬀ’s ergodic theorem implies n 1 fk (S n X1 , S n X2 ) → fk d(µ × µ) = 2−2mk −2 > 0. n m=1 Y ×Y Thus for almost all pairs (X1 , X2 ) we have fk (S n−mk −1 X1 , S n−mk −1 X2 ) = 1 for inﬁnitely many n’s. In particular, for almost every pair (X1 , X2 ) we can ﬁnd an increasing sequence {nk }k such that the assumptions of Lemma 12.2.6 are satisﬁed. This completes the proof of the theorem.

Having proved generic results, we now consider special cases. We start with the periodic case.

12.2 Entropy of Binary Shifts

221

Proposition 12.2.8. Suppose X ⊂ N is periodic. Then hτ (σX ) = log 2. Proof. Let p be the period of −X ∪ X. Put w = s0 sp . Then w lies in the center of A(X). Since either w or iw is a symmetry, and τ (w) = 0, the restriction n of σX to the C∗ -algebra generated by σX (w) is the Bernoulli shift with weights ( 21 , 12 ). Hence hτ (σX ) ≥ log 2. On the other hand, by Proposition 12.2.1 we have hτ (σX ) ≤ ht(σX ) ≤ log 2. Thus the proof is complete.

We next consider binary shifts corresponding to ﬁnite sets. It turns out that these systems admit several dynamical characterizations. Recall that an automorphism α of a C∗ -algebra A is asymptotically abelian if limn [αn (a), b] = 0 for all a, b ∈ A. It is asymptotically abelian with locality if there is a dense ∗-subalgebra A such that for each pair a, b ∈ A we have [αn (a), b] = 0 for suﬃciently large n, and C ∗ (a, b) is ﬁnite dimensional. Recall also that a W∗ -dynamical system (M, τ, α) is called an entropic Ksystem if lim hτ (γ; αn ) = Hτ (γ) n→∞

for any channel γ: B → M . Theorem 12.2.9. Suppose X ⊂ N is nonperiodic. Then the following four conditions are equivalent: (i) X is ﬁnite; (ii) (A(X), σX ) is asymptotically abelian; (iii) (A(X), σX ) is asymptotically abelian with locality; (iv) (M (X), τX , σX ) is an entropic K-system. -∞ Proof. Suppose X is ﬁnite, say X ⊂ {1, 2, . . . , r}. Let A = n=1 A[−n,n] , where A[−n,n] is generated by s−n , . . . , sn . Then A is dense in A(X). If a, b ∈ A[−n,n] , m let m > 2n + r. Then [σX (a), b] = 0, proving (A(X), σX ) is asymptotically abelian with locality. Thus (i) implies (iii). n Clearly (iii) implies (ii). Assume (ii). Since [s0 , σX (s0 )] = [s0 , sn ] = 2 if n ∈ X, it is immediate from asymptotic abelianness that there exists r ∈ N such that n ∈ / X for n > r. Thus X ⊂ {1, . . . , r}, and (i) follows. We conclude by showing (i)⇔(iv). Assume X ⊂ {1, . . . , r}. Let M0 be the weak operator closure of ∪n A[−n,0] in M (X). Since M (X) is a factor and −n n (M0 ) ∩ σX (M0 ), we conclude that (M (X), τX , σX ) is an A[−n+r+1,n] ⊂ σX entropic K-system by Theorem 4.3.2. Note however that we have the following factorization property, which makes the proof of Theorem 4.3.2 in this case m very simple. If a ∈ A[−n,n] and b ∈ σX (A[−n,n] ) with m ≥ 2n + r + 1, then ab = ba and τ (ab) = τ (a)τ (b). Conversely, assume (M (X), τX , σX ) is an entropic K-system. Let C be a ﬁnite dimensional abelian subalgebra of A(X). Since (2m−1)n

n Hτ (C, σX (C), . . . , σX

n (C)) mHτ (C, σX (C)),

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12 Binary Shifts

n n n so that 2hτ (C; σX ) ≤ Hτ (C, σX (C)), we conclude Hτ (C, σX (C)) → 2Hτ (C). ∗ n Let C = C (s0 ) = C1 + Cs0 . Then Hτ (C) = log 2, so Hτ (C, σX (C)) ≥ n log 2. If sn and s0 commute then C and σX (C) are independent, so that n Hτ (C, σX (C)) = 2 log 2. On the other hand, if they anticommute, then C and n n σX (C) generate the algebra of 2 × 2 matrices, so that Hτ (C, σX (C)) ≤ log 2. To summarize, log 4, if n ∈ / X, n Hτ (C, σX (C)) = log 2, if n ∈ X. n Since Hτ (C, σX (C)) → 2Hτ (C) = log 4, we conclude that there exists r such n that Hτ (C, σX (C)) = log 4 whenever n > r. Thus X ⊂ {1, . . . , r}.

Note that we could alternatively use Theorem 4.3.5 to show that (iv) implies that the system (M (X), τX , σX ) is strongly asymptotically abelian, and then deduce ﬁniteness of X by the same argument as for the implication (ii)⇒(i) above. Our next result shows that for X ﬁnite, dynamical entropy coincides with both topological entropy and mean entropy. Theorem 12.2.10. Let X be a ﬁnite nonempty subset of N. Then hτ (σX ) = ht(σX ) = lim

n→∞

1 1 Hτ (An ) = log 2. n 2

m km (An ), . . . , σX (An ) are τ Proof. If X ⊂ {1, . . . , r}, then the algebras An , σX independent for m ≥ n + r + 1. By Proposition 10.4.5 we conclude that

hτ (σX ) = lim

n→∞

1 Hτ (An ). n

On the other hand, by Theorem 12.1.4(i) we have Hτ (An ) = log rank An . Since rank An = 2n/2 for inﬁnitely many n’s by Theorem 12.1.4(iii), we get hτ (σX ) = lim

n→∞

1 1 1 Hτ (An ) = lim log rank An = log 2. n→∞ n n 2

Since we always have ht(σX ) ≤ lim sup n→∞

we also get ht(σX ) =

1 2

1 log rank An , n

log 2.

Remark that by Theorem 12.2.10 the algebra A(X) is UHF, hence τX is the unique tracial state on A(X). By Theorem 11.4.2, if X is ﬁnite, there exist invariant states on A(X) diﬀerent from τX . Then by Theorem 9.3.3 we have hϕ (σX ) < 12 log 2 for any such state. Corollary 12.2.11. Suppose Xi ⊂ N, i = 1, 2, are ﬁnite. Then hτX1 ⊗τX2 (σX1 ⊗ σX2 ) = log 2 = hτX1 (σX1 ) + hτX2 (σX2 ).

12.2 Entropy of Binary Shifts

223

Proof. Let Ain ⊂ A(Xi ) be the algebra generated by s1 , . . . , sn ∈ A(Xi ), i = 1, 2. Then hτX1 ⊗τX2 (σX1 ⊗ σX2 ) ≤ lim inf n→∞

= lim

n→∞

1 Hτ ⊗τ (A1n ⊗ A2n ) n X1 X2

1 (HτX1 (A1n ) + HτX2 (A2n )). n

On the other hand, by Theorem 3.2.2(iv), hτX1 ⊗τX2 (σX1 ⊗ σX2 ) ≥ hτX1 (σX1 ) + hτX2 (σX2 ).

Thus the result follows from the previous theorem. We next show that even when X is inﬁnite the entropy can often be

1 2

log 2.

Lemma 12.2.12. Suppose X is nonperiodic and either contained in the even numbers or the odd numbers, then hτ (σX ) = 12 log 2. Proof. Assume ﬁrst X ⊂ {1, 3, 5, . . .}. Let C0 = C ∗ (s0 ). Since s2i s2j = s2j s2i 2n 2 for all i, j ∈ Z, the algebras σX (C0 ), n ∈ Z, are independent. Thus σX acts 1 1 ∗ as the Bernoulli shift with weights ( 2 , 2 ) on the abelian C -algebra C they 2 generate, so that hτ |C (σX |C ) = log 2. Hence hτ (σX ) =

1 1 1 2 2 ) ≥ hτ |C (σX |C ) = log 2, hτ (σX 2 2 2

which together with Proposition 12.2.1 proves the lemma for X contained in the odd numbers. Suppose next X ⊂ {2, 4, 6, . . .}. Then s2i+1 s2j = s2j s2i+1 for all i, j ∈ Z. Letting ti = s2i s2i+1 it follows that ti tj = tj ti for all i, j ∈ Z. Thus the C∗ 2 algebra D = C ∗ (ti | i ∈ Z) is abelian, and σX |D is the Bernoulli shift with 1 1 weights ( 2 , 2 ). Now the argument is completed as in the odd case.

Let us say that a W∗ -dynamical system (M, τ, σ) has completely positive entropy if hτ (γ; σ) > 0 for any nontrivial channel γ: B → M (that is, γ(B) = C1). Clearly, an entropic K-system has completely positive entropy. Although by Theorem 12.2.9 the system (M (X), τX , σX ) is an entropic Ksystem only if X is ﬁnite, we shall next show that it can have completely positive entropy even if X is inﬁnite. Proposition 12.2.13. Suppose q ≥ 3 is an odd integer and X ⊂ {q n | n ≥ 0}. Then (M (X), τX , σX ) has completely positive entropy. m ) ≤ mhτ (γ; σX ) and Hτ (γ) > 0 for nontrivial γ, it Proof. Since hτ (γ; σX suﬃces to prove that 2q lim hτ (γ; σX ) = Hτ (γ). n

n→∞

224

12 Binary Shifts

2kq Note that if x, y ∈ A[−m,m] and m ≤ (q n − 1)/2 then x and σX (y) commute for any k ∈ N. Indeed, otherwise there exist i, j such that |i|, |j| ≤ m and q l = 2kq n + i − j ∈ X for some l. It follows that (2k − 1)q n + 1 ≤ q l ≤ (2k + 1)q n − 1, so that 2k − 1 < q l−n < 2k + 1, which is impossible as q is an odd integer. For any ε > 0 we can ﬁnd a ﬁnite partition of unity 1 = i xi in A[−m,m] for some m such that n

Hτ (γ) < Hτ (γ; {xi }i ) + ε. 2q 2kq (xi1 ), . . . , σX (xik ) are indepenThen for q n ≥ 2m + 1 the elements xi0 , σX dent, so that n

n

2kq 2q 2kq 2q (xik )}) ◦ γ, . . . , σX ◦ γ; {xi0 σX (xi1 ) . . . σX Hτ (γ, σX n n 2q 2kq η(τ (xi0 σX (xi1 ) . . . σX (xik ))) + (k + 1) S(τ (γ(·)xi ), τ ◦ γ) = n

i0 ,...,ik

= (k + 1)

n

n

n

i

(η(τ (xi )) + S(τ (γ(·)xi ), τ ◦ γ)) > (k + 1)(Hτ (γ) − ε),

i 2q ) > Hτ (γ) − ε. Since ε was arbitrary and we always have whence hτ (γ; σX n 2q

hτ (γ; σX ) ≤ Hτ (γ), the proof is complete. n

We thus get examples of systems which have completely positive entropy, but are not entropic K-systems. It turns out that among them there are uncountably many nonisomorphic ones. Proposition 12.2.14. Let q ≥ 2 be an integer and Xi ⊂ {q n | n ≥ 0}, i = 1, 2. Suppose the set X1 ∩ (N\X2 ) is inﬁnite. Then (M (X1 ), τX1 , σX1 ) and (M (X2 ), τX2 , σX2 ) are nonconjugate. / X2 , Proof. Let n1 < n2 < . . . be such that q nk ∈ X1 ∩ (N\X2 ). Since q nk ∈ we have [q nk − q nk −1 + 1, q nk + q nk −1 − 1] ⊂ N\X2 . Thus the result follows from Lemma 12.2.6.

If we combine the above proposition with Lemma 12.2.12 and Proposition 12.2.13 we obtain the following. Theorem 12.2.15. There is an uncountable family of pairwise nonconjugate systems (M (X), τX , σX ) with completely positive entropy such that hτX (σX ) = 1

2 log 2.

12.3 Notes The study of binary shifts was initiated by Powers [171]. One should be aware that in the literature one usually understands by a binary shift the one-sided

12.3 Notes

225

shift. The criterion for simplicity of the C∗ -algebra of a bitstream was obtained by Price [173]. The fact that the dimensions cn of the centers of the algebras An have the form described in Theorem 12.1.4 was proved by Powers and Price [172]. The observation that any sequence {cn }n with properties as in Theorem 12.1.4 arises from a bitstream was made by Price [175]. In our exposition of these results we also beneﬁted from the work of Vik [220]. Proposition 12.1.6 is due to Enomoto, Nagisa, Watatani and Yoshida [64]. Some of the results for binary shifts can be generalized to automorphisms of the shift type on twisted group C∗ -algebras, see e.g. [39], [63], [38]. The entropy of the binary shifts corresponding to ﬁnite sets, Theorem 12.2.10, was computed by Choda [40]. A larger class of binary shifts with entropy 12 log 2 was exhibited by Price [174]. The ﬁrst example of a binary shift with zero entropy was given by Narnhofer, Størmer and Thirring [128]; see [69] for many such examples. It was the ﬁrst example of a system for which the tensor product formula for the entropy fails. The fact that generically a binary shift has zero entropy, Theorem 12.2.3, was stated by Narnhofer and Thirring [134]. As was proved already by Powers [171], one-sided binary shifts corresponding to diﬀerent nonperiodic sets are nonisomorphic. On the other hand, Lemma 12.2.6, proved by Golodets and Størmer [80], seems to be the only known suﬃcient condition for nonisomorphism of two-sided binary shifts. The observation that it implies that generically binary shifts are nonisomorphic, Theorem 12.2.7, is new. The rest of Sect. 12.2 is based on the work of Golodets and Størmer [80]. We have computed entropies of a large class of binary shifts, and the only values we have obtained are 0, 12 log 2, log 2. The following problem thus naturally presents itself. Does there exist X ⊂ N such that hτ (σX ) = 0, 12 log 2, log 2? Let (M (X), τX , σX ) be a binary shift. Consider the von Neumann subaln gebra M0 of M (X) generated by sn , n ≤ 0. Then M0 ⊂ σX (M0 ), ∪n σX (M0 ) n is dense in M (X), ∩n σX (M0 ) = C1. Thus (M (X), τX , σX ) is what we called in Chap. 4 an algebraic K-system. Therefore there exist algebraic K-systems with zero entropy. Theorem 12.2.15 shows that on the hyperﬁnite II1 -factor there exist uncountably many nonisomorphic systems with the same ﬁnite completely positive entropy. On the other hand, there are no examples of nonisomorphic entropic K-systems. In this respect a challenging open problem is whether binary shifts corresponding to diﬀerent ﬁnite subsets are isomorphic. In Chap. 13 we shall construct uncountably many nonisomorphic entropic K-systems on the hyperﬁnite III1 -factor.

13 Bogoliubov Automorphisms

In this chapter we shall consider one of the basic models of quantum statistical mechanics, a system of noninteracting fermions. Our goal is to compute the topological entropy of such a system, as well as the dynamical entropy with respect to a natural class of states.

13.1 Canonical Anticommutation Relations In the present section we collect basic facts about the CAR-algebra, the algebra of the canonical anticommutation relations. We refer the reader to [30, Chapter 5.2] for more details. Let H be a Hilbert space. The CAR-algebra A(H) over H is a unital C∗ algebra generated by elements a(f ) and a∗ (f ), f ∈ H, such that the mapping H f → a∗ (f ) is linear(1) , a(f )∗ = a∗ (f ) and a∗ (f )a(g) + a(g)a∗ (f ) = (f, g)1, a(f )a(g) + a(g)a(f ) = 0 for any f, g ∈ H. If H is one-dimensional, H = Cf , ||f || = 1, then the algebra A(Cf ) is isomorphic to Mat2 (C), with matrix units deﬁned by e11 (f ) = a(f )a∗ (f ), e22 (f ) = a∗ (f )a(f ), e12 (f ) = a(f ), e21 (f ) = a∗ (f ). (13.1) More generally, let H = Cf ⊕ K, f = 1. Then the self-adjoint unitary V (f ) = a(f )a∗ (f ) − a∗ (f )a(f ) anticommutes with a∗ (f ) and commutes with a∗ (g) for g ∈ K. Hence the map a∗ (g) → a∗ (g)V (f ) extends to an embedding of A(K) into A(H) such that the image commutes with A(Cf ) ∼ = Mat2 (C), and we thus get an isomorphism 1

In the literature the opposite notation for the operators a(f ) and a∗ (f ) is sometimes used, but we follow the conventions of [30].

228

13 Bogoliubov Automorphisms

A(Cf ⊕ K) ∼ = Mat2 (C) ⊗ A(K). It follows that if H has dimension n then A(H) ∼ = Mat2 (C)⊗n ,

(13.2)

(13.3)

2n

so A(H) is a full matrix algebra of dimension 2 . Explicitly, if f1 , . . . , fn is an (k) orthonormal basis in H then the matrix units eij in the k-th factor Mat2 (C) are given by (k)

(k)

e21 = a∗ (fk )V (f1 ) . . . V (fk−1 ), e12 = a(fk )V (f1 ) . . . V (fk−1 ), (k)

(k)

e11 = a(fk )a∗ (fk ), e22 = a∗ (fk )a(fk ).

(13.4)

Note that as a∗ (f ) is a partial isometry when f = 1, we have a∗ (f ) = 1, and hence a∗ (g) = g (13.5) for any vector g. Let U be a unitary operator on H. By the universal property of the CARalgebra, it deﬁnes an automorphism αU of A(H), αU (a(f )) = a(U f ), called a Bogoliubov automorphism. Let now A be an operator on H, 0 ≤ A ≤ 1. It deﬁnes a state ωA on A(H) by the formula ωA (a∗ (f1 ) . . . a∗ (fn )a(gm ) . . . a(g1 )) = δnm det((Afi , gj ))i,j , called a quasi-free state. If H = Cf , f = 1, and A = λ1, then the quasi-free state ωA on A(Cf ) ∼ = Mat2 (C) has density matrix 1−λ 0 . 0 λ More generally, if H = Cf ⊕ K, f = 1, and Af = λf , then under the isomorphism (13.2) we have 1−λ 0 ωA = Tr · ⊗ ωA|K . 0 λ It follows that if H has dimension n and f1 , . . . , fn is an orthonormal basis consisting of eigenvectors of A, Afk = λk fk , then under the isomorphism (13.3) we get n . 1 − λk 0 ωA = . (13.6) Tr · 0 λk k=1

This gives a way to prove existence of quasi-free states: if A has pure point spectrum prove that the above product state is indeed the required quasifree state, then approximate an arbitrary quasi-free state by quasi-free states corresponding to operators with pure point spectrum. Finally note that if two operators A and U commute then the state ωA is αU -invariant.

13.2 Topological Entropy

229

13.2 Topological Entropy Let U be a unitary operator on a separable Hilbert space H. In this section we compute the topological entropy of the Bogoliubov automorphism deﬁned by U . To formulate the result recall, see C.7, that there exists a unique decomposition H = Hs ⊕ Ha such that Us = U |Hs has singular spectral measure and Ua = U |Ha has absolutely continuous spectral measure with respect to the Lebesgue measure on T. Furthermore, we have a direct integral decomposition ⊕ ⊕ Ha = Hz dµ(z), Ua = z dµ(z), T

T

where µ is the normalized Lebesgue measure on T. Denote by mU the multiplicity function of Ua , mU (z) = dim Hz . Theorem 13.2.1. With the above notation, for the Bogoliubov automorphism αU of the CAR-algebra A(H) we have ht(αU ) = (log 2) mU (z)dµ(z). T

In particular, ht(αU ) = 0 if U has singular spectral measure. Consider the simplest case when the unitary U is the bilateral shift, so it has homogeneous Lebesgue spectrum of multiplicity one. Then the theorem asserts that ht(αU ) = log 2. To prove this let {fn }n∈Z be an orthonormal basis such that U fn = fn+1 . Then according to (13.3) and (13.4) the unital C∗ -subalgebra generated by a∗ (fn )a(fn ), n ∈ Z, is isomorphic to C({0, 1}Z ), and the restriction of αU to this subalgebra is just the shift automorphism with entropy log 2. Hence ht(αU ) ≥ log 2. On the other hand, if Ω is a ﬁnite subset of A(Hk,l ), where Hk,l is the m space spanned by fn , n = k, . . . , l, then αU (Ω) ⊂ A(Hk,l+m ), so that for any δ > 0 and n ∈ N n−1 rcp(Ω ∪ αU (Ω) ∪ . . . ∪ αU (Ω), δ) ≤ rank A(Hk,l+n−1 ) = 2l−k+n .

Hence ht(Ω, δ; αU ) ≤ log 2. So indeed ht(αU ) = log 2. Consider now the case of singular spectral measure. Then ht(αU ) = 0. To show this we need a couple of technical lemmas. Lemma 13.2.2. Let U be a unitary operator with singular spectral measure, P a ﬁnite rank projection. Then for any ε > 0 there exists n0 ∈ N such that for n ≥ n0 there is a projection Pn such that rank Pn ≤ εn and (1 − Pn )U k P < ε for k = 0, . . . , n.

230

13 Bogoliubov Automorphisms

Proof. Note that if P = Q1 + Q2 and Qin is such that rank Qin ≤ εn/2 and (1 − Qin )U k Qi < ε/2, i = 1, 2, then the projection Pn = Q1n ∨ Q2n satisﬁes the conditions of the lemma. Thus it suﬃces to prove the lemma for rank one projections. Replacing H by a subspace we may further assume that P is the projection onto the space spanned by a cyclic vector for U . Then we can identify the Hilbert space with L2 (T, ν), where ν is a singular probability measure, such that P becomes the projection onto the constant functions and U becomes the operator of multiplication by the function z. √ Choose N ∈ N such that 2/N < ε. Since ν is singular, there exist m ∈ N and disjoint arcs I1 , . . . , Im on T of length not greater than 1/N 2 m each such that ε2 ν(T\(I1 ∪ . . . ∪ Im )) < . 2 Consider now any n of the form n = N ml, l ∈ N. Subdivide each arc Is into l arcs Ist , 1 ≤ t ≤ l, of length not greater than 1/N 2 ml each. Let Pn be the projection onto the subspace spanned by the characteristic functions of these arcs. Then rank Pn ≤ ml = n/N . On each arc Ist choose a point zst . Then for w ∈ Ist and k = 0, . . . , n we have k |≤ |wk − zst

n N 2 ml

=

1 , N

so that 2 ε2 1 k k k )1Ist 22 < zst 1Ist = ν (T\ (∪s,t Ist )) + (z k − zst + 2 < ε2 . z − 2 N s,t s,t 2

Hence z k − Pn z k 2 < ε, that is, (1 − Pn )U k P < ε. Now for any n ∈ N set Pn = P([n/N m]+1)N m . Then (1 − Pn )U k P < ε for k = 0, . . . , n, and rank Pn ≤

/ n 0 1 m , +1 m≤n + Nm N n

so that rank Pn < εn if n is suﬃciently large.

Lemma 13.2.3. For any m ∈ N and δ > 0 there exists ε > 0 such that if P and Q are projections in B(H) such that rank P = m and (1 − Q)P < ε, then for any x in the unit ball of A(P H) ⊂ A(H) there exists y in the unit ball of A(QH) ⊂ A(H) such that x − y < δ. Proof. This follows from norm-continuity of the map g → a∗ (g). In fact, by (13.5) we know that

13.2 Topological Entropy

231

a∗ (g) − a∗ (Qg) = (1 − Q)g for any g ∈ H.

Let U be a unitary operator with singular spectral measure. Let Ω be a ﬁnite subset of the unit ball of A(P H) for a projection P of rank m. Then k αU (Ω) is contained in the unit ball of A(U k P U −k H). For δ > 0 choose ε < δ in accordance with Lemma 13.2.3. Choose a projection Pn according to k Lemma 13.2.2. Then αU (Ω) ⊂δ A(Pn H) for k = 0, . . . , n. Consequently n−1 rcp(Ω ∪ αU (Ω) ∪ . . . ∪ αU (Ω), δ) ≤ rank A(Pn H) = 2rank Pn ≤ 2εn ≤ 2δn ,

whence ht(Ω, δ; αU ) ≤ δ log 2. Thus ht(αU ) = 0. It turns out that the general result follows essentially from the above cases, thanks to the following characterization of the Lebesgue integral. Proposition 13.2.4. Let h be a function deﬁned on the set of all unitary operators on a separable Hilbert space such that: (i) h(U ) depends only on the conjugacy class of U ; (ii) if Ua ∼ = W N , where W is the bilateral shift and N ∈ N, then h(U ) = N ; (iii) if P is a projection commuting with U then h(U |P H ) ≤ h(U ); (iv) if {Pn }n is a sequence of projections commuting with U such that Pn → 1 strongly, then h(U |Pn H ) → h(U ); (v) h(U n ) = nh(U ) for any n ∈ N. Then h(U ) = mU (z)dµ(z).

T

Proof. Denote T mU (z)dµ(z) by µ(U ). Let us also write U % V if U is conjugate to the restriction of V to an invariant subspace. In view of (iv) it suﬃces to consider unitary operators with bounded multiplicity function mU . Moreover, using (iv) and regularity of the measure we may assume that the sets m−1 U ({k}), k ≥ 0, are closed. Then we can ﬁnd a sequence {U (n)}n of unitary operators such that Us ∼ = U (n)s , mU ≤ mU (n) , µ(U (n)) → µ(U ), and mU (n) is the sum of the characteristic functions of pn arcs of length 2π/qn each, where qn ∈ N. Observe next that if I is a measurable subset of T such that the map I z → z k is injective with image J, then the operator of multiplication by the function z → z k on L2 (I, dµ) is conjugate to the operator of multiplication by the function z → z on L2 (J, dµ). It follows that U (n)qan ∼ = W pn , where W is the bilateral shift, that is, the operator of multiplication by the function z on L2 (T, dµ). More generally, the above observation implies that for any unitary operator V we have mV k (z) = mV (w). (13.7) w: wk =z

232

13 Bogoliubov Automorphisms

∼ W pn . Let V (n) be a unitary operator such that We have Uaqn % U (n)qan = pn ∼ qn pn V (n) = U . Since V (n)a % W pn , by (13.7) we have mV (n) ≤ 1. Therefore V (n)pn ∼ = U qn % U (n)qn , V (n)a % W and U (n)qan ∼ = W pn . Applying µ we get pn pn µ(V (n)) = µ(U ) ≤ µ(U (n)) = . qn qn Since µ(U (n)) → µ(U ), by the above inequality we get pn /qn → µ(U ) and µ(V (n)) → 1 (unless µ(U ) = 0, in which case we have h(U ) = 0, as h(U ) ≤ h(U (n)) = pn /qn ). Since mV (n) ≤ 1, it follows that mV (n) → 1 in measure. Then by (iv) and (ii) we get h(V (n)a ) → 1. Since V (n)a % V (n) % W ⊕V (n)s , it follows that h(V (n)) → 1. Since h(U ) =

pn h(V (n)), qn

we conclude that pn /qn → h(U ). Thus h(U ) = µ(U ).

Proof of Theorem 13.2.1. We have to check that the function which takes the value (log 2)−1 ht(αU ) at U satisﬁes the conditions of Proposition 13.2.4. Properties (i), (iii) and (v) are immediate consequences of general properties of topological entropy. Property (iv) follows from the fact that any ﬁnite subset in A(H) can be approximated by a ﬁnite subset in A(Pn H) for all suﬃciently large n’s. It remains to prove (ii). So assume that U is such that there exists an orthonormal basis {fi,n | i = 1, . . . , N, n ∈ Z} in Ha such that U fi,n = fi,n+1 . Then, similarly to the argument after the formulation of Theorem 13.2.1, the C∗ -algebra generated by a∗ (fi,n )a(fi,n ), i = 1, . . . , N , n ∈ Z, is isomorphic to C({0, 1}Z )⊗N , and the restriction of αU to this subalgebra is the tensor product of the shift automorphisms, so that ht(αU ) ≥ N log 2. Denote by Pk,l the projection onto the space spanned by fi,n , i = 1, . . . , N , n = k, . . . , l. Let P be a ﬁnite rank projection in B(Hs ), Ω a ﬁnite subset of the unit ball of A(P Hs ⊕ Pk,l Ha ), and δ > 0. Choose ε < δ in accordance with Lemma 13.2.3 applied to m = rank P + rank Pk,l . For n ∈ N ﬁnd a projection j Pn ∈ B(Hs ) according to Lemma 13.2.2. Then αU (Ω) is contained in the unit j j −j ball of A(Us P Us Hs ⊕Pk+j,l+j Ha ), so that αU (Ω) ⊂δ A(Pn Hs ⊕Pk,l+n−1 Ha ) for j = 0, . . . , n − 1. Consequently n−1 rcp(Ω ∪ αU (Ω) ∪ . . . ∪ αU (Ω), δ) ≤ 2rank Pn +rank Pk,l+n−1 ≤ 2δn+N (l−k+n) ,

whence ht(Ω, δ; αU ) ≤ (δ + N ) log 2. Thus ht(αU ) ≤ N log 2, and property (ii) is proved. This completes the proof of the theorem.

The following example of an asymptotically abelian system with nontracial states of maximal entropy was promised in Chap. 9. Let us ﬁrst introduce one more notion. The even CAR-algebra is the ﬁxed point subalgebra A(H)e of the CAR-algebra A(H) for the Bogoliubov automorphism α−1

13.3 Classical Bernoullian Subsystems

233

corresponding to the operator −1. It is generated by operators of the form a# (f )a# (g), where a# (f ) is any of the operators a∗ (f ) and a(f ). Example 13.2.5. It easy to see that the restriction α of the Bogoliubov automorphism αU to the even CAR-algebra is asymptotically abelian if and only if (U n f, g) → 0 as n → ∞ for any f, g ∈ H. If in addition U has singular spectrum then by Theorem 13.2.1 we have ht(α) = 0, while there are many nontracial α-invariant states (for example, quasi-free states corresponding to scalars λ ∈ (0, 1/2)). Unitaries with such properties can be obtained using Riesz products. We shall brieﬂy recall the construction. Let q > 3 be a real number, {nk }∞ k=1 a sequence of positive integers such ∞ that nk+1 ≥ qnk , {a } a sequence of real numbers such that ak ∈ (−1, 1), k k=1 ak → 0 as k → ∞, k a2k = ∞. Then the sequence of measures n 1 $ (1 + ak cos nk t) dt 2π k=1

on [0, 2π] converges weakly∗ to a probability measure µ with Fourier coeﬃcients ⎧ ∞ $ ak |εk | ⎪ ⎨ , if n = εk nk with εk ∈ {−1, 0, 1}, int µ ˆ(n) = µ(e ) = k=1 2 k ⎪ ⎩ 0, otherwise. The measure µ is singular by [238, Theorem V.7.6]. We see also that µ ˆ(n) → 0 as |n| → ∞. Thus the operator U of multiplication by eit on L2 ([0, 2π], dµ) has the desired properties.

13.3 Classical Bernoullian Subsystems In this section we begin the computation of the dynamical entropy of Bogoliubov automorphisms with respect to quasi-free states. Consider ﬁrst the case when U is the bilateral shift, so there exists an orthonormal basis {fn }n∈Z such that U fn = fn+1 , and A = λ1, 0 ≤ λ ≤ 1, is a scalar operator. We shall write ωλ instead of ωλ1 for the quasi-free state corresponding to λ1. Then by (13.3)-(13.6) the C∗ -algebra generated by a∗ (fn )a(fn ), n ∈ Z, is isomorphic to C({0, 1}Z ), and by restricting ωλ and αU to this subalgebra we obtain the classical Bernoulli system with weights (1−λ, λ) and entropy η (λ)+η (1 − λ). Since this subalgebra lies in the centralizer of the state, we get hωλ (αU ) ≥ η (λ) + η (1 − λ) . In fact, it is not diﬃcult to see that equality holds, which allows us to conclude along the lines of the proof of Theorem 13.2.1 that

234

13 Bogoliubov Automorphisms

hωλ (αU ) = (η (λ) + η (1 − λ))

T

mU (z)dµ(z)

for any unitary U . Since we shall in the next section prove a more general formula without relying on this particular case, we are not going into details. Our goal in this section is rather to prove an analogue of the above inequality when A is close to a scalar, which is the main technical result we shall need later. Proposition 13.3.1. For given ε > 0 and C, 0 < C < 1, there exists δ > 0 such that if Sp A ⊂ (λ0 −δ, λ0 +δ) for some λ0 ∈ (0, C), and the spectrum of U has a homogeneous Lebesgue component, so that there exists a unit vector f such that {U n f }n∈Z is an orthonormal system, then hωA (αU ) ≥ η (λ0 ) + η (1 − λ0 ) − ε. The same inequality is true for the restrictions of ωA and αU to the even CAR-algebra. Consider the matrix units eij (f ) deﬁned by (13.1). Let P be the algebra spanned by p1 = e11 (f ) = a(f )a∗ (f ) and p2 = 1 − p1 . For n ∈ N we want to estimate n−1 n−1 HωA (P, αU (P), . . . , αU (P); {pi0 αU (pi1 ) . . . αU (pin−1 )}i0 ,...,in−1 =1,2 ).

For this we need ﬁrst of all to estimate the action of the modular group of ωA on p1 and p2 . Assuming that Ker A = Ker(1 − A) = 0 set B=

A . 1−A

Then ωA is a KMS−1 -state for the one-parameter group {σt = αB it }t of Bogoliubov automorphisms. If A has pure point spectrum this follows from (13.3)-(13.6), since the product-state in (13.6) is a KMS−1 -state for the oneparameter group n . (1 − λk )it 0 , Ad 0 λit k k=1

which is exactly the above group of Bogoliubov automorphisms. In the general case we can approximate A by operators with pure point spectrum and argue by continuity. Another possibility is to use the explicit description of the GNSrepresentation, see [30, Example 5.2.20]. It follows that the extension of σt to πωA (A(H)) is the modular group of the state. Lemma 13.3.2. If Sp A ⊂ (0, 1) and 1/2 1/2 A λ f− f < δ 1−A 1−λ

13.3 Classical Bernoullian Subsystems

235

for some δ > 0 and f , ||f || = 1, where λ = (Af, f ), then √ 1/2 1/2 λj σ−i/2 (ekj (f )) − λk ekj (f )ωA ≤ 2(λj λk )1/4 δ 1/2 for k, j = 1, 2, where λ1 = 1 − λ, λ2 = λ. Proof. We shall write ekj for ekj (f ). By deﬁnition we have ωA (e11 ) = λ1 , ωA (e22 ) = λ2 , ωA (e12 ) = ωA (e21 ) = 0. By the KMS-condition, for any operators x and y, ωA (σ−i/2 (x)∗ σ−i/2 (y)) = ωA (σi/2 (x∗ )σ−i/2 (y)) = ωA (σ−i/2 (y)σ−i/2 (x∗ )) = ωA (yx∗ ). Using also that ωA (σ−i/2 (x)∗ y) = ωA (σi/2 (x∗ )y) = ωA (x∗ σ−i/2 (y)), we get 1/2

1/2

||λj σ−i/2 (ekj ) − λk ekj ||2ωA = 2(λj λk )1/2 ((λj λk )1/2 − ωA (ejk σ−i/2 (ekj ))). So we must prove that ωA (ejk σ−i/2 (ekj )) is close to (λj λk )1/2 to within δ. Since λ1 − ωA (e11 σ−i/2 (e11 )) = ωA (e11 σ−i/2 (e22 )) = λ2 − ωA (e22 σ−i/2 (e22 )) and ωA (e12 σ−i/2 (e21 )) = ωA (e21 σ−i/2 (e12 )) by the KMS-condition, we must show that ωA (e11 σ−i/2 (e22 )) is close to zero and ωA (e21 σ−i/2 (e12 )) is close to λ1/2 (1 − λ)1/2 to within δ. λ A and β = . Then Let B = 1−A 1−λ σ−i/2 (e21 ) = σ−i/2 (a∗ (f )) = a∗ (B 1/2 f ), so

σ−i/2 (e21 ) − β 1/2 e21 = B 1/2 f − β 1/2 f < δ.

Hence, since e11 e21 = 0, |ωA (e11 σ−i/2 (e22 ))| = |ωA (e11 (σ−i/2 (e21 ) − β 1/2 e21 )σ−i/2 (e12 ))| < δ and since ωA (e11 ) = 1 − λ, |ωA (e12 σ−i/2 (e21 )) − λ1/2 (1 − λ)1/2 | = |ωA (e12 (σ−i/2 (e21 ) − β 1/2 e21 ))| < δ, which proves the lemma.

The previous lemma will allow us to estimate the correction term of the entropy. It will also allow us to estimate the classical term thanks to the following approximate factorization result.

236

13 Bogoliubov Automorphisms

Lemma 13.3.3.Let {eij }ni,j=1 be a system of matrix units in a von Neumann algebra M with k ekk = 1, and let ω be a faithful normal state on M . Then for any x ∈ M commuting with the matrix units we have √ 1/2 ω 1/2 |ω(ekk x) − λk ω(x)| ≤ 2 ||λj σ−i/2 (ekj ) − λk ekj ||ω ||x||# ω, j ∗ ∗ 1/2 and λk = ω(ekk ). where ||x||# ω = (ω(x x) + ω(xx ))

Proof. We may assume that ω is the vector state deﬁned by a cyclic and separating vector ξ. Let J be the corresponding modular conjugation. Then ω(ekk x) = (ekk xξ, ξ) = (ejk xξ, ejk ξ) = (Jejk ξ, Jejk xξ). Using that x commutes with ejk , and JxJ commutes with ekj , we also have (ekj ξ, Jejk xξ) = (ekj , JxJJejk ξ) = (ekj Jx∗ ξ, Jejk ξ) and

(ekj Jx∗ ξ, ekj ξ) = (ekj ξ, ekj Jxξ) = (ejj ξ, Jxξ) = (xξ, Jejj ξ).

It follows that 1/2

1/2

1/2

λj ω(ekk x) = λj ((λj Jejk − λk ekj )ξ, Jejk xξ) 1/2

1/2

1/2

+λk (ekj Jx∗ ξ, (λj Jejk − λk ekj )ξ) + λk (xξ, Jejj ξ). ω Since σ−i/2 (y)ξ = Jy ∗ ξ for y ∈ M , we then get

|λj ω(ekk x) − λk (xξ, Jejj ξ)| 1/2

1/2

1/2

1/2

ω (ekj ) − λk ekj ||ω ≤ (λj ||x||ω + λk ||x∗ ||ω )||λj σ−i/2 √ 1/2 ω 1/2 ≤ 2||x||# ω ||λj σ−i/2 (ekj ) − λk ekj ||ω .

Summing up the above inequalities over all j’s we obtain the desired estimate.

Proof of Proposition 13.3.1. First choose δ1 > 0 such that |η (λ) + η (1 − λ) − η(λ0 ) − η(1 − λ0 )|

−n . 3 j=1

(13.9)

Turning to the classical term, by Lemma 4.3.4 there exists ε1 > 0 such that if |ωA (pi x) − ωA (pi )ωA (x)| ≤ ε1 ||x||, i = 1, 2, (13.10) for any x commuting with p1 and p2 , then 2

η(ωA (pi xj )) ≥

i=1 j∈J

2

η(ωA (pi )) +

i=1

η(ωA (xj )) −

j∈J

ε 3

for any partition of unity 1 = j∈J xj commuting with p1 and p2 . By Lemmas 13.3.2 and 13.3.3 applied to {ekj (f )}2k,j=1 , there exists δ3 ∈ (0, δ2 ) such that if Sp A ⊂ (λ0 −δ3 , λ0 +δ3 ), λ0 ∈ (0, C), then (13.10) is satisﬁed. Applying the above inequality recursively we then get 2

n−1 η(ωA (pi0 αU (pi1 ) . . . αU (pin−1 )))

i0 ,...,in−1 =1

≥n

2 i=1

η(ωA (pi )) − (n − 1)

ε ε = n(η (λ) + η (1 − λ)) − (n − 1) , 3 3

(13.11)

where λ = (Af, f ) ∈ (λ0 − δ3 , λ0 + δ3 ). It follows from (13.8), (13.9) and (13.11) that if we take δ = δ3 then for any n ∈ N n−1 n−1 HωA (P, αU (P), . . . , αU (P); {pi0 αU (pi1 ) . . . αU (pin−1 )})

238

13 Bogoliubov Automorphisms

> n(η (λ0 ) + η (1 − λ0 )) − (3n − 1)

ε 3

as long as Sp A ⊂ (λ0 − δ, λ0 + δ) for some λ0 ∈ (0, C). Hence hωA (αU ) ≥ hωA (P; αU ) ≥ η (λ0 ) + η (1 − λ0 ) − ε. Since P is a subalgebra of the even CAR-algebra, the same inequalities hold for the restrictions of αU and ωA to A(H)e .

13.4 Dynamical Entropy In this section we shall prove a formula for the entropy of a Bogoliubov automorphism of the CAR-algebra with respect to a quasi-free state. To formulate the result consider a unitary U on a separable Hilbert space H and an operator A, 0 ≤ A ≤ 1, commuting with U . Let Ua = U |Ha be the part of U with absolutely continuous spectral measure. Then, see App. C, there exists a direct integral decomposition ⊕ ⊕ ⊕ Ha = Hz dµ(z), Ua = z dµ(z), A|Ha = Az dµ(z), T

T

T

where µ is the normalized Lebesgue measure on the torus T. Theorem 13.4.1. With the above notation, for the Bogoliubov automorphism αU of the CAR-algebra A(H) and the quasi-free state ωA deﬁned by A we have hωA (αU ) = Tr(η (Az ) + η (1 − Az ))dµ(z). T

A particular case, where one can get an explicit direct integral decomposition, is given in the following corollary. Corollary 13.4.2. Let I be an open subset of R, ω a real locally absolutely continuous function on I, ρ a measurable function on I, 0 ≤ ρ ≤ 1. Let U and A be the operators on L2 (I, dx) of multiplication by the functions eiω and ρ, respectively. Then 1 hωA (αU ) = (η (ρ(x)) + η (1 − ρ(x)))|ω (x)|dx. 2π I Proof. Consider the sets I1 = {x ∈ I | ω (x) = 0} and I2 = {x ∈ I | ω (x) = 0}. For any compact subset X ⊂ I1 the set ω(X) has measure zero. Hence the spectrum of the restriction of U to L2 (X, dx) has measure zero. It follows that

13.4 Dynamical Entropy

239

the restriction of U to L2 (I1 , dx) has singular spectral measure and thus does not contribute to the entropy. On the other hand, if x ∈ I2 then ω(y) = ω(x) for any y close to x, y = x, so that for every z ∈ T the set Y (z) = {x ∈ I2 | eiω(x) = z} is at most countable. For any f ∈ L1 (I2 , |ω (x)|dx) we have f (x)|ω (x)|dx = 2π f (x)dµ(z), T x∈Y (z)

I2

see e.g. [67, Theorem 3.2.3] for the case when ω is Lipschitzian, and use [67, Theorem 3.1.8] and [67, Theorem 2.10.43] to extend the result to the general case. It follows that if for a function f and z ∈ T we put fz = (2π)1/2 |ω |−1/2 f |Y (z) , then the map f → (fz )z deﬁnes an isometry between the spaces L2 (I2 , dx) ⊕ and T 2 (Y (z))dµ(z). Since I\(I1 ∪ I2 ) has measure zero, we thus obtain a direct integral decomposition of Ua and A|Ha such that Az is the operator of multiplication by ρ|Y (z) . Hence Tr(η (Az ) + η (1 − Az ))dµ(z) = (η (ρ(x)) + η (1 − ρ(x)))dµ(z) T

T x∈Y (z)

1 = 2π

(η (ρ(x)) + η (1 − ρ(x)))|ω (x)|dx,

I2

which proves the corollary.

Turning to the proof of Theorem 13.4.1 we shall ﬁrst establish the lower bound for the entropy. The main technical step in this direction has been done in the previous section, and our goal now is to extend that result to all unitaries with absolutely continuous spectrum. This extension is based on the following two results, which provide tools for changing the spectrum of a unitary operator without changing the entropy. Let us ﬁrst make a few remarks about the even CAR-algebra. If K and L are mutually orthogonal subspaces of H then A(K) and A(L)e commute, and the C∗ -algebra they generate decomposes into the tensor product A(K) ⊗ A(L)e . Indeed, it suﬃces to consider the case when K is ﬁnite dimensional, and then this is true as A(K) is a full matrix algebra. If in addition K and L are invariant subspaces for an operator A, 0 ≤ A ≤ 1, then ωA |A(K)⊗A(L)e = ωA |A(K) ⊗ ωA |A(L)e . If moreover H = K ⊕ L, the subalgebra A(K) ⊗ A(L)e of A(H) is the ﬁxed point algebra for the involutive Bogoliubov automorphism α1⊕−1 preserving ωA . In particular, there exists an ωA -preserving conditional expectation

240

13 Bogoliubov Automorphisms

onto A(K) ⊗ A(L)e , and by monotonicity and superadditivity of entropy, Theorem 3.2.2(v),(iv), we get hωA (αU ) ≥ hωA |A(K)⊗ωA |A(L)e (αU |A(K) ⊗ αU |A(L)e ) ≥ hωA |A(K) (αU |A(K) ) + hωA |A(L)e (αU |A(L)e ).

(13.12)

Lemma 13.4.3. Let Un be a unitary operator on a Hilbert space Hn , n ∈ N, and {zn }∞ on the space n=1 ⊂ T. Consider two unitary operators U and U ∞ H = ⊕n=1 Hn , ∞

∞

n=1

n=1

U = ⊕ Un and U = ⊕ zn Un . Then hω (αU ) = hω (αU ) for any αU - and αU -invariant state ω on A(H). The same holds for the restrictions of the automorphisms to the even CARalgebra A(H)e . Proof. Consider the unitary operator V = ⊕∞ n=1 zn 1. For each n ∈ N choose an increasing sequence {Hnk }∞ of ﬁnite-dimensional subspaces of Hn such k=1 that ∪k Hnk is dense in Hn . Set Kn = H1n ⊕ . . . ⊕ Hnn . Then Kn is ﬁnite-dimensional, Kn ⊂ Kn+1 , and ∪n Kn is dense in H. Since k V Kn = Kn and αU = αV ◦ αU = αU ◦ αV , we have αU (A(Kn )) = k αU (A(Kn ) for any k ∈ Z, whence hω (A(Kn ); αU ) = hω (A(Kn ); αU ). As {A(Kn )}n is an increasing sequence with dense union in A(H), it follows that hω (αU ) = hω (αU ). Similarly, using the algebras A(Kn )e instead of A(Kn ), we get the result for the even CAR-algebra.

Lemma 13.4.4. Let X1 , X2 be measurable subsets of T such that µ(X1 ) = µ(X2 ) > 0. Then there exist a countable measurable partition X1 = ∞ n=1 Yn ∞ of X1 and a sequence of numbers {zn }∞ n=1 ⊂ T such that X2 = n=1 zn Yn modulo a set of measure zero. Proof. It suﬃces to prove that there exists a subset Y ⊂ X1 of positive measure and z ∈ T such that zY ⊂ X2 . Then an application of Zorn’s lemma ﬁnishes the proof. Denoting by 1X1 and 1X2 the characteristic functions of the sets X1 and X2 we have 1X1 (w)1X2 (zw)dµ(w)dµ(z) = µ(X1 )µ(X2 ) > 0. T×T

Hence there exists z such that 1X1 (w)1X2 (zw)dµ(w) > 0, T

that is, zX1 ∩ X2 has positive measure. Now we can extend Proposition 13.3.1 to arbitrary unitaries.

13.4 Dynamical Entropy

241

Proposition 13.4.5. For ε > 0 and C, 0 < C < 1, choose δ > 0 as in Proposition 13.3.1. Let U be a unitary with absolutely continuous spectral measure and a direct integral decomposition ⊕ ⊕ H= Hz dµ(z), U = z dµ(z). T

T

Set X = {z ∈ T | Hz = 0}. Let A, 0 ≤ A ≤ 1 be an operator commuting with U such that Sp A ⊂ (λ0 − δ, λ0 + δ) for some λ0 ∈ (0, C). Then hωA (αU ) ≥ µ(X)(η (λ0 ) + η (1 − λ0 ) − ε). The same inequality is true for the entropy of the restriction of αU to A(H)e . −1 Proof. Let {nk }∞ k=1 ⊂ N be a sequence such that µ(X) = k nk . By Lemma 13.4.4 there exist a measurable partition X = ∞ k,m=1 Xkm and a set {zkm }∞ k,m=1 ⊂ T such that for all k ∈ N 2 3 1 1 = exp 2πi 0, zkm Xkm nk m ⊕ modulo sets of measure zero. Let Hkm = Xkm Hz dµ(z) be the spectral subspace of U corresponding to the set Xkm . Set Hk = ⊕m Hkm , and deﬁne a unitary operator Uk on Hk by Uk = ⊕ zkm U |Hkm . m

By construction the multiplicity function of Uk is positive on exp (2πi[0, n−1 k ]), so that Uknk has Lebesgue spectrum. By Lemma 13.4.3 and Proposition 13.3.1 we get 1 hω |A(Hk )e (αU nk |A(Hk )e ) k nk A 1 (η (λ0 ) + η (1 − λ0 ) − ε). ≥ nk

hωA |A(Hk )e (αU |A(Hk )e ) =

By (13.12) for any n ∈ N we get hωA (αU ) ≥ ≥

n

hωA |A(Hk )e (αU |A(Hk )e ) k=1 n k=1

1 nk

(η (λ0 ) + η (1 − λ0 ) − ε).

Letting n → ∞ we obtain the required estimate. The same proof works for the restriction of the automorphism to the even part of the algebra.

242

13 Bogoliubov Automorphisms

We are now ready to proof a half of Theorem 13.4.1. Proof of the lower bound for the entropy in Theorem 13.4.1. Since hωA (αU ) ≥ hωA |A(Ha ) (αU |A(Ha ) ) by (13.12), to prove the result it suﬃces to consider unitaries with absolutely continuous spectral measure. Assuming that U is such a unitary, let us ﬁrst show that if hωA (αU ) < ∞ then Az has pure point spectrum for almost all z ∈ T. Fix δ0 ∈ (0, 1/2) and take ε ∈ (0, η(δ0 )). Let δ be as in the formulation of Proposition 13.4.5 with C = 1 − δ0 . For any Borel subset X of R, let 1X (A) be the spectral projection of A corresponding to X. Then ⊕ 1X (A) = 1X (Az )dµ(z). T

Deﬁne a measurable function ϕX on T, 1, if 1X (Az ) = 0, ϕX (z) = 0, otherwise. By Proposition 13.4.5 if X is a Borel subset of (λ0 − δ, λ0 + δ) for some λ0 ∈ (δ0 , 1 − δ0 ) then hωA (αU |A(1X (A)H)e ) ≥ (η (λ0 ) + η (1 − λ0 ) − ε) ϕX (z)dµ(z) T

≥ η(1 − δ0 )

T

ϕX (z)dµ(z),

(13.13)

where we have used that η (λ0 ) + η (1 − λ0 ) ≥ η (δ0 ) + η (1 − δ0 ) in the second inequality. Let t0 = δ0 < t1 < . . . < tm = 1 − δ0 , tk − tk−1 < δ. Then, similarly to the proof of Proposition 13.4.5, using (13.12) we obtain from (13.13) the inequality m hωA (αU ) ≥ η(1 − δ0 ) ϕ(tk−1 ,tk ] (z)dµ(z). T k=1

If we let max(tk − tk−1 ) → 0 then the integrand in the expression above converges to inﬁnity at every point z ∈ T such that (δ0 , 1 − δ0 ) ∩ Sp Az is inﬁnite. Therefore if hωA (αU ) < ∞, then the intersection (δ0 , 1 − δ0 ) ∩ Sp Az is ﬁnite for almost all z ∈ T. Since δ0 is arbitrary, it follows that Az has pure point spectrum for almost all z ∈ T. Thus to get the lower bound for the entropy we may assume that Az has pure point spectrum for almost all z ∈ T, since otherwise both the entropy and the integral in the formulation of the theorem are inﬁnite. But then, see C.6, we have N H = ⊕ L2 (Xn , dµ), n=1

where Xn is a measurable subset of T, N ∈ N ∪ {∞}, and U and A act on L2 (Xn , dµ) as multiplications by functions z and λn (z), respectively. We must prove that

13.4 Dynamical Entropy

hωA (αU ) ≥

N n=1

243

(η (λn (z)) + η (1 − λn (z)))dµ(z).

Xn

Using (13.12) once again, we see that it suﬃces to estimate hωA (αU |A(H)e ) assuming N = 1. As in the proof above, ﬁxing δ0 > 0, ε > 0 and choosing t0 = δ0 < t1 < . . . < tm = 1 − δ0 , we get that hωA (αU |A(H)e ) ≥

m k=1

{z∈X1 | tk−1

A Series of Modern Surveys in Mathematics

Editorial Board 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 Managing Editor R. Remmert, Münster

Volume 50

Sergey Neshveyev Erling Størmer

Dynamical Entropy in Operator Algebras

123

Sergey Neshveyev Erling Størmer Department of Mathematics University of Oslo P. B. 1053 Blindern 0316 Oslo, Norway e-mail: [email protected] [email protected]

Library of Congress Control Number: 2006928835

Mathematics Subject Classiﬁcation (2000): 46L55, 28D20

ISSN 0071-1136 ISBN-10 3-540-34670-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-34670-8 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors using a Springer LATEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

46/3100YL - 5 4 3 2 1 0

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 lattice systems, that is rather straightforward, since in those models the mean entropy deﬁnition in the classical case can be extended to C∗ -algebras by replacing partitions by local algebras. But in more general cases the C∗ -algebras generated by ﬁnite dimensional C∗ -algebras can easily be inﬁnite dimensional, so the mean entropy cannot be used as a deﬁnition. In order to deﬁne dynamical entropy for automorphisms of C∗ -algebras one has to rewrite the classical 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 and Narnhofer and Thirring [50] extended the theory to states of C∗ -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 dynamical entropy and show its main properties, and follow this up in Chap. 5 with a deﬁnition, due to Sauvageot and Thouvenot [188], inspired by the classical 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; the ﬁrst contains chapters of general nature, while we in the second part 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. Ocneanu, 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Relative Entropy for Matrix Algebras . . . . . . . . . . . . . . . . . . . . . . 2.2 Von Neumann Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Relative Entropy for General C*-algebras . . . . . . . . . . . . . . . . . . . 2.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 21 26 30

3

Dynamical Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mutual Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Entropy of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Type I Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 48 54 57

4

Maximality of Entropy and Commutativity . . . . . . . . . . . . . . . . 4.1 Maximal Entropy of Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Independent Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Entropic K-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 66 69 74

5

Dynamical Abelian Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Entropy via Stationary Couplings . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Zero Entropy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 86 91

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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

In this chapter we include some background material on entropy in the classical commutative case. We essentially include all the results from the classical theory which will be needed later, proving those which will be used repeatedly and leaving more diﬃcult and rarely used ones without proof. In the last 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 = {X1 , . . . , Xn } be a ﬁnite measurable partition of X. The entropy of ξ is H(ξ) = η(µ(X1 )) + . . . + η(µ(Xn )), where η is the function deﬁned by −t log t, if t > 0, η(t) = 0, if t = 0, the logarithm being considered with base e(1) . We shall write Hµ (ξ) instead 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 1

In the probabilistic literature the logarithm with base 2 is often used.

4

1 Classical Dynamical Systems

number of measurable sets. Then X/ζ is again a Lebesgue space. The algebra L∞ (X/ζ, µ) can be identiﬁed with the von Neumann subalgebra of L∞ (X, µ) consisting of functions measurable with respect to a complete σalgebra Bζ . This way we get one-to-one correspondences between measurable partitions (considered up to a set of measure zero), von Neumann subalgebras of L∞ (X, µ) and complete σ-subalgebras of B. Let Eζ : L∞ (X) → L∞ (X/ζ) be the µ-preserving conditional expectation, that is, Eζ is determined by the condition X Eζ (f )g dµ = X f g dµ for f ∈ L∞ (X) and g ∈ L∞ (X/ζ). The conditional entropy is deﬁned as H(ξ|ζ) =

n k=1

X

η(Eζ (1Xk )(x))dµ(x) = −

n k=1

Xk

log Eζ (1Xk )(x)dµ(x),

where 1Y denotes the characteristic function of a subset Y ⊂ X. We shall often write Hµ (ξ|ζ) instead of H(ξ|ζ). If ζ = {Yl }m l=1 then Eζ (f ) =

m µ(f 1Yl ) 1Yl , µ(Yl ) l=1

where µ(g) =

X

g dµ for g ∈ L1 (X, µ), and therefore µ(Xk ∩ Yl ) µ(Yl ). η H(ξ|ζ) = µ(Yl ) k,l

In particular, if ν = {X} is the trivial partition then H(ξ|ν) = H(ξ). For measurable partitions ξ and ζ we write ξ ≺ ζ if L∞ (X/ξ) ⊂ L∞ (X/ζ). If ξ = {Xk }nk=1 and ζ = {Yl }m l=1 this means that up to a set of measure zero each Xk is the union of some of the Yl ’s. If {ξi }i∈I is a collection of measurable partitions then we denote by ∨i∈I ξi the measurable partition (deﬁned up to a set of measure zero) corresponding to the von Neumann algebra generated by L∞ (X/ξi ), i ∈ I. If ξ = {Xk }nk=1 and ζ = {Yl }m l=1 then the partition ξ ∨ ζ consists of the sets Xk ∩ Yl . Proposition 1.1.1. Conditional entropy has the following properties: (i) H(ξ|ζ) ≥ 0, with equality if and only if ξ ≺ ζ; (ii) H(ξ ∨ ϑ|ζ) = H(ξ|ζ) + H(ϑ|ξ ∨ ζ); (iii) H(ξ|ζ) is increasing in ξ and decreasing in ζ. Proof. It is clear that H(ξ|ζ) ≥ 0. Assume H(ξ|ζ) = 0. Let ξ = {Xk }nk=1 . Since the function η takes the value 0 only at 0 and 1, it follows that Eζ (1Xk ) = 1Yk for some Yk ⊂ X. We have µ(Yk ) = µ(Eζ (1Xk )) = µ(Xk ), and

1.1 Measure Entropy

5

µ(Yk ∩ Xk ) = µ(1Yk 1Xk ) = µ(1Yk Eζ (1Xk )) = µ(Yk ). Hence 1Xk = 1Yk ∈ L∞ (X/ζ), so that ξ ≺ ζ. Conversely, if ξ ≺ ζ then Eζ (1Xk ) = 1Xk , so that η(Eζ (1Xk )) = 0 and hence H(ξ|ζ) = 0. Thus (i) is proved. To prove (ii) assume ξ = {Xk }nk=1 . One can easily check that for any f ∈ L∞ (X) we have Eξ∨ζ (f ) =

n Eζ (f 1Xk ) 1Xk . Eζ (1Xk )

k=1

Hence log Eξ∨ζ (f ) =

n

(log Eζ (f 1Xk ) − log Eζ (1Xk ))1Xk ,

k=1

whence for ϑ = {Yl }m l=1 we get 1Yl log Eξ∨ζ (1Yl ) = 1Xk ∩Yl log Eζ (1Xk ∩Yl ) − 1Xk log Eζ (1Xk ). l

k,l

k

Integrating over X we get (ii). To show (iii) ﬁrst note that if ξ ≺ ϑ then ξ ∨ ϑ = ϑ, so it follows from (ii) that H(ξ|ζ) ≤ H(ϑ|ζ). Assume now that ζ ≺ ϑ. Then Eζ = Eζ ◦ Eϑ . Since the function η is concave, by Jensen’s inequality for any measurable Y ⊂ X we get Eζ (η(Eϑ (1Y ))) ≤ η((Eζ ◦ Eϑ )(1Y )) = η(Eζ (1Y )), whence

η(Eϑ (1Y ))dµ ≤

X

η(Eζ (1Y ))dµ. X

This implies that H(ξ|ϑ) ≤ H(ξ|ζ).

From parts (ii) and (iii) in the above proposition we see that if ξ, ϑ, ζ and ε are measurable partitions then H(ξ ∨ ϑ|ζ ∨ ε) ≤ H(ξ|ζ) + H(ϑ|ε).

(1.1)

In particular, entropy is subadditive, i.e., H(ξ ∨ ϑ) ≤ H(ξ) + H(ϑ). Let now T be a measure-preserving invertible transformation of a Lebesgue space (X, µ). In such a case we say that (X, µ, T ) is a dynamical system. It will sometimes be convenient to assume that T is a homeomorphism of a compact metric space. More precisely, if (X, µ, T ) is a dynamical system then by a topological model for it we mean a homeomorphism S of a compact metric space Y together with an S-invariant probability measure ν such that (X, µ, T ) and (Y, ν, S) are isomorphic. Topological models always exist: take

6

1 Classical Dynamical Systems

the spectrum of any weakly operator dense invariant separable C∗ -subalgebra of L∞ (X, µ) for Y . Let (X, µ, T ) be a dynamical system. For a ﬁnite measurable partition ξ = {X1 , . . . , Xm } set 1 H(ξ ∨ T ξ ∨ . . . ∨ T n−1 ξ), n→∞ n

h(ξ; T ) = lim

where T ξ = {T X1 , . . . , T Xm }. Note that the limit exists by virtue of subadditivity of entropy and the following standard result. Lemma 1.1.2. Let {an }∞ n=1 be a sequence of real numbers such that an+m ≤ an + am for any n, m ∈ N. Then the limit of the sequence {an /n}n exists and coincides with the inﬁmum. Proof. Fix m ∈ N. Any n ∈ N can be written as n = km + l for some k and l, 0 ≤ l < m. By assumption an ≤ kam + al . It follows that

an am ≤ . m n→∞ n Since this is true for every m, we get the result. lim sup

In fact with a bit more work one can show that monotonically decreasing. Indeed, we have

k ∞ {H(∨n−1 k=0 T ξ)/n}n=1

is

n−1 k k H(∨n−1 k=0 T ξ) = H(ξ) + H(∨k=1 T ξ|ξ) k −1 = H(ξ) + H(∨n−2 ξ) k=0 T ξ|T n−1

= ... =

l H(ξ| ∨−1 l=−k T ξ).

(1.2)

k=0 l Since the sequence {H(ξ| ∨−1 l=−k T ξ)}k is monotonically decreasing by Propok sition 1.1.1(iii), the sequence {H(∨n−1 k=0 T ξ)/n}n is also monotonically decreasing. Furthermore, by the martingale convergence theorem for any f ∈ L1 (X, µ) we have E∨−1 T k ξ (f ) → E∨−1 T k ξ (f ) a.e. So denoting the partik=−n

k − we get tion ∨−1 k=−∞ T ξ by ξ

k=−∞

k − h(ξ; T ) = lim H(ξ| ∨−1 k=−n T ξ) = H(ξ|ξ ). n→∞

(1.3)

The entropy of T is now deﬁned as h(T ) = sup h(ξ; T ), ξ

where the supremum is taken over all ﬁnite measurable partitions of (X, µ). We shall write hµ (ξ; T ) and hµ (T ) instead of h(ξ; T ) and h(T ) when we want to stress that we deal with measure µ. We need a more practical way of computing entropy than taking supremum over all ﬁnite partitions.

1.1 Measure Entropy

7

Lemma 1.1.3. Let ξ and ζ be ﬁnite measurable partitions. Denote by ζ ± the partition ∨k∈Z T k ξ. Then h(ξ; T ) ≤ h(ζ; T ) + H(ξ|ζ ± ). Proof. We have n−1 k n−1 k n−1 k n−1 k k H(∨n−1 k=0 T ξ) ≤ H(∨k=0 T (ξ ∨ ζ)) = H(∨k=0 T ζ) + H(∨k=0 T ξ| ∨k=0 T ζ)

and by (1.1) n−1 k k H(∨n−1 k=0 T ξ| ∨k=0 T ζ) ≤

n−1

H(T k ξ|T k ζ) = nH(ξ|ζ).

k=0

Therefore h(ξ; T ) ≤ h(ζ; T ) + H(ξ|ζ). Since also

h(∨nk=−n T k ζ; T )

= h(ζ; T ) for any n, we then get

h(ξ; T ) ≤ h(ζ; T ) + H(ξ| ∨nk=−n T k ζ). Letting n → ∞ we obtain the result.

±

A measurable partition ξ is called generating if ξ is the partition of X into points. From the above lemma we then get the following result. Theorem 1.1.4. (Kolmogorov-Sinai) Let ξ be a ﬁnite generating partition for a dynamical system (X, µ, T ). Then h(T ) = h(ξ; T ).

The following example was motivating for the whole theory. Example 1.1.5. Let ν be a probability measure on {1, . . . , n}, ν({k}) = λk . Let X = {1, . . . , n}Z , µ the product measure on X and T the shift to the right. The system (X, µ, T ) is called the Bernoulli shift with weights λ1 , . . . , λn . Consider the partition ξ = {X1 , . . . , Xn }, where Xk is the cylindrical set consisting of x = {xn }n∈Z ∈ X such that x0 = k. Then ξ is generating and we get h(T ) = h(ξ; T ) = k η(λk ). The following two theorems are fundamental results on Bernoulli shifts. Theorem 1.1.6. (Ornstein) Bernoulli shifts with the same entropy are isomorphic.

Recall that a system (X, µ, T ) is called ergodic if any T -invariant measurable subset of X has measure either zero or one. If ξ is a T -invariant measurable partition then the system (X/ξ, µ, T ) is called a factor system of (X, µ, T ).

8

1 Classical Dynamical Systems

Theorem 1.1.7. (Sinai) Let (X, µ, T ) be an ergodic dynamical system with entropy h(T ) > 0. Then for any ﬁnite h, 0 < h ≤ h(T ), there exists a factor system which is isomorphic to the Bernoulli shift with entropy h.

Let us now list the basic properties of entropy. Theorem 1.1.8. For any dynamical system (X, µ, T ) we have: (i) h(T n ) = |n|h(T ) for n ∈ Z; (ii) if (Y, ν, S) is a factor system then h(T ) ≥ h(S); (iii) if Y ⊂ X is T -invariant then h(T ) = µ(Y )h(T |Y ) + µ(Y c )h(T |Y c ), where we consider the measures µ(Y )−1 µ|Y and µ(Y c )−1 µ|Y c on Y and Y c = X\Y , respectively; (iv) if (Y, ν, S) is another dynamical system then h(T × S) = h(T ) + h(S).

We shall next show properties of H(ξ) which will be useful later. Proposition 1.1.9. For ﬁnite measurable partitions we have: (i) if ξ = {X1 , . . . , Xn } then H(ξ) ≤ log n, and equality holds if and only if µ(Xk ) = 1/n for all k; (ii) H(ξ ∨ ζ ∨ ϑ) + H(ζ) ≤ H(ξ ∨ ζ) + H(ζ ∨ ϑ); (iii) H(ξ ∨ ζ) ≤ H(ξ) + H(ζ), and equality holds if and only if ξ and ζ are independent, that is, µ(f g) = µ(f )µ(g) for f ∈ L∞ (X/ξ) and g ∈ L∞ (X/ζ); n n n (iv) k=1 H(ξk ) − H(∨nk=1 ξk ) ≤ k=1 H(ζk ) − H(∨k=1 ζk ) if ξk ≺ ζk for k = 1, . . . , n. Proof. Since η is strictly concave,

η(µ(Xk )) ≤ nη

µ(Xk )

k

k

n

= log n,

moreover, equality holds if and only if all µ(Xk )’s are equal, that is, µ(Xk ) = 1/n. Thus (i) is proved. We have by Proposition 1.1.1(ii) H(ξ ∨ ζ ∨ ϑ) − H(ξ ∨ ζ) = H(ϑ|ξ ∨ ζ) and H(ζ ∨ ϑ) − H(ζ) = H(ϑ|ζ). Since H(ϑ|ζ) ≥ H(ϑ|ξ ∨ ζ), this proves (ii). Since H(ξ ∨ ζ) = H(ξ) + H(ζ|ξ), to prove (iii) we have to show that H(ζ|ξ) = H(ζ) if and only if ξ and ζ are independent. Let ζ = {X1 , . . . , Xn }. Since η is strictly concave, H(ζ|ξ) = η(Eξ (1Xk ))dµ ≤ η Eξ (1Xk )dµ = η(µ(Xk )), k

X

k

X

k

1.2 Topological Entropy

9

and equality holds if and only if Eξ (1Xk ) is constant a.e. for all k, that is, Eξ (1Xk ) = µ(Xk ). But the latter condition means exactly that µ(1Xk g) = µ(Xk )µ(g) for any g ∈ L∞ (X/ξ), that is, ξ and ζ are independent. We have by (1.1) H(∨k ζk | ∨k ξk ) ≤ H(ζk |ξk ). k

On the other hand, assuming ξk ≺ ζk , we have H(∨k ζk | ∨k ξk ) = H(∨k ζk ) − H(∨k ξk ) and H(ζk |ξk ) = H(ζk ) − H(ξk ). This gives (iv).

It is useful to note that subadditivity of entropy is equivalent to the following number inequality. Lemma 1.1.10. Let {λi1 ...in }i1 ∈I1 ,...,in ∈In be a ﬁnite set of nonnegative num (k) bers such that i1 ,...,in λi1 ...in = 1. Set λik = i1 ,...,ik−1 ,ik+1 ,...,in λi1 ...in . Then n (k) η(λi1 ...in ) ≤ η(λik ) k=1 ik ∈Ik

i1 ,...,in

Proof. Apply the subadditivity inequality H(∨k ξk ) ≤ k H(ξk ) to the partitions ξk of I1 × . . . × In consisting of the sets I1 × . . . × Ik−1 × {ik } × Ik+1 × In , k = 1, . . . , n, and the measure µ deﬁned by µ({(i1 , . . . , in )}) = λi1 ...in .

1.2 Topological Entropy Let T be a homeomorphism of a compact separable space X. We call (X, T ) a topological dynamical system. Let U = {Ui }i be an open cover of X. Denote by N (U) the minimal number of elements of U needed to cover X. If U = {Ui }i and V = {Vj }j are two open covers, we denote by U ∨ V the open cover formed by the sets Ui ∩ Vj . We obviously have N (U ∨ V) ≤ N (U)N (V). Deﬁne 1 h(U; T ) = lim log N (U ∨ T U ∨ . . . ∨ T n−1 U). n→∞ n The quantity htop (T ) = sup h(U; T ), U

where the supremum is taken over all open covers of X, is called the topological entropy of T . An equivalent deﬁnition is given in terms of separating and spanning sets. Let d be a compatible metric on X. Take ε > 0. A subset F of X is called (n, ε)-spanning with respect to T if for every x ∈ X there exists y ∈ F such that

10

1 Classical Dynamical Systems

max

0≤k≤n−1

d(T k x, T k y) < ε,

and it is called (n, ε)-separated if for any x, y ∈ F , x = y, we have max

0≤k≤n−1

d(T k x, T k y) ≥ ε.

Let spn (ε) be the minimal cardinality of an (n, ε)-spanning set, and srn (ε) the maximal cardinality of an (n, ε)-separated set. If U is an open cover with Lebesgue number ε (that is, any open ball of radius ε is contained in one of the elements of U), and V an open cover with diameter less than ε (that is, the diameter of every element in V is less than ε), then one can check that n−1 k k N (∨n−1 k=0 T U) ≤ spn (ε) ≤ srn (ε) ≤ N (∨k=0 T V).

It follows that 1 1 log spn (ε) = lim lim sup log spn (ε) ε↓0 n→∞ n n 1 1 = lim lim inf log srn (ε) = lim lim sup log srn (ε). n→∞ ε↓0 ε↓0 n→∞ n n

htop (T ) = lim lim inf ε↓0 n→∞

This gives an intuitive interpretation of topological entropy as a measure of the asymptotic distortion of the iterates of T along orbits. The main properties of topological entropy are as follows. Theorem 1.2.1. For every topological dynamical system (X, T ) we have: (i) htop (T n ) = |n|htop (T ) for n ∈ Z; (ii) if (X, T ) → (Y, S) is a factor map then htop (T ) ≥ htop (S); (iii) if Y ⊂ X is a closed T -invariant subset then htop (T ) ≥ htop (T |Y ), and if in addition Y is open then htop (T ) = max{htop (T |Y ), htop (T |X\Y )}; (iv) if (Y, S) is another topological dynamical system then htop (T × S) = htop (T ) + htop (S); (v) if U is an open cover such that the diameter of ∨nk=−n T k U with respect to some compatible metric tends to zero as n → ∞ then htop (T ) = h(U; T ).

The relation between measure and topological entropy is given by the variational principle. Theorem 1.2.2. For every topological dynamical system (X, T ) we have htop (T ) = sup hµ (T ), µ

where the supremum is taken over all T -invariant probability measures.

1.3 Entropy via Partitions of Unity

11

A topological dynamical system (X, T ) is called uniquely ergodic if there exists a unique T -invariant probability measure µ. Then by the variational principle hµ (T ) = htop (T ). In Sect. 1.1 we remarked that any dynamical system has a topological model. The following theorem gives a much more precise result for ergodic systems. Theorem 1.2.3. (Jewett-Krieger) Every ergodic dynamical system has a uniquely ergodic topological model.

Finishing our discussion of connections between measure entropy and topological entropy we shall give an equivalent deﬁnition of measure entropy in terms of spanning sets. Let (X, T ) be a topological dynamical system, and µ a T -invariant probability measure, which for simplicity we assume to be ergodic. A subset F of X is called (n, ε, δ)-spanning for (T, µ) if the set of x ∈ X such that there exists y ∈ F with max d(T k x, T k y) < ε 0≤k≤n−1

has measure greater than δ. Denote by spn (ε; δ) the minimal cardinality of an (n, ε, δ)-spanning set. Then for any δ ∈ (0, 1) hµ (T ) = lim lim inf ε↓0 n→∞

1 log spn (ε; δ). n

The proof can be obtained from the following theorem. Theorem 1.2.4. (Shannon-McMillan-Breiman) Let (X, µ, T ) be an ergodic dynamical system, ξ a ﬁnite measurable partition of X. For x ∈ X and n ∈ N k denote by In (x) the element of the partition ∨n−1 k=0 T ξ containing x. Then the convergence log µ(In (x)) − → h(ξ; T ) n holds in mean and almost everywhere.

1.3 Entropy via Partitions of Unity We shall in the present section rewrite the deﬁnition of H(∨nk=1 ξk ) in a form which is more suitable for generalization to the noncommutative case. Since the algebra generated by noncommuting ﬁnite dimensional algebras can be inﬁnite dimensional, we shall consider H(∨nk=1 ξk ) as a function of n partitions. Consider the case n = 1, so we are dealing with one partition ξ = {Xk }k . Then 1 = k 1Xk is a partition of unity in L∞ (X/ξ), and

12

1 Classical Dynamical Systems

H(ξ) =

η(µ(1Xk )).

k

This expression makes perfect sense for ﬁnite dimensional noncommutative algebras. However, in the noncommutative setting it is diﬃcult to form new partitions of unity consisting of projections from given noncommuting ones. Hence a natural question is how we can express H(ξ) using arbitrary ﬁnite decompositions 1 = i fi , fi ∈ L∞ (X/ξ), fi ≥ 0. The quantity i η(µ(fi )) can be arbitrarily large. So we have to compensate for the fact that the fi ’s are not necessarily projections. The function η(fi ) is then a natural ingredient of a correction term as it vanishes exactly when fi is a projection. Lemma 1.3.1. We have H(ξ) = sup

(η(µ(fi )) − µ(η(fi ))),

i

where the supremum is taken over all ﬁnite families {fi }i of positive functions in L∞ (X/ξ) with sum 1. Proof. We just have to check that the supremum is not larger thanH(ξ). Let n ∞ ξ = {X } and f ∈ L (X/ξ), 0 ≤ f ≤ 1, k i i i fi = 1. Then fi = k λik 1Xk k=1 with i λik = 1. Using that η(x + y) ≤ η(x) + η(y) for any positive numbers x and y we compute: (η(µ(fi )) − µ(η(fi ))) = η(µ(fi )) − µ(η(λik 1Xk )) i

i

≤

i,k

(η(µ(λik 1Xk )) − µ(η(λik 1Xk )))

i,k

=

(η(λik µ(Xk )) − η(λik )µ(Xk ))

i,k

=

η(µ(Xk )),

k

which proves the lemma.

Given a partition of unity 1 = i fi in L∞ (X) we can form a partition of unity in L∞ (X/ξ) by applying the conditional expectation Eξ . Thus H(ξ) is the supremum of (η(µ(fi )) − µ(η(Eξ (fi )))) i

over all partitions of unity 1 =

i

fi in L∞ (X).

Note now that

f (log f − log µ(f ))dµ.

η(µ(f )) − µ(η(f )) = X

1.3 Entropy via Partitions of Unity

13

For arbitrary positive functions f and g in L1 (X, µ) the quantity S(f, g) = f (log f − log g)dµ X

is called the relative entropy of the measures f dµ and g dµ. (k)

Next we turn to the case of n partitions ξ1 , . . . , ξn . Let ξk = {Xjk }jk ∈Jk . Then H(∨k ξk ) = η(µ(1X (1) . . . 1X (n) )). j1

j1 ,...,jn

jn

In other words, we consider the partitions of unity consisting of projections 1X (k) and form a new partition consisting of products of these projk

jections. This is not easy to generalize to the noncommutative case. So instead of trying to form a new partition of unity we consider all possible ﬁnite partitions 1 = i1 ,...,in fi1 ...in and think of them as coming from n parti (k) (k) tions 1 = ik fik , k = 1, . . . , n, wherefik is the sum of the fi1 ...in ’s over i1 , . . . , ik−1 , ik+1 , . . . , in . We then take i1 ,...,in η(µ(fi1 ...in )) as the ﬁrst approximation of the entropy and deﬁne the correction term as the sum of the correction terms for the above n partitions of unity. We thus arrive at the following expression. Proposition 1.3.2. Let ξ1 , . . . , ξn be ﬁnite measurable partitions of X. Then ⎧ ⎫ n ⎨ ⎬ (k) η(µ(fi1 ...in )) − µ(η(Eξk (fik ))) , H(∨nk=1 ξk ) = sup ⎩ ⎭ i1 ,...,in

k=1 ik

where the supremum is taken over all ﬁnite families {fi1 ...in }i1 ,...,in of positive functions in L∞ (X) with sum 1. Proof. Let ξ = ∨nk=1 ξk . Let 1 = i1 ,...,in fi1 ...in be a partition of unity. Since µ ◦ Eξ = µ, by Lemma 1.3.1 H(ξ) ≥ η(µ(fi1 ...in )) − µ(η(Eξ (fi1 ...in ))). i1 ,...,in

i1 ,...,in

Apply Lemma 1.1.10 to the numbers Eξ (fi1 ...in )(x), x ∈ X, and then integrate over X. We then see that the above expression is not smaller than i1 ,...,in

η(µ(fi1 ...in )) −

n

(k)

µ(η(Eξ (fik ))).

k=1 ik

As was already used in the proof of Proposition 1.1.1(iii), since ξk ≺ ξ and therefore Eξk = Eξk ◦ Eξ , it follows from Jensen’s inequality that µ(η(Eξ (f ))) ≤ µ(η(Eξk (f ))) for any f ≥ 0. Hence the above is expression is not smaller than

14

1 Classical Dynamical Systems

i1 ,...,in

η(µ(fi1 ...in )) −

n

(k)

µ(η(Eξk (fik ))).

k=1 ik

Thus the left side of the formula in the proposition majorizes the right side. (k) For the opposite inequality assume ξk = {Xik }ik and let fi1 ...in = 1X (1) ∩...∩X (n) . i1

(k)

in

(k)

Then fik = 1X (k) , so that η(Eξk (fik )) = 0. Thus the right side with these ik

fi1 ...in ’s equals H(ξ).

1.4 Notes As our basic reference on entropy we refer the reader to the book by Glasner [72], which contains all the theorems we stated without proofs. For more on Lebesgue spaces, introduced by Rohlin in [181], one can consult [1] or [184]. Entropy was introduced by Kolmogorov [111], who was inspired by Shannon’s information theory [195] and motivated by the isomorphism problem for Bernoulli shifts. The deﬁnition was improved by Sinai [199]. The isomorphism theorem for Bernoulli shifts with the same entropy was proved by Ornstein [150]. He developed a powerful theory of ﬁnitely determined processes giving several dynamical characterizations of Bernoulli shifts. A more constructive proof was given by Keane and Smorodinsky [104], see also [53] for an exposition of their proof. Theorem 1.1.7, due to Sinai, was proved earlier [200], and then it stated that any Bernoulli shift with entropy h ≤ h(T ) can be obtained as a factor system. The notion of topological entropy in terms of open coverings was introduced by Adler, Konheim and McAndrew [2]. The deﬁnition in terms of spanning and separated sets was given by Bowen [29]. The variational principle, Theorem 1.2.2, was proved by Goodwyn [82] and Dinaburg [55]. A stronger version of the variational principle has been obtained by Blanchard, Glasner and Host [26], [72, Chapter 17]. Theorem 1.2.3 is due to Jewett [96] and Krieger [113]. The deﬁnition of measure entropy in terms of spanning sets was proposed by Katok [99], see also [166] for a proof of equivalence of that deﬁnition to the usual one. Theorem 1.2.4 was obtained by Shannon [195], McMillan [121] and Breiman [31] who established convergence in measure, in mean and ﬁnally almost everywhere, respectively. The formula for entropy in the last section, obtained by Connes and Størmer [51], will be the basis for the deﬁnition of entropy in the noncommutative setting in Chap. 3.

2 Relative Entropy

Relative entropy of measures has appeared naturally in the expression for entropy of partitions given in Proposition 1.3.2. In the present chapter we shall deﬁne relative entropy for states on C∗ -algebras and show its properties needed in the sequel. We shall mainly consider ﬁnite dimensional C∗ -algebras.

2.1 Relative Entropy for Matrix Algebras Let A be a ﬁnite dimensional C∗ -algebra. Denote by TrA the canonical trace on A, so TrA (p) = 1 for every minimal projection p ∈ A. The algebra A is a direct sum of full matrix algebras, and the restriction of TrA to each summand is the usual matricial trace. We shall write Tr instead of TrA if no confusion can arise. Each positive linear functional ϕ on A is of the form ϕ = Tr(· Qϕ ) for a unique positive element Qϕ ∈ A called the density operator. Deﬁnition 2.1.1. If ϕ and ψ are positive linear functionals on A, then the relative entropy of ϕ and ψ is Tr(Qϕ (log Qϕ − log Qψ )), if ϕ ≤ λψ for some λ > 0, S(ϕ, ψ) = +∞, otherwise. Since A is ﬁnite dimensional, ϕ ≤ λψ for some λ > 0 if and only if supp ϕ ≤ supp ψ, that is, ϕ(a) = 0 for a positive a ∈ A as soon as ψ(a) = 0. Two identities are immediate from the deﬁnition: S(λ1 ϕ, λ2 ψ) = λ1 S(ϕ, ψ) + λ1 ϕ(1) log S(ϕ, ψ) =

λ1 if λ1 , λ2 > 0, λ2

d Tr(Qtϕ Q1−t ψ )|t=1 if ϕ ≤ λψ. dt

We next list the main properties of relative entropy.

(2.1) (2.2)

16

2 Relative Entropy

Theorem 2.1.2. We have: (i) S(ϕ, ψ) ≥ 0 for states ϕ and ψ, and equality holds if and only if ϕ = ψ; (ii) S(ϕ, ψ) is decreasing in ψ, i.e., S(ϕ, ψ1 ) ≥ S(ϕ, ψ2 ) if ψ1 ≤ ψ2 ; (iii) S(ϕ, ψ) is a lower semicontinuous function of (ϕ, ψ); moreover, it is continuous on the sets {(ϕ, ψ) | ϕ ≤ λψ}, λ > 0; (iv) S(ϕ, ψ) is a convex function of (ϕ, ψ); (v) if α: B → A is a unital Schwarz map, then S(ϕ ◦ α, ψ ◦ α) ≤ S(ϕ, ψ); (vi) if ϕ and ψ are states on A, B is a C∗ -subalgebra of A, and E: A → B is a ψ-preserving conditional expectation, then S(ϕ, ψ) = S(ϕ|B , ψ|B ) + S(ϕ, ϕ ◦ E); n (vii)if ϕ = i=1 ϕi , then i S(ϕi , ψ) = S(ϕ, ψ) + i S(ϕi , ϕ). Recall that if A and B are C∗ -algebras, a linear map α: B → A is said to be a Schwarz map if α(b∗ b) ≥ α(b)∗ α(b) for each b ∈ B. Then clearly α is positive. The map α is completely positive if α ⊗ id: B ⊗ Matn (C) → A ⊗ Matn (C) is positive for all natural numbers n, where Matn (C) denotes the complex n × n matrices. See App. A, where we have collected some basic facts on such maps. In particular, unital completely positive maps are Schwarz maps by A.2. Recall also that if B ⊂ A, then a conditional expectation of A onto B is a unital positive map E: A → B such that E(bac) = bE(a)c for all b, c ∈ B and a ∈ A. Then E is completely positive, see A.13. If A is ﬁnite dimensional, there always exists a unique TrA -preserving conditional expectation EB of A onto B. However, for an arbitrary state ψ on A a ψ-preserving conditional expectation may not exist. Parts (iv) and (v) of Theorem 2.1.2 require nontrivial estimates. In both cases the proof will be based on (2.2) and the following operator inequality. Proposition 2.1.3. Let x1 and x2 be bounded positive operators on a Hilbert space K, and yi , zi , i = 1, 2, bounded positive operators on a Hilbert space H such that x1 x2 = x2 x1 , y1 y2 = y2 y1 , z1 z2 = z2 z1 . Suppose v: K → H is a bounded operator such that xi ≥ v ∗ (yi + zi )v, i = 1, 2. Then for t ∈ [0, 1].

xt1 x1−t ≥ v ∗ (y1t y21−t + z1t z21−t )v 2

2.1 Relative Entropy for Matrix Algebras

17

Proof. If we can show the proposition for xi replaced by xi + ε1 for ε > 0 then by continuity the conclusion of the proposition follows when ε → 0. We may thus assume x1 and x2 are invertible. We ﬁrst consider the case t = 1/2. Let 1/2 1/2

1/2 1/2

1/2 1/2

x = x1 x2 , y = y1 y2 , z = z1 z2 . Put c = v ∗ (y + z)v, and let ζ, ξ ∈ K. Then by using the Cauchy-Schwarz inequality twice together with the assumption of the proposition we have |(cξ, ζ)| = |((y + z)vξ, vζ)| 1/2

1/2

1/2

1/2

≤ |(y1 vξ, y2 vζ)| + |(z1 vξ, z2 vζ)| ≤ (y1 vξ, vξ)1/2 (y2 vζ, vζ)1/2 + (z1 vξ, vξ)1/2 (z2 vζ, vζ)1/2 ≤ ((y1 + z1 )vξ, vξ)1/2 ((y2 + z2 )vζ, vζ)1/2 ≤ (x1 ξ, ξ)1/2 (x2 ζ, ζ)1/2 . −1/2

Replacing ξ by x1

−1/2

ξ and ζ by x2 −1/2

|(x2 −1/2

−1/2

cx1

ζ in the above inequality, we get

ξ, ζ)| ≤ ξ ζ.

−1/2

−1/2

−1/2

Hence x2 cx1 ≤ 1. In particular, the spectrum of x2 cx1 is contained in the unit disc. Recall that the nonzero parts of the spectra of the operators ab and ba coincide (indeed, if ab − 1 is invertible, then ba − 1 is invertible and (ba − 1)−1 = b(ab − 1)−1 a − 1). Since x1 and x2 commute, we thus have −1/4 −1/4 −1/4 −1/4 x2 cx1 x2 )\{0} −1/2 −1/2 Sp(x2 cx1 )\{0}.

Sp(x−1/2 cx−1/2 )\{0} = Sp(x1 =

Hence x−1/2 cx−1/2 ≤ 1. Thus c ≤ x, so that 1/2 1/2

v ∗ (y1 y2

1/2 1/2

1/2 1/2

+ z1 z2 )v ≤ x1 x2 .

Assuming that the proposition is true for exponents s and t, apply the t 1−t t 1−t s 1−s , y1s y21−s , z1s z21−s instead of above inequality to xt1 x1−t 2 , y1 y2 , z1 z2 , x1 x2 x1 , y1 , z1 , x2 , y2 , z2 . Then we see that the proposition holds for the exponent (s + t)/2. Hence it holds for all dyadic numbers and by continuity, for every t ∈ [0, 1].

Proof of Theorem 2.1.2. Note that for any numbers x, y ≥ 0 we have x(log x − log y) ≥ x − y,

(2.3)

and equality holds only if x = y, which is just another way of writing the inequality log t ≤ t − 1 for t = y/x. Now let ϕ and ψ be states, and Qϕ = λ p and Q = ψ i i i j µj qj the spectral decompositions. It follows that

18

2 Relative Entropy

λi (log λi − log µj )Tr(pi qj ) ≥ (λi − µj )Tr(pi qj ). Summing over i and j and assuming that ϕ(1) = ψ(1) = 1 we get S(ϕ, ψ) ≥ 0. Moreover, if equality holds then (λi − µj )pi qj = 0, so that summing over i and j again, we get Qϕ = Qψ . Thus (i) is proved. Part (ii) is a consequence of operator monotonicity of log, see B.4. To prove (iii) assume ϕn → ϕ, ψn → ψ. If ψ is faithful, then ψn is also faithful for suﬃciently large n, and the equality S(ϕ, ψ) = −Tr(η(Qϕ )) − ϕ(log Qψ ), where η(t) = −t log t, shows that S(ϕn , ψn ) → S(ϕ, ψ). In general, by (ii) we have for ε > 0 S(ϕn , ψn ) ≥ S(ϕn , ψn + εTr) → S(ϕ, ψ + εTr), and since S(ϕ, ψ + εTr) = −Tr(η(Qϕ )) − ϕ(log(Qψ + ε1)) S(ϕ, ψ) as ε → 0, we conclude that lim inf S(ϕn , ψn ) ≥ S(ϕ, ψ). n

If in addition ϕn ≤ λψn , then 0 ≤ S(ϕn , ψn ) − S(ϕn , ψn + εTr) = ϕn (log(Qψn + ε1) − log Qψn ) ≤ λψn (log(Qψn + ε1) − log Qψn ) → λψ(log(Qψ + ε1) − log Qψ ), and since the latter expression tends to zero as ε → 0, we see that S(ϕn , ψn ) → S(ϕ, ψ). To prove (iv), by (2.2) it suﬃces to show (for x = 1) that for x ∈ A and t ∈ [0, 1] Tr(x∗ at xb1−t ) is a concave function of (a, b) ∈ A+ × A+ .

(2.4)

For this consider the Hilbert space H = L2 (A, Tr), that is, the space A with inner product (Λ(y), Λ(z)) = Tr(z ∗ y), where we denote by Λ(y) the element y ∈ A viewed as a vector in H. Assume a = λa1 + (1 − λ)a2 and b = λb1 + (1 − λ)b2 . Deﬁne operators on H by x1 Λ(y) = Λ(ay), y1 Λ(y) = Λ(λa1 y), z1 Λ(y) = Λ((1 − λ)a2 y), x2 Λ(y) = Λ(yb), y2 Λ(y) = Λ(λyb1 ), z2 Λ(y) = Λ((1 − λ)yb2 ). Then x1 x2 = x2 x1 , y1 y2 = y2 y2 , z1 z2 = z2 z1 . We have yi + zi = xi , i = 1, 2, so by Proposition 2.1.3 applied to v = 1 xt1 x1−t ≥ y1t y21−t + z1t z21−t . 2

2.1 Relative Entropy for Matrix Algebras

19

Hence for x ∈ A Tr(x∗ at xb1−t ) = (Λ(xb1−t ), Λ(at x)) = (xt1 x1−t 2 Λ(x), Λ(x)) ≥ ((y1t y21−t + z1t z21−t )Λ(x), Λ(x)) 1−t ∗ t = λTr(x∗ at1 xb1−t 1 ) + (1 − λ)Tr(x a2 xb2 ), and (2.4) is proved. Similarly, by virtue of (2.2) in order to prove (v) it suﬃces to show (for x = 1) that for x ∈ B and t ∈ [0, 1] 1−t ∗ t TrB (x∗ Qtϕ◦α xQ1−t ψ◦α ) ≥ TrA (α(x) Qϕ α(x)Qψ ).

(2.5)

Deﬁne a linear operator v: L2 (B, TrB ) → L2 (A, TrA ) by vΛ(y) = Λ(α(y)), y ∈ B. For y ∈ B, z ∈ A deﬁne x1 Λ(y) = Λ(Qϕ◦α y), y1 Λ(z) = Λ(Qϕ z), x2 Λ(y) = Λ(yQψ◦α ), y2 Λ(z) = Λ(zQψ ), We have

(x1 Λ(y), Λ(y)) = TrB (y ∗ Qϕ◦α y) = ϕ(α(yy ∗ )), (v ∗ y1 vΛ(y), Λ(y)) = TrA (α(y)∗ Qϕ α(y)) = ϕ(α(y)α(y)∗ ).

Since α is a Schwarz map, it follows that x1 ≥ v ∗ y1 v. Similarly we have x2 ≥ v ∗ y2 v. Hence by Proposition 2.1.3, with z1 = z2 = 0, ∗ t 1−t (xt1 x1−t 2 Λ(x), Λ(x)) ≥ (v y1 y2 vΛ(x), Λ(x)),

which is exactly (2.5). To prove (vi), ﬁrst of all note that both sides of the equality are simultaneously either ﬁnite or inﬁnite. Indeed, if supp ϕ ≤ supp ψ, then obviously supp ϕ|B ≤ supp ψ|B . Furthermore, if 0 ≤ a ≤ supp ψ then by A.13 the support s(E(a)) of E(a) is not smaller than s(a), so ϕ(E(a)) > 0 if ϕ(a) > 0, and thus supp ϕ ≤ supp ϕ ◦ E. Conversely, suppose supp ϕ|B ≤ supp ψ|B and supp ϕ ≤ supp ϕ ◦ E. Then if ϕ(a) > 0 for some a ≥ 0, we have ϕ(E(a)) > 0, hence ψ(a) = ψ(E(a)) > 0, so that supp ϕ ≤ supp ψ. Thus we may assume that ϕ ≤ λψ, ϕ ≤ λϕ ◦ E. Let F : A → B be any faithful conditional expectation, and ω a faithful state on A. Then using (iii) we may replace E by Eε = (1 − ε)E + εF and ψ by ((1 − ε)ψ + εω) ◦ Eε , and thus assume that ψ and E are faithful. Replacing further ϕ by (1 − ε)ϕ + εψ, we may assume that ϕ is faithful as well. We claim that

20

2 Relative Entropy −t t Qtϕ◦E Q−t ψ = Qϕ|B Qψ|B .

Then, applying ϕ to both sides and taking the derivative at t = 0, we get S(ϕ|B , ψ|B ) on the right side and ϕ(log Qϕ◦E − log Qψ ) = S(ϕ, ψ) − S(ϕ, ϕ ◦ E) on the left side, proving (vi). It remains to prove the claim. In other words, we must prove that −t t Qtϕ Q−t ψ = Qϕ|B Qψ|B

(2.6)

if ϕ = ϕ ◦ E and ψ = ψ ◦ E. This is a well-known general property of Connes’ Radon-Nikodym cocycle, but an elementary proof of our particular case is possible. We shall reduce the proof of (2.6) to the case when B is abelian. The identity t −t t −t −1 Qtϕ Q−t ψ = Qϕ Qω (Qψ Qω ) for a state ω shows that to prove (2.6) for arbitrary ϕ and ψ, it suﬃces to consider some ﬁxed ψ. Thus we can assume that ψ|B is a trace. Choose a maximal abelian subalgebra C of B containing Qϕ|B . Since Qψ|B is in the center of B, we also have Qψ|B ∈ C. The algebra C is generated by minimal projections p1 , . . . , pm in B. Then EC (b) = i pi bpi is the unique conditional expectation of B onto C. It is ϕ|B - and ψ|B -preserving. Thus to prove (2.6) it is enough to prove the analogous statement for EC ◦ E: A → C and EC : B → C. In other words, we may assume that B = C is abelian. The conditional expectation E: A → B is then of the form E(a) = i . Let Qi be the density i ϕi (a)pi , where ϕi is a state with supp ϕi = p operator of ϕi . Then Qϕ = i ϕ(pi )Qi and Qψ = i ψ(pi )Qi , so that Qtϕ Q−t ψ =

ϕ(pi )t p = Qtϕ|B Q−t ψ|B , t i ψ(p ) i i

and the proof of (vi) is complete. Part (vii) is immediate as Qϕ = i Qϕi .

Remark 2.1.4. (i) Concerning Theorem 2.1.2(i), a more careful analysis shows that S(ϕ, ψ) ≥

1 ϕ − ψ2 2

for any states ϕ and ψ. (ii) If ϕ ≤ ψ are positive linear functionals, then S(ϕ, ψ) ≤ 0, and equality holds if and only if the support p of ϕ commutes with Qψ and Qϕ = pQψ . Indeed, S(ϕ, ψ) ≤ S(ϕ, ϕ) = 0 by Theorem 2.1.2(ii). Assume S(ϕ, ψ) = 0. As

2.2 Von Neumann Entropy

21

−1 −1 Qψ ≥ Qϕ , we have by B.3. So using the integral ϕ + t) ∞ (Qψ +−1t) ≤ (Q−1 formula log a = 0 ((1+t) −(a+t) ) dt as in B.4, we get from S(ϕ, ψ) = 0 that p((Qϕ + t)−1 − (Qψ + t)−1 )p = 0

for any t > 0. In other words, p commutes with (Qψ + t)−1 and p(Qψ + t)−1 = p(Qϕ + t)−1 , so pQψ = pQϕ = Qϕ .

2.2 Von Neumann Entropy A natural generalization of the entropy of a partition is the following notion. Deﬁnition 2.2.1. The entropy, or von Neumann entropy, of a positive linear functional ϕ on a ﬁnite dimensional C∗ -algebra A is S(ϕ) = Tr(η(Qϕ )) = −S(ϕ, Tr), where η(t) = −t log t. Before we proceed to the study of this notion, recall that the centralizer of a positive linear functional ϕ is the set Aϕ of all x ∈ A such that ϕ(xy) = ϕ(yx) for any y ∈ A. This is just the algebra of elements commuting with Qϕ . By the rank of A we mean the dimension of a maximal abelian subalgebra of A. Then rank A = TrA (1). Theorem 2.2.2. We have: (i) 0 ≤ S(ϕ) ≤ log rank A for any state ϕ; moreover, S(ϕ) = 0 if and only if ϕ is pure, and S(ϕ) = log rank A if and only if ϕ is the normalized canonical trace; (ii) the entropy is continuous, concave, and S(ϕ + ψ) ≤ S(ϕ) + S(ψ); in particular, if ϕ, ψ are states and 0 < λ < 1, then S(λϕ + (1 − λ)ψ) ≤ λS(ϕ) + (1 − λ)S(ψ) + η(λ) + η(1 − λ); (iii) if ϕ = i λi ϕi is a convex combination of states, then λi S(ϕi , ϕ), S(ϕ) ≥ i

and equality holds if and only if ϕi is pure for every λi = 0; (iv) if ϕ is a state on A and ψ is a state on B, then S(ϕ ⊗ ψ) = S(ϕ) + S(ψ);

22

2 Relative Entropy

(v) if ϕ is a positive linear functional on A ⊗ B ⊗ C, then S(ϕ) + S(ϕ|B ) ≤ S(ϕ|A⊗B ) + S(ϕ|B⊗C ); (vi) if ϕ is a state on A ⊗ B, then |S(ϕ) − S(ϕ|B )| ≤ S(ϕ|A ); (vii) if B is a maximal abelian subalgebra of A, then S(ϕ|B ) ≥ S(ϕ), and equality holds if and only if B is in the centralizer Aϕ of ϕ. Most of the properties of von Neumann entropy are simple consequences of properties of relative entropy. In order to prove (vi) we shall need the following result. Lemma 2.2.3. We have: (i) if ϕ is a pure state on the algebra Matn (C) ⊗ Matm (C) ∼ = Matnm (C), then S(ϕ|Matn (C) ) = S(ϕ|Matm (C) ); (ii) every state on Matn (C) ⊗ 1 ⊂ Matn (C) ⊗ Matn (C) extends to a pure state on Matn (C) ⊗ Matn (C). Proof. If ϕ is a pure state on Matn (C) ⊗ Matm (C), then the density operator Qϕ is a projection of rank one, so it is the projection onto the line spanned by a unit vector i,j λij ei ⊗ ej ∈ n2 ⊗ m 2 . Thus Qϕ =

¯ kl eik ⊗ ejl , λij λ

i,j,k,l

where {eik }i,k are the matrix units. The density matrices of ϕ|Matn (C) and ϕ|Matm (C) are (id ⊗ Trm )(Qϕ ) and (Tr n ⊗ id)(Qϕ ), respectively, where Trn is the usual trace on Matn (C). Set T = i,j λij eij ∈ Matn×m (C). Then (Trn ⊗ id)(Qϕ ) = (T ∗ T ) , the transpose of the matrix T ∗ T , and (id⊗Trm )(Qϕ ) = T T ∗ . Since the nonzero eigenvalues of T ∗ T and T T ∗ are the same (counting with multiplicities), we get S(ϕ|Matn (C) ) = S(ϕ|Matm (C) ), and (i) is proved. If ϕ is a state non Matn (C), we may assume that its density matrix is diagonal, Qϕ = i=1 λi eii . Then 1/2 1/2 λi λj eij ⊗ eij P = i,j

is a rank one projection in Matn (C) ⊗ Matn (C) such that Qϕ = (id ⊗ Trn )(P ). Thus the pure state with density matrix P has the property that its restriction to Matn (C) ⊗ 1 coincides with ϕ.

2.2 Von Neumann Entropy

23

Proof of Theorem 2.2.2. If ϕ isa state, and {λi }i are the eigenvalues of Qϕ , then i λi = 1 and S(ϕ) = i η(λi ). We see that S(ϕ) = 0 if and only if λi = 1 for some i and λj = 0 for j = i, that is, ϕ is a pure state. The rest of (i) follows from the corresponding properties of entropy of partitions, Proposition 1.1.9(i). As S(ϕ) = −S(ϕ, Tr), the entropy is concave by Theorem 2.1.2(iv). Since η is continuous, the entropy is also continuous. To prove the ﬁrst inequality in (ii) we use operator monotonicity of log, B.4, so S(ϕ + ψ) = −ϕ(log(Qϕ + Qψ )) − ψ(log(Qϕ + Qψ )) ≤ −ϕ(log Qϕ ) − ψ(log Qψ ) = S(ϕ) + S(ψ). This implies the second inequality in (ii), since S(λϕ) = λS(ϕ) + η(λ), if ϕ is a state. To prove (iii) note that λi S(ϕi , ϕ) = λi ϕi (log Qϕi − log Qϕ ) = S(ϕ) − λi S(ϕi ). i

i

i

Since S(ϕi ) ≥ 0, and S(ϕi ) = 0 if and only if ϕi is pure, we get (iii). Part (iv) is immediate. To prove (v) note that S(ϕ|A⊗B ) − S(ϕ|B ) = −S(ϕ|A⊗B , TrA ⊗ ϕ|B ) and similarly S(ϕ) − S(ϕ|B⊗C ) = −S(ϕ, TrA ⊗ ϕ|B⊗C ). Since S(ϕ, TrA ⊗ϕ|B⊗C ) ≥ S(ϕ|A⊗B , TrA ⊗ϕ|B ) by Theorem 2.1.2(v) (applied to the inclusion mapping A ⊗ B → A ⊗ B ⊗ C), we get (v). Part (v) implies in particular that if ϕ is a state on A ⊗ B, then S(ϕ) ≤ S(ϕ|A ) + S(ϕ|B ). So to prove (vi) it is enough to show that S(ϕ|B ) ≤ S(ϕ) + S(ϕ|A ).

(2.7)

Note that if D is a ﬁnite dimensional C∗ -algebra, then there exists an embedding D → Matn (C) with n = rank D such that TrD = Trn |D , where Trn is the usual trace on Matn (C). This allows us to assume without loss of generality that A and B are full matrix algebras. Let C ∼ = A ⊗ B. By Lemma 2.2.3(ii)

24

2 Relative Entropy

there exists a pure state ϕ˜ on A⊗B⊗C extending ϕ. Then S(ϕ| ˜ B⊗C ) = S(ϕ| ˜ A) by Lemma 2.2.3(i). Hence (2.7) follows from (v) applied to the state ϕ. ˜ To prove (vii) note that as B is maximal abelian in A, we have TrA |B = TrB . In particular, if EB : A → B is the trace preserving conditional expectation, then Qϕ◦EB = Qϕ|B . Hence S(ϕ|B ) − S(ϕ) = ϕ(log Qϕ − log Qϕ|B ) = S(ϕ, ϕ ◦ EB ). Of course, this also follows from Theorem 2.1.2(vi). Hence S(ϕ|B ) ≥ S(ϕ). If Qϕ ∈ B, then Qϕ|B = Qϕ as TrA |B = TrB , so S(ϕ|B ) = S(ϕ). Conversely, if S(ϕ|B ) = S(ϕ), then S(ϕ, ϕ◦EB ) = 0. By Theorem 2.1.2(i) we get ϕ = ϕ◦EB , so Qϕ = Qϕ◦EB = Qϕ|B ∈ B, that is, B is in the centralizer of ϕ.

Part (vii) of the above theorem implies that if a subalgebra B of A contains a maximal abelian subalgebra of A, then S(ϕ|B ) ≥ S(ϕ). The following lemma gives a useful estimate of the diﬀerence. Lemma 2.2.4. Let e1 , . . . , en be projections in A with sum 1, ϕ a state on A. Let B = ⊕ni=1 Bi , where Bi = ei Aei . Then η(ϕ(ei )). S(ϕ|B ) − S(ϕ) ≤

i

Proof. Consider a decomposition ϕ = k λk ϕk of ϕ into a convex combination of pure states. Then by monotonicity of relative entropy, Theorem 2.1.2(v), and Theorem 2.2.2(iii) we have S(ϕ) = λk S(ϕk , ϕ) ≥ λk S(ϕk |B , ϕ|B ) k

= S(ϕ|B ) −

k

λk S(ϕk |B ) = S(ϕ|B ) −

i

k

λk S(ϕk |Bi ).

k

The restriction of ϕk to Bi = ei Aei is a scalar multiple of a pure state, whence S(ϕk |Bi ) = η(ϕk (ei )). By concavity of η we then get λk S(ϕk |Bi ) = λk η(ϕk (ei )) ≤ η λk ϕk (ei ) = η(ϕ(ei )), k

k

k

which completes the proof of the lemma.

In Chap. 9 we shall use the following form of positivity of relative entropy, called the thermodynamic inequality. Proposition 2.2.5. If ϕ is a state on a ﬁnite dimensional C∗ -algebra A and H a self-adjoint element in A, then we have S(ϕ) + ϕ(H) ≤ log Tr(eH ), and equality holds if and only if Qϕ = Tr(eH )−1 eH .

2.2 Von Neumann Entropy

25

Proof. Let ψ be the state with density operator Tr(eH )−1 eH . Then S(ϕ, ψ) = −S(ϕ) − ϕ(log Qψ ) = −S(ϕ) − ϕ(H) + log Tr(eH ). So the result follows from Theorem 2.1.2(i).

Finishing our discussion of von Neumann entropy we prove the following estimate. Proposition 2.2.6. If ϕ and ψ are states on a ﬁnite dimensional C∗ -algebra A, d = rank A, and ϕ − ψ ≤ ε ≤ 1/e, then |S(ϕ) − S(ψ)| ≤ ε log d + η(ε). For this we need an elementary inequality 1 . (2.8) 2 To show this note that as the function f (x) = η(x+(s−t))−η(x) is decreasing, we have for x taking the increasing values 0, t, 1 − (s − t) |η(s) − η(t)| ≤ η(s − t) when s, t ∈ [0, 1], 0 ≤ s − t ≤

η(s − t) ≥ η(s) − η(t) ≥ −η(1 − (s − t)). If 1/e ≤ s − t ≤ 1/2, then η(1 − (s − t)) ≤ η(s − t) as η is decreasing on [1/e, 1], so (2.8) is proved in this case. On the other hand, if s − t ≤ 1/e, then using that η(x) + x is increasing on [0, 1], we get η(s) − η(t) ≥ t − s. But t − s ≥ −η(s − t) whenever 0 ≤ s − t ≤ 1/e, so (2.8) follows in this case too. We also need the following well-known inequality. Recall that the trace norm is deﬁned by x1 = Tr(|x|). Then if a and b are positive trace-class operators, λ1 ≥ λ2 ≥ . . . and µ1 ≥ µ2 ≥ . . . the eigenvalues of a and b, respectively, we have a − b1 ≥ |λi − µi |. (2.9) i

Indeed, let x and y be positive operators such that xy = 0 and x − y = a − b. Then a − b1 = Tr(x) + Tr(y). Let c = b + x = a + y, and let ν1 ≥ ν2 ≥ . . . be the eigenvalues of c. Then νi ≥ λi , µi , hence |λi − µi | ≤ 2νi − λi − µi . Thus |λi − µi | ≤ (2νi − λi − µi ) = Tr(2c − a − b) = Tr(x) + Tr(y) = a − b1 , i

i

and (2.9) is proved. Proof of Proposition 2.2.6. Let λ1 ≥ . . . ≥ λd and µ1 ≥ . . . ≥ µ d be the eigenvalues of Qϕ and Qψ , respectively. Put δi = |λi − µi |, δ = i δi . By (2.8) |S(ϕ) − S(ψ)| = (η(λi ) − η(µi )) ≤ |η(λi ) − η(µi )| ≤ η(δi ). i i i Then i η(δi ) = δ i η(δi /δ) − δ log δ ≤ δ log d + η(δ). By (2.9) we have δ ≤ ϕ − ψ, so δ ≤ ε ≤ 1/e and η(δ) ≤ η(ε), proving the proposition.

26

2 Relative Entropy

2.3 Relative Entropy for General C*-algebras A few times, notably in Chap. 5, we shall need relative entropy to be deﬁned for arbitrary C∗ -algebras. The shortest way of introducing it is the following variational expression. Let ϕ and ψ be positive linear functional on a unital C∗ -algebra A. Then ∞ dt , (ϕ(y(t)∗ y(t)) + t−1 ψ(x(t)x(t)∗ )) S(ϕ, ψ) = sup sup ϕ(1) log n − t n∈N x 1/n (2.10) where the second supremum is taken over all step functions x: (1/n, ∞) → A with ﬁnite range, and where y(t) = 1 − x(t). It suﬃces to consider functions x with values in a ﬁxed weakly dense subspace of A containing the unit. It is not at all obvious that in the ﬁnite dimensional case the above expression gives the same value of relative entropy as before. Since all our computations will be done by a reduction to the ﬁnite dimensional case anyway, instead of developing the whole theory we conﬁne ourselves to listing the main properties of relative entropy, see Notes at the end of the chapter for references. Theorem 2.3.1. For positive linear functionals on unital C∗ -algebras we have: λ1 if λ1 , λ2 > 0; (i) S(λ1 ϕ, λ2 ψ) = λ1 S(ϕ, ψ) + λ1 ϕ(1) log λ2 (ii) S(ϕ, ψ) ≥ 0 for states ϕ and ψ, and equality holds if and only if ϕ = ψ; (iii) S(ϕ, ψ) is decreasing in ψ, i.e., S(ϕ, ψ1 ) ≥ S(ϕ, ψ2 ) if ψ1 ≤ ψ2 ; (iv) S(ϕ, ψ) is a weakly∗ lower semicontinuous function of (ϕ, ψ); (v) S(ϕ, ψ) is a convex function of (ϕ, ψ); (vi) if α: B → A is a unital Schwarz mapping, then S(ϕ ◦ α, ψ ◦ α) ≤ S(ϕ, ψ); (vii) if M is a von Neumann algebra, ϕ and ψ are normal states on M , N a von Neumann subalgebra of M , and E: M → N a ψ-preserving faithful normal conditional expectation, then S(ϕ, ψ) = S(ϕ|N , ψ|N ) + S(ϕ, ϕ ◦ E); (viii) if ϕ = i=1 ϕi , then i S(ϕi , ψ) = S(ϕ, ψ) + i S(ϕi , ϕ); (ix) if ψ is a state on A, and ψ¯ its normal extension to πψ (A) , where πψ : A → B(Hψ ) is the GNS-representation, then ¯ ¯ S(ϕ, ¯ ψ), if ϕ = ϕ¯ ◦ πψ for a normal linear functional ϕ, S(ϕ, ψ) = +∞, otherwise; n

(x) if τ is a ﬁnite trace on A, a and b positive elements in A such that a ≤ λb for some λ > 0, then S(τ (· a), τ (· b)) = τ (a(log a − log b)).

2.3 Relative Entropy for General C*-algebras

27

We ﬁnish the chapter with several corollaries of Theorem 2.3.1 (or of Theorem 2.1.2 in the ﬁnite dimensional case). Corollary 2.3.2. If {ϕi }i∈I and {ψi }i∈I are ﬁnite sets of positive linear functionals then ϕi , ψi ≤ S(ϕi , ψi ). S i

i

i

Proof. This follows from the scaling (i) and convexity (v) properties in Theorem 2.3.1.

Corollary 2.3.3. Let A be a C∗ -algebra and E: A → B be a conditional expectation onto a subalgebra B, ϕ and ψ positive linear functionals on A such that ϕ ◦ E = ϕ and ψ ◦ E = ψ. Then S(ϕ|B , ψ|B ) = S(ϕ, ψ). Proof. This is obtained by applying the monotonicity property of relative entropy, Theorem 2.3.1(vi), to E: A → B and to the inclusion map B → A.

Corollary 2.3.4. Let B be a C∗ -subalgebra of a C∗ -algebra A, ψ a state on A, πψ the GNS-representation deﬁned by ψ. Assume πψ (B) is weakly operator dense in πψ (A). Then S(ϕ, ψ) = S(ϕ|B , ψ|B ) for any positive linear functional ϕ ≤ ψ. Proof. We can identify πψ |B with the GNS-representation deﬁned by ψ|B . Then the result follows from Theorem 2.3.1(ix).

Corollary 2.3.5. Let ϕ and ψ be positive linear functionals on a C∗ -algebra A, {γi : Ai → A}i and {θi : A → Ai }i two nets of unital completely positive maps such that ϕ ◦ γi ◦ θi → ϕ and ψ ◦ γi ◦ θi → ψ weakly∗ . Then S(ϕ ◦ γi , ψ ◦ γi ) → S(ϕ, ψ). Proof. By lower semicontinuity, Theorem 2.3.1(iv), S(ϕ, ψ) ≤ lim inf S(ϕ ◦ γi ◦ θi , ψ ◦ γi ◦ θi ). i

On the other hand, by monotonicity, Theorem 2.3.1(vi), S(ϕ ◦ γi ◦ θi , ψ ◦ γi ◦ θi ) ≤ S(ϕ ◦ γi , ψ ◦ γi ) ≤ S(ϕ, ψ). Hence S(ϕ ◦ γi , ψ ◦ γi ) → S(ϕ, ψ).

A related but easier result is the following. Corollary 2.3.6. If {Ai }i is an increasing net of unital C∗ -subalgebras of a C∗ -algebra A such that ∪i Ai is norm-dense, then S(ϕ|Ai , ψ|Ai ) S(ϕ, ψ) for any ϕ and ψ.

28

2 Relative Entropy

Proof. This follows immediately from the variational expression (2.10), since it suﬃces to consider step functions with values in ∪i Ai .

Recall that a unital C∗ -algebra is called nuclear if there exist two nets {γi : Ai → A}i and {θi : A → Ai }i of unital completely positive maps with ﬁnite dimensional C∗ -algebras Ai such that (γi ◦ θi )(a) − a → 0 for every a ∈ A. Then by Corollary 2.3.5 it follows that S(ϕ, ψ) = limi S(ϕ ◦ γi , ψ ◦ γi ) for any ϕ and ψ. This equality could be used as a deﬁnition of relative entropy for nuclear algebras, which would in fact be suﬃcient for us in most cases. Corollary 2.3.7. Let A be a separable C∗ -algebra, (X, µ) a Lebesgue space, X x → ϕx and X x → ψx measurable maps into the state space of A, the state space being considered with the weak∗ topology. Consider the states ϕ and ψ on A ⊗ L∞ (X, µ) deﬁned by ϕ(a ⊗ f ) = ϕx (a)f (x)dµ(x), ψ(a ⊗ f ) = ψx (a)f (x)dµ(x). X

X

S(ϕx , ψx )dµ(x).

Then S(ϕ, ψ) = X

Proof. Note that if A is ﬁnite dimensional, then TrA ⊗ µ is a ﬁnite trace on A ⊗ L∞ (X, µ), and the result can be deduced from Theorem 2.3.1(x). Using Corollary 2.3.5 one can extend the result to nuclear C∗ -algebras. In general we can argue as follows. Elements of the algebraic tensor product A L∞ (X, µ) can be considered as A-valued functions on X. Hence any step function (1/n, ∞) → A L∞ (X, µ) deﬁnes a family of step functions (1/n, ∞) → A indexed by elements of X. By the variational expression (2.10) this implies the inequality ≤. On the other hand, if ξ is a ﬁnite measurable partition of X then S(ϕ, ψ) ≥ S(ϕ|A⊗L∞ (X/ξ,µ) , ψ|A⊗L∞ (X/ξ,µ) ) S(ϕ|A⊗C1Z ) , ψ|A⊗C1Z ) = Z∈ξ

=

S Z

Z∈ξ

=

S

= X

Z

µ(Z)S µ(Z)

Z∈ξ

ψy dµ(y)

ϕy dµ(y), −1

ϕy dµ(y), µ(Z)

−1

Z

−1

µ(Z(x))

ψy dµ(y)

Z −1

ϕy dµ(y), µ(Z(x)) Z(x)

ψy dµ(y) dµ(x),

Z(x)

where Z(x) is the element of ξ containing x. Choose an increasing sequence {ξn }n of ﬁnite measurable partitions such that ∨n ξn is the partition into points. For x ∈ X denote by Zn (x) the element of ξn containing x. By the martingale convergence theorem for any f ∈ L1 (X, µ) we have

2.3 Relative Entropy for General C*-algebras

29

µ(Zn (x))−1 Zn (x) f (y)dµ(y) → f (x) for a.e. x ∈ X. It follows that −1

µ(Zn (x))

−1

Zn (x)

ϕy dµ(y) → ϕx and µ(Zn (x))

Zn (x)

ψy dµ(y) → ψx

weakly∗ for a.e. x ∈ X. Hence the above inequality together with weak∗ lower semicontinuity of relative entropy implies the inequality ≥ in the corollary.

Corollary 2.3.8. Let {ϕij }i∈I,j∈J be a ﬁnite linear functionals set of positive on a C∗ -algebra. Put ϕ = i,j ϕij , ϕIi = j ϕij and ϕJj = i ϕij . Then

S(ϕij , ϕ) ≥

i,j

S(ϕIi , ϕ) +

i

S(ϕJj , ϕ).

j

Proof. Using Theorem 2.3.1(viii) and Corollary 2.3.2 we compute J J S(ϕij , ϕ) − S(ϕj , ϕ) = S(ϕij , ϕ) − S(ϕj , ϕ) i,j

j

j

=

i,j

≥

i

S(ϕij , ϕJj ) ⎞ ⎛ S⎝ ϕij , ϕJj ⎠

i

=

j

j

S(ϕIi , ϕ),

i

which proves the corollary.

Property (x) in Theorem 2.3.1 could also be used as a deﬁnition of entropy for C∗ -algebras with ﬁnite trace. To prove the main properties of entropy such as convexity and monotonicity one could then use the same technique as in the ﬁnite dimensional case. In particular, the following three corollaries do not require the full strength of Theorem 2.3.1 and can be deduced from Proposition 2.1.3. The ﬁrst corollary is the noncommutative extension of Lemma 1.1.10. In Chap. 4 we shall prove a stronger result. Corollary 2.3.9. Let {xi1 ...in }i1 ,...,in be aﬁnite set of positive elements in a C∗ -algebra with a ﬁnite trace τ . Assume i1 ,...,in xi1 ...in = 1. For each k put (k) xik = i1 ,...,ik−1 ,ik+1 ,...,in xi1 ...in . Then i1 ,...,in

τ (η(xi1 ...in )) ≤

k

ik

(k)

τ (η(xik )).

30

2 Relative Entropy

Proof. For n = 2 the result follows by applying Corollary 2.3.8 to ϕi1 i2 = τ (· xi1 i2 ) and using Theorem 2.3.1(x). The general case is proved by induction.

Corollary 2.3.10. Let τ be a ﬁnite trace on a C∗ -algebra A, a and b positive elements in A such that a ≤ λb for some λ > 0. Then τ (a(log a − log b)) ≥ τ (a) − τ (b). Proof. By monotonicity, Theorem 2.3.1(vi), S(τ (· a), τ (· b)) ≥ S(τ (· a)|C1 , τ (· b)|C1 ) = τ (a)(log τ (a) − log τ (b)). On the other hand, τ (a)(log τ (a) − log τ (b)) ≥ τ (a) − τ (b) by (2.3).

In a similar way we get the following Peierls-Bogoliubov inequality. Corollary 2.3.11. Let τ be a ﬁnite trace on a C∗ -algebra A, a and b selfadjoint elements in A. Then log τ (ea+b ) − log τ (eb ) ≥

τ (aeb ) . τ (eb )

In particular, | log τ (ea+b ) − log τ (eb )| ≤ a. Proof. Applying S(ϕ, ψ) ≥ S(ϕ|C1 , ψ|C1 ) = ϕ(1)(log ϕ(1) − log ψ(1)) to ϕ = τ (· eb ) and ψ = τ (· ea+b ) we get the ﬁrst inequality. It follows that log τ (ea+b ) − log τ (eb ) ≥ −a. Applying this to −a and a + b instead of a and b we get the second inequality.

2.4 Notes Our main sources during the preparation of this chapter were the books by Simon [198] and Ohya and Petz [147]. Chapter 5 of the latter book contains in particular proofs of all the properties listed in Theorem 2.3.1. Relative entropy of measures was introduced by Kullback and Leibler [114]. The deﬁnition for normal states on ﬁnite von Neumann algebras was given by Umegaki [218]. One should be aware that in the literature several different conventions are used. Sometimes one uses the opposite order of the

2.4 Notes

31

arguments or the opposite sign. The statement (2.4) is the famous Lieb concavity. It was conjectured by Wigner and Yanase [235] on the basis of remarks of Dyson and proved by Lieb [116]. As we saw, it implies one of the main properties of relative entropy, joint convexity. The monotonicity property, Theorem 2.1.2(v) and Theorem 2.3.1(vi), was proved by Uhlmann [217]. Our deduction of both properties from Proposition 2.1.3 essentially follows Uhlmann’s paper using simpliﬁcations made by Simon [198]. The conditional expectation property, Theorem 2.1.2(vi) and Theorem 2.3.1(vii), is due to Hiai, Ohya and Tsukada [89] in a weaker form and to Petz [158], [159] in general. Although in our exposition entropy of states appears after entropy of partitions, it was introduced by von Neumann [145] more than twenty years earlier than the latter [195]. Of course, both notions have their origin in works of Boltzmann and Gibbs on thermodynamics. Among the properties of von Neumann entropy the one which attracted most attention was strong subadditivity, Theorem 2.2.2(v). It was ﬁrst proved by Lieb and Ruskai [117] using the Lieb concavity. The triangle inequality, Theorem 2.2.2(vi), was established by Araki and Lieb [11]. Proposition 2.2.6 is due to Fannes [66]. The notion of relative entropy was extended to normal states on arbitrary von Neumann algebras by Araki [9], [10]. If ϕ and ψ are normal states on a von Neumann algebra, ϕ ≤ λψ, then, [158], S(ϕ, ψ) = i

d ϕ((Dϕ: Dψ)t )|t=0 , dt

where (Dϕ: Dψ)t is Connes’ Radon-Nikodym cocycle. This is an extension of (2.2). A deﬁnition for states on arbitrary ∗-algebras was given by Uhlmann [217]. Later Kosaki [112], using ideas of Pusz and Woronowicz [176], obtained the variational expression (2.10). The inequality in Corollary 2.3.8, sometimes called the marginal inequality, was ﬁrst used by Connes [49], and the inequality in Corollary 2.3.9 by Connes and Størmer [51]. Finally note that in view of Theorem 2.2.2(iii) one can deﬁne the entropy of a state ϕ on a C∗ -algebra as the supremum of i λi S(ϕi , ϕ) over all convex decompositions ϕ = i λi ϕi . It turns out that if S(ϕ) < ∞, then πϕ (A) is discrete, that is, it is a direct sum of algebras B(H), see [147, Chapter 6].

3 Dynamical Entropy

As we already mentioned in Chap. 1, one of the main problems in deﬁning noncommutative entropy is that two ﬁnite dimensional subalgebras of a noncommutative algebra can generate an inﬁnite dimensional subalgebra. To overcome this diﬃculty we consider the entropy of the join of n partitions as a function of the partitions and try to ﬁnd its noncommutative analogue for each n ∈ N. In the present chapter we shall deﬁne such an analogue and study the dynamical invariant arising from it.

3.1 Mutual Entropy In Chap. 1 we succeeded in writing a formula for H(ξ1 ∨. . .∨ξn ) which does not use joins of the partitions ξi , see Proposition 1.3.2. This formula makes sense for ﬁnite von Neumann algebras. Namely, let N be a von Neumann algebra and τ a faithful normal trace on N . Let N1 , . . . , Nn be ﬁnite dimensional subalgebras of N , and let ENk : N → Nk denote the τ -preserving conditional expectation. Then the mutual entropy of N1 , . . . , Nn ⊂ N with respect to τ is ⎧ ⎫ n ⎨ ⎬ (k) η(τ (xi1 ...in )) − τ η ENk (xik ) , Hτ (N1 , . . . , Nn ) = sup ⎩ ⎭ i1 ,...,in

k=1 ik

where η(t) = −t log t, and the supremum is taken over all ﬁnite partitions of unity xi1 ...in , xi1 ...in ≥ 0, 1= i1 ∈I1 ,...,in ∈In (k) xik

= i1 ,...,ik−1 ,ik+1 ,...,in xi1 ...in . in N , where In the case of one algebra B ⊂ N , the deﬁnition says that Hτ (B) is the supremum of

34

3 Dynamical Entropy

η(τ (xi )) −

i

τ (η(EB (xi ))) = −

i

τ (xi )τ (η(EB (τ (xi )−1 xi ))).

(3.1)

i

We shall see soon that this supremum is just S(τ |B ). This deﬁnition does not make sense if τ is not a trace, since in general a state preserving conditional expectation N → Nk may not exist. However, by Theorem 2.3.1(x) we know that if x ∈ Nk then τ (η(x)) = −S(τ (·x)|Nk , τ |Nk ). Recalling that the map x → τ (·x) establishes a one-to-one correspondence between positive elements x ∈ N , x ≤ 1, and positive linear functionals ϕ ≤ τ on N , we can thus rewrite the deﬁnition of the mutual entropy as ⎧ ⎫ ⎨ ⎬ (k) Hτ (N1 , . . . , Nn ) = sup η(ϕi1 ...in (1)) + S(ϕik |Nk , τ |Nk ) , ⎩ ⎭ i1 ,...,in

ik

k

where the supremum is taken over all ﬁnite decompositions of τ into a sum (k) of positive linear functionals, and ϕik = i1 ,...,ik−1 ,ik+1 ,...,in ϕi1 ...ik . This formula already makes sense for arbitrary states. We shall now extend the deﬁnition to channels. Let A be a unital C∗ algebra. By a channel in A we mean a unital completely positive map γ: B → A, where B is a ﬁnite dimensional C∗ -algebra. The reason to consider channels is that in general a C∗ -algebra may have too few ﬁnite dimensional subalgebras. Deﬁnition 3.1.1. Let A be a unital C∗ -algebra, γk : Ak → A, 1 ≤ k ≤ n, a collection of channels. The mutual entropy Hϕ (γ1 , . . . , γn ) of γ1 , . . . , γn with respect to a state ϕ on A is the supremum of (k) Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) = η(ϕi1 ...in (1)) + S(ϕik ◦ γk , ϕ ◦ γk ) i1 ,...,in

k

ik

(k)

over all ﬁnite decompositions ϕ = i1 ,...,in ϕi1 ...in , where ϕik is obtained by taking the sum of the elements ϕi1 ...in with the k-th index equal to ik . We call Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) the mutual entropy of γ1 , . . . , γn with respect to the decomposition ϕ = i1 ,...,in ϕi1 ...in . Note that since S(λψ, ϕ) = λS(ψ, ϕ) − ψ(1)η(λ), it can also be written as (k) (k) (k) η(ϕi1 ...in (1)) − η(ϕik (1)) + ϕik (1)S(ϕˆik ◦ γk , ϕ ◦ γk ), i1 ,...,in

k

ik

k

ik

(3.2) ˆ where for a positive linear functional ψ we denote by ψ the state ψ(1)−1 ψ. The last expression can equivalently be written as (k) (k) η(ϕi1 ...in (1)) − η(ϕik (1)) + Hϕ (γk ; {ϕik }). (3.3) i1 ,...,in

k

ik

k

ik

3.1 Mutual Entropy

35

In the case when the Ai ’s are subalgebras of A and the γi ’s are the inclusion maps, we write Hϕ (A1 , . . . , An ) instead of Hϕ (γ1 , . . . , γn ). To get a better understanding of the deﬁnition, consider the case of one channel γ: B → A. Then Hϕ (γ) is the supremum of the quantities η(ϕi (1)) + S(ϕi ◦ γ, ϕ ◦ γ). (3.4) i

i

For a state ψ on B, call a decomposition ψ = i ψi orthogonal if the density matrices of the ψi ’s are mutually orthogonal. Equivalently, there exists a partition of unity 1 = i pi consisting of projections in the centralizer of ψ such that ψi = ψ(·pi ). By Remark 2.1.4(ii) this is also the same as requiring S(ψi , ψ) = 0 for all i. Thus the ﬁrst summand in (3.4) is a classical entropy term, while the second one is a correction term which compensates for nonorthogonality of the decomposition ϕ ◦ γ = i ϕi ◦ γ. The quantity S(ϕi ◦ γ, ϕ ◦ γ) = S(ϕi ◦ γ) − S(ϕ ◦ γ) − i

i

=

i

η(ϕi (1)) +

ϕi (1)S(ϕˆi ◦ γ) − S(ϕ ◦ γ) (3.5)

i

is also called the entropy defect of the decomposition ϕ ◦ γ = i ϕi ◦ γ. In the case of several channels we similarly have the classical term i1 ,...,in η(ϕi1 ...in (1)) plus the sum of the correction terms corresponding to (k) the decompositions ϕ ◦ γk = ik ϕik ◦ γk , 1 ≤ k ≤ n. If A is a von Neumann algebra and ϕ a faithful normal state, then instead of decompositions of ϕ we can talk about partitions of unity in A. To show this we ﬁrst need to recall a few facts from modular theory, see e.g. [30, Chapter 2.5] or [205] for details. We may assume A ⊂ B(H) and ϕ is a vector state, ϕ(a) = (aξϕ , ξϕ ), where ξϕ is a cyclic and separating vector in H. The anti-linear map Aξϕ aξϕ → a∗ ξϕ 1/2

is closable. Denote by Sϕ its closure. Let Sϕ = Jϕ ∆ϕ be the polar decomposition, where Jϕ is an anti-linear isometry and ∆ϕ a positive operator, which is in general unbounded. Then Jϕ2 = 1, ∆ϕ is nonsingular and Jϕ ξϕ = ∆ϕ ξϕ = ξϕ . The operators Jϕ and ∆ϕ are called the modular conjugation and the modular operator deﬁned by ϕ. The formula −it σtϕ (a) = ∆it ϕ a∆ϕ , a ∈ A,

deﬁnes a one-parameter automorphism group of A called the modular group of ϕ. The modular group measures how far the state ϕ is from a tracial state. Namely, the centralizer Aϕ of ϕ, that is, the set of all elements a ∈ A such

36

3 Dynamical Entropy

that ϕ(ab) = ϕ(ba) for any b ∈ A, coincides with the ﬁxed point algebra of the modular group. If B is a von Neumann subalgebra of A, then a normal ϕ-preserving conditional expectation A → B exists if and only if σtϕ (B) = B for any t ∈ R. The most important property of Jϕ is that Jϕ AJϕ = A , the commutant of A in B(H). It follows that any positive linear functional ψ ≤ ϕ can be written in the form ψ = (·Jϕ xJϕ ξϕ , ξϕ ) = (·Jϕ xξϕ , ξϕ ) for a unique element x ∈ A, 0 ≤ x ≤ 1. Since 1/2 −1/2 ξϕ , Jϕ xξϕ = Jϕ Sϕ xξϕ = ∆1/2 ϕ xξϕ = ∆ϕ x∆ϕ ϕ (x)). Thus any by slightly abusing notation we can write ψ = ϕ(· σ−i/2 partition of unity 1 = j xj in A gives rise to a decomposition ϕ = ϕ ϕ(· σ (x )), and vice versa. We shall write Hϕ (γ1 , . . . , γn ; {xj1 ...jn }) inj j −i/2 ϕ (xj1 ...jn ))}). stead of Hϕ (γ1 , . . . , γn ; {ϕ(· σ−i/2

Yet another equivalent way of describing decompositions of states is as follows. Let C be a ﬁnite dimensional abelian C∗ -algebra, C1 , . . . , Cn ⊂ C subalgebras, µ a state on C, and P : A → C a unital positive map such that ϕ = µ ◦ P . Then we say that (C, µ, {Ck }nk=1 , P ) is an abelianmodel for (A, ϕ, {γk }nk=1 ). An abelian model deﬁnes a decomposition ϕ = ϕi1 ...in by (1)

(n)

ϕi1 ...in (a) = µ(P (a)pi1 . . . pin ),

(3.6)

(k)

where {pik }ik is the set of atoms of Ck . We call Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) the entropy of the abelian model. By deﬁnition (k) Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) = S(µ|∨k Ck ) + S(µ((P ◦ γk )(·)pik ), ϕ ◦ γk ). k

ik

(3.7) Conversely, starting with a decomposition of ϕ one can easily construct an abelian model deﬁning this decomposition: take C = C(I1 × . . . × In ), Ck = C(Ik ), then (3.6) determines µ and P . Note that if C is a ﬁnite dimensional abelian C∗ -algebra, C1 , . . . , Cn are subalgebras of C, and µ is a state on C, then (C, µ, {Ck }nk=1 , idC ) is an abelian model for (C, µ, {Ck }nk=1 ). The entropy of this model is S(µ|∨k Ck ). The setting of abelian models makes it explicit that in our deﬁnition of mutual entropy we ﬁrst project a noncommutative C∗ -algebra into a commutative one, and then compute the entropy of the images of channels under this projection. More precisely, we have the following. Lemma 3.1.2. The mutual entropy Hϕ (γ1 , . . . , γn ) is the supremum of the entropies Hµ (P ◦ γ1 , . . . , P ◦ γn ) over all unital completely positive maps P : A → C of A into abelian C∗ -algebras and over all states µ such that µ ◦ P = ϕ.

3.1 Mutual Entropy

37

Proof. The inequality Hϕ (γ1 , . . . , γn ) ≥ Hµ (P ◦ γ1 , . . . , P ◦ γn ) will follow from Proposition 3.1.3(ii) below. On the other hand, if (C, µ, {Ck }nk=1 , P ) is an abelian model for (A, ϕ, {γk }nk=1 ), then (C, µ, {Ck }nk=1 , idC ) is an abelian model for (C, µ, {P ◦ γk }nk=1 ). By deﬁnition, see (3.7), the entropies of these two abelian models coincide, so that the entropy of the abelian model (C, µ, {Ck }nk=1 , P ) for (A, ϕ, {γk }nk=1 ) is not larger than Hµ (P ◦γ1 , . . . , P ◦γn ). Thus we get the result.

The main properties of mutual entropy are the following. Proposition 3.1.3. We have: (i) if θk : Bk → Ak , 1 ≤ k ≤ n, are unital completely positive maps, then Hϕ (γ1 ◦ θ1 , . . . , γn ◦ θn ) ≤ Hϕ (γ1 , . . . , γn ); (ii) if θ: A → B is a unital completely positive map and ψ a state on B, then Hψ (θ ◦ γ1 , . . . , θ ◦ γn ) ≤ Hψ◦θ (γ1 , . . . , γn ); in particular, Hψ (θ ◦γ1 , . . . , θ ◦γn ) = Hψ◦θ (γ1 , . . . , γn ) if θ is an isomorphism; (iii) if k < n, then Hϕ (γ1 , . . . , γk ) ≤ Hϕ (γ1 , . . . , γn ), and Hϕ (γ1 , . . . , γn ) ≤ Hϕ (γ1 , . . . , γk ) + Hϕ (γk+1 , . . . , γn ); (iv) Hϕ (γ1 , . . . , γn ) depends only on the set {γ1 , . . . , γn }; in other words, it is invariant under permutations of γ1 , . . . , γn , and if γn−1 = γn , then Hϕ (γ1 , . . . , γn ) = Hϕ (γ1 , . . . , γn−1 ). Proof. Part (i) follows from monotonicity of relative entropy, Theorem 2.1.2(v). Part (ii) is immediate as any decomposition of the state ψ deﬁnes a decomposition of ψ ◦ θ. The inequality Hϕ (γ1 , . . . , γk ) ≤ Hϕ (γ1 , . . . , γn ) follows from the fact that any decomposition ϕ = ϕi1 ...ik can be considered as a decomposition ϕ = ϕi1 ...in with single-point indexsets Ik+1 , . . . , In . Given adecomposition ϕ = ϕi1 ...in , we can construct two decompositions ϕ = ϕi1 ...ik and ϕ = ϕik+1 ...in , which are more transparently described in terms of abelian models: given an abelian model (C, µ, {Cj }nj=1 , P ) for (A, ϕ, {γj }nj=1 ) we get the abelian models (C, µ, {Cj }kj=1 , P ) and (C, µ, {Cj }nj=k+1 , P ) for (A, ϕ, {γj }kj=1 ) and (A, ϕ, {γj }nj=k+1 ), respectively. Then the subadditivity inequality in part (iii) is a consequence of the classical subadditivity S(µ|∨nj=1 Cj ) ≤ S(µ|∨kj=1 Cj ) + S(µ|∨nj=k+1 Cj ). It remains to prove part (iv). The invariance under permutations is obvious. Assume now that γn−1 = γn . By (iii) we already know that

38

3 Dynamical Entropy

Hϕ (γ1 , . . . , γn−1 ) ≤ Hϕ (γ1 , . . . , γn ). To make the proof of the opposite inequality more transparent, consider ﬁrst the case n = 2. Then given any decomposition ϕ = i1 ,i2 ϕi1 i2 we have Hϕ (γ1 ; {ϕi1 i2 }) ≥ Hϕ (γ1 , γ1 ; {ϕi1 i2 }) by Corollary 2.3.8, since the classical terms on both sides of the above inequality coincide. Consider now the general case. Any = ϕi1 ...in can be decomposition ϕ considered as a decomposition ϕ = ϕi1 ...in−1 with Ik = Ik for k < n − 1 = In−1 ×In . In other words, any abelian model (C, µ, {Ck }nk=1 , P ) for and In−1 n−1 (A, ϕ, {γk }nk=1 ) deﬁnes an abelian model (C, µ, {Ck }n−1 k=1 , P ) for (A, ϕ, {γk }k=1 ) with Ck = Ck for k < n − 1 and Cn−1 = Cn−1 ∨ Cn . Comparing the entropies of these abelian models, we see that the classical terms as well as the ﬁrst n−2 correction terms coincide. Thus Corollary 2.3.8 applied to I = In−1 and J = In shows that the entropy of (C, µ, {Ck }n−1 k=1 , P ) is not smaller than the entropy of (C, µ, {Ck }nk=1 , P ). It follows that Hϕ (γ1 , . . . , γn−1 ) ≥ Hϕ (γ1 , . . . , γn ).

So far we have not given any estimates for Hϕ . Lemma 3.1.4. For any channel γ: B → A, we have 0 ≤ Hϕ (γ) ≤ S(ϕ ◦ γ). If, moreover, A is a von Neumann algebra, ϕ a faithful normal state and the image of γ does not consist only of scalars, then Hϕ (γ) > 0. Together with Proposition 3.1.3(iii) this implies that Hϕ (γk ) ≤ S(ϕ ◦ γk ), 0 ≤ Hϕ (γ1 , . . . , γn ) ≤ k

k

and if Im γk ⊂ B ⊂ A for 1 ≤ k ≤ n, then by Proposition 3.1.3(i),(iv) Hϕ (γ1 , . . . , γn ) ≤ Hϕ (B, . . . , B ) = Hϕ (B) ≤ S(ϕ|B ). n

Proof of Lemma 3.1.4. The entropy Hϕ (γ) is the supremum of the quantities (see (3.2)) ϕi (1)S(ϕˆi ◦ γ, ϕ ◦ γ) = S(ϕ ◦ γ) − ϕi (1)S(ϕˆi ◦ γ). i

i

The left hand side of the above equality shows that Hϕ (γ) ≥ 0, while the right hand side implies Hϕ (γ) ≤ S(ϕ ◦ γ). If A is a von Neumann algebra and ϕ is normal and faithful, we may assume that A ⊂ B(H) and ϕ is given vector ξ. by a cyclic and separating Then any partition of unity 1 = x in the commutant A of A deﬁnes i i a decomposition ϕ = ϕ with ϕ = (· x ξ, ξ). If H (γ; {ϕ }) = 0, then i i ϕ i i i S(ϕˆi ◦ γ, ϕ ◦ γ) = 0 for all i with ϕi (1) = 0. By Theorem 2.1.2(i) it follows that ϕ ◦ γ = ϕˆi ◦ γ. In other words, if Hϕ (γ) = 0 and a ∈ Im γ, then

3.1 Mutual Entropy

ϕ(a) =

39

(axξ, ξ) (xξ, ξ)

for any x ∈ A , 0 ≤ x ≤ 1, x = 0. Thus ((a − ϕ(a)1)xξ, ξ) = 0. Since ξ is cyclic for A , this shows that a = ϕ(a)1.

As opposed to the commutative case, Hϕ (B) for B ⊂ A cannot be determined just by the restriction of ϕ to B. For example, if the state ϕ on A is pure, then Hϕ (B) = 0, since there are no nontrivial decompositions of ϕ into a convex combination of states. On the other hand, as we shall see in a moment, Hϕ|B (B) = S(ϕ|B ), which is positive unless ϕ|B is pure. We have however the following simple but useful result. Lemma 3.1.5. If the images of the channels γk : Ak → A, 1 ≤ k ≤ n, are contained in a C∗ -subalgebra B ⊂ A, and there exists a ϕ-preserving conditional expectation E: A → B, then Hϕ|B (γ1 , . . . , γn ) = Hϕ (γ1 , . . . , γn ). Proof. Clearly Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) = Hϕ|B (γ1 , . . . , γn ; {ϕi1 ...in |B }). Since any decomposition of ϕ|B can be extended to a decomposition of ϕ by composing positive functionals on B with E, we get the result.

In general it is rather diﬃcult to compute Hϕ , or at least to ﬁnd a way of producing optimal decompositions. The following proposition gives one of the few examples when an explicit computation is possible. Proposition 3.1.6. Let A1 , . . . , An ⊂ B ⊂ A. Suppose there exist mutually commuting abelian subalgebras Ck ⊂ Ak , 1 ≤ k ≤ n, such that ∨k Ck is maximal abelian in the centralizer of ϕ|B . Suppose also that there exists a ϕ-preserving conditional expectation E: A → B. Then Hϕ (A1 , . . . , An ) = Hϕ (B) = S(ϕ|B ). Proof. Let F : B → C = ∨k Ck be a ϕ-preserving conditional expectation. Then (C, ϕ|C , {Ck }k , F ◦ E) is an abelian model for (A, ϕ, {Ck }k ). Since F ◦ E is the identity map on Ck ⊂ C, the entropy of this model is the same as the entropy of the model (C, ϕ|C , {Ck }k , idC ) for (C, ϕ|C , {Ck }k ). The latter entropy is S(ϕ|C ). Thus Hϕ (C1 , . . . , Cn ) ≥ S(ϕ|C ) = S(ϕ|B ), where the equality follows from Theorem 2.2.2(vii). On the other hand, we have Hϕ (C1 , . . . , Cn ) ≤ Hϕ (A1 , . . . , An ) ≤ Hϕ (B) ≤ S(ϕ|B ). Hence the above inequalities are in fact equalities.

In particular, in the abelian case we recover the classical quantities H(∨k ξk ), which also follows from Proposition 1.3.2. Further properties of mutual entropy are given in the following proposition.

40

3 Dynamical Entropy

Proposition 3.1.7. We have: (i) if πϕ : A → B(Hϕ ) is the GNS-representation and ϕ¯ is the normal extension of ϕ to πϕ (A) , then Hϕ (γ1 , . . . , γn ) = Hϕ¯ (πϕ ◦ γ1 , . . . , πϕ ◦ γn ); (ii) if ψ is another state on A and 0 ≤ λ ≤ 1, then Hλϕ+(1−λ)ψ (γ1 , . . . , γn ) ≥ λHϕ (γ1 , . . . , γn ) + (1 − λ)Hψ (γ1 , . . . , γn ) − (n − 1)(η(λ) + η(1 − λ)); (iii) if ψ is a state on a C∗ -algebra B, 0 ≤ λ ≤ 1 and θk : Bk → B, 1 ≤ k ≤ n, are channels, then on A ⊕ B Hλϕ⊕(1−λ)ψ (γ1 ⊕ θ1 , . . . , γn ⊕ θn ) = λHϕ (γ1 , . . . , γn ) + (1 − λ)Hψ (θ1 , . . . , θn ) + η(λ) + η(1 − λ); (iv) under the assumptions of part (iii), Hϕ⊗ψ (γ1 ⊗ θ1 , . . . , γn ⊗ θn ) ≥ Hϕ (γ1 , . . . , γn ) + Hψ (θ1 , . . . , θn ) on A ⊗ B, and equality holds if B is abelian and the θk ’s are injective homomorphisms; (v) for channels γk : Ak ⊗ Bk → A, 1 ≤ k ≤ n, Hϕ (γ1 , . . . , γn ) ≤ Hϕ (γ1 |A1 , . . . , γn |An ) + 2 S(ϕ ◦ γk |Bk ). k

As we shall see in Chaps. 11 and 12, the inequality in (iv) can be strict. Proof of Proposition 3.1.7. Part (i) follows from the fact that for any ψ ≤ ϕ there exists a unique normal positive linear functional ψ¯ on πϕ (A) such that ψ = ψ¯ ◦ πϕ , so that any decomposition of ϕ comes from a decomposition of ϕ. ¯ To prove (ii), given abelian models (C, µ, {Ck }k , P ) for (A, ϕ, {γk }k ) and (D, ν, {Dk }k , Q) for (A, ψ, {γk }k ), we can form the abelian model (C ⊕ D, λµ ⊕ (1 − λ)ν, {Ck ⊕ Dk }k , P ⊕ Q) for (A, λϕ + (1 − λ)ψ, {γk }k ). In the simplest case n= 1 this means that starting with decompositions ϕ = i ϕi and ψ = j ψj we consider the decomposition λϕ + (1 − λ)ψ = λϕi + (1 − λ)ψj . i

The classical entropy term is then η(λϕi (1)) + η((1 − λ)ψj (1)) i

j

j

3.1 Mutual Entropy

=λ

η(ϕi (1)) + (1 − λ)

i

41

η(ψj (1)) + η(λ) + η(1 − λ).

j

On the other hand, the correction term is by the ﬁrst equality in (3.5) S(λϕ ◦ γ + (1 − λ)ψ ◦ γ) − i S(λϕi ◦ γ) − j S((1 − λ)ψj ◦ γ) = S(λϕ◦γ +(1−λ)ψ ◦γ)−

λS(ϕi ◦γ)−

i

(1−λ)S(ψj ◦γ)−η(λ)−η(1−λ).

j

Recalling that by Theorem 2.2.2(ii) von Neumann entropy is concave, we see that by adding the above expressions for the classical entropy term and the correction term, part (ii) is true for n = 1. For an arbitrary n we similarly get one summand η(λ) + η(1 − λ) from the classical term and n summands −η(λ) − η(1 − λ) from the correction terms. Thus (ii) is proved. To prove (iii) we repeat the computation in (ii), but now for the channels γk ⊕ θk : Ak ⊕ Bk → A ⊕ B, 1 ≤ k ≤ n, and the states ϕ ⊕ 0 and 0 ⊕ ψ. This time, for n = 1, the correction term is S(λϕ ◦ γ ⊕ (1 − λ)ψ ◦ θ) − i S(λϕi ◦ γ) − j S((1 − λ)ψj ◦ θ) = λS(ϕ ◦ γ) + (1 − λ)S(ψ ◦ θ) −

λS(ϕi ◦ γ) −

i

(1 − λ)S(ψj ◦ θ).

j

Thus we do not get n summands −η(λ) − η(1 − λ), which proves the inequality ≥ in (iii). To provethe opposite inequality, starting with a decomposition λϕ ⊕ (1 − λ)ψ = ω , we get decompositions ϕ = ϕi1 ...in and i ...i 1 n ψ = ψi1 ...in such that ωi1 ...in = λϕi1 ...in ⊕ (1 − λ)ψi1 ...in . The correction terms for the ﬁrst decomposition are just convex combinations of the correction terms for the other two decompositions. To estimate the classical term we use the inequality S(λµ + (1 − λ)ν) ≤ λS(µ) + (1 − λ)S(ν) + η(λ) + η(1 − λ) from Theorem 2.2.2(ii). This proves the inequality ≤ in (iii). ϕi1 ...in and ψ = Turning to (iv), note that any decompositions ϕ = ψj1 ...jn give rise to a decomposition ϕi1 ...in ⊗ ψj1 ...jn . ϕ⊗ψ = (i1 ,j1 ),...,(in ,jn )

This proves the inequality in (iv). Assume now that B is abelian, Bk ⊂ B and θk : Bk → B is the inclusion map, 1 ≤ k ≤ n. Since there exists a ψ-preserving conditional expectation B → ∨k Bk , by Lemma 3.1.5 we may without loss of generality assume that B = ∨k Bk . Let {pj }rj=1 be the atoms of B, {χj }rj=1 the corresponding characters. The algebra A ⊗ B can be identiﬁed with ⊕rj=1 A. Under this identiﬁcation ϕ ⊗ ψ = ⊕j ψ(pj )ϕ, and

42

3 Dynamical Entropy

γk (a) ⊗ θk (b) = (χ1 (b)γk (a), . . . , χr (b)γk (a)). Thus the channel γk ⊗ θk factorizes through the channel ⊕j γk : ⊕j Ak → ⊕j A. By part (iii) (or rather by its obvious generalization to direct sums of r algebras) and Proposition 3.1.3(i), we get Hϕ⊗ψ (γ1 ⊗ θ1 , . . . , γn ⊗ θn ) ≤ H⊕j ψ(pj )ϕ (⊕j γ1 , . . . , ⊕j γn ) η(ψ(pj )) ψ(pj )Hϕ (γ1 , . . . , γn ) + = j

j

= Hϕ (γ1 , . . . , γn ) + S(ψ), which completes the proof of (iv), as S(ψ) = Hψ (B1 , . . . , Bn ) by Proposition 3.1.6. It remains to prove (v). So assume we have channels γk : Ak ⊗ Bk → A, 1 ≤ k ≤ n. Any abelian model for (A, ϕ, {γk }k ) can be considered as an abelian model for (A, ϕ, {γk |Ak }k ). The classical terms for the entropies of these models are the same. So we have to compare the correction terms. In view of the second equality in (3.5) we have to prove that if ω = i ωi is a decomposition of a state on Ak ⊗ Bk , then ωi (1)S(ˆ ωi ) ≤ S(ω|Ak ) − ωi (1)S(ˆ ωi |Ak ) + 2S(ω|Bk ). S(ω) − i

i

It suﬃces to show that each of the quantities ωi (1)(S(ˆ ωi |Ak ) − S(ˆ ωi )) S(ω) − S(ω|Ak ) and i

does not exceed S(ω|Bk ). By Theorem 2.2.2(vi) we have |S(ω) − S(ω|Ak )| ≤ S(ω|Bk ) and |S(ˆ ωi ) − S(ˆ ωi |Ak )| ≤ S(ˆ ωi |Bk ). It remains to note that ωi (1)S(ˆ ωi |Bk ) ≤ S(ω|Bk ) i

by concavity of von Neumann entropy, Theorem 2.2.2(ii).

Our next goal is to establish continuity of the entropy Hϕ (γ1 , . . . , γn ) as a function of γ1 , . . . , γn . For this we need to show that there exist models of a ﬁxed size. The idea is quite simple. Given a decomposition ϕ = ϕi1 ...in , we deﬁne a coarser decomposition by collecting indices i, k ∈ Ij such that the (j) (j) states ϕˆi and ϕˆk are close to each other. We shall formulate this procedure in a more reﬁned form, which will be useful later. We shall ﬁrst describe one more way of encoding decompositions of states. If we have a ﬁnite decomposition ϕ = ϕi1 ...in , we can deﬁne a state λ on A ⊗ C(X), where X = I1 × . . . × In , by

3.1 Mutual Entropy

43

λ(a ⊗ 1{(i1 ,...,in )} ) = ϕi1 ...in (a), where 1Z denotes the characteristic function of the set Z. If ξk is the partition of X with atoms I1 × . . . × {ik } × . . . × In , then the original decomposition can be written as λ(· ⊗ 1X1 ∩...∩Xn ). (3.8) ϕ= X1 ∈ξ1 ,...,Xn ∈ξn

More generally, let (X, µ) be a probability space. Deﬁnition 3.1.8. A coupling of (A, ϕ) with (X, µ) is a state λ on the tensor product A ⊗ L∞ (X, µ) such that λ|A = ϕ and λ|L∞ (X,µ) = µ. If X = S(A) is the state space of A, and µ is a regular Borel measure on X, then a coupling is called canonical if λ(a ⊗ f ) = ψ(a)f (ψ) dµ(ψ). X

Thus a canonical coupling is determined by a probability measure µ on S(A) with barycenter ϕ, that is, ϕ = ψ dµ(ψ). Note that for any a ∈ A the functional λ(a ⊗ ·) is normal on L∞ (X, µ), since if a is positive then it is dominated by a scalar multiple of µ. It follows that if X is a compact space and µ is a regular Borel measure on X, then every coupling λ of (A, ϕ) with (X, µ) is determined by its restriction to A ⊗ C(X). Moreover, any state λ on A⊗C(X) such that λ|A = ϕ and λ|C(X) = µ extends uniquely to a coupling of (A, ϕ) with (X, µ). Now if ξ1 , . . . , ξn are ﬁnite measurable partitions of (X, µ), we can deﬁne a decomposition of ϕ by (3.8). For any channels γ1 , . . . , γn we denote their mutual entropy with respect to this decomposition by Hλ (γ1 , . . . , γn ; ξ1 , . . . , ξn ). Thus by deﬁnition Hλ (γ1 , . . . , γn ; ξ1 , . . . , ξn ) = Hµ (∨k ξk ) + S(λ(γk (·) ⊗ 1Z ), ϕ ◦ γk ). k Z∈ξk

(3.9) By virtue of (3.2), the right hand side can be written as Hµ (∨k ξk )− Hµ (ξk )+ µ(Z)S(µ(Z)−1 λ(γk (·)⊗1Z ), ϕ◦γk ). (3.10) k

k Z∈ξk

Putting ϕZ = µ(Z)−1 λ(· ⊗ 1Z ), we can also write this expression as ⎛ ⎞ ⎝S(ϕ ◦ γk ) − Hµ (ξk ) + µ(Z)S(ϕZ ◦ γk )⎠ . (3.11) Hµ (∨k ξk ) − k

k

Note that (3.3) now becomes

Z∈ξk

44

3 Dynamical Entropy

Hλ (γ1 , . . . , γn ; ξ1 , . . . , ξn ) = Hµ (∨k ξk ) −

Hµ (ξk ) +

k

Hλ (γk ; ξk ). (3.12)

k

coupling in an obvious Any decomposition ϕ = i ϕi deﬁnes a canonical way: take the measure on X = S(A) to be ϕ (1)δ ˆi is the i ϕ ˆi , where δϕ i δ-measure concentrated at the point ϕˆi . To put it diﬀerently, given a coupling λ of (A, ϕ) with (Y, µ), and a ﬁnite measurable partition ξ of Y , we can construct a map f : Y → X which sends an atom Z of ξ to the state ϕZ = µ(Z)−1 λ(·⊗1Z ). Consider the image µ = f∗ (µ) of the measure µ, and put λ = λ◦(idA ⊗f ∗ ), where f ∗ : L∞ (X, µ ) → L∞ (Y, µ) is the map induced by f . Then λ is the canonical coupling deﬁned by µ . In ∗ deed, for any bounded Borel function g on X we have f (g) = Z∈ξ g(ϕZ )1Z , whence λ (a ⊗ g) = g(ϕZ )λ(a ⊗ 1Z ). Z

On the other hand, g(ψ)ψ(a) dµ (ψ) = g(ψZ )ψZ (a)µ(Z) = g(ψZ )λ(a ⊗ 1Z ). X

Z

Z

Thus the coupling λ is canonical. Note further that if ζ is a ﬁnite Borel partition of X, then the decompositions of ϕ deﬁned by the pairs (λ, f −1 (ζ)) and (λ , ζ) are the same, so that Hλ (γ; f −1 (ζ)) = Hλ (γ; ζ) for any channel γ: B → A. In particular, if ζ is such that f −1 (ζ) = ξ, then Hλ (γ; ξ) = Hλ (γ; ζ). But in fact any suﬃciently ﬁne ζ gives Hλ (γ; ζ) close to Hλ (γ; ξ). Lemma 3.1.9. Let notation be as above. For ε > 0 choose δ > 0 such that |S(ψ1 ) − S(ψ2 )| < ε for states ψ1 and ψ2 on B as soon as ψ1 − ψ2 < δ. Assume ζ is a ﬁnite Borel partition of S(A) such that ϕ1 ◦ γ − ϕ2 ◦ γ < δ as soon as ϕ1 and ϕ2 are in the same atom of ζ. Then |Hλ (γ; ξ) − Hλ (γ; ζ)| < ε. Proof. For W ∈ ζ, set ϕW = µ (W )−1 λ (· ⊗ 1W ). Then ϕW = µ (W )−1

Z∈ξ:Z⊂f −1 (W )

λ(· ⊗ 1Z ) =

Z∈ξ:Z⊂f −1 (W )

µ(Z) ϕZ . µ(f −1 (W ))

By assumption, the states ϕZ with Z ⊂ f −1 (W ) are δ-close to each other when composed with γ. It follows that (ϕW − ϕZ ) ◦ γ < δ, and thus |S(ϕW ◦ γ) − S(ϕZ ◦ γ)| < ε for Z ∈ ξ, Z ⊂ f −1 (W ). Since

3.1 Mutual Entropy

S(ϕ ◦ γ) − Hλ (γ; ζ) =

45

µ(f −1 (W ))S(ϕW ◦ γ)

W

by (3.11), and similarly S(ϕ ◦ γ) − Hλ (γ; ξ) =

µ(Z)S(ϕZ ◦ γ) =

Z

µ(Z)S(ϕZ ◦ γ),

W Z⊂f −1 (W )

we see that |Hλ (γ; ξ) − Hλ (γ; ζ)| < ε.

Note that a partition ζ satisfying the conditions of the previous lemma can be chosen such that the number |ζ| of elements of ζ is not bigger than the number of balls of diameter δ which are needed to cover the state space of B. Thus, what we have proved, is that if we want to compute Hϕ (γ) up to ε, it suﬃces to consider canonical couplings and one Borel partition of S(A) of size depending only on dim B and ε. More generally, if λ is a coupling of (A, ϕ) with (Y, µ), and ξ1 , . . . , ξn are ﬁnite measurable partitions of Y , we can deﬁne a measure µ on X n , where X = S(A), and a coupling λ of (A, ϕ) with (X n , µ ) as follows. Let fk : Y → X be the map deﬁned by λ and ξk , so fk maps an atom Z of ξk to the state µ(Z)−1 λ(·⊗1Z ) ∈ X. Then the map f = (f1 , . . . , fn ): Y → X n is measurable, and we set µ = f∗ (µ) and λ = λ ◦ (id ⊗ f ∗ ). The properties of the coupling λ are summarized in the following proposition. Proposition 3.1.10. Under the above assumptions, ﬁx ε > 0 and choose δ > 0 such that |S(ψ1 ) − S(ψ2 )| < ε for states ψ1 and ψ2 on Ak as soon as ψ1 − ψ2 < δ, 1 ≤ k ≤ n. For each k let ζk be a ﬁnite Borel partition of X such that ϕ1 ◦ γk − ϕ2 ◦ γk < δ as soon as ϕ1 and ϕ2 are in the same atom of ζk . Then (i) fk−1 (ζk ) ≺ ξk ; (ii) the coupling λk = λ ◦ (id ⊗ pr∗k ) = λ ◦ (id ⊗ fk∗ ) of (A, ϕ) with (X, µk ), where prk : X n → X is the projection onto the k-th factor and µk = (prk )∗ (µ ) = (fk )∗ (µ), is canonical and |Hλ (γk ; ξk ) − Hλk (γk ; ζk )| < ε; −1 (iii) Hλ (γ1 , . . . , γn ; ξ1 , . . . , ξn ) < Hλ (γ1 , . . . , γn ; pr−1 1 (ζ1 ), . . . , prn (ζn )) + nε.

Proof. Part (i) is obvious as by deﬁnition fk maps the atoms of ξk to points. Part (ii) follows from Lemma 3.1.9 and the discussion preceding it. −1 To prove (iii) note that since Hλ (γ1 , . . . , γn ; pr−1 1 (ζ1 ), . . . , prn (ζn )) is −1 −1 −1 nothing but Hλ (γ1 , . . . , γn ; f1 (ζ1 ), . . . , fn (ζn )), and Hλ (γk ; fk (ζk )) equals Hλk (γk ; ζk ), in view of (ii) and equality (3.12), it suﬃces to show that Hµ (∨k ξk ) −

k

Hµ (ξk ) ≤ Hµ (∨k fk−1 (ζk )) −

Hµ (fk−1 (ζk )).

k

Since fk−1 (ζk ) ≺ ξk , this is true by Proposition 1.1.9(iv).

46

3 Dynamical Entropy

The previous proposition in its full generality will be used in subsequent chapters. What is important for us at the moment, is that it implies existence of models of a ﬁxed size. Namely, given ε > 0 and d ∈ N there exists r > 0 such that for any n ∈ N and any channels γk : Ak → A, 1 ≤ k ≤ n, such that dim Ak ≤ d, there exist a coupling λ of (A, ϕ) with a probability space (Y, µ), and ﬁnite measurable partitions ζ1 , . . . , ζn such that |ζk | ≤ r, 1 ≤ k ≤ n, and Hϕ (γ1 , . . . , γn ) < Hλ (γ1 , . . . , γn ; ζ1 , . . . , ζn ) + nε. Explicitly, ﬁx ε < ε. Then take r to be the number of balls of diameter δ which are needed to cover the state space of a C∗ -algebra B with dim B ≤ d, where δ is such that |S(ψ1 ) − S(ψ2 )| < ε for states ψ1 and ψ2 on B as soon as ψ1 − ψ2 < δ. Take a coupling Λ and partitions ξ1 , . . . , ξn such that HΛ (γ1 , . . . , γn ; ξ1 , . . . , ξn ) is close to Hϕ (γ1 , . . . , γn ) up to ε−ε . Then applying the procedure described above we get λ and ζ1 , . . . , ζn . For channels γ, γ : B → A, put γ − γ ϕ =

sup

γ(x) − γ (x)ϕ ,

x∈B,x≤1

where aϕ = ϕ(a∗ a)1/2 . Proposition 3.1.11. For every ε > 0 and d ≥ 1 there exists δ > 0 such that for any C∗ -algebra A with a state ϕ, n ∈ N and channels γk , γk : Ak → A such that dim Ak ≤ d and γk − γk ϕ < δ, 1 ≤ k ≤ n, we have |Hϕ (γ1 , . . . , γn ) − Hϕ (γ1 , . . . , γn )| < nε. Proof. By Proposition 3.1.10 and the discussion following it we can ﬁnd r ∈ N depending only on ε and d, and a decomposition ϕ = i1 ,...,in ∈I ϕi1 ...in with |I| ≤ r such that Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) > Hϕ (γ1 , . . . , γn ) −

nε . 2

To estimate the diﬀerence between the mutual entropies of γ1 , . . . , γn and γ1 , . . . , γn with respect to the decomposition ϕ = i1 ,...,in ϕi1 ...in we have to estimate the diﬀerences between the correction terms. Thus, what we need to prove is the following: there exists δ > 0such that if γ, γ : B → A are channels, dim B ≤ d, γ − γ ϕ < δ and ϕ = i∈I ϕi with |I| ≤ r, then ε S(ϕi ◦ γ, ϕ ◦ γ) − S(ϕi ◦ γ , ϕ ◦ γ ) < . (3.13) 2 i i We have S(ϕi ◦ γ, ϕ ◦ γ) − S(ϕi ◦ γ , ϕ ◦ γ ) i

i

3.1 Mutual Entropy

= S(ϕ ◦ γ) − S(ϕ ◦ γ ) −

47

ϕi (1)(S(ϕˆi ◦ γ) − S(ϕˆi ◦ γ )).

i

Choose δ0 > 0 such that |S(ψ1 ) − S(ψ2 )| < ε/6 for states ψ1 and ψ2 on B as soon as ψ1 − ψ2 < δ0 and dim B ≤ d. If δ < δ0 , then as |ϕ(x)| ≤ xϕ , we have ϕ ◦ γ − ϕ ◦ γ ≤ γ − γ ϕ < δ0 , so that

ε . 6 Set I0 = {i ∈ I | ϕi (1) > (δ/δ0 )2 }. Then for i ∈ I0 we have |S(ϕ ◦ γ) − S(ϕ ◦ γ )|

0 there exists a ﬁnite index set I depending only on ε and the dimensions of the Ak ’s such that for any state ϕ there is a decomposition ϕ = i1 ,...,in ∈I ϕi1 ...in with Hϕ (γ1 , . . . , γn ; {ϕi1 ...in }) > Hϕ (γ1 , . . . , γn ) − nε. Now if {ϕj }j∈J is a net converging to ϕ we can choose decompositions of the ϕj ’s as above, and passing to a subnet may assume that these decompositions converge to a decomposition of ϕ. Since the map (ω, ψ) → S(ω◦γk , ψ◦γk ) is continuous on the set {ω ≤ ψ} by Theorem 2.1.2(iii), it follows that Hϕ (γ1 , . . . , γn ) ≥ lim sup Hϕj (γ1 , . . . , γn ) − nε. j

Since ε > 0 was arbitrary, we get the result.

48

3 Dynamical Entropy

3.2 Entropy of Dynamical Systems We are now ready to deﬁne entropy of dynamical systems. By a C∗ -dynamical system we mean a triple (A, ϕ, α), where A is a unital C∗ -algebra, ϕ a state on A, α a ϕ-preserving automorphism of A. If A is a von Neumann algebra and the state is normal, we call (A, ϕ, α) a W∗ -dynamical system. Deﬁnition 3.2.1. The entropy of α with respect to a channel γ: B → A is 1 Hϕ (γ, α ◦ γ, . . . , αn−1 ◦ γ). n→∞ n

hϕ (γ; α) = lim

The dynamical entropy hϕ (α) is the supremum of hϕ (γ; α) for all possible γ’s. Note that if we denote Hϕ (γ, α ◦ γ, . . . , αn−1 ◦ γ) by an , then an+m ≤ an + am by properties (ii) and (iii) in Proposition 3.1.3. Hence the limit in the above deﬁnition exists by Lemma 1.1.2. Note also that the deﬁnition makes sense for any unital completely positive ϕ-preserving map α. Theorem 3.2.2. Let (A, ϕ, α) be a C∗ -dynamical system. Then (i) if β: A → B is an isomorphism, then hϕ◦β −1 (β ◦ α ◦ β −1 ) = hϕ (α); (ii) if πϕ : A → B(H) is the GNS-representation and ϕ¯ and α ¯ are the normal state and the automorphism of πϕ (A) deﬁned by ϕ and α, respectively, then hϕ¯ (¯ α) = hϕ (α); (iii) hϕ (α−1 ) = hϕ (α), and hϕ (αn ) ≤ |n|hϕ (α) for any n ∈ Z; (iv) for any C∗ -dynamical system (B, ψ, β), hλϕ⊕(1−λ)ψ (α ⊕ β) = λhϕ (α) + (1 − λ)hψ (β) and hϕ⊗ψ (α ⊗ β) ≥ hϕ (α) + hψ (β); (v) if B is an α-invariant subalgebra of A and there exists a ϕ-preserving conditional expectation A → B, then hϕ|B (α|B ) ≤ hϕ (α). Proof. Part (i) follows immediately from the deﬁnitions. ¯ ) = hϕ (γ; α) for any channel Proposition 3.1.7(i) shows that hϕ¯ (πϕ ◦ γ; α γ: B → A. By Proposition 3.1.11 we also know that if two channels γ, γ : B → πϕ (A) are close in the · ϕ¯ -seminorm, then hϕ¯ (γ; α ¯ ) and hϕ¯ (γ ; α ¯ ) are close. Thus to prove (ii) we have to show that for any channel γ: B → πϕ (A) there exists a channel γ˜ : B → A such that γ − πϕ ◦ γ˜ ϕ¯ is arbitrarily small. This is possible by A.10 and A.11, and (ii) is proved. By Proposition 3.1.3(ii) we have Hϕ (γ, α ◦ γ, . . . , αn ◦ γ) = Hϕ (α−n ◦ γ, α−n+1 ◦ γ, . . . , γ).

3.2 Entropy of Dynamical Systems

49

It follows that hϕ (α−1 ) = hϕ (α). If n ≥ 1, we have Hϕ (γ, αn ◦ γ, . . . , α(m−1)n ◦ γ) ≤ Hϕ (γ, α ◦ γ, . . . , αmn−1 ◦ γ), whence hϕ (γ; αn ) ≤ nhϕ (γ; α). Thus (iii) is proved. Part (iv) follows from Proposition 3.1.7(iii)-(iv), while (v) is a consequence of Lemma 3.1.5.

To proceed further we need to restrict ourselves to systems satisfying the following approximation property. We say that a net of channels {γi : Ai → A}i is ϕ-approximating if there exists a net {θi : A → Ai }i of unital completely positive maps such that (γi ◦ θi )(x) − xϕ → 0 for any x ∈ A. If γ: B → A is an arbitrary channel, then γi ◦ θi ◦ γ − γϕ → 0. Hence by Proposition 3.1.11 lim hϕ (γi ◦ θi ◦ γ; α) = hϕ (γ; α). i

On the other hand, hϕ (γi ◦ θi ◦ γ; α) ≤ hϕ (γi ; α) ≤ hϕ (α) by Proposition 3.1.3(i). We thus get the following weak analogue of the Kolmogorov-Sinai theorem. Theorem 3.2.3. If {γi : Ai → A}i is a ϕ-approximating net for a C∗ -dynamical system (A, ϕ, α), then hϕ (α) = lim hϕ (γi ; α).

i

Approximating nets exist for nuclear C∗ -algebras and injective von Neumann algebras, and in fact existence of a ϕ-approximating net is close to πϕ (A) being injective. If {Ai }i is an increasing net of ﬁnite dimensional C∗ -subalgebras of a C∗ -algebra A with ∪i Ai norm-dense in A, then the net {Ai → A}i is ϕ-approximating (take arbitrary conditional expectations A → Ai for θi ’s). If A is a von Neumann algebra, ϕ is normal and ∪i Ai is dense in A in the strong operator topology, then the net {Ai → A}i is again ϕ-approximating, but this is not so simple, see e.g. [91]. Note that we do not really need this fact, since what is important for Theorem 3.2.3, is that any channel into A can be approximated by channels into ∪i Ai , which is easy by A.10. We also have the following related approximation result. Proposition 3.2.4. Let (A, ϕ, α) be a C∗ -dynamical system, {Ai }i∈I an increasing net of α-invariant C∗ -subalgebras of A such that ∪i πϕ (Ai ) is strongly operator dense in πϕ (A). Then hϕ (α) ≤ lim inf i hϕ|Ai (α|Ai ). Proof. By A.10 and A.11 any channel into A can be approximated in the · ϕ -seminorm by channels into ∪i Ai . But if γ: B → Ai is a channel, then hϕ (γ; α) ≤ hϕ|Ai (γ; α|Ai ) by Theorem 3.1.3(ii) applied to the inclusion map θ: Ai → A.

We now show how approximation properties translate into properties of dynamical entropy.

50

3 Dynamical Entropy

Theorem 3.2.5. Let (A, ϕ, α) be a C∗ -dynamical system having a ϕ-approximating net. Then (i) hϕ (αn ) = |n|hϕ (α) for any n ∈ Z; moreover, if {αt }t∈R is a ϕ-preserving one-parameter automorphism group of A such that α1 = α, then hϕ (αt ) = |t|hϕ (α) for any t ∈ R; (ii) if ϕ = λω + (1 − λ)ψ, where ω and ψ are α-invariant states, then hϕ (α) ≥ λhω (α) + (1 − λ)hψ (α); (iii) if (B, ψ, β) is an abelian C∗ -dynamical system, then hϕ⊗ψ (α ⊗ β) = hϕ (α) + hψ (β); (iv) if ψ is a state on a C∗ -algebra B, and there exists a ψ-approximating net, then hϕ⊗ψ (α ⊗ idB ) = hϕ (α). In Sect. 3.3 we shall extend (iii) to type I algebras. Proof of Theorem 3.2.5. We already know from Theorem 3.2.2(iii) that hϕ (αn ) ≤ |n|hϕ (α). To prove the opposite inequality we may by the same theorem assume n ≥ 1. Let {γi : Ai → A}i be a ϕ-approximating net, and θi : A → Ai , i ∈ I, be maps as in the deﬁnition of such a net. Fix a channel γ: B → A and δ > 0, and choose i ∈ I such that γi ◦ θi ◦ αk ◦ γ − αk ◦ γϕ < δ for 0 ≤ k ≤ n − 1. Then αln ◦ γi ◦ θi ◦ αk ◦ γ − αln+k ◦ γϕ < δ for any l ≥ 0 and 0 ≤ k ≤ n − 1. It follows by Propositions 3.1.11 and 3.1.3(i),(iv) that Hϕ (γ, α ◦ γ, . . . , αnm−1 ◦ γ) ≤ Hϕ ({αln ◦ γi ◦ θi ◦ αk ◦ γ}0≤l≤m−1,0≤k≤n−1 ) + nmε ≤ Hϕ (γi , αn ◦ γi , . . . , αn(m−1) ◦ γi ) + nmε, where ε = ε(δ, dim B) → 0 as δ → 0. Thus hϕ (γ; α) ≤ n−1 hϕ (γi ; αn ) + ε, so that hϕ (αn ) ≥ nhϕ (α). Assume now we have a ﬂow {αt }t with α1 = α. To prove that t−1 hϕ (αt ) is independent of t > 0, it suﬃces to show that t2 hϕ (αt1 ) ≤ t1 hϕ (αt2 ) for any t1 , t2 > 0. Replacing {αt }t by {αt2 t }t , we just have to show that hϕ (αt ) ≤ thϕ (α1 ) for any t > 0. The proof is similar to the argument above. Since the set {αs (a) | 0 ≤ s ≤ 1} is compact for any a ∈ A, given a channel γ: B → A and δ > 0 we can ﬁnd i ∈ I such that γi ◦ θi ◦ αs ◦ γ − αs ◦ γϕ < δ for 0 ≤ s ≤ 1. [lt]

Then α1 ◦ γi ◦ θi ◦ αlt−[lt] ◦ γ − αtl ◦ γϕ < δ for any l ≥ 0, where [lt] denotes the integer part of lt. Hence

3.2 Entropy of Dynamical Systems [(n−1)t]

Hϕ (γ, αt ◦ γ, . . . , αtn−1 ◦ γ) ≤ Hϕ (γi , α1 ◦ γi , . . . , α1

51

◦ γi ) + nε,

so that in the limit hϕ (γ; αt ) ≤ thϕ (γi ; α1 ) + ε. Thus (i) is proved. By Proposition 3.1.7(ii) we have hϕ (α) ≥ λhω (α) + (1 − λ)hψ (α) − η(λ) − η(1 − λ). We can apply this inequality to αn . By virtue of (i), dividing by n and letting n → ∞, we get (ii). Let (B, ψ, β) be an abelian C∗ -dynamical system. By Proposition 3.2.2(ii), in proving (iii) we may assume that B is a von Neumann algebra. Then B is in particular an inductive limit of ﬁnite dimensional abelian C∗ -algebras, so there exists a ψ-approximating net {γk }k consisting of injective homomorphisms. Then {γi ⊗ γk }(i,k) is a (ϕ ⊗ ψ)-approximating net, and (iii) follows from Proposition 3.1.7(iv). To prove (iv), note that hϕ⊗ψ (α ⊗ idB ) ≥ hϕ (α) by Theorem 3.2.2(v). To prove the opposite inequality consider ﬁrst the case when B is ﬁnite dimensional. If {γi }i is a ϕ-approximating net, then the net {γi ⊗ idB }i is (ϕ ⊗ ψ)-approximating. By Proposition 3.1.7(v) we have hϕ⊗ψ (γi ⊗ idB ; α ⊗ idB ) ≤ hϕ (γi ; α) + 2S(ψ), so that hϕ⊗ψ (α ⊗ idB ) ≤ hϕ (α) + 2S(ψ). As in the proof of (ii), applying this inequality to αn we conclude that hϕ⊗ψ (α ⊗ idB ) ≤ hϕ (α). Consider now the general case. Let {γk : Bk → B}k be a ψ-approximating net. Then {γi ⊗γk }(i,k) is a (ϕ⊗ψ)-approximating net. By Proposition 3.1.3(ii) and ﬁnite dimensionality of the Bk ’s we have hϕ⊗ψ (γi ⊗ γk ; α ⊗ idB ) ≤ hϕ⊗(ψ◦γk ) (γi ⊗ idBk ; α ⊗ idBk ) ≤ hϕ (α). Hence hϕ⊗ψ (α ⊗ idB ) ≤ hϕ (α).

Similarly to the classical case, the general theory we have developed so far was ﬁrst applied to compute entropy of noncommutative Bernoulli shifts. We next discuss these systems. Example 3.2.6. (i) Noncommutative Bernoulli shifts. Let B be a ﬁnite dimensional C∗ algebra, ψ a state on B, (M, ϕ) = (B, ψ)⊗Z the inﬁnite W∗ -tensor product (that is, if A = B ⊗Z is the inﬁnite C∗ -tensor product, then M is the weak operator closure of A in the GNS-representation corresponding to the state ψ ⊗Z ), and α the shift to the right on M . Then hϕ (α) = S(ψ). To see this, let πn : B → M be the canonical homomorphism onto the n-th factor, so that α(πn (b)) = πn+1 (b). Let B[n,m] be the algebra generated by πk (B) for n ≤ k ≤ m. If D is a maximal abelian subalgebra in the centralizer of ψ, then the algebras πk (D), n ≤ k ≤ m, mutually commute and generate a

52

3 Dynamical Entropy

maximal abelian subalgebra in the centralizer of the restriction of ϕ to B[n,m] . By Proposition 3.1.6 it follows that for any n, m ∈ N we have Hϕ (B[−n,n] , . . . , αm (B[−n,n] )) = S(ϕ|B[−n,n+m] ) = (m + 2n + 1)S(ψ). Thus hϕ (B[−n,n] ; α) = S(ψ), and hϕ (α) = S(ψ) by Theorem 3.2.3. We remark that the same result is true if B is inﬁnite dimensional and we use the deﬁnition of entropy of a state mentioned in Notes to Chap. 2. (ii) Bernoulli shifts on the hyperﬁnite II1 -factor. Consider the same system as above, but assume that B ∼ = Matn (C) and ψ is faithful. Let N = Mϕ be the centralizer of ϕ, so β = α|N is an automorphism and τ = ϕ|N is a faithful normal β-invariant trace on N . Then hτ (β) = S(ψ). To see this we can repeat the above argument with B[n,m] replaced by N ∩ B[n,m] . Another possibility is to argue as follows. Denote by C the von Neumann subalgebra of M generated by πn (D), n ∈ Z. There exist ϕ-preserving conditional expectations M → N and N → C. Hence hτ |C (β|C ) ≤ hτ (β) ≤ hϕ (α). But (C, τ |C , β|C ) is the classical Bernoulli shift with entropy S(ψ). Since also hϕ (α) = S(ψ), we see that hτ (β) = S(ψ). Observe next that N is the hyperﬁnite II1 -factor. The hyperﬁniteness is clear. We shall give two proofs of the factoriality. The ﬁrst one is shorter, but the second one can be applied in more general situations. The group S∞ of ﬁnite permutations of Z acts naturally on M . Denote this action by α: S∞ → Aut(M ). If x, y ∈ ∪n B[−n,n] , then ϕ(αg (x)y) = ϕ(x)ϕ(y) for g outside a ﬁnite subset of S∞ . It follows that for any x, y ∈ M we have ϕ(αg (x)y) → ϕ(x)ϕ(y) as g goes to inﬁnity, meaning that g eventually leaves every ﬁnite subset of S∞ . In particular, the ﬁxed point algebra M S∞ is trivial. On the other hand, since any automorphism of a full matrix algebra is inner, for each g ∈ S∞ there exists ug ∈ ∪n B[−n,n] such that αg = Ad ug . Since ϕ is αg -invariant, we have ug ∈ Mϕ = N . Hence if z is in the center of N , then z ∈ M S∞ . Therefore z is a scalar. For the second proof note that M , being an inﬁnite tensor product of factors with respect to a product-state, is a factor. Thus it suﬃces to check that the center of N is contained in the center of M . We shall prove that the relative commutant N ∩ M is contained in the center of M . Let z ∈ N ∩ M , x ∈ B[−m,m] , x = 0. We want to prove that xz = zx. Since σtϕ |B[−m,m] has pure point spectrum, without loss of generality we may assume that σtϕ (x) = λit x for t ∈ R and some λ > 0. Then σtϕ (αn (x∗ )x) = αn (x∗ )x, so αn (x∗ )x ∈ N . Hence αn (x∗ )xz = zαn (x∗ )x, and thus αn (xx∗ )xz = αn (x)zαn (x∗ )x. We claim that αn (xx∗ ) → ϕ(xx∗ )1 in the weak operator topology and [αn (x), z] → 0 in the strong operator topology as n → ∞. Thus letting n → ∞ in the equality above we get ϕ(xx∗ )xz on the left hand side and ϕ(xx∗ )zx on the right hand side, whence xz = zx.

3.2 Entropy of Dynamical Systems

53

It remains to prove the claims. If y ∈ ∪k B[−k,k] , then ϕ(αn (xx∗ )y) = ϕ(xx∗ )ϕ(y) if n is suﬃciently large. Thus any weak operator limit point a of the sequence {αn (xx∗ )}n has the property ϕ(ay) = ϕ(xx∗ )ϕ(y). Hence a = ϕ(xx∗ )1, and the ﬁrst claim is proved. To prove the second claim, note 1/2 that αn (x)ξϕ = Jϕ ∆ϕ αn (x∗ )ξϕ = λ−1/2 Jϕ αn (x∗ )ξϕ , where we use the same notation as on page 35. Then the equality [αn (x), y]ξϕ = αn (x)yξϕ − yαn (x)ξϕ = αn (x)yξϕ − λ−1/2 Jϕ αn (x∗ )Jϕ yξϕ shows that to prove that [αn (x), y] → 0 strongly for any y ∈ M , it suﬃces to consider y lying in a strongly dense subspace. But if y ∈ ∪k B[−k,k] then [αn (x), y] = 0 for all n suﬃciently large. This completes our second proof of factoriality of N . It follows from the discussion prior to Lemma 3.1.5 that the entropy of a system can be smaller than the entropy of a subsystem. Bernoulli shifts provide a simple example. If in Example 3.2.6 we took an arbitrary maximal abelian subalgebra D of B, then we would get an abelian system (C, ϕ|C , α|C ) with entropy S(ψ|D ), which is strictly larger than S(ψ) unless D is in the centralizer of ψ. Nevertheless if we have a conditional expectation onto a subalgebra commuting with an automorphism, then any state on the subalgebra extends to a state on the algebra such that the entropy of the system becomes at least as large as the entropy of the subsystem. The following result is a simple but useful extension of this fact. Proposition 3.2.7. Let α be an automorphism of a unital C∗ -algebra A, B ⊂ A an α-invariant unital C∗ -subalgebra, ψ an α-invariant state on B. Assume there exists a ψ-approximating net. Then there exists an α-invariant state ϕ on A such that ϕ|B = ψ and hϕ (α) ≥ hψ (α|B ). Proof. Let {γi : Bi → B}i be a ψ-approximating net, and θi : B → Bi , i ∈ I, be maps as in the deﬁnition of such a net. By Arveson’s extension theorem, A.8, we can extend each θi to a unital completely positive map θ¯i : A → Bi . Consider the GNS-representation π: B → B(H) corresponding to ψ, and let ψ¯ and β be the normal state and the automorphism of π(B) deﬁned by ψ and α|B , respectively. Let Φ: A → π(B) be any pointwise weak operator limit point of {π ◦γi ◦ θ¯i }i . Then Φ is a unital completely positive map, and Φ(b)ξψ = π(b)ξψ for any b∈ B. Replacing further Φ by a pointwise weak operator limit point n−1 of {n−1 k=0 β k ◦ Φ ◦ α−k }, we may also assume that β ◦ Φ = Φ ◦ α. Then ¯ ϕ = ψ ◦ Φ is an α-invariant state extending ψ. Moreover, if ω is a positive linear functional on B such that ω ≤ ψ, then ω extends to a normal positive functional ω ¯ on π(B) , and ω ¯ ◦ Φ extends ω. Hence any decomposition of ψ extends to a decomposition of ϕ, so that hϕ (α) ≥ hψ (α|B ).

A related result will be proved in Chap. 5.

54

3 Dynamical Entropy

3.3 Type I Algebras In this section we consider type I algebras. Not surprisingly the theory of dynamical entropy in this case reduces completely to abelian systems. Recall that a type I von Neumann algebra is a direct sum of algebras of ¯ the form A⊗B(H) with A abelian. If the algebra is σ-ﬁnite, that is, every set of mutually orthogonal projections is at most countable, then one can equivalently say that the algebra is a direct integral of algebras B(H), see App. C. Theorem 3.3.1. Let (M, ϕ, α) be a W∗ -dynamical system with M a type I von Neumann algebra. Let Z = Z(M ) be the center of M . Then hϕ (α) = hϕ|Z (α|Z ). More generally, if (B, ψ, β) is a C∗ -dynamical system having a ψ-approximating net, then hψ⊗ϕ (β ⊗ α) = hψ (β) + hϕ|Z (α|Z ). Proof. Since the modular group acts trivially on the center, there exists a ϕ-preserving conditional expectation M → Z, and hence the inequality ≥ follows from Theorem 3.2.2(iv),(v). To prove the opposite inequality consider ﬁrst the case when M ∼ = Z ⊗ Matn (C). The proof in this case is similar to that of Theorem 3.2.5(iv). Let {γi }i be a ψ-approximating net and {γk }k be a ϕ|Z -approximating net. Then the net {γi ⊗ γk ⊗ idMatn (C) }(i,k) is (ψ ⊗ ϕ)approximating. By Proposition 3.1.7(v) we have hψ⊗ϕ (γi ⊗ γk ⊗ idMatn (C) ; β ⊗ α) ≤ hψ⊗(ϕ|Z) (γi ⊗ γk ; β ⊗ α|Z ) + 2 log n, so that hψ⊗ϕ (β ⊗ α) ≤ hψ⊗(ϕ|Z) (β ⊗ α|Z ) + 2 log n. Applying this inequality to β m and αm we conclude that hψ⊗ϕ (β ⊗ α) ≤ hψ⊗(ϕ|Z) (β ⊗ α|Z ) = hψ (β) + hϕ|Z (α|Z ), where the equality follows from Theorem 3.2.5(iii). Consider next the case when M is ﬁnite, so M is a (possibly inﬁnite) direct sum of algebras Zn ⊗Matn (C) with Zn abelian. The required inequality would follow from the previous case if we could use an inﬁnite analogue of Theorem 3.2.2(iv). In other words, we have to check that if (N, ϕ, γ) is a W∗ dynamical system and zn , n ∈ N, are γ-invariant central projections in N such that n zn = 1, then hϕ (γ) = λn hϕn (γ|N zn ), n

where λn = ϕ(zn ) and ϕn = λ−1 n ϕ|N zn . Set Mn = N (z1 + . . . + zn ) + C(1 − z1 − . . . − zn ).

3.3 Type I Algebras

55

Then ∪n Mn is strongly operator dense in N . Since there exist ϕ-preserving conditional expectations N → Mn , by Theorem 3.2.2(v) and Proposition 3.2.4 we get hϕ|Mn (γ|Mn ) → hϕ (γ). Since hϕ|Mn (γ|Mn ) =

n

λk hϕk (γ|N zk )

k=1

by Theorem 3.2.2(iv), we get the result. Consider now the general case. Replacing M by pM p, where p is the support of ϕ, we may assume that ϕ is faithful, in particular M is σﬁnite. We claim that there exists an increasing sequence {pn }∞ n=1 of ﬁnite α-invariant projections which belongs to the centralizer Mϕ of ϕ and is such that 1 − pn ϕ → 0. Indeed, identify Z with L∞ (X, µ), so that ϕ|Z is deﬁned by µ. Then, see C.3 for the notation, we have a direct integral decomposition ⊕ ⊕ M= B(Hx )dµ(x), ϕ = ϕx dµ(x). X

X

Let Qx be the density operator of the state ϕx , so ϕx = Tr(· Qx ). Let pn (x) be the spectral projection of Qx corresponding to the interval [1/n, 1]. Then pn (x) is a ﬁnite projection in B(Hx ) belonging to the centralizer of ϕx , and 1 − pn (x)ϕx → 0. Hence

⊕

pn =

pn (x)dµ(x), n ∈ N,

X

are the required projections. Set Mn = pn M pn + C(1 − pn ). Then a → pn apn + ϕ(a(1 − pn ))(1 − pn ) is a ϕ-preserving conditional expectation onto Mn . Denote it by En . Since 1 − pn ϕ → 0, we have a − En (a)ϕ → 0 for any a ∈ M . It follows that hψ⊗ϕ (β ⊗ α) = lim hψ⊗(ϕ|Mn ) (β ⊗ α|Mn ). n

The algebra Mn is ﬁnite and of type I. Denote by Zn its center. By the previous case hψ⊗(ϕ|Mn ) (β ⊗ α|Mn ) = hψ (β) + hϕ|Zn (α|Zn ). To ﬁnish the proof, it thus suﬃces to check that hϕ|Zn (α|Zn ) ≤ hϕ|Z (α|Z ) for any n. If pn = 1, there is nothing to prove. Otherwise let zn ∈ Z be the central support of pn . Then Zn = Zpn ⊕ C(1 − pn ) is isomorphic to Zzn ⊕ C. Under this isomorphism the state ϕ|Zn becomes ϕ(pn )ψn ⊕ ϕ(1 − pn ), where ψn = ϕ(pn )−1 ϕ(· pn )|Zzn . Hence hϕ|Zn (α|Zn ) = ϕ(pn )hψn (α|Zzn ). On the other hand, consider the state ϕn = ϕ(zn )−1 ϕ|Zzn . Since Zzn + C(1 − zn ) is an α-invariant subalgebra of Z, we have hϕ|Z (α|Z ) ≥ ϕ(zn )hϕn (α|Zzn ). Hence we just have to prove that

56

3 Dynamical Entropy

hψn (α|Zzn ) ≤

ϕ(zn ) hϕ (α|Zzn ). ϕ(pn ) n

Since ψn ≤ ϕ(pn )−1 ϕ(zn )ϕn , and the function η(t) = −t log t is monotone for small t, this follows easily from the classical deﬁnition of entropy.

Recall that a C∗ -algebra is of type I if its second dual is a type I von Neumann algebra. It follows from Theorem 3.3.1 that inner automorphisms of type I C∗ -algebras have zero entropy. It turns out this is a characteristic property of the type. Theorem 3.3.2. Let A be a unital C∗ -algebra. Then A is of type I if and only if for every unitary u ∈ A and (Ad u)-invariant state ϕ on A we have hϕ (Ad u) = 0. Proof. One implication follows from Theorem 3.3.1. So assuming that A is not of type I we have to show that there exists an inner automorphism with nonzero entropy. By Glimm’s theorem, see e.g. [156, Corollary 6.7.4], there exists a unital C∗ -subalgebra B of A and an ideal I ⊂ B such that B/I is a UHF-algebra. Assume there exists a unitary v ∈ B/I such that hϕ (Ad v) > 0 for a state ϕ on B/I. Since the unitary group of an AF-algebra is connected, the unitary v lifts to a unitary u in B by [24, Proposition 3.4.5]. Let ψ be the state on B obtained by composing ϕ with the quotient map B → B/I. Then hψ (Ad u|B ) = hϕ (Ad v) by Theorem 3.2.2(ii). By Proposition 3.2.7 we can extend the state ψ to a state ϕ on A such that hϕ (Ad u) ≥ hψ (Ad u|B ) > 0. Thus it remains to show that UHF-algebras have inner automorphisms with positive entropy. This will be done in the next lemma.

Lemma 3.3.3. Let R be the hyperﬁnite II1 -factor, u ∈ R a unitary with nonatomic spectral measure, A ⊂ R a strongly operator dense UHF-subalgebra. Then there exists a unitary v ∈ R such that vuv ∗ ∈ A. In particular, there exist inner automorphisms of A having positive entropy with respect to the tracial state. Proof. Let τ be the tracial state on R. We are going to use the fact that if p and q are projections in R such that p − q2 = p − qτ is small and τ (p) = τ (q), then there exists a partial isometry w such that w∗ w = p, ww∗ = q and w −p2 is small, see e.g. [214, Lemma XIV.2.1]. Here is a sketch of the proof. Let qp = w1 |qp| be the polar decomposition. Then p − w1 2 is small. Let w2 be any partial isometry such that w2∗ w2 = p − w1∗ w1 and w2 w2∗ = q − w1 w1∗ . Put w = w1 + w2 . Let F ⊂ [0, 1] be the set of values of τ on projections in A. Since A is UHF, if s, t ∈ F and s+t ≤ 1, then s+t ∈ F. Choose an increasing sequence {ξn }∞ n=1 of ﬁnite partitions of T into intervals such that if Y ∈ ξn then the length of Y is at most 2−n , and if pY is the spectral projection of u corresponding to Y then τ (pY ) ∈ F.

3.4 Notes

57

Fix ε > 0. The projections pY , Y ∈ ξ1 , can be approximated arbitrarily close in the L2 -norm by mutually orthogonal projections qY , Y ∈ ξ1 , in A such that τ (pY ) = τ (qY ). It follows that there exists a unitary w1 ∈ R such that w1 − 12 < ε/2 and qY = w1∗ pY w1 ∈ A for Y ∈ ξ1 . If Z ∈ ξ2 and Z ⊂ Y for some Y ∈ ξ1 , we can approximate w1 pZ w1∗ by projections in qY AqY ⊂ A and repeat the above argument. By continuing this process we obtain a sequence of unitaries wn ∈ R, n ∈ N, such that wn − 12 < 2−n ε and vn pY vn∗ ∈ A for Y ∈ ξn , where vn = wn . . . w1 , and if Z ⊂ Y for some Z ∈ ξn+1 and Y ∈ ξn then vn+1 pZ vn+1 ≤ vn pY vn∗ . The sequence {vn }∞ n=1 converges in the strong operator topology to a unitary v ∈ R. Moreover, 1 − v2 < ε. We claim that vuv ∗ ∈ A. To show this choose a point tY ∈ Y ⊂ T for each Y ∈ ξn , n ∈ N, and set un = Y ∈ξn tY pY . Then u − un ≤ 2−n . Since ∗ |tZ − tY | ≤ 2−m and vn pZ vn∗ ≤ vm pY vm for Z ⊂ Y if Z ∈ ξn and Y ∈ ξm , we also have ∗ vn un vn∗ − vm um vm ≤ 2−m , n ≥ m, ∗ whence vn uvn∗ − vm um vm ≤ 2−n + 2−m , so that ∗ ∗ ≤ lim inf vn un vn∗ − vm um vm ≤ 2−m . vuv ∗ − vm um vm n→∞

∗ vm um vm

∈ A, we see that vuv ∗ ∈ A. Since Since hτ |A (Ad vuv ∗ |A ) = hτ (Ad vuv ∗ ) = hτ (Ad u), to prove the second part of the lemma it suﬃces to show that there exists a unitary u ∈ R with nonatomic spectral measure such that hτ (Ad u) > 0. Take any ergodic dynamical system (X, µ, T ) with positive entropy. Then L∞ (X, µ) Z is the hyperﬁnite II1 -factor, see [214, Chapters XIII and XIV]. For the canonical unitary u in the crossed product we have hτ (Ad u) ≥ hτ (Ad u|L∞ (X,µ) ) = hµ (T ) > 0 (the ﬁrst inequality is actually equality, as will be shown in Chap. 8). This completes the proof of the lemma and thus also of Theorem 3.3.2.

It is worth mentioning that there are shorter but less elementary ways of ﬁnishing the proof of Theorem 3.3.2. For example, one could use a result of Voiculescu [222] stating that the crossed product of Mat2 (C)⊗Z by the shift automorphism embeds into the UHF-algebra ⊗n∈N Matn (C).

3.4 Notes Dynamical entropy for automorphisms preserving a tracial state was introduced by Connes and Størmer [51]. Later Connes [49] emphasized the role of relative entropy and showed how the deﬁnition can be extended to arbitrary W∗ -dynamical system. The formula for Hϕ (B) was also independently obtained by Narnhofer and Thirring [129]. The deﬁnition of dynamical entropy was extended to C∗ -dynamical systems by Connes, Narnhofer and Thirring [50], who introduced the concept of an abelian model and gave a detailed treatment of the foundations of the theory. The material in

58

3 Dynamical Entropy

Sects. 3.1 and 3.2 is almost entirely based on their paper. A few properties of entropy observed later include the superadditivity inequality in Proposition 3.1.7(iv) [212], with equality in the case when one of the factors is abelian [141], the apparently new inequality in Proposition 3.1.7(v), the formula for the entropy of a ﬂow [130], Theorem 3.2.5(iii) [141] and Theorem 3.2.5(iv) [79], and Proposition 3.2.7, which is essentially from [34]. The concept of a coupling was introduced by Sauvageot and Thouvenot [188]. It will play a central role in Chap. 5, where we shall give an equivalent deﬁnition of dynamical entropy. The notion of an approximating net is borrowed from the Ohya and Petz book [147]. It allows us to simultaneously discuss nuclear C∗ -algebras and injective von Neumann algebras. Theorem 3.3.1 is due to the authors [141], and Theorem 3.3.2 to Brown [34]. As we already mentioned, the problem of ﬁnding optimal decompositions of a state is quite complicated, see [18], [19] for some results in the ﬁnite dimensional case. Though mutual and dynamical entropies share most properties of their abelian counterparts, there are some properties which are still missing. E.g. if {γi }i∈I is a ﬁnite collection of channels, and I = J ∪ K, it is to the best of our knowledge unknown whether the strong subadditivity inequality Hϕ ({γi }i∈I ) + Hϕ ({γi }i∈J∩K ) ≤ Hϕ ({γi }i∈J ) + Hϕ ({γi }i∈K ) holds. It is also unknown whether the entropy hϕ (α) is an aﬃne function of the state, though this is stated in [50]. There is one aspect of the theory which may be viewed as a drawback, namely, that it is in a sense too abelian, the entropy being deﬁned by abelian models. We shall see later that the entropy tends to be the smaller the less abelian the system is. Similarly to abelian systems the notion of entropy can be extended to actions of arbitrary discrete amenable groups. Namely, let α: G → Aut(A) be an action of a countable amenable group G on A by ϕ-preserving automorphisms. Then for any channel γ: B → A the function m deﬁned on ﬁnite subsets of G by m(F ) = Hϕ ({αg ◦ γ}g∈F ) for F = ∅, m(∅) = 0, is positive, nondecreasing, left-invariant and subadditive, i.e., m(F1 ∪ F2 ) ≤ m(F1 ) + m(F2 ). By [120, Theorem 6.1] it follows that if {Fn }n is a sequence of ﬁnite sets such that |Fn g ∩ Fn | → 1 as n → ∞, |Fn | then the sequence {m(Fn )/|Fn |}n converges, and its limit does not depend on the choice of {Fn }n . Denote this limit by hϕ (γ; αG ). Then we can deﬁne

3.4 Notes

59

hϕ (αG ) by taking supremum over all channels γ into A. Note that if we knew in addition that m is strongly subadditive, by [124, Proposition 3.1.9 and Remark 3.1.7] we would have hϕ (γ; αG ) =

inf

F ⊂G,F =∅

Hϕ ({αg ◦ γ}g∈F ) . |F |

So far the entropy for actions of groups on noncommutative algebras has been considered for very few models, such as shifts on UHF-algebras [92], [102] and Bogoliubov actions [155], [21], [149], [78]. Although the Connes-Størmer-Narnhofer-Thirring deﬁnition of entropy is arguably the most successful extension of the classical theory, it is not the only one available. The ﬁrst deﬁnition of entropy for noncommutative systems is due to Emch [59]. It is as follows. Let (M, ϕ, α) be a W∗ -dynamical system with a faithful normal state ϕ. Then the entropy is deﬁned (compare with (1.3)) as the supremum of the quantities lim ϕ(η(En (pi ))), n→∞

i∈I

where the pi ’s are projections in the centralizer Mϕ of ϕ such that i pi = 1, and En is the ϕ-preserving conditional expectation onto the von Neumann algebra generated by α−k (pi ), i ∈ I, k = 1, . . . , n. Very little is known about this entropy. Note also that it depends only on the restriction of the automorphism to the centralizer of the state, which can be quite small. We shall see later examples of systems with trivial centralizer and inﬁnite dynamical entropy. A diﬀerent approach was suggested by Lindblad [119] and developed by Alicki and Fannes [4]. Let (A, ϕ, α) be a C∗ -dynamical system, and A0 ⊂ A an α-invariant unital ∗-subalgebra. By an operational partition of unity in A0 one means a ﬁnite subset X = {xk }nk=1 of A0 such that k x∗k xk = 1. Such a partition deﬁnes a completely positive map θX : Matn (C) → A by θX (eij ) = x∗i xj . Denote by H[ϕ, X] the entropy of the state ϕ ◦ θX . Note that the density matrix of ϕ◦θX is (ϕ(x∗j xi ))i,j . If Y = {yl }m l=1 is another operational partition of unity, we can deﬁne a new partition X ◦ Y consisting of the elements xk yl . Then put h[ϕ, α, X] = lim sup n→∞

1 H[ϕ, αn−1 (X) ◦ αn−2 (X) ◦ . . . ◦ X]. n

Finally deﬁne the entropy h[ϕ, α, A0 ] of the C∗ -dynamical system (A, ϕ, α) with respect to A0 as the supremum of h[ϕ, α, X] over all operational partitions of unity in A0 . Computability of h[ϕ, α, A0 ] depends very much on the choice of A0 , and in this respect most choices of A0 give too large subalgebras. As a result one can hardly use this entropy as an invariant of a dynamical system. On the other hand, in most examples there is a natural choice of A0 .

60

3 Dynamical Entropy

E.g. for models of quantum statistical mechanics this is the algebra of local observables. We refer the reader to the Alicki and Fannes book [5] for interesting examples. There is also a completely diﬀerent class of entropies based on the idea of approximation of ﬁnite sets. We shall discuss this topic in Chap. 6.

4 Maximality of Entropy and Commutativity

Since our deﬁnition of entropy was based on abelian models, one may expect that the entropy is the larger the more amount of commutativity is present in the system. In this chapter we shall prove several results in this direction. We shall mainly consider tracial states, where proofs are simpler and available results are stronger.

4.1 Maximal Entropy of Subalgebras If N1 , . . . , Nm are ﬁnite dimensional subalgebras of a C∗ -algebra then in general the C∗ -algebra N = ∨k Nk they generate is inﬁnite dimensional. But if N is ﬁnite dimensional, a natural question is, under which conditions on the relative positions of the Nk ’s the entropy of N coincides with the mutual entropy of N1 , . . . , Nm . It follows from Proposition 3.1.6 that if τ is a tracial state and N1 , . . . , Nm contain pairwise commuting abelian subalgebras A1 , . . . , An , respectively, such that ∨k Ak is maximal abelian in N then Hτ (N1 , . . . , Nm ) = Hτ (N ). In the present section we show the converse to this result. Theorem 4.1.1. Let N be a ﬁnite dimensional C∗ -algebra with a faithful tracial state τ , N1 , . . . , Nm ⊂ N . Suppose Hτ (N1 , . . . , Nm ) = Hτ (N ). Then there exist mutually commuting abelian subalgebras Ak ⊂ Nk such that ∨k Ak is maximal abelian in N . The key point in the proof of this theorem is the following reﬁnement of the subadditivity inequality from Corollary 2.3.9. Proposition 4.1.2. Let 1 = i1 ,...,im xi1 ...im be a ﬁnite partition of unity in a C∗ -algebra with a tracial state τ . Then

62

4 Maximality of Entropy and Commutativity

k

(k)

τ (η(xik )) −

ik

τ (η(xi1 ...im )) ≥

i1 ,...,im

1 (k) (l) [(xik )1/2 , (xil )1/2 ]22 , max 2 k=l i ,i k

l

where x2 = τ (x∗ x)1/2 . We shall postpone the proof of this inequality. Note that if the elements of the partition lie in a ﬁnite dimensional C∗ -algebra N , then it can be written as (k) (k) η(τ (xik )) − Hτ (N ; {xik }) k

ik

⎛

⎞

−⎝

η(τ (xi1 ...im )) − Hτ (N ; {xi1 ...im })⎠

i1 ,...,im

≥

1 (k) (l) [(xik )1/2 , (xil )1/2 ]22 . max 2 k=l i ,i k

(4.1)

l

Proof of Theorem 4.1.1. Let X = S(N ) be the state space of N , {ξ (n) }n a sequence of ﬁnite Borel partitions of X with diameter tending to zero as n → ∞. By Proposition 3.1.10 we can ﬁnd a sequence {λ(n) }n of states on N ⊗ C(X m ) such that λ(n) |N = τ and (n)

(i) the coupling λk = λ(n) ◦ (id ⊗ pr∗k ) is canonical, where prk : X m → X is the projection onto the k-th factor, 1 ≤ k ≤ m; (n) (n) ), . . . , pr−1 )) → Hτ (N ) as n → ∞. (ii) Hλ(n) (N1 , . . . , Nm ; pr−1 m (ξ 1 (ξ Choosing a subsequence, we may assume that {λ(n) }n converges to a state λ. Let µ(n) (resp. µ) be the measure on X m deﬁned by λ(n) |C(X m ) (resp. (n) (n) by λ|C(X m ) ), µk = (prk )∗ (µ(n) ), µk = (prk )∗ (µ), and λk = λ(n) ◦ (id ⊗ pr∗k ). By subadditivity of the function ξ → Hµ(n) (ξ) − Hλ(n) (N ; ξ) = S(λ(n) (· ⊗ 1Z ), τ ), Z∈ξ (n) ), we have see Corollary 2.3.8, applied to ξ = ∨k pr−1 k (ξ (n) (n) )) − Hλ(n) (N ; ∨k pr−1 )) Hµ(n) (∨k pr−1 k (ξ k (ξ

≤

k

(Hµ(n) (ξ (n) ) − Hλ(n) (N ; ξ (n) )). k

k

Using (3.12), monotonicity of relative entropy, Theorem 2.1.2(v), and the above inequality we then get (n) (n) ), . . . , pr−1 )) Hλ(n) (N1 , . . . , Nm ; pr−1 m (ξ 1 (ξ

4.1 Maximal Entropy of Subalgebras (n) = Hµ(n) (∨k pr−1 )) − k (ξ

Hµ(n) (ξ (n) ) + k

k

(n) )) − ≤ Hµ(n) (∨k pr−1 k (ξ

k

Hµ(n) (ξ (n) ) + k

k

k

63

Hλ(n) (Nk ; ξ (n) ) k

Hλ(n) (N ; ξ (n) ) k

(n) ≤ Hλ(n) (N ; ∨k pr−1 )) k (ξ ≤ Hτ (N ).

It follows that the above inequalities become close to equalities as n → ∞. In other words, Hλ(n) (N ; ξ (n) ) − Hλ(n) (Nk ; ξ (n) ) → 0, 1 ≤ k ≤ m, k

and

(4.2)

k

(Hµ(n) (ξ (n) ) − Hλ(n) (N ; ξ (n) )) k

k

k

(n) (n) −(Hµ(n) (∨k pr−1 )) − Hλ(n) (N ; ∨k pr−1 )) → 0. k (ξ k (ξ

(4.3)

sa Let aϕ ∈ N+ be the density operator of a state ϕ on N with respect to τ , (n) so that ϕ = τ (· aϕ ). Since the coupling λk is canonical, the decomposition (n) (n) of τ deﬁned by λk and ξ (n) consists of the functionals Z ϕ dµk (ϕ), Z ∈ ξ (n) . (n) In other words, the partition of unity in N deﬁned by λk and ξ (n) is (n) aϕ dµk (ϕ). 1= Z∈ξ (n)

Z

Hence, see (3.1), Hλ(n) (Nk ; ξ (n) ) is equal to k

−

(n) µk (Z)τ

(n) (n) −1 η ENk µk (Z) aϕ dµk (ϕ) . Z

Z∈ξ (n)

Since aϕ depends continuously on ϕ, and the diameters of the atoms of ξ (n) (n) tend to zero, this expression becomes close to − X τ (η(ENk (aϕ )))dµk (ϕ) as n → ∞, so that Hλ(n) (Nk ; ξ (n) ) → − τ (η(ENk (aϕ )))dµk (ϕ). (4.4) k

X

Similarly, Hλ(n) (N ; ξ (n) ) → − k

τ (η(aϕ ))dµk (ϕ). X

Thus (4.2) implies that τ (η(ENk (aϕ )))dµk (ϕ) = τ (η(aϕ ))dµk (ϕ). X

X

64

4 Maximality of Entropy and Commutativity

Since η(ENk (aϕ )) ≥ ENk (η(aϕ )) by strict operator concavity of η, see B.2 and B.4 , we conclude that η(ENk (aϕ )) = ENk (η(aϕ )) for ϕ ∈ supp µk . By B.5 this implies that aϕ ∈ Nk for ϕ ∈ supp µk . Denote by Bk the C∗ -subalgebra of Nk generated by aϕ for ϕ ∈ supp µk . (k) (n) Then (4.3) and inequality (4.1) applied to {xik } = { Z aϕ dµk (ϕ)} imply that for k = l 1/2 1/2 2 (n) (n) aϕ dµk (ϕ) , aψ dµl (ψ) → 0. Z W (n) Z,W ∈ξ

2

(n) (n) Since µk (Z)−1 Z aϕ dµk (ϕ) becomes arbitrarily close to aψ for ψ ∈ Z, as n → ∞ the above expression gets close to 1/2 2 (n) (n) [a1/2 ϕ , aψ ]2 dµk (ϕ)dµl (ψ), X×X

so that in the limit we obtain 1/2 2 [a1/2 ϕ , aψ ]2 dµk (ϕ)dµl (ψ) = 0. X×X

Thus aϕ aψ = aψ aϕ for ϕ ∈ supp µk and ψ ∈ supp µl . Therefore the algebras B1 , . . . , Bm mutually commute. Analogously to (4.4) we have Hλ(n) (Bk ; ξ (n) ) → − τ (η(EBk (aϕ )))dµk (ϕ). k

X

But aϕ = ENk (aϕ ) = EBk (aϕ ) for ϕ ∈ supp µk , so that Hλ(n) (Nk ; ξ (n) ) − Hλ(n) (Bk ; ξ (n) ) → 0. k

k

(n) (n) It follows that Hλ(n) (B1 , . . . , Bm ; pr−1 ), . . . , pr−1 )) becomes close to m (ξ 1 (ξ (n) −1 (n) Hλ(n) (N1 , . . . , Nm ; pr−1 (ξ ), . . . , pr (ξ )) as n → ∞. Since the latter exm 1

pression converges to Hτ (N ), it follows that Hτ (B1 , . . . , Bm ) = Hτ (N ). Let Ak be a maximal abelian subalgebra of Bk . Since B1 , . . . , Bm mutually commute, by Proposition 3.1.6 Hτ (B1 , . . . , Bm ) = S(τ |∨k Ak ), hence Hτ (N ) = S(τ |∨k Ak ). Let A be a maximal abelian subalgebra of N containing ∨k Ak . Then S(τ ) = S(τ |A ) by Theorem 2.2.2(vii). Thus S(τ |∨k Ak ) =

4.1 Maximal Entropy of Subalgebras

65

S(τ |A ). Since the atoms of ∨k Ak are sums of the atoms of A, and η(x + y) < η(x)+η(y) for any numbers x, y > 0, this is possible only when ∨k Ak = A.

In order to prove Proposition 4.1.2 we shall ﬁrst strengthen the GoldenThompson inequality Tr(ea eb ) ≥ Tr(ea+b ). Lemma 4.1.3. For self-adjoint elements a and b in a C∗ -algebra with a tracial state τ we have 1 τ (ea eb ) − τ (ea+b ) ≥ [ea/2 , eb/2 ]22 . 2 Proof. By the Trotter-Lie formula, see e.g. [177, Theorem VIII.29], (ea/2 eb/2 )2 → ea+b in norm as n → ∞. n

n

n

Note that since |ea/2 eb/2 |2 = eb/2 ea/2 n

n

n

|ea/2 eb/2 |2 = (eb/2 ea/2 n

n

n

n

n−1

eb/2 )2 n

n−1

n−1

n

eb/2 , we have n

= eb/2 (ea/2

n−1

eb/2

n−1

)2

n−1

e−b/2 . n

Using the generalized H¨older inequality x1 . . . xp 1 ≤ x1 p . . . xp p , see B.6, we then get τ ((ea/2 eb/2 )2 ) ≤ (ea/2 eb/2 )2 1 n n n ≤ ea/2 eb/2 22n n n n = τ (|ea/2 eb/2 |2 ) n

n

n

n

= τ ((ea/2

n

n−1

n

eb/2

n−1

)2

n−1

).

Thus τ ((ea/2 eb/2 )2 ) τ (ea+b ). In particular, τ (ea+b ) ≤ τ ((ea/2 eb/2 )2 ). It remains to note that n

n

n

τ (ea eb ) − τ ((ea/2 eb/2 )2 ) =

1 a/2 b/2 2 [e , e ]2 . 2

Proof of Proposition 4.1.2. Consider ﬁrst the case m = 2. So we have a partition of unity 1 = i1 ∈I1 ,i2 ∈I2 xi1 i2 . Replacing xi1 i2 by (1 − ε)xi1 i2 +

ε 1, |I1 ||I2 |

we may assume that the xi1 i2 ’s are invertible. Then by Corollary 2.3.10 and Lemma 4.1.3 we have (1) (2) τ (η(xi1 )) + τ (η(xi2 )) − τ (η(xi1 i2 )) i1

i2

=

i1 ,i2 (1)

τ (xi1 i2 (log xi1 i2 − log elog xi1

i1 ,i2

≥

i1 ,i2

(1)

(τ (xi1 i2 ) − τ (elog xi1

(2) 2

+log xi

(2) 2

+log xi

))

))

66

4 Maximality of Entropy and Commutativity

= 1− =

(1)

τ (elog xi1

(2) 2

+log xi

)

i1 ,i2 (1)

(1) (2)

(τ (xi1 xi2 ) − τ (elog xi1

(2) 2

+log xi

))

i1 ,i2

≥

1 (1) (2) [(xi1 )1/2 , (xi2 )1/2 ]22 . 2 i ,i 1

2

Consider now the case m ≥ 3. It is clear that it suﬃces to prove the inequality for k = 1 and l = 2. Put xi1 ...im and zi3 ...im = xi1 ...im , y i1 i2 = i3 ,...,im (1)

(1)

(2)

i1 ,i2

(2)

(n)

(n)

so that yi1 = xi1 , yi2 = xi2 and zin = xin for n = 3, . . . , m. Then τ (η(xi1 ...im )) ≤ τ (η(yi1 i2 )) + τ (η(zi3 ...im )) i1 ,...,im

i1 ,i2

≤

τ (η(yi1 i2 )) +

i3 ,...,im m

(n)

τ (η(xin )),

n=3 in

i1 ,i2

whence m n=1 in

(n)

τ (η(xin ))−

τ (η(xi1 ...im )) ≥

i1 ,...,im

2

(n)

τ (η(yin ))−

n=1 in

Therefore the result follows from the case m = 2.

τ (η(yi1 i2 )).

i1 ,i2

4.2 Independent Algebras Let us now consider the problem of maximizing Hτ (N1 , . . . , Nm ) without assuming that N1 , . . . , Nm generate a ﬁnite dimensional algebra. By subadditivity we have Hτ (N1 , . . . , Nm ) ≤ Hτ (N1 ) + . . . + Hτ (Nm ). In the commutative case, by Proposition 1.1.9(iii), the equality holds exactly when N1 , . . . , Nm are mutually independent. For a C∗ -algebra A and a faithful tracial state τ on A let us say that C∗ -subalgebras A1 , . . . , Am of A are τ independent if they mutually commute and τ (a1 . . . am ) = τ (a1 ) . . . τ (am ) for ai ∈ Ai . Let N1 , . . . , Nm be ﬁnite dimensional C∗ -subalgebras of A. Assume there exist maximal abelian subalgebras Ai ⊂ Ni such that A1 , . . . , Am are τ -independent. Since

4.2 Independent Algebras

Hτ (A1 , . . . , Am ) ≤ Hτ (N1 , . . . , Nm ) ≤

Hτ (Nk ) =

k

and Hτ (A1 , . . . , Am ) = Hτ (∨k Ak ) =

67

Hτ (Ak )

k

Hτ (Ak ),

k

we conclude that Hτ (N1 , . . . , Nm ) = Hτ (N1 ) + . . . + Hτ (Nm ). One may expect that conversely, if the above equality holds, then there exist τ -independent maximal abelian subalgebras Ak ⊂ Nk . This is indeed the case by Theorem 4.1.1 and Proposition 1.1.9(iii) if we in addition assume that N1 , . . . , Nm ⊂ N and Hτ (N1 , . . . , Nm ) = Hτ (N ). We do not know whether this is true in general. We have, however, the following weaker result. Theorem 4.2.1. Let A1 , . . . , Am be ﬁnite dimensional abelian subalgebras of a von Neumann algebra N with a faithful normal trace τ . Suppose Hτ (A1 , . . . , Am ) = Hτ (A1 ) + . . . + Hτ (Am ). Then the algebras A1 , . . . , Am are τ -independent. Note that as soon as the mutual commutativity is established, the independence becomes a purely classical statement, Proposition 1.1.9(iii). We shall use that for abelian algebras any state has a unique pure state decomposition, and any optimal model arises from this decomposition. Lemma 4.2.2. For any d ∈ N and ε > 0, there exists δ > 0 such that if A is an abelian C∗ -algebra of dimension d with pure states ϕ1 , . . . , ϕd , and τ = i∈I λi ψi is a ﬁnite convex combination of states with

λi S(ψi , τ ) > S(τ ) − δ,

i

then there exists a partition I = I(0) . . . I(d) of the index set I such that λi < ε, and ψi − ϕk < ε for i ∈ I(k) and k = 1, . . . , d. i∈I(0)

Proof. Since the von Neumann entropy is a continuous function which is zero only on pure states, we can chose δ0 depending on d and ε such that a state ψ on A is ε-close to a pure state as soon as S(ψ) < δ0 . If λi S(ψi ) = S(τ ) − λi S(ψi , τ ) < δ, i

i

then putting I(0) = {i ∈ I | S(ψi ) ≥ δ0 }, we get

68

4 Maximality of Entropy and Commutativity

λi

0. The system is called uniformly mixing if for any measurable A ⊂ X and any ﬁnite measurable partition ζ we have lim sup{|µ(A ∩ B) − µ(A)µ(B)| : B is (∨∞ k=n ζ)-measurable} = 0.

n→∞

One can then check that H(ξ| ∨∞ k=n ζ) → H(ξ) for any ﬁnite measurable partition ξ, which in view of (1.3) implies h(ξ; T n ) → H(ξ). The latter property is in fact equivalent to uniform mixing. It has a direct generalization to noncommutative systems. Deﬁnition 4.3.1. A W∗ -dynamical system (M, ϕ, α) with faithful state ϕ and M = C is called an entropic K-system if for any channel γ: A → M we have hϕ (γ; αn ) → Hϕ (γ) as n → ∞. Note that since hϕ (γ; αn ) ≤ nhϕ (γ; α), Lemma 3.1.4 implies that for any γ with Im γ = C1 we have hϕ (γ; α) > 0. In particular, hϕ (α) > 0. The simplest examples of K-systems in the classical case are Bernoulli shifts. If ξ is the standard generating partition of a Bernoulli shift then the partition ζ = ∨0k=−∞ T k ξ has the following properties: ζ ≺ T ζ, the algebra ∪n∈Z L∞ (X/T n ζ) is weakly operator dense in L∞ (X), and ∩n∈Z L∞ (X/T n ζ) = C1. It turns out that existence of a partition ζ with such properties is yet another equivalent form of the K-property. This is not the case for noncommutative systems. Nevertheless we have the following result. Theorem 4.3.2. Let (M, ϕ, α) be a W∗ -dynamical system with faithful ϕ and M = C. Suppose there exists a von Neumann subalgebra M0 of M such that (i) M0 ⊂ α(M0 ); (ii) ∪n∈N (α−n (M0 ) ∩ αn (M0 )) is weakly operator dense in M ; (iii) ∩n∈Z αn (M0 ) = C1. Then (M, ϕ, α) is an entropic K-system. For the proof we need two lemmas. Lemma 4.3.3. Let M be a von Neumann algebra, {Mn }∞ n=1 a decreasing sequence of von Neumann subalgebras, ϕ a normal faithful state on M . Assume ∩n Mn = C1. Then for any a ∈ M we have |ϕ(ab) − ϕ(a)ϕ(b)| → 0 as n → ∞, bϕ + b∗ ϕ b∈Mn \{0} sup

where xϕ = ϕ(x∗ x)1/2 .

70

4 Maximality of Entropy and Commutativity

Proof. We may assume that M ⊂ B(H), and ϕ is the state deﬁned by a cyclic and separating vector ξ ∈ H, ϕ(x) = (xξ, ξ). Assume the conclusion of the lemma is false. Then there exist ε > 0, a ∈ M and a sequence {bn }n such that bn ∈ Mn , bn ϕ + b∗n ϕ = 1 and |ϕ(abn ) − ϕ(a)ϕ(bn )| ≥ ε. Replacing {Mn }n by a subsequence we may assume that the bounded sequences {bn ξ}n and {b∗n ξ}n are weakly convergent. Denote by ζ and ϑ their weak limits. Then |(aζ, ξ) − (aξ, ξ)(ζ, ξ)| ≥ ε. (4.5) On the other hand, if x ∈ Mm then for n ≥ m we have (xbn ξ, ξ) = (xξ, b∗n ξ), whence (xζ, ξ) = (xξ, ϑ). Since ∩m Mm = C1, the algebra ∪m Mm is weakly operator dense in B(H). It follows that the above identity holds for all x ∈ B(H). Then, denoting the projection onto Cξ by p, for any x ∈ B(H) we get

(pζ, xξ) = (x∗ pζ, ξ) = (x∗ pξ, ϑ) = (x∗ ξ, ϑ) = (x∗ ζ, ξ) = (ζ, xξ). Hence pζ = ζ, so that ζ ∈ Cξ. But this contradicts (4.5).

Lemma 4.3.4. For any m ∈ N and ε > 0 there exists δ > 0 satisfying the ∗ ∗ following property. m If A is a C -algebra, B ⊂ A a C -subalgebra, ϕ a state on A, and 1 = i=1 xi a partition of unity in B ∩ A such that |ϕ(xi y) − ϕ(xi )ϕ(y)| ≤ δ||y|| for any y ∈ B, i = 1, . . . , m, then

η(ϕ(xi yj )) >

i,j

i

η(ϕ(xi )) +

η(ϕ(yj )) − ε

j

for any ﬁnite partition of unity 1 = j yj in B. m m Proof. Let δ > 0 be such that | i=1 η(λi ) − i=1 η(µi )| < ε/2 for probability distributions (λ1 , . . . , λm ) and (µ1 , . . . , µm ) as soon as m

|λi − µi | ≤ 2δε−1 m log m.

i=1

For every element y of the form y = j ±yj we have −1 ≤ y ≤ 1, and consequently |ϕ(xi y) − ϕ(xi )ϕ(y)| ≤ δ. Hence |ϕ(xi yj ) − ϕ(xi )ϕ(yj )| ≤ δ, j

and thus

4.3 Entropic K-systems

71

|ϕ(xi yj ) − ϕ(xi )ϕ(yj )| ≤ mδ.

i,j

Consider the set J consisting of the indices j such that |ϕ(xi yj ) − ϕ(xi )ϕ(yj )| ≤ 2ϕ(yj )ε−1 mδ log m. i

Since mδ ≥

j ∈J /

we have

j ∈J /

j ∈J /

|ϕ(xi yj ) − ϕ(xi )ϕ(yj )| > 2ε−1 mδ log m

ϕ(yj ),

j ∈J /

i

ϕ(yj ) < ε/2 log m. Hence

ϕ(x y ) ε i j − ϕ(yj ) η(ϕ(xi )) ≤ ϕ(yj ) log m < , η ϕ(yj ) 2 i i j ∈J /

since neither sum over i is larger than log m. On the other hand, by our choice of δ, for every j ∈ J we have ε ϕ(x y ) i j − η η(ϕ(xi )) < . 2 ϕ(y ) j i i It follows that j

ϕ(xi yj ) − ϕ(yj ) η η(ϕ(xi )) > −ε, ϕ(yj ) i i

which is exactly what we need.

Proof of Theorem m 4.3.2. Let γ: A → M be a channel. There exists a partition of unity 1 = i=1 xi in M such that Hϕ (γ; {xi }) is arbitrarily close to Hϕ (γ). Since ∪N (α−N (M0 ) ∩ αN (M0 )) is strongly operator dense in M , we may assume that the xi ’s belong to α−N (M0 ) ∩ αN (M0 ) for some N ∈ N (note that one can think of a partition of unity as a unital completely positive map Cm → M , so the claim follows from A.10). Then αn (xi ) commutes with xj for n ≥ 2N . For n ≥ 2N and k ∈ N we want to estimate Hϕ (γ, αn ◦ γ, . . . , α(k−1)n ◦ γ; {xi1 αn (xi2 ) . . . α(k−1)n (xik )}) = η(ϕ(xi1 αn (xi2 ) . . . α(k−1)n (xik ))) − k η(ϕ(xi )) + kHϕ (γ; {xi }), i1 ,...,ik

i

see (3.3). To prove the theorem it suﬃces to show that for any ε > 0 the above expression is (kε)-close to kHϕ (γ; {xi }) when n is suﬃciently large and k is arbitrary.

72

4 Maximality of Entropy and Commutativity

Choose δ according to Lemma 4.3.4. Since the algebras αl (M0 ) have trivial intersection, by Lemma 4.3.3 there exists n0 ≤ −N such that |ϕ(xi y) − ϕ(xi )ϕ(y)| ≤ δ||y|| for y ∈ αn0 (M0 ), i = 1, . . . , n. Then by Lemma 4.3.4 for any l ∈ Z and any partition of unity 1 = j yj in αn0 +ln (M0 ) we get η(ϕ(yj αln (xi ))) > η(ϕ(xi )) + η(ϕ(yj )) − ε. i,j

i

j

Since xi1 αn (xi2 ) . . . α(l−1)n (xil ) ∈ αN −n+ln (M0 ) ⊂ αn0 +ln (M0 ) if n ≥ N −n0 , applying recursively the above inequality we obtain η(ϕ(xi )) − (k − 1)ε, η(ϕ(xi1 αn (xi2 ) . . . α(k−1)n (xik ))) > k i1 ,...,ik

i

which completes the proof of the theorem.

The simplest examples of entropic K-systems are noncommutative Bernoulli shifts. We shall see more interesting examples in Chaps. 11 and 13. It is worth noticing that the condition (ii) in the above theorem can not be weakened to the requirement that ∪n αn (M0 ) is weakly operator dense in M . In fact, as we shall see in Chap. 12, we can then even get a system with zero entropy. Our next goal is to show that an entropic K-system is necessarily asymptotically abelian, at least when the state is tracial. Theorem 4.3.5. Let (N, τ, α) be an entropic K-system, where τ is a faithful normal trace. Then the system is strongly asymptotically abelian and strongly mixing. In other words, for any x, y ∈ N we have [αn (x), y] → 0 in the strong operator topology and τ (αn (x)y) → τ (x)τ (y) as n → ∞. Proof of Theorem 4.3.5. Let A be a ﬁnite dimensional abelian subalgebra of N . Since Hτ (A, αn (A), . . . , α(2m−1)n (A)) ≤ mHτ (A, αn (A)), so that 2hτ (A; αn ) ≤ Hτ (A, αn (A)), it follows from our assumptions that Hτ (A, αn (A)) converges to 2Hτ (A). We thus need an approximate version of Theorem 4.2.1. A standard trick in such cases is to use ultraproducts. So let ω be a free ultraﬁlter on N. Consider the ultraproduct N ω . By deﬁnition this is the quotient of the algebra ∞ (N, N ) of bounded sequences in N by the ideal Iω consisting of sequences (xn )n such that xn 2 → 0 as n → ω. This is a von Neumann algebra with a faithful normal trace τω deﬁned by

4.3 Entropic K-systems

73

τω ((xn )n ) = lim τ (xn ). n→ω

We identify N with the subalgebra of N ω consisting of constant sequences. Consider the automorphism θ of N ω deﬁned by θ((xn )n ) = (αn (xn ))n . We claim that Hτω (A, θ(A)) = 2Hτ (A). Indeed, by Proposition 3.1.10 and the discussion following it, for any ε > 0 there exists a ﬁnite index set I such that for each n there exists a partition of unity 1 = i,j∈I x(n)ij in N such that Hτ (A, αn (A); {x(n)ij }) is ε-close to Hτ (A, αn (A)) (in fact, if we consider only suﬃciently large n, then by the proof of Theorem 4.2.1 we may even take I consisting of d + 1 elements, where d is the dimension of A). Then yij = (x(n)ij )n , i, j ∈ I, form a partition of unity in N ω such that Hτω (A, θ(A); {yij }) = lim Hτ (A, αn (A); {x(n)ij }) ≥ 2Hτ (A) − ε. n→ω

Thus we can apply Theorem 4.2.1 and conclude that A and θ(A) are τω independent. To conclude that N and θ(N ) are τω -independent we need the following lemma. Lemma 4.3.6. Let M be a von Neumann algebra with a faithful normal trace τ , N ⊂ M , and θ a τ -preserving automorphism of M . Assume that for any commuting projections p and q in N we have θ(p)q = qθ(p) and τ (θ(p)q) = τ (p)τ (q). Then the algebras N and θ(N ) are τ -independent. Proof. We shall ﬁrst check that τ (xθ(y)) = τ (x)τ (y) for x, y ∈ N . Since every self-adjoint operator is approximated by a linear combination of its spectral projections, we have θ(a)b = bθ(a) and τ (θ(a)b) = τ (a)τ (b) for any commuting self-adjoint elements a, b ∈ N . Let A ⊂ N be a maximal abelian von Neumann subalgebra of N containing a. Since θ(a) commutes with any element in N commuting with a, the same is true for EN (θ(a)). Hence EN (θ(a)) lies in A. On the other hand, τ (EN (θ(a))b) = τ (θ(a)b) = τ (a)τ (b). for every b ∈ A. It follows that EN (θ(a)) is a scalar, so that τ (xθ(a)) = τ (x)τ (a) for every x ∈ N .

74

4 Maximality of Entropy and Commutativity

To prove that N and θ(N ) commute, it is enough to check that θ(p) and q commute for any projections p and q in N . Since [p + q, θ(p + q)] = 0, [p, θ(p)] = 0 and [q, θ(q)] = 0, we get [p, θ(q)] = [θ(p), q]. Using this identity we compute τ (θ(p)qθ(p)q) = τ (θ(p)qθ(p)) + τ (θ(p)q[θ(p), q]) = τ (θ(p)q) + τ (θ(p)q[p, θ(q)]) = τ (θ(p)q) + τ (θ(p)qpθ(q)) − τ (θ(p)qθ(q)p) = τ (θ(p)q) + τ (θ(q)θ(p)qp) − τ (θ(p)θ(q)qp) = τ (p)τ (q) + τ (qp)τ (qp) − τ (pq)τ (qp) = τ (p)τ (q), whence [θ(p), q]22 = τ (qθ(p)q) − τ (θ(p)qθ(p)q) − τ (qθ(p)qθ(p)) + τ (θ(p)qθ(p)) = 0. Thus θ(p)q = qθ(p).

Returning to the proof of the theorem, we conclude that N and θ(N ) are τω -independent. That is, for any x, y ∈ N we get [αn (x), y]2 → 0 and τ (αn (x)y) → τ (x)τ (y) as n → ω. Since ω was an arbitrary free ultraﬁlter, we conclude that these convergences hold as n → ∞.

4.4 Notes Theorem 4.1.1 was proved by Haagerup and Størmer [83]. Theorem 4.2.1 is contained in the work of Benatti and Narnhofer [15]. Curiously enough it has apparently never been formulated explicitly. K-systems (called also Kolmogorov systems, or systems regular in the sense of Kolmogorov) were introduced by Kolmogorov [111], who called them quasiregular. The main deﬁnition is the last one we mentioned. So (X, µ, T ) is a K-system if there exists a measurable partition ζ such that ζ ≺ T ζ, ∨n T n ζ is the partition of (X, µ) into points, and the partition ∧n T n ζ corresponding to ∩n L∞ (X/T n ζ) is trivial. Equivalent forms of the K-property were obtained by Rohlin and Sinai [183]. For some time the only known examples of Ksystems were Bernoulli shifts. It was Ornstein who gave the ﬁrst example of a K-system which is not a Bernoulli shift [151]. Ornstein and Shields constructed an uncountable family of pairwise nonisomorphic K-systems with

4.4 Notes

75

the same entropy [152]. A simple example of a non-Bernoullian K-system was given by Kalikow [98]. Katok found examples of smooth non-Bernoullian Ksystems [100]. The ﬁrst appearance of K-systems in the quantum setting is the work of Emch [59], [60], [61]. One says that a W∗ -dynamical system (M, ϕ, α) is an algebraic K-system, the name suggested by Narnhofer and Thirring [132], if there exists a W∗ -subalgebra M0 ⊂ M such that M0 ⊂ α(M0 ), ∪n αn (M0 ) is dense in M , and ∩n αn (M0 ) = C1. As was already mentioned, in Chap. 12 we shall see examples of algebraic K-systems with zero entropy. The notion of entropic K-system was proposed by Narnhofer and Thirring [131]. A stronger result than that formulated in Theorem 4.3.2 was stated by Narnhofer [126]. Explicitly Theorem 4.3.2 appeared in the work of Golodets and Neshveyev [76]. Lemma 4.3.3 is due to Powers [170]. Theorem 4.3.5 was proved by Benatti and Narnhofer [15]. An attempt to capture the K-property by imposing certain hyperbolicity conditions was made by Emch, Narnhofer, Sewell and Thirring [62]. See [135], [136] for results in this direction.

5 Dynamical Abelian Models

In Chap. 3 we used abelian models to deﬁne mutual entropy of channels into A. It is then natural to consider dynamical abelian models, that is, abelian dynamical systems (X, µ, T ) together with equivariant maps A → L∞ (X, µ). In this chapter we shall prove that there always exist enough such models to compute the entropy.

5.1 Entropy via Stationary Couplings As in Chap. 3 we prefer to work with states on A ⊗ L∞ (X, µ) rather than with maps A → L∞ (X, µ). Then by a stationary coupling of (A, ϕ, α) with (X, µ, T ) we mean an (α ⊗ β)-invariant state λ on A ⊗ L∞ (X, µ), where β(f ) = f ◦ T −1 , such that λ|A = ϕ and λ|L∞ (X,µ) = µ. Given a stationary coupling λ, a channel γ into A and a ﬁnite measurable partition ξ of X, we have (see (3.12)) Hλ (γ, α ◦ γ, . . . , αn−1 ◦ γ; ξ, T ξ, . . . , T n−1 ξ) k = Hµ (∨n−1 k=0 T ξ) − nHµ (ξ) + nHλ (γ; ξ).

(5.1)

This motivates the following deﬁnition of entropy. Deﬁnition 5.1.1. The Sauvageot-Thouvenot entropy of a C∗ -dynamical system (A, ϕ, α) is hST ϕ (α) = sup{hµ (ξ; T ) − Hµ (ξ) + Hλ (A; ξ)}, where the supremum is taken over all stationary couplings λ of (A, ϕ, α) with abelian dynamical systems (X, µ, T ) and over all ﬁnite measurable partitions ξ of X. To make sense of the expression

78

5 Dynamical Abelian Models

Hλ (A; ξ) = Hµ (ξ) +

S(λ(· ⊗ 1Z ), ϕ)

Z∈ξ

we need relative entropy to be deﬁned for arbitrary C∗ -algebras. However, essentially all we have to know in order to deal with, say, nuclear C∗ -algebras is that if {γi }i is a ϕ-approximating net, then Hλ (γi ; ξ) Hλ (A; ξ) by Corollary 2.3.5. Let us ﬁrst observe that in some cases just existence of nontrivial couplings, that is, couplings diﬀerent from product-states, implies positivity of entropy. Proposition 5.1.2. If a C∗ -dynamical system (A, ϕ, α) has a nontrivial stationary coupling with a classical Bernoulli shift then hϕ (α) > 0. Proof. Let λ be a nontrivial stationary coupling of (A, ϕ, α) with a Bernoulli shift (X, µ, T ). Let ξ be the standard generating partition for (X, µ, T ). Since λ is assumed to be nontrivial, there exist a self-adjoint element a ∈ A and an element Y ∈ ∨nk=−n T k ξ such that λ(a ⊗ 1Y ) = ϕ(a)µ(Y ). Since hϕ (αm ) ≤ mhϕ (α) for any m ∈ N, it suﬃces to show that hϕ (α2n+1 ) > 0. Hence replacing α by α2n+1 , ξ by ∨nk=−n T k ξ and T by T 2n+1 we may assume n = 0. Since hµ (ξ; T ) = Hµ (ξ), by virtue of (5.1) and (3.10) for any channel γ into A we get hϕ (γ; α) ≥ Hλ (γ; ξ) = µ(Z)S(µ(Z)−1 λ(γ(·) ⊗ 1Z ), ϕ ◦ γ). Z∈ξ

If a ∈ Im γ, then by assumption µ(Y )−1 λ(γ(·) ⊗ 1Y ) = ϕ ◦ γ, and by Theorem 2.1.2(i) we conclude that hϕ (γ; α) > 0.

In the next section we shall prove a partial converse to the above proposition. The quantity Hµ (ξ) − Hλ (A; ξ), which is the entropy defect of the decomposition of ϕ deﬁned by λ and ξ, should be thought of as an analogue of the conditional entropy of L∞ (X/ξ) with respect to A in (A ⊗ L∞ (X, µ), λ). We shall now show that all the main properties of the classical conditional entropy are satisﬁed, which allows us to deal eﬃciently with our new deﬁnition of entropy. For γ: B → A denote by Hλ,γ (ξ) the entropy defect of the abelian model deﬁned by λ and ξ, so by (3.9) Hλ,γ (ξ) = Hµ (ξ) − Hλ (γ; ξ) = − S(λ(γ(·) ⊗ 1Z ), ϕ ◦ γ). (5.2) Z∈ξ

This is well-deﬁned for an arbitrary unital completely positive mapping γ: B → A, not only for ﬁnite dimensional B’s. More generally, for any measurable (not necessarily ﬁnite) partition ζ of Y , denote by iζ the embedding L∞ (Y /ζ) → L∞ (Y ), and put

5.1 Entropy via Stationary Couplings

Hλ,γ (ξ|ζ) = −

79

S(λ(γ(·) ⊗ iζ (·)1Z ), λ ◦ (γ ⊗ iζ )).

Z∈ξ

Equivalently, using the diagonal embedding Y → Y × Y , for any coupling λ of (A, ϕ) with (Y, µ) we can deﬁne a coupling of (A ⊗ L∞ (Y, µ), λ) with (Y, µ), which we again denote by λ, so that a ⊗ f ⊗ g → λ(a ⊗ f g). Then Hλ,γ (ξ|ζ) = Hλ,γ⊗iζ (ξ). If ζ is ﬁnite then Hλ,γ (ξ|ζ) = − S(λ(γ(·) ⊗ 1Z∩W ), λ(γ(·) ⊗ 1W )). Z∈ξ,W ∈ζ

Proposition 5.1.3. For any unital completely positive map γ: B → A we have: (i) Hλ (γ; ξ) is increasing in γ and ξ, that is, Hλ (γ ◦ ρ; ξ) ≤ Hλ (γ; ϑ) for any unital completely positive mapping ρ into B and any ﬁnite partition ϑ ξ; (ii) 0 ≤ Hλ (γ; ξ) ≤ Hµ (ξ); (iii) the function λ → Hλ (γ; ξ) is convex; (iv) Hλ,γ (ξ|ζ) is increasing in ξ and decreasing in γ and ζ; moreover, if {ζn }n is an increasing sequence of measurable partitions with ∨n ζn = ζ, then Hλ,γ (ξ|ζn ) Hλ,γ (ξ|ζ); (v) 0 ≤ Hλ,γ (ξ|ζ) ≤ Hµ (ξ|ζ); (vi) Hλ,γ (ξ ∨ ϑ|ζ) = Hλ,γ (ξ|ζ) + Hλ,γ (ϑ|ξ ∨ ζ); in particular, Hλ,γ (ξ ∨ ϑ) ≤ Hλ,γ (ξ) + Hλ,γ (ϑ). Proof. Essentially all these statements appeared in one or another form in Chap. 3, but we shall spell out the proofs for convenience. The fact that Hλ (γ; ξ) is increasing in γ follows from monotonicity of relative entropy, Theorem 2.3.1(vi). That it is increasing in ξ can be deduced either from convexity of relative entropy or from (v) and (vi) as follows. If ϑ ξ, then Hλ,γ (ϑ) = Hλ,γ (ξ) + Hλ,γ (ϑ|ξ) ≤ Hλ,γ (ξ) + Hµ (ϑ|ξ) = Hλ,γ (ξ) + Hµ (ϑ) − Hµ (ξ), so Hλ (γ; ξ) ≤ Hλ (γ; ϑ) by (5.2). The ﬁrst inequality in (ii) follows from S(ψ, ω) ≤ 0 for ψ ≤ ω, Theorem 2.3.1(iii). On the other hand, S(ψ, ω) ≥ 0 if ψ and ω are states, Theorem 2.3.1(ii). Hence −S(λ(γ(·)⊗1Z ), ϕ◦γ) = −S(µ(Z)−1 λ(γ(·)⊗1Z ), ϕ◦γ)+η(µ(Z)) ≤ η(µ(Z)), which gives the second inequality in (ii). Part (iii) follows from (3.10) and convexity of relative entropy, Theorem 2.3.1(v).

80

5 Dynamical Abelian Models

Since Hλ,γ (ξ|ζ) = Hλ,γ⊗iζ (ξ), it follows from (i) that Hλ,γ (ξ|ζ) = Hµ (ξ) − Hλ (γ ⊗ iζ ; ξ) is decreasing in γ and ζ. Since Hλ,γ (ϑ|ξ ∨ ζ) ≥ 0 by (ii), it will follow from (vi) that Hλ,γ (ξ|ζ) is increasing in ξ. The convergence statement in (iv) follows from Corollaries 2.3.6 and 2.3.4. We have already mentioned that positivity of Hλ,γ follows from (ii). Since Hλ,γ (ξ|ζ) is decreasing in γ, to prove the second inequality in (v) we may assume that B = C. Then Hλ,γ (ξ|ζ) = − S(µ(·1Z )|L∞ (X/ζ) , µ|L∞ (X/ζ) ). Z∈ξ

Since µ(·1Z ) = µ(·Eζ (1Z )) on L∞ (X/ζ), where Eζ is the µ-preserving conditional expectation onto L∞ (X/ζ), by Theorem 2.3.1(x) we have S(µ(·1Z )|L∞ (X/ζ) , µ|L∞ (X/ζ) ) = − η(Eζ (1Z ))dµ. X

Hence Hλ,γ (ξ|ζ) = Hµ (ξ|ζ). To prove (vi) ﬁrst notice that Hλ,γ (ϑ|ξ ∨ ζ) = Hλ,γ⊗iζ (ϑ|ξ), or in other words, Hλ,γ⊗iξ∨ζ (ϑ) = Hλ,γ⊗iξ ⊗iζ (ϑ). This follows from Corollary 2.3.4, since the homomorphism L∞ (X/ξ) ⊗ L∞ (X/ζ) → L∞ (X/ξ ∨ ζ), f ⊗ g → f g, has dense image in the strong operator topology. It follows that in proving (vi) we may replace γ by γ ⊗ iζ and thus assume that ζ is trivial. By Theorem 2.3.1(viii) we then have S(λ(γ(·) ⊗ 1Z∩W ), ϕ ◦ γ) Z∈ϑ

= S(λ(γ(·) ⊗ 1W ), ϕ ◦ γ) +

S(λ(γ(·) ⊗ 1Z∩W ), λ(γ(·) ⊗ 1W ))

Z∈ϑ

for W ∈ ξ. Hence Hλ,γ (ξ ∨ ϑ) = Hλ,γ (ξ) + Hλ,γ (ϑ|ξ). The following formula for

hST ϕ (α)

is reminiscent of (1.3).

Proposition 5.1.4. For any C∗ -dynamical system (A, ϕ, α) we have − − hST ϕ (α) = sup{Hµ (ξ|ξ ) − Hλ,A (ξ|ξ )},

where the supremum is taken over all stationary couplings λ of (A, ϕ, α) with abelian dynamical systems (Y, µ, T ) and over all ﬁnite measurable partitions ξ. Proof. As Hλ,A (ξ|ξ − ) ≤ Hλ,A (ξ) = Hµ (ξ) − Hλ (A; ξ) and Hµ (ξ|ξ − ) = hµ (ξ; T ), the entropy hST ϕ (α) is not bigger than the supremum.

5.1 Entropy via Stationary Couplings

81

To prove the opposite inequality, ﬁx a stationary coupling λ. First of all note that using Proposition 5.1.3(iv),(vi) the same argument as in the proof of (1.3) shows that Hλ,A (ξ|ξ − ) = lim

n→∞

1 k Hλ,A (∨n−1 k=0 T ξ), n

and hence Hµ (ξ|ξ − ) − Hλ,A (ξ|ξ − ) = lim

n→∞

1 k Hλ (A; ∨n−1 k=0 T ξ). n

(5.3)

Fix ε > 0 and choose n ∈ N such that 1 k − − Hλ (A; ∨n−1 k=0 T ξ) > Hµ (ξ|ξ ) − Hλ,A (ξ|ξ ) − ε n and

1 k Hµ (∨n−1 k=0 T ξ) < hµ (ξ; T ) + ε. n Consider the space Y = Y × (Z/nZ) with the transformation T (x, k) = (T x, k + 1) and the measure µ given by the product of µ with the counting measure. By taking the tensor product of λ with the counting measure, we get a stationary coupling λ of (A, ϕ, α) with (Y , µ , T ). Let ξ be the partition of Y consisting of the elements Y × {1, . . . , n − 1} and Z × {0}, where Z runs k through the elements of ∨n−1 k=0 T ξ. It is easy to check that 1 1 1 n−1 k +η 1 − and hµ (ξ ; T ) = hµ (ξ; T ), Hµ (ξ ) = Hµ (∨k=0 T ξ) + η n n n so that

1 1 −η 1− > −2ε n n if n is suﬃciently large. On the other hand, if ϕ = i ϕi is the decomposition k deﬁned by λ and ∨n−1 k=0 T ξ, then 1 1 ϕ ϕi + 1 − ϕ= n n i hµ (ξ ; T ) − Hµ (ξ ) > −ε − η

is the decomposition deﬁned by λ and ξ . Recalling that we denote by ϕˆi the state ϕi (1)−1 ϕi , by virtue of (3.10) we get ϕi (1) 1 S(ϕ, ϕ) S(ϕˆi , ϕ) + 1 − Hλ (A; ξ ) = n n i = It follows that

1 k − − Hλ (A; ∨n−1 k=0 T ξ) > Hµ (ξ|ξ ) − Hλ,A (ξ|ξ ) − ε. n

82

5 Dynamical Abelian Models

hµ (ξ ; T ) − Hµ (ξ ) + Hλ (A; ξ ) > Hµ (ξ|ξ − ) − Hλ,A (ξ|ξ − ) − 3ε,

and the proof is complete. We can now formulate the main result of the chapter.

∗ Theorem 5.1.5. We have hST ϕ (α) ≥ hϕ (α) for any C -dynamical system (A, ϕ, α). If the system has a ϕ-approximating net, then hST ϕ (α) = hϕ (α).

Proof. By (5.1) we have hϕ (γ; α) ≥ hµ (ξ; T ) − Hµ (ξ) + Hλ (γ; ξ). If {γi }i is a ϕ-approximating net, then Hλ (γi ; ξ) Hλ (A; ξ) by Corollary 2.3.5. Hence hST ϕ (α) ≤ hϕ (α). To prove the opposite inequality in general we shall ﬁnd a channel γ: B → A, a sequence {λn }n of nonstationary couplings of (A, ϕ, α) with (Y, T, µn ), where Y is a compact metric space, and a partition ξ of Y consisting of clopen sets such that 1 Hλ (γ, α ◦ γ, . . . , αn−1 ◦ γ; ξ, T ξ, . . . , T n−1 ξ) > hϕ (α) − ε. n n Then to geta stationary coupling it is natural to take a weak∗ limit point of the n−1 states n−1 k=0 λn ◦(αk ⊗β k ), n ∈ N, on A⊗C(Y ). However, relative entropy is convex, which makes it impossible in general to estimate the entropy of the coupling we get. The key observation allowing to overcome this problem is that although Hλ (γ; ξ) is a convex function of λ, it is almost aﬃne on canonical couplings, see Deﬁnition 3.1.8. Lemma 5.1.6. Let γ: B → A be a channel. For ε > 0 choose δ > 0 such that |S(ψ1 ) − S(ψ2 )| < ε whenever ψ1 and ψ2 are states on B satisfying ψ1 − ψ2 < δ. Let ζ be a ﬁnite Borel partition of the state space X of A for which ω1 ◦ γ − ω2 ◦ γ < δ when ω1 and ω2 belong to the same element of ζ.Then if λi is a canonical coupling of (A, ϕ) with (X, µi ), i ∈ I, and λ = i αi λi is a convex combination, we have αi Hλi (γ; ζ) < ε. Hλ (γ; ζ) − i

Proof. For any canonical coupling Λ we have by (3.11) HΛ (γ; ζ) = S(ϕ ◦ γ) − ν(Z)S ν(Z)−1 ω ◦ γ dν(ω) , Z∈ζ

Z

where ν = Λ|C(X) . Let µ = i αi µi , and set ϕZ = µ(Z)−1 Z ω dµ(ω). Then by the choice of ζ we have ω ◦ γ − ϕZ ◦ γ < δ for any ω ∈ Z. Hence

5.1 Entropy via Stationary Couplings

83

HΛ (γ; ζ) − S(ϕ ◦ γ) + ν(Z)S(ϕZ ◦ γ) < ε. Z∈ζ In other words, if we put f (Λ) = S(ϕ ◦ γ) − Z∈ζ ν(Z)S(ϕZ ◦ γ), then f is an aﬃne function on the space of canonical couplings such that f (λ) = Hλ (γ, ζ) and |HΛ (γ, ζ) − f (Λ)| < ε for any Λ. This gives the result.

Returning to the proof of Theorem 5.1.5, ﬁx h < hϕ (α) and ε > 0. Choose a channel γ: B → A such that hϕ (γ; α) > h. Choose also a ﬁnite Borel partition ζ of X = S(A) as in Lemma 5.1.6. +∞ Put Z = −∞ X, and deﬁne a homeomorphism T of Z by ˆ (ωn−1 ) = ωn−1 ◦ α−1 for ω = (ωn )n ∈ Z. (T ω)n = α Let β be the corresponding automorphism of C(Z), so β(f ) = f ◦ T −1 . Let prk : Z → X be the projection onto the k-th factor. Set ξ = pr−1 0 (ζ). k Then T k ξ = pr−1 (ˆ α (ζ)). By Proposition 3.1.10 applied to the channels αk ◦γ k k and the partitions α ˆ (ζ) of X, 0 ≤ k ≤ n − 1, there exists a coupling λn of (A, ϕ) with (Z, µn ) such that 1 Hλ (γ, α ◦ γ, . . . , αn−1 ◦ γ; ξ, T ξ, . . . , T n−1 ξ) > h − ε n n n−1

(more precisely, we ﬁrst get a state on A ⊗ C( k=0 X), and then extend it arbitrarily to A ⊗ C(Z)). In other words, by (5.1) and (5.2), n−1 1 k Hµn (∨n−1 Hλn ,αk ◦γ (T k ξ) > h − ε. k=0 T ξ) − n k=0

By a computation similar to (1.2) this can be written as n−1 1 j k (Hµn (T k ξ| ∨k−1 j=0 T ξ) − Hλn ,αk ◦γ (T ξ)) > h − ε. n k=0

For any ﬁxed p ∈ N the contribution of the summands with k < p becomes k−1 j j negligible as n → ∞. On the other hand, if k ≥ p then ∨k−1 k−p T ξ ≺ ∨j=0 T ξ, k−1 j j T ξ) ≤ Hµn (T k ξ| ∨k−1 so that Hµn (T k ξ| ∨j=0 k−p T ξ). Hence n−1 1 j k (Hµn (T k ξ| ∨k−1 k−p T ξ) − Hλn ,αk ◦γ (T ξ)) > h − 2ε n k=0

for any ﬁxed p and any n suﬃciently large. By Proposition 5.1.3(vi), j −Hλn ,αk ◦γ (T k ξ) ≤ −Hλn ,αk ◦γ (∨kk−p T j ξ) + Hλn ,αk ◦γ (∨k−1 k−p T ξ), k−1 j j j k so that using Hµn (T k ξ| ∨k−1 k−p T ξ) = Hµn (∨k−p T ξ) − Hµn (∨k−p T ξ) we get

84

5 Dynamical Abelian Models n−1 1 j (Hλn (αk ◦ γ; ∨kk−p T j ξ) − Hλn (αk ◦ γ; ∨k−1 k−p T ξ)) > h − 2ε n

(5.4)

k=0

for any ﬁxed p and any n suﬃciently large. Our goal is to replace ∨kk−p T j ξ by T k ξ in the ﬁrst summand to be able to use Lemma 5.1.6. Set an,p =

n−1 1 Hλn (αk ◦ γ; ∨kk−p T j ξ). n k=0

We have 0 ≤ an,p ≤ Hϕ (γ). Thus there exists an inﬁnite subset I ⊂ N such that the sequence {an,p }n∈I converges as I n → ∞ for any p ∈ N. Denote its limit by ap . By Proposition 5.1.3(i) we have an,p ≥ an,r if p ≥ r, so ap ≥ ar . Let a∞ be the limit of {ap }p as p → ∞. Fix r ∈ N such that ar > a∞ − ε. As ap ≤ a∞ , for any ﬁxed p and any n ∈ I suﬃciently large we have an,r > an,p − ε. Thus (5.4) implies that n−1 1 k−1 j (Hλn (αk ◦ γ; ∨kk−r T j ξ) − Hλn (αk ◦ γ; ∨k−p T ξ)) > h − 3ε n

(5.5)

k=0

for any ﬁxed p and any n ∈ I suﬃciently large. Now apply Proposition 3.1.10 to the coupling λn , the channels αk ◦ γ, the partitions ∨kk−r T j ξ of Z and the partitions α ˆ k (ζ) of X, 0 ≤ k ≤ n − 1. Thus we get a Borel map f : Z → Z such that for 0 ≤ k ≤ n − 1 we have (i) f −1 (T k ξ) ≺ ∨kk−r T j ξ; (ii) if we set λn = λn ◦ (id ⊗ f ∗ ), then |Hλn (αk ◦ γ; ∨kk−r T j ξ) − Hλn (αk ◦ γ; T k ξ)| < ε; (iii) the coupling λn ◦ (id ⊗ pr∗k ) is canonical. (Again, to be more precise, we have maps fk : Z → X for 0 ≤ k ≤ n − 1, but then we take arbitrary Borel maps Z → X for all other k’s.) k−1 j j By virtue of (i) we have f −1 (∨k−1 k−p+r T ξ) ≺ ∨k−p T ξ if k ≥ p − r > 0 and k ≤ n − 1. By Proposition 5.1.3(i) we thus have j k −1 j Hλn (αk ◦ γ; ∨k−1 (∨k−1 k−p T ξ) ≥ Hλn (α ◦ γ; f k−p+r T ξ)) j = Hλn (αk ◦ γ; ∨k−1 k−p+r T ξ)

if 0 < p − r ≤ k ≤ n − 1, so that n−1 n−1 1 1 j j Hλn (αk ◦ γ; ∨k−1 T ξ) > Hλn (αk ◦ γ; ∨k−1 k−p+r T ξ) − ε k−p n n k=0

k=0

for p > r and n suﬃciently large, since the contribution of the summands with k < p − r becomes negligible as n → ∞. Thus property (ii) of the map f and inequality (5.5) imply

5.1 Entropy via Stationary Couplings

85

n−1 1 k−1 (Hλn (αk ◦ γ; T k ξ) − Hλn (αk ◦ γ; ∨k−p+r T j ξ)) > h − 5ε, n k=0

or equivalently, n−1 1 j (Hλn ◦(αk ⊗β k ) (γ; ξ) − Hλn ◦(αk ⊗β k ) (γ; ∨−1 −p+r T ξ)) > h − 5ε n k=0

for any ﬁxed p > r and any n ∈ I suﬃciently large. Set λn =

n−1 1 λn ◦ (αk ⊗ β k ). n k=0

ˆ ∗ )−k , The couplings λn ◦ (αk ⊗ β k ) ◦ (id ⊗ pr∗0 ) = λn ◦ (id ⊗ pr∗k ) ◦ (α−1 ⊗ α ∗ where α ˆ (g) = g ◦ α ˆ for g ∈ C(X), are canonical for 0 ≤ k ≤ n − 1 by property (iii) of f and the fact that the automorphism α−1 ⊗ α ˆ ∗ of A ⊗ C(X) −1 preserves the set of canonical couplings. Since ξ = pr0 (ζ), by Lemma 5.1.6 and Proposition 5.1.3(iii) we get j Hλn (γ; ξ) − Hλn (γ; ∨−1 −p+r T ξ) > h − 6ε.

Now replace Z by the spectrum Y of a separable β-invariant C∗ -algebra consisting of bounded Borel functions on Z and containing the characteristic functions of the elements of ξ. Denote the homeomorphism of Y deﬁned by T and the restriction of λn to A ⊗ C(Y ) by the same symbols. We can think of ξ as a partition of Y consisting of clopen sets. Let λ be a weak∗ limit point of {λn }n∈I . Since the von Neumann entropy is continuous, we have j Hλ (γ; ξ) − Hλ (γ; ∨−1 −k T ξ) ≥ h − 6ε

for any k ∈ N. Hence j Hλ (γ; ∨0−k T j ξ) − Hλ (γ; ∨−1 −k T ξ) ≥ h − 6ε,

equivalently, with µ = λ|C(Y ) , −1 j j 0 j Hµ (∨0−k T j ξ) − Hµ (ξ| ∨−1 −k T ξ) − Hλ,γ (∨−k T ξ) + Hλ,γ (∨−k T ξ) ≥ h − 6ε,

which by Proposition 5.1.3(vi) can be written as −1 j j Hµ (ξ| ∨−1 −k T ξ) − Hλ,γ (ξ| ∨−k T ξ) ≥ h − 6ε.

Letting k → ∞, by Proposition 5.1.3(iv) we get Hµ (ξ|ξ − ) − Hλ,γ (ξ|ξ − ) ≥ h − 6ε. Since Hλ,γ (ξ|ξ − ) ≥ Hλ,A (ξ|ξ − ) by Proposition 5.1.3(iv), using Proposition 5.1.4 we conclude that hST ϕ (α) ≥ h − 6ε, which completes the proof of the theorem.

86

5 Dynamical Abelian Models

In Proposition 3.2.7 we proved that a state on a nuclear subalgebra of a C∗ -algebra can be extended to a state on the algebra in an entropy-increasing way. As our ﬁrst application of existence of abelian models we shall now prove another similar result. Proposition 5.1.7. Let α be an automorphism of a nuclear C∗ -algebra A, B ⊂ A an α-invariant C∗ -subalgebra, ψ an α-invariant state on B. Then for every h < hψ (α|B ) there exists an α-invariant state ϕ on A such that ϕ|B = ψ and hϕ (α) > h. Proof. Let λ be a stationary coupling of (B, ψ, α|B ) with (X, µ, T ), and ξ a measurable partition of (X, µ) such that hµ (ξ; T ) − Hµ (ξ) + Hλ (B; ξ) > h. Extend λ to an (α ⊗ β)-invariant state Λ on A ⊗ L∞ (X, µ), where β is the automorphism of L∞ (X, µ) deﬁned by T , and put ϕ = Λ|A . By Proposition 5.1.3(i) we have HΛ (A; ξ) ≥ Hλ (B; ξ), hence hϕ (α) = hST

ϕ (α) > h. If (X/ξ, µ, T ) is a factor system of (X, µ, T ) such that the map X → X/ξ is ﬁnite-to-one almost everywhere, then it is not diﬃcult to check that both systems have the same entropy. The following is a noncommutative analogue of this result. Proposition 5.1.8. Let (M, ϕ, α) be a W∗ -dynamical system having a ϕapproximating net, N ⊂ M an α-invariant von Neumann subalgebra, E: M → N a ϕ-preserving faithful normal conditional expectation commuting with α. Assume there exists a constant c > 0 such that E(a) ≥ ca for any positive a ∈ M . Then hϕ (α) = hϕ|N (α|N ). Proof. Using the conditional expectation E, any stationary coupling of the system (N, ϕ|N , α|N ) with an abelian system can be extended to a coupling of (M, ϕ, α) with the same system. This implies hϕ (α) ≥ hϕ|N (α|N ). Conversely, let λ be a stationary coupling of (M, ϕ, α) with (X, µ, T ), and ξ a measurable partition of (X, µ). For any normal state ψ on M we have ψ ◦ E ≥ cψ. By Theorem 2.3.1(vii) we then get S(ψ, ϕ) − S(ψ|N , ϕ|N ) = S(ψ, ψ ◦ E) ≤ S(ψ, cψ) = − log c. It follows that Hλ (M ; ξ) − Hλ (N ; ξ) ≤ − log c, and hence hϕ (α) ≤ hϕ|N (α|N ) − log c. Applying this to αn , dividing by n and letting n → ∞ we get hϕ (α) ≤ hϕ|N (α|N ).

5.2 Zero Entropy Systems Let (A, ϕ, α) be a C∗ -dynamical system. Consider the space of the GNSrepresentation deﬁned by ϕ. On this space the automorphism α is implemented by the unitary U deﬁned by U aξϕ = α(a)ξϕ for a ∈ A. We say that

5.2 Zero Entropy Systems

87

the system has singular spectrum if the spectral measure of U is singular with respect to the Lebesgue measure on T. If the space is nonseparable, we mean by this that the spectral measure of the restriction of U to any separable invariant subspace is singular. Equivalently, for any vector ξ the measure νξ on T such that (U n ξ, ξ) = T z n dνξ (z) for n ∈ Z is singular. Theorem 5.2.1. Let (A, ϕ, α) be a C∗ -dynamical system with singular spectrum. Then hϕ (α) = hST ϕ (α) = 0. Proof. We shall ﬁrst reduce the proof to the case when A separable. Choose an increasing net {Ai }i of separable α-invariant C∗ -subalgebras with dense union. Let λ be a stationary coupling of (A, ϕ, α) with an abelian system (X, µ, T ), and ξ a ﬁnite measurable partition of (X, µ). By Corollary 2.3.6 it follows that Hλ (A; ξ) = lim Hλ (Ai ; ξ). i

≤ So it is enough to show that the enHence tropy of α|Ai is zero for every i. Since the systems (Ai , ϕ|Ai , α|Ai ) still have singular spectrum, we see that is suﬃces to prove the theorem for separable C∗ -algebras. By Proposition 5.1.4 we have to show that hST ϕ (α)

lim inf i hST ϕ|Ai (α|Ai ).

Hµ (ξ|ξ − ) = Hλ,A (ξ|ξ − )

(5.6)

for every stationary coupling λ with (X, µ, T ) and every ﬁnite measurable partition ξ. We shall do this in three steps, ﬁrst assuming that (X, µ, T ) is a Bernoulli shift, then that it is ergodic, and ﬁnally in general. So assume that (X, µ, T ) is a Bernoulli shift. We claim that then λ is trivial, λ = ϕ ⊗ µ. Let β be the automorphism of L∞ (X, µ) deﬁned by T , β(f ) = f ◦ T −1 for f ∈ L∞ (X, µ). Denote by V the canonical unitary on L2 (X, µ) implementing β, V f = f ◦ T −1 for f ∈ L2 (X, dµ). Let ζ be the standard generating partition for (X, µ, T ). Then, as we already remarked in Sect. 4.3, ζ − ≺ T ζ − , the algebra ∪n∈Z L∞ (X/T n ζ − ) is weakly operator dense in L∞ (X, µ), and ∩n∈Z L∞ (X/T n ζ − ) = C1. It follows that if we set L = L2 (X/T ζ − , dµ) L2 (X/ζ − , dµ) then L2 (X, dµ) C = ⊕ V n L. n∈Z

Thus on the orthogonal complement of the constants the unitary V has homogeneous Lebesgue spectrum, see C.7. Now let H be the space of the GNSrepresentation deﬁned by λ, and v ∈ H the corresponding cyclic vector. Let W be the unitary on H implementing α ⊗ β. According to C.7 the space H decomposes into a direct sum Hs ⊕ Ha such that the restriction of W to Hs has singular spectral measure, and the restriction of W to Ha has absolutely continuous spectral measure. The restriction of W to Av can be identiﬁed with U , while the restriction of W to L∞ (X, µ)v can be identiﬁed with V . It

88

5 Dynamical Abelian Models

follows that av ∈ Hs for any a ∈ A, and f v ∈ Ha for any f ∈ L∞ (X, µ) with µ(f ) = 0. Hence av and f v are orthogonal, that is, λ(a ⊗ f¯) = 0. For any f ∈ L∞ (X, µ) we then have λ(a ⊗ f ) = ϕ(a)µ(f ) + λ(a ⊗ (f − µ(f )1)) = ϕ(a)µ(f ), and our claim is proved. Assume now that (X, µ, T ) is ergodic. Replacing (X, µ, T ) by the factor system (X/ξ ± , µ, T ), where ξ ± = ∨n∈Z T n ξ, we may assume that ξ is a generating partition. By Sinai’s Theorem 1.1.7 there exists a Bernoullian factor system with the same entropy. In other words, there exists a ﬁnite measurable partition ζ such that the partitions T n ζ, n ∈ Z, are mutually independent, and hµ (T ) = Hµ (ζ). Since ξ is generating and, as we already observed in the proof of Proposition 5.1.4, Hλ,A (ϑ|ϑ− ) = lim

n→∞

1 k Hλ,A (∨n−1 k=0 T ϑ) n

for any ﬁnite ϑ, the same proof as that of Lemma 1.1.3 shows that Hλ,A (ζ|ζ − ) ≤ Hλ,A (ξ|ξ − ) + Hλ,A (ζ|ξ ± ) = Hλ,A (ξ|ξ − ). On the other hand, by the case of Bernoullian systems considered above, we have Hλ,A (ζ|ζ − ) = Hµ (ζ|ζ − ) = hµ (T ) = Hµ (ξ|ξ − ). Since we always have Hλ,A (ξ|ξ − ) ≤ Hµ (ξ|ξ − ) by Proposition 5.1.3(iv), we conclude that (5.6) holds. Consider the general case. As before, let β be the automorphism of L∞ (X, µ) deﬁned by T , β(f ) = f ◦ T −1 for f ∈ L∞ (X, µ). Let ϑ be the measurable partition such that L∞ (X/ϑ) = L∞ (X)β , the ﬁxed point algebra of β. Then, see C.1-C.4, the space H of the GNS-representation deﬁned by λ ⊕ decomposes into a direct integral X/ϑ Hy dµ(y), elements of A⊗L∞ (X, µ) are decomposable, and a normal extension Λ of λ to the commutant of L∞ (X/ϑ) ⊕ decomposes into a direct integral Λ = X/ϑ Λy dµ(y). Without loss of generality we may assume that X and X/ϑ are compact metric spaces, X → X/ϑ is continuous, and T is a homeomorphism. Then ⊕ by C.5 the representation of A ⊗ C(X) decomposes into a direct integral X/ϑ πy dµ(y) of representations. Set λy = Λy ◦ πy , ϕy = λy |A and µy = λy |C(X) . The state λy is (α ⊗ β)invariant for a.e. y ∈ X/ϑ, so that λy is a stationary coupling of (A, ϕy , α) with (X, µy , T ). ∗ n a ∈ A and ν (resp. νy ) is the nmeasure on T such that ϕ(a α (a)) = If n ∗ n z dν(z) (resp. ϕy (a α (a)) = T z dνy (z)), then ν = X/ϑ νy dµ(y). Since T ν is singular by assumption, it follows that νy is singular for a.e. y. Thus the system (A, ϕy , α) has singular spectrum for a.e. y. Next note that (X, µy , T ) is ergodic for a.e. y. We shall give a sketch of the argument, see e.g. [72, Theorem 3.42] for details. By Birkhoﬀ’s ergodic

5.2 Zero Entropy Systems

89

n−1 −1

−k theorem, for any f ∈ C(X), the sequence {n x)}∞ n=1 converges k=0 f (T ∞ ∞ to E(f )(x) for a.e. x ∈ X, where E: L (X) → L (X/ϑ) is the µ-preserving conditional expectation. Since the operator E(f ) is diagonal, it follows that for n−1 f ∈ C(X) and a.e. y ∈ X/ϑ the sequence {n−1 k=0 πy (β k (f ))}n converges to a scalar operator in the strong operator topology. But if for a ﬁxed y this convergence holds for a countable dense subset of functions f ∈ C(X), then (X, µy , T ) is ergodic. We can thus apply the previous case to the couplings λy and conclude that

Hµy (ξ|ξ − ) = Hλy ,A (ξ|ξ − ) for any ﬁnite Borel partition ξ of X and a.e. y ∈ X/ϑ. To prove that (5.6) holds, it is then enough to show that Hµ (ξ|ξ − ) = Hµy (ξ|ξ − )dµ(y) and Hλ,A (ξ|ξ − ) = Hλy ,A (ξ|ξ − )dµ(y). X/ϑ

X/ϑ

We claim that

Hλy ,A (ξ|ξ − )dµ(y) = Hλ,A (ξ|ξ − ∨ ϑ).

(5.7)

X/ϑ

Since Hλy ,A (ξ| ∨nk=1 T −k ξ) Hλy ,A (ξ|ξ − ), and similarly Hλ,A (ξ|ξ − ∨ ϑ) is the limit of Hλ,A (ξ|ϑ ∨ ∨nk=1 T −k ξ), it suﬃces to check that Hλy ,A (ξ|ζ)dµ(y) = Hλ,A (ξ|ζ ∨ ϑ) X/ϑ

for any ﬁnite Borel partition ζ. By Proposition 5.1.3(vi) it then suﬃces to check that Hλy ,A (ζ)dµ(y) = Hλ,A (ζ|ϑ) X/ϑ

for any ζ. By (5.2) for this, in turn, it is enough to check that for any Borel subset Z ⊂ X we have S(λy (· ⊗ ·1Z ), λy )dµ(y) = S(λ(· ⊗ ·1Z ), λ). (5.8) X/ϑ

˜ and λ ˜ Z on A ⊗ C(X) ⊗ C(X/ϑ) deﬁned by Consider the states λ ˜ ˜ Z (a ⊗ f ⊗ g) = λ(a ˜ ⊗ f 1Z ⊗ g). λ(a ⊗ f ⊗ g) = λy (a ⊗ f )g(y)dµ(y) and λ X/ϑ

˜ Z , λ), ˜ Then by Corollary 2.3.7 the left hand side of (5.8) coincides with S(λ ˜ ˜ while the right hand side is S(λZ |A⊗C(X) , λ|A⊗C(X) ). Since the homomorphism A ⊗ C(X) ⊗ C(X/ϑ) → A ⊗ C(X), a ⊗ f ⊗ g → a ⊗ f g, which is also a

90

5 Dynamical Abelian Models

conditional expectation, preserves both states, both sides indeed coincide by Theorem 2.3.1(ix) or Corollary 2.3.3. Thus (5.7) is established. Any ﬁnite measurable partition ζ ≺ ϑ of (X, µ) is T -invariant. By Proposition 5.1.3(vi) it follows that Hλ,A (ξ ∨ ζ|ξ − ∨ ζ − ) = Hλ,A (ξ ∨ ζ|ξ − ∨ ζ) = Hλ,A (ξ|ξ − ∨ ζ). On the other hand, 1 k Hλ,A (∨n−1 k=0 T ξ) n 1 k − − ≤ lim Hλ,A (∨n−1 k=0 T (ξ ∨ ζ)) = Hλ,A (ξ ∨ ζ|ξ ∨ ζ ). n→∞ n

Hλ,A (ξ|ξ − ) = lim

n→∞

Consequently

Hλ,A (ξ|ξ − ) = Hλ,A (ξ|ξ − ∨ ζ).

Choosing an increasing sequence of ﬁnite measurable partitions ζn , n ∈ N, such that ∨n ζn = ϑ, we conclude that Hλ,A (ξ|ξ − ) = Hλ,A (ξ|ξ − ∨ ϑ). Together with (5.7) this shows that Hλ,A (ξ|ξ − ) = X/ϑ Hλy ,A (ξ|ξ − )dµ(y). A similar argument gives Hµ (ξ|ξ − ) = X/ϑ Hµy (ξ|ξ − )dµ(y), which completes the proof of the theorem.

Proposition 5.1.2 implies that if a system has a nontrivial stationary coupling with a classical Bernoulli shift, then hST ϕ (α) > 0. The proof of the above theorem shows that the converse would be true if we could show that if the system does not have nontrivial couplings with Bernoulli shifts, and ϕ = X ϕx dµ(x), then the same is true for (A, ϕx , α) for almost every x. This is obviously true if we do not have nontrivial integral decompositions of ϕ into α-invariant states. Thus we get the following. Theorem 5.2.2. Let (A, ϕ, α) be a C∗ -dynamical system with separable A. Assume ϕ is ergodic, that is, ϕ is an extremal α-invariant state. Then hST ϕ (α) > 0 if and only if there exists a nontrivial stationary coupling with a classical Bernoulli shift.

We shall now introduce a class of systems which do not have nontrivial couplings with abelian systems. Deﬁnition 5.2.3. A system (A, α) consisting of a unital C∗ -algebra A and an automorphism α is called asymptotically highly anticommutative if there exists a set {xi }i∈I of elements of A which together with the unit spans a dense subspace of A and satisﬁes the following property. For any i ∈ I, ε > 0 and n ∈ N there exist m1 , . . . , mn ∈ N such that αmk (x∗i )αml (xi )+αml (xi )αmk (x∗i ) < ε for all k, l ≤ n with k = l.

5.3 Notes

91

Proposition 5.2.4. If (A, α) is asymptotically highly anticommutative then there exists a unique α-invariant state on A. Proof. Let ϕ be an α-invariant state, and {xi }i∈I a set as in the deﬁnition of an asymptotically highly anticommutative system. Fix i ∈ I. Let ε > 0 and mk m1 , . . . , mn ∈ N be such that αmk (x∗i )αml (xi ) + αml (x (x∗i ) < ε for i )α n −1 mk all k, l ≤ n with k = l. Consider the element x = n (xi ). Then k=1 α ∗ ∗ x x + xx equals n 1 mk ∗ ml 1 mk ∗ ∗ α (x x + x x ) + (α (xi )α (xi ) + αml (xi )αmk (x∗i )), i i i i n2 n2 k=l

k=1

whence x∗ x + xx∗ ≤

2xi 2 (n − 1)ε + . n n

Since |ϕ(xi )| = |ϕ(x)| ≤

1 1 (ϕ(x∗ x)1/2 + ϕ(xx∗ )1/2 ) ≤ √ ϕ(x∗ x + xx∗ )1/2 , 2 2

it follows that ϕ(xi ) = 0. Since {1} ∪ {xi }i∈I spans a dense subspace of A, the condition ϕ(xi ) = 0 completely determines the state.

The above proof implies the following. Corollary 5.2.5. If (A, ϕ, α) is an asymptotically highly anticommutative C∗ dynamical system, then any stationary coupling with an abelian dynamical system is trivial. In particular, hϕ (α) = hST ϕ (α) = 0. Proof. Let λ be a stationary coupling of (A, ϕ, α) with (X, µ, T ). Then the same estimate as in the proof of the previous proposition applied to xi ⊗ f instead of xi shows that λ(xi ⊗f ) = 0 = ϕ(xi )µ(f ). Together with the equality λ(1 ⊗ f ) = µ(f ) this implies λ = ϕ ⊗ µ.

We shall see examples of asymptotically highly anticommutative systems in Chaps. 11 and 12. In Chap. 14 we shall consider a system which is not asymptotically highly anticommutative, but which still has only trivial stationary couplings with abelian systems.

5.3 Notes The notion of a stationary coupling was introduced by Sauvageot and Thouvenot [188], who obtained all the main results of the present chapter. In the classical case stationary couplings are called joinings. They were introduced by Furstenberg [68]. See [72], [184] for an exposition of ergodic theory from the theory of joinings point of view.

92

5 Dynamical Abelian Models

Propositions 5.1.7 and 5.1.8 are due to the authors [140], [141]. Theorem 5.2.1, proved by Sauvageot and Thouvenot [188], is an extension to the noncommutative case of a result of Rohlin and Sinai [183]. In the classical case the result states a bit more: if a system has positive entropy then the spectrum has a countably multiple Lebesgue component. Asymptotically highly anticommutative systems were introduced by Narnhofer, Størmer and Thirring [128], who proved Proposition 5.2.4 and Corollary 5.2.5. The name was coined later by Narnhofer and Thirring [134].

6 Topological Entropy

In this chapter we shall study the extension of topological entropy to C∗ algebras. The main problem is to ﬁnd the right analogue of the cardinality N (U) of a minimal subcover for an open cover U. If X is a totally disconnected compact space and U a clopen partition of X, then N (U) is the rank of the abelian algebra spanned by the characteristic functions of the elements of U. In the general case instead of the algebra generated by the characteristic functions of the elements of U we shall consider ﬁnite collections of functions which are almost constant on the elements of U. It turns out that instead of N (U) one can then take the ranks of completely positive approximations of these collections.

6.1 Rank of a Completely Positive Approximation Let A be a unital C∗ -algebra, Ω a ﬁnite subset of A. Deﬁnition 6.1.1. For δ > 0 denote by rcpA (Ω, δ) the minimal number n ∈ N for which there exist a unital injective homomorphism π: A → B for some C∗ -algebra B and unital completely positive maps θ: A → Matn (C) and γ: Matn (C) → B such that (γ ◦ θ)(a) − π(a) < δ for any a ∈ Ω. Put rcpA (Ω, δ) = +∞ if no such approximation exists. We shall write rcp(Ω, δ) instead of rcpA (Ω, δ) when no confusion can arise. If D is a ﬁnite dimensional C∗ -algebra, then D can be imbedded into Matn (C), where n is the rank of D. A unital completely positive map D → B can be extended to a unital completely positive map Matn (C) → B by composing it with the trace preserving conditional expectation Matn (C) → D. So we could deﬁne rcpA (Ω, δ) as the minimal number n ∈ N for which there exist an injective homomorphism π: A → B for some C∗ -algebra B and unital completely positive maps θ: A → D and γ: D → B such that n = rank D and (γ ◦ θ)(a) − π(a) < δ for any a ∈ Ω.

94

6 Topological Entropy

Observe next that there is no need to consider all possible embeddings of A. Lemma 6.1.2. In computing rcpA (Ω, δ) it is enough to consider one faithful representation π: A → B(H). It also suﬃces to take a nuclear embedding π: A → B, if one exists. Recall that a homomorphism π: A → B is called nuclear if there exist two nets {θi : A → Matni (C)}i∈I and {γi : Matni (C) → B}i∈I of unital completely positive maps such that (γi ◦ θi )(a) − π(a) → 0 for any a ∈ A. The algebras admitting nuclear embeddings are said to be nuclearly embeddable, or exact. If rcp(Ω, δ) is ﬁnite for any Ω ⊂ A and any δ > 0, then Lemma 6.1.2 implies that any faithful representation π: A → B(H) is nuclear, so that A is exact. If A is nuclear, then the identity map A → A is nuclear, so we can take B = A. At least one reason to consider the class of exact C∗ -algebras is that subalgebras of nuclear algebras are not necessarily nuclear, but always exact. The class of exact algebras is quite large. It is obviously closed under taking subalgebras and minimal tensor products. In Chap. 8 we shall show that it is closed under taking crossed products by actions of amenable locally compact groups, while in Chap. 14 we shall prove that reduced free products of exact algebras are exact. Quotients of exact algebras are also exact. This is nontrivial, and will not be used in the sequel. Proof of Lemma 6.1.2. Denote by rcp(π, Ω, δ) the quantity deﬁned exactly as rcp(Ω, δ) but using a ﬁxed embedding π: A → B. Let π: A → B(H) be a faithful representation. Given an arbitrary embedding π0 : A → B, by Arveson’s extension theorem A.8 there exists a unital completely positive map Φ: B → B(H) such that Φ ◦ π0 = π. Given a unital completely positive map γ: Matn (C) → B we can now consider Φ◦γ: Matn (C) → B(H). Since (Φ◦γ ◦θ)(a)−π(a) = Φ((γ ◦θ)(a)−π0 (a)), we get rcp(π, Ω, δ) ≤ rcp(π0 , Ω, δ). Similarly, assume π: A → B is a nuclear embedding. Let {θi : A → Matni (C)}i∈I and {γi : Matni (C) → B}i∈I be nets of unital completely positive maps such that (γi ◦ θi )(a) − π(a) → 0 for any a ∈ A. Let π0 : A → B0 be another embedding, and let θ0 : A → Matn (C) and γ0 : Matn (C) → B0 be unital completely positive maps such that (γ0 ◦θ0 )(a)−π0 (a) < δ for a ∈ Ω. Again by Arveson’s extension theorem there exist unital completely positive maps θ¯i : B0 → Matni (C), i ∈ I, such that θ¯i ◦ π0 = θi . Consider the maps γi ◦ θ¯i ◦ γ0 : Matn (C) → B. We have (γi ◦ θ¯i ◦ γ0 ◦ θ0 )(a) − π(a) ≤ (γi ◦ θ¯i ◦ γ0 ◦ θ0 )(a) − (γi ◦ θ¯i ◦ π0 )(a) + (γi ◦ θ¯i ◦ π0 )(a) − π(a). ≤ (γ0 ◦ θ0 )(a) − π0 (a) + (γi ◦ θi )(a) − π(a). Since (γi ◦ θi )(a) − π(a) → 0, we see that for suﬃciently large i we get (γi ◦θ¯i ◦γ0 ◦θ0 )(a)−π(a) < δ for any a ∈ Ω. Thus rcp(π, Ω, δ) ≤ rcp(π0 , Ω, δ).

6.1 Rank of a Completely Positive Approximation

95

In order to obtain a better understanding of the deﬁnition consider the abelian case. Lemma 6.1.3. Let X be a compact space, Ω ⊂ C(X) a ﬁnite subset, and δ > 0. Let U be a ﬁnite open cover of X such that |f (x) − f (y)| < δ for every f ∈ Ω and x, y ∈ X lying in the same element of U. Then rcp(Ω, δ) ≤ N (U). Proof. Let V be a subcover of U with minimal number N = N (U) of elements. Let V = {V1 , . . . , VN }, and for each i choose xi ∈ Vi . Denote the subset of X consisting of the xi ’s by Y . Let further {χi }N i=1 be a partition of unity subordinate to V, that is, supp χi ⊂ Vi and i χi = 1. We set B = C(Y ), and deﬁne θ: C(X) → B, θ(f ) = f |Y , and γ: B → C(X), γ(g) =

N

g(xi )χi .

i=1

If f ∈ Ω then for any x ∈ X |f (x) − ((γ ◦ θ)(f ))(x)| ≤

N

χi (x)|f (x) − f (xi )|

i=1

=

χi (x)|f (x) − f (xi )| < δ.

i : x∈Vi

Hence rcp(Ω, δ) ≤ rank B = N (U).

We next list some properties of rcp. Lemma 6.1.4. For every unital C∗ -algebra A we have: (i) rcp(Ω, δ) = rcp(Ω ∪ Ω ∗ ∪ {1}, δ); (ii) if Ω ⊂ Ω0 then rcp(Ω, δ) ≤ rcp(Ω0 , δ); (iii) if Ω ⊂δ0 Ω0 , meaning that for any a ∈ Ω there is a0 ∈ Ω0 such that a − a0 < δ0 , then rcp(Ω, δ + 2δ0 ) ≤ rcp(Ω0 , δ); (iv) if each a ∈ Ω is a linear combination of at most k elements of Ω0 with coeﬃcients of modulus not greater than l, then rcp(Ω, klδ) ≤ rcp(Ω0 , δ); (v) if Ω consists of unitary operators and contains the unit, then rcp(Ω 2 , 4δ) ≤ rcp(Ω, δ) ≤ rcp(Ω 2 , δ), where Ω 2 = {uv | u, v ∈ Ω}; (vi) if α is an automorphism of A, then rcp(α(Ω), δ) = rcp(Ω, δ); (vii)if β: A → C is an injective unital homomorphism, then rcpC (β(Ω), δ) = rcpA (Ω, δ). Proof. Parts (i)-(iv) are straightforward. For example, part (iii) follows from the inequality (γ ◦ θ)(a) − π(a) ≤ (γ ◦ θ)(a0 ) − π(a0 ) + (γ ◦ θ)(a − a0 ) + π(a − a0 ).

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6 Topological Entropy

To prove (v) note that the second inequality is trivial as Ω ⊂ Ω 2 . To prove the other inequality suppose A ⊂ B, and let θ: A → Matn (C) and γ: Matn (C) → B be unital completely positive maps such that (γ ◦ θ)(a) − a < δ for a ∈ Ω. Put Φ = γ ◦ θ. Then for u ∈ Ω Φ(u)Φ(u)∗ − 1 ≤ (Φ(u) − u)Φ(u)∗ + u(Φ(u)∗ − u∗ ) < 2δ, and similarly Φ(u)Φ(v) − uv < 2δ and Φ(v)∗ Φ(v) − 1 < 2δ when furthermore v ∈ Ω. Thus by A.5 we have Φ(uv) − uv < Φ(uv) − Φ(u)Φ(v) + 2δ ≤ Φ(uu∗ ) − Φ(u)Φ(u)∗ 1/2 Φ(v ∗ v) − Φ(v)∗ Φ(v)1/2 + 2δ = 1 − Φ(u)Φ(u)∗ 1/2 1 − Φ(v)∗ Φ(v)1/2 + 2δ < (2δ)1/2 (2δ)1/2 + 2δ = 4δ. It follows that rcp(Ω 2 , 4δ) ≤ rcp(Ω, δ). Part (vi) is a particular case of (vii). To prove (vii), by Lemma 6.1.2 it suﬃces to take a faithful representation π: C → B(H) and check that rcp(π ◦ β, Ω, δ) = rcp(π, β(Ω), δ). This follows again from Arveson’s extension theorem, which implies that every unital completely positive map A → Matn (C) extends to C.

6.2 Topological Entropy We are now in position to deﬁne topological entropy. Deﬁnition 6.2.1. Let A be a unital C∗ -algebra, and α an automorphism of A. Let 1 log rcpA (Ω ∪ α(Ω) ∪ . . . ∪ αn−1 (Ω), δ), n ht(Ω; α) = sup ht(Ω, δ; α),

ht(Ω, δ; α) = lim sup n→∞

δ>0

ht(α) = sup ht(Ω; α), Ω

where the last supremum is taken over all ﬁnite subsets Ω of A. The quantity ht(α) is called the topological entropy of α. Note that if δ1 < δ2 , then rcp(Ω, δ1 ) ≥ rcp(Ω, δ2 ). So the supremum in the deﬁnition of ht(Ω; α) can be replaced by the limit as δ → 0. We know that rcp(Ω, δ) is ﬁnite for any δ > 0 and any ﬁnite Ω ⊂ A if and only if A is exact. Thus, even though our deﬁnition of topological entropy makes sense for arbitrary C∗ -algebras, it is inﬁnity for any automorphism of a nonexact algebra.

6.2 Topological Entropy

97

Theorem 6.2.2. Let α be an automorphism of an exact C∗ -algebra A. Then (i) if β: A → C is an isomorphism then ht(α) = ht(β ◦ α ◦ β −1 ); (ii) ht(αn ) = |n|ht(α) for every n ∈ Z; if {αt }t∈R is a one-parameter automorphism group of A such that α1 = α, then ht(αt ) = |t|ht(α) for t ∈ R; (iii) if B is an α-invariant unital subalgebra of A, then ht(α) ≥ ht(α|B ); (iv) if β is an automorphism of a unital exact C∗ -algebra B, then ht(α ⊗ β) ≤ ht(α) + ht(β); in particular, ht(α ⊗ idB ) = ht(α); (v) under the assumptions of (iv), ht(α ⊕ β) = max{ht(α), ht(β)}. Proof. Parts (i) and (iii) are obvious. By Lemma 6.1.4(vi) we have m−1 −k k (Ω), δ), rcp(∪m−1 k=0 α (Ω), δ) = rcp(∪k=0 α

which shows ht(α) = ht(α−1 ). Thus to prove the ﬁrst part of (ii) we may n(m−1) k nk α (Ω), so that assume n > 0. We have ∪m−1 k=0 α (Ω) ⊂ ∪k=0 n(m−1) k

nk rcp(∪m−1 k=0 α (Ω), δ) ≤ rcp(∪k=0

α (Ω), δ),

which implies ht(Ω, δ; αn ) ≤ n ht(Ω, δ; α), and thus ht(αn ) ≤ n ht(α). For the reverse inequality note that if Ω ⊂ A, then Ω0 = ∪n−1 k=0 (Ω) satisﬁes m−1 ! k=0

[m/n]

αk (Ω) ⊂

!

αnk (Ω0 ).

k=0

Thus we have n ht(Ω, δ; α) ≤ ht(Ω0 , δ; αn ). Since Ω is an arbitrary ﬁnite subset of A, it follows that n ht(α) ≤ ht(αn ). Let now {αt }t∈R be a one-parameter automorphism group of A such that α1 = α. As in the proof of Theorem 3.2.5(i), to prove that t−1 ht(αt ) is independent of t > 0, it suﬃces to check that ht(αt ) ≤ t ht(α). Given a ﬁnite subset Ω ⊂ A and δ > 0 ﬁnd a ﬁnite Ω0 ⊂ A such that αs (Ω) ⊂δ Ω0 for 0 ≤ s ≤ 1. Then by Lemma 6.1.4(iii) [(n−1)t] k

k rcp(∪n−1 k=0 αt (Ω), 3δ) ≤ rcp(∪k=0

α (Ω0 ), δ),

so that ht(Ω, 3δ; αt ) ≤ t ht(Ω0 , δ; α) ≤ t ht(α). Hence ht(αt ) ≤ t ht(α). To prove the inequality in (iv) note that by Lemma 6.1.4(iii),(iv) in order to compute the entropy it suﬃces to consider ﬁnite subsets of A ⊗ B of the form Ω = Ω1 ⊗ Ω2 , where Ω1 and Ω2 are ﬁnite subsets of the unit balls of A and B, respectively. Since Matk (C) ⊗ Matm (C) ∼ = Matkm (C), it is then clear that

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6 Topological Entropy

n−1 j n−1 j j rcp(∪n−1 j=0 (α ⊗ β) (Ω1 ⊗ Ω2 ), 2δ) ≤ rcp(∪j=0 α (Ω1 ), δ) rcp(∪j=0 β (Ω2 ), δ).

This in turn gives ht(Ω1 ⊗ Ω2 , 2δ; α ⊗ β) ≤ ht(Ω1 , δ; α) + ht(Ω2 , δ; β), and hence we get the desired conclusion. In particular, ht(α ⊗ idB ) ≤ ht(α) if B is exact, since ht(idB ) = 0 in this case. Since A ⊂ A ⊗ B, we also have the opposite inequality. It remains to prove (v). By considering direct sums of unital completely positive maps it is clear that if Ω1 ⊂ A and Ω2 ⊂ B, then rcp(Ω1 ⊕ Ω2 , δ) ≤ rcpA (Ω1 , δ) + rcpB (Ω2 , δ). Since log(s + t) ≤ log 2 + max{log s, log t} for any positive numbers s, t, we conclude that ht(Ω1 ⊕ Ω2 , δ; α ⊕ β) ≤ max{ht(Ω1 , δ; α), ht(Ω2 , δ; β)}. If A ⊂ B(H) and B ⊂ B(K), then A ⊕ B ⊂ B(H ⊕ K). Now given θ: A ⊕ B → Matn (C) and γ: Matn (C) → B(H ⊕ K), ﬁx a state ϕ on A and ˜ A → Matn (C) deﬁned by θ(a) ˜ = θ(a⊕ϕ(a)1), and the composition consider θ: γ˜ : Matn (C) → B(H) of γ with the compression map B(H ⊕ K) → B(H). It ˜ δ), where Ω ˜ consists of the elements follows that rcpA (Ω, δ) ≤ rcpA⊕B (Ω, a ⊕ ϕ(a)1, a ∈ Ω. Thus ht(α) ≤ ht(α ⊕ β), and analogously ht(β) ≤ ht(α ⊕ β).

Recall that by Theorem 3.2.2(iv) the inequality for the dynamical entropy of a tensor product automorphism goes the opposite way. We next show two Kolmogorov-Sinai type results. We need the following simple lemma. Lemma 6.2.3. Let Ω ⊂ A be ﬁnite, k ∈ N, α ∈ Aut(A), and Ωk = ∪kj=−k αj (Ω). Then ht(Ωk , δ; α) = ht(Ω, δ; α). j −k (∪n+2k−1 αj (Ω)), by Lemma 6.1.4(vi) we have Proof. Since ∪n−1 j=0 α (Ωk ) = α j=0 n+2k−1 j j rcp(∪n−1 α (Ω), δ). j=0 α (Ωk ), δ) = rcp(∪j=0

Taking the logarithm, dividing by n and letting n → ∞, we get the result.

Theorem 6.2.4. Let A be a C∗ -algebra and α ∈ Aut(A). Suppose {Ωλ }λ∈Λ is an increasing net of ﬁnite subsets of A such that the linear span of ∪λ∈Λ,n∈Z αn (Ωλ ) is dense in A. Then ht(α) = lim ht(Ωλ ; α). λ

Proof. By Lemma 6.1.4(iii),(iv) in order to compute the entropy it suﬃces to consider ﬁnite subsets of ∪n∈Z ∪λ∈Λ αn (Ωλ ). Given such a subset Ω we can ﬁnd λ0 and n ∈ N such that Ω ⊂ ∪nk=−n αk (Ωλ0 ). Then by the previous lemma and Lemma 6.1.4(ii) ht(Ω, δ; α) ≤ ht(∪nk=−n αk (Ωλ0 ), δ; α) = ht(Ωλ0 , δ; α),

6.2 Topological Entropy

so that ht(Ω; α) ≤ ht(Ωλ ; α) for any λ λ0 .

99

If we consider sets consisting of unitary elements, we get a much more satisfactory result. Lemma 6.2.5. Assume Ω consists of unitary operators and contains the unit. Put Ω n = {u1 . . . un | ui ∈ Ω}. Then ht(Ω; α) = ht(Ω n ; α). Proof. By Lemma 6.1.4(v) we have ht(Ω; α) = ht(Ω 2 ; α). Thus by iteration k k ht(Ω; α) = ht(Ω 2 ; α) for all k ∈ N. Since Ω ⊂ Ω n ⊂ Ω 2 for suﬃciently large k the lemma follows from Lemma 6.1.4(ii).

Theorem 6.2.6. Let A be a unital C∗ -algebra and α ∈ Aut(A). Suppose {Ωλ }λ∈Λ is an increasing net of ﬁnite sets of unitary operators in A such that the ∗-algebra generated by ∪λ∈Λ,n∈Z αn (Ωλ ) is dense in A. Then ht(α) = lim ht(Ωλ ; α). λ

Proof. By Lemma 6.1.4(i) we may assume 1 ∈ Ωλ and Ωλ∗ = Ωλ for any λ ∈ Λ. Set Ωλ,n = (∪nk=−n αk (Ωλ ))n . Deﬁne an order on the set {(λ, n) | λ ∈ Λ, n ∈ N} by letting (λ1 , n1 ) (λ2 , n2 ) if λ1 λ2 and n1 ≥ n2 . Then {Ωλ,n }λ,n is an increasing net, and ∪λ,n Ωλ,n spans a dense subspace of A. Hence ht(α) = lim ht(Ωλ,n ; α). (λ,n)

Since ht(Ωλ,n ; α) = ht(Ωλ ; α) by Lemmas 6.2.3 and 6.2.5, the proof of the theorem is complete.

In the abelian case the entropy of a homeomorphism with respect to an invariant probability measure is majorized by the topological entropy of the homeomorphism. We now show a noncommutative analogue of this result. Proposition 6.2.7. Let (A, ϕ, α) be a C∗ -dynamical system with A exact. Then hϕ (α) ≤ ht(α). Proof. Assume A ⊂ B(H). Let N be a ﬁnite dimensional C∗ -algebra and γ: N → A be a channel. Fix δ > 0 and choose a ﬁnite subset ω of N such that its convex hull contains the unit ball. Put Ω = γ(ω). Let n ∈ N, and θ: A → Matm (C) and ρ: Matm (C) → B(H) be unital completely positive maps such that (ρ ◦ θ)(a) − a < δ for a ∈ αk (Ω), 0 ≤ k ≤ n − 1. Then ρ ◦ θ ◦ αk ◦ γ − αk ◦ γ < δ for 0 ≤ k ≤ n − 1. Let ϕ = i1 ,...,in ϕi1 ...in be a ﬁnite decomposition. Extend each ϕi1 ...in to a positive linear functional ψi1 ...in on B(H), and set ψ = i1 ,...,in ψi1 ...in . We obviously have

100

6 Topological Entropy

Hϕ (γ, . . . , αn−1 ◦ γ; {ϕi1 ...in }) = Hψ (γ, . . . , αn−1 ◦ γ; {ψi1 ...in }). On the other hand, by Proposition 3.1.11 |Hψ (γ, . . . , αn−1 ◦ γ) − Hψ (ρ ◦ θ ◦ γ, . . . , ρ ◦ θ ◦ αn−1 ◦ γ)| < nε for some ε depending only on δ and dim N . Since Hψ (ρ ◦ θ ◦ γ, . . . , ρ ◦ θ ◦ αn−1 ◦ γ) ≤ S(ψ ◦ ρ) ≤ log m by Proposition 3.1.3(i),(iv) and Lemma 3.1.4, we thus get Hϕ (γ, . . . , αn−1 ◦ γ; {ϕi1 ...in }) < log m + nε. It follows that hϕ (γ; α) ≤ ht(Ω, δ; α) + ε ≤ ht(α) + ε. Since ε = ε(δ, dim N ) can be made arbitrarily small, we obtain the desired inequality.

Remark 6.2.8. For the entropy hST introduced in Chap. 5, which always majorizes the dynamical entropy and coincides with it in the nuclear case by Theorem 5.1.5, we also have hST ϕ (α) ≤ ht(α). Indeed, assume A ⊂ B(H). We may also assume that α extends to an automorphism α ¯ of B(H) (represent A α Z faithfully and take α ¯ = Ad u, where u is the canonical unitary in the crossed product). Let λ be a stationary coupling of (A, ϕ, α) with (X, µ, T ). ¯ on B(H)⊗C(X), where β(f ) = f ◦T −1 Extend λ to an (¯ α⊗β)-invariant state λ ¯ for f ∈ C(X). Set ϕ¯ = λ|B(H) . By deﬁnition of hST we have to prove that S(λ(· ⊗ 1Y ), ϕ) ≤ ht(α) hµ (ξ; T ) + Y ∈ξ

for any ﬁnite measurable partition ξ of X. Fix ε > 0. Since relative entropy is weakly∗ lower semicontinuous by Theorem 2.3.1(iv), there exist δ > 0 and a ﬁnite subset Ω ⊂ A such that for Y ∈ ξ S(λ(· ⊗ 1Y ), ϕ) < S(ω, ψ) +

ε |ξ|

as soon as |ω(a) − λ(a ⊗ 1Y )| < δ and |ψ(a) − ϕ(a)| < δ for a ∈ Ω. Let n ∈ N, and θ: A → Matm (C) and γ: Matm (C) → B(H) be unital completely positive maps such that (γ ◦ θ)(a) − a < δ for a ∈ αk (Ω), 0 ≤ k ≤ n − 1. Then using monotonicity of relative entropy, Theorem 2.3.1(vi), we get (see (3.9)) k Hλ¯ (γ, . . . , γ; ξ, T ξ, . . . , T n−1 ξ) − Hµ (∨n−1 k=0 T ξ)

6.2 Topological Entropy

=

n−1

101

¯ S(λ(γ(·) ⊗ 1T k Y ), ϕ¯ ◦ γ)

k=0 Y ∈ξ

=

n−1

¯ α−k ◦ γ)(·) ⊗ 1Y ), ϕ¯ ◦ γ) S(λ((¯

k=0 Y ∈ξ

≥

n−1

¯ α−k ◦ γ ◦ θ ◦ αk )(·) ⊗ 1Y ), ϕ¯ ◦ γ ◦ θ ◦ αk ) S(λ((¯

k=0 Y ∈ξ

>n

S(λ(· ⊗ 1Y ), ϕ) − nε.

Y ∈ξ

On the other hand, Hλ¯ (γ, . . . , γ; ξ, . . . , T n−1 ξ) ≤ Hϕ¯ (γ) ≤ S(ϕ¯ ◦ γ) ≤ log m. It follows that n−1 k k log rcp(∪n−1 k=0 α (Ω), δ) > Hµ (∨k=0 T ξ) + n

S(λ(· ⊗ 1Y ), ϕ) − nε.

Y ∈ξ

Dividing by n and letting n → ∞ we get the desired conclusion, as ε can be made arbitrarily small. We can now prove that our deﬁnition of topological entropy reduces to the classical one in the abelian case. Proposition 6.2.9. Let T : X → X be a homeomorphism of a compact metric space X, and let α be the automorphism of C(X) induced by T via α(f ) = f ◦ T −1 . Then ht(α) = htop (T ). Proof. Let Ω be a ﬁnite subset of C(X), δ > 0, and let U be an open cover of X such that |f (x) − f (y)| < δ for any f ∈ Ω and any x, y ∈ X lying in the j same element of U. Let n ∈ N. Then if f ∈ ∪n−1 j=0 α (Ω) and x, y belong to the n−1 j same element of Un = ∨j=0 T U we have |f (x) − f (y)| < δ. By Lemma 6.1.3 we get j rcp(∪n−1 j=0 α (Ω), δ) ≤ N (Un ), whence ht(Ω, δ; α) ≤ h(U; T ). It follows that ht(α) ≤ htop (T ). The opposite inequality is more involved. By Proposition 6.2.7, we have ht(α) ≥ hϕ (α) for every α-invariant state ϕ on C(X). In other words, ht(α) ≥ hµ (T ) for every T -invariant probability measure µ. Thus by the variational principle, Theorem 1.2.2, we have htop (T ) = sup hµ (T ) ≤ ht(α), µ

and the proof is complete.

We consider next two simple examples of computations of topological entropy.

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6 Topological Entropy

Example 6.2.10. (i) Let α be an automorphism of A = K(H) + C1 ⊂ B(H), where K(H) is the algebra of compact operators on a Hilbert space H. Then ht(α) = 0. It suﬃces to prove that ht(Ω; α) = 0 for any ﬁnite Ω ⊂ pB(H)p, where k p is a ﬁnite rank projection. Let pn = ∨n−1 k=0 α (p). Then pn is a ﬁnite rank n−1 k projection such that ∪k=0 α (Ω) ⊂ pn B(H)pn . So if we put Bn = pn B(H)pn + C(1 − pn ), and take θ: A → Bn to be a conditional expectation (e.g. θ(a) = pn apn +ϕ(a)(1−pn ), where ϕ is a state with support ≤ 1−pn ) and γ: Bn → A to be the embedding map, then for any δ > 0 we get k rcp(∪n−1 k=0 α (Ω), δ) ≤ rank Bn = Tr(pn ) + 1 ≤ nTr(p) + 1.

Hence ht(Ω, δ; α) = 0. (ii) Let B be a ﬁnite dimensional C∗ -algebra, A = B ⊗Z the inﬁnite C∗ -tensor product, α the shift automorphism of A. Then ht(α) = log rank B. The inequality ≤ follows easily from rank B ⊗n = (rank B)n . On the other hand, if τ0 is the normalized canonical trace on B and τ = τ0⊗Z , then by Example 3.2.6(i) we have hτ (α) = S(τ0 ) = log rank B. Since hτ (α) ≤ ht(α), this gives the opposite inequality. We ﬁnish the chapter with some technical remarks. Remark 6.2.11. (i) Even when dealing with unital algebras it is sometimes convenient to consider nonunital maps. Deﬁne rcp (Ω, δ) exactly as rcp(Ω, δ) but by requiring θ and γ to be contractive completely positive maps instead of unital ones (whether we require π: A → B to be unital or not is not important as we can always replace B by π(1)Bπ(1)), and then deﬁne ht (α). We obviously have rcp (Ω, δ) ≤ rcp(Ω, δ). But if 1 ∈ Ω, so x = (γ ◦ θ)(1) is close to 1, we can consider the completely positive map γ˜ = x−1/2 γ(·)x−1/2 , which is close to γ. Then by A.12 we can modify θ and γ˜ to unital completely positive maps keeping γ˜ ◦ θ unchanged. This shows that if 1 ∈ Ω, then ht (Ω; α) = ht(Ω; α), so that ht (α) = ht(α). (ii) Let A be a nonunital C∗ -algebra, α an automorphism of A. One way to deﬁne the topological entropy of α is to consider the algebra A∼ obtained by adjoining unit to A, extend α to an automorphism α ˜ of A∼ , and deﬁne ht(α) = ht(˜ α). Another possibility is to deﬁne ht(α) using rcp from the previous remark, that is, by considering contractive completely positive maps. The result will be the same. Indeed, by A.4 any contractive completely positive maps A → Matn (C) and Matn (C) → B extend to unital completely positive maps A∼ → Matn (C)∼ ∼ = Matn (C) ⊕ C and Matn (C)∼ → B ∼ . It is then easy to check that all the main properties of the topological entropy remain true for nonunital C∗ -algebras. (iii) Lemma 6.1.4(vii) shows that rcpA (Ω, δ) depends only on the position

6.3 Notes

103

of Ω in the C∗ -algebra generated by Ω. Using Arveson’s extension theorem for operator systems we can conclude that it actually depends only on the position of Ω in the operator system spanned by Ω, Ω ∗ and 1. More precisely, recall that an operator system is a closed self-adjoint subspace X of a unital C∗ -algebra containing the unit of the algebra. The notion of a completely positive map makes sense for operator systems. Then a unital completely positive map θ: X → Y is called a complete order embedding if it is injective and the inverse map θ(X) → X is completely positive. Equivalently, θ is completely isometric, that is, the map θ ⊗ id: X ⊗ Matn (C) → Y ⊗ Matn (C) is isometric for all n ∈ N. Let Ω be a ﬁnite subset of an operator system X. Deﬁne rcpX (Ω, δ) by considering approximate factorizations of complete order embeddings of X into C∗ -algebras through ﬁnite dimensional C∗ -algebras. Then if θ: X → A is a complete order embedding, we have rcpA (θ(Ω), δ) = rcpX (Ω, δ). In particular, we get the following strengthening of Theorem 6.2.2(iii). If θ: B → A is a complete order embedding of B into A, and α ◦ θ = θ ◦ β, then ht(β) ≤ ht(α).

6.3 Notes The notion of topological entropy presented here was introduced by Voiculescu [227] for nuclear C∗ -algebras and extended by Brown to exact algebras [32] (we refer the reader to the book by Wassermann [232] for more information on exact C∗ -algebras). Most of the results in this chapter are contained in those papers. An important exception is property (v) in Lemma 6.1.4 implying Theorem 6.2.6, which was observed by Ozawa [153]. As we proved in Chap. 3, inner automorphisms of type I algebras have zero dynamical entropy. This fact and Example 6.2.10(i) suggest that the topological entropy of inner automorphisms of type I C∗ -algebras is zero. This is, however, known only in a few cases, e.g. for homogeneous C∗ -algebras [33]. As we noted in Remark 6.2.11(i), instead of unital completely positive maps we can deal with contractive completely positive maps. Pop and Smith [167] showed that we can equally well consider completely contractive maps, that is, maps θ: A → B such that θ ⊗ id: A ⊗ Matn (C) → B ⊗ Matn (C) is contractive for any n ∈ N. The reason is brieﬂy that if we have a factorization of an algebra through a matrix algebra via completely contractive maps such that their composition almost preserves the unit, then we can construct a contractive completely positive factorization through the same matrix algebra [204]. It is, however, unknown whether it is possible instead to use completely bounded maps with a universal bound on the norms. Another open problem is how topological entropy behaves in quotients. In other words, if α is an automorphism of A leaving a two-sided ideal I ⊂ A invariant, and α ¯ is the automorphism of B = A/I deﬁned by α, the question is

104

6 Topological Entropy

whether the inequality ht(¯ α) ≤ ht(α) holds. If there exists a unital completely positive splitting σ: B → A such that α ◦ σ = σ ◦ α ¯ , then this is true by Remark 6.2.11(iii). It is shown in [36] that if we in addition assume that there exists an approximate unit {ei }i in I such that α(ei ) = ei for any i, then moreover ht(α) = max{ht(¯ α), ht(α|I )}. As in the case of dynamical entropy, the deﬁnition of topological entropy used in this chapter is not the only one possible. Similarly to the LindbladAlicki-Fannes entropy discussed in Notes to Chap. 3, there is an approach based on partitions of unity due to Hudetz [93], [94], [14]. Let us say that a ﬁnite set X = {xi }i of operators is an operator cover if i x∗i xi is invertible. ¯ (X) the cardinality of a minimal operator subcover of X. Given Denote by N two operator covers X = {xi }i and Y = {yj } we can form a cover X ∨ Y consisting of the elements yj xi (in Chap. 3 we denoted this operation by Y ◦ X). One of the unpleasant features of this notion is that if we consider ¯ (X (n) ) can grow exponentially fast as X (n) = X ∨ . . . ∨ X (n times) then N n → ∞. Still it is possible to manufacture a notion of entropy using it, see the above mentioned papers for details. Thomsen [215] proposed a diﬀerent operation of joining operator covers. He requires operator covers to be in addition self-adjoint sets and calls them partitions. Given n partitions X1 , . . . , Xn he deﬁnes X1 ∨ . . . ∨ Xn as the partition consisting of the elements xσ(1) . . . xσ(n) , where xi ∈ Xi and σ ∈ Sn . As was the case for the Lindblad-Alicki-Fannes entropy, in either approach one cannot consider all possible operator covers or partitions in a C∗ -algebra, since this leads to an essentially uncomputable invariant. So Hudetz introduces a notion of appropriately approximated C∗ -algebras, while Thomsen works with local C∗ -algebras. There are a number of dynamical invariants whose deﬁnitions follow the same pattern as that of topological entropy, the main diﬀerence being the way a ﬁnite set Ω is approximated and what kind of a rank-type function is used, see e.g. [44], [110], [109]. We would like to mention one of them, which was introduced by Voiculescu [227]. Let (A, ϕ, α) be a C∗ -dynamical system. For a ﬁnite Ω ⊂ A and δ > 0 denote by rcpϕ (Ω, δ) the minimal n ∈ N for which there exist unital completely positive maps θ: A → Matn (C) and γ: Matn (C) → πϕ (A) such that ϕ ◦ γ ◦ θ = ϕ and (γ ◦ θ)(a) − πϕ (a)ϕ < δ for any a ∈ Ω, where xϕ = ϕ(x∗ x)1/2 . Then we deﬁne the completely positive approximation entropy hcpaϕ (α) just as we deﬁned the topological entropy ht(α). Most of the results on ht(α) can be proved in roughly the same way for hcpaϕ (α). In particular, if (B, ψ, β) is another C∗ -dynamical system, then hcpaϕ⊗ψ (α ⊗ β) ≤ hcpaϕ (α) + hcpaψ (β). Since we have the opposite inequality for the entropy hϕ (α), we get the equality hϕ⊗ψ (α ⊗ β) = hϕ (α) + hψ (β)

6.3 Notes

105

whenever hcpaϕ (α) = hϕ (α) and hcpaψ (β) = hψ (β). If (X, T ) is a topological dynamical system, µ a T -invariant probability measure on X, α(f ) = f ◦ T −1 and ϕ(f ) = f dµ for f ∈ C(X), then hµ (T ) ≤ hcpaϕ (α) ≤ htop (T ). If the system is ergodic, then hµ (T ) = hcpaϕ (α). This can be deduced either from the above inequalities and the Jewett-Krieger theorem, Theorem 1.2.3, or using the Shannon-McMillan-Breiman theorem, Theorem 1.2.4. Without the ergodicity assumption the equality hµ (T ) = hcpaϕ (α) is not always true. To see this, observe that if (A, ϕ, α) = (A1 ⊕A2 , λϕ1 ⊕(1−λ)ϕ2 , α1 ⊕α2 ), 0 < λ < 1, then hcpaϕ (α) = maxi hcpaϕi (αi ), while hϕ (α) = λhϕ1 (α1 )+(1−λ)hϕ2 (α2 ). To get a similar invariant which coincides with hϕ (α) for arbitrary abelian systems, we should have used exp(S(ϕ ◦ γ)) in place of n in the deﬁnition of rcpϕ . Voiculescu’s motivation for introducing completely positive approximation entropy hcpaϕ (α) was that the equality hϕ (α) = hcpaϕ (α), when it holds, should be thought of as a weak form of the Shannon-McMillan-Breiman theorem. For a result in this direction see [142]. The idea of using some sort of approximation of ﬁnite sets of an algebra to deﬁne entropy was apparently ﬁrst used in the work of Connes and Størmer [52]. Their deﬁnition is as follows. Let M ⊂ B(H) be a von Neumann algebra, ξ a cyclic and separating tracial vector, and τ = ωξ the trace on M . For δ > 0 and a ﬁnite set Ω of unitaries in M deﬁne hτδ (Ω) = inf{S(ϕ) | ϕ ∈ B(H)∗ , ϕ|M = τ, S(ϕ, ϕ ◦ Ad u) < δ for all u ∈ Ω}. Then deﬁne the entropy of a τ -preserving automorphism α of M by the same procedure as before: take lim sup as n → ∞ of the quantities 1 τ h (Ω ∪ α(Ω) ∪ . . . ∪ αn−1 (Ω)), n δ and then supremum over δ > 0 and ﬁnite sets of unitaries Ω ⊂ M . The only fact which is known about this entropy is that it coincides with the Kolmogorov-Sinai entropy for abelian M . Voiculescu himself introduced another type of approximation entropy, called the perturbation-theoretic entropy [223], [224], [225]. To deﬁne − it recall that the Macaev ideal C∞ (H) is the space of compact operators T on H such that ∞ sn (T ) T − < ∞, ∞ = n n=1 where s1 (T ) ≥ s2 (T ) ≥ . . . are the eigenvalues of |T |. For a ﬁnite set Ω = {x1 , . . . , xn } of operators on H put − (Ω) = lim inf max [a, xi ]− k∞ ∞, a

1≤i≤n

106

6 Topological Entropy

where the lim inf is taken over the net of ﬁnite rank operators a, 0 ≤ a ≤ 1, with its natural order. Now if (A, ϕ, α) is a C∗ -dynamical system, and U is the canonical unitary implementing α in the GNS-representation, we deﬁne − HP (U, πϕ (A)) = sup k∞ (Ω ∪ {U }),

where the supremum is taken over all ﬁnite subsets Ω of πϕ (A) generating a ﬁnite dimensional C∗ -algebra. It is shown in [223] that if (X, µ, T ) is an ergodic dynamical system, and UT is the unitary on L2 (X, µ) implementing T , then 1 h(T ) ≤ HP (UT , L∞ (X, µ)) ≤ 6h(T ). 2 It is also known that for noncommutative systems this entropy is quite different from the dynamical entropy considered in Chap. 3. E.g. the dynamical entropy of a free shift to be considered in Chap. 14 is zero, while the perturbation-theoretic entropy is inﬁnite [223]. It is interesting to note [224] that the perturbation-theoretic entropy allows us to deﬁne the entropy of an action of a ﬁnitely generated group G with a ﬁxed set S of generators on a von Neumann algebra M . We do not have to assume that G is amenable or that the action preserves a state. Namely, if we consider M in its standard form, M ⊂ B(H), then any automorphism of M is implemented by a canonical unitary. If us , s ∈ S, are the unitaries corresponding to the generators of the group, we can then deﬁne HP (S, M ) = HP ({us }s , M ). See [148] for an estimate of this entropy for the action of a free group on its boundary, which may suggest that perturbation-theoretic entropy is related to entropy of pseudogroups [71], [229].

7 Dynamics on the State Space

Throughout this chapter we ﬁx a unital separable C∗ -algebra A and an automorphism α of A. Denote by X = S(A) the state space of A, and by T the homeomorphism of X deﬁned by T ϕ = ϕ ◦ α−1 . It would be too simple (and actually quite disappointing) if the entropy of α could be computed by looking at (X, T ). Nevertheless it is a natural problem to investigate relations between entropic properties of (A, α) and (X, T ). We shall see that either both systems have positive entropy or both have zero entropy.

7.1 Measure Entropy We ﬁrst consider measure-theoretic entropy. Theorem 7.1.1. Let notation be as above and let ϕ be an α-invariant state on A. Assume hµ (T ) = 0 for any T -invariant probability measure µ on X with barycenter ϕ. Then hϕ (α) = 0. In particular, if htop (T ) = 0 then hϕ (α) = 0 for every α-invariant state ϕ. This is a consequence of the inequality hϕ (α) ≤ hST ϕ (α) from Theorem 5.1.5 and the following result of independent interest, which in particular implies that hST ϕ (α) is not larger than the supremum of hµ (T ) over all T -invariant probability measures µ on X with barycenter ϕ. Proposition 7.1.2. We have − − hST ϕ (α) = sup{Hµ (ξ|ξ ) − Hλ,A (ξ|ξ )},

where the supremum is taken over all stationary canonical couplings λ of (A, ϕ, α) with (X, µ, T ) and over all ﬁnite measurable partitions ξ of (X, µ). For the proof we need the following Pinsker formula, which is a reﬁnement of the inequality from Lemma 1.1.3.

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7 Dynamics on the State Space

Lemma 7.1.3. If (Y, ν, S) is a dynamical system, then for any two ﬁnite measurable partitions ξ and ζ of (Y, ν) we have k h(ξ ∨ ζ; S) = h(ζ; S) + H(ξ|ξ − ∨ ζ ± ), where ζ ± = ∨∞ −∞ S ζ.

Proof. Observe ﬁrst that by a computation similar to (1.3) we have H(ξ|ξ − ∨ ζ ± ) = lim

n→∞

1 k ± H(∨n−1 k=0 S ξ|ζ ). n

(7.1)

n−1 k k Denote the partitions ∨n−1 k=0 S ξ and ∨k=0 S ζ by ξn and ζn , respectively. By Lemma 1.1.3,

h(ξn ∨ ζn ; S n ) ≤ h(ζn ; S n ) + H(ξn |ζ ± ). Since h(ξn ∨ ζn ; S n ) = nh(ξ ∨ ζ; S) and h(ζn ; S n ) = nh(ζ; S), dividing the above inequality by n, letting n → ∞ and using (7.1) we get the inequality ≤ in the lemma. On the other hand, H(ξn ∨ ζn ) = H(ζn ) + H(ξn |ζn ) ≥ H(ζn ) + H(ξn |ζ ± ). Dividing by n, letting n → ∞ and using (7.1) again, we get the inequality ≥ in the lemma.

Proof of Proposition 7.1.2. Let Λ be a stationary coupling of (A, ϕ, α) with (Y, ν, S). We can desintegrate the state Λ over (Y, ν). That is, there exists a measurable map Y y → Λy ∈ X such that Λy (a)f (y) dν(y) for a ∈ A, f ∈ L∞ (Y, ν). Λ(a ⊗ f ) = Y

Existence of such a map can be deduced from C.3-C.5, by ﬁrst decomposing the space of the GNS-representation deﬁned by Λ into a direct integral of Hilbert spaces over (Y, ν), and then decomposing the representation of A and a normal extension of Λ to the algebra of decomposable operators into corresponding direct integrals (we used a similar argument in the proof of Theorem 5.2.1). Let µ be the image of ν under the map y → Λy . Since Λ is invariant, we have ΛSy = Λy ◦ α−1 for a.e. y ∈ Y . Since ν is S-invariant, it follows that µ is T -invariant. Hence µ deﬁnes a stationary canonical coupling λ, λ(a ⊗ f ) = ψ(a)f (ψ)dµ(ψ) = Λy (a)f (Λy ) dν(y) for a ∈ A, f ∈ C(X). X ∞

Y

Identify L (X, µ) with a subalgebra of L∞ (Y, ν) using the map y → Λy . Let εX be the measurable partition of Y such that L∞ (X) = L∞ (Y /εX ). Then λ is just the restriction of Λ to A ⊗ L∞ (Y /εX ). By Proposition 5.1.4 it suﬃces to show that λ is as good as Λ for computing the entropy, that is, the supremum

7.1 Measure Entropy

109

sup{Hν (ξ|ξ − ) − HΛ,A (ξ|ξ − )} over ﬁnite partitions ξ of Y will not change if we only consider ξ ≺ εX . Let ξ be a ﬁnite partition of Y . Since by (5.3) Hν (ξ|ξ − ) − HΛ,A (ξ|ξ − ) = lim

n→∞

1 k HΛ (A; ∨n−1 k=0 S ξ), n

by virtue of Proposition 5.1.3(i), we see that the expression Hν (ξ|ξ − ) − HΛ,A (ξ|ξ − ) is increasing in ξ, so that Hν (ξ|ξ − ) − HΛ,A (ξ|ξ − ) ≤ Hν (ξ ∨ ϑ|ξ − ∨ ϑ− ) − HΛ,A (ξ ∨ ϑ|ξ − ∨ ϑ− ) for any ﬁnite partition ϑ ≺ εX . By the Pinsker formula, Lemma 7.1.3, Hν (ξ ∨ ϑ|ξ − ∨ ϑ− ) = Hν (ϑ|ϑ− ) + Hν (ξ|ξ − ∨ ϑ± ), k where ϑ± = ∨∞ −∞ S ϑ. Using Proposition 5.1.3(iv),(vi) we similarly get

HΛ,A (ξ ∨ ϑ|ξ − ∨ ϑ− ) = HΛ,A (ϑ|ϑ− ) + HΛ,A (ξ|ξ − ∨ ϑ± ). Therefore Hν (ξ|ξ − ) − HΛ,A (ξ|ξ − ) ≤ Hν (ϑ|ϑ− ) − HΛ,A (ϑ|ϑ− ) +Hν (ξ|ξ − ∨ ϑ± ) − HΛ,A (ξ|ξ − ∨ ϑ± ). Thus it suﬃces to show that Hν (ξ|ξ − ∨ ϑ± ) − HΛ,A (ξ|ξ − ∨ ϑ± ) can be made arbitrarily small for some ϑ ≺ εX . If {ϑn }n is an increasing sequence of ﬁnite partitions such that ∨n ϑn = εX , then − − ± − Hν (ξ|ξ − ∨ ϑ± n ) Hν (ξ|ξ ∨ εX ) and HΛ,A (ξ|ξ ∨ ϑn ) HΛ,A (ξ|ξ ∨ εX )

by Proposition 5.1.3(iv). Hence it is enough to prove that Hν (ξ|ζ) = HΛ,A (ξ|ζ) for any ζ εX . For this, in turn, it is enough to check that S(ν(·1Z )|L∞ (Y /ζ) , ν|L∞ (Y /ζ) ) = S(Λ(· ⊗ ·1Z )|A⊗L∞ (Y /ζ) , Λ|A⊗L∞ (Y /ζ) ) for any measurable Z ⊂ Y . This is indeed the case by Corollary 2.3.3, since the map E: A ⊗ L∞ (Y /ζ) → L∞ (Y /ζ), E(a ⊗ f )(y) = Λy (a)f (y), is a conditional expectation which preserves any functional of the form a ⊗ f → Λ(a ⊗ f 1Z ) = Λy (a)f (y) dν(y), Z ⊂ Y, Z

on A ⊗ L∞ (Y /ζ).

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7 Dynamics on the State Space

7.2 Topological Entropy It is natural to conjecture that if hST ϕ (α) = 0 for any α-invariant state ϕ then htop (T ) = 0. We do not know whether this is true. We have however the following result. Theorem 7.2.1. If ht(α) = 0 then htop (T ) = 0. The proof is based on the following combinatorial statement. Proposition 7.2.2. For given ε > 0 and a > 0, there exist n0 ∈ N and b > 0 such that if n ≥ n0 and Φ: Matm (C)∗ → n∞ is a linear contractive map such that the image of the unit ball contains at least ean real elements that are ε-separated, then m ≥ ebn . Recall that we say that a subset of a metric space is ε-separated if the distance between any two diﬀerent elements of the set is at least ε. We shall postpone the proof of Proposition 7.2.2 and ﬁrst use it to deduce Theorem 7.2.1. Proof of Theorem 7.2.1. Assume A ⊂ B(H). Choose a relatively compact subset Ω0 of the unit ball of A which consists of self-adjoint elements and spans a dense subspace of A (e.g. take a sequence which converges to zero and spans a dense subspace of A). Then the weak∗ topology on X is deﬁned by the metric d(ϕ, ψ) = sup |ϕ(y) − ψ(y)|. y∈Ω0

Assume htop (T ) > a0 > 0. Recalling the deﬁnition of the topological entropy in terms of separated sets from Sect. 1.2, this means that for any suﬃciently small ε > 0 and for all n large enough there exist at least ea0 n points of X that are (n, 4ε)-separated with respect to T −1 , that is, they are 4ε-separated in the metric dn (ϕ, ψ) =

max

0≤k≤n−1

d(T −k ϕ, T −k ψ) =

max

0≤k≤n−1

d(ϕ ◦ αk , ψ ◦ αk ).

Choose a ﬁnite subset Ω ⊂ Ω0 such that d(ϕ, ψ) < max |ϕ(y) − ψ(y)| + ε for ϕ, ψ ∈ X. y∈Ω

Given n ∈ N and a unital completely positive map σ: A → Matm (C), deﬁne the map k Φ: Matm (C)∗ → n|Ω| ∞ , Φ(f ) = {(f ◦ σ ◦ α )(y)}0≤k≤n−1,y∈Ω ,

where we ﬁx a certain order on the set {0, . . . , n − 1} × Ω. Assume there exists a unital completely positive map γ: Matm (C) → B(H) such that (γ ◦ σ ◦ αk )(y) − αk (y) < ε for 0 ≤ k ≤ n − 1, y ∈ Ω.

7.2 Topological Entropy

111

Extend each state ϕ on A to a state ϕ˜ on B(H). Then |(ϕ˜ ◦ γ ◦ σ ◦ αk )(y) − (ϕ ◦ αk )(y)| < ε. If two states ϕ and ψ are (n, 4ε)-separated with respect to T −1 , there exists k, 0 ≤ k ≤ n − 1, such that d(ϕ ◦ αk , ψ ◦ αk ) ≥ 4ε. Hence there exists y ∈ Ω such that |(ϕ ◦ αk )(y) − (ψ ◦ αk )(y)| > 3ε, whence |(ϕ˜ ◦ γ ◦ σ ◦ αk )(y) − (ψ˜ ◦ γ ◦ σ ◦ αk )(y)| > ε, ˜ so that Φ(ϕ◦γ) ˜ and Φ(ψ◦γ) are ε-separated. Thus the image of the unit ball of Matm (C)∗ under Φ contains at least ea0 n real elements that are ε-separated. Hence by Proposition 7.2.2 applied to a = a0 /|Ω| we can ﬁnd b > 0 such that m ≥ ebn if n is suﬃciently large. In other words, rcp(Ω ∪ α(Ω) ∪ . . . αn−1 (Ω), ε) ≥ ebn , so that ht(α) ≥ b > 0.

In order to prove Proposition 7.2.2 we shall compare the type 2 constants of the dual spaces. Recall that for a Banach space X the type 2 constant T2 (X ) is deﬁned as the inﬁmum of the positive numbers T such that for any N ∈ N and x1 , . . . , xN ∈ X we have N 2 N 1 2 ε x ≤ T xk 2 . (7.2) k k 2N ε1 ,...,εN =±1

k=1

k=1

Lemma 7.2.3. We have: √ (i) T2 (n1 ) ≥ n; √ (ii) there is C > 0 such that T2 (Matn (C)) ≤ C log n (for n ≥ 2). Proof. To prove the ﬁrst inequality it suﬃces to take N = n and the canonical basis of n1 for xk ’s. To prove the second inequality consider the Schatten p-class Sp (n), that is, the space Matn (C) with the norm xp = Tr(|x|p )1/p . We claim that there exists a constant C such that √ (7.3) T2 (S2m (n)) ≤ C m for any m, n ∈ N. Assuming this and using that x ≤ x2m ≤ n1/2m x, we get from (7.2) √ T2 (Matn (C)) ≤ C n1/2m m. Applying this to m = [log n], the integer part of log n, we obtain the second inequality of the lemma.

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7 Dynamics on the State Space

It remains to prove (7.3). We shall prove that T2 (S2m (n)) ≤

(2m)! 2m m!

1/2m ,

√ which implies (7.3) by the Stirling formula m! ∼ (m/e)m 2πm. Consider the measure with weights ( 12 , 12 ) on the two-point set {−1, 1}. Consider then the space Y = {−1, 1}N with the product measure µ, and let rk : Y → {±1} be the projection onto the k-th coordinate. Then, by H¨ older’s inequality, 1 2N

ε1 ,...,εN =±1

N 2 εk xk

2m

k=1

2 N = rk (t)xk dµ(t) Y k=1 2m ⎞1/m ⎛ 2m N rk (t)xk dµ(t)⎠ . ≤⎝ Y 2m

k=1

∗ m Since x2m 2m = Tr((x x) ), the m-th power of the right hand side is equal to N k1 ,...,k2m =1

Y

rk1 (t) . . . rk2m (t) dµ(t)Tr(x∗k1 xk2 . . . x∗k2m−1 xk2m ).

Since the random variables r1 , . . . , rN are independent, and one depending on whether l is odd or even, the integral rk1 (t) . . . rk2m (t) dµ(t)

Y

rkl dµ is zero or

Y

is zero unless each k, 1 ≤ k ≤ N , appears in the sequence k 1 , . . . , k2m an even number of times, and then the integral is 1. Denoting by the summation over the indices satisfying the latter property, we thus have to show that m (2m)! 2 ∗ ∗ xk 2m . (7.4) Tr(xk1 xk2 . . . xk2m−1 xk2m ) ≤ m 2 m! k

k1 ,...,k2m

Apply the generalized Horn inequality Tr(|y1 . . . yp |) ≤

n

sj (y1 ) . . . sj (yp )

j=1

(see e.g. [198, Theorem 1.15] for p = 2, the same proof works in general), where sj (yk ) is the j-th s-number of yk ∈ Matn (C), i.e., s1 (yk ) ≥ . . . ≥ sn (yk ) are the eigenvalues of |yk |. Using that sj (xk ) = sj (x∗k ), we then see that the left hand side of (7.4) is not larger than

7.2 Topological Entropy

j

k1 ,...,k2m

113

sj (xk1 ) . . . sj (xk2m ).

Since the number of multi-indices (k1 , . . . , k2m ) containing each k, 1 ≤ k ≤ N , 2mk times is equal to (2m)!((2m1 )! . . . (2mN )!)−1 , the above expression equals

j

m1 ,...,mN m1 +...+mN =m

(2m)! sj (x1 )2m1 . . . sj (xN )2mN . (2m1 )! . . . (2mN )!

By virtue of the inequality (2mk )! ≥ 2mk mk !, the last expression is not larger than (2m)! m! sj (x1 )2m1 . . . sj (xN )2mN m 2 m! j m ! . . . m ! 1 N m1 ,...,mN m1 +...+mN =m

m (2m)! = m sj (xk )2 2 m! j k 1/m m (2m)! 2m ≤ m sj (xk ) 2 m! j k

(by the triangle inequality for N m) m (2m)! xk 22m , = m 2 m! k

which establishes (7.4) and thus completes the proof of the lemma.

We shall also need the following form of the combinatorial Sauer-Shelah lemma. For a set I we denote by 2I the set of all subsets of I. If J ⊂ I and A ⊂ 2I , we denote by A(J) the subset of 2J consisting of all sets of the form H ∩ J, H ∈ A. Lemma 7.2.4. For every c > 0 there exist d > 0 and n0 ∈ N such that if I is a ﬁnite set, |I| = n ≥ n0 , and A ⊂ 2I satisﬁes |A| ≥ ecn , then there exists J ⊂ I such that |J| ≥ dn and A(J) = 2J . Proof. We shall prove the following more precise result. If m ≤ n is such that |A| >

m−1 k=0

n , k

then there exists J such that |J| = m and A(J) = 2J . To show that this is indeed enough, it suﬃces to check that there exists d > 0 such that the than ecn for suﬃciently large n and m = "above # sum " n # is " msmaller # " [dn] # + 1. Since n m n k ≤ m k for k ≤ m, the above sum is smaller than 2 m , and by the

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7 Dynamics on the State Space

Stirling formula we get 2dn+1

n [dn]+1

≤ 2c(d)n for suﬃciently large n, with

c(d) → 0 as d → 0. So we just have to choose d such that c(d) ≤ c. The proof of the above statement is by induction on n. Since the result is obvious for m = 1 and m = n, assume 1 < m < n. Fix any point p ∈ I, and set Ip = I\{p}, Ap = A(Ip ). m−1 " n−1 # If |Ap | > k=0 , the result is true by induction. So assume k |Ap | ≤

m−1 k=0

n−1 k

.

Deﬁne B = {H ⊂ Ip | H ∈ A, H ∪ {p} ∈ A}. Then every element in B has two preimages in A under the map A H → H ∩ Ip ∈ Ap , while every element in Ap \B has one preimage. Therefore |A| = |Ap \B| + 2|B| = |Ap | + |B|. Using " n # " n−1 # n−1 that k − k = k−1 for k ≥ 1, we then get |B| = |A| − |Ap | >

m−1 k=0

m−2 m−1 n − 1 n n − 1 = . − k k k k=0

k=0

By induction there exists Jp ⊂ Ip such that |Jp | = m−1 and B(Jp ) = 2Jp . Put J = Jp ∪ {p}. Then by deﬁnition of B, for any H ⊂ Jp there exists H ⊂ Ip such that H ∩ Jp = H and both H and H ∪ {p} belong to A. This shows that A(J) = 2J .

This lemma allows us to replace the assumption on the number of εseparated elements in Proposition 7.2.2 by the assumption that the image of the unit ball under the map Φ in the proposition contains a ball of a ﬁxed radius. Lemma 7.2.5. For given ε > 0 and a > 0, there exist n0 ∈ N, d > 0 and δ > 0 such that if n ≥ n0 and Ω is a symmetric and convex subset of the unit ball of n∞ (R) containing at least ean elements that are ε-separated, then there exists a subset J ⊂ {1, . . . , n} such that |J| ≥ dn and the image of Ω under the projection πJ : n∞ (R) → ∞ (J; R) contains the ball of radius δ. Proof. Let Y ⊂ Ω be an ε-separated set, |Y | ≥ ean . Fix N ∈ N to be speciﬁed later, and for each k, 0 ≤ k < N , consider the set Uk ⊂ R deﬁned by Uk =

M !

[mε + kδ0 , mε + (k + 1)δ0 ),

m=−M

where M = 1 + [1/ε] and δ0 = ε/N . Let Yk be the subset of Y consisting of elements y = (y1 , . . . , yn ) such that the set Jk (y) = {j | yj ∈ Uk } contains no more than n/N elements. Since the sets Uk are disjoint and cover the

7.2 Topological Entropy

115

interval [−1, 1], the sets Jk (y), 0 ≤ k ≤ N − 1, are disjoint, and their union −1 is {1, . . . , n}. Hence |Jk (y)| ≤ n/N for some k. Therefore ∪N k=0 Yk = Y . Fix k such that |Yk | ≥ |Y |/N . For all m, |m| ≤ M , and I ⊂ {1, . . . , n} with |I| > n(N − 1)/N consider Am,I ⊂ 2I deﬁned as follows. A subset H ⊂ I is in Am,I if there exists y ∈ Yk such that yj ≥ mε + (k + 1)δ0 , if j ∈ H, yj < mε + kδ0 , if j ∈ I\H, yj ∈ [mε + kδ0 , mε + (k + 1)δ0 ) ⊂ Uk , if j ∈ {1, . . . , n}\I. For any m and y ∈ Yk there exists exactly one set I such that y deﬁnes an element of Am,I . In other words, for each m we have a map from Yk into the disjoint union I Am,I . If two points y, z ∈ Yk have the same images under these maps, that is, they deﬁne the same element of 2Im for each m (for some uniquely deﬁned Im ), then they cannot be ε-separated, and hence y = z. It follows that $ |Yk | ≤ |Am,I | . m

I

1)/N . Equivalently, Let cn (N ) be the number of I’s such that |I| > n(N − cn (N ) is the number of J’s with |J| ≤ n/N , so cn (N ) = j≤n/N ( nj ). Then

M $ m=−M

|Am,I |

≤

2M +1 cn (N ) max |Am,I | .

I

m,I

Since |Yk | ≥ |Y |/N , we get −1

max |Am,I | ≥ cn (N ) m,I

|Y | N

1/(2M +1)

≥ cn (N )−1 N −1/(2M +1) ean/(2M +1) .

(7.5) We have already remarked in the proof of Lemma 7.2.4 that cn (N ) does not grow faster than ec(N )n as n → ∞, with c(N ) → 0 as N → ∞. Thus we can choose N and c > 0 such that the right hand side of (7.5) is not smaller than ecn for suﬃciently large n. Fix m and I, |I| > n(N − 1)/N , such that |Am,I | ≥ ecn . By Lemma 7.2.4 there exists J ⊂ I such that |J| ≥ dn and Am,I (J) = 2J . In other words, for any H ⊂ J there is y ∈ Y satisfying yj ≥ mε + (k + 1)δ0 , if j ∈ H, yj < mε + kδ0 , if j ∈ J\H. Since the set πJ (Ω) is convex, this implies (e.g. by the Hahn-Banach theorem) that it contains all elements y ∈ ∞ (J; R) such that mε + kδ0 ≤ yj ≤ mε + (k + 1)δ0 , j ∈ J.

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7 Dynamics on the State Space

In other words, it contains a translation of the ball of radius δ0 /2 in ∞ (J; R), namely, Bδ0 /2 (y) = y + Bδ0 /2 (0) with yj = mε + kδ0 + δ0 /2 for j ∈ J. Since πJ (Ω) is also symmetric, it follows that it contains 12 (y + Bδ0 /2 (0) − y) = Bδ0 /4 (0), the ball of radius δ = δ0 /4.

Proof of Proposition 7.2.2. By Lemma 7.2.5 we can ﬁnd J ⊂ {1, . . . , n} such that |J| ≥ dn and the image of the unit ball of Matm (C)∗ under the map πJ ◦ Φ contains the ball of radius δ in ∞ (J; R). Decomposing elements of ∞ (J) into real and imaginary parts, we conclude that the image of the unit ball contains the ball of radius δ0 = δ/2 in ∞ (J). Then the dual map S = (πJ ◦ Φ)∗ : 1 (J) → Matm (C) has the property δ0 y1 ≤ Sy ≤ y1 . Hence the type 2 constants satisfy T2 (Matm (C)) ≥ δ0 T2 (1 (J)). By Lemma 7.2.3 this gives % % √ C log m ≥ δ0 |J| ≥ δ0 dn, so that m ≥ e(δ0 /C)

2

dn

.

Although we have proved that ht(α) is positive as soon as htop (T ) is positive, our proof does not provide any estimates for ht(α). This is not a drawback of the proof, as in fact htop (T ) can take only two values, zero and inﬁnity. Proposition 7.2.6. We have either htop (T ) = 0 or htop (T ) = ∞. Proof. As in the proof of Theorem 7.2.1, let the metric on X be given by d(ϕ, ψ) = sup |ϕ(y) − ψ(y)|. y∈Ω0

If htop (T ) > a > 0, there exists ε > 0 such that for all n suﬃciently large −1 there exists ean points that are (n, ε)-separated with respect to T . Fix r ∈ N and choose ε > 0 and λ1 , . . . , λr > 0 such that j λj = 1 and ελi − 2

r

λj > ε for i = 1, . . . , r − 1.

j=i+1

Let En be an (n, ε)-separated set. Consider the set En consisting of states of the form r λj ϕj , where ϕj ∈ En for j = 1, . . . , r. j=1

We claim that |En | = |En |r and the set En is (n, ε )-separated. This would imply that htop (T ) ≥ ra, so that htop (T ) = ∞, as r was arbitrary.

7.2 Topological Entropy

117

Let (ϕ1 , . . . , ϕr ) and (ψ1 , . . . , ψr ) be diﬀerent r-tuples in Enr , and let i be the minimal number such that ϕi = ψi . As ϕi and ψi are (n, ε)-separated, there exist y ∈ Ω0 and k, 0 ≤ k ≤ n − 1, such that |(ϕi ◦ αk )(y) − (ψi ◦ αk )(y)| ≥ ε. Then, since ϕj = ψj for j < i, the distance between T −k ( j λj ϕj ) and T −k ( j λj ψj ) is not smaller than # " k k λj (ϕj ◦ α )(y) − (ψj ◦ α )(y) j r

≥ λi |(ϕi ◦ αk )(y) − (ψi ◦ αk )(y)| −

λj (|(ϕj ◦ αk )(y)| + |(ψj ◦ αk )(y)|)

j=i+1

≥ ελi − 2

r

λj > ε ,

j=i+1

and the proof is complete.

Remark 7.2.7. The above proof actually shows that the topological entropy of any aﬃne homeomorphism of a convex compact subset of a locally convex space is either zero or inﬁnity. We ﬁnish this chapter with the following suﬃcient condition for positivity of entropy, which is implicit in the proof of Theorem 7.2.1. Recall ﬁrst that a subset I of N is said to have positive upper density if lim sup n→∞

|I ∩ {1, . . . , n}| > 0. n

Proposition 7.2.8. Suppose there exist a self-adjoint element a ∈ A and a subset I = {n1 < n2 < . . .} ⊂ N of positive upper density such that the map Γ : 1 → A deﬁned by Γ (ek ) = αnk (a), k ∈ N, where {ek }k denotes the canonical basis of 1 , is an isomorphism of 1 onto a closed subspace of A. Then ht(α) > 0. Proof. Assume A ⊂ B(H). We may also assume that a ≤ 1, so that Γ is a contraction. Let δ > 0 be such that Γ (x) ≥ δx1 for x ∈ 1 . For each n ∈ N set In = I ∩ {1, . . . , n}. Denote by Γn the restric|I | tion of Γ to 1 n . If unital completely positive maps σ: A → Matm (C) and γ: Matm (C) → B(H) are such that (γ ◦ σ ◦ αk )(a) − αk (a) < δ/2 for k = 1, . . . , n, then Γn − γ ◦ σ ◦ Γn < δ/2. It follows that (γ ◦ σ ◦ Γn )(x) ≥

δ |I | x1 for x ∈ 1 n . 2

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7 Dynamics on the State Space

Since γ is a contraction, the same inequality holds for the contraction |I | σ ◦ Γn : 1 n → Matm (C). The same argument as in the proof of Proposition 7.2.2 then shows that C

% δ% log m ≥ |In |, 2

2

so m ≥ e(δ/2C) |In | . Since |In | ≥ dn for certain suﬃciently large n’s and d > 0 by assumption, we conclude that ht({a}, δ/2; α) ≥

δ 2C

2 d > 0,

and the proof is complete. {an }∞ n=1

∗

is a sequence of self-adjoint elements in the unit ball of a C If algebra A, then in order to check that the map 1 → A, en → an , is an isomorphism onto a closed subspace it suﬃces to ﬁnd ε > 0 such that for any n ∈ N and ε1 , . . . , εn = ±1 there exists a state ϕε1 ...εn satisfying εk ϕε1 ...εn (ak ) ≥ ε, k = 1, . . . , n. Indeed, if λ1 , . . . , λn ∈ R and εk is the sign of λk , then ϕε1 ...εn (λ1 a1 + . . . + λn an ) ≥ ε(|λ1 | + . . . + |λn |), so that λ1 a1 + . . . + λn an ≥ ε(λ1 , . . . , λn )1 . By the proof of the above proposition it then follows that there exist δ > 0 and d > 0 such that rcp({a1 , . . . , an }, δ) ≥ edn for all suﬃciently large n ∈ N. Note however that in this case the same conclusion can be made without relying on the estimates of the type 2 constants from Lemma 7.2.3. Indeed, consider the channels γk : C2 → A deﬁned by γk (p1 ) =

1 1 (1 + ak ), γk (p2 ) = (1 − ak ), 2 2

where p1 = (1, 0), p2 = (0, 1). For n ∈ N choose ϕε1 ...εn as above and put ϕ=

1 2n

ϕε1 ...εn .

ε1 ,...,εn =±1

Then Hϕ (γ1 , . . . , γn ; {2−n ϕε1 ...εn }) =

n k=1

By the choice of the ϕε1 ...εn ’s,

Hϕ (γk ; {2−n ϕ(k) εk }).

7.3 Notes (k)

2−n+1 (ϕ1 ◦ γk )(p1 ) ≥

119

1 1 (k) (1 + ε) and 2−n+1 (ϕ−1 ◦ γk )(p1 ) ≤ (1 − ε). 2 2

Note that if µ1 and µ2 are two states on C2 such that µ1 (p1 ) ≥ (1 + ε)/2 and µ2 (p1 ) ≤ (1 − ε)/2, and µ = (µ1 + µ2 )/2, then µ − µi ≥ |µ(p1 ) − µi (p1 )| =

1 ε |µ1 (p1 ) − µ2 (p1 )| ≥ . 2 2

Hence there exists ε > 0 depending only on ε such that 1 1 S(µ1 , µ) + S(µ2 , µ) ≥ ε . 2 2 It follows that Hϕ (γ1 , . . . , γn ) ≥ nε . Then by an argument similar to that in the proof of Proposition 6.2.7, for any d < ε we can ﬁnd δ > 0 such that rcp({a1 , . . . , an }, δ) ≥ edn for any n ∈ N. Example 7.2.9. Let u be the shift to the right on 2 (Z). Denote by α the automorphism Ad u of B(H). Let a ∈ B(H) be a self-adjoint operator such that for any n ∈ N and ε1 , . . . , εn = ±1 there exists m ∈ Z such that (aem−k , em−k ) = (αk (a)em , em ) = εk for k = 1, . . . , n, where {ek }k∈Z is the canonical basis in 2 (Z). Then for any α-invariant C∗ subalgebra A of B(H) containing a we have ht(α|A ) > 0.

7.3 Notes Proposition 7.1.2, of which Theorem 7.1.1 is an obvious consequence, was proved by Sauvageot [190]. The Pinsker formula, Lemma 7.1.3, appeared in [164]. In the abelian case Theorem 7.2.1 was proved by Glasner and Weiss [73]. They gave two diﬀerent proofs of the result, one measure-theoretic, and one using geometry of the Banach spaces 1 and ∞ . The second proof was extended to noncommutative systems by Kerr [107]. The estimate of the type 2 constants of matrix algebras in Lemma 7.2.3 is attributed in [107] to TomczakJaegermann, who obtained the estimate of the type 2 constants of the Schatten ideals in [216]. Inequality (7.3) can be extended to noninteger m ≥ 1 by interpolation [216]. Lemma 7.2.4, or more precisely the statement established in the course of its proof, is a famous result of Perles and Sauer [187], ˇ Shelah [196] and Vapnik and Cervonenkis [219]. Lemma 7.2.5 is apparently well-known to specialists on geometric Banach space theory. We follow Glasner and Weiss [73] in its proof. Proposition 7.2.6 was proved by Sigmund [197]

120

7 Dynamics on the State Space

in the abelian case. As was observed in [107], the same proof works for noncommutative systems. Proposition 7.2.8 is due to Kerr and Li [108]. In the abelian case the converse statement is also true [108], which extends a result of Glasner and Weiss [73]. Example 7.2.9 is from [107]. In the abelian case the converse to Theorem 7.2.1 clearly holds, since the spectrum of an abelian algebra can be identiﬁed with the pure state space. To put it diﬀerently, if A is a C∗ -algebra and X = S(A) is its state space, then we can deﬁne an injective unital completely positive map θ: A → C(X) by θ(a)(ϕ) = ϕ(a). If α is an automorphism of A, T ϕ = ϕ ◦ α−1 , and α ˆ is the automorphism of C(X) deﬁned by T , α ˆ (f ) = f ◦ T −1 , then α ˆ ◦ θ = θ ◦ α. If A is abelian then θ is a complete order embedding, so by Remark 6.2.11(iii) we have ht(α) ≤ ht(ˆ α) = htop (T ). For general A the map θ is far from being a complete order embedding. In fact, the inverse map θ−1 : θ(A) → A is not even completely bounded unless A is subhomogeneous, that is, a subalgebra of C(Y ) ⊗ Matn (C) for a compact set Y and n ∈ N [95], [203]. Kerr and Li [108] introduced the contractive approximation entropy hc(α), where they use isometric embeddings instead of injective homomorphisms or complete order embeddings, and contractive factorizations through abelian ﬁnite dimensional C∗ -algebras instead of contractive completely positive factorizations through matrix algebras. One of the advantages of this entropy is that the converse to Theorem 7.2.1 holds [108], as one can see using the ˜ where X ˜ is the unit ball in A∗ , and the isometric embedding of A into C(X), ˜ deﬁned by α is fact that the topological entropy of the homeomorphism of X zero as soon as htop (T ) = 0 [107]. The price one pays is that for noncommutative systems contractive approximation entropy behaves very diﬀerently from the Voiculescu-Brown topological entropy. E.g. if α is the shift on Matm (C)⊗Z then, as we know, ht(α) = log m, while hc(α) = +∞ [108]. It remains an open question whether the converse to Theorem 7.2.1 is true for exact or nuclear algebras. In this respect it would be interesting to investigate automorphisms of noncommutative tori, see Chap. 11, where we have positive topological entropy, but generically only one invariant state with zero dynamical entropy.

8 Crossed Products

In this chapter we shall prove that canonical extensions of automorphisms to crossed products have the same entropy as the original systems. This is interesting and nontrivial already for crossed products by Z, but we shall work with larger classes of groups and with twisted crossed products as this does not require signiﬁcant additional eﬀorts.

8.1 Crossed Products by Discrete Amenable Groups Let G be a discrete group, and β: G → Aut(A) an action of G on a unital C∗ -algebra A. Let also ω be a T-valued 2-cocycle on G, so ω(h, k)ω(gh, k)ω(g, hk)ω(g, h) = 1 for g, h, k ∈ G. Note that if e ∈ G is the unit element then the cocycle identity implies that ω(g, e) = ω(e, e) = ω(e, k) for any g, k ∈ G (put h = k = e and g = h = e), and ω(g, g −1 ) = ω(g −1 , g) for any g ∈ G (put h = g −1 and k = g). Consider the full crossed product Aβ,ω G, which is the universal unital C∗ -algebra generated by a copy of A and by unitaries ug , g ∈ G, such that ug a = βg (a)ug and ug uh = ω(g, h)ugh . Note that ue = ω(e, e)1 and u∗g = ω(g, g −1 )ω(e, e)ug−1 for any g ∈ G. Assume A ⊂ B(H). Then we can deﬁne a representation π of A β,ω G on H ⊗ 2 (G) as follows: π(a)(ξ ⊗ δh ) = βh−1 (a)ξ ⊗ δh , π(ug )(ξ ⊗ δh ) = ω(g, h)ξ ⊗ δgh ,

(8.1)

where δg ∈ 2 (G) is deﬁned by δg (h) = 0 for h = g and δg (g) = 1. The reduced crossed product A r,β,ω G is by deﬁnition the quotient of the full crossed product by the kernel of π, which is independent of the embedding of A into B(H). Recall that if G is amenable then the full and reduced crossed

122

8 Crossed Products

product coincide, which we shall actually prove as a byproduct. If ω ≡ 1 we write A β G and A r,β G for the crossed products. If G = Z then the action of G on A is given by one automorphism β of A, and the crossed product A β Z is generated by A and a unitary u such that ua = β(a)u. We say that u ∈ A β Z is the canonical unitary implementing β. Let α be an automorphism of A commuting with βg for every g ∈ G. Then α extends to an automorphism α ¯ of A r,β,ω G such that α(u ¯ g ) = ug . Using the canonical conditional expectation E: A r,β,ω G → A deﬁned by E(aug ) = 0 for g = e, we can also extend any state ϕ on A to a state ϕ¯ on the crossed product. If ϕ is α-invariant, then ϕ¯ is α ¯ -invariant. Theorem 8.1.1. Let G be a discrete amenable group, ω ∈ Z 2 (G; T) a 2cocycle, β: G → Aut(A) an action of G on A, and α an automorphism of A commuting with β. Then for the automorphism α ¯ of A β,ω G we have (i) ht(¯ α) = ht(α); (ii) if ϕ is an α- and β-invariant state on A such that there exists a ϕapproximating net, then hϕ¯ (¯ α) = hϕ (α). If A ⊂ B(H) is a von Neumann algebra, we can also deﬁne the von Neumann algebra crossed product of A by G, which is by deﬁnition the von Neumann algebra generated by A r,β,ω G in B(H ⊗ 2 (G)). Since the restriction of an automorphism of a von Neumann algebra to a dense C∗ -subalgebra gives the same dynamical entropy, part (ii) is of course also true for W∗ -dynamical systems and von Neumann algebra crossed products. The most interesting particular case of the above theorem is formulated in the following corollary. Corollary 8.1.2. Let α be an automorphism of a unital C∗ -algebra A. Let u ∈ A α Z be the canonical unitary implementing α. Then (i) ht(Ad u) = ht(α); (ii) if ϕ is an α-invariant state on A such that there exists a ϕ-approximating net, then hϕ¯ (Ad u) = hϕ (α).

To prove Theorem 8.1.1 we shall approximate the system (A r,β,ω G, α ¯) by ampliﬁcations of the system (A, α). Let f be a positive ﬁnitely supported function on G such that f 1 = h f (h) = 1. Denote by F the support of f . We shall construct two unital completely positive maps Φf : A r,β,ω G → A ⊗ B(2 (F )) and Ψf : A ⊗ B(2 (F )) → A r,β,ω G. To deﬁne Φf assume A ⊂ B(H), identify 2 (F ) with a subspace of 2 (G), and let PF : H ⊗ 2 (G) → H ⊗ 2 (F ) be the orthogonal projection. Then consider the representation π of A r,β,ω G on H ⊗ 2 (G) deﬁned above, and set

8.1 Crossed Products by Discrete Amenable Groups

123

Φf (x) = PF π(x)PF for x ∈ A r,β,ω G. A priori we have Φf (x) ∈ B(H) ⊗ B(2 (F )), but the equality ω(g, h)(βh−1 g−1 (a)ξ ⊗ δgh ), if gh ∈ F , PF π(aug )(ξ ⊗ δh ) = 0, otherwise, see (8.1), shows that Φf (x) ∈ A ⊗ B(2 (F )). More precisely, we have Φf (aug ) = ω(g, h)βh−1 g−1 (a) ⊗ egh,h ,

(8.2)

h∈F ∩g −1 F

where {es,t }s,t∈F is the set of matrix units in B(2 (F )) corresponding to the basis {δh }h∈F . In particular, βh−1 (a) ⊗ eh,h . Φf (a) = h∈F

To deﬁne Ψf note that for any elements xs ∈ A r,β,ω G, s ∈ F , we have a completely positive map A ⊗ B(2 (F )) → A r,β,ω G, a ⊗ es,t → xs ax∗t . This is proved similarly to A.9. Namely, identify A r,β,ω G with (A r,β,ω G) ⊗ Ceh,h ⊂ A ⊗ B(2 (F )) for some ﬁxed h ∈ F , and write the map as A ⊗ B(2 (F )) x → V xV ∗ ∈ (A r,β,ω G) ⊗ Ceh,h , where V = s∈F xs ⊗ eh,s . Apply this construction to xs = f (s)1/2 us , s ∈ F , to get Ψf . Thus Ψf (a ⊗ es,t ) = f (s)1/2 us a(f (t)1/2 ut )∗ = f (s)1/2 f (t)1/2 βs (a)us u∗t . Since h f (h) = 1, the map Ψf is unital.

(8.3)

Proposition 8.1.3. For every positive ﬁnitely supported function f on G such that f 1 = 1, the unital completely positive maps Φf and Ψf have the following properties: ¯ = (α ⊗ id) ◦ Φf and Ψf ◦ (α ⊗ id) = α ¯ ◦ Ψf ; (i) Φf ◦ α (ii) if ϕ is a G-invariant state, then (ϕ ⊗ ϕf ) ◦ Φf = ϕ¯ and ϕ¯ ◦ Ψf = ϕ ⊗ ϕf , where ϕf is the state on B(2 (F )) given by ϕf (es,t ) = δs,t f (s); (iii) (Ψf ◦ Φf )(aug ) = ( h∈G f (gh)1/2 f (h)1/2 )aug , so that 1/2

(Ψf ◦ Φf )(aug ) − aug ≤ g f − f 1 a, where g f (h) = f (g −1 h).

124

8 Crossed Products

Proof. Parts (i) and (ii) are immediate consequences of (8.2) and (8.3). To check (iii) we compute ⎛ ⎞ (Ψf ◦ Φf )(aug ) = Ψf ⎝ ω(g, h)βh−1 g−1 (a) ⊗ egh,h ⎠ h∈F ∩g −1 F

=

f (gh)1/2 f (h)1/2 ω(g, h)augh u∗h

h∈F ∩g −1 F

=

1/2

f (gh)

1/2

f (h)

aug .

h∈G

Then the inequality in (iii) follows from f (gh)1/2 f (h)1/2 − 1 = |(f 1/2 , g f 1/2 ) − (f 1/2 , f 1/2 )| h∈G

≤ g f 1/2 − f 1/2 2 1/2

≤ g f − f 1 , where the last inequality holds, as (x−y)2 ≤ |x2 −y 2 | for any positive numbers x and y.

Since the group G is amenable, there exists a net {fi }i of positive ﬁnitely supported functions on G such that fi 1 = 1 and g fi − fi 1 → 0 for any g ∈ G. Then (Ψfi ◦ Φfi )(x) → x in norm for any x ∈ A r,β,ω G. ¯f from the full Note that for full crossed products we can deﬁne a map Φ crossed into A⊗B(2 (F )) by composing Φf with the quotient map Aβ,ω G → Ar,β,ω G, and then deﬁne a map Ψ¯f into the full crossed product by the same ¯f → id pointwise. Since by deﬁnition formula as for Ψf . Then we get Ψ¯fi ◦ Φ i ¯ Φf factorizes through the reduced crossed product, it follows that full and reduced crossed products by amenable groups coincide. To complete the proof of the theorem we need the following approximation result similar to Proposition 3.2.4. Lemma 8.1.4. Let α be an automorphism of a C∗ -algebra B, {Bi }i a net of C∗ -algebras together with automorphisms αi ∈ Aut(Bi ), and {Φi : B → Bi }i and {Ψi : Bi → B}i two nets of equivariant (that is, Φi ◦ α = αi ◦ Φi and α ◦ Ψi = Ψi ◦ αi ) unital completely positive maps such that (Ψi ◦ Φi )(b) → b in norm for any b ∈ B. Then (i) ht(α) ≤ lim inf i ht(αi ); (ii) if ϕ is an α-invariant state on B such that ϕ ◦ Ψi ◦ Φi = ϕ for any i, then hϕ (α) ≤ lim inf i hϕ◦Ψi (αi ). Proof. Let B ⊂ B(H), Bi ⊂ B(Hi ). By Arveson’s extension theorem, A.8, we can extend Ψi : Bi → B to a unital completely positive map Ψ¯i : B(Hi ) →

8.2 Generalizations

125

B(H). Then given unital completely positive maps γ: Bi → Matn (C) and θ: Matn (C) → B(Hi ) we can consider γ ◦ Φi : B → Matn (C) and Ψ¯i ◦ θ: Matn (C) → B(H), and estimate (Ψ¯i ◦ θ ◦ γ ◦ Φi )(b) − b ≤ Ψ¯i ((θ ◦ γ ◦ Φi )(b) − Φi (b)) + (Ψi ◦ Φi )(b) − b ≤ (θ ◦ γ)(Φi (b)) − Φi (b) + (Ψi ◦ Φi )(b) − b. It follows that if Ω ⊂ B is a ﬁnite subset, then ht(Ω, 2ε; α) ≤ ht(Φi (Ω), ε; αi ) ≤ ht(αi ) as soon as (Ψi ◦ Φi )(b) − b < ε for b ∈ Ω. Thus (i) is proved. Similarly, if γ: A → B is a channel, then by Proposition 3.1.3(ii) hϕ (Ψi ◦ Φi ◦ γ; α) ≤ hϕ◦Ψi (Φi ◦ γ; αi ) ≤ hϕ◦Ψi (αi ). On the other hand, by Proposition 3.1.11 hϕ (γ; α) ≤ hϕ (Ψi ◦ Φi ◦ γ; α) + ε if Ψi ◦ Φi ◦ γ − γ is suﬃciently small. This proves (ii).

Proof of Theorem 8.1.1. If {fi }i is a net of positive ﬁnitely supported functions on G such that fi 1 = 1 and g fi − fi 1 → 0 for any g ∈ G, then by Proposition 8.1.3, the discussion following it and Lemma 8.1.4(i) we have ht(¯ α) ≤ lim inf ht(α ⊗ idB(2 (Fi )) ), i

where Fi is the support of fi . By Theorem 6.2.2(iv), ht(α ⊗ id) = ht(α). Thus we get ht(¯ α) ≤ ht(α). On the other hand, the opposite inequality holds by Theorem 6.2.2(iii). The case of dynamical entropy is similar. The inequality hϕ¯ (¯ α) ≥ hϕ (α) holds by Theorem 3.2.2(v). In order to get the opposite inequality we just have to recall that by Theorem 3.2.5(iv), if there exists a ϕ-approximating net, then hϕ⊗ψ (α ⊗ id) = hϕ (α) for any state ψ on a ﬁnite dimensional C∗ algebra.

8.2 Generalizations In this section we shall discuss several generalizations of Theorem 8.1.1. First of all, what are the minimal assumptions on G for the equality ht(¯ α) = ht(α) to be true? Since by our deﬁnition of topological entropy an algebra is exact if and only if the entropy of the identity automorphism is

126

8 Crossed Products

zero, for ω ≡ 1 the equality implies that A r,β G is exact as soon as A is exact. But then the reduced group C∗ -algebra Cr∗ (G) ⊂ A r,β G is also exact. Groups with this property are called exact. We shall use an equivalent definition of exactness saying that there exists an amenable action of G on a compact space. That is, there are an action γ: G → Aut(B) of G on an abelian unital C∗ -algebra B (in fact, we can always take the action by translations on ∞ (G))and a net {fi }i of positive ﬁnitely supported functions G → B such that h fi (h) = 1 and lim |fi (gh) − γg (fi (h))| = 0 for any g ∈ G. i h∈G

Note that if B = C1, then we get the usual deﬁnition of amenability of G. The class of exact groups is however much bigger than the class of amenable groups. For example, free groups are exact (in Chap. 14 we shall prove a more general result). Theorem 8.2.1. Let G be a discrete exact group, ω ∈ Z 2 (G; T) a 2-cocycle, β: G → Aut(A) an action of G on A, and α an automorphism of A commuting with β. Then for the automorphism α ¯ of A r,β,ω G we have ht(¯ α) = ht(α). Proof. The proof is a slight modiﬁcation of that of Theorem 8.1.1, so we will be sketchy. Assume ﬁrst that there exists a unital C∗ -subalgebra B of the center of A such that the action β|B of G on B is amenable and α| B is trivial. Given a positive ﬁnitely supported function f : G → B such that h f (h) = 1 we can deﬁne unital completely positive maps Φf and Ψf by formulas (8.2) and (8.3), so, with F = supp f , ω(g, h)βh−1 g−1 (a) ⊗ egh,h , Φf (aug ) = h∈F ∩g −1 F

Ψf (a ⊗ es,t ) = f (s)1/2 us a(f (t)1/2 ut )∗ = f (s)1/2 βst−1 (f (t)1/2 )βs (a)us u∗t . Then (Ψf ◦ Φf )(aug ) =

f (gh)1/2 βg (f (h)1/2 )aug ,

h

and as f (gh)1/2 βg (f (h)1/2 ) − 1 = (f (gh)1/2 − βg (f (h))1/2 )βg (f (h))1/2 h h 1/2 ≤ |f (gh) − βg (f (h))| , h

we see that the maps Φf and Ψf have properties similar to those in Proposition 8.1.3(i),(iii). Hence ht(¯ α) = ht(α).

8.2 Generalizations

127

In the general case, given an amenable action γ of G on a unital abelian C∗ -algebra B, we can conclude that the extension of the automorphism id ⊗ α of B ⊗A to (B ⊗A)r,γ⊗β,ω G has entropy not greater than ht(id⊗α) = ht(α). As A r,β,ω G is a subalgebra of (B ⊗ A) r,γ⊗β,ω G, we get ht(¯ α) ≤ ht(α), whereas the opposite inequality always holds.

Note that the previous argument does not allow us to prove an analogue of Theorem 8.1.1(ii) for exact groups. The problem is that if a group acts amenably and preserves a state, then it is amenable. Thus if G is exact but nonamenable, there is no G-invariant state on B. In particular, there is no canonical extension of a state on A to a state on (B ⊗ A) r,γ⊗β G. Moreover, there exists no conditional expectation (B ⊗ A) r,γ⊗β G → A r,β G, since otherwise the restriction to B of the composition of ϕ¯ with such a conditional expectation would be a G-invariant state. So even if we manage to get an estimate of the dynamical entropy of the extension of idB ⊗ α to the crossed product with respect to some invariant state, it will not give us an immediate estimate of the entropy of α. ¯ We next turn to nondiscrete groups. Then the crossed products are nonunital, but the notion of topological entropy still makes sense by Remark 6.2.11(ii). Recall the deﬁnition of a crossed product. Let G be a locally compact group, ω: G × G → T a continuous 2-cocycle, β: G → Aut(A) a strongly continuous action of G on a C∗ -algebra A. Fix a left-invariant Haar measure 1 on G. Consider the Banach ∗-algebra Lω (G, A) consisting of functions f : G → A such that G f (g)dg < ∞, with product (f1 ∗ω f2 )(g) = ω(h, h−1 g)f1 (h)βh (f2 (h−1 g))dh G

and involution f ∗ (g) = ω(g, g −1 )ω(e, e)∆(g −1 )βg (f (g −1 )∗ ), where ∆ is the modular function of G. Then the full crossed product algebra A β,ω G is the enveloping C∗ -algebra of L1ω (G, A). The algebra A is in general a subalgebra not of the crossed product, but of its multiplier algebra. Namely, for a ∈ A and f ∈ L1ω (G, A) we deﬁne (af )(g) = af (g) and (f a)(g) = f (g)βg (a). Similarly we have unitary elements ug , g ∈ G, in the multiplier algebra deﬁned by (ug f )(h) = ω(g, g −1 h)βg (f (g −1 h)) and (f ug )(h) = ω(hg −1 , g)∆(g −1 )f (hg −1 ). If A ⊂ B(H) then similarly to (8.1) deﬁne a representation π of A β,ω G on L2 (G, H) ∼ = H ⊗ L2 (G). Its extension to the multiplier algebra is given by (π(a)ξ)(h) = βh−1 (a)ξ(h), (π(ug )ξ)(h) = ω(g, g −1 h)ξ(g −1 h), so that for f ∈ L1ω (G, A) we have

(8.4)

128

8 Crossed Products

π(f ) =

π(f (g))π(ug )dg. G

Then A r,β,ω G is the quotient of the full crossed product by the kernel of this representation. Again, this kernel is independent of the choice of the embedding of A into B(H), and if G is amenable then the kernel is trivial. Finally, if α is an automorphism of A commuting with the automorphisms βg , g ∈ G, we can deﬁne an automorphism α ¯ of A r,β,ω G such that (¯ α(f ))(g) = α(f (g)) for f ∈ L1ω (G, A). Theorem 8.2.2. Let G be a locally compact amenable group, ω ∈ Z 2 (G; T) a continuous 2-cocycle, β: G → Aut(A) a strongly continuous action of G on a C∗ -algebra A, and α an automorphism of A commuting with β. Then for the automorphism α ¯ of A β,ω G we have ht(¯ α) = ht(α). Proof. The proof is again similar to that of Theorem 8.1.1, but the details become a bit more involved. This time the maps Φf and Ψf will be deﬁned for positive compactly supported functions f on G, so that Φf : A r,β,ω G → A ⊗ K(L2 (F )) and Ψf : A ⊗ K(L2 (F )) → A r,β,ω G, where F = supp f , and K(L2 (F )) denotes the algebra of compact operators on L2 (F ). To deﬁne these maps represent A faithfully on a Hilbert H, and consider the representation of the crossed product on H0 = L2 (G, H) as above. Let PF : H0 → L2 (F, H) be the orthogonal projection, and set Φf (x) = PF π(x)PF . Then, using (8.4), similarly to (8.2) we have (Φf (aug )ξ)(h) = ω(g, g −1 h)βh−1 (a)ξ(g −1 h) for ξ ∈ L2 (F, H).

(8.5)

To deﬁne Ψf we shall consider A⊗K(L2 (F )) acting on H0 ⊗L2 (F ) = L2 (F, H0 ) rather than on H ⊗L2 (F ), and deﬁne a map B(L2 (F, H0 )) → B(H0 ). Consider the operator Vf : H0 → L2 (F, H0 ), (Vf ξ)(s) = f (s)1/2 π(us )∗ ξ. The adjoint operator is given by ∗ Vf ζ = f (s)1/2 π(us )ζ(s) ds. F

In particular, Vf∗ Vf is just the scalar operator f 1 , so that if f 1 = 1 we get a unital completely positive map Ψ˜f : B(L2 (F, H0 )) → B(H0 ), Ψ˜f (x) = Vf∗ xVf . To see that Ψ˜f (π(A) ⊗ K(L2 (F ))) ⊂ π(A r,β,ω G), consider an integral operator T on L2 (F ) with continuous kernel k ∈ C(F × F ), in other words

8.2 Generalizations

129

(T ξ)(s) =

k(s, t)ξ(t) dt. F

Then a simple computation yields (compare with (8.3)) ˜ Ψf (π(a)⊗T ) = f (s)1/2 f (t)1/2 k(s, t)π(βs (a)us u∗t )ds dt ∈ π(Ar,β,ω G). F ×F

Since integral operators are dense in the algebra of compact operators, we conclude that Ψ˜f (π(A) ⊗ K(L2 (F ))) ⊂ π(A r,β,ω G). Hence there exists a contractive completely positive map Ψf : A ⊗ K(L2 (F )) → A r,β,ω G such that (π ◦ Ψf )(a ⊗ T ) = Ψ˜f (π(a) ⊗ T ) for a ∈ A and T ∈ K(L2 (F )). Consider the representation ρ of A⊗K(L2 (F )) on L2 (F, H0 ) = H0 ⊗L2 (F ) deﬁned by ρ(a ⊗ T ) = π(a) ⊗ T . Then, denoting the extension of ρ to the multiplier algebra by the same letter, (8.5) can be written as (ρ(Φf (aug ))ζ)(s) = ω(g, g −1 s)π(βs−1 (a))ζ(g −1 s) for ζ ∈ L2 (F, H0 ), s ∈ F. For ξ ∈ H0 we then compute π((Ψf ◦ Φf )(aug ))ξ = Vf∗ ρ(Φf (aug ))Vf ξ = f (s)1/2 π(us )(ρ(Φf (aug ))Vf ξ)(s) ds F f (s)1/2 ω(g, g −1 s)π(us )π(βs−1 (a))(Vf ξ)(g −1 s) ds = F f (s)1/2 f (g −1 s)1/2 ω(g, g −1 s)π(us βs−1 (a)u∗g−1 s )ξ ds = F f (s)1/2 f (g −1 s)1/2 π(aug )ξ ds, = F

so that exactly as in Proposition 8.1.3(iii) 1/2 −1 1/2 f (s) f (g s) ds aug . (Ψf ◦ Φf )(aug ) = F

By an analogue of Lemma 8.1.4(i) for contractive completely positive maps instead of unital ones, this allows us to conclude that ht(¯ α) ≤ ht(α). If G is nondiscrete, A is not a subalgebra of the crossed product, so we need an additional argument for ht(α) ≤ ht(¯ α). This has nothing to do with amenability and is done as follows. It suﬃces to construct two nets {ϕi : A → A r,β,ω G}i and {ψi : A r,β,ω G → A}i of equivariant completely positive contractive maps such that (ψi ◦ ϕi )(a) − a → 0 for any a ∈ A. Choose a net {fi }i ⊂ L1 (G) of functions such that fi 1 = 1, and the support of fi is eventually contained in an arbitrarily small neighbourhood of the unit element e ∈ G. Put xi = G fi (g)ug dg. Then set ϕi (a) = xi ax∗i . For ζ ∈ L2 (G), ζ2 = 1, consider the Fubini map

130

8 Crossed Products

ψζ = id ⊗ ωζ : B(H ⊗ L2 (G)) → B(H), where ωζ is the state on B(L2 (G)) given by ωζ (T ) = (T ζ, ζ). We shall see in a moment that ψζ (π(A r,β,ω G)) ⊂ A. We claim that if βh−1 (a) − a < ε for h ∈ supp ζ, then (ψζ ◦ π ◦ ϕi )(a) − a < ε for suﬃciently large i. Thus, replacing {ϕi }i by a subnet, an appropriate net {ψi }i can be chosen from the maps ψζ ◦ π, ζ ∈ L2 (G). To prove our claim consider the C∗ -subalgebra Cr∗ (G, ω) = C1 r,ω G of the multiplier algebra of Ar,β,ω G. Consider the representation π ¯ of Cr∗ (G, ω) 2 −1 −1 on L (G) given by (¯ π (ug )ϑ)(h) = ω(g, g h)ϑ(g h). Thus π(x) = 1 ⊗ π ¯ (x) for x ∈ Cr∗ (G, ω). The equality ψζ (π(aug )) = ω(g, g −1 h)ζ(g −1 h)ζ(h)βh−1 (a)dh shows that if βh−1 (a) − a < ε for h ∈ supp ζ, then π (ug ))a < ε, ψζ (π(aug )) − ωζ (¯ π (xi x∗i ))a < ε. Hence ψζ (π(axi x∗i )) − a < ε so that ψζ (π(axi x∗i )) − ωζ (¯ for i large enough, since π ¯ (xi x∗i )ζ → ζ. On the other hand, fi (g)(βg (a) − a)ug dg → 0, [xi , a] = G

axi x∗i

so becomes arbitrarily close to xi ax∗i = ϕi (a). This proves our claim and thus completes the proof of the theorem.

Note that similarly to the case of a discrete group the maps Φf and Ψf can be deﬁned for full crossed products. This is clearly true for Φf . To construct Ψf it suﬃces to take a faithful representation of the full crossed product on a Hilbert space H0 and use the same formulas as in the proof of the theorem. Thus, exactly as for discrete groups, we may conclude that if G is amenable then the full and reduced crossed products coincide. Moreover, then the identity map of A β,ω G approximately factorizes through the algebras A ⊗ K(H), where K(H) is the algebra of compact operators on H. So if in addition A is nuclear then A β,ω G is also nuclear, and if A is exact then by an argument similar to the proof of Lemma 6.1.2 the algebra A β,ω G is exact.

8.3 Notes The question whether for an ergodic dynamical system (X, µ, T ) the dynamical entropy of the inner automorphism of L∞ (X, µ)Z deﬁned by the canonical unitary implementing T coincides with hµ (T ), was raised by Størmer [208].

8.3 Notes

131

Voiculescu [227] answered this question positively by constructing the maps Φf and Ψf from Sect. 8.1. These maps were also independently introduced by Sinclair and Smith [201] for diﬀerent purposes. The observation made in the proof of Theorem 8.2.2 that a similar construction works for crossed products by arbitrary locally compact groups, seems to be new. Remark that the composition of the maps Ψf and Φf is a map of the form aug → ϕ(g)aug , where ϕ is a ﬁnitely supported positive deﬁnite function. Such maps were considered already by Zeller-Meier [237]. But the fact that they have nice factorizations through tensor products of the algebra and matrix algebras had apparently not been observed before Voiculescu [227] and Sinclair and Smith [201]. The result of Voiculescu was independently extended to general C∗ dynamical systems by Brown [32] in the case of topological entropy and by Golodets and Neshveyev [79] in the case of dynamical entropy. A similar result is true for other types of dynamical invariants, see e.g. [35], [110]. Essentially, one has to check that such invariants (i) have the approximation property stated in Lemma 8.1.4, (ii) do not change if we tensor an automorphism with the trivial automorphism of a full matrix algebra. For equivalence of various notions of exactness for discrete groups we refer the reader to the work of Anantharaman-Delaroche [6]. Theorem 8.2.1 was obtained by Choda [47] and Germain [70]. Theorem 8.2.2, in the case of the trivial cocycle, is due to Smith and Pop [167], whose proof is based on completely contractive factorizations of crossed products and the Takesaki-Takai duality. Smith and Pop prove, in fact, a slightly stronger result. Namely, assume that α does not commute with β, but there exists a continuous automorphism γ of G such that α ◦ βg = βγ(g) ◦ α. Assume also that the cocycle ω is γ-invariant. Then there exists an automorphism α ¯ of A r,β,ω G such that α ¯ (aug ) = α(a)uγ(g) . Then, assuming that the group {γ n }n∈Z is relatively compact in Aut(G), we have ht(¯ α) = ht(α). In order to show this one just has to use γ-invariant functions in the constructions of the maps involved, which is possible by the relative compactness assumption, and instead of the automorphism α ⊗id of A⊗K(L2 (F )) use the automorphism α ⊗Ad U , where U is the unitary on L2 (G) deﬁned by γ, so (U ξ)(h) = ξ(γ −1 (h)). Without the assumption that {γ n }n∈Z is relatively compact it is natural to ask whether the inequality ht(¯ α) ≤ ht(α) + ht(αγ ) holds, where αγ is the automorphism of Cr∗ (G, ω) deﬁned by γ. Instead of ht(αγ ) one can try to use approximation entropies of automorphisms of discrete groups, e.g. the ones introduced by Choda [45] and Brown and Germain [37]. Very little is known about this problem, see [79], [45], [75] for some examples and partial results. In Chap. 11 we shall see that the inequality above can be strict already for discrete abelian groups and ω ≡ 1. The results of the present chapter show that the entropy of inner automorphisms is a computable invariant. Moreover, this invariant tells something

132

8 Crossed Products

about the position of the algebra generated by the unitary deﬁning the automorphism. E.g. it is shown in [207] and [141] that if hϕ (Ad u) > 0 for a unitary u in a hyperﬁnite factor, then u is not contained in any Cartan subalgebra. See [33], [143], [144] for some other results in this direction. It thus seems natural to pose the following question. Given a von Neumann algebra M , a normal state ϕ on M , and a von Neumann subalgebra N of the centralizer Mϕ of the state ϕ, is it possible to deﬁne a computable entropic invariant of the action of the unitary group of N on M ?

9 Variational Principle

The variational principle formulated in Chap. 1 states that for topological dynamical systems the topological entropy is the supremum of the measure entropies over all invariant measures. More generally, given a continuous function f one introduces the notion of pressure and proves that it can be obtained as the supremum of the quantities hµ (T ) + f dµ. The measures where the supremum is attained are called equilibrium, as they indeed are equilibrium states in gas lattice models. In this form, using mean entropy and a formulation of equilibrium in terms of the KMS condition, the variational principle was extended to quantum spin lattice system. The notion of mean entropy depends on the choice of a sequence of ﬁnite subsystems and so does not make sense in general. It is natural to replace it by dynamical entropy, and our goal is to prove a variational principle in this setting for certain asymptotically abelian systems. In particular, we shall prove that under certain assumptions the topological entropy is the supremum of the dynamical entropies, and if the supremum is attained at some state, then this state must be tracial.

9.1 Pressure In order to deﬁne pressure we follow the setup of Chap. 6 of the deﬁnition of topological entropy. Let A be a unital C∗ -algebra, α an automorphism of A, H ∈ Asa a selfadjoint element. For δ > 0 and a ﬁnite subset Ω ⊂ A denote by P (H, Ω; δ) the inﬁmum of the quantities log TrB (e−θ(H) ), where B is a ﬁnite dimensional C∗ -algebra and θ: A → B is a unital completely positive map such that there exist a unital injective homomorphism π: A → C for some C∗ -algebra C and a unital completely positive map γ: B → C with (γ ◦ θ)(a) − π(a) < δ for any a ∈ Ω. Put P (H, Ω; δ) = +∞ if no such B and θ exist. As before, TrB denotes the canonical trace on B, i.e., the trace which takes the value 1 on each minimal projection.

134

9 Variational Principle

By the argument in the proof of Lemma 6.1.2 it suﬃces to consider one ﬁxed faithful representation π: A → B(H). If A is nuclear we may also take C = A and π the identity map. Note also that if H = 0 then P (H, Ω; δ) = log rcp(Ω, δ). Now put ⎞ ⎛ n−1 n−1 ! 1 ⎝ j Pα (H, Ω; δ) = lim sup P α (H), αj (Ω); δ ⎠ , n→∞ n j=0 j=0 Pα (H, Ω) = sup Pα (H, Ω; δ). δ>0

Deﬁnition 9.1.1. The pressure of α at H is Pα (H) = sup Pα (H, Ω), Ω

where the supremum is taken over all ﬁnite subsets of Ω ⊂ A. Note that it suﬃces to consider ﬁnite subsets Ω of a set spanning a dense subspace of A. It is clear also that if H = 0 then the pressure coincides with the topological entropy ht(α). Proposition 9.1.2. For any α-invariant state ϕ of A we have Pα (H) ≥ hϕ (α) − ϕ(H). Proof. Assume A ⊂ B(H). Let N be a ﬁnite dimensional C∗ -algebra and γ: N → A be a channel. Fix δ > 0. Choose a ﬁnite subset Ω0 of N such that its convex hull contains the unit ball. Put Ω = {H} ∪ γ(Ω0 ). Let B be a ﬁnite dimensional C∗ -algebra, and θ: A → B and ψ: B → B(H) be unital completely positive maps such that (ψ ◦ θ)(a) − a < δ for a ∈ αk (Ω), 0 ≤ k ≤ n − 1. Then ψ ◦ θ ◦ αk ◦ γ − αk ◦ γ < δ for 0 ≤ k ≤ n − 1. Let ϕ = i1 ,...,in ϕi1 ...in be a ﬁnite decomposition. Extend each ϕi1 ...in to a positive linear functional ϕ˜i1 ...in on B(H), and set ϕ˜ = i1 ,...,in ϕ˜i1 ...in . We obviously have Hϕ (γ, . . . , αn−1 ◦ γ; {ϕi1 ...in }) = Hϕ˜ (γ, . . . , αn−1 ◦ γ; {ϕ˜i1 ...in }). On the other hand, by Proposition 3.1.11 |Hϕ˜ (γ, . . . , αn−1 ◦ γ) − Hϕ˜ (ψ ◦ θ ◦ γ, . . . , ψ ◦ θ ◦ αn−1 ◦ γ)| < nε for some ε depending only on δ and dim N . Since Hϕ˜ (ψ ◦ θ ◦ γ, . . . , ψ ◦ θ ◦ αn−1 ◦ γ) ≤ S(ϕ˜ ◦ ψ) by Proposition 3.1.3(i),(iv) and Lemma 3.1.4, we get

9.1 Pressure

135

Hϕ (γ, . . . , αn−1 ◦ γ; {ϕi1 ...in }) ≤ S(ϕ˜ ◦ ψ) + nε. By Proposition 2.2.5 we have log TrB (e−K ) ≥ S(ω) − ω(K) for any state ω and any self-adjoint K. Therefore n−1 n−1 j S(ϕ˜ ◦ ψ) ≤ log TrB e−θ( j=0 α (H)) + ϕ˜ (ψ ◦ θ) αj (H) j=0 n−1 n−1 j αj (H) + nδ ≤ log TrB e−θ( j=0 α (H)) + ϕ j=0 j −θ ( n−1 α (H) ) + nϕ(H) + nδ. j=0 = log TrB e

P P P

Thus

⎞ ⎛ n−1 n−1 ! 1 1 αj (H), αj (Ω); δ ⎠ + ϕ(H) + δ + ε. Hϕ ({αj ◦ γ}0≤j≤n−1 ) ≤ P ⎝ n n j=0 j=0

It follows that hϕ (γ; α) ≤ Pα (H, Ω) + ϕ(H), hence hϕ (α) ≤ Pα (H) + ϕ(H).

Remark 9.1.3. We have chosen the minus sign e−θ(H) in the deﬁnition of P (H, Ω; δ) because of its use in physical applications [30], rather than the plus sign used in ergodic theory [231]. If A is abelian, A = C(X), and Pαcl (H) denotes the pressure as deﬁned in [231], see also Notes at the end of the chapter, then Pα (H) = Pαcl (−H). The inequality ≤ can be proved similarly to the proof of Lemma 6.1.3. The converse inequality follows from Proposition 9.1.2 and the classical variational principle. We list some properties of the function H → Pα (H) on Asa . Proposition 9.1.4. We have: (i) if H ≤ K then Pα (H) ≥ Pα (K); (ii) Pα (H + c1) = Pα (H) − c for c ∈ R; (iii) Pα (H) is either inﬁnite for all H or is ﬁnite valued; (iv) if Pα is ﬁnite valued then |Pα (H) − Pα (K)| ≤ ||H − K||; k−1 (v) for k ∈ N, Pαk ( j=0 αj (H)) = kPα (H); (vi) Pα (H + α(K) − K) = Pα (H). Proof. Let θ: A → D be a unital completely positive map. If H ≤ K, then by the Peierls-Bogoliubov inequality, Corollary 2.3.11, n−1 j n−1 j log TrB e−θ( j=0 α (H)) ≥ log TrB e−θ( j=0 α (K)) .

P

Thus (i) follows. Since

P

136

9 Variational Principle

P

log TrB e−θ(

n−1 j=0

αj (H+c1))

P

= log TrB e−θ(

n−1 j=0

αj (H))

− nc,

we get (ii). By (i) and (ii) we have Pα (H) ≥ Pα (||H||) = Pα (0) − ||H|| = ht(α) − ||H||, and similarly Pα (H) ≤ ht(α) + ||H||. This shows (iii). By the Peierls-Bogoliubov inequality, Corollary 2.3.11, 1 n−1 n−1 log TrB e−θ( j=0 αj (H)) − 1 log TrB e−θ( j=0 αj (K)) ≤ ||H − K||, n n

P

P

and (iv) follows. Let Ω be a ﬁnite subset of A, and k ∈ N. Given n ∈ N choose m ∈ N such k−1 j that mk ≤ n < (m + 1)k. Set Hk = j=0 αj (H) and Ωk = ∪k−1 j=0 α (Ω). Then, using the Peierls-Bogoliubov inequality once again, n−1 j m−1 jk log TrB e−θ( j=0 α (H)) ≥ log TrB e−θ( j=0 α (Hk )) − k||H||.

P

Similarly

P

P

log TrB e−θ(

n−1 j=0

αj (H))

P

≤ log TrB e−θ(

m j=0

αjk (Hk ))

+ k||H||.

n−1 j jk m jk Since ∪m−1 j=0 α (Ωk ) ⊂ ∪j=0 α (Ω) ⊂ ∪j=0 α (Ωk ), it follows that ⎛ ⎞ k−1 k−1 ! 1 Pα (H, Ω; δ) = Pαk ⎝ αj (H), αj (Ω); δ ⎠ , k j=0 j=0

k−1 and hence Pα (H) = k1 Pαk ( j=0 αj (H)), which proves (v). k−1 Set Hk = j=0 αj (H) and Hk =

k−1

αj (H + α(K) − K) = Hk + αk (K) − K.

j=0

Then by (iv) and (v) we have |Pα (H) − Pα (H + α(K) − K)| = Thus (vi) follows.

2||K|| 1 |Pαk (Hk ) − Pαk (Hk )| ≤ . k k

It follows from Proposition 9.1.2 that if ϕ is an α-invariant state then −ϕ(H) ≤ Pα (H) for any H ∈ Asa . It is interesting that the converse is also true.

9.1 Pressure

137

Proposition 9.1.5. Suppose ht(α) < ∞. Let ϕ be a self-adjoint linear functional on A. Then ϕ is an α-invariant state if and only if −ϕ(H) ≤ Pα (H) for all H ∈ Asa . Proof. As we already remarked, in one direction the result follows from the inequality −ϕ(H) ≤ Pα (H) − hϕ (α) for α-invariant states, Proposition 9.1.2. Conversely, if −ϕ(H) ≤ Pα (H) for all H ∈ Asa then by Proposition 9.1.4(vi) 1 1 1 −ϕ(α(H) − H) = − ϕ(α(nH) − nH) ≤ Pα (α(nH) − nH) = Pα (0). n n n Since Pα (0) = ht(α) < ∞, letting n → ∞ we get ϕ(H) ≤ ϕ(α(H)). Applying this also to −H we see that ϕ is α-invariant. Furthermore, by Proposition 9.1.4(i),(ii) 1 1 1 −ϕ(H) = − ϕ(nH) ≤ Pα (nH) ≤ ht(α) + ||H|| −→ ||H||, n→∞ n n n so that ||ϕ|| ≤ 1. For c ∈ R we have −cϕ(1) ≤ Pα (c1) = ht(α) − c.

Hence ϕ(1) = 1, and ϕ is a state.

Deﬁnition 9.1.6. We say that an α-invariant state ϕ is an equilibrium state at H if Pα (H) = hϕ (α) − ϕ(H). If ϕ is an equilibrium state then, by Proposition 9.1.2, hϕ (α) − ϕ(H) = sup(hψ (α) − ψ(H)), ψ

where the sup is taken over all α-invariant states. Even though we cannot prove in general that the pressure is a convex function, this is the case in our main examples. The notion of equilibrium state is then closely related to diﬀerentiability of the pressure. Recall that if F is a real convex continuous function on a real Banach space X, then a linear functional f on X is called a subgradient, or tangent functional, of F at x ∈ X if F (x + y) − F (x) ≥ f (y) for any y ∈ X. We shall usually identify self-adjoint linear functionals on A with real linear functionals on Asa . Proposition 9.1.7. Suppose ht(α) < ∞ and the pressure is a convex function on Asa . Then (i) if ϕ is an equilibrium state at H then −ϕ is a subgradient for the pressure at H; (ii) if −ϕ is a subgradient for the pressure at H then ϕ is an α-invariant state.

138

9 Variational Principle

Proof. Assume ϕ is an equilibrium state at H. Let K ∈ Asa . Then by Proposition 9.1.2 Pα (H + K) − Pα (H) ≥ (hϕ (α) − ϕ(H + K)) − (hϕ (α) − ϕ(H)) = −ϕ(K), so −ϕ is a subgradient. Assume now that ϕ is a subgradient at H. If K ∈ Asa then by Proposition 9.1.4(vi) −ϕ(α(K) − K) ≤ Pα (H + α(K) − K) − Pα (H) = 0. Applying this also to −K we see that ϕ is α-invariant. Now note that ||ϕ|| ≤ 1 by Proposition 9.1.4(iv). By Proposition 9.1.4(ii) we have also −c ≥ −cϕ(1) for any c ∈ R. Hence ϕ(1) = 1, and ϕ is a state.

9.2 The variational principle We shall prove the variational principle for the following class of systems. Deﬁnition 9.2.1. A system (A, α) consisting of a unital C∗ -algebra A and an automorphism α of A is called asymptotically abelian with locality if there is a dense α-invariant ∗-subalgebra A of A such that for each pair a, b ∈ A the C∗ -algebra generated by a and b is ﬁnite dimensional, and for some p = p(a, b) ∈ N we have [αj (a), b] = 0 whenever |j| ≥ p. We call elements of A for local operators and ﬁnite dimensional C∗ subalgebras of A for local algebras. We may assume that 1 ∈ A. Since each ﬁnite dimensional C∗ -algebra is singly generated, an easy induction argument shows that the C∗ -algebra generated by a ﬁnite set of local operators is ﬁnite dimensional. In particular, if A is separable then A is an AF-algebra. Note also that for each local algebra N there is p ∈ N such that [αj (a), b] = 0 for all a, b ∈ N whenever |j| ≥ p. Theorem 9.2.2. Let (A, α) be asymptotically abelian with locality, and H ∈ Asa . Then Pα (H) = sup{hϕ (α) − ϕ(H)}, ϕ

where the supremum is taken over all α-invariant states of A. In particular, the topological entropy satisﬁes ht(α) = sup hϕ (α). ϕ

Consider ﬁrst the case when there exists a ﬁnite dimensional C∗ -subalgebra N of A such that H ∈ N , αj (N ) commutes with N for j = 0, and the C∗ -algebra generated by αj (N ), j ∈ Z, coincides with A.

9.2 The variational principle

139

Lemma 9.2.3. Under the above assumptions there exists an α-invariant state ϕ such that n−1 j 1 Pα (H) = hϕ (α) − ϕ(H) = lim log Tr∨n−1 αj (N ) e− j=0 α (H) . j=0 n→∞ n

P

Proof. Of course, this is essentially an abelian situation, so the result can be deduced from the classical variational principle. But we shall give a selfcontained argument. First note that if A1 and A2 are commuting ﬁnite dimensional C∗ -algebras, and ai ∈ Ai , ai ≥ 0, i = 1, 2, then TrA1 ∨A2 (a1 a2 ) ≤ TrA1 (a1 )TrA2 (a2 ), since if pi is a minimal projection in Ai , i = 1, 2, then p1 p2 is either zero or minimal in A1 ∨ A2 . Hence the sequence & ' n−1 j log Tr∨n−1 αj (N ) e− j=0 α (H)

P

j=0

n

is subadditive, so by Lemma 1.1.2 the limit in the formulation of the lemma exists and coincides with the inﬁmum. We denote it by P˜α (H). It is clear that P˜α (H) ≥ Pα (H). Consider M = N ⊗Z , the inﬁnite C∗ -tensor product of N with itself. Denote the n-th factor of M by Nn . Let β be the shift to the right on M , and π: M → A the homomorphism which intertwines β with α, and identiﬁes N0 with N . Set I = Ker π. For each n ∈ N let Mn = N0 ⊗ . . . ⊗ Nn−1 , In = I ∩ Mn , πn = π|Mn . j Consider the state fn on πn (Mn ) = ∨n−1 j=0 α (N ) with density operator

Tr∨n−1 αj (N ) e−

P

n−1 j=0

αj (H)

−1

e−

P

n−1 j=0

αj (H)

.

j=0

Then fn ◦ πn is a state on Mn , and identifying M with Mn⊗Z we denote the corresponding product-state on M by ψn . The state ψn is β n -invariant. Set n−1 ϕn = n−1 j=0 ψn ◦ β j . Then ϕn is β-invariant. Using concavity of entropy, Theorem 3.2.5(ii), and that hψn (β n ) = S(fn ◦ πn ) by Example 3.2.6(i), we obtain n−1 1 1 1 1 hψn ◦β j (β n ) = hψn (β n ) = S(fn ◦ πn ) hϕn (β n ) ≥ 2 n n j=0 n n ⎛ ⎞ n−1 1 n−1 j 1 1 αj (H)⎠ = S(fn ) = log Tr∨n−1 αj (N ) e− j=0 α (H) + fn ⎝ j=0 n n n j=0 ⎛ ⎞ n−1 1 αj (H)⎠ ≥ P˜α (H) + fn ⎝ n j=0

hϕn (β) =

P

140

9 Variational Principle

⎛ ⎞ n−1 1 β j (H)⎠ = P˜α (H) + ϕn (H). = P˜α (H) + ψn ⎝ n j=0 Let ϕ˜ be any weak∗ limit point of the sequence {ϕn }n . Then ϕ˜ is β-invariant. Let D be a maximal abelian subalgebra of N0 containing H. Then D is in the centralizer of the state ϕn . Hence, using Proposition 3.1.6, by the same reasoning as in Example 3.2.6(i) we get 1 Hϕn (D, β(D), . . . , β k−1 (D)). k∈N k

hϕn (β) = hϕn (D; β) = inf

By Proposition 3.1.12 the map ψ → hψ (D; β) is weak∗ upper semicontinuous. Hence hϕ˜ (β) ≥ P˜α (H) + ϕ(H). ˜ Now observe that ϕ˜ is zero on I. Indeed, if x ∈ In then β j (x) ∈ Im for j = 0, . . . , m − n and m ≥ n, whence |ϕm (x)| ≤

m−1 1 n−1 |(ψm ◦ β j )(x)| ≤ ||x||, m j=m−n+1 m

so ϕ(x) ˜ = 0. Thus ϕ˜ deﬁnes a state ϕ on A. We have hϕ (α) = hϕ˜ (β) ≥ P˜α (H) + ϕ(H), where the ﬁrst equality follows from Theorem 3.2.2(ii). Since by Proposition 9.1.2, hϕ (α) − ϕ(H) ≤ Pα (H) ≤ P˜α (H), the proof of the lemma is complete.

We shall reduce the general case to the case considered above by replacing α by its powers. For this suppose that N is a local subalgebra of A, and H ∈ N . Choose p ≥ 1 such that αj (N ) commutes with N whenever |j| ≥ p. k−p j j For k ≥ p set Mk = ∨k−p j=0 α (N ), Hk = j=0 α (H). Then Hk ∈ Mk , and αjk (Mk ) commutes with Mk for j = 0. Lemma 9.2.4. For every ﬁnite subset Ω of N we have Pα (H, Ω) ≤ lim inf k→∞

1 (Hk ). P k jk k α |∨j∈Z α (Mk )

Proof. The idea of the proof is to reduce to the situation of Lemma 9.2.3 by showing that the contribution of the indices in the intervals [jk − p + 1, jk − 1], j ∈ N, becomes negligible for large k. Fix δ > 0. Choose m0 ∈ N such that 2(p − 1)||a|| < δ for a ∈ Ω. m0

9.2 The variational principle

141

Take k ≥ m0 + p. Let n ∈ N. Then (m − 1)k ≤ n < mk for some m ∈ N. Set jk i m B0 = ∨m j=0 α (Mk ) = ∨i∈X α (N ), where X = ∪j=0 [jk, (j + 1)k − p]. Put B = B0 ⊕ . . . ⊕ B0 . m0

Choose a conditional expectation E: A → B0 , and deﬁne unital completely positive maps ψ: B → A and θ: A → B by ψ(b1 , . . . , bm0 ) =

m0 1 α−i+1 (bi ), m0 i=1

θ(a) = (E(a), (E ◦ α)(a), . . . , (E ◦ αm0 −1 )(a)). / B0 }. Then Let a ∈ A, and put S = {i | 0 ≤ i ≤ m0 − 1, αi (a) ∈ (ψ ◦ θ)(a) − a =

1 −i ((α ◦ E ◦ αi )(a) − a). m0 i∈S

Assume now that a = αl (b) for some b ∈ Ω and l, 0 ≤ l ≤ n − 1. Then l < mk, and hence [l, l + m0 − 1] is contained in Y = [0, (m + 1)k − p]. The set X is a union of m + 1 intervals of length k − p ≥ m0 , and it is obtained from Y by removing m open intervals of length p. Hence [l, l + m0 − 1] can intersect at most one of the latter intervals. It follows that S contains at most p − 1 elements. Hence ||(ψ ◦ θ)(a) − a|| ≤

2(p − 1)||a|| < δ. m0

Consequently ⎛ ⎞ n−1 n−1 ! P⎝ αj (H), αj (Ω); δ ⎠ ≤ log TrB e−θ(

P

j=0

n−1 j=0

αj (H))

.

j=0

Denoting by #I the number of integer points in a set I ⊂ R, note that for Xi = [i, i + n − 1], 0 ≤ i ≤ m0 − 1, we have #(Xi X) ≤ #(Y \Xi ) + #(Y \X) = ((m + 1)k − p + 1 − n) + m(p − 1) ≤ (m + 1)k − p + 1 − (m − 1)k + m(p − 1) < mp + 2k. Since αj (H) ∈ B0 for j ∈ X, we then get ⎛ ⎛ ⎞ ⎞ n−1 m j jk j j (E ◦ αi ) ⎝ ⎠ ⎝ ⎠ α (H) α (H) − α (Hk ) = E α (H) − j=0 j=0 j∈Xi j∈X

142

9 Variational Principle

⎛ ⎞ j j ⎝ ⎠ α (H) = E α (H) − ≤ (mp + 2k)||H||. j∈X\Xi j∈Xi \X By the Peierls-Bogoliubov inequality, Corollary 2.3.11, we obtain n−1 j i m jk TrB0 e−(E◦α )( j=0 α (H)) ≤ e(mp+2k)||H|| TrB0 e− j=0 α (Hk ) ,

P

so

P

TrB e−θ(

n−1 j=0

P

αj (H))

≤ m0 e(mp+2k)||H|| TrB0 e−

P

m j=0

αjk (Hk )

.

Taking the logarithm, dividing by n and letting n → ∞, we get 1 p||H|| 1 + lim log Tr∨m−1 αjk (Mk ) e− j=0 k k m→∞ m p||H|| 1 + Pαk |∨j∈Z αjk (Mk ) (Hk ), = k k

Pα (H, Ω; δ) ≤

P

m−1 j=0

αjk (Hk )

where the last equality follows from Lemma 9.2.3. Since this is true for any k ≥ m0 + p, we get the result.

Proof of Theorem 9.2.2. The inequality ≥ has been proved in Proposition 9.1.2. Since the pressure is continuous by Proposition 9.1.4(iv), to prove the converse inequality it suﬃces to consider local H. Then by Lemma 9.2.4 we have only to show that if H is contained in a local algebra N then with k−p Hk = j=0 αj (H) as in Lemma 9.2.4 sup(hϕ (α) − ϕ(H)) ≥ lim inf k→∞

ϕ

1 (Hk ). P k jk k α |∨j∈Z α (Mk )

By Lemma 9.2.3, for each k ∈ N there exists an αk -invariant state ψk on ∨j∈Z αjk (Mk ) such that hψk (αk |∨j∈Z αjk (Mk ) ) − ψk (Hk ) = Pαk |∨j∈Z αjk (Mk ) (Hk ). By Proposition 3.2.7 we can extend ψk to an αk -invariant state ϕ˜k on A such that hϕ˜k (αk ) ≥ hψk (αk |∨j∈Z αjk (Mk ) ). Set ϕk =

k−1 1 ϕ˜k ◦ αj . Then, as in the proof of Lemma 9.2.3, k j=0

hϕk (α) ≥ Since

1 1 hϕ˜k (αk ) ≥ hψk (αk |∨j∈Z αjk (Mk ) ). k k

9.2 The variational principle

ϕk (H) =

143

k−1 1 1 p−1 p−1 1 ϕ˜k (αj (H)) ≤ ϕ˜k (Hk ) + ||H|| = ψk (Hk ) + ||H||, k j=0 k k k k

we get 1 1 (p − 1)||H|| hψ (αk |∨j∈Z αjk (Mk ) ) − ψk (Hk ) − k k k k (p − 1)||H|| 1 , = Pαk |∨j∈Z αjk (Mk ) (Hk ) − k k

hϕk (α) − ϕk (H) ≥

and the proof is complete.

Corollary 9.2.5. If (A, α) is asymptotically abelian with locality then the pressure Pα (H) is a convex function of H. Proof. Indeed, it is the supremum of the aﬃne functions H → hϕ (α) − ϕ(H).

Corollary 9.2.6. If (A1 , α1 ) and (A2 , α2 ) are asymptotically abelian with locality then ht(α1 ⊗ α2 ) = ht(α1 ) + ht(α2 ). Proof. If ϕi is an αi -invariant state, i = 1, 2, then by Theorem 3.2.2(iv), Proposition 6.2.7 and Theorem 6.2.2(iv), used in that order, hϕ1 (α1 ) + hϕ2 (α2 ) ≤ hϕ1 ⊗ϕ2 (α1 ⊗ α2 ) ≤ ht(α1 ⊗ α2 ) ≤ ht(α1 ) + ht(α2 ). Taking the supremum over ϕi we get the conclusion.

In the next section we shall need an explicit formula for the pressure, which is a consequence of our proof of the variational principle. Corollary 9.2.7. Assume A is separable. Let N be a local algebra. Then there exist a sequence {An }n of local algebras containing N and three sequences {pn }n , {mn }n , {kn }n of positive integers such that (i) αp (An ) commutes with An whenever |p| ≥ pn ; pn (ii) → 0 as n → ∞; kn j 1 log Tr∨j∈In αj (An ) e− j∈In α (H) for all H ∈ N sa , (iii) Pα (H) = lim n→∞ kn mn mn −1 [jkn , (j + 1)kn − pn ] ∩ Z. where In = ∪j=0

P

Proof. Let {An }n be an increasing sequence of local algebras containing N such that ∪n An is dense in A, Ωn a ﬁnite subset of An such that the linear span of Ωn is An . Let {pn }n be a sequence satisfying condition (i). By Lemma 9.2.4 ⎛ ⎞ k−p n 1 Pα (H, Ωn ) ≤ lim inf Pαk |∨j∈Z αjk (An,k ) ⎝ αj (H)⎠ for H ∈ N sa , k→∞ k j=0

144

9 Variational Principle

n j where An,k = ∨k−p j=0 α (An ). On the other hand, by Theorem 9.2.2 and its proof ⎛ ⎞ k−p n 1 Pα (H) ≥ lim sup Pαk |∨j∈Z αjk (An,k ) ⎝ αj (H)⎠ for H ∈ N sa , n ∈ N. k→∞ k j=0

Choose a countable dense subset X of N sa . Since Pα (H, Ωn ) Pα (H) for any H ∈ X, we can ﬁnd a sequence {kn }n such that condition (ii) is satisﬁed and ⎛ ⎞ kn −pn 1 αj (H)⎠ for H ∈ X. Pα (H) = lim Pαkn |∨j∈Z αjkn (An,kn ) ⎝ n→∞ kn j=0 Since by Lemma 9.2.3

⎛

Pαkn |∨j∈Z αjkn (An,kn ) ⎝

kn −pn

⎞ αj (H)⎠

j=0

1 − log Tr∨j∈In,m αj (An ) e m→∞ m

= lim

P

j∈In,m

αj (H)

,

where In,m = ∪m−1 j=0 [jkn , (j + 1)kn − pn ], we can choose a sequence {mn }n such that condition (iii) is satisﬁed for all H ∈ X. But then it is satisﬁed for all H ∈ N sa by Proposition 9.1.4(iv) and the Peierls-Bogoliubov inequality, Corollary 2.3.11.

9.3 KMS-states Let A be a ﬁnite dimensional C∗ -algebra, H ∈ Asa a self-adjoint element. By the thermodynamic inequality, Proposition 2.2.5, for any state ϕ on A we have S(ϕ) − ϕ(H) ≤ log Tr(e−H ), and equality holds if and only if Qϕ = Tr(e−H )−1 e−H , in which case ϕ is called the Gibbs state. Consider the one-parameter automorphism group σ of A deﬁned by σt (a) = eitH ae−itH . Then one can check that if A is a full matrix algebra then the Gibbs state is the unique state with the property ϕ(ab) = ϕ(bσi (a)). In this section we shall show that equilibrium states have a similar property. Let A be a C∗ -algebra, σ a strongly continuous one-parameter automorphism group of A. Recall that an element a ∈ A is called an entire analytic element for σ if the map R t → σt (a) ∈ A extends to an entire analytic function. Let β ∈ R. Then a state ϕ on A is called a σ-KMSβ -state if

9.3 KMS-states

145

ϕ(ab) = ϕ(bσiβ (a)) for any entire analytic element a ∈ A and any b ∈ A. If β = 0 then ϕ is automatically σ-invariant by [30, Proposition 5.3.3]. Furthermore, by [30, Corollary 5.3.9 and Theorem 5.3.10] the normal state on πϕ (A) deﬁned by ϕ is faithful, and its modular group is given by σtϕ (πϕ (a)) = πϕ (σ−βt (a)). In particular, if ϕ is faithful and β = 0 then there exists at most one oneparameter group σ for which ϕ is a KMSβ -state. We shall use the following form of the KMS-condition which does not involve analytic continuation of σ and hence is better suited for passing to limits. By [30, Proposition 5.3.12] a state ϕ is a σ-KMSβ if and only if ˆ (9.1) f (t)ϕ(aσt (b))dt = fˆ(t + iβ)ϕ(σt (b)a)dt R

R

for all a, b ∈ A and f ∈ D, where D is the space of compactly supported C ∞ -functions, and fˆ(t) = (2π)−1/2 R f (x)e−ixt dx is the Fourier transform of f . Return now to our setting of asymptotically abelian systems with locality. Let H be a local self-adjoint element. Our ﬁrst goal is to show that there is a natural dynamics associated with H. For a subset I ⊂ Z and a local operator a put δH,I (a) = [αj (H), a]. j∈I

Then δH,I is a derivation on the algebra of local operators. Put σtH,I = exp(itδH,I ). We shall write δH for δH,Z , and σtH for σtH,Z . It will follow from the next lemma that σ H,I is a well-deﬁned strongly continuous one-parameter automorphism group of A. Lemma 9.3.1. Let r ∈ N, and let cn be the number of n-tuples (k1 , . . . , kn ) ∈ Zn such that min |kj − ki | ≤ r for j = 1, . . . , n, (9.2) 0≤i<j

where k0 = 0. Put c0 = 1. Then the series convergence.

∞ cn n z has inﬁnite radius of n! n=0

Proof. For m ∈ N let Kn (m) denote the set of n-tuples (k1 , . . . , kn ) such that condition (9.2) is satisﬁed, and furthermore max ki = min ki + m − 1,

0≤i≤n

0≤i≤n

146

9 Variational Principle

that is, the minimal interval containing k0 , . . . , kn has length m−1. Let cn (m) denote the number of elements inKn (m) for n ≥ 1, and put c0 (1) = 1 and ∞ c0 (m) = 0 for m ≥ 2. Then cn = m=1 cn (m). We assert that the following recurrence relation holds: cn+1 (m) = mcn (m) + 2

r

cn (m − k),

k=1

with the convention that cn (l) = 0 for l ≤ 0. To show this let (k1 , . . . , kn ) ∈ ∪m−1 k=0 Kn (m − k). Let [a, b] be the minimal interval containing k0 , . . . , kn . We want to see how many choices there are for kn+1 such that (k1 , . . . , kn+1 ) ∈ Kn+1 (m). If (k1 , . . . , kn ) ∈ Kn (m) then we can take any integer in [a, b] for kn+1 , so there are m choices. If (k1 , . . . , kn ) ∈ Kn (m − k) with 1 ≤ k ≤ r, there are two choices, kn+1 = a − k and kn+1 = b + k. And if k > r, no kn+1 with the required properties exists. This computation proves the recurrence formula. By virtue of (9.2) we have cn (m) = 0 for m > nr + 1. Thus by the recurrence formula max cn+1 (m) ≤ (nr + 1 + 2r) max cn (m), m

m

and consequently max cn+1 (m) ≤ m

n $

(1 + r(k + 2)).

k=0

Hence the series f (x, y) =

∞ ∞ cn (m) n m x y n! n=0 m=1

deﬁnes an analytic function in a neighborhood of (0, 0) ∈ C2 . We have, using the recurrence formula, ∞ ∞ ∂f cn+1 (m) n m = x y ∂x n! n=0 m=1

=

∞ ∞

m

n=0 m=1 ∞ ∞

∞ ∞ r cn (m − k) n m cn (m) n m x y +2 x y n! n! n=0 m=1 k=1

cn (m) n m = m yk . x y + 2f (x, y) n! n=0 m=1 r

k=1

On the other hand, y

∞ ∞ ∂f cn (m) n m m = x y . ∂y n! n=0 m=1

9.3 KMS-states

147

We thus get a partial diﬀerential equation ∂f ∂f yk . −y = 2f (x, y) ∂x ∂y r

k=1

Working in a neighbourhood of a point (0, y0 ) with y0 = 0 consider the change of variables x = u + v, y = ev−u . Then

∂x ∂y = 1 and = −y, and the above equation can be written as ∂u ∂u ∂ log f ∂ yk = −2 . ∂u ∂u k r

k=1

Hence log f = g(v) − 2

r yk k=1

k

=g

x + log y 2

−2

r yk

k

k=1

for a function g. To ﬁnd g let x = 0. Since f (0, y) = y, we get g

log y 2

= log y + 2

r yk

k

k=1

that is, g(v) = 2v + 2

r e2kv k=1

k

,

.

We thus have log f (x, y) = x + log y + 2

r y k ekx

k

k=1

whence

f (x, y) = y exp x + 2

r yk k=1

k

−2

r yk k=1

k

,

(ekx − 1) .

We see that f is analytic on C2 . In particular, the series has inﬁnite radius of convergence.

∞ cn n z = f (z, 1) n! n=0

Proposition 9.3.2. Let (A, α) be asymptotically abelian with locality, H ∈ Asa a local self-adjoint element. Then the series σzH,I (a) =

∞ (iz)n n δ (a) n! H,I n=0

148

9 Variational Principle

converges absolutely in norm for any z ∈ C and any local operator a. Moreover, if N is a local algebra containing H, Ω ⊂ C a compact subset and ε > 0, then (i) there exists a ﬁnite set I0 ⊂ Z such that σzH (a) − σzH,I (a) ≤ εa for any I ⊃ I0 , z ∈ Ω and a ∈ N ; (ii) there is δ > 0 such that if K ∈ N sa with H − K < δ then σzH (a) − σzK (a) ≤ εa for all z ∈ Ω and a ∈ N . Proof. Let N be a local algebra containing H. Let r ∈ N be so large that αn (N ) commutes with N for |n| > r. For a ∈ N , n (a) = [αkn (H), [. . . [αk1 (H), a] . . .]]. δH k1 ,...,kn ∈Z

The commutator in the sum can be nonzero only when the n-tuple (k1 , . . . , kn ) satisﬁes condition (9.2), hence with cn as in Lemma 9.3.1 we have n δH (a) ≤ cn 2n Hn a. ∞ n It follows from that lemma that the series n=0 (z n /n!)δH (a) converges in norm for any z ∈ C. n If I ⊂ Z then in the formula for δH,I (a) we sum over a smaller set, so convergence holds in this case too. Moreover, for any m ∈ N, if I contains n n Z ∩ [−mr, mr] then δH,I (a) = δH (a) for n ≤ m, whence

σzH (a) − σzH,I (a) ≤ 2

cn 2n |z|n Hn a. n! n>m

This implies part (i). This also implies that to show (ii) it suﬃces to prove a similar statement for σ H,I and σ K,I for someﬁnite I. But then it is immediate, since σzH,I (a) = eizHI ae−izHI , where HI = j∈I αj (H).

We can now formulate the main result of the present section. Theorem 9.3.3. Let (A, α) be asymptotically abelian with locality. Assume ht(α) < ∞. If H is a local self-adjoint operator in A and ϕ is an equilibrium state at H then ϕ is a σ H -KMS1 -state. In particular, if ht(α) = hϕ (α) then ϕ is a trace. By Corollary 9.2.5 the pressure Pα is a convex function of H, so that Proposition 9.1.7 shows that Theorem 9.3.3 will follow from the following more general result.

9.3 KMS-states

149

Theorem 9.3.4. If −ϕ is a subgradient of Pα at H then ϕ is a σ H -KMS1 state. The proof is based on Corollary 9.2.7 which, as we shall see, implies that for generic H a subgradient of the pressure at H can be obtained as a limit of Gibbs states on local algebras. Note that if B is a ﬁnite dimensional C∗ -algebra, H0 ∈ B sa , and ϕ is the state with density operator Qϕ = TrB (e−H0 )−1 TrB (e−H0 ) then for any H ∈ B sa we have d log TrB (e−H0 −tH )|t=0 = −TrB (e−H0 )−1 TrB (He−H0 ) = −ϕ(H), dt so that −ϕ is the gradient of the function H → log TrB (e−H ) at H0 . Note also that this function is convex, being the supremum of the aﬃne functions H → S(ψ) − ψ(H) by the thermodynamic inequality, Proposition 2.2.5. Lemma 9.3.5. Let N be a local algebra, H ∈ N sa , −ϕ ∈ N ∗ a subgradient of (Pα )|N sa at H. Let E: A → N be a conditional expectation. Then for every function f in the space D of C ∞ -functions with compact support and a, b ∈ N we have fˆ(t)ϕ(aE(σtH (b)))dt − fˆ(t + i)ϕ(E(σtH (b))a)dt R

≤ ||a||

R

R

(|fˆ(t)| + |fˆ(t + i)|)||σtH (b) − E(σtH (b))||dt.

Proof. We may assume that A is separable. Indeed, for every K ∈ Asa there exists a sequence {Ωn }∞ n=1 of ﬁnite subsets of A such that Pα (H, Ωn ) Pα (H). Then for any α-invariant C∗ -subalgebra A0 containing K and the Ωn ’s we have Pα (K) = Pα|A0 (K). It follows that there exists a separable α-invariant C∗ -subalgebra A0 containing N such that (A0 , α|A0 ) is asymptotically abelian with locality and Pα|A0 (K) = Pα (K) for a dense family of elements K in N sa . But then the equality Pα|A0 (K) = Pα (K) holds for all K ∈ N sa by continuity. Therefore we may replace A by A0 and thus assume that A is separable. First consider the case when (Pα )|N sa has a unique subgradient at H. With the notation of Corollary 9.2.7 consider the state fn on ∨j∈In αj (An ) with density operator

Tr∨j∈In αj (An ) e−

P

j∈In

αj (H)

−1

e−

P

Then deﬁne a positive linear functional ϕn on N by ϕn (x) =

1 fn (αj (x)). kn mn j∈In

j∈In

αj (H)

.

150

9 Variational Principle

Note that ||ϕn || = ϕn (1) ≤ 1. As we remarked before the formulation of the lemma, the functional −fn is the gradient of the convex function x → log Tr∨j∈In αj (An ) (e−x ) on (∨j∈In αj (An ))sa at the point j∈In αj (H). Hence −ϕn is the gradient of the convex function j 1 log Tr∨j∈In αj (An ) e− j∈In α (x) N sa x → kn mn

P

at H. It follows that any limit point of the sequence {−ϕn }n is a subgradient of (Pα )|N sa at H. Since the latter is unique by assumption, ϕn → ϕ as n → ∞. Since fn is a σ H,In -KMS1 -state, by (9.1) for every j ∈ In we have

R

fˆ(t)fn (αj (a)σtH,In (αj (b)))dt =

R

fˆ(t + i)fn (σtH,In (αj (b))αj (a))dt. (9.3)

Note that σtH,In (αj (b)) = αj (σtH,In −j (b)). Fix q ∈ N, and put n −1 In,q = ∪m j=0 [jkn + q, (j + 1)kn − pn − q] ∩ Z,

which is nonempty for suﬃciently large n. Then In − j contains [−q, q] ∩ Z for j ∈ In,q . By Proposition 9.3.2(i), if q is large enough then σtH,In −j (b) is arbitrarily close to σtH (b) for every j ∈ In,q and t in a ﬁxed compact subset of R. But then σtH,In (αj (b)) − αj (E(σtH (b))) is arbitrarily close to αj (σtH (b) − E(σtH (b))). It follows that for all suﬃciently large n ∈ N 1 H,I fˆ(t) fn αj (a)σt n (αj (b)) − αj (a)αj (E(σtH (b))) dt kn mn R j∈In,q ≤ ||a||

R

|fˆ(t)| ||σtH (b) − E(σtH (b))||dt + ε(q),

where ε(q) → 0 as q → ∞. Since |In,q |/|In | → 1 as n → ∞, letting n → ∞ we may replace averaging over the set In,q by averaging over In , and then obtain 1 H,In j j H ˆ ˆ (α (b)) dt − f (t)ϕ(aE(σt (b)))dt f (t)fn α (a)σt kn mn R j∈In R ≤ ||a||

R

|fˆ(t)| ||σtH (b) − E(σtH (b))||dt + εn ,

→ ∞. Since an analogous estimate holds for the integral with εn → 0 as n ˆ(t + i)ϕ(E(σ H (b))a)dt, we obtain the conclusion of the lemma by virtue f t R of (9.3). To ﬁnish the proof we need a few facts about diﬀerentiability of convex functions, see e.g. [179, Section 25]. Let F be a convex function on Rn . For

9.4 Notes

151

any x ∈ Rn the function F is diﬀerentiable at x if and only if it has a unique subgradient at x, and then this subgradient coincides with the gradient of F at x. The points where F is diﬀerentiable form a dense Gδ -set of full measure. Finally, for any x ∈ Rn the set of subgradients of F at x coincides with the closed convex hull of subgradients which are limits of sequences of the form {fn (xn )}n such that xn → x, F is diﬀerentiable at xn , and fn (xn ) is the gradient of F at xn . Returning to the proof of the lemma, we conclude that ϕ lies in the closed convex hull of linear functionals ϕ˜ for which there exist sequences {Hn }n ⊂ N sa and {ϕn }n ⊂ N ∗ such that Hn → H, ϕn → ϕ, ˜ and −ϕn is the unique subgradient of (Pα )|N sa at Hn . Since for ϕn the lemma is already proved (for Hn instead of H), using Proposition 9.3.2(ii) we conclude that the conclusion of the lemma is true for ϕ. ˜ But then it is true for any functional in the closed convex hull of the ϕ’s. ˜

Proof of Theorem 9.3.4. If −ϕ is a subgradient of Pα at H then −ϕ|N is a subgradient of (Pα )|N sa at H for any local algebra N containing H. Thus applying Lemma 9.3.5 to an increasing sequence of local algebras we conclude that the equality H ˆ f (t)ϕ(aσt (b))dt = fˆ(t + i)ϕ(σtH (b)a)dt R

R

holds for all f ∈ D and all local a, b, hence for all a, b ∈ A. But this is one of the equivalent forms of the KMS-condition.

Remark 9.3.6. Under the assumptions of Theorem 9.3.3, any weak∗ limit point of a sequence on which the supremum in the variational principle is attained is a subgradient of the pressure, hence a σ H -KMS1 state. In order to deﬁne the dynamics σ H we need some assumptions on α, such as asymptotic abelianness with locality. On the other hand, the question when hϕ (α) = ht(α) < ∞ forces ϕ to be tracial makes sense in general. In Chap. 13 we shall give an example of an asymptotically abelian system (A, α) with zero topological entropy and many nontracial invariant states. In Chaps. 11 and 12 we shall see examples of an automorphism α with a unique invariant state ϕ such that 0 = hϕ (α) < ht(α).

9.4 Notes Let B be a ﬁnite dimensional C∗ -algebra, A = B ⊗Z the inﬁnite C∗ -tensor product, α the shift to the right on A. For a subset Λ ⊂ Z denote by AΛ the C∗ -subalgebra of A generated by the factors corresponding to the elements of Λ. Given an α-invariant state ϕ on A, the quantity s(ϕ) = lim

n→∞

1 S(ϕ|A[0,n−1] ) n

152

9 Variational Principle

is called the mean entropy of ϕ. We always have hϕ (α) ≤ s(ϕ). By an interaction potential one understands a map Φ which associates a self-adjoint element Φ(X) ∈ AX to every X ⊂ Z. We assume that Φ is translationally invariant, Φ(n + X) = Φ(X), and that the series Φ(X) |X|

H=

X0

is absolutely convergent. For a ﬁnite subset Λ ⊂ Z deﬁne the local Hamiltonian by H(Λ) = Φ(X). X⊂Λ

The pressure (at inverse temperature β = 1) is deﬁned by P (Φ) = lim

n→∞

1 log TrA[0,n−1] (e−H([0,n−1]) ). n

Then the variational principle of Ruelle [185] (classical lattices, B is abelian) and Robinson [180] (quantum lattices, B is a full matrix algebra) states that P (Φ) = sup{s(ϕ) − ϕ(H)}. ϕ

Moreover, as was proved by Lanford and Robinson [115], see also [30] for a detailed exposition, under suitable conditions on the potential, the limit σt (x) = lim (Ad eitH([−n,n]) )(x) n→∞

deﬁnes a one-parameter automorphism group of A, and if the supremum above is attained at a state ϕ, then ϕ is a σ-KMS1 -state. Note that if Φ has ﬁnite range in the sense that there exist only ﬁnitely many sets X 0 such that Φ(X) = 0, then the above σ coincides with the dynamics σ H deﬁned in Sect. 9.3 by a result of Araki [7], who also proved that local operators are σ H -analytic, Proposition 9.3.2 and Lemma 9.3.1. In the classical case there is a more general result subsuming also the variational principle for topological entropy. Let (X, T ) be a topological dynamical system, and f ∈ C(X). For an open cover U of X put f (x) q(T, f, U) = inf , inf e V

V ∈V

x∈V

where the ﬁrst inﬁmum is taken over all ﬁnite subcovers V of U, and then denoting f + f ◦ T + . . . + f ◦ T n−1 by Sn f deﬁne P (T, f ) = sup lim sup U

n→∞

1 −k U). log q(T, Sn f, ∨n−1 k=0 T n

9.4 Notes

153

Then P (T, f ) = supµ {hµ (T ) + X f dµ}. This was proved by Ruelle [186] under additional assumptions, and by Walters [230] in general, see [231] for details. It is therefore only natural to try to generalize the variational principle for lattice systems to C∗ -dynamical systems using dynamical entropy instead of mean entropy. The ﬁrst attempt in this direction was made by Narnhofer [125]. She studied the question whether a state ϕ on a lattice system such that hϕ (α)−ϕ(H) = sup{hψ (α)−ψ(H)} satisﬁes the KMS-condition. Moriya [123] approached this problem diﬀerently by proving that sup{hψ (α) − ψ(H)} = sup{s(ψ) − ψ(H)}. ψ

ψ

This implies in particular that the pressure P (Φ) deﬁned above coincides with the pressure Pα (H). The pressure Pα (H) for nuclear C∗ -algebras was introduced by the authors [140], who obtained the main results of this chapter. But of course most of the arguments are already found in the above mentioned papers. The notion of pressure was extended to exact C∗ -algebras by Kerr and Pinzari [110]. For further results and relevant examples of systems see [105], [110], [213], [77], [3]. There are a number of nonasymptotically abelian systems where the topological entropy is the supremum of the dynamical entropies. E.g. Choda [43] showed that both the topological entropy and the dynamical entropy with respect to a certain state of the canonical endomorphism of the Cuntz algebra On are equal to log n; see [27], [165], [110] for generalizations. Nevertheless in all known examples there always exists an asymptotically abelian system around which completely determines the entropy. Although, as we mentioned above, for lattice systems we can use both dynamical entropy and mean entropy in the variational principle, it is unknown whether the equality hϕ (α) = s(ϕ) always holds. In order to prove it one probably needs a better understanding not only of dynamical entropy, but also of mean entropy. E.g. if ϕ is pure, we clearly have hϕ (α) = 0, but it is still an open problem whether s(ϕ) = 0. The equality hϕ (α) = s(ϕ) holds for states derived from quasi-free states (see Chap. 13), Markov states [160], [154] (which under the assumption of faithfulness can be described as the states such that σtϕ (A0 ) ⊂ A[−1,1] [137], [76]), and of course the states coming from diagonal maximal abelian subalgebras (that is, obtained by composing a state on D = C ⊗Z , where C is a maximal abelian subalgebra of B, with the trace preserving conditional expectation A → D). It is stated in [50] that if Φ is a ﬁnite range interaction then hϕ (α) = s(ϕ) for the state ϕ such that s(ϕ) − βϕ(H) = sup{s(ψ) − βψ(H)} for suﬃciently small β > 0 (such ϕ is known to be unique [8]). However, we were unable to follow the proof. A related open problem is to ﬁnd conditions on automorphisms of C∗ -algebras under which the map ϕ → hϕ (α) is weakly∗ upper semicontinuous.

Part II

Special Topics

10 Relative Entropy and Subfactors

Conditional entropy is, as we saw in Chap. 1, an important concept in the classical theory. In the noncommutative case it was introduced under the name of relative entropy as a tool to take care of approximation of entropy. We shall in the present chapter develop the theory of relative entropy and show its relationship to subfactors of II1 -factors. Then we shall show a formula analogous to the classical formula h(T ) = H(ξ|ξ − ). Finally we shall give applications to the canonical shift on the tower of relative commutants deﬁned by an inclusion of II1 -factors and for shifts on the Jones projections.

10.1 Relative Entropy Let M be a von Neumann algebra, ϕ a normal state on M , P and Q von Neumann subalgebras of M . Deﬁnition 10.1.1. The relative entropy of P and Q with respect to ϕ is Hϕ (P |Q) = sup (S(ϕi |P , ϕ|P ) − S(ϕi |Q , ϕ|Q )), i

where the supremum is taken over all ﬁnite decompositions ϕ = into a sum of positive linear functionals.

i

ϕi of ϕ

Note that if ϕ = τ is a trace, which is the case we are mainly interested in, any ψ ≤ τ has the form ψ = τ (·x). Hence by Theorem 2.3.1(x) the relative entropy can be written as Hτ (P |Q) = sup (τ (η(EQ (xi ))) − τ (η(EP (xi )))), (10.1) i

where the supremum is taken over all ﬁnite partitions of unity 1 = i xi in M , and EP and EQ are the τ -preserving conditional expectations on P and Q, respectively. One often suppresses τ in the notation for relative entropy. The main properties of relative entropy are as follows.

158

10 Relative Entropy and Subfactors

Theorem 10.1.2. We have: (i) Hϕ (P |Q) ≥ 0, and if ϕ is a faithful tracial state then Hϕ (P |Q) = 0 if and only if P ⊂ Q; (ii) Hϕ (P |R) ≤ Hϕ (P |Q) + Hϕ (Q|R); (iii) Hϕ (P |Q) is increasing in P and decreasing in Q; (iv) if Q ⊂ P and there is a ϕ-preserving faithful normal conditional expectation E: P → Q, then Hϕ (P |Q) = sup S(ϕi |P , ϕi ◦ E), i

where the supremum is taken over all ﬁnite decompositions ϕ =

i

ϕi .

Proof. Taking the trivial decomposition ϕ = ϕ we see that Hϕ (P |Q) ≥ 0. If P ⊂ Q then Hϕ (P |Q) = 0 by monotonicity of relative entropy S, Theorem 2.3.1(vi). Let now ϕ = τ be a faithful tracial state. If P is not a subset of Q, there exists a projection e ∈ P such that e ∈ / Q. Then by (10.1) Hτ (P |Q) ≥ τ (η(EQ (e))) + τ (η(EQ (1 − e))). The element EQ (e) is not a projection (this follows e.g. by A.13, as if EQ (e) is a projection then e ≤ s(EQ (e)) = EQ (e), where s(a) is the support of a self-adjoint element a, and hence e = EQ (e) by faithfulness of EQ ). Therefore τ (η(EQ (e))) > 0 and similarly τ (η(EQ (1 − e))) > 0. This completes the proof of (i). Part (ii) is immediate from the deﬁnition of Hϕ (P |Q). Part (iii) follows from monotonicity of relative entropy S, while part (iv) follows from Theorem 2.3.1(vii).

Let us show next that relative entropy coincides with conditional entropy in the abelian case. So let M = L∞ (X, µ), ξ and ζ be measurable partitions of X such that ξ is ﬁnite, P = L∞ (X/ξ) and Q = L∞ (X/ζ). We want to show that Hτ (P |Q) = Hµ (ξ|ζ). If p1 , . . . , pn are the atoms of P , by (10.1) we have Hτ (P |Q) ≥ τ (η(EQ (pi ))). i

The latter expression is exactly Hµ (ξ|ζ). Thus Hτ (P |Q) ≥ Hµ (ξ|ζ). It suﬃces to prove the opposite inequality for ﬁnite ζ. Indeed, if {ζn }n is an increasing sequence of ﬁnite measurable partitions such that ∨n ζn = ζ then Hτ (P |L∞ (X/ζn )) ≥ Hτ (P |Q) ≥ Hµ (ξ|ζ), and Hµ (ξ|ζn ) Hµ (ξ|ζ) by the martingale convergence theorem. Thus if Hτ (P |L∞ (X/ζn )) = Hµ (ξ|ζn ), we conclude that Hτ (P |Q) = Hµ (ξ|ζ).

10.1 Relative Entropy

159

So assume ζ is ﬁnite. Then Hτ (P |Q) ≤ Hτ (P ∨ Q|Q) and Hµ (ξ|ζ) = Hµ (ξ ∨ ζ|ζ). It follows that to prove the inequality Hτ (P |Q) ≤ Hµ (ξ|ζ) we may assume that ζ ≺ ξ, so Q ⊂ P . In this case it is enough to consider partitions of unity in P . Then part (iv) of Theorem 10.1.2 and convexity of relative entropy of positive functionals, Corollary 2.3.2, show that it suﬃces to consider partitions 1 = i xi such that each xi is a scalar multiple of a minimal projection in P . Since S(λψ, λϕ) = λS(ψ, ϕ), we then see that this is the same as to consider one partition 1 = i pi , where p1 , . . . , pn are the minimal projections in P . But then clearly Hτ (P |Q) = Hµ (ξ|ζ). Next we want to show that in some cases the computation of relative entropy can be reduced to the ﬁnite dimensional case. For this we need the following notion. If P , Q, R are von Neumann subalgebras of a von Neumann algebra M , and EP : M → P , EQ : M → Q, ER : M → R are conditional expectations, then we say that P ∪ R

⊂ M ∪ ⊂ Q

is a commuting square if ER = EQ ◦EP = EP ◦EQ . Equivalently, R = P ∩Q and EQ (P ) ⊂ R, EP (Q) ⊂ R. Remark that if the conditional expectations preserve a faithful normal state ϕ, then is suﬃces to check that, say, ER = EQ ◦EP . Indeed, if M ⊂ B(H) and the state ϕ is deﬁned by a cyclic and separating vector ξ ∈ H, then ER (x)ξ = eR xξ for x ∈ M , where eR ∈ B(H) is the projection onto Rξ. Then ER = EQ ◦ EP implies that eR = eQ eP . But then eR = eP eQ , and so ER = EP ◦ E Q . Proposition 10.1.3. Let M be a von Neumann algebra with a faithful normal state ϕ, N a von Neumann subalgebra. Suppose {Mn }n and {Nn }n are increasing sequences of von Neumann subalgebras of M such that Nn ⊂ Mn , M = (∪n Mn ) and N = (∪n Nn ) . Suppose there exist ϕ-preserving conditional expectations EMn : M → Mn and ENn : M → Nn such that Mn ∪ Nn

⊂ ⊂

Mn+1 ∪ Nn+1

is a commuting square for every n. Then Hϕ (M |N ) = limn Hϕ (Mn |Nn ). Proof. If ψ ≤ ϕ then S(ψ|Mn , ϕ|Mn ) S(ψ, ϕ) by Corollary 2.3.5, and a similar convergence holds for Nn and N . This implies that Hϕ (M |N ) ≤ lim inf Hϕ (Mn |Nn ). n

160

10 Relative Entropy and Subfactors

Since ENn+1 ◦ EMn = ENn for all n, we have ENn+k+1 ◦ EMn = ENn+k+1 ◦ EMn+k ◦ EMn = ENn+k ◦ EMn , so by induction ENn+k ◦EMn = ENn for all k ∈ N. Next note that {ENn }n converges in the pointwise strong operator topology to a ϕ-preserving conditional expectation EN : M → N . It follows that for any n Mn ∪ Nn

⊂ M ∪ ⊂ N

is a commuting square. In other words, EN |Mn = ENn |Mn . Hence for any positive linear functional ψ we have S(ψ, ψ ◦ EN ) ≥ S(ψ|Mn , ψ ◦ EN |Mn ) = S(ψ|Mn , ψ ◦ ENn |Mn ). By Theorem 10.1.2(iv) it follows that Hϕ (M |N ) ≥ Hϕ (Mn |Nn ), which completes the proof.

From now onwards we shall only consider tracial states, and our next goal is to compute the relative entropy Hτ (M |N ) when N ⊂ M are ﬁnite dimensional (and τ is a trace on M ). Let Z(M ) and Z(N ) be the centers of M and N , respectively, m

n

l=1

k=1

M = ⊕ Ml , N = ⊕ N k , where Ml ∼ = Matml (C) and Nk ∼ = Matnk (C). Let wl and zk be the central projections in M and N such that Ml = M wl and Nk = N zk . Let akl be the multiplicity of Nk wl in Ml . Thus if tl denotes the trace of a minimal projection in Ml and sk that of a minimal projection in Nk , nk akl tl and τ (zk ) = nk sk = nk akl tl . τ (wl ) = ml tl = k

l

Theorem 10.1.4. With the above notation the relative entropy Hτ (M |N ) is (2Hτ (M ) − Hτ (Z(M ))) − (2Hτ (N ) − Hτ (Z(N ))) + nk akl tl log ckl , k,l

where ckl = min

( nk ,1 . akl

In particular, if M is abelian we have Hτ (M |N ) = Hτ (M ) − Hτ (N ), which we already know, and if M = Matm (C) and N = Matn (C) then

10.1 Relative Entropy

Hτ (M |N ) = 2 log m − 2 log n + log min

n2 ,1 m

(

161

& m' = min log m, 2 log . n

A straightforward computation of the formula in the theorem shows that it is equivalent to the formula ml tl log tl + ml tl log ml + nk sk log sk Hτ (M |N ) = − l

−

l

nk sk log nk +

k

k

nk akl tl log ckl .

(10.2)

k,l

Note that if τ = i λi ϕi is a decomposition of τ into a convex combination of states, then S(λi ϕi , τ ) = S(τ ) − λi S(ϕi ) − η(λi ), i

i

i

S(λi ϕi |N , τ |N ). It follows that λi (S(ϕi |N ) − S(ϕi )) , (10.3) Hτ (M |N ) = sup Hτ (M ) − Hτ (N ) +

and we have a similar expression for

i

i

where the supremum is taken over all decompositions τ = i λi ϕi of τ into a convex combination of states. This is the expression we shall use in the proof of the theorem. Note also that instead of ﬁnite convex decompositions we can use integral decompositions of τ . Proof of Theorem 10.1.4. We ﬁrst show that the left side is majorized by the right side in equation (10.2). Consider an integral decomposition τ = ϕ dµ(x). By (10.3) we have to estimate X x Hτ (M ) − Hτ (N ) + (S(ϕx |N ) − S(ϕx ))dµ(x). (10.4) X

By Theorem 2.2.2(ii) we have S(ϕx |N zk ) ≤ S(ϕx |N ) = S(ϕx |N zk wl ). k

k,l

For ﬁxed k and l consider the state ψ = ϕx (zk wl )−1 ϕx on zk M wl zk . By Theorem 2.2.2(i) we have S(ψ|N zk wl ) − S(ψ) ≤ S(ψ|N zk wl ) ≤ log nk . On the other hand, since zk M wl zk ∼ = N zk wl ⊗Matakl (C), by Theorem 2.2.2(vi) we have S(ψ|N zk wl ) − S(ψ) ≤ log akl .

162

10 Relative Entropy and Subfactors

Using S(λψ) = λS(ψ) + η(λ) with λ = ϕx (zk wl ), we therefore get S(ϕx |N zk wl ) − S(ϕx |zk M wl zk ) ≤ ϕx (zk wl ) log min{nk , akl }. Finally, from Lemma 2.2.4 applied to the state ϕx (wl )−1 ϕx on M wl , we obtain ϕx (zk wl ) . S(ϕx |zk M wl zk ) − S(ϕx |M wl ) ≤ ϕx (wl ) η ϕx (wl ) k

k

Thus, since l S(ϕx |M wl ) = S(ϕx ), the integral in (10.4) is estimated from above by ϕx (zk wl ) dµ(x). ϕx (zk wl ) log(min{nk , akl })dµ(x) + ϕx (wl )η ϕx (wl ) X X k,l

k,l

Since X ϕx dµ(x) = τ and τ (zk wl ) = nk akl tl , the ﬁrst summand in the above expression is equal to nk akl tl log min{nk , akl }. k,l

On the other hand, using concavity of η and that τ (wl ) = ml tl , we can estimate the second summand by ϕx (wl ) ϕx (zk wl ) ϕx (zk wl ) ϕx (wl ) dµ(x) ≤ ml tl η ml tl η dµ(x) ϕx (wl ) ml tl ϕx (wl ) X m l tl X nk akl tl = ml tl η ml tl nk akl = −tl nk akl log . ml Thus we estimate (10.4) by nk akl Hτ (M ) − Hτ (N ) + nk akl tl log(min{akl , nk }) − nk akl tl log ml k,l

k,l

= Hτ (M ) − Hτ (N ) + −

nk akl tl log akl

k,l

nk akl tl (log akl + log nk − log ml )

= Hτ (M ) − Hτ (N ) +

nk akl tl log ckl +

k,l

k,l

−

nk akl tl log ckl

k,l

nk sk log nk +

k

where we used that sk =

ml tl log ml ,

l

l

akl tl and ml =

k

nk akl . Since

10.1 Relative Entropy

Hτ (M ) = −

ml tl log tl and Hτ (N ) =

l

163

nk sk log sk ,

k

we have shown that the left side of equation (10.2) is majorized by the right side. In order to prove the opposite inequality choose a pure state ϕl of M wl such that nk akl ϕl (zk wl ) = , ml ϕl |N zk wl is a trace if nk < akl , ϕl |(N T M )zk wl is a trace if nk ≥ akl . nk akl l Explicitly ϕl can be constructed as follows. Identify m 2 with ⊕k (2 ⊗ 2 ), and let {ei }i∈N denote the standard basis in 2 . Put

ξl =

) k

nk akl min{nk , akl }ml

1/2

min{nk ,akl }

ei ⊗ ei .

i=1

Then let ϕl = ωξl be the vector state deﬁned by ξl . Let Kl denote the subalgebra of ⊕k zk M wl zk ⊂ M wl consisting of elements ⊕k xk such that xk ∈ N zk wl if nk < akl , and xk ∈ (N ∩ M )zk wl if nk ≥ akl . Then ϕl is a pure state on M wl such that its restriction to Kl coincides with the restriction of the unique tracial state τl on M wl to Kl . Furthermore, the map El : M wl → Kl deﬁned by El (x) = (Ad u)(x)dµl (u), U (Kl ∩M wl )

where µl is the normalized Haar measure on the unitary group U (Kl ∩ M wl ) of Kl ∩ M wl , is a τl -preserving conditional expectation. It follows that ml tl ϕl ◦ Ad u dµl (u). τ= l

U (Kl ∩M wl )

Thus by (10.3) Hτ (M |N ) ≥ Hτ (M ) − Hτ (N ) +

ml tl S(ϕl |N zk wl ).

(10.5)

k,l

The restricition of ϕl to zk M wl zk is a scalar multiple of a pure state. Hence by Lemma 2.2.3(i) ml ml ϕl |N zk wl = S ϕl |(N ∩M )zk wl = log min{nk , akl }, S nk akl nk akl whence

164

10 Relative Entropy and Subfactors

S(ϕl |N zk wl ) =

nk akl nk akl nk akl (log ckl + log akl ) − log . ml ml ml

Thus the right side of (10.5) equals Hτ (M ) − Hτ (N ) +

nk akl tl (log ckl + log akl ) −

k,l

k,l

nk akl tl log

nk akl , ml

which is exactly what we need.

10.2 Index of Subfactors Our goal in this section is to show that relative entropy is related to index of subfactors. We shall brieﬂy recall some basic facts of the latter theory referring the reader to [214, Chapter XIX] for more details. Let M be a II1 -factor with trace τ . If M acts on a Hilbert space H then H can be embedded into L2 (M ) ⊗ K for a suﬃciently large Hilbert space K. Let ¯ p ∈ (M ⊗ C1) = M ⊗B(K) be the projection onto H. Then the dimension of H relative to M is deﬁned by dimM (H) = (τ ⊗ Tr)(p), where τ is the unique tracial state on M ⊂ B(L2 (M )) and Tr is the canonical trace on B(K). The dimension can take any value in (0, +∞], and if H is countably generated in the sense that it is a direct sum of at most countably many cyclic subspaces for M , then dimM (H) is a complete invariant of the unitary equivalence class of the representation M → B(H). The dimension dimM (H) is ﬁnite if and only if the commutant M of M in B(H) is a II1 -factor. In the latter case dimM (H) =

τ ([M ξ]) , τ ([M ξ])

(10.6)

where ξ ∈ H is any nonzero vector, τ is the unique tracial state on M ⊂ B(H), and [M ξ] denotes the projection onto the subspace M ξ. If N is a subfactor of M , the index of N in M is [M : N ] = dimN (L2 (M )). If M acts on H then dimN (H) = [M : N ] dimM (H). It follows that if N ⊂ M ⊂ L then [L: N ] = [L: M ][M : N ]. It follows also that if [M : N ] < ∞ and N is represented on H then this representation extends to a representation of M on H. Indeed, decomposing H into a direct sum of cyclic subspaces we may assume that dimN (H) ≤ 1. Choose a representation of M on a Hilbert space H such that dimM (H ) = [M : N ]−1 dimN (H). Then dimN (H ) = dimN (H).

10.2 Index of Subfactors

165

Hence the representations of N on H and H are equivalent. Since the representation of N on H extends to a representation of M , the same is true for H. Using Proposition 10.1.3 and Theorem 10.1.4 it is easy to check that if R is the hyperﬁnite II1 -factor then Hτ (R ⊗ Matn (C)|R ⊗ 1) = 2 log n. On the other hand, [R ⊗ Matn (C): R ⊗ 1] = n2 . So we see that in this case the relative entropy is the logarithm of the index. This is a particular case of a far more general result. For simplicity of the presentation we stick to the irreducible case, that is, when N ∩ M = C1. Theorem 10.2.1. Let N ⊂ M be II1 -factors such that [M : N ] < ∞ and N ∩ M = C1. Then Hτ (M |N ) = log[M : N ]. To prove this theorem we need two properties of ﬁnite index subfactors. The ﬁrst one is the Pimsner-Popa inequality stating that EN (x) ≥ [M : N ]−1 x for any x ∈ M+ , where EN : M → N is the trace preserving conditional expectation, see [214, Theorem XIX.4.14]. Proposition 10.2.2. If N ⊂ M is a ﬁnite index subfactor, then Hτ (M |N ) ≤ log[M : N ]. Proof. Let λ = [M : N ]−1 . Then by the Pimsner-Popa inequality ϕ ◦ EN ≥ λϕ for any positive linear functional ϕ on M . Since the relative entropy S is decreasing in the second variable, Theorem 2.3.1(iii), we then have S(ϕ, ϕ ◦ EN ) ≤ S(ϕ, λϕ) = −ϕ(1) log λ. By Theorem 10.1.2(iv) it follows that Hτ (M |N ) ≤ − log λ = log[M : N ].

The second property of subfactors which we need, is existence of special projections. Lemma 10.2.3. If N ⊂ M is a ﬁnite index subfactor then there exists a projection q ∈ M such that EN (q) = [M : N ]−1 1. Moreover, any other projection with this property is of the form vqv ∗ , where v is a unitary in N . Proof. See [214, Theorem XIX.4.12]. The existence will also essentially be shown in Lemma 10.4.1 below.

Proof of Theorem 10.2.1. Let λ = [M : N ]−1 and q be the projection from Lemma 10.2.3. The idea of the proof is to consider the state ϕ = λ−1 τ (· q)

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10 Relative Entropy and Subfactors

and the decomposition τ = U (N ) ϕ ◦ Ad u du, which formally holds because of EN ∩M (q) = λ1. For II1 -factors the above decomposition does not make sense, but it is still possible to get an approximate version of it. Let x = q − λ1. Then τ (x) = 0. Let Kx = conv{vxv ∗ : v ∈ U (N )}, where the closure is in the weak operator topology. Then Kx is a convex wcompact set in M , and τ (y) = 0 for all y ∈ Kx . The weak operator closure of a bounded convex set coincides with the closure with respect to the L2 -norm. Thus there exists a unique y0 ∈ Kx such that y0 2 = inf{y2 : y ∈ Kx }. But vy0 v ∗ ∈ Kx for all v ∈ U (N ), and vy0 v ∗ 2 = y0 2 , so that vy0 v ∗ = y0 , i.e., y0 ∈ N ∩ M . Since N ∩ M = C1, it follows that y0 is a scalar, so y0 = 0. Hence for any ﬁxed ε > 0 there are unitaries v1 , . . . , vn ∈ N such that n 1 ∗ vi qvi − λ1 < ε2 λ. n i=1

2

Let y = (λn)−1 i vi qvi∗ . Then 0 ≤ y ≤ λ−1 1 and y − 12 ≤ ε2 . Let p be the spectral projection of y corresponding to the interval [0, 1 + ε]. Set xi = ((1 + ε)λn)−1 vi qvi∗ ∧ p. Then i xi ≤ ((1 + ε)λn)−1 i pvi qvi∗ p = (1 + ε)−1 yp ≤ 1. Hence Hτ (M |N ) ≥ τ (η(EN (xi ))−η(xi )) = ((1+ε)λn)−1 τ (η(EN (vi qvi∗ ∧p))). i

i

Using operator monotonicity of log as in the proof of Theorem 2.2.2(ii), we get τ (η(e + f )) ≤ τ (η(e)) + τ (η(f )) for any positive e and f . Hence τ (η(EN (vi qvi∗ ∧ p))) ≥ τ (η(EN (vi qvi∗ ))) − τ (η(EN (vi qvi∗ − vi qvi∗ ∧ p))) ≥ η(λ) − η(τ (vi qvi∗ − vi qvi∗ ∧ p)), where in the second inequality we used that EN (q) = λ1 and that η ◦ τ ≥ τ ◦ η by concavity of η. Thus Hτ (M |N ) ≥ −(1+ε)−1 log λ−((1+ε)λn)−1 η(τ (vi qvi∗ −vi qvi∗ ∧p)). (10.7) i

Now recall that τ (e ∨ f ) − τ (f ) = τ (e) − τ (e ∧ f ) for any projections e and f . Hence

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167

τ (vi qvi∗ − vi qvi∗ ∧ p) ≤ 1 − τ (p). Since y(1 − p) ≥ (1 + ε)(1 − p), we have y − 122 ≥ τ ((y − 1)2 (1 − p)) ≥ ε2 τ (1 − p), so that 1 − τ (p) ≤ ε−2 y − 122 ≤ ε2 . Thus τ (vi qvi∗ − vi qvi∗ ∧ p) ≤ ε2 . Since η is increasing on [0, e−1 ], for ε small enough we obtain η(τ (vi qvi∗ − vi qvi∗ ∧ p)) ≤ η(ε2 ). From (10.7) we then get Hτ (M |N ) ≥ −(1 + ε)−1 log λ − ((1 + ε)λ)−1 η(ε2 ). Letting ε → 0 we conclude that Hτ (M |N ) ≥ − log λ = log[M : N ]. The opposite inequality follows from Proposition 10.2.2.

Remark 10.2.4. What we actually used in the proof is not the irreducibility N ∩ M = C1, but that EN ∩M (q) is a scalar. Subfactors satisfying the latter condition are called extremal, and in fact the equality Hτ (M |N ) = log[M : N ] holds if and only if N ⊂ M is extremal [162, Corollary 4.5]. Note that if [M : N ] < 4 then automatically N ∩ M = C1, see [214, Corollary XIX.2.10]. Recall also that the possible values of the index is the set {4 cos2 π/n | n ≥ 3} ∪ [4, +∞], see [214, Theorem XIX.2.22]. Finally note that with some extra work the above theorem can be extended to arbitrary subfactors [162, Theorem 4.5]. In that case Hτ (M |N ) = ∞ whenever N ∩ M has a diﬀuse part. If N ∩ M is atomic, and {fk }k are minimal projections in N ∩ M with sum 1, then Hτ (M |N ) = 2 η(τ (fk )) + τ (fk ) log[fk M fk : fk N fk ]. k

k

10.3 Generators and Relative Entropy In this section we shall prove an analogue of the formula h(T ) = H(ξ|ξ − ), see (1.3) and Theorem 1.1.4. Before we state the main result we introduce some notation. Let (M, τ, α) be a W∗ -dynamical system, where τ is a faithful normal trace. Let {An }∞ n=1 be an increasing sequence of ﬁnite dimensional C∗ -subalgebras of M . We say that {An }∞ n=1 is a generating sequence for α if

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(i) α(An ) ⊂ An+1 , n ∈ N; (ii) the von Neumann subalgebra generated by αk (An ), k ∈ Z, n ∈ N, coincides with M ; 1 (iii) hτ (α) = lim Hτ (An ). n→∞ n We say {An }n satisﬁes the commuting square condition if An+2 An+1 ⊂ ∪ ∪ α(An ) ⊂ α(An+1 ) is a commuting square with respect to the τ -preserving conditional expectak tions for every n ∈ N. Remark that if M is abelian and An = ∨n−1 k=0 α (A1 ), then this condition is satisﬁed if and only if A1 corresponds to the generating partition of a Markov process. Write An in the form An = ⊕ Mkn , l∈Kn

∼ where = Matmnk (C). Let (ankl )k∈Kn−1 ,l∈Kn be the inclusion matrix for α(An−1 ) ⊂ An . Denote by Z(An ) the center of An . Mkn

Theorem 10.3.1. Let (M, τ, α) be a W∗ -dynamical system, where τ is a faithful normal trace and M is of type II1 . Suppose {An }∞ n=1 is a generating sequence for α satisfying the commuting square condition. Assume also that ankl n−1 < ∞. n,k,l mk

hτ (α) < ∞ and sup

1 Hτ (Z(An )) exists, and with R = (∪n An ) we have n→∞ n

Then the limit lim

hτ (α) =

1 1 1 Hτ (R|α(R)) + lim Hτ (Z(An )). n→∞ 2 2 n

If we combine this theorem with Theorem 10.2.1, we immediately obtain the following corollary. Corollary 10.3.2. If we in the above theorem add the assumptions that α(R) ∩ R = C1 and limn n1 Hτ (Z(An )) = 0, then hτ (α) = 12 log[R: α(R)].

To prove the theorem we need some preparation. Lemma 10.3.3. Fix r > 0. Let fn be the central projection in An such that An fn = ⊕mnk ≤r Mkn . Then {fn }n is a decreasing sequence converging strongly to zero.

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169

Proof. First note that R = (∪n An ) is of type II1 . Indeed, let z be the central projection in R corresponding to the sum of the type Ik components of R with k ≤ r, and assume z = 0. Since α(R) ⊂ R, and type II and type Ik algebras can not be embedded into a type Il algebra with l < k, we have z ≤ α(z). As τ is α-invariant and faithful, it follows that z = α(z). The maximal number of mutually equivalent orthogonal projections in Rz is not larger than r. By the deﬁnition of a generating sequence, the sequence {α−n (R)}∞ n=1 is increasing with union dense in M . Since z = α−n (z) is central in each α−n (R), the projection z is central in M , and the maximal number of mutually equivalent orthogonal projections in M z is not larger than r. In other words, M z is a sum of type Ik algebras with k ≤ r. But this contradicts our assumption on M . Since a type Ik algebra can not be embedded into a type Il algebra with l < k, it is clear that {fn }n is a decreasing sequence. Let f be its strong limit. Since fn is central in An , the projection f is central in R. Then Rf = (∪n An f ) , which by the same argument as above contradicts the fact that R is of type II1 unless f = 0.

Denote by tnk the trace of a minimal projection in Mkn ⊂ An . Since Hτ (An−1 ) = Hτ (α(An−1 )) and Hτ (Z(An−1 )) = Hτ (Z(α(An−1 ))), by Theorem 10.1.4 we have Hτ (An |α(An−1 )) = (2Hτ (An ) − Hτ (Z(An ))) − (2Hτ (An−1 ) − Hτ (Z(An−1 ))) mn−1 ankl tnl log cnkl , (10.8) + k k,l

/ankl , 1}. where cnkl = min{mn−1 k Lemma 10.3.4. Suppose C = sup N

Then

N 1 n−1 n n mk akl tl log ankl < ∞. N n=2 k,l

N 1 n−1 n n mk akl tl log cnkl = 0. N →∞ N n=2

lim

k,l

Proof. Since by assumption of the theorem there is c > 0 such that c ≤ cnkl ≤ 1 for all k, l and n, it suﬃces to prove that lim N

N 1 N n=2

mn−1 ankl tnl = 0. k

n−1 k,l:an kl >mk

Fix r > 0. Let fn be as in Lemma 10.3.3. Then N 1 N n=2

k,l:mn−1 ≤r k

mn−1 ankl tnl = k

N 1 N n=2

k:mn−1 ≤r k

mn−1 tn−1 = k k

N 1 τ (fn−1 ), N n=2

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and the latter expression tends to zero by Lemma 10.3.3. On the other hand, N 1 N n=2

mn−1 ankl tnl k

−1

≤ (log r)

n−1 k,l:an >r kl >mk

N 1 n−1 n n mk akl tl log ankl N n=2 k,l

−1

≤ C(log r)

.

Since we can take an arbitrary large r, this proves the lemma.

Proof of Theorem 10.3.1.Let us check that the assumption of the previous n−1 n n n n a = m and , we have lemma is satisﬁed. Since k mn−1 kl l l akl tl = tl k

mnl mn−1 k k,l = mnl tnl log mnl − mn−1 tn−1 log mn−1 k k k

mn−1 ankl tnl log ankl ≤ k

k,l

mn−1 ankl tnl log k

l

k

= Hτ (An ) − Hτ (Z(An )) − Hτ (An−1 ) + Hτ (Z(An−1 )). Hence, with A0 = C1, N 1 1 1 1 n−1 n n mk akl tl log ankl ≤ Hτ (AN ) − Hτ (Z(AN )) ≤ Hτ (AN ). N n=1 N N N k,l

Since the sequence { N1 Hτ (AN )}N is bounded by assumption of the theorem, it follows of Lemma 10.3.4 is satisﬁed. Hence if we put that then assumption n n Cn = k,l mn−1 a t log c , we get kl l kl k N 1 Cn → 0. N n=1

By equation (10.8) we have N N 2 1 1 1 Hτ (An |α(An−1 )) = Hτ (AN ) − Hτ (Z(AN )) + Cn . N n=1 N N N n=1

Since {An }n satisﬁes the commuting square condition, by Proposition 10.1.3 Hτ (An |α(An−1 )) → Hτ (R|α(R)). Since N1 Hτ (AN ) → hτ (α) by assumption, it follows that limN exists and 1 Hτ (R|α(R)) = 2hτ (α) − lim Hτ (Z(AN )). N N Thus 1 1 1 hτ (α) = Hτ (R|α(R)) + lim Hτ (Z(AN )). 2 2 N N

1 N Hτ (Z(AN ))

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171

10.4 The Canonical Shift In the present section we shall apply the results of the preceding sections to an interesting automorphism, called the canonical shift, deﬁned on the tower of relative commutants associated with an inclusion N ⊂ M of II1 -factors of ﬁnite index. In order to introduce it we need a few more facts from index theory, again see [214, Chapter XIX] for more details. Let τ be the trace on M , ξ ∈ L2 (M ) the cyclic vector deﬁning τ . Denote by eN the projection onto N ξ, and by M1 = !M, eN " the von Neumann subalgebra of B(L2 (M )) generated by M and eN . Since eN xeN = EN (x)eN = eN EN (x) for x ∈ M , we have {eN } ∩ M = N . In other words, N is generated by M and eN . So if we denote by J the modular conjugation deﬁned by τ , Jxξ = x∗ ξ, so that JM J = M and JeN J = eN , then JM1 J = N . It follows that M1 is a II1 -factor and dimM1 (L2 (M )) = dimN (L2 (M )) =

1 1 = , 2 dimN (L (M )) [M : N ]

where the second equality follows from (10.6). Hence [M1 : M ] = [M : N ]. Furthermore, by (10.6) the unique tracial state τ on N has the property τ (eN ) = [M : N ]−1 . The unique tracial state on M1 , which we again denote by τ , is such that τ (x) = τ (Jx∗ J). Hence τ (eN ) = [M : N ]−1 . Since N is a factor, τ (ab) = τ (a)τ (b) for a ∈ N and b ∈ N ∩ M1 . Hence, for x ∈ M , τ (xEM (eN )) = τ (xeN ) = τ (EN (x)eN ) = τ (EN (x))τ (eN ) = [M : N ]−1 τ (x), so we also have EM (eN ) = [M : N ]−1 1, where EM : M1 → M is the trace preserving conditional expectation. Thus starting with a ﬁnite index inclusion N ⊂ M we construct a II1 factor M1 and a projection eN ∈ M1 such that [M1 : M ] = [M : N ], M1 = !M, eN ", N = {eN } ∩ M and EM (eN ) = [M : N ]−1 1. This is called the basic construction. Iterating it we get the Jones tower of II1 -factors M−1 = N ⊂ M0 = M ⊂ M1 ⊂ . . . ⊂ Mk ⊂ . . . and projections ek ∈ Mk+1 such that Mk+1 = !Mk , ek ", k ≥ 0, e0 = eN . There is also a downward construction. So there exists a projection e−1 and a subfactor N−1 ⊂ N such that N−1 ⊂ N ⊂ !N, e−1 " = M is the basic construction. Since EN (e−1 ) = [M : N ]−1 1, by Lemma 10.2.3 the projection e−1 is deﬁned up to conjugation by a unitary in N . Since N−1 = {e−1 } ∩ N , the subfactor N−1 is also deﬁned up to conjugation by a unitary in N . Denoting N−1 by M−2 and iterating the downward construction we get a tunnel . . . ⊂ M−k ⊂ . . . ⊂ M−1 ⊂ M0

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10 Relative Entropy and Subfactors

and projections e−k ∈ M−k+1 , k ≥ 1. Using that ek xek = ek EMk−1 (x) = EMk−1 (x)ek for x ∈ Mk and EMk (ek ) = [M : N ]−1 1 one then checks that the projections ek , k ∈ Z, satisfy the relations ek ek±1 ek = [M : N ]−1 ek , ek ej = ej ek if |k − j| ≥ 2. Since [Mk+1 : Mk ] = [M : N ] < ∞, by the discussion at the beginning of Sect. 10.2 we can inductively construct a representation of ∪k Mk on L2 (M ) extending the representation of M1 . This representation is not unique, but as soon as we ﬁx a tunnel there exists a canonical one. Lemma 10.4.1. With the above notation there exists a unique extension of 2 the representation of M1 on L2 (M ) to a representation of ∪∞ k=1 Mk on L (M ) such that ek = Je−k J for k ≥ 1. In this representation JMk J = M−k for every k ∈ Z. Proof. Since Mk+1 = !Mk , ek ", the condition ek = Je−k J completely determines the representation. We therefore only have to prove its existence. We shall do this by constructing the required representation of Mk by induction on k. For k = 1 we have a representation by deﬁnition, and M1 = JM−1 J. So assume the representation is well-deﬁned for some k, and Mj = JM−j J for j = 1, . . . , k, where we identify Mj with its image under the representation. Since [Mk+1 : Mk ] < ∞, the identity map Mk → Mk ⊂ B(L2 (M )) extends to a representation π: Mk+1 → B(L2 (M )). Let f = Jπ(ek )J. Denote [M : N ]−1 by λ. We assert that f ∈ M−k+1 and EM−k (f ) = λ1. Indeed, since ek ∈ Mk−1 ∩ Mk+1 , we have π(ek ) ∈ Mk−1 , so that f ∈ JMk−1 J = M−k+1 . Then EM−k (f )e−k+1 = e−k+1 f e−k+1 = Jek−1 JJπ(ek )JJek−1 J = Jπ(ek−1 ek ek−1 )J = λJek−1 J = λe−k+1 , so that EM−k (f ) = λ1. But then by Lemma 10.2.3 there exists a unitary u ∈ M−k such that e−k = uf u∗ = J(JuJπ(ek )Ju∗ J)J. Since JuJ ∈ JM−k J = Mk , we thus see that x → JuJπ(x)Ju∗ J is a representation of Mk+1 which coincides with π on Mk and maps ek onto Je−k J. Therefore it is the required representation of Mk+1 . Since M−k−1 = {e−k } ∩ M−k , we have J, Je−k J" = !Mk , ek " = Mk+1 , J = !JM−k JM−k−1

which completes the proof of the induction step.

J = Mn , it follows that M−n ⊂ M0 ⊂ Mn is the basic conSince JM−n struction for each n ∈ N. Hence, more generally, Mk−n ⊂ Mk ⊂ Mk+n is the basic construction for any k ∈ Z and n ∈ N. Consider now the AF-algebra A∞ obtained by taking the inductive limit of the algebras Mk ∩ Ml as k → −∞ and l → +∞ (note that Mk ∩ Ml is ﬁnite dimensional by [214, Corollary XIX.2.9]). By the above lemma we get

10.4 The Canonical Shift

173

a mirroring γ0 , the involutive anti-automorphism of A∞ deﬁned by γ0 (x) = Jx∗ J for x ∈ ∪k,l (Mk ∩ Ml ). It has the properties γ0 (Mk ∩ Ml ) = M−l ∩ M−k and γ0 (ek ) = e−k . On the other hand, we can consider . . . ⊂ Mk ⊂ Mk+1 ⊂ . . . as the tower and the tunnel associated with the inclusion M0 ⊂ M1 , and get an anti-automorphism γ1 of A∞ such that γ1 (Mk ∩ Ml ) = M−l+2 ∩ M−k+2 and γ1 (ek ) = e−k+2 . The canonical shift for the inclusion N ⊂ M is the automorphism Γ = γ1 ◦ γ0 of A∞ . It has the properties Γ (Mk ∩ Ml ) = Mk+2 ∩ Ml+2 and Γ (ek ) = ek+2 .

Since each algebra Mk has a unique trace, the C∗ -algebra A∞ has a canonical tracial state, which we continue to denote by τ . Our next goal is to show that Γ is τ -preserving. By [214, Proposition XIX.4.19] in the ﬁnite depth case, which we shall consider below, the trace τ is the unique tracial state on A∞ . Hence in that case it is γ0 - and γ1 -invariant. However, this is not true in general. To prove that nevertheless the trace is Γ -invariant we need to compare the constructed representations of ∪k Mk on L2 (M0 ) and L2 (M1 ). To simplify the notation we consider ∪k Mk as a subalgebra of B(L2 (M1 )), and denote by π the representation of ∪k Mk on L2 (M0 ). Lemma 10.4.2. If we identify L2 (M0 ) with e1 L2 (M1 ), then the representation π satisﬁes π(x) = λ−k e1 . . . ek xek+1 ek . . . e1 for x ∈ Mk , k ≥ 1, where λ = [M : N ]−1 . Proof. Using the relations ei ei−1 ei = λei , i = 1, . . . , k, we get ek+1 ek . . . e1 e1 . . . ek ek+1 = λk ek+1 . Since ek+1 x = xek+1 for x ∈ Mk , it follows that the expression in the formulation of the lemma deﬁnes a homomorphism. For x ∈ M0 we have π(x) = xe1 . Since Mk is generated by M0 and e0 , . . . , ek−1 , all that remains to show is that the equality in the formulation holds for x = e0 , . . . , ek−1 . In other words, π(ek−1 ) = ek+1 e1 for k ≥ 2, and π(e0 ) = λ−1 e1 e0 e2 e1 . Denote by J1 and J0 the modular conjugations on L2 (M1 ) and L2 (M0 ), respectively. Since J0 = J1 |e1 L2 (M1 ) , for k ≥ 2 we have π(ek−1 ) = J0 π(e−k+1 )J0 = J1 e−k+1 e1 J1 = ek+1 e1 , proving the ﬁrst identity. To prove the second note that if ξ1 is the cyclic trace vector in L2 (M1 ), then

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10 Relative Entropy and Subfactors

(λ−1 e1 e0 e2 e1 ξ1 , ξ1 ) = λ−1 (e2 ξ1 , e0 ξ1 ) = λ−1 (J1 e0 J1 ξ1 , e0 ξ1 ) = λ−1 (e0 ξ1 , e0 ξ1 ) = λ−1 τ (e0 ) = 1. Since λ−1 e1 e0 e2 e1 is a projection, we conclude that ξ1 = λ−1 e1 e0 e2 e1 ξ1 . Since e1 e0 e2 e1 ∈ M−1 , it follows that the projection [M−1 ξ1 ] is majorized −1 by λ e1 e0 e2 e1 . On the other hand, since M1 is generated by M0 and e0 , we have e0 M1 e0 = e0 M−1 . Hence [M−1 ξ1 ] = [M−1 e1 e2 e0 e1 ξ1 ] = [e1 e2 e0 M−1 ξ1 ] = [e1 e2 e0 M1 e0 ξ1 ]. Since e2 ξ1 = J1 e0 J1 ξ1 = e0 ξ1 and e2 ∈ M1 , the above projection equals [e1 e2 e0 M1 e2 ξ1 ] = [e1 e2 e0 M1 ξ1 ] = [e1 e2 e0 L2 (M1 )] ≥ λ−1 e1 e0 e2 e1 . By deﬁnition of e0 = eN we get π(e0 ) = [M−1 ξ1 ] = λ−1 e1 e0 e2 e1 , completing the proof of the lemma.

Proposition 10.4.3. The canonical shift Γ is trace preserving. Proof. We use the same notation as in the proof of the previous lemma. Since x = γ0 (γ1 (Γ (x))), π ◦ γ0 = J0 π(·)∗ J0 and J0 = J1 |e1 L2 (M1 ) , for any x ∈ A∞ we have π(x) = J1 π(γ1 (Γ (x∗ )))J1 . ∩ M−k . If x ∈ Mk ∩ Ml with k ≤ −1 and l ≥ 1, then γ1 (Γ (x)) = γ0 (x) ∈ M−l Using Lemma 10.4.2 and that J1 ei J1 = e−i+2 , we conclude that the above expression equals

λk J1 e1 . . . e−k J1 Γ (x)J1 e−k+1 e−k . . . e1 J1 = λk e1 e0 . . . ek+2 Γ (x)ek+1 . . . e0 e1 . Thus λk e1 e0 . . . ek+2 Γ (x)ek+1 . . . e0 e1 = π(x) = λ−l e1 . . . el xel+1 . . . e1 .

(10.9)

Applying the trace to the left side and using that ei ei+1 ei = λei and Γ (x) ∈ Mk+2 ∩ Ml+2 we get λk τ (Γ (x)ek+1 . . . e0 e1 e0 . . . ek+1 ) = τ (Γ (x)ek+1 ) = λτ (Γ (x)). Similarly applying τ to the right side of (10.9) we get λτ (x), so that τ (Γ (x)) = τ (x).

Remark 10.4.4. If x ∈ Mk ∩ Ml then multiplying (10.9) by ek+1 . . . e0 from the left and by e0 . . . ek+1 from the right, we conclude that Γ (x) is an element in Mk+2 ∩ Ml+2 such that Γ (x)ek+1 = λk−l ek+1 . . . el xel+1 . . . ek+1 .

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175

Since Mk+2 ⊂ B(L2 (M1 )) is a factor and ek+1 ∈ Mk+2 , the homomorphism Mk+2 a → aek+1 is faithful. Hence Γ (x) is uniquely determined by the above identity. In particular, the restriction of Γ to ∪n≥0 (N ∩ Mn ) ⊂ A∞ is independent of the choice of the tunnel. Often it is this restriction which is called the canonical shift. It follows also that if for some n ∈ Z we consider

. . . ⊂ Mk ⊂ Mk+1 ⊂ . . . as the tower and the tunnel for Mn ⊂ Mn+1 and deﬁne the corresponding mirrorings γn and γn+1 , then Γ = γn+1 ◦ γn . To compute the entropy of Γ we shall use Theorem 10.3.1. We have to check that its assumptions are satisﬁed. To show that {M ∩ M2n }n is a generating sequence for Γ we prove the next general proposition. Proposition 10.4.5. Let (P, τ, σ) be a W∗ -dynamical system, where τ is a normal tracial state. Let {Pn }n be an increasing sequence of ﬁnite dimensional C∗ -subalgebras such that σ(Pn ) ⊂ Pn+1 and P is generated by σ k (Pn ), k ∈ Z, n ∈ N. Assume there exists a sequence {kn }n of natural numbers such that kn → 1 as n → ∞; (i) n (ii) the algebras σ mkn (Pn ), m ∈ N, are mutually τ -independent. Then {Pn }n is a generating sequence for σ. Recall that we say that A1 , . . . , An are τ -independent if they mutually commute, and τ (a1 . . . an ) = τ (a1 ) . . . τ (an ) for ai ∈ Ai . Proof of Proposition 10.4.5. Since σ k (Pl ) ⊂ σ −[n/2] (Pn ) for −[n/2] ≤ k ≤ [n/2] − l, the algebra ∪n σ −[n/2] (Pn ) is dense in P . Hence, by the KolmogorovSinai type theorem, Theorem 3.2.3, hτ (σ) = lim hτ (Pn ; σ). n→∞

Since σ i (Pk ) ⊂ Pk+i , for any n ∈ N we have Hτ (Pk , σ(Pk ), . . . , σ n (Pk )) ≤ Hτ (Pn+k ), whence hτ (Pk ; σ) ≤ lim inf n→∞

1 Hτ (Pn ). n

On the other hand, since the algebras σ mkn (Pn ), m ∈ Z, are independent, we have

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hτ (Pn ; σ kn ) = Hτ (Pn ). Hence hτ (σ) = Since

1 1 1 hτ (σ kn ) ≥ hτ (Pn ; σ kn ) = Hτ (Pn ). kn kn kn

kn → 1, it follows that n hτ (σ) ≥ lim sup n→∞

1 Hτ (Pn ). n

1 Thus the limit lim Hτ (Pn ) exists and equals hτ (σ), that is, {Pn }n is a genn n erating sequence.

The proposition implies that {M ∩ M2n }n is a generating sequence for the canonical shift Γ . Indeed, since Γ mn (M ∩ M2n ) = M2mn ∩ M2(m+1)n , we mn see that Γ (M ∩ M2n ) commutes with M ∩ M2n for any m ≥ 1. Moreover, since M2n is a factor, we have τ (xy) = τ (x)τ (y) for any x ∈ M2n and y ∈ ∪k (M2n ∩M2n+k ). Thus the algebras Γ mn (M ∩M2n ), m ∈ N, are independent, and we can apply the previous proposition with kn = n. Next note that the von Neumann algebra generated by ∪n (M ∩ M2n ) in the GNS-representation of A∞ deﬁned by τ has type II1 , since already the von Neumann algebra generated by the projections en , n ≥ 1, is a II1 -factor by [214, Theorem XIX.3.1]. The generating sequence {M ∩ M2n }n satisﬁes the commuting square condition, that is, M ∩ M2n ⊂ M ∩ M2n+2 ∪ ∪ M2 ∩ M2n ⊂ M2 ∩ M2n+2 is a commuting square for any n ∈ N. Indeed, the trace preserving conditional expectation M ∩ M2n+2 → M ∩ M2n is the restriction of the conditional expectation EM2n : M2n+2 → M2n . But then we obviously have EM2n (M2 ∩ M2n+2 ) ⊂ M2 ∩ M2n . Next we check that the entropy is ﬁnite. Proposition 10.4.6. We have hτ (Γ ) ≤ ht(Γ ) ≤ log[M : N ]. Proof. If Ω is a ﬁnite subset of M ∩Mk , then Γ n (Ω) ⊂ M ∩M2n+k . It follows that 1 ht(Ω; Γ ) ≤ lim sup log rank M ∩ M2n . n→∞ n By [214, Proposition XIX.2.8] if p1 , . . . , pm are projections in M ∩ M2n with sum 1, then

10.4 The Canonical Shift

[pi M2n pi : M pi ] Hence

i

i

τ (pi )

177

= [M2n : M ].

τ (pi )−1 ≤ [M2n : M ], so by the Cauchy-Schwarz inequality m=

m

τ (pi )1/2 τ (pi )−1/2 ≤ [M2n : M ]1/2 .

i=1

Therefore rank M ∩ M2n ≤ [M2n : M ]1/2 = [M : N ]n , and the proof of the proposition is complete.

Finally, the multiplicities for the inclusion M ∩ Mn−1 ⊂ M ∩ Mn are bounded by [M : N ] (in fact, by [M : N ]1/2 ). To see this let p be a minimal projection in M ∩Mn−1 . Then q = pen is a minimal projection in M ∩Mn+1 . Indeed, since en Mn+1 en = Mn−1 en , we have en (M ∩Mn+1 )en = (M en ) ∩en Mn+1 en = (M en ) ∩Mn−1 en = (M ∩Mn−1 )en , so that q(M ∩ Mn+1 )q = p(M ∩ Mn−1 )pen = Cq. Furthermore, if f is a projection in M ∩ Mn and f ≤ p, then qf q = en f en = EMn−1 (f )en = 0. Therefore if p majorizes a minimal projection f ∈ M ∩ Mn then the central support of f in M ∩ Mn+1 majorizes q. In other words, if (aij )i,j and (bjk )j,k are the inclusion matrices for M ∩ Mn−1 ⊂ M ∩ Mn and M ∩ Mn ⊂ M ∩ Mn+1 , respectively, i0 corresponds p and k0 corresponds to q, then bjk0 = 0 as soon as ai0 j = 0. If {zk }k is the set of minimal central projections in M ∩ Mn+1 , then denoting by sk the trace of a minimal projection majorized by zk , we get ai0 j bjk sk ≥ ai0 j bjk0 τ (q) = [M : N ]−1 ai0 j bjk0 τ (p). τ (p) = j,k

It follows that ai0 j ≤ [M : N ] (in fact, it is known that ai0 j = bjk0 , so that ai0 j ≤ [M : N ]1/2 ). Since . . . ⊂ M2k−2 ⊂ M2k ⊂ . . . can be considered as the tower and the tunnel associated with M−2n ⊂ M−2n+2 , we conclude that the multiplicities for the inclusion M−2n ∩ M−2 ⊂ M−2n ∩ M are bounded by [M : N ]2 . Since γ0 (M2 ∩ M2n ) = M−2n ∩ M−2 and γ0 (M ∩ M2n ) = M−2n ∩ M , it follows that the multiplicities for the inclusion Γ (M ∩ M2n−2 ) ⊂ M ∩ M2n are also bounded by [M : N ]2 . We have thus checked that all the assumptions of Theorem 10.3.1 are satisﬁed, so we get our ﬁrst and most general result on the entropy of Γ . Theorem 10.4.7. Let Γ be the canonical shift for the inclusion N ⊂ M of II1 -factors of ﬁnite index. Let R = πτ (∪∞ n=1 (M ∩ M2n )) . Then 1 1 1 Hτ (M ∩ M2n ) = Hτ (R|Γ (R)) + lim Hτ (Z(M ∩ M2n )). n→∞ 2n n→∞ n 2

hτ (Γ ) = lim

178

10 Relative Entropy and Subfactors

Combining this theorem and Proposition 10.4.6 we obtain the following estimates. Corollary 10.4.8. We have 1 Hτ (R|Γ (R)) ≤ hτ (Γ ) ≤ ht(Γ ) ≤ log[M : N ]. 2

In particular, if Hτ (R|Γ (R)) = 2 log[M : N ], then both the topological entropy and the dynamical entropy of Γ with respect to τ are equal to log[M : N ]. By [169, Theorem 5.3.1] the condition Hτ (R|Γ (R)) = 2 log[M : N ] is one of several equivalent characterizations of extremality of N ⊂ M and strong amenability of the standard invariant of the inclusion. The simplest case when this condition is satisﬁed, is that of ﬁnite depth inclusions. A ﬁnite index inclusion N ⊂ M of II1 -factors is said to have ﬁnite depth if sup dim Z(M ∩ Mn ) < ∞. n

Then, by the discussion prior to [214, Proposition XIX.4.19], there exists n0 such that if G denotes the matrix of the inclusion M ∩ M2n0 ⊂ M ∩ M2n0 +1 , then the inclusions M ∩ M2n ⊂ M ∩ M2n+1 and M ∩ M2n+1 ⊂ M ∩ M2n+2 are given by the matrices G and Gt , respectively, for any n ≥ n0 . Moreover, the matrix GGt is primitive, and if s denotes the vector formed by the values of the trace on minimal projections in M ∩ M2n0 , then GGt s = [M : N ]s. We shall now show that this allows us to compute the entropy without relying on the deep results of Popa [169]. Recall that a matrix B ∈ Matr (R) with nonnegative coeﬃcients is called primitive if there exists m ∈ N such that B m has only positive entries. By the Perron-Frobenius theorem, see e.g. [193, Section 1.1], the spectral radius β of B is an eigenvalue of B, called the Perron-Frobenius eigenvalue, and the following conditions are satisﬁed: (i) there is an eigenvector ξ of B with eigenvalue β whose coordinates are all positive; similarly, there is an eigenvector ζ of B ∗ with eigenvalue β and positive coordinates; (ii) if ζ is normalized such that (ξ, ζ) = 1 then β −n B n → P as n → +∞, where P = (·, ζ)ξ is the projection onto Rξ along ζ ⊥ . These properties imply ∩n∈N B n (Rr+ ) = R+ ξ. Indeed, assume ϑn ∈ Rr+ , n ≥ 0, are such that ϑ0 = β −n B n ϑn . Since ζ has positive coordinates, there exists ε > 0 such that P ϑ = (ϑ, ζ)ξ ≥ εϑ for ϑ ∈ Rr+ . Then if ε < ε and n is such that β −n B n − P ≤ ε , we get β −n B n ϑ ≥ (ε − ε )ϑ. It follows that the sequence {ϑn }n is bounded. But then the vectors β −n B n ϑn become arbitrarily close to P ϑn as n → ∞. Hence ϑ0 ∈ R+ ξ. It follows that ξ is the unique (up to a scalar factor) eigenvector of B with nonnegative coordinates. In particular, if B = GGt corresponds to a ﬁnite depth inclusion N ⊂ M , then the Perron-Frobenius eigenvalue is [M : N ].

10.4 The Canonical Shift

179

Proposition 10.4.9. Let A = lim An be an AF-algebra such that for every n −→ the inclusion An ⊂ An+1 is deﬁned by a ﬁxed primitive matrix B. Then there exists a unique tracial state τ on A, and lim

n→∞

1 Hτ (An ) = log β, n

where β is the Perron-Frobenius eigenvalue of B. Proof. Let An = ⊕rk=1 Matmnk (C). A tracial state τ on A is deﬁned by a sequence of vectors sn ∈ Rr+ , n ∈ N, such that k s1k m1k = 1 and sn = Bsn+1 . Let of B normalized by the condition ξ be1 the Perron-Frobenius eigenvector n −n+1 ξ m = 1. Then we can take s = β ξ and obtain a tracial state on A. k k k Conversely, given a tracial state τ on A, we have sn ∈ ∩m B m (Rr+ ). Hence by the discussion before the formulation of the proposition the vector sn is a scalar multiple of ξ for any n, and the scalar is completely determined by the condition τ (1) = 1. Then we have Hτ (An ) = −

r

mnk snk log snk = −

k=1

r

mnk snk log(β −n+1 ξk )

k=1

= (n − 1) log β −

r

mnk snk log ξk .

k=1

Since

k

mnk snk = 1, dividing by n and letting n → ∞ we get the result.

Applying this proposition to An = M ∩ M2n for a ﬁnite depth inclusion, we then get 1 lim Hτ (M ∩ M2n ) = log[M : N ]. n→∞ n We also obviously have lim

n→∞

1 Hτ (Z(M ∩ M2n )) = 0. n

Thus combining Proposition 10.4.6 and Theorem 10.4.7 we obtain the following result. Theorem 10.4.10. Let N ⊂ M be an inclusion of II1 -factors of ﬁnite index and ﬁnite depth. Let R = πτ (∪∞ n=1 (M ∩ Mn )) . Then 1 Hτ (R|Γ (R)) = hτ (Γ ) = ht(Γ ) = log[M : N ]. 2

Remark that by [214, Corollary XIX.4.9] any inclusion of index < 4 has ﬁnite depth.

180

10 Relative Entropy and Subfactors

10.5 Shifts on Temperley-Lieb Algebras π Let λ be a real number such that λ−1 ∈ {4 cos2 n+1 | n ≥ 3} ∪ [4, ∞). Consider ∗ the universal C -algebra A generated by a sequence of projections {ek }k∈Z such that ek ek±1 ek = λek , ek ej = ej ek if |k − j| ≥ 2.

As we saw in the previous section, a representation of this algebra arises naturally from an inclusion of II1 -factors with index λ−1 . We also saw that there exists a trace τ on A such that τ (wek ) = λτ (w)

(10.10)

for any k ∈ Z and any w in the algebra generated by projections ej with j < k. For n ≥ 2 denote by An the C∗ -algebra generated by e1 , . . . , en−1 , and set A0 = A1 = C1. An easy induction argument shows that An = An−1 + An−1 en−1 An−1 , n ≥ 2. In particular, the trace is completely determined by (10.10). The dimension and the trace vectors mn and sn of the algebras An , n ∈ N, are described as follows, see § 3 of Chapter XIX in [214] for a proof. If λ−1 ≥ 4 then n n [n/2] , snk = λk Pn−2k (λ), An ∼ − = ⊕ Matmnk (C), mnk = k k−1 k=0 where {Pn }∞ n=0 is the sequence of polynomials such that P0 = P1 = 1, Pn+1 (t) = Pn (t) − tPn−1 (t), n ≥ 1. π with n ≥ 3 then for any k ≥ 0 the On the other hand, if λ−1 = 4 cos2 n+1 embeddings An+2k−1 → An+2k+1 are given by a ﬁxed primitive matrix with the Perron-Frobenius eigenvalue λ−1 . Denote by θλ the automorphism of A deﬁned by θλ (ek ) = ek+1 .

Theorem 10.5.1. We have: 1 (i) hτ (θλ ) = − log λ, if λ−1 ≤ 4; 2 (ii) hτ (θλ ) = η(α) + η(1 − α), where α =

1+

√

1 − 4λ , if λ−1 ≥ 4. 2

Proof. The algebras θλnm (An ), m ∈ Z, are mutually τ -independent. By Proposition 10.4.5 it follows that {An }n is a generating sequence, that is, hτ (θλ ) = lim

n→∞

If λ−1 < 4, then by Proposition 10.4.9

1 Hτ (An ). n

10.5 Shifts on Temperley-Lieb Algebras

181

1 Hτ (A2n ) = log λ. n→∞ n lim

Hence hτ (θλ ) = − 12 log λ. It remains to consider the case λ−1 ≥ 4. Assume ﬁrst that λ−1 > 4. Set √ √ 1 + 1 − 4λ 1 − 1 − 4λ α= and β = 1 − α = . 2 2 Then using that αβ = λ we check that Pn (λ) =

αn+1 − β n+1 , n ≥ 0. α−β

k k 2n−2k (1 − (β/α)2n−2k+1 )(1 − β/α)−1 , we see Since s2n k = λ P2n−2k (λ) = λ α that the diﬀerence k 2n−2k log s2n ) = log s2n k − log(λ α k − ((2n − k) log α + k log β)

is uniformly bounded. Since Hτ (A2n ) = −

n

2n m2n k sk

log s2n k

and

k=0

n

2n m2n k sk = 1,

k=0

the limit of Hτ (A2n )/2n coincides with the limit of n α 2n k m2n . k sk β 2n k=0 (10.11) = (αβ)k (α2n−2k+1 − β 2n−2k+1 )/(α − β), we have

1 2n 2n mk sk ((2n − k) log α + k log β) = − log α + log − 2n n

k=0

Since s2n k n

2n m2n k sk

k=0

n n k α 2n 2n−k k k β 2n k 2n−k k mk α β mk α β = − . 2n α−β 2n α − β 2n k=0

As α > β, m2n k ≤ n

" 2n # k

k=0

and 4αβ = 4λ < 1, k ≤ (αβ)n 2n n

k 2n−k m2n k α β

k=0

k=0

2n k

≤ (4αβ)n → 0.

On the other hand, n 2n k=0

k

α

n 2n − 1 k α2n−k β k β = 2n k−1 k=1 n−1 2n − 1 α2n−1−k β k → β, =β k

2n−k k

k=0

182

10 Relative Entropy and Subfactors

since

2n−1 " 2n−1 # k=n

k

α2n−1−k β k ≤ α−1 (αβ)n

2n−1 " 2n−1 # k=n

k

→ 0. Similarly

n n 2n k 2n k−1 α2n−k β k α2n−k β k = lim n→∞ k−1 k−1 2n n→∞ 2n k=0 k=1 n−1 β 2n k β2 α2n−k β k = lim = . n→∞ α k 2n α lim

k=0

Therefore, since m2n k = lim

n

n→∞

" 2n # k

2n m2n k sk

k=0

−

2n k−1

,

n k α 2n−k k k m2n β = lim k α 2n α − β n→∞ 2n k=0 2 β α β− = β, = α−β α

so that from (10.11) we get that α hτ (θλ ) = − log α + log β = η(α) + η(β). β Finally consider the case λ−1 = 4. Then Pn (λ) =

n+1 , 2n

−2n whence s2n (2n − 2k + 1), and k =2

1 1 2n 2n 2n log(2n − 2k + 1) m2n mk sk log s2n Hτ (A2n ) = − . k sk k = log 2 − 2n 2n 2n n

n

k=0

k=0

Letting n → ∞ we get hτ (θ1/4 ) = log 2.

Denote by M the von Neumann subalgebra of πτ (A) generated by en , n ≤ −1, and by R the von Neumann subalgebra generated by en , n ≥ 1. By [214, Theorem XIX.3.1] both M and R are the hyperﬁnite II1 -factors, N = θλ−1 (M ) ⊂ M is a subfactor of index λ−1 , and N ⊂ M ⊂ θλ (M ) = !M, e0 " is the basic construction. It follows that the automorphism θλ2 of πτ (A) is (the extension to the weak operator closure of) the canonical shift associated with the inclusion N ⊂ M . It is clear that An ⊂ M ∩ Mn for any n ≥ 0. It can be shown that if −1 λ ≤ 4 then An = M ∩ Mn . For λ−1 < 4 the inclusion N ⊂ M has ﬁnite depth, and from Theorem 10.4.10 we again obtain hτ (θλ ) =

1 1 1 hτ (θλ2 ) = log[M : N ] = − log λ. 2 2 2

10.6 Notes

183

On the other hand, if λ−1 > 4 then it can be shown that already M ∩M1 is diﬀerent from A1 = C1. It is not diﬃcult to check that the generating sequence {An }n satisﬁes the commuting square condition. Hence by Theorem 10.3.1 hτ (θλ ) =

1 Hτ (R|θλ (R)). 2

Therefore Hτ (R|θλ (R)) = 2(η(α) + η(β)). This equality, as well as the entropy hτ (θλ ), can also be obtained from the embedding A → Mat2 (C)⊗Z , % k+1 k+1 k k ek → (1 − λ)ek11 ⊗ ek+1 λ(1 − λ)(ek12 ⊗ ek+1 22 + λe22 ⊗ e11 + 21 + e21 ⊗ e12 ), where {ekij }2i,j=1 are the matrix units in the k-th factor Mat2 (C). It can be shown, see [162], that in the GNS-representation of Mat2 (C)⊗Z corresponding to the product-state α 0 ψ = ϕ⊗Z , Qϕ = , 0 β the weak operator closure of A coincides with the centralizer of the state. Thus θλ is one of the Bernoulli shifts on the hyperﬁnite factor considered in Example 3.2.6(ii). Moreover, the von Neumann algebra generated by the projections en , n ≥ 1, coincides with the centralizer of ψ on πψ (Mat2 (C)⊗N ) . In particular, there exists a projection p ∈ R which is mapped onto e111 under the above embedding, and then τ (p) = α, p ∈ θλ (R) ∩ R, pRp = θλ (R)p, (1 − p)R(1 − p) = θλ (R)(1 − p). It should be remarked that the projections en , n ≤ −1, do not generate the centralizer of ψ on πψ (⊗n≤0 Mat2 (C)) .

10.6 Notes The notion of relative entropy for subalgebras of a von Neumann algebra with a normal tracial state was introduced by Connes and Størmer [51], who used it as a tool to prove continuity of mutual entropy. The deﬁnition was extended to arbitrary states by Connes [49]. To see that relative entropy is relevant for estimates of mutual entropy, observe that for ﬁnite dimensional C∗ -subalgebras P1 , . . . , Pn and Q1 , . . . , Qn of M we have Hϕ (P1 , . . . , Pn ) ≤ Hϕ (Q1 , . . . , Qn ) +

n

Hϕ (Pk |Qk ).

k=1

It was Pimsner and Popa [162] who undertook a serious study of this notion and discovered its connection with Jones’ index of II1 -factors. Sects. 10.1 and 10.2 are almost entirely based on their paper. For some examples of

184

10 Relative Entropy and Subfactors

computations of relative entropy see [22], [103], [233]. The results of Pimsner and Popa were extended to type III factors by Hiai [86]. Theorem 10.3.1 was proved by Størmer [210] extending results of Choda [40] and Hiai [88]. See [81] for further discussion of generating sequences. The canonical shift was introduced by Pimsner and Popa [162] and Ocneanu [146]. One usually deﬁnes the canonical shift a bit diﬀerently. As we remarked after the proof of Lemma 10.4.1, for any n ∈ N the algebras M−1 ⊂ Mn ⊂ M2n+1 form a basic construction. Pimsner and Popa [163] found an explicit formula for the corresponding Jones projection in M2n+1 . In other words, they found an explicit representation of M2n+1 on L2 (Mn ) extending the representation of Mn+1 . Using this representation we can deﬁne an antiautomorphism of M−1 ∩M2n+1 , which for the moment we denote by γn . Then one checks that γn+2 ◦γn+1 and γn+1 ◦γn agree on M−1 ⊂ Mn ⊂ M2n+1 , ∩M so there exists a well-deﬁned endomorphism of ∪n (M−1 2n+1 ). To see that we get the same endomorphism as the one deﬁned in the present chapter, by Remark 10.4.4 it suﬃces to check that γn = γn on M−1 ∩ M2n+1 , or even 2 better, the representation of M2n+1 on L (Mn ) alluded to above coincides with the representation deﬁned as in Lemma 10.4.1. Since these representations are determined by the images of en+1 , . . . , e2n , and γn (en+k ) = en−k by construction, we just have to check that γn (en+k ) = en−k for k = 1, . . . , n. This is indeed the case, see e.g. [54]. Alternatively one can use the identity in Remark 10.4.4 and an explicit formula for the canonical shift, see e.g. [23]. The entropy of the shifts on the Temperley-Lieb algebras, Theorem 10.5.1, was computed by Pimsner and Popa [162] for all λ = 1/4. For the case λ < 1/4 instead of the elementary but tedious computations presented above they proved that the shifts are isomorphic to the Bernoulli shifts on the hyperﬁnite factor considered in Example 3.2.6(ii); see [191] for a more general result. The missing case λ = 1/4 was treated by Choda [40] and Yin [236]. The latter paper also contains the computation of the entropy for λ < 1/4, which we used. For arbitrary subfactors the study of entropic properties of the canonical shift was initiated by Choda [41]. In particular, she obtained the inequalities from Corollary 10.4.8 and computed the entropy in the case of a ﬁnite depth subfactor, Theorem 10.4.10. The most general result, Theorem 10.4.7, was proved by Hiai [88]. For similar results in the type III case see [48], [87]. An argument similar to the proof of Proposition 10.4.5 appeared already in the paper of Connes and Størmer [51] and had been used by several authors until it was formalized by Choda [40], [41]. Proposition 10.4.9 was discovered by so many people that it should probably be considered a folklore. It was successfully applied by Choda [40], [41] to a number of models.

11 Systems of Algebraic Origin

Given an automorphism T of a discrete abelian group preserving a 2-cocycle, we deﬁne an automorphism αT of the corresponding twisted group C∗ -algebra. If the cocycle is trivial, we get an abelian dynamical system deﬁned by the ˆ Dynamical properties of such automorphisms are dual automorphism Tˆ of G. well understood. In this chapter we shall study entropic properties of αT for general cocycles.

11.1 Twisted Group C*-algebras For a discrete abelian group G and a 2-cocycle ω ∈ Z 2 (G; T) consider the twisted group C∗ -algebra C ∗ (G, ω). By deﬁnition it is the universal C∗ -algebra generated by unitaries ug , g ∈ G, such that ug uh = ω(g, h)ug+h . We have considered a more general class of algebras in Chap. 8. In particular, we know that C ∗ (G, ω) is nuclear and the regular representation C ∗ (G, ω) → B(2 (G)), ug δh = ω(g, h)δg+h , is faithful, where δg ∈ 2 (G) is deﬁned by δg (h) = 0 for h = g and δg (g) = 1. Recall also that ω(g, 0) = ω(0, g) = ω(0, 0) for any g ∈ G. The vector δ0 deﬁnes the canonical trace τ on C ∗ (G, ω) such that τ (ug ) = 0 for g = 0. Since δ0 is cyclic and the regular representation is faithful, the trace τ is faithful. For any 2-cocycle ω the cocycle ρω (g, h) = ω(g, h)ω(h, g) is a skewsymmetric bicharacter, that is, ρω (g, g) = 1 and ρω (g1 + g2 , h) = ρω (g1 , h)ρω (g2 , h), ρω (h, g1 + g2 ) = ρω (h, g1 )ρω (h, g2 ). Indeed, using that ug uh = ρω (g, h)uh ug in C ∗ (G, ω) we get

186

11 Systems of Algebraic Origin

ug1 +g2 uh = ρω (g1 + g2 , h)uh ug1 +g2 , and on the other hand ug1 +g2 uh = ω(g1 , g2 )ug1 ug2 uh = ω(g1 , g2 )ρω (g1 , h)ρω (g2 , h)uh ug1 ug2 = ρω (g1 , h)ρω (g2 , h)uh ug1 +g2 , so that ρω (g1 + g2 , h) = ρω (g1 , h)ρω (g2 , h). Note also that any bicharacter is a 2-cocycle. It is now easy to describe when the twisted group C∗ -algebra is simple. Theorem 11.1.1. For any ω ∈ Z 2 (G; T) the following conditions are equivalent: (i) the C∗ -algebra C ∗ (G, ω) is simple; (ii) τ is the unique tracial state on C ∗ (G, ω); (iii) the skew-symmetric bicharacter ρω is nondegenerate. Recall that nondegeneracy of ρω means by deﬁnition that for any h = 0 there exists g such that ρω (g, h) = 1. In other words, the subgroup of the dual ˆ formed by the characters ρω (g, ·), g ∈ G, has trivial annihilator in G, group G ˆ so that it is dense in G. Proof of Theorem 11.1.1. Since ug uh u∗g = ρω (g, h)uh , if ρω is nondegenerate then ϕ(uh ) = 0 for every tracial state ϕ. Thus (iii)⇒(ii). If h = 0 is in the kernel of ρω , so that ρω (g, h) = 1 for any g ∈ G, then uh belongs to the center of C ∗ (G, ω). Hence C ∗ (G, ω) is nonsimple. Moreover, then πτ (C ∗ (G, ω)) has nontrivial center and consequently possesses a normal trace diﬀerent from τ . Thus (i)⇒(iii) and (ii)⇒(iii). ˆ on C ∗ (G, ω), It remains to prove (iii)⇒(i). Consider the dual action α of G so ˆ h ∈ G. αχ (uh ) = !χ, h"uh for χ ∈ G, Since ρω is nondegenerate, any character of G can be approximated in the topology of pointwise convergence by characters ρω (g, ·). Since (Ad ug )(uh ) = ρω (g, h)uh , we conclude that the automorphism αχ is approximately inner for ˆ in the sense that it can be approximated by inner automorphisms any χ ∈ G in the pointwise norm topology. Hence any ideal in C ∗ (G, ω) is αχ -invariant. Since τ (·)1 = αχ (·)dχ, ˆ G

it follows that if a ≥ 0 lies in an ideal I, then either 1 ∈ I or τ (a) = 0, and hence a = 0, as τ is faithful. Therefore C ∗ (G, ω) is simple.

In particular, if G is countable and ρω is nondegenerate, then the weak operator closure W ∗ (G, ω) = πτ (C ∗ (G, ω)) of C ∗ (G, ω) is a II1 -factor. This factor is hyperﬁnite since C ∗ (G, ω) is nuclear.

11.2 Estimates of Topological Entropy

187

If ρω is degenerate, the argument in the proof of Theorem 11.1.1 essentially gives a description of the ideal structure of C ∗ (G, ω). The following statement will be suﬃcient for our purposes. Lemma 11.1.2. Let H be the kernel of ρω , that is, H consists of the elements h ∈ G such that ρω (g, h) = 1 for every g ∈ G. Then the center of C ∗ (G, ω) is generated by the elements uh , h ∈ H, so it is isomorphic to C ∗ (H, ω|H ). Note that it follows from Lemma 11.1.3 below that C ∗ (H, ω|H ) is isomorˆ but the isomorphism is noncanonical. phic to C(H), Proof of Lemma 11.1.2. It is clear that C ∗ (H, ω|H ) is contained in the center. On the other hand, the map E deﬁned by E(x) = αχ (x)dχ G/H

is a conditional expectation onto C ∗ (H, ω|H ). Since (Ad ug )(uh ) = ρω (g, h)uh , the autoand the characters ρω (g, ·), g ∈ G, form a dense subgroup of G/H, morphism αχ is approximately inner for any χ ∈ G/H. Hence E(x) = x for any element x in the center, so the center is contained in C ∗ (H, ω|H ).

Finally remark that the map ω → ρω deﬁnes an embedding of H 2 (G; T) into the group of skew-symmetric bicharacters of G. Lemma 11.1.3. If ρω (g, h) = 1 for every g, h ∈ G, then ω is a coboundary, that is, there exists a function G g → zg ∈ T such that ω(g, h) = zg zh z¯g+h . Proof. By assumption the C∗ -algebra C ∗ (G, ω) is abelian. Choose a character f on C ∗ (G, ω). Then applying it to the identity ug uh = ω(g, h)ug+h we see that ω is the coboundary of G g → f (ug ).

Notice that in the notation of the lemma the elements vg = z¯g ug satisfy ˆ the relations vg vh = vg+h , so that C ∗ (G, ω) ∼ = C ∗ (G) ∼ = C(G).

11.2 Estimates of Topological Entropy Let T be an automorphism of a discrete countable abelian group G. If a cocycle ω ∈ Z 2 (G; T) is T -invariant, then T deﬁnes an automorphism αT of C ∗ (G, ω) such that αT (ug ) = uT g . We shall write αT,ω for αT and ug,ω for ug when we want to stress that we consider an automorphism of C ∗ (G, ω). ˆ so that !Tˆχ, g" = !χ, T −1 g". Consider the dual automorphism Tˆ of G, ˆ Then Tˆ is a homeomorphism of the compact space G. Theorem 11.2.1. For the automorphism αT of C ∗ (G, ω) we have 1 htop (Tˆ) ≤ ht(αT ) ≤ htop (Tˆ). 2

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11 Systems of Algebraic Origin

ˆ deﬁned by Tˆ, so that If ω ≡ 1 then αT is the automorphism of C(G) ht(αT ) = htop (Tˆ). Thus the upper bound in the theorem is optimal. In Example 11.2.7 below we shall see that the lower bound is also optimal. Proof of the lower bound in Theorem 11.2.1. Let ω ¯ be the cocycle deﬁned by ω ¯ (g, h) = ω(g, h). The elements ug,ω ⊗ug,¯ω , g ∈ G, generate a C∗ -subalgebra D ˆ Under this isoof C ∗ (G, ω)⊗C ∗ (G, ω ¯ ), which is isomorphic to C ∗ (G) ∼ = C(G). ˆ morphism the automorphism αT,ω ⊗αT,¯ω becomes the automorphism of C(G) deﬁned by Tˆ. Using that the topological entropy is subadditive and monotone we then get ht(αT,ω ) + ht(αT,¯ω ) ≥ ht(αT,ω ⊗ αT,¯ω ) ≥ ht((αT,ω ⊗ αT,¯ω )|D ) = htop (Tˆ). Thus we just have to prove that ht(αT,ω ) = ht(αT,¯ω ). ¯ ) can Assume ﬁrst that ω(g, h) = ω(h, g) for any g, h ∈ G. Then C ∗ (G, ω be identiﬁed with the opposite algebra C ∗ (G, ω)op , since for any g, h ∈ G we have uh,¯ω ug,¯ω = ω ¯ (h, g)ug+h,¯ω = ω(g, h)ug+h,¯ω . Hence it suﬃces to prove that if α is an automorphism of a C∗ -algebra and we consider it as an auop tomorphism αop of the opposite algebra, then ht(α) ). By A.1 a n= ht(α ∗ map γ: A → B is completely positive if and only if i,j bi γ(a∗i aj )bj ≥ 0 for any n and a1 , . . . , an ∈ A, b1 , . . . , bn ∈ B. It follows that any completely positive map A → B remains completely positive being considered as a map Aop → B op . Consequently, if Ω is a ﬁnite subset of a C∗ -algebra A, then rcpA (Ω, δ) = rcpAop (Ω, δ). Hence ht(α) = ht(αop ). In the general case consider the cocycle ω op (g, h) = ω(h, g), so that ∗ C (G, ω op ) can be identiﬁed with C ∗ (G, ω)op . Then it is enough to show that ht(αT,ωop ) = ht(αT,¯ω ). Since ρωop = ρω¯ , by Lemma 11.1.3 the cocycles ω op and ω ¯ are cohomologous, so that there exists a function G g → zg ∈ T such that ω ¯ (g, h) = ω op (g, h)zg zh z¯g+h . ¯ ) → C ∗ (G, ω op ) deﬁned by ug,¯ω → We thus get an isomorphism γ: C ∗ (G, ω zg ug,ωop . Even though this isomorphism is not equivariant, the elements αT,¯ω (ug,¯ω ) and (γ −1 ◦αT,ωop ◦γ)(ug,¯ω ) diﬀer only by a scalar factor of modulus one. It follows that for any ﬁnite set Ω consisting of elements ug,¯ω we have ht(Ω, δ; αT,¯ω ) = ht(Ω, δ; γ −1 ◦ αT,ωop ◦ γ). Hence ht(αT,¯ω ) = ht(γ −1 ◦ αT,ωop ◦ γ) = ht(αT,ωop ), and the proof of the lower bound for ht(αT ) is complete.

To obtain the upper bound we need a way to compute htop (Tˆ) by looking at the action of T on G. For a ﬁnite set F ⊂ G let h(F ; T ) = lim

n→∞

1 log |F + T (F ) + . . . + T n−1 (F )|. n

Then deﬁne h(T ) = supF ⊂G h(F ; T ).

11.2 Estimates of Topological Entropy

189

Lemma 11.2.2. We have ht(αT ) ≤ h(T ). Proof. To estimate ht(αT ) we shall use the maps Φf and Ψf from Chap. 8. Let X be a ﬁnite subset of G. Fix δ > 0. Since G is amenable, there exists a ﬁnitely supported positive function f on G such that g f − f 1 < δ 2 for g ∈ X, where g f (h) = f (h − g), and f 1 = 1. Let F be the support of f . For n ∈ N consider the function f (n) = f ∗ (f ◦ T −1 ) ∗ . . . ∗ (f ◦ T −n+1 ). Notice that for any h ∈ G and 0 ≤ k ≤ n − 1 we can write h (f

(n)

) = f ∗ . . . ∗ (f ◦ T −k+1 ) ∗ h (f ◦ T −k ) ∗ (f ◦ T −k−1 ) ∗ . . . ∗ (f ◦ T −n+1 ).

Then for g ∈ X and 0 ≤ k ≤ n − 1 we have T k g (f (n) ) − f (n) 1 ≤ f 1 . . . T k g (f ◦ T −k ) − f ◦ T −k 1 . . . f ◦ T −n+1 1 = (g f ) ◦ T −k − f ◦ T −k 1 = g f − f 1 < δ 2 . Let Ω = {ug | g ∈ X}. Using the maps Φf (n) and Ψf (n) from Chap. 8 we then get by Proposition 8.1.3(iii) that rcp(Ω ∪ αT (Ω) ∪ . . . ∪ αTn−1 (Ω), δ) ≤ |supp f (n) |. Since the support of f (n) is contained in F + T (F ) + . . . + T n−1 (F ), we thus obtain ht(Ω, δ; αT ) ≤ h(F ; T ). Hence ht(αT ) ≤ h(T ).

To ﬁnish the proof of Theorem 11.2.1 it is now suﬃcient to prove the following. Theorem 11.2.3. We have h(T ) = htop (Tˆ). For this we need yet another formula for the entropy of automorphisms of compact groups. Theorem 11.2.4. Let S be a continuous automorphism of a compact separable group Γ . Then htop (S) = hµ (S) = sup lim sup − V

n→∞

1 log µ(V ∩ S −1 (V ) ∩ . . . ∩ S −n+1 (V )), n

where µ is the Haar measure on Γ and the supremum is taken over all neighbourhoods V of the unit element e ∈ Γ . Proof. Since hµ (S) ≤ htop (S), it suﬃces to prove that htop (S) ≤ sup lim sup − V

n→∞

1 log µ(V ∩S −1 (V )∩. . .∩S −n+1 (V )) ≤ hµ (S). (11.1) n

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11 Systems of Algebraic Origin

Choose a left Γ -invariant metric d on Γ . For a ﬁxed ε > 0 let V = Bε/2 (e), the open ball of radius ε/2 with center at e. If x and y are (n, ε)-separated with respect to S, then the sets x(V ∩S −1 (V )∩. . .∩S −n+1 (V )) and y(V ∩S −1 (V )∩ . . . ∩ S −n+1 (V )) are disjoint. Indeed, otherwise for each k = 0, . . . , n − 1 we could ﬁnd a, b ∈ V such that xS −k (a) = yS −k (b), or equivalently, S k (x)a = S k (y)b. But then d(S k (x), S k (y)) < ε, so that x and y are not (n, ε)-separated. Hence if srn (ε) is the maximal number of (n, ε)-separated points, then srn (ε)µ(V ∩ S −1 (V ) ∩ . . . ∩ S −n+1 (V )) ≤ 1, which implies the ﬁrst inequality in (11.1). To prove the second inequality ﬁx a neighbourhood V of e. Let ε > 0 be such that Bε (e) ⊂ V . Choose a measurable partition ξ = {X1 , . . . , Xm } such that the diameter of each atom is smaller than ε. If x ∈ Xi0 ∩ S −1 (Xi1 ) ∩ . . . ∩ S −n+1 (Xin−1 ), then Xik ⊂ Bε (S k (x)) ⊂ S k (x)V for k = 0, . . . , n − 1, so that Xi0 ∩ S −1 (Xi1 ) ∩ . . . ∩ S −n+1 (Xin−1 ) ⊂ x(V ∩ S −1 (V ) ∩ . . . ∩ S −n+1 (V )) and hence µ(Xi0 ∩ S −1 (Xi1 ) ∩ . . . ∩ S −n+1 (Xin−1 )) ≤ µ(V ∩ S −1 (V ) ∩ . . . ∩ S −n+1 (V )). Taking the logarithm, multiplying by µ(Xi0 ∩ S −1 (Xi1 ) ∩ . . . ∩ S −n+1 (Xin−1 )) and summing over all multi-indices (i0 , . . . , in−1 ) we get Hµ (ξ ∨ S −1 ξ ∨ . . . ∨ S −n+1 ξ) ≥ − log µ(V ∩ S −1 (V ) ∩ . . . ∩ S −n+1 (V )). Dividing by n and letting n → ∞ we get the second inequality in (11.1).

Note that instead of the supremum over all neighbourhoods we can take the limit along a basis of neighbourhoods. Before we return to the computation of the entropy of αT , we shall illustrate this theorem by a computation of the entropy of toral automorphisms. Example 11.2.5. Let T ∈ GLn (Z). Then T deﬁnes an automorphism ST of the n-dimensional torus Tn ∼ = Rn /Zn . Then log |λi |, htop (ST ) = i:|λi |>1

where Sp T = {λ1 , . . . , λn } (counting with multiplicities). To see this consider the quotient map π: Rn → Tn . Let U be a relatively compact neighbourhood of 0 ∈ Rn such that π is injective on U ∪ T (U ). Let V = π(U ). If x ∈ U is such that π(T (x)) = ST (π(x)) ∈ V , then T (x) ∈ U . It follows that V ∩ ST−1 (V ) ∩ . . . ∩ ST−n+1 (V ) = π(U ∩ T −1 (U ) ∩ . . . ∩ T −n+1 (U )). Hence

11.2 Estimates of Topological Entropy

htop (ST ) = lim lim sup − U

n→∞

191

1 log µ(U ∩ T −1 (U ) ∩ . . . ∩ T −n+1 (U )), n

where µ is now the Lebesgue measure on Rn . Since for any two relatively compact neighbourhoods U1 , U2 of 0 ∈ Rn we have c−1 U1 ⊂ U2 ⊂ cU1 for some c > 0, the lim sup above is actually the same for all U ’s. Replacing Rn by Cn , which simply doubles the lim sup, we may assume that T is an operator on a complex vector space. By considering the normal Jordan form of T , we can reduce the computation to the case when T ∈ GLm (C) has only one eigenvalue λ. If we write T in the normal Jordan form then the matrix ST S −1 can be made arbitrarily close to the matrix λ1 by a suitable choice of a diagonal matrix S. In other words, for any ε > 0, we can introduce a norm on Cm such that T ±1 − λ±1 1 < ε. Take the unit ball in this norm for U . Then T −k (U ) is contained in the ball of radius (|λ|−1 + ε)k and contains the ball of radius (|λ| + ε)−k . So if |λ| + ε ≤ 1 then U ⊂ T −1 (U ) and the lim sup is zero, and if |λ| + ε ≥ 1 then it lies between −2m log(|λ|−1 + ε) and 2m log(|λ| + ε). It follows that the lim sup is zero when |λ| < 1, and it is 2m log |λ| when |λ| ≥ 1. We can now prove one half of Theorem 11.2.3. Lemma 11.2.6. We have h(T ) ≤ htop (Tˆ). Proof. It is clear that h(F ; T ) is nondecreasing in F . Thus in proving that h(F ; T ) ≤ htop (Tˆ) we may assume that F = −F and 0 ∈ F . For m ∈ N set F (m) = F + . . . + F . m

By replacing, if necessary, F by a larger set we may also assume that |F (m) | → 1 as m → ∞. |F (m+1) |

(11.2)

Indeed, consider the group generated by F . It is a ﬁnite product G1 × . . . × Gn of cyclic groups. We may then assume that F is of the form F1 × . . . × Fn . Thus it suﬃces to prove the claim for cyclic groups. But then it is obvious: if the group is ﬁnite, replace F by the group itself, and if the group is isomorphic to Z, take a suﬃciently large symmetric interval. We assert now that for any n, m ∈ N we have |F + T (F ) + . . . + T n−1 (F )| |F (m+1) |2n , (1F (m) ∗ 1F (m) ∗ 1T (F (m) ) ∗ 1T (F (m) ) ∗ . . . ∗ 1T n−1 (F (m) ) ∗ 1T n−1 (F (m) ) )(0) (11.3) where 1X denotes the characteristic function of a set X ⊂ G. Indeed, denote by N the denominator in the above expression. It is exactly the number of ≤

192

11 Systems of Algebraic Origin

elements (g1 , . . . , g2n ) ∈ F (m) × F (m) × . . . × T n−1 (F (m) ) × T n−1 (F (m) ) such that g1 + . . . + g2n = 0. For every such (g1 , . . . , g2n ) and an element h = h1 + . . . + hn ∈ F + T (F ) + . . . + T n−1 (F ) consider the element (h1 + g1 , g2 , . . . , hn + g2n−1 , g2n ) of F (m+1) ×F (m+1) ×. . .×T n−1 (F (m+1) )×T n−1 (F (m+1) ). This way we get N diﬀerent elements representing h, which proves (11.3). Fix m and consider the function f = |F (m) |−2 1F (m) ∗ 1F (m) on G. Then taking the logarithm of the both sides of (11.3) and dividing by n we get 1 log |F + T (F ) + . . . + T n−1 (F )| n |F (m+1) | 1 . (11.4) log(f ∗ (f ◦ T −1 ) ∗ . . . ∗ (f ◦ T −n+1 ))(0) + 2 log n |F (m) | Denote by ϕ the Fourier transform of f , so ϕ(χ) = g !χ, g"f (g). Then, (m) ˆ such that = −F (m) , ϕ is a positive continuous function on G since F ϕ(e) = g f (g) = 1. Fix ε > 0 and put ≤−

ˆ | ϕ(χ) > 1 − ε}. V = {χ ∈ G ˆ and Then V is an open neighbourhood of the unit e ∈ G, (f ∗ (f ◦ T −1 ) ∗ . . . ∗ (f ◦ T −n+1 ))(0) =

ˆ G

ϕ(χ)(ϕ ◦ Tˆ−1 )(χ) . . . (ϕ ◦ Tˆ−n+1 )(χ)dµ(χ) ≥ (1 − ε)n µ

n−1 *

k ˆ T (V ) .

k=0

This and (11.4) then imply that 1 log |F + T (F ) + . . . + T n−1 (F )| n n−1 * 1 |F (m+1) | k Tˆ (V ) − log(1 − ε) + 2 log , ≤ − log µ n |F (m) | k=0 whence by Theorem 11.2.4 |F (m+1) | . h(F ; T ) ≤ htop (Tˆ) − log(1 − ε) + 2 log |F (m) | In view of (11.2) and since ε could be taken arbitrarily small, we get h(F ; T ) ≤

htop (Tˆ). By virtue of Lemma 11.2.2 the previous lemma completes the proof of Theorem 11.2.1.

11.2 Estimates of Topological Entropy

193

Proof of Theorem 11.2.3. By Lemmas 11.2.2 and 11.2.6 we have ht(αT ) ≤ h(T ) ≤ htop (Tˆ). ˆ deﬁned by Tˆ, so that But if ω ≡ 1 then αT is the automorphism of C(G) ht(αT ) = htop (Tˆ). Hence h(T ) = htop (Tˆ).

Examples of nontrivial invariant cocycles can be obtained as follows. Let R = Z[t, t−1 ] be the ring of Laurent polynomials over Z. For g ∈ R deﬁne g˜(t) = g(t−1 ). Let f ∈ Z[t], f = 1, be a polynomial with nonzero constant term and no cyclotomic factors, that is, its roots are not roots of unity. Consider the abelian group G = R/(f˜) ⊕ R/(f ). Let T be the automorphism of G of multiplication by t ∈ R. Then for any character χ of the abelian group R/(f ) we have a pairing (g1 , g2 ) → χ(˜ g1 g2 )

(11.5)

of R/(f˜) with R/(f ), and a T -invariant bicharacter ω on G deﬁned by ˜ 1 g2 ). ω((g1 , g2 ), (h1 , h2 )) = χ(h The corresponding bicharacter ρω is then ˜ 1 g2 − g˜1 h2 ). ρω ((g1 , g2 ), (h1 , h2 )) = χ(h This bicharacter is nondegenerate if and only if the pairing (11.5) is nonde˜2 g˜1 , it suﬃces to check nondegeneracy in the second generate. Since g˜1 g2 = g˜ ), which is dual to the automorvariable. Let S be the automorphism of R/(f phism of multiplication by t. Then the right hand side of (11.5) can be written as χ(g1 (T −1 )g2 ) = !g1 (S)χ, g2 ". Hence (11.5) is nondegenerate if and only if ). By Lemma 11.3.2 the group generated by the S-orbit of χ is dense in R/(f below and our assumption on f , the automorphism S is ergodic, so the orbit of χ is dense for almost every χ. Note that if the leading coeﬃcient and the constant term of f are equal to 1, then G ∼ = Z2n , where n is the degree of f . Then by Example 11.2.5, if λ1 , . . . , λn are the roots of f , we have 1 ˆ log |λj | = 2 log |f (e2πis )|ds, htop (T ) = 2 j:|λj |>1

0

1 where the second equality follows from the fact that 0 log |λ − e2πis |ds is zero 1 if |λ| ≤ 1, and it is log |λ| if |λ| ≥ 1. The number m(f ) = 0 log |f (e2πis )|ds is called the logarithmic Mahler measure of f , and the formula htop (S) = m(f ) holds in fact for arbitrary f . It follows from Theorem 11.2.1 that

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11 Systems of Algebraic Origin

m(f ) ≤ ht(αT ) ≤ 2m(f ). Note also that if we denote R/(f˜) by G1 and R/(f ) by G2 , then the ˆ 1 . Using this homomorphism pairing (11.5) deﬁnes a homomorphism G2 → G ˆ 1 . Then C ∗ (G, ω) ∼ we can deﬁne the action by right translations of G2 on G = ∗ ˆ ˆ C(G1 ) G2 . In particular C(G1 ) is a subalgebra of C (G, ω), which gives an alternative way of obtaining the lower bound for ht(αT ). Example 11.2.7. Let f = 2. Then G = R/(2) ⊕ R/(2) ∼ = ⊕ (Z/2Z ⊕ Z/2Z), n∈Z

T is the shift to the right. We have htop (Tˆ) = 2 log 2, so that log 2 ≤ ht(αT ) ≤ 2 log 2. A character on ⊕n∈Z Z/2Z is just a sequence χ = (χn )n of numbers χn = ±1. Consider the simplest nontrivial character, namely, χ0 = −1 and χn = 1 for n = 0. In other words, if we consider χ as a character on R = Z[t, t−1 ] then χ(1) = −1 and χ(tn ) = 1 for n ∈ Z\{0}. Then the pairing (11.5) is $ ⊕ (Z/2Z ⊕ Z/2Z) (g1 , g2 ) → (−1)g1 (n)g2 (n) . n∈Z

n∈Z

It follows that not only the bicharacter ρω is nondegenerate, but its restriction to the subgroup n G(n) = ⊕ (Z/2Z ⊕ Z/2Z) ⊂ G i=1

is nondegenerate for any n ∈ N. It follows by Theorem 11.1.1 that the C∗ subalgebra C ∗ (G(n) , ω|G(n) ) of C ∗ (G, ω) is a simple C∗ -algebra of dimension 4n , hence it is isomorphic to Mat2n (C). But then ht(αT ) ≤ lim sup n→∞

1 log rank C ∗ (G(n) , ω|G(n) ) = log 2. n

Thus ht(αT ) = log 2. This example shows that the lower bound in Theorem 11.2.1 is optimal.

11.3 K-systems Keeping the notation of the previous section, it is known that if the dual automorphism Tˆ is ergodic then Tˆ, considered as a transformation preserving the Haar measure, is a Bernoulli shift, see Notes at the end of the chapter. Although we cannot prove such a result for the automorphism αT of C ∗ (G, ω), we shall show that if the cocycle ω is in a sense asymptotically trivial then the powers of αT can be approximated by shifts on UHF-algebras. As we shall see,

11.3 K-systems

195

this implies that we get an entropic K-system, in particular hτ (αT ) > 0. Notice that since by Theorem 11.2.4 the topological entropy of Tˆ coincides with the ˆ Theorem 11.2.1 measure entropy with respect to the Haar measure µ on G, implies that hτ (αT ) ≤ hµ (Tˆ) = htop (Tˆ). However, since the dynamical entropy is superadditive we can not obtain a lower bound for hτ (αT ) analogous to that in Theorem 11.2.1. We start with the following classical characterization of ergodicity. Extend αT to W ∗ (G, ω) = πτ (C ∗ (G, ω)) and continue to denote the extended automorphism by αT . Lemma 11.3.1. The following conditions are equivalent: (i) the automorphism αT of W ∗ (G, ω) is ergodic; (ii) the automorphism T of G is aperiodic, that is, the orbit of any nonzero element is inﬁnite. Proof. If T is not aperiodic, thenthere exist n ∈ N and g =

0 such that n−1 T n g = g. Then the element x = k=0 αTk (ug ) of W ∗ (G, ω) is αT -invariant and not a scalar, since the vector xδ0 =

n−1

ω(0, 0)δT k g ∈ 2 (G)

k=0

is nonzero and orthogonal to δ0 . Thus αT is not ergodic. Conversely, assume that T is aperiodic. The automorphism αT is implemented by the unitary U on 2 (G) deﬁned by U δg = δT g . To prove ergodicity of αT it suﬃces to show that if U ξ = ξ then ξ is a scalar multiple of the trace vector δ0 . This is indeed the case, since if ξ = g ag δg then aT n g = ag for every n ∈ Z and g ∈ G, which is possible only when ag = 0 for every g = 0.

In particular, Tˆ is ergodic if and only if T is aperiodic. As we already mentioned, then Tˆ is a Bernoulli shift. At the level of G the Bernoullicity manifests itself in the following property of T . Proposition 11.3.2. Let G be a discrete abelian group, T an aperiodic endomorphism of G, Y a ﬁnite subset of G containing 0 ∈ G. Then there exists m ∈ N such that whenever y1 , . . . , yn ∈ Y and m1 , . . . , mn ∈ N satisfy mk+1 − mk ≥ m, k = 1, . . . , n − 1, then n k=1

only if y1 = . . . = yn = 0.

T mk yk = 0

(11.6)

196

11 Systems of Algebraic Origin

Proof. First consider the case when G is ﬁnitely generated. Then the periodic part of G is ﬁnite. Since T acts on it aperiodically, the periodic part is trivial, so G ∼ = ZN for some N ∈ N. Then T is deﬁned by a nondegenerate matrix with integral coeﬃcients, which we denote by the same letter T . It is clear that the aperiodicity is equivalent to the fact that T has no roots of unity as eigenvalues. Let P be the spectral projection of T corresponding to the part of the spectrum lying outside the closed unit disc. Thus V0 = Ker P ⊂ CN is the sum of the root spaces of T corresponding to the eigenvalues lying in the closed unit disc, and V1 = P (CN ) is the sum of the root spaces corresponding to the eigenvalues lying outside the closed unit disc. We claim that ZN ∩ V0 = 0. Indeed, if W is the linear span of ZN ∩ V0 ∼ = Zl , then T |W is represented by an l by l matrix with integral coeﬃcients, all of whose eigenvalues lie in the unit disc, hence (if l = 0) on the unit circle, since the determinant is a nonzero integer. But then the eigenvalues are roots of unity by the Dirichlet theorem, see e.g. [178, 10.1.B], which contradicts our assumptions. It follows that W = 0. It follows that (11.6) is equivalent to n

T mk P yk = 0.

(11.7)

k=1

Since the eigenvalues of T |V1 have absolute values greater than 1, writing T |V1 in the normal Jordan form we conclude that there exist δ > 0 and C > 0 such that T −k |V1 ≤ C(1 − δ)k for all k ∈ N. Choose C1 > 0 such that C1−1 ≤ P y ≤ C1 for y ∈ Y \{0}. Now choose m ∈ N such that ∞ CC1 (1 − δ)k < C1−1 . k=m

Then if mk+1 − mk ≥ m, we have n−1 n−1 CC1 (1 − δ)mn −mk < C1−1 . T mk −mn P yk ≤ k=1

k=1

So if (11.7) holds then P yn < C1−1 , and consequently yn = 0 and n−1 mk P yk = 0. Thus we recursively get yn = yn−1 = . . . = y1 = 0. k=1 T Consider now the general case and prove the proposition by induction on |Y |. Let H0 be the group generated by T k (Y ), k = 0, 1, . . . , ∞. Set Hk = T k (H0 ), H∞ = ∩k Hk , Y = Y ∩ H∞ . Suppose Y =Y . Since Y is ﬁnite, there exists l ∈ N such that Y = Y ∩Hl . n We claim that if k=1 T mk yk ∈ H∞ and mk+1 −mk ≥ l, then y1 , . . . , yn ∈ Y . If g ∈ H∞ ⊂ T (H0 ), then g = T h for a unique element h, since Ker T = 0. As g ∈ T k (H0 ), we conclude that h ∈ T k−1 (H0 ) for any k ∈ N, so that

11.3 K-systems

197

h ∈ H∞ . In other words, T deﬁnes an automorphism of H∞ . It follows that n n mk −m1 mk −m1 T y ∈ H . Hence y ∈ H + T yk , so y1 ∈ Y ∩Hl = k ∞ 1 ∞ k=1 k=2 n Y ⊂ H∞ and therefore k=2 T mk yk ∈ H∞ . Continuing this process we get that y1 , . . . , yn ∈ Y . In particular, if (11.6) holds with mk+1 − mk ≥ l, then y1 , . . . , yn ∈ Y . Since |Y | < |Y |, we may then apply the inductive assumption. On the other hand, if Y = Y , then Y ⊂ H1 . Hence there exists l ∈ N such that if H is the group generated by Y, T (Y ), . . . , T l (Y ), then Y ⊂ T (H), so that H ⊂ T (H). Then H is a ﬁnitely generated group, T −1 is a aperiodic endomorphism of H, and (11.6) can be written as well-deﬁned n −1 mn −mk (T ) yk = 0. Thus we can apply the case of a ﬁnitely generated k=1 group to the endomorphism T −1 of H, the elements yn , . . . , y1 ∈ Y and the numbers 0, mn − mn−1 , . . . , mn − m1 .

Using this proposition we shall show that under an additional assumption on the cocycle the powers of αT can be approximated by shifts on UHFalgebras. Theorem 11.3.3. Let T be an aperiodic automorphism of a discrete abelian group G, ω ∈ Z 2 (G; T) a T -invariant cocycle such that ∞

|1 − ρω (T n g, h)| < ∞ for any g, h ∈ G.

n=1

Then for any ﬁnite subset Ω ⊂ C ∗ (G, ω) and ε > 0 there exist a full matrix algebra B and n0 ∈ N such that for any n ≥ n0 there exist trace preserving unital completely positive maps Φ: C ∗ (G, ω) → A = B ⊗Z and Ψ : A → C ∗ (G, ω) such that (i) α ◦ Φ = Φ ◦ αTn and αTn ◦ Ψ = Ψ ◦ α, where α is the shift to the right on A; (ii) Φ(Ω) ⊂ε B0 and (Ψ ◦ Φ)(a) − a < ε for any a ∈ Ω, where B0 ∼ = B is the 0-th factor of A. Before we turn to the proof, let us show that this theorem implies a strong form of positivity of entropy. Recall that according to Deﬁnition 4.3.1 we say that a W∗ -dynamical system (M, ϕ, α) with M = C1 and faithful ϕ is an entropic K-system if hϕ (γ; αn ) → Hϕ (γ) as n → ∞ for any channel γ. Corollary 11.3.4. Under the assumptions of Theorem 11.3.3 the system (W ∗ (G, ω), τ, αT ) is an entropic K-system (assuming G = 0). In particular, hτ (αT ) > 0. Proof. Let γ: C → W ∗ (G, ω) be a channel. By deﬁnition H τ (γ) is the supremum of Hτ (γ; {xi }) over all ﬁnite partitions of unity 1 = i xi in W ∗ (G, ω). Any partition of unity in W ∗ (G, ω) can be approximated in the strong operator topology by a partition of unity in C ∗ (G, ω) (this follows from A.10, since a partition of unity can be considered as a unital completely positive map from

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11 Systems of Algebraic Origin

an abelian algebra into W ∗ (G, ω)). Thus it suﬃces to consider partitions of unity in C ∗ (G, ω). Fix a partition Ω = {xi }i such that Hτ (γ; {xi }) is δ-close to Hτ (γ) for some δ > 0. Then ﬁx ε > 0 and choose n0 and B according to Theorem 11.3.3. Let n ≥ n0 and let Φ and Ψ be the corresponding maps. Let E: A → B0 be a conditional expectation. Put bi = E(Φ(xi )), and then for k ∈ N let yi1 ...ik = Ψ (bi1 α(bi2 ) . . . αk−1 (bik )). Thus for each k we get a partition of unity in C ∗ (G, ω). Since Ψ is assumed to be trace preserving, and the unique trace on A is the product trace, we have τ (yi1 ...ik ) = τ (yi1 ) . . . τ (yik ). (l)

(l−1)n

Since also yil = Ψ (αl−1 (bil )) = αT see (3.3), we obtain (k−1)n

Hτ (γ, αTn ◦ γ, . . . , αT

(yil ) for l = 1, . . . , k, by deﬁnition,

◦ γ; {yi1 ...ik }) = kHτ (γ; {yi }).

Hence hτ (γ; αTn ) ≥ Hτ (γ; {yi }). Since bi − Φ(xi ) ≤ ε by assumption on Φ, we also have yi − xi ≤ 2ε. Thus we could choose ε such that Hτ (γ; {yi }) is δ-close to Hτ (γ; {xi }). Then hτ (γ; αTn ) is (2δ)-close to Hτ (γ) for any n ≥ n0 . Hence hτ (γ; αTn ) → Hτ (γ) as n → ∞.

Remark that by Theorem 4.3.5 the convergence ρω (T n g, h) → 1 as n → ∞ is a necessary condition for (W ∗ (G, ω), τ, αT ) to be a K-system. Proof of Theorem 11.3.3. Without loss of generality we may assume that Ω consists of the elements ug for g in a ﬁnite subset of G. By Proposition 8.1.3(iii) we can ﬁnd a ﬁnite subset X ⊂ G such that the maps ΦX : : C ∗ (G, ω) → B(2 (X)) and ΨX : B(2 (X)) → C ∗ (G, ω) from Chap. 8 corresponding to the function f = 1X satisfy (ΨX ◦ ΦX )(a) − a

0 such that there exists a trigonometric polynomial f satisfying (i) |f (z)| ≤ 1 for any z ∈ T; (ii) |f (z)| ≤ δ for any z ∈ T such that |z + 1| ≥ ε; (iii) | T f (z)dµ(z)| > δ 1/(N −1) . Let f (z) = |l|<m al z l . Our assumption on g implies that g has inﬁnite N −1 order. Hence by Proposition 11.3.2 there exists M0 such that i=1 li T ki g = 0 for any k1 , . . . , kN −1 such that ki+1 − ki ≥ M0 and (l1 , . . . , lN −1 ) = (0, . . . , 0) such that |li | < m. Assuming that M ≥ M0 choose m1 , . . . , mN −1 satisfying (11.13). Then by the same argument as before we get N −1 n−1 N −1 1 $ −1 f (ρω (T k g, T mi g)) → aN = f (z)dµ(z) . 0 n T i=1 k=0

Hence for inﬁnitely many k’s N −1 $

|f (ρω (T k g, T mi g))| > δ,

i=1

so that |1 + ρω (T g, T can be taken for mN . k

mi

g)| < ε for i = 1, . . . , N − 1. Any such k ≥ mN −1 + M

Finishing our discussion of entropic properties of the automorphisms αT , we shall show that the ergodic zero entropy systems are exactly those which are uniquely ergodic.

11.4 Zero Entropy Systems

203

Theorem 11.4.2. Let T be an aperiodic automorphism of G, ω ∈ Z 2 (G; T) a T -invariant cocycle. Then the following conditions are equivalent: (i) hτ (αT ) > 0; (ii) there exists an αT -invariant state diﬀerent from τ . Proof. Let ϕ be an αT -invariant state, ϕ = τ . Using ϕ we shall construct a nontrivial stationary coupling with a Bernoulli shift. Consider the positive ˆ deﬁned by unital map P : C ∗ (G, ω) → C(G) ˆ P (x)(χ) = ϕ(αχ (x)) for x ∈ C ∗ (G, ω), χ ∈ G, where αχ is the dual automorphism, so αχ (ug ) = !χ, g"ug . Since τ (·)1 = ˆ ˆ αχ (·)dµ(χ), where µ is the Haar measure on G, we have µ ◦ P = τ . G ˆ Consider the automorphism αTˆ of C(G) deﬁned by the dual automorphism Tˆ. Then αTˆ (P (a))(χ) = P (a)(Tˆ−1 χ) = ϕ(αTˆ−1 χ (a)) = ϕ((αT−1 ◦ αχ ◦ αT )(a)) = ϕ((αχ ◦ αT )(a)) = P (αT (a))(χ), so that αTˆ ◦ P = P ◦ αT . Since ϕ = τ , there exists g = 0 such that ϕ(ug ) = 0. Then P (ug ) = 0, so that P = τ (·)1. Fix a self-adjoint element a ∈ C ∗ (G, ω) such that P (a) = τ (a)1. Let ε = P (a) − τ (a)1. By Theorem 11.3.3 applied to the trivial cocycle there exist n ∈ N, a full matrix algebra B and a unital completely ˆ → A = B ⊗Z such that α ◦ Φ = Φ ◦ αn , where α is the positive map Φ: C(G) Tˆ shift to the right on A, ψ ◦ Φ = µ, where ψ is the unique tracial state on A, Φ(P (a)) − a0 < ε/4 for some self-adjoint element a0 in the 0-th factor B0 of A, and Φ(P (a)) − τ (a)1 ≥ ε/2. Let C be an abelian C∗ -subalgebra of B0 containing a0 . Put D = C ⊗Z , an let E: A → D be the ψ-preserving conditional expectation. Consider the map Θ = E ◦ Φ ◦ P : C ∗ (G, ω) → D. Then ψ ◦ Θ = τ and α ◦ Θ = Θ ◦ αTn . Moreover, Θ(a) = τ (a)1, since otherwise in view of Φ(P (a)) − a0 < ε/4 we get τ (a)1 − a0 < ε/4, so that Φ(P (a)) − τ (a)1 < ε/2, which is a contradiction. It follows that the formula λ(x ⊗ y) = ψ(Θ(x)y) deﬁnes a nontrivial stationary coupling of (C ∗ (G, ω), τ, αTn ) with the classical Bernoulli shift (D, ψ|D , α|D ). By Proposition 5.1.2 we get hτ (αTn ) > 0, and hence hτ (αT ) > 0. Note that we could make the proof a bit shorter if we used the fact that ˆ µ, Tˆ) is already a Bernoulli shift. the system (G, Conversely, let hτ (αT ) > 0. Then there exists a nontrivial coupling with an abelian system. Thus there exists an abelian C∗ -algebra B, an automorphism β of B, an (αT ⊗ β)-invariant state λ on C ∗ (G, ω) ⊗ B and a nonzero element g ∈ G such that λ(ug ⊗ · ) = 0. Then there exists a unitary v ∈ B such

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11 Systems of Algebraic Origin

that λ(ug ⊗ v) = 0. Multiplying v by a scalar we may also assume that λ(ug ⊗ (v + v ∗ )) = 0. By Proposition 11.3.2 applied to the set {±2g, ±g, 0} there exists n ∈ N such that if N

ai T in g = 0

i=1

for some N ∈ N and a1 , . . . , aN ∈ {−2, −1, 0, 1, 2}, then ai g = 0 for all i. We claim that there exists a unital completely positive map Φ: C ∗ (G, ω) → C ∗ (G, ω) ⊗ B such that Φ ◦ αTn = (αTn ⊗ β n ) ◦ Φ and ⎧ 1 ⎪ ⎨ ug ⊗ v, if 2g = 0, 2 Φ(ug ) = 1 ⎪ ⎩ ug ⊗ (v + v ∗ ), if 2g = 0. 2 Assuming that the claim is proved we then get an αTn -invariant state ψ = λ◦Φ such that ψ(ug ) = 0, so ψ = τ . Consider ϕ=

n−1 1 ψ ◦ αTk . n k=0

Then ϕ is an αT -invariant state. If ϕ = τ , then ψ ≤ nτ and hence ψ extends to a normal αTn -invariant state on W ∗ (G, ω). Since T n is aperiodic, the automorphism αTn of W ∗ (G, ω) is ergodic by Lemma 11.3.1. Hence ψ = τ , which is a contradiction. Thus ϕ = τ . It remains to construct the map Φ. For k, l ∈ Z such that k < l and ε = (εk , . . . , εl ) ∈ {0, 1}l−k+1 set u(ε) = uεTkkn g . . . uεTlln g and v(ε) = β kn (v εk ) . . . β ln (v εl ). Consider the element Tk,l ∈ C ∗ (G, ω) ⊗ C ∗ (G, ω) ⊗ B deﬁned by Tk,l = 2−(l−k+1)/2 u(ε)∗ ⊗ u(ε) ⊗ v(ε), εk ,...,εl =0,1

and deﬁne a completely positive map Φk,l : C ∗ (G, ω) → C ∗ (G, ω) ⊗ B by ∗ Φk,l (a) = (τ ⊗ id ⊗ id)(Tk,l (a ⊗ 1 ⊗ 1)Tk,l ) ∗ −(l−k+1) τ (au(ε )u(ε) )u(ε)u(ε )∗ ⊗ v(ε)v(ε )∗ . =2 ε,ε

l If a = uh then τ (au(ε )u(ε)∗ ) = 0 unless h = i=k (εi − εi )T in g. In the latter case u(ε)u(ε )∗ = zuh for a scalar z ∈ T, so that u(ε )u(ε)∗ = z¯u∗h and thus τ (uh u(ε )u(ε)∗ )u(ε)u(ε )∗ = uh and v(ε)v(ε )∗ =

l $ i=k

β in (v εi −εi ).

11.4 Zero Entropy Systems

Therefore

⎛ Φk,l (uh ) = 2−(l−k+1) uh ⊗ ⎝

l $ ε,ε

205

⎞ β in (v

εi −εi

)⎠ ,

i=k

l where the summation is over all ε and ε such that h = i=k (εi − εi )T in g. By ourchoice of n, if 2g = 0, a representation of an element h in the l form h = i=k ai T in g, ai ∈ {−1, 0, 1}, is unique if it exists. In this case the numbers εi and εi in {0, 1} such that εi − εi = ai are uniquely determined if ai = 0, and we have exactly two choices εi = εi = 0 and εi = εi = 1 for them if ai = 0. Hence l $ −N in ai Φk,l (uh ) = 2 uh ⊗ β (v ) , i=k

where N is the number of i’s such that ai = 0. If 2g = 0, then the numbers ai are determined only up to a sign, and we similarly get ⎛ ⎞ $ Φk,l (uh ) = 2−N uh ⊗ ⎝ β in (v + v −1 )⎠ . i:ai =0

l If no representation of h in the form i=k ai T in g exists, then Φk,l (uh ) = 0. In particular, we see that Φk,l is a unital map, and for any ﬁxed h ∈ G the image Φk,l (uh ) of uh is the same for all suﬃciently small k ∈ Z and suﬃciently large l ∈ Z. Since also (αTn ⊗ β n ) ◦ Φk,l = Φk+1,l+1 ◦ αTn , letting k → −∞ and l → +∞ we get the required map Φ.

Remark 11.4.3. If the assumptions of Theorem 11.3.3 are satisﬁed, invariant states diﬀerent from the trace can be obtained from product-states on UHFalgebras. Such states can be thought of as noncommutative analogues of Riesz products. Explicitly, let g ∈ G, and assume 2g = 0. Consider the map Φ from Theorem 11.3.3 corresponding to the set X = {0, g}. On B(2 (X))⊗Z consider the product state deﬁned by the density matrix 1/2 ω(0, 0)/2 . ω(0, 0)/2 1/2 Let ψ be the composition of this state with the map Φ. Using (11.10) and that ΦX (ug ) = ω(0, 0)eg0 we then get ∞ 1 $ 1 ρω (T kn g, g) . + ψ(ug ) = 2 2 2 k=1

So if n is suﬃciently large, ψ is an αTn -invariant state diﬀerent from the trace.

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11 Systems of Algebraic Origin

Using the method of the proof of the previous theorem the same state can be obtained as the limit of the states ψk,l (a) = 2−(l−k+1) τ (u(ε)au(ε )∗ ). εk ,...,εl =0,1 ε ,...,ε =0,1 k l

In fact, the map Φ from the previous theorem can be obtained as the composition of the map Φ and a modiﬁcation of the map Ψ from Theorem 11.3.3. Although those maps were constructed under additional assumptions on the cocycle, their composition makes sense in general.

11.5 Automorphisms of Noncommutative Tori Let G = Z2 . Consider the Z-valued skew-symmetric form σ deﬁned by σ(g, h) = g1 h2 − g2 h1 . For θ ∈ R set ωθ (g, h) = eπiθσ(g,h) . The algebra Aθ = C ∗ (Z2 , ωθ ) is called the irrational rotation algebra. Since σ is nondegenerate, by Theorem 11.1.1 the algebra Aθ is simple if θ is irrational. Denote by uθ and vθ the unitaries in C ∗ (Z2 , ωθ ) corresponding to the elements (1, 0) and (0, 1), respectively. Then uθ vθ = e2πiθ vθ uθ . One can alternatively deﬁne Aθ as the universal C∗ -algebra generated by unitaries uθ and vθ satisfying the above relation. Any matrix T ∈ SL2 (Z) preserves the form σ and hence deﬁnes an automorphism of Aθ , which we denote by αT,θ . We also denote by ρθ the bicharacter deﬁned by ωθ , and by τθ the canonical trace on Aθ . Let {λ, λ−1 } be the spectrum of T . Assume |λ| > 1. Then it is known that λ is irrational (otherwise λ, being a root of a monic polynomial with integral coeﬃcients, must be an integer, which is impossible as λ + λ−1 = Tr(T ) is an integer). For any n ∈ Z we have Tn =

1 λn λ−n n −1 −n (λ −T )+λ (T −λ)) = (1−λT )+ (λ (T −λ), λ−1 − λ 1 − λ2 λ−1 − λ

as can be checked on the eigenvectors of T . Hence for any g, h ∈ Z2 there exists C > 0 such that for any n ∈ N n σ(T n g, h) − λ (σ(g, h) − λσ(T g, h)) ≤ C|λ|−n . (11.14) 2 1−λ We need the following classical result of Weyl. Lemma 11.5.1. For any real number λ, |λ| > 1, the sequence {eiθλ }∞ n=1 is uniformly distributed on T for almost all θ ∈ R. n

11.5 Automorphisms of Noncommutative Tori

207

Proof. Fix m ∈ Z, m = 0. For n ∈ N put 1 imθλk . e n n

Vn (θ) =

k=1

We have to prove that Vn (θ) → 0 as n → ∞ for almost all θ ∈ R. Let a, b ∈ R, a < b. We have b b n k j 1 2 |Vn (θ)| dθ = 2 eimθ(λ −λ ) dθ n a a k,j=1

=

k j k j 1 eimb(λ −λ ) − eima(λ −λ ) b−a + 2 . n n im(λk − λj )

k=j

Since |λk − λj | ≥ |λ|k+1 − |λ|k for j > k, k=j

n−1 n−1 n 1 1 1 2n ≤ 2n ≤ . ≤ 2 k j k+1 k k+1 k |λ − λ | |λ| − |λ| |λ| − |λ| (|λ| − 1)2 k=1 j=k+1

k=1

b

Thus a |Vn (θ)|2 dθ ≤ C/n for a constant C depending on λ, m, a and b. In particular, b ∞ |Vk2 (θ)|2 dθ < ∞. a k=1

It follows that Vk2 (θ) → 0 as k → ∞ for almost all θ ∈ [a, b]. If this convergence holds for some θ, then for k 2 ≤ n < (k + 1)2 we have 2 (k+1) (k+1)2 l l 1 eimθλ − eimθλ |Vn (θ)| = n l=1

≤

l=n+1

(k + 1) (k + 1)2 − n |V(k+1)2 (θ)| + . n n

Hence Vn (θ) → 0 as n → ∞.

2

We are now in position to apply the results of the previous sections to the automorphisms of noncommutative tori. Theorem 11.5.2. For any T ∈ SL2 (Z) with Sp T = {λ, λ−1 } and |λ| > 1 we have: (i) 12 log |λ| ≤ ht(αT,θ ) ≤ log |λ| for any θ ∈ R; (ii) if θ ∈ Q then hτθ (αT,θ ) = ht(αT,θ ) = log |λ|; (iii) if θ lies in the additive group generated by 1 and λm (1 − λ2 ), m ∈ Z, then (πτθ (Aθ ) , τθ , αT,θ ) is an entropic K-system; in particular, hτθ (αT,θ ) > 0 for any θ ∈ Q(λ) = Qλ + Q; (iv) hτθ (αT,θ ) = 0 for almost all θ ∈ R, and for any such θ the trace τθ is the unique αT,θ -invariant state on Aθ .

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11 Systems of Algebraic Origin

Proof. Part (i) follows from Theorem 11.2.1 and Example 11.2.5. For any n ∈ N we have an embedding An2 θ → Aθ given by un2 θ → unθ , vn2 θ → vθn . In particular, if θ is rational then we have an embedding of C(T2 ) into Aθ , so that the entropy hτθ (αT,θ ) is at least as large as in the classical case, where it is log |λ| by Example 11.2.5. This proves (ii). If θ = λm (1 − λ2 ), then by virtue of (11.14) for any g and h we have |θσ(T n g, h) − λn+m σ(g, h) + λn+m+1 σ(T g, h)| ≤ C|λ|m−n . Since λn+m = Tr(T n+m ) − λ−n−m and λn+m+1 = Tr(T n+m+1 ) − λ−n−m−1 , and the numbers Tr(T n+m ) and Tr(T n+m+1 ) are integers, we see that there exist integers mn such that |θσ(T n g, h) − mn | ≤ C0 |λ|−n for some new constant C0 . Hence |1 − ρθ (T n g, h)| = |1 − e2πiθσ(T

n

g,h)

| ≤ 2πC0 |λ|−n .

∞ Therefore n=1 |1 − ρθ (T n g, h)| < ∞. The same is true if we assume that θ lies in the group H generated by 1 and λm (1−λ2 ), m ∈ Z. By Corollary 11.3.4 we then conclude that (πτθ (Aθ ) , τθ , αT,θ ) is an entropic K-system. Since λ ∈ Qλ2 + Q and 1, λ2 ∈ H, for any θ ∈ Q(λ) we can ﬁnd n ∈ N such that n2 θ ∈ H. Then, since An2 θ embeds into Aθ , hτθ (αT,θ ) ≥ hτn2 θ (αT,n2 θ ) > 0, which completes the proof of (iii). Remark that in fact by Proposition 5.1.8 we have hτθ (αT,θ ) = hτn2 θ (αT,n2 θ ). To prove the ﬁrst statement in (iv) note that if {an }n and {bn }n are two sequences of real numbers such that bn → 0, then {ei(an +bn ) }n is uniformly distributed on T if and only if {eian }n is uniformly distributed. So by (11.14) and Theorem 11.4.1 it suﬃces to check that the sequence {e2πiθλ

n

(1−λ2 )−1 (σ(g,h)−λσ(T g,h)) ∞ }n=1

is uniformly distributed for all g, h = 0 and almost all θ. But this is true by Lemma 11.5.1, since σ(g, h) − λσ(T g, h) = 0. Indeed, since λ is irrational the latter expression is zero only if σ(g, h) = σ(T g, h) = 0. But then both g and T g are scalar multiples of h, which is impossible as T has no rational eigenvalues. The last statement in (iv) follows from Theorem 11.4.2.

Finally we give an example of systems for which the tensor product formula for the entropy fails. Choose any θ such that hτθ (αT,θ ) = 0. Then the same argument as in the proof of the lower bound in Theorem 11.2.1 shows that also hτ−θ (αT,−θ ) = 0, while hτθ ⊗τ−θ (αT,θ ⊗ αT,−θ ) ≥ log |λ|.

11.6 Notes

209

11.6 Notes Theorem 11.1.1 is a result of Slawny [202]. The idea of using the diagonal subalgebra of C ∗ (G, ω) ⊗ C ∗ (G, ω ¯ ) to estimate the entropy of the tensor product automorphism is due to Narnhofer; it was ﬁrst , employed in [128] for binary shifts, which corresponds to the group G = Z Z/2Z, see Chap. 12. As was observed by Kerr and Li [108] in the context of higher dimensional noncommutative tori, it allows one to estimate not only the topological entropy of αT , but also the topological entropy of αχ ◦ αT ˆ We use a similar argument in the proof of the lower bound in for χ ∈ G. Theorem 11.2.1 to show that ht(αT,ω ) depends only on the cohomology class of ω. The upper bound in Theorem 11.2.1 was obtained by Voiculescu [227] for G = Zn , and by Brown and Germain [37] in general. Theorem 11.2.3 is due to Peters [157]. Theorem 11.2.4 was proved by Bowen [29]. The fact that the Haar measure is the measure of maximal entropy for automorphisms of compact groups was established earlier by Berg [20]. The entropy of automorphisms of the two-dimensional torus was computed already by Sinai [199]. For general compact groups it was done by Juzvinskii [97]. For more information on entropic properties of automorphisms (and, more generally, automorphic actions of Zn ) of compact groups we refer the reader to the book by Schmidt [192]. The examples of cocycles at the end of Sect. 11.2 are due to Golodets and Neshveyev [79]. They were used to construct automorphisms α of the hyperﬁnite II1 -factor with a ﬁxed Cartan subalgebra A such that hτ (α) = t and hτ |A (α|A ) = s for any given t and s, 0 ≤ s ≤ t ≤ +∞. In this respect an interesting open problem is to ﬁnd an automorphism of a II1 -factor with entropy larger than the supremum of the entropies of the restrictions of the automorphism to invariant abelian subalgebras. Existence of such an automorphism was announced in [74], but the details have never appeared. The discussion before Example 11.2.7 shows that ,the example can be interpreted as follows. There exist an action of G = Z Z/2Z on X = {0, 1}Z and an automorphism α of C(X) G such that ht(α) = ht(α|C(X) ) = ht(α|C ∗ (G) ) = log 2. In particular, ht(α) < ht(α|C(X) ) + ht(α|C ∗ (G) ). Such an example was promised in Notes to Chap. 8. It was proved by Benatti, Narnhofer and Sewell [17] that for a ﬁxed T ∈ SL2 (Z) the automorphism αT,θ of Aθ can be asymptotically abelian only for a countable number of θ’s. Later Narnhofer observed that the automorphism has indeed a strong form of asymptotic abelianness for certain irrational θ’s in the quadratic ﬁeld generated by the eigenvalues of T , and suggested that we then get an entropic K-system [127]. The proof of the more general result in Sect. 11.3 is an elaboration on the work of Neshveyev [138]. Note that Lemma 11.3.1 is a classical result of Halmos [84], while the argument in the

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11 Systems of Algebraic Origin

proof of Proposition 11.3.2 essentially goes back to Rohlin [182], who proved that ergodic automorphisms of compact abelian group have the K-property. Later, using powerful Ornstein’s machinery, it was proved that any ergodic automorphism of a compact separable group is Bernoullian (here one should allow the Bernoulli shift with inﬁnite entropy). The result was obtained by Katznelson [101] for tori, and by Lind [118] and Miles and Thomas [122] in general. It was Narnhofer who suggested that a suﬃciently chaotic behavior of the cocycle should lead to zero dynamical entropy. Such results were independently obtained by Narnhofer and Thirring [134] for binary shifts and noncommutative tori and by Sauvageot [189] for noncommutative tori. Our exposition in Sect. 11.4 is based on the unpublished work of Sauvageot [189]. Lemma 11.5.1 is a classical result of Weyl [234]. Finally remark that the results of this chapter are valid with minor changes ˆ Some of these automorfor the automorphisms of the form αχ ◦ αT , χ ∈ G. a b ∈ SL2 (Z) phisms may look even more natural than αT . E.g. if T = c d then with an appropriate choice of χ we get the automorphism of Aθ such that uθ → uaθ vθc , vθ → ubθ vθd .

12 Binary Shifts

A rich source of C∗ -dynamical systems is obtained from bitstreams, i.e. sequences of 0’s and 1’s. Given such a sequence we shall construct a C∗ -algebra, which in the interesting cases is the UHF-algebra of type 2∞ , and an automorphism, called a binary shift. Such a system is an example of systems considered in Chap. 11. Diﬀerent bitstreams can give rise to quite diﬀerent C∗ -dynamical systems, and in this chapter we shall study their entropic properties.

12.1 The C*-algebra of a Bitstream With every bitstream {εn }∞ n=1 , where εn ∈ {0, 1}, we associate a subset X of N by X = {n ∈ N | εn = 1}. Conversely each subset X ⊂ N gives rise to a bitstream in this way. Let , Z/2Z. Denote by gi the element 1 in the i-th summand (Z/2Z)i G = Z of G. Deﬁne a bicharacter ω on G by −1, if j − i ∈ X, ω(gi , gj ) = +1, otherwise, so that

⎞ ⎛ gi , gj ⎠ = ω⎝ i∈I

j∈J

$

ω(gi , gj ).

i∈I,j∈J

As in Chap. 11 consider the algebra A(X) = C ∗ (G, ω), which is the universal C∗ -algebra generated by unitaries ug , g ∈ G, such that ug uh = ω(g, h)ug+h . Recall that the canonical trace τ is deﬁned by τ (ug ) = 0 for g = 0. Let si = ugi . Then si is a symmetry, that is, a self-adjoint unitary. We have −ugi +gj , if j − i ∈ X, si sj = ω(gi , gj )ugi +gj = ugi +gj , otherwise.

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12 Binary Shifts

Thus, if χ denotes the characteristic function of X, we have si sj = (−1)χ(|i−j|) sj si . We could alternatively deﬁne A(X) = C ∗ (G, ω) as the universal C∗ -algebra generated by symmetries sn , n ∈ Z, such that the above relation is satisﬁed. Let G[n,m] = ⊕m i=n (Z/2Z)i ⊂ G. We shall write Gn instead of G[1,n] . Let An = C ∗ (Gn , ω|Gn ) ⊂ C ∗ (G, ω). So An is the C∗ -algebra generated by s1 , . . . , sn . Our goal in this section is to determine the structure of the C∗ algebras A(X) and A+ (X) = ∪∞ n=1 An . From Chap. 11 we know that for this we have to study the bicharacter ω(g, h)ω(h, g), which can be written as (−1)B(g,h) , where B: G × G → Z/2Z is the skew-symmetric bilinear form given by 1, if |j − i| ∈ X, B(gi , gj ) = 0, otherwise. We consider G as a vector space over the ﬁeld F2 = Z/2Z. Then gi , i ∈ Z, form a basis. Let Hn denote the kernel of the restriction of B to Gn , so Hn = {h ∈ Gn | B(h, g) = 0 for every g ∈ Gn }. By Lemma 11.1.2 the center Z(An ) of An coincides with C ∗ (Hn , ω|Hn ). Set cn = dimF2 Hn . Thus Z(An ) is an abelian algebra of dimension 2cn . Note that c1 = 1. Denote by T the shift to the right on G, so T (G[n,m] ) = G[n+1,m+1] . The form B is clearly T -invariant. Lemma 12.1.1. For each n ≥ 1 we have one of the following possibilities: (i) cn+1 = cn + 1 and Hn = Gn ∩ Hn+1 = T −1 (Hn+1 ) ∩ Hn+1 , and if in addition Hn = 0 then Hn+1 = Hn + T (Hn ); (ii) cn+1 = cn − 1 and Hn+1 = Hn ∩ T (Hn ), and if in addition Hn+1 = 0 then Hn = Hn+1 + T −1 (Hn+1 ). Proof. Let us ﬁrst show that if V is a ﬁnite dimensional vector space, W a subspace of V of codimension one, and F a skew-symmetric bilinear form on V , then dim Ker F = dim Ker(F |W ) ± 1. Fix a vector e ∈ V \W . Consider the functional F (·, e) on Ker(F |W ). Assume ﬁrst that this functional is nonzero, so there exists h ∈ Ker(F |W ) such that F (h, e) = 0. Then F (h, g + e) = F (h, e) = 0 for every g ∈ W , which implies that Ker F contains no elements in V \W , so that Ker F ⊂ W . But then Ker F is exactly the kernel of the nonzero functional F (·, e) on Ker(F |W ). So Ker F ⊂ Ker(F |W ) and dim Ker F = dim Ker(F |W ) − 1. Assume now that F (h, e) = 0 for every h ∈ Ker(F |W ). Then Ker(F |W ) ⊂ Ker F . Since F deﬁnes a nondegenerate form on W/Ker(F |W ), there exists f ∈ W such that F (g, e) = F (g, f ) for every g ∈ W . It follows that f − e ∈ Ker F (note that F (e, f − e) = F (e, f ) = F (f, f ) = 0). Thus

12.1 The C*-algebra of a Bitstream

213

the inclusion Ker(F |W ) ⊂ Ker F is proper. In particular, dim Ker F ≥ dim Ker(F |W )+1. On the other hand, we always have W ∩Ker F ⊂ Ker(F |W ), so that dim Ker F − 1 ≤ dim Ker(F |W ). Hence dim Ker F = dim Ker(F |W ) + 1 and W ∩ Ker F = Ker(F |W ). We have thus proved that cn+1 = cn ± 1. Assume cn+1 = cn + 1. It follows from our considerations that this implies Hn = Gn ∩ Hn+1 . Applying this to the inclusion T (Gn ) ⊂ Gn+1 instead of Gn ⊂ Gn+1 we can also conclude that T (Hn ) = T (Gn ) ∩ Hn+1 . Hence Hn ⊂ T −1 (Hn+1 ) ∩ Hn+1 . Since T −1 (Hn+1 ) ∩ Hn+1 ⊂ G[1,n] , the opposite inclusion is obvious. In particular, Hn +T (Hn ) ⊂ Hn+1 . If Hn = 0 then Hn is a proper subspace of Hn + T (Hn ), whence Hn+1 = Hn + T (Hn ). Assume now cn+1 = cn − 1. As we have seen, this implies Hn+1 ⊂ Hn . Applying this to T (Gn ) ⊂ Gn+1 we also conclude Hn+1 ⊂ T (Hn ). Hence Hn+1 ⊂ Hn ∩ T (Hn ). The opposite inclusion is obvious. In particular, Hn+1 + T −1 (Hn+1 ) ⊂ Hn . If Hn+1 = 0 then Hn+1 is a proper subspace of Hn+1 + T −1 (Hn+1 ). Hence Hn = Hn+1 + T −1 (Hn+1 ).

From the previous lemma we get further restrictions on the cn ’s. Lemma 12.1.2. If cn−1 = cn+1 = cn + 1 for some n ≥ 2 then cn = 0. Proof. Since cn = cn−1 − 1, by Lemma 12.1.1(ii) we have Hn = Hn−1 ∩ T (Hn−1 ) ⊂ G[2,n−1] . On the other hand, since cn = cn+1 − 1, by Lemma 12.1.1(i) we have Hn Hn+1 . Since we always have T −1 (Hn+1 ) ∩ Gn ⊂ Hn , we get T −1 (Hn ) ∩ Gn Hn . Since T −1 (Hn ) ⊂ G[1,n−2] ⊂ Gn , we see that T −1 (Hn ) ⊂ Hn . This possible only when Hn = 0.

⊂ ⊂ is

It follows that the sequence {cn }∞ n=1 increases by one to some point, then it decreases by one to zero, and so on. We shall next study the case when the sequence only increases after some n. We say a subset X of N is periodic if the subset −X ∪ X of Z is periodic. In terms of the form B this means that there exists m ≥ 1 such that B(T m g, h) = B(g, h) for g, h ∈ G. Otherwise we say X is nonperiodic. Lemma 12.1.3. The following conditions are equivalent: (i) X is periodic; (ii) there exists n ≥ 0 such that cn+k = k for any k ≥ 0. Here we put for convenience G0 = 0 and c0 = 0.

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Proof of Lemma 12.1.3. If B(T m g, h) = B(g, h) for g, h ∈ G, then g1 + gm+1 is in the kernel of B, so cn ≥ 1 for all n ≥ m + 1. Hence there exists a largest number n ≤ m such that cn = 0, and then cn+k = k for any k ≥ 1 by Lemmas 12.1.1 and 12.1.2. Conversely, assume (ii) is satisﬁed. Then Hn+1 is one-dimensional, and Hn+k is spanned by Hn+1 , T (Hn+1 ), . . . , T k−1 (Hn+1 ) by Lemma 12.1.1(i). In particular, the kernel H∞ of the restriction of B to G∞ = ∪n Gn is spanned by T k (Hn+1 ), k ≥ 0, so that T (H∞ ) ⊂ H∞ . Hence T induces a linear operator T¯ on G∞ /H∞ . Since Hn+k is k-dimensional and has trivial intersection with Gn (otherwise cn = 0), we have Gn+k = Gn ⊕ Hn+k for any k ∈ N. Hence G∞ = Gn ⊕ H∞ . Thus G∞ /H∞ is a ﬁnite set. It follows that there exist k ≥ 0 and m ≥ 1 such that T¯k+m = T¯k . Hence B(T k+m g, h) = B(T k g, h) for any g, h ∈ G∞ . Since B is T -invariant, we conclude that B(T m g, h) = B(g, h) for all g, h ∈ G, so that X is periodic.

Let us now summarize properties of the sequence {An }n . ∞ Theorem 12.1.4. There exist sequences {cn }∞ n=0 and {dn }n=1 of nonnegatiN ve integers and a strictly increasing sequence {nl }l=1 of even integers, where N ∈ N ∪ {∞}, such that ∼ Mat2dn (C) ⊗ C2cn , and the trace τ takes the value 2−cn −dn on every (i) An = minimal projection in An ; (ii) n = cn + 2dn for any n ≥ 1; (iii) n1 = 0, and if ml = (nl+1 − nl )/2 then k, if 0 ≤ k < ml , cnl +k = 2ml − k, if ml ≤ k ≤ 2ml ,

Z(Anl +k ) ⊂ Z(Anl +k+1 ) for 0 ≤ k ≤ ml − 1 and Z(Anl +k ) ⊃ Z(Anl +k+1 ) for ml ≤ k ≤ 2ml − 1. Furthermore, N < ∞ if and only if X is periodic. Proof. Let Gn be a subspace of Gn such that Gn = Gn ⊕ Hn . Since ug uh = uh ug for g ∈ Gn and h ∈ Hn , we have An ∼ = C ∗ (Gn , ω|Gn ) ⊗ C ∗ (Hn , ω|Hn ), and the canonical trace on An is the tensor product of the canonical traces. By Lemma 12.1.1 the numbers n and cn have the same parity, so Gn has even dimension 2dn = n − cn , which also follows from nondegeneracy of the skew-symmetric form B on Gn . By Lemma 11.1.2 the nondegeneracy implies also that C ∗ (Gn , ω|Gn ) is simple, so it is a full matrix algebra of dimension 22dn . Thus C ∗ (Gn , ω|Gn ) ∼ = Mat2dn (C). The algebra Z(An ) = C ∗ (Hn , ω|Hn ) is an abelian C∗ -algebra of dimension 2cn . Moreover, it is a tensor product of cn twisted group C∗ -algebras of the

12.1 The C*-algebra of a Bitstream

215

group Z/2Z. Since the latter algebras have minimal projections of trace 1/2, we conclude that every minimal projection in Z(An ) has trace 2−cn . We have thus proved (i) and (ii). The rest of the theorem follows from Lemmas 12.1.1–12.1.3.

Corollary 12.1.5. Let X ⊂ N. Then Mat2m (C) ⊗ C({0, 1}N ), A+ (X) ∼ = ⊗∞ n=1 Mat2 (C),

if X is periodic, if X in nonperiodic.

The same is true for A(X). Proof. In the notation of Theorem 12.1.4 if X is nonperiodic then N = ∞, so that A2n ∼ = Mat2n (C) for an inﬁnite number of n’s. Hence A+ (X) is the UHF-algebra of type 2∞ . If X is periodic, then there exists n = 2m such that cn+k = k for all k ≥ 0. Then An ∼ = Mat2m (Z) and An+k ∼ = An ⊗ Z(An+k ). Since Z(An+k ) ⊂ Z(An+k+1 ), we get A+ (X) ∼ = An ⊗ (∪k Z(An+k )) ∼ = Mat2m (C) ⊗ C({0, 1}N ). To see that the same is true for A(X), denote by A˜2n the algebra generated by s−n+1 , . . . , sn , and by A˜2n+1 the algebra generated by s−n , . . . , sn . Then ˜ {A˜n }∞ n=1 is an increasing sequence, and A(X) = ∪n An . It is then not diﬃcult to see that all the properties of the sequence {An }n stated in Theorem 12.1.4 remain true for the sequence {A˜n }n . So the same proof as the one for A+ (X) gives the result for A(X).

For the rest of the section we discuss further properties of the sequence {An }n , which are of independent interest, but are not used later. Note ﬁrst that the proof of Lemma 12.1.3 gives a more precise information about the center of A+ (X) for periodic X than that formulated in the above corollary. Namely, let n = 2m be such that cn+k = k for k ≥ 0. Then in the notation of Lemma 12.1.3 the space H∞ is spanned by T k h, k ≥ 0, where h is the unique nonzero element in Hn+1 . Hence the center Z = C ∗ (H∞ , ω|H∞ ) of A+ (X) is isomorphic to C({0, 1}N ) in such a way that the endomorphism α|Z becomes the usual shift endomorphism of C({0, 1}N ). It follows also that the kernel H of B is spanned by T k h, k ∈ Z. Thus the center Z(X) of A(X) is isomorphic to C({0, 1}Z ) in such a way that the automorphism α|Z(X) becomes the shift automorphism of C({0, 1}Z ). Let us call a word in A every nonempty product w = si1 . . . sik with i1 < . . . < ik . Then we can reformulate the above remark as follows. If X is periodic then there exists a unique word w ∈ A+ (X) such that the center Z+ (X) of A+ (X) is generated by αk (w), k ≥ 0, and then the center Z(X) of A(X) is generated by αk (w), k ∈ Z. If w = si1 . . . sik , then i1 = 1 (otherwise α−1 (w) would be in the center), ik = 2m + 1 for some m ≥ 0, and A+ (X) ∼ = Mat2m (C) ⊗ Z+ (X) and A(X) ∼ = Mat2m (C) ⊗ Z(X).

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12 Binary Shifts

The sequences {cn }n and {dn }n in Theorem 12.1.4 are determined by {nl }l . Any sequence {nl }l can arise this way (assuming that it is strictly increasing, consists of even numbers, and n1 = 0). In other words, the restrictions on {cn }n given by Lemmas 12.1.1 and 12.1.2 are the only ones. To see this note that H1 , . . . , Hn depend only on X∩[1, n−1]. Hence we can construct X inductively. So let {cn }n be the sequence deﬁned by {nl }N l=1 according to part (iii) of Theorem 12.1.4, and assume we have constructed Xn ⊂ {1, . . . , n − 1} such that for the corresponding form Bn we have dim Hk = ck for k ≤ n. We have to set either Xn+1 = Xn or Xn+1 = Xn ∪ {n} such that for the corresponding form Bn+1 we get dim Hn+1 = cn+1 . If cn = 0, then cn+1 = 1 and dim Hn+1 = 1 independently of how we deﬁne Xn+1 . Similarly, if cn > 0 and cn = cn−1 − 1, then cn+1 = cn − 1 and dim Hn+1 = cn − 1 independently of the deﬁnition. So we may assume that for some m < n we have ck = k − m for m ≤ k ≤ n. Let h ∈ Hm+1 be the unique nonzero element. By Lemma 12.1.1 the space Hn is spanned by h, T h, . . . , T n−m−1 h. Since Hm = 0, we have h = g1 + h for some h ∈ G[2,m+1] . Since Bn+1 (g, gn+1 ) = Bn+1 (T −1 g, gn ) = Bn (T −1 g, gn ) for g ∈ G[2,n+1] , we have T h, . . . , T n−m−1 h ∈ Hn+1 independently of how we deﬁne Xn+1 . On the other hand, since Bn+1 (h, gn+1 ) = Bn+1 (g1 , gn+1 ) + Bn (T −1 h , gn ), we see that we can arrange so that we have both h ∈ Hn+1 or h ∈ / Hn+1 . In the ﬁrst case Hn ⊂ Hn+1 , so that dim Hn+1 = cn + 1 by Lemma 12.1.1. In the second case Hn is not contained in Hn+1 , so that dim Hn+1 = cn − 1. So whatever cn+1 is, we can deﬁne Xn+1 such that dim Hn+1 = cn+1 . The above argument shows also that if N = ∞ then for inﬁnitely many n’s it does not matter how we deﬁne Xn+1 from Xn . Thus there exist uncountably many sets X giving the same sequence {nl }N l=1 . In the next section we shall see that an interesting class of systems is obtained when X is ﬁnite. We ﬁnish this section by showing that in this case the sequence {cn }n is periodic. Proposition 12.1.6. Let X ⊂ N be ﬁnite and nonempty. Then there exists m ∈ N such that cn+m = cn for any n ≥ 0. n Proof. Let d be the largest number in X. If g = j=1 ξj gj , ξj ∈ F2 , then n g ∈ Hn if and only if B(g, gi ) = 0 for i = 1, . . . , n, that is, j=1 χ(|j−i|)ξj = 0, where χ is the characteristic function of X. Letting ξ−d+1 = . . . = ξ0 = ξn+1 = . . . = ξn+d = 0 we can write this condition as ξi−d +

d−1 k=−d+1

χ(|k|)ξi+k + ξi+d = 0

(12.1)

12.2 Entropy of Binary Shifts

217

for i = 1, . . . , n. Consider (12.1) for all i ∈ Z as a system of equations. If ξ−d+1 , . . . , ξn+d satisfy (12.1) for i = 1, . . . , n then all other ξj ’s are uniquely determined. Thus there is a bijection between Hn and the set of solutions ξ = (ξi )i∈Z of (12.1) such that ξ−d+1 = . . . = ξ0 = ξn+1 = . . . = ξn+d = 0. ¯ = ∞ Z/2Z. Let T¯ We consider sequences ξ = (ξi )i∈Z as elements of G −∞ ¯ so (T¯ξ)i = ξi−1 . Then equations (12.1) can be be the shift to the right on G, written as p(T¯)ξ = 0, where p(t) ∈ F2 [t] is the polynomial deﬁned by p(t) = t2d +

d−1

χ(|k|)td−k + 1.

k=−d+1

¯ To summarize, there is an isomorphism between Hn and the space of ξ ∈ G ¯ such that p(T )ξ = 0 and ξ−d+1 = . . . = ξ0 = ξn+1 = . . . = ξn+d = 0. The ring F2 [t]/(p(t)) is ﬁnite. The element t is invertible in this ring, so it is of ﬁnite order. In other words, there exists m ∈ N such that tm + 1 lies in the ideal generated by p(t). But then if p(T¯)ξ = 0 we have T¯m ξ = ξ, that is, ξi+m = ξi . Hence for solutions of p(T¯)ξ = 0 instead of the condition ξ−d+1 = . . . = ξ0 = ξn+1 = . . . = ξn+d = 0 we can equally well consider the condition ξ−d+1 = . . . = ξ0 = ξn+m+1 = . . . = ξn+m+d = 0. But the latter condition determines Hn+m . Thus we get an isomorphism ben+m tween Hn+m and Hn . Explicitly, the isomorphism maps j=1 ξj gj ∈ Hn+m n to j=1 ξj gj ∈ Hn .

In particular, if X is ﬁnite and nonempty, the sequence {cn }n is bounded. If d is the largest number in X, then using that B(g, gn+d ) = 1 for any g ∈ Gn \Gn−1 , we see that if cn−1 = 0 then Hn , which is nonzero and contained in Gn \Gn−1 , is not contained in Hn+d . It follows that the sequence 1 = cn , cn+1 , . . . , cn+d cannot be increasing. Hence ck ≤ d for any k ∈ N.

12.2 Entropy of Binary Shifts In the notation of the previous section let X be a subset of N associated with a bitstream, and let A(X) be the C∗ -algebra generated by symmetries sn , n ∈ Z, satisfying the commutation relations si sj = (−1)χ(|i−j|) sj si , where χ is the characteristic function of X. Let also τX be the canonical tracial state on A(X). The shift sn → sn+1 deﬁnes an automorphism σX of A(X) which is called the binary shift corresponding to X. In the present section we shall study entropic properties of such automorphisms. We start with a simple estimate. Proposition 12.2.1. We have 21 log 2 ≤ ht(σX ) ≤ log 2. If X is nonperiodic and ϕ is a σX -invariant state on A(X) then hϕ (σX ) ≤ 12 log 2. Proof. The estimates for the topological entropy follow from Theorem 11.2.1. They will also follow from the argument below and Proposition 12.2.4.

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As in the previous section denote by An the C∗ -algebra generated by s1 , . . . , sn . Then by the by now standard argument, see e.g. the proof of Proposition 10.4.5, we have hϕ (σX ) ≤ lim inf n

1 1 log rank An and ht(σX ) ≤ lim sup log rank An . n n n

By Theorem 12.1.4 the rank of An is not greater than 2n , and if X is nonperiodic it is equal to 2n/2 for an inﬁnite number of n’s. This gives the required upper estimates for ht(σX ) and hϕ (σX ).

Our next goal is to show that generically a binary shift has zero entropy and diﬀerent bitstreams give rise to nonconjugate systems. , As in the previous section let G = Z Z/2Z and let ωX = ω denote the bicharacter on G deﬁned by X. Let T denote the shift to the right on G, so that σX is the automorphism of A(X) = C ∗ (G, ωX ) deﬁned by T . We shall ∞ identify a subset X of N with the point (1X (i))i≥1 in Y = i=1 {0, 1}, where 1X is the characteristic function of X. Let µ denote the product measure on Y with weights ( 12 , 12 ). Lemma 12.2.2. Let ρX (g, h) = ωX (g, h)ωX (h, g) for g, h ∈ G. Then the sequence {ρX (T n g, h)}∞ n=1 is uniformly distributed for g, h = 0 for µ-almost every X. Recall that a sequence {an }n such that an = ±1 is said to be uniformly n distributed if n−1 m=1 am → 0 as n → ∞. Proof of Lemma 12.2.2. Let f denote the function on Y deﬁned by f (y) = (−1)y1 , y = (y1 , y2 , . . .) ∈ Y. Let S be the shift to the left on Y , S(y1 , y2 , . . .) = (y2 , y3 , . . .). Then if i > j we have ρX (gi , gj ) = (f ◦ S i−j−1 )(X), where as before gi = 1 in the i-th summand (Z/2Z)i . Let g = gi1 + . . . + gik , i1 < . . . < ik , and h = gj1 + . . . + gjl , j1 < . . . < jl , be given elements in G. Replacing g by T n0 g for some large n0 we may assume is > jt for all s, t. Let F denote the function on Y deﬁned by F (X) = ρX (g, h). Then $ (f ◦ S is −jt −1 )(X). F (X) = 1sk,1tl

It follows that ρX (T n g, h) = (F ◦ S n )(X). By Birkhoﬀ’s ergodic theorem, see e.g. [72, Theorem 3.41], for almost all X we have n 1 (F ◦ S m )(X) → F dµ. n m=1 Y

12.2 Entropy of Binary Shifts

219

The function f takes the value ±1 with probability 12 . Hence Y f dµ = 0, and similarly Y f ◦ S ik −j1 −1 dµ = 0. Now F is a product of f ◦ S ik −j1 −1 and a function depending onlyon the ﬁrst ik − j1 − 1 coordinates. Since µ is a product measure, we have Y F dµ = 0, whence {ρX (T n g, h)}∞ n=1 is uniformly distributed.

If we combine the above lemma with Theorems 11.4.1 and 11.4.2 we obtain the following result. Theorem 12.2.3. For almost all subsets X of N the binary shift σX has entropy hτ (σX ) = 0 with respect to the trace τ = τX . For any such X the trace τX is the unique σX -invariant state on A(X).

Though almost every binary shift has zero entropy, we shall see soon that in many cases the entropy is 12 log 2. Similarly to the discussion at the end of Sect. 11.5, we conclude that binary shifts provide many examples of systems for which the tensor product formula for the entropy fails. More precisely, we have the following result. Proposition 12.2.4. If X is nonperiodic then hτ ⊗τ (σX ⊗ σX ) = log 2. Proof. Similarly to the proof of Proposition 12.2.1 we have hτ ⊗τ (σX ⊗ σX ) ≤ lim inf n

1 log rank An ⊗ An = log 2. n

The argument from the proof of Theorem 11.2.1 for the converse inequality now goes as follows. Let D denote the C∗ -subalgebra of A(X) ⊗ A(X) generated by the symmetries sn ⊗ sn , n ∈ Z. Then D is abelian, and the system (D, (τ ⊗ τ )|D , (σX ⊗ σX )|D ) is isomorphic to the Bernoulli shift with weights ( 12 , 12 ), hence it has entropy log 2. Thus hτ ⊗τ (σX ⊗ σX ) ≥ h(τ ⊗τ )|D (σX ⊗ σX |D ) = log 2.

Denote by M (X) the weak operator closure of A(X) in the GNS-representation corresponding to τ = τX . Denote the extensions of τX and σX to M (X) by the same symbols. Note that if X is nonperiodic then by Corollary 12.1.5 the algebra M (X) is the hyperﬁnite II1 -factor. Remark 12.2.5. In analogy with the classical situation if (M, τ, σ) is a W∗ dynamical system we can say that a σ-invariant von Neumann subalgebra Mπ is a Pinsker algebra if it satisﬁes the following conditions: (i) hτ |Mπ (σ|Mπ ) = 0, (ii) if B is a ﬁnite dimensional C∗ -subalgebra of Mπ such that hτ (B; σ) = 0 then B ⊂ Mπ . In the abelian case such an algebra always exists. In the noncommutative case, however, this is false. Indeed, by Theorem 12.2.3 there exists a set X such that hτ (σX ) = 0. Then Hτ (B; σX ) = 0 for all ﬁnite dimensional C∗ -subalgebras B of M (X). Consider now the W∗ -dynamical system

220

12 Binary Shifts

¯ (X), τ ⊗ τ, σX ⊗ σX ). Then hτ ⊗τ (σX ⊗ σX ) = log 2 by Proposi(M (X)⊗M tion 12.2.4. For each ﬁnite dimensional C∗ -subalgebra B of M (X) we have hτ ⊗τ (B ⊗ C; σX ⊗ σX ) = 0. Hence, if the Pinsker algebra for the system ¯ (X), τ ⊗ τ, σX ⊗ σX ) existed, it would contain M (X) ⊗ C, and by (M (X)⊗M ¯ (X), a contrasymmetry C ⊗ M (X). Thus it would coincide with M (X)⊗M diction. Lemma 12.2.6. Let X1 , X2 ⊂ N. Suppose there exist sequences {nk }∞ k=1 ⊂ X1 and {mk }∞ k=1 ⊂ N such that nk , mk → ∞ and Z ∩ [nk − mk , nk + mk ] ⊂ N\X2 . Then the systems (M (X1 ), τX1 , σX1 ) and (M (X2 ), τX2 , σX2 ) are nonconjugate. Proof. Let s1n and s2n be the symmetries corresponding to X1 and X2 , respectively. Then we have nk 1 nk 1 1 σX (sn )sn = −s1n σX (sn ) 1 1 nk 1 (sn ), s1n ]2 = 2. On the other hand, we have for all n and k, so that [σX 1 nk 2 2 nk 2 (si )sj = s2j σX (si ) σX 2 2

whenever |i − j| ≤ mk , from which it follows that nk (x), y]2 = 0 for any x, y ∈ M (X2 ). lim [σX 2

k→∞

Thus σX1 and σX2 are nonconjugate.

Theorem 12.2.7. For almost all pairs (X1 , X2 ) (with respect to the measure µ × µ on Y × Y ) the W∗ -dynamical systems (M (X1 ), τX1 , σX1 ) and (M (X2 ), τX2 , σX2 ) are nonconjugate. Proof. Choose a strictly increasing sequence {mk }k of natural numbers. For each k ∈ N let fk be the function on Y × Y deﬁned by = 1, yl = 0 for 1 ≤ l ≤ 2mk + 1, 1, if ym k +1 fk (y , y ) = 0, otherwise. As before let S be the shift to the left on Y . Then for n ≥ mk + 1 the conditions n ∈ X1 and Z ∩ [n − mk , n + mk ] ⊂ N\X2 mean exactly that fk (S n−mk −1 X1 , S n−mk −1 X2 ) = 1. For almost every pair (X1 , X2 ) Birkhoﬀ’s ergodic theorem implies n 1 fk (S n X1 , S n X2 ) → fk d(µ × µ) = 2−2mk −2 > 0. n m=1 Y ×Y Thus for almost all pairs (X1 , X2 ) we have fk (S n−mk −1 X1 , S n−mk −1 X2 ) = 1 for inﬁnitely many n’s. In particular, for almost every pair (X1 , X2 ) we can ﬁnd an increasing sequence {nk }k such that the assumptions of Lemma 12.2.6 are satisﬁed. This completes the proof of the theorem.

Having proved generic results, we now consider special cases. We start with the periodic case.

12.2 Entropy of Binary Shifts

221

Proposition 12.2.8. Suppose X ⊂ N is periodic. Then hτ (σX ) = log 2. Proof. Let p be the period of −X ∪ X. Put w = s0 sp . Then w lies in the center of A(X). Since either w or iw is a symmetry, and τ (w) = 0, the restriction n of σX to the C∗ -algebra generated by σX (w) is the Bernoulli shift with weights ( 21 , 12 ). Hence hτ (σX ) ≥ log 2. On the other hand, by Proposition 12.2.1 we have hτ (σX ) ≤ ht(σX ) ≤ log 2. Thus the proof is complete.

We next consider binary shifts corresponding to ﬁnite sets. It turns out that these systems admit several dynamical characterizations. Recall that an automorphism α of a C∗ -algebra A is asymptotically abelian if limn [αn (a), b] = 0 for all a, b ∈ A. It is asymptotically abelian with locality if there is a dense ∗-subalgebra A such that for each pair a, b ∈ A we have [αn (a), b] = 0 for suﬃciently large n, and C ∗ (a, b) is ﬁnite dimensional. Recall also that a W∗ -dynamical system (M, τ, α) is called an entropic Ksystem if lim hτ (γ; αn ) = Hτ (γ) n→∞

for any channel γ: B → M . Theorem 12.2.9. Suppose X ⊂ N is nonperiodic. Then the following four conditions are equivalent: (i) X is ﬁnite; (ii) (A(X), σX ) is asymptotically abelian; (iii) (A(X), σX ) is asymptotically abelian with locality; (iv) (M (X), τX , σX ) is an entropic K-system. -∞ Proof. Suppose X is ﬁnite, say X ⊂ {1, 2, . . . , r}. Let A = n=1 A[−n,n] , where A[−n,n] is generated by s−n , . . . , sn . Then A is dense in A(X). If a, b ∈ A[−n,n] , m let m > 2n + r. Then [σX (a), b] = 0, proving (A(X), σX ) is asymptotically abelian with locality. Thus (i) implies (iii). n Clearly (iii) implies (ii). Assume (ii). Since [s0 , σX (s0 )] = [s0 , sn ] = 2 if n ∈ X, it is immediate from asymptotic abelianness that there exists r ∈ N such that n ∈ / X for n > r. Thus X ⊂ {1, . . . , r}, and (i) follows. We conclude by showing (i)⇔(iv). Assume X ⊂ {1, . . . , r}. Let M0 be the weak operator closure of ∪n A[−n,0] in M (X). Since M (X) is a factor and −n n (M0 ) ∩ σX (M0 ), we conclude that (M (X), τX , σX ) is an A[−n+r+1,n] ⊂ σX entropic K-system by Theorem 4.3.2. Note however that we have the following factorization property, which makes the proof of Theorem 4.3.2 in this case m very simple. If a ∈ A[−n,n] and b ∈ σX (A[−n,n] ) with m ≥ 2n + r + 1, then ab = ba and τ (ab) = τ (a)τ (b). Conversely, assume (M (X), τX , σX ) is an entropic K-system. Let C be a ﬁnite dimensional abelian subalgebra of A(X). Since (2m−1)n

n Hτ (C, σX (C), . . . , σX

n (C)) mHτ (C, σX (C)),

222

12 Binary Shifts

n n n so that 2hτ (C; σX ) ≤ Hτ (C, σX (C)), we conclude Hτ (C, σX (C)) → 2Hτ (C). ∗ n Let C = C (s0 ) = C1 + Cs0 . Then Hτ (C) = log 2, so Hτ (C, σX (C)) ≥ n log 2. If sn and s0 commute then C and σX (C) are independent, so that n Hτ (C, σX (C)) = 2 log 2. On the other hand, if they anticommute, then C and n n σX (C) generate the algebra of 2 × 2 matrices, so that Hτ (C, σX (C)) ≤ log 2. To summarize, log 4, if n ∈ / X, n Hτ (C, σX (C)) = log 2, if n ∈ X. n Since Hτ (C, σX (C)) → 2Hτ (C) = log 4, we conclude that there exists r such n that Hτ (C, σX (C)) = log 4 whenever n > r. Thus X ⊂ {1, . . . , r}.

Note that we could alternatively use Theorem 4.3.5 to show that (iv) implies that the system (M (X), τX , σX ) is strongly asymptotically abelian, and then deduce ﬁniteness of X by the same argument as for the implication (ii)⇒(i) above. Our next result shows that for X ﬁnite, dynamical entropy coincides with both topological entropy and mean entropy. Theorem 12.2.10. Let X be a ﬁnite nonempty subset of N. Then hτ (σX ) = ht(σX ) = lim

n→∞

1 1 Hτ (An ) = log 2. n 2

m km (An ), . . . , σX (An ) are τ Proof. If X ⊂ {1, . . . , r}, then the algebras An , σX independent for m ≥ n + r + 1. By Proposition 10.4.5 we conclude that

hτ (σX ) = lim

n→∞

1 Hτ (An ). n

On the other hand, by Theorem 12.1.4(i) we have Hτ (An ) = log rank An . Since rank An = 2n/2 for inﬁnitely many n’s by Theorem 12.1.4(iii), we get hτ (σX ) = lim

n→∞

1 1 1 Hτ (An ) = lim log rank An = log 2. n→∞ n n 2

Since we always have ht(σX ) ≤ lim sup n→∞

we also get ht(σX ) =

1 2

1 log rank An , n

log 2.

Remark that by Theorem 12.2.10 the algebra A(X) is UHF, hence τX is the unique tracial state on A(X). By Theorem 11.4.2, if X is ﬁnite, there exist invariant states on A(X) diﬀerent from τX . Then by Theorem 9.3.3 we have hϕ (σX ) < 12 log 2 for any such state. Corollary 12.2.11. Suppose Xi ⊂ N, i = 1, 2, are ﬁnite. Then hτX1 ⊗τX2 (σX1 ⊗ σX2 ) = log 2 = hτX1 (σX1 ) + hτX2 (σX2 ).

12.2 Entropy of Binary Shifts

223

Proof. Let Ain ⊂ A(Xi ) be the algebra generated by s1 , . . . , sn ∈ A(Xi ), i = 1, 2. Then hτX1 ⊗τX2 (σX1 ⊗ σX2 ) ≤ lim inf n→∞

= lim

n→∞

1 Hτ ⊗τ (A1n ⊗ A2n ) n X1 X2

1 (HτX1 (A1n ) + HτX2 (A2n )). n

On the other hand, by Theorem 3.2.2(iv), hτX1 ⊗τX2 (σX1 ⊗ σX2 ) ≥ hτX1 (σX1 ) + hτX2 (σX2 ).

Thus the result follows from the previous theorem. We next show that even when X is inﬁnite the entropy can often be

1 2

log 2.

Lemma 12.2.12. Suppose X is nonperiodic and either contained in the even numbers or the odd numbers, then hτ (σX ) = 12 log 2. Proof. Assume ﬁrst X ⊂ {1, 3, 5, . . .}. Let C0 = C ∗ (s0 ). Since s2i s2j = s2j s2i 2n 2 for all i, j ∈ Z, the algebras σX (C0 ), n ∈ Z, are independent. Thus σX acts 1 1 ∗ as the Bernoulli shift with weights ( 2 , 2 ) on the abelian C -algebra C they 2 generate, so that hτ |C (σX |C ) = log 2. Hence hτ (σX ) =

1 1 1 2 2 ) ≥ hτ |C (σX |C ) = log 2, hτ (σX 2 2 2

which together with Proposition 12.2.1 proves the lemma for X contained in the odd numbers. Suppose next X ⊂ {2, 4, 6, . . .}. Then s2i+1 s2j = s2j s2i+1 for all i, j ∈ Z. Letting ti = s2i s2i+1 it follows that ti tj = tj ti for all i, j ∈ Z. Thus the C∗ 2 algebra D = C ∗ (ti | i ∈ Z) is abelian, and σX |D is the Bernoulli shift with 1 1 weights ( 2 , 2 ). Now the argument is completed as in the odd case.

Let us say that a W∗ -dynamical system (M, τ, σ) has completely positive entropy if hτ (γ; σ) > 0 for any nontrivial channel γ: B → M (that is, γ(B) = C1). Clearly, an entropic K-system has completely positive entropy. Although by Theorem 12.2.9 the system (M (X), τX , σX ) is an entropic Ksystem only if X is ﬁnite, we shall next show that it can have completely positive entropy even if X is inﬁnite. Proposition 12.2.13. Suppose q ≥ 3 is an odd integer and X ⊂ {q n | n ≥ 0}. Then (M (X), τX , σX ) has completely positive entropy. m ) ≤ mhτ (γ; σX ) and Hτ (γ) > 0 for nontrivial γ, it Proof. Since hτ (γ; σX suﬃces to prove that 2q lim hτ (γ; σX ) = Hτ (γ). n

n→∞

224

12 Binary Shifts

2kq Note that if x, y ∈ A[−m,m] and m ≤ (q n − 1)/2 then x and σX (y) commute for any k ∈ N. Indeed, otherwise there exist i, j such that |i|, |j| ≤ m and q l = 2kq n + i − j ∈ X for some l. It follows that (2k − 1)q n + 1 ≤ q l ≤ (2k + 1)q n − 1, so that 2k − 1 < q l−n < 2k + 1, which is impossible as q is an odd integer. For any ε > 0 we can ﬁnd a ﬁnite partition of unity 1 = i xi in A[−m,m] for some m such that n

Hτ (γ) < Hτ (γ; {xi }i ) + ε. 2q 2kq (xi1 ), . . . , σX (xik ) are indepenThen for q n ≥ 2m + 1 the elements xi0 , σX dent, so that n

n

2kq 2q 2kq 2q (xik )}) ◦ γ, . . . , σX ◦ γ; {xi0 σX (xi1 ) . . . σX Hτ (γ, σX n n 2q 2kq η(τ (xi0 σX (xi1 ) . . . σX (xik ))) + (k + 1) S(τ (γ(·)xi ), τ ◦ γ) = n

i0 ,...,ik

= (k + 1)

n

n

n

i

(η(τ (xi )) + S(τ (γ(·)xi ), τ ◦ γ)) > (k + 1)(Hτ (γ) − ε),

i 2q ) > Hτ (γ) − ε. Since ε was arbitrary and we always have whence hτ (γ; σX n 2q

hτ (γ; σX ) ≤ Hτ (γ), the proof is complete. n

We thus get examples of systems which have completely positive entropy, but are not entropic K-systems. It turns out that among them there are uncountably many nonisomorphic ones. Proposition 12.2.14. Let q ≥ 2 be an integer and Xi ⊂ {q n | n ≥ 0}, i = 1, 2. Suppose the set X1 ∩ (N\X2 ) is inﬁnite. Then (M (X1 ), τX1 , σX1 ) and (M (X2 ), τX2 , σX2 ) are nonconjugate. / X2 , Proof. Let n1 < n2 < . . . be such that q nk ∈ X1 ∩ (N\X2 ). Since q nk ∈ we have [q nk − q nk −1 + 1, q nk + q nk −1 − 1] ⊂ N\X2 . Thus the result follows from Lemma 12.2.6.

If we combine the above proposition with Lemma 12.2.12 and Proposition 12.2.13 we obtain the following. Theorem 12.2.15. There is an uncountable family of pairwise nonconjugate systems (M (X), τX , σX ) with completely positive entropy such that hτX (σX ) = 1

2 log 2.

12.3 Notes The study of binary shifts was initiated by Powers [171]. One should be aware that in the literature one usually understands by a binary shift the one-sided

12.3 Notes

225

shift. The criterion for simplicity of the C∗ -algebra of a bitstream was obtained by Price [173]. The fact that the dimensions cn of the centers of the algebras An have the form described in Theorem 12.1.4 was proved by Powers and Price [172]. The observation that any sequence {cn }n with properties as in Theorem 12.1.4 arises from a bitstream was made by Price [175]. In our exposition of these results we also beneﬁted from the work of Vik [220]. Proposition 12.1.6 is due to Enomoto, Nagisa, Watatani and Yoshida [64]. Some of the results for binary shifts can be generalized to automorphisms of the shift type on twisted group C∗ -algebras, see e.g. [39], [63], [38]. The entropy of the binary shifts corresponding to ﬁnite sets, Theorem 12.2.10, was computed by Choda [40]. A larger class of binary shifts with entropy 12 log 2 was exhibited by Price [174]. The ﬁrst example of a binary shift with zero entropy was given by Narnhofer, Størmer and Thirring [128]; see [69] for many such examples. It was the ﬁrst example of a system for which the tensor product formula for the entropy fails. The fact that generically a binary shift has zero entropy, Theorem 12.2.3, was stated by Narnhofer and Thirring [134]. As was proved already by Powers [171], one-sided binary shifts corresponding to diﬀerent nonperiodic sets are nonisomorphic. On the other hand, Lemma 12.2.6, proved by Golodets and Størmer [80], seems to be the only known suﬃcient condition for nonisomorphism of two-sided binary shifts. The observation that it implies that generically binary shifts are nonisomorphic, Theorem 12.2.7, is new. The rest of Sect. 12.2 is based on the work of Golodets and Størmer [80]. We have computed entropies of a large class of binary shifts, and the only values we have obtained are 0, 12 log 2, log 2. The following problem thus naturally presents itself. Does there exist X ⊂ N such that hτ (σX ) = 0, 12 log 2, log 2? Let (M (X), τX , σX ) be a binary shift. Consider the von Neumann subaln gebra M0 of M (X) generated by sn , n ≤ 0. Then M0 ⊂ σX (M0 ), ∪n σX (M0 ) n is dense in M (X), ∩n σX (M0 ) = C1. Thus (M (X), τX , σX ) is what we called in Chap. 4 an algebraic K-system. Therefore there exist algebraic K-systems with zero entropy. Theorem 12.2.15 shows that on the hyperﬁnite II1 -factor there exist uncountably many nonisomorphic systems with the same ﬁnite completely positive entropy. On the other hand, there are no examples of nonisomorphic entropic K-systems. In this respect a challenging open problem is whether binary shifts corresponding to diﬀerent ﬁnite subsets are isomorphic. In Chap. 13 we shall construct uncountably many nonisomorphic entropic K-systems on the hyperﬁnite III1 -factor.

13 Bogoliubov Automorphisms

In this chapter we shall consider one of the basic models of quantum statistical mechanics, a system of noninteracting fermions. Our goal is to compute the topological entropy of such a system, as well as the dynamical entropy with respect to a natural class of states.

13.1 Canonical Anticommutation Relations In the present section we collect basic facts about the CAR-algebra, the algebra of the canonical anticommutation relations. We refer the reader to [30, Chapter 5.2] for more details. Let H be a Hilbert space. The CAR-algebra A(H) over H is a unital C∗ algebra generated by elements a(f ) and a∗ (f ), f ∈ H, such that the mapping H f → a∗ (f ) is linear(1) , a(f )∗ = a∗ (f ) and a∗ (f )a(g) + a(g)a∗ (f ) = (f, g)1, a(f )a(g) + a(g)a(f ) = 0 for any f, g ∈ H. If H is one-dimensional, H = Cf , ||f || = 1, then the algebra A(Cf ) is isomorphic to Mat2 (C), with matrix units deﬁned by e11 (f ) = a(f )a∗ (f ), e22 (f ) = a∗ (f )a(f ), e12 (f ) = a(f ), e21 (f ) = a∗ (f ). (13.1) More generally, let H = Cf ⊕ K, f = 1. Then the self-adjoint unitary V (f ) = a(f )a∗ (f ) − a∗ (f )a(f ) anticommutes with a∗ (f ) and commutes with a∗ (g) for g ∈ K. Hence the map a∗ (g) → a∗ (g)V (f ) extends to an embedding of A(K) into A(H) such that the image commutes with A(Cf ) ∼ = Mat2 (C), and we thus get an isomorphism 1

In the literature the opposite notation for the operators a(f ) and a∗ (f ) is sometimes used, but we follow the conventions of [30].

228

13 Bogoliubov Automorphisms

A(Cf ⊕ K) ∼ = Mat2 (C) ⊗ A(K). It follows that if H has dimension n then A(H) ∼ = Mat2 (C)⊗n ,

(13.2)

(13.3)

2n

so A(H) is a full matrix algebra of dimension 2 . Explicitly, if f1 , . . . , fn is an (k) orthonormal basis in H then the matrix units eij in the k-th factor Mat2 (C) are given by (k)

(k)

e21 = a∗ (fk )V (f1 ) . . . V (fk−1 ), e12 = a(fk )V (f1 ) . . . V (fk−1 ), (k)

(k)

e11 = a(fk )a∗ (fk ), e22 = a∗ (fk )a(fk ).

(13.4)

Note that as a∗ (f ) is a partial isometry when f = 1, we have a∗ (f ) = 1, and hence a∗ (g) = g (13.5) for any vector g. Let U be a unitary operator on H. By the universal property of the CARalgebra, it deﬁnes an automorphism αU of A(H), αU (a(f )) = a(U f ), called a Bogoliubov automorphism. Let now A be an operator on H, 0 ≤ A ≤ 1. It deﬁnes a state ωA on A(H) by the formula ωA (a∗ (f1 ) . . . a∗ (fn )a(gm ) . . . a(g1 )) = δnm det((Afi , gj ))i,j , called a quasi-free state. If H = Cf , f = 1, and A = λ1, then the quasi-free state ωA on A(Cf ) ∼ = Mat2 (C) has density matrix 1−λ 0 . 0 λ More generally, if H = Cf ⊕ K, f = 1, and Af = λf , then under the isomorphism (13.2) we have 1−λ 0 ωA = Tr · ⊗ ωA|K . 0 λ It follows that if H has dimension n and f1 , . . . , fn is an orthonormal basis consisting of eigenvectors of A, Afk = λk fk , then under the isomorphism (13.3) we get n . 1 − λk 0 ωA = . (13.6) Tr · 0 λk k=1

This gives a way to prove existence of quasi-free states: if A has pure point spectrum prove that the above product state is indeed the required quasifree state, then approximate an arbitrary quasi-free state by quasi-free states corresponding to operators with pure point spectrum. Finally note that if two operators A and U commute then the state ωA is αU -invariant.

13.2 Topological Entropy

229

13.2 Topological Entropy Let U be a unitary operator on a separable Hilbert space H. In this section we compute the topological entropy of the Bogoliubov automorphism deﬁned by U . To formulate the result recall, see C.7, that there exists a unique decomposition H = Hs ⊕ Ha such that Us = U |Hs has singular spectral measure and Ua = U |Ha has absolutely continuous spectral measure with respect to the Lebesgue measure on T. Furthermore, we have a direct integral decomposition ⊕ ⊕ Ha = Hz dµ(z), Ua = z dµ(z), T

T

where µ is the normalized Lebesgue measure on T. Denote by mU the multiplicity function of Ua , mU (z) = dim Hz . Theorem 13.2.1. With the above notation, for the Bogoliubov automorphism αU of the CAR-algebra A(H) we have ht(αU ) = (log 2) mU (z)dµ(z). T

In particular, ht(αU ) = 0 if U has singular spectral measure. Consider the simplest case when the unitary U is the bilateral shift, so it has homogeneous Lebesgue spectrum of multiplicity one. Then the theorem asserts that ht(αU ) = log 2. To prove this let {fn }n∈Z be an orthonormal basis such that U fn = fn+1 . Then according to (13.3) and (13.4) the unital C∗ -subalgebra generated by a∗ (fn )a(fn ), n ∈ Z, is isomorphic to C({0, 1}Z ), and the restriction of αU to this subalgebra is just the shift automorphism with entropy log 2. Hence ht(αU ) ≥ log 2. On the other hand, if Ω is a ﬁnite subset of A(Hk,l ), where Hk,l is the m space spanned by fn , n = k, . . . , l, then αU (Ω) ⊂ A(Hk,l+m ), so that for any δ > 0 and n ∈ N n−1 rcp(Ω ∪ αU (Ω) ∪ . . . ∪ αU (Ω), δ) ≤ rank A(Hk,l+n−1 ) = 2l−k+n .

Hence ht(Ω, δ; αU ) ≤ log 2. So indeed ht(αU ) = log 2. Consider now the case of singular spectral measure. Then ht(αU ) = 0. To show this we need a couple of technical lemmas. Lemma 13.2.2. Let U be a unitary operator with singular spectral measure, P a ﬁnite rank projection. Then for any ε > 0 there exists n0 ∈ N such that for n ≥ n0 there is a projection Pn such that rank Pn ≤ εn and (1 − Pn )U k P < ε for k = 0, . . . , n.

230

13 Bogoliubov Automorphisms

Proof. Note that if P = Q1 + Q2 and Qin is such that rank Qin ≤ εn/2 and (1 − Qin )U k Qi < ε/2, i = 1, 2, then the projection Pn = Q1n ∨ Q2n satisﬁes the conditions of the lemma. Thus it suﬃces to prove the lemma for rank one projections. Replacing H by a subspace we may further assume that P is the projection onto the space spanned by a cyclic vector for U . Then we can identify the Hilbert space with L2 (T, ν), where ν is a singular probability measure, such that P becomes the projection onto the constant functions and U becomes the operator of multiplication by the function z. √ Choose N ∈ N such that 2/N < ε. Since ν is singular, there exist m ∈ N and disjoint arcs I1 , . . . , Im on T of length not greater than 1/N 2 m each such that ε2 ν(T\(I1 ∪ . . . ∪ Im )) < . 2 Consider now any n of the form n = N ml, l ∈ N. Subdivide each arc Is into l arcs Ist , 1 ≤ t ≤ l, of length not greater than 1/N 2 ml each. Let Pn be the projection onto the subspace spanned by the characteristic functions of these arcs. Then rank Pn ≤ ml = n/N . On each arc Ist choose a point zst . Then for w ∈ Ist and k = 0, . . . , n we have k |≤ |wk − zst

n N 2 ml

=

1 , N

so that 2 ε2 1 k k k )1Ist 22 < zst 1Ist = ν (T\ (∪s,t Ist )) + (z k − zst + 2 < ε2 . z − 2 N s,t s,t 2

Hence z k − Pn z k 2 < ε, that is, (1 − Pn )U k P < ε. Now for any n ∈ N set Pn = P([n/N m]+1)N m . Then (1 − Pn )U k P < ε for k = 0, . . . , n, and rank Pn ≤

/ n 0 1 m , +1 m≤n + Nm N n

so that rank Pn < εn if n is suﬃciently large.

Lemma 13.2.3. For any m ∈ N and δ > 0 there exists ε > 0 such that if P and Q are projections in B(H) such that rank P = m and (1 − Q)P < ε, then for any x in the unit ball of A(P H) ⊂ A(H) there exists y in the unit ball of A(QH) ⊂ A(H) such that x − y < δ. Proof. This follows from norm-continuity of the map g → a∗ (g). In fact, by (13.5) we know that

13.2 Topological Entropy

231

a∗ (g) − a∗ (Qg) = (1 − Q)g for any g ∈ H.

Let U be a unitary operator with singular spectral measure. Let Ω be a ﬁnite subset of the unit ball of A(P H) for a projection P of rank m. Then k αU (Ω) is contained in the unit ball of A(U k P U −k H). For δ > 0 choose ε < δ in accordance with Lemma 13.2.3. Choose a projection Pn according to k Lemma 13.2.2. Then αU (Ω) ⊂δ A(Pn H) for k = 0, . . . , n. Consequently n−1 rcp(Ω ∪ αU (Ω) ∪ . . . ∪ αU (Ω), δ) ≤ rank A(Pn H) = 2rank Pn ≤ 2εn ≤ 2δn ,

whence ht(Ω, δ; αU ) ≤ δ log 2. Thus ht(αU ) = 0. It turns out that the general result follows essentially from the above cases, thanks to the following characterization of the Lebesgue integral. Proposition 13.2.4. Let h be a function deﬁned on the set of all unitary operators on a separable Hilbert space such that: (i) h(U ) depends only on the conjugacy class of U ; (ii) if Ua ∼ = W N , where W is the bilateral shift and N ∈ N, then h(U ) = N ; (iii) if P is a projection commuting with U then h(U |P H ) ≤ h(U ); (iv) if {Pn }n is a sequence of projections commuting with U such that Pn → 1 strongly, then h(U |Pn H ) → h(U ); (v) h(U n ) = nh(U ) for any n ∈ N. Then h(U ) = mU (z)dµ(z).

T

Proof. Denote T mU (z)dµ(z) by µ(U ). Let us also write U % V if U is conjugate to the restriction of V to an invariant subspace. In view of (iv) it suﬃces to consider unitary operators with bounded multiplicity function mU . Moreover, using (iv) and regularity of the measure we may assume that the sets m−1 U ({k}), k ≥ 0, are closed. Then we can ﬁnd a sequence {U (n)}n of unitary operators such that Us ∼ = U (n)s , mU ≤ mU (n) , µ(U (n)) → µ(U ), and mU (n) is the sum of the characteristic functions of pn arcs of length 2π/qn each, where qn ∈ N. Observe next that if I is a measurable subset of T such that the map I z → z k is injective with image J, then the operator of multiplication by the function z → z k on L2 (I, dµ) is conjugate to the operator of multiplication by the function z → z on L2 (J, dµ). It follows that U (n)qan ∼ = W pn , where W is the bilateral shift, that is, the operator of multiplication by the function z on L2 (T, dµ). More generally, the above observation implies that for any unitary operator V we have mV k (z) = mV (w). (13.7) w: wk =z

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13 Bogoliubov Automorphisms

∼ W pn . Let V (n) be a unitary operator such that We have Uaqn % U (n)qan = pn ∼ qn pn V (n) = U . Since V (n)a % W pn , by (13.7) we have mV (n) ≤ 1. Therefore V (n)pn ∼ = U qn % U (n)qn , V (n)a % W and U (n)qan ∼ = W pn . Applying µ we get pn pn µ(V (n)) = µ(U ) ≤ µ(U (n)) = . qn qn Since µ(U (n)) → µ(U ), by the above inequality we get pn /qn → µ(U ) and µ(V (n)) → 1 (unless µ(U ) = 0, in which case we have h(U ) = 0, as h(U ) ≤ h(U (n)) = pn /qn ). Since mV (n) ≤ 1, it follows that mV (n) → 1 in measure. Then by (iv) and (ii) we get h(V (n)a ) → 1. Since V (n)a % V (n) % W ⊕V (n)s , it follows that h(V (n)) → 1. Since h(U ) =

pn h(V (n)), qn

we conclude that pn /qn → h(U ). Thus h(U ) = µ(U ).

Proof of Theorem 13.2.1. We have to check that the function which takes the value (log 2)−1 ht(αU ) at U satisﬁes the conditions of Proposition 13.2.4. Properties (i), (iii) and (v) are immediate consequences of general properties of topological entropy. Property (iv) follows from the fact that any ﬁnite subset in A(H) can be approximated by a ﬁnite subset in A(Pn H) for all suﬃciently large n’s. It remains to prove (ii). So assume that U is such that there exists an orthonormal basis {fi,n | i = 1, . . . , N, n ∈ Z} in Ha such that U fi,n = fi,n+1 . Then, similarly to the argument after the formulation of Theorem 13.2.1, the C∗ -algebra generated by a∗ (fi,n )a(fi,n ), i = 1, . . . , N , n ∈ Z, is isomorphic to C({0, 1}Z )⊗N , and the restriction of αU to this subalgebra is the tensor product of the shift automorphisms, so that ht(αU ) ≥ N log 2. Denote by Pk,l the projection onto the space spanned by fi,n , i = 1, . . . , N , n = k, . . . , l. Let P be a ﬁnite rank projection in B(Hs ), Ω a ﬁnite subset of the unit ball of A(P Hs ⊕ Pk,l Ha ), and δ > 0. Choose ε < δ in accordance with Lemma 13.2.3 applied to m = rank P + rank Pk,l . For n ∈ N ﬁnd a projection j Pn ∈ B(Hs ) according to Lemma 13.2.2. Then αU (Ω) is contained in the unit j j −j ball of A(Us P Us Hs ⊕Pk+j,l+j Ha ), so that αU (Ω) ⊂δ A(Pn Hs ⊕Pk,l+n−1 Ha ) for j = 0, . . . , n − 1. Consequently n−1 rcp(Ω ∪ αU (Ω) ∪ . . . ∪ αU (Ω), δ) ≤ 2rank Pn +rank Pk,l+n−1 ≤ 2δn+N (l−k+n) ,

whence ht(Ω, δ; αU ) ≤ (δ + N ) log 2. Thus ht(αU ) ≤ N log 2, and property (ii) is proved. This completes the proof of the theorem.

The following example of an asymptotically abelian system with nontracial states of maximal entropy was promised in Chap. 9. Let us ﬁrst introduce one more notion. The even CAR-algebra is the ﬁxed point subalgebra A(H)e of the CAR-algebra A(H) for the Bogoliubov automorphism α−1

13.3 Classical Bernoullian Subsystems

233

corresponding to the operator −1. It is generated by operators of the form a# (f )a# (g), where a# (f ) is any of the operators a∗ (f ) and a(f ). Example 13.2.5. It easy to see that the restriction α of the Bogoliubov automorphism αU to the even CAR-algebra is asymptotically abelian if and only if (U n f, g) → 0 as n → ∞ for any f, g ∈ H. If in addition U has singular spectrum then by Theorem 13.2.1 we have ht(α) = 0, while there are many nontracial α-invariant states (for example, quasi-free states corresponding to scalars λ ∈ (0, 1/2)). Unitaries with such properties can be obtained using Riesz products. We shall brieﬂy recall the construction. Let q > 3 be a real number, {nk }∞ k=1 a sequence of positive integers such ∞ that nk+1 ≥ qnk , {a } a sequence of real numbers such that ak ∈ (−1, 1), k k=1 ak → 0 as k → ∞, k a2k = ∞. Then the sequence of measures n 1 $ (1 + ak cos nk t) dt 2π k=1

on [0, 2π] converges weakly∗ to a probability measure µ with Fourier coeﬃcients ⎧ ∞ $ ak |εk | ⎪ ⎨ , if n = εk nk with εk ∈ {−1, 0, 1}, int µ ˆ(n) = µ(e ) = k=1 2 k ⎪ ⎩ 0, otherwise. The measure µ is singular by [238, Theorem V.7.6]. We see also that µ ˆ(n) → 0 as |n| → ∞. Thus the operator U of multiplication by eit on L2 ([0, 2π], dµ) has the desired properties.

13.3 Classical Bernoullian Subsystems In this section we begin the computation of the dynamical entropy of Bogoliubov automorphisms with respect to quasi-free states. Consider ﬁrst the case when U is the bilateral shift, so there exists an orthonormal basis {fn }n∈Z such that U fn = fn+1 , and A = λ1, 0 ≤ λ ≤ 1, is a scalar operator. We shall write ωλ instead of ωλ1 for the quasi-free state corresponding to λ1. Then by (13.3)-(13.6) the C∗ -algebra generated by a∗ (fn )a(fn ), n ∈ Z, is isomorphic to C({0, 1}Z ), and by restricting ωλ and αU to this subalgebra we obtain the classical Bernoulli system with weights (1−λ, λ) and entropy η (λ)+η (1 − λ). Since this subalgebra lies in the centralizer of the state, we get hωλ (αU ) ≥ η (λ) + η (1 − λ) . In fact, it is not diﬃcult to see that equality holds, which allows us to conclude along the lines of the proof of Theorem 13.2.1 that

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13 Bogoliubov Automorphisms

hωλ (αU ) = (η (λ) + η (1 − λ))

T

mU (z)dµ(z)

for any unitary U . Since we shall in the next section prove a more general formula without relying on this particular case, we are not going into details. Our goal in this section is rather to prove an analogue of the above inequality when A is close to a scalar, which is the main technical result we shall need later. Proposition 13.3.1. For given ε > 0 and C, 0 < C < 1, there exists δ > 0 such that if Sp A ⊂ (λ0 −δ, λ0 +δ) for some λ0 ∈ (0, C), and the spectrum of U has a homogeneous Lebesgue component, so that there exists a unit vector f such that {U n f }n∈Z is an orthonormal system, then hωA (αU ) ≥ η (λ0 ) + η (1 − λ0 ) − ε. The same inequality is true for the restrictions of ωA and αU to the even CAR-algebra. Consider the matrix units eij (f ) deﬁned by (13.1). Let P be the algebra spanned by p1 = e11 (f ) = a(f )a∗ (f ) and p2 = 1 − p1 . For n ∈ N we want to estimate n−1 n−1 HωA (P, αU (P), . . . , αU (P); {pi0 αU (pi1 ) . . . αU (pin−1 )}i0 ,...,in−1 =1,2 ).

For this we need ﬁrst of all to estimate the action of the modular group of ωA on p1 and p2 . Assuming that Ker A = Ker(1 − A) = 0 set B=

A . 1−A

Then ωA is a KMS−1 -state for the one-parameter group {σt = αB it }t of Bogoliubov automorphisms. If A has pure point spectrum this follows from (13.3)-(13.6), since the product-state in (13.6) is a KMS−1 -state for the oneparameter group n . (1 − λk )it 0 , Ad 0 λit k k=1

which is exactly the above group of Bogoliubov automorphisms. In the general case we can approximate A by operators with pure point spectrum and argue by continuity. Another possibility is to use the explicit description of the GNSrepresentation, see [30, Example 5.2.20]. It follows that the extension of σt to πωA (A(H)) is the modular group of the state. Lemma 13.3.2. If Sp A ⊂ (0, 1) and 1/2 1/2 A λ f− f < δ 1−A 1−λ

13.3 Classical Bernoullian Subsystems

235

for some δ > 0 and f , ||f || = 1, where λ = (Af, f ), then √ 1/2 1/2 λj σ−i/2 (ekj (f )) − λk ekj (f )ωA ≤ 2(λj λk )1/4 δ 1/2 for k, j = 1, 2, where λ1 = 1 − λ, λ2 = λ. Proof. We shall write ekj for ekj (f ). By deﬁnition we have ωA (e11 ) = λ1 , ωA (e22 ) = λ2 , ωA (e12 ) = ωA (e21 ) = 0. By the KMS-condition, for any operators x and y, ωA (σ−i/2 (x)∗ σ−i/2 (y)) = ωA (σi/2 (x∗ )σ−i/2 (y)) = ωA (σ−i/2 (y)σ−i/2 (x∗ )) = ωA (yx∗ ). Using also that ωA (σ−i/2 (x)∗ y) = ωA (σi/2 (x∗ )y) = ωA (x∗ σ−i/2 (y)), we get 1/2

1/2

||λj σ−i/2 (ekj ) − λk ekj ||2ωA = 2(λj λk )1/2 ((λj λk )1/2 − ωA (ejk σ−i/2 (ekj ))). So we must prove that ωA (ejk σ−i/2 (ekj )) is close to (λj λk )1/2 to within δ. Since λ1 − ωA (e11 σ−i/2 (e11 )) = ωA (e11 σ−i/2 (e22 )) = λ2 − ωA (e22 σ−i/2 (e22 )) and ωA (e12 σ−i/2 (e21 )) = ωA (e21 σ−i/2 (e12 )) by the KMS-condition, we must show that ωA (e11 σ−i/2 (e22 )) is close to zero and ωA (e21 σ−i/2 (e12 )) is close to λ1/2 (1 − λ)1/2 to within δ. λ A and β = . Then Let B = 1−A 1−λ σ−i/2 (e21 ) = σ−i/2 (a∗ (f )) = a∗ (B 1/2 f ), so

σ−i/2 (e21 ) − β 1/2 e21 = B 1/2 f − β 1/2 f < δ.

Hence, since e11 e21 = 0, |ωA (e11 σ−i/2 (e22 ))| = |ωA (e11 (σ−i/2 (e21 ) − β 1/2 e21 )σ−i/2 (e12 ))| < δ and since ωA (e11 ) = 1 − λ, |ωA (e12 σ−i/2 (e21 )) − λ1/2 (1 − λ)1/2 | = |ωA (e12 (σ−i/2 (e21 ) − β 1/2 e21 ))| < δ, which proves the lemma.

The previous lemma will allow us to estimate the correction term of the entropy. It will also allow us to estimate the classical term thanks to the following approximate factorization result.

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13 Bogoliubov Automorphisms

Lemma 13.3.3.Let {eij }ni,j=1 be a system of matrix units in a von Neumann algebra M with k ekk = 1, and let ω be a faithful normal state on M . Then for any x ∈ M commuting with the matrix units we have √ 1/2 ω 1/2 |ω(ekk x) − λk ω(x)| ≤ 2 ||λj σ−i/2 (ekj ) − λk ekj ||ω ||x||# ω, j ∗ ∗ 1/2 and λk = ω(ekk ). where ||x||# ω = (ω(x x) + ω(xx ))

Proof. We may assume that ω is the vector state deﬁned by a cyclic and separating vector ξ. Let J be the corresponding modular conjugation. Then ω(ekk x) = (ekk xξ, ξ) = (ejk xξ, ejk ξ) = (Jejk ξ, Jejk xξ). Using that x commutes with ejk , and JxJ commutes with ekj , we also have (ekj ξ, Jejk xξ) = (ekj , JxJJejk ξ) = (ekj Jx∗ ξ, Jejk ξ) and

(ekj Jx∗ ξ, ekj ξ) = (ekj ξ, ekj Jxξ) = (ejj ξ, Jxξ) = (xξ, Jejj ξ).

It follows that 1/2

1/2

1/2

λj ω(ekk x) = λj ((λj Jejk − λk ekj )ξ, Jejk xξ) 1/2

1/2

1/2

+λk (ekj Jx∗ ξ, (λj Jejk − λk ekj )ξ) + λk (xξ, Jejj ξ). ω Since σ−i/2 (y)ξ = Jy ∗ ξ for y ∈ M , we then get

|λj ω(ekk x) − λk (xξ, Jejj ξ)| 1/2

1/2

1/2

1/2

ω (ekj ) − λk ekj ||ω ≤ (λj ||x||ω + λk ||x∗ ||ω )||λj σ−i/2 √ 1/2 ω 1/2 ≤ 2||x||# ω ||λj σ−i/2 (ekj ) − λk ekj ||ω .

Summing up the above inequalities over all j’s we obtain the desired estimate.

Proof of Proposition 13.3.1. First choose δ1 > 0 such that |η (λ) + η (1 − λ) − η(λ0 ) − η(1 − λ0 )|

−n . 3 j=1

(13.9)

Turning to the classical term, by Lemma 4.3.4 there exists ε1 > 0 such that if |ωA (pi x) − ωA (pi )ωA (x)| ≤ ε1 ||x||, i = 1, 2, (13.10) for any x commuting with p1 and p2 , then 2

η(ωA (pi xj )) ≥

i=1 j∈J

2

η(ωA (pi )) +

i=1

η(ωA (xj )) −

j∈J

ε 3

for any partition of unity 1 = j∈J xj commuting with p1 and p2 . By Lemmas 13.3.2 and 13.3.3 applied to {ekj (f )}2k,j=1 , there exists δ3 ∈ (0, δ2 ) such that if Sp A ⊂ (λ0 −δ3 , λ0 +δ3 ), λ0 ∈ (0, C), then (13.10) is satisﬁed. Applying the above inequality recursively we then get 2

n−1 η(ωA (pi0 αU (pi1 ) . . . αU (pin−1 )))

i0 ,...,in−1 =1

≥n

2 i=1

η(ωA (pi )) − (n − 1)

ε ε = n(η (λ) + η (1 − λ)) − (n − 1) , 3 3

(13.11)

where λ = (Af, f ) ∈ (λ0 − δ3 , λ0 + δ3 ). It follows from (13.8), (13.9) and (13.11) that if we take δ = δ3 then for any n ∈ N n−1 n−1 HωA (P, αU (P), . . . , αU (P); {pi0 αU (pi1 ) . . . αU (pin−1 )})

238

13 Bogoliubov Automorphisms

> n(η (λ0 ) + η (1 − λ0 )) − (3n − 1)

ε 3

as long as Sp A ⊂ (λ0 − δ, λ0 + δ) for some λ0 ∈ (0, C). Hence hωA (αU ) ≥ hωA (P; αU ) ≥ η (λ0 ) + η (1 − λ0 ) − ε. Since P is a subalgebra of the even CAR-algebra, the same inequalities hold for the restrictions of αU and ωA to A(H)e .

13.4 Dynamical Entropy In this section we shall prove a formula for the entropy of a Bogoliubov automorphism of the CAR-algebra with respect to a quasi-free state. To formulate the result consider a unitary U on a separable Hilbert space H and an operator A, 0 ≤ A ≤ 1, commuting with U . Let Ua = U |Ha be the part of U with absolutely continuous spectral measure. Then, see App. C, there exists a direct integral decomposition ⊕ ⊕ ⊕ Ha = Hz dµ(z), Ua = z dµ(z), A|Ha = Az dµ(z), T

T

T

where µ is the normalized Lebesgue measure on the torus T. Theorem 13.4.1. With the above notation, for the Bogoliubov automorphism αU of the CAR-algebra A(H) and the quasi-free state ωA deﬁned by A we have hωA (αU ) = Tr(η (Az ) + η (1 − Az ))dµ(z). T

A particular case, where one can get an explicit direct integral decomposition, is given in the following corollary. Corollary 13.4.2. Let I be an open subset of R, ω a real locally absolutely continuous function on I, ρ a measurable function on I, 0 ≤ ρ ≤ 1. Let U and A be the operators on L2 (I, dx) of multiplication by the functions eiω and ρ, respectively. Then 1 hωA (αU ) = (η (ρ(x)) + η (1 − ρ(x)))|ω (x)|dx. 2π I Proof. Consider the sets I1 = {x ∈ I | ω (x) = 0} and I2 = {x ∈ I | ω (x) = 0}. For any compact subset X ⊂ I1 the set ω(X) has measure zero. Hence the spectrum of the restriction of U to L2 (X, dx) has measure zero. It follows that

13.4 Dynamical Entropy

239

the restriction of U to L2 (I1 , dx) has singular spectral measure and thus does not contribute to the entropy. On the other hand, if x ∈ I2 then ω(y) = ω(x) for any y close to x, y = x, so that for every z ∈ T the set Y (z) = {x ∈ I2 | eiω(x) = z} is at most countable. For any f ∈ L1 (I2 , |ω (x)|dx) we have f (x)|ω (x)|dx = 2π f (x)dµ(z), T x∈Y (z)

I2

see e.g. [67, Theorem 3.2.3] for the case when ω is Lipschitzian, and use [67, Theorem 3.1.8] and [67, Theorem 2.10.43] to extend the result to the general case. It follows that if for a function f and z ∈ T we put fz = (2π)1/2 |ω |−1/2 f |Y (z) , then the map f → (fz )z deﬁnes an isometry between the spaces L2 (I2 , dx) ⊕ and T 2 (Y (z))dµ(z). Since I\(I1 ∪ I2 ) has measure zero, we thus obtain a direct integral decomposition of Ua and A|Ha such that Az is the operator of multiplication by ρ|Y (z) . Hence Tr(η (Az ) + η (1 − Az ))dµ(z) = (η (ρ(x)) + η (1 − ρ(x)))dµ(z) T

T x∈Y (z)

1 = 2π

(η (ρ(x)) + η (1 − ρ(x)))|ω (x)|dx,

I2

which proves the corollary.

Turning to the proof of Theorem 13.4.1 we shall ﬁrst establish the lower bound for the entropy. The main technical step in this direction has been done in the previous section, and our goal now is to extend that result to all unitaries with absolutely continuous spectrum. This extension is based on the following two results, which provide tools for changing the spectrum of a unitary operator without changing the entropy. Let us ﬁrst make a few remarks about the even CAR-algebra. If K and L are mutually orthogonal subspaces of H then A(K) and A(L)e commute, and the C∗ -algebra they generate decomposes into the tensor product A(K) ⊗ A(L)e . Indeed, it suﬃces to consider the case when K is ﬁnite dimensional, and then this is true as A(K) is a full matrix algebra. If in addition K and L are invariant subspaces for an operator A, 0 ≤ A ≤ 1, then ωA |A(K)⊗A(L)e = ωA |A(K) ⊗ ωA |A(L)e . If moreover H = K ⊕ L, the subalgebra A(K) ⊗ A(L)e of A(H) is the ﬁxed point algebra for the involutive Bogoliubov automorphism α1⊕−1 preserving ωA . In particular, there exists an ωA -preserving conditional expectation

240

13 Bogoliubov Automorphisms

onto A(K) ⊗ A(L)e , and by monotonicity and superadditivity of entropy, Theorem 3.2.2(v),(iv), we get hωA (αU ) ≥ hωA |A(K)⊗ωA |A(L)e (αU |A(K) ⊗ αU |A(L)e ) ≥ hωA |A(K) (αU |A(K) ) + hωA |A(L)e (αU |A(L)e ).

(13.12)

Lemma 13.4.3. Let Un be a unitary operator on a Hilbert space Hn , n ∈ N, and {zn }∞ on the space n=1 ⊂ T. Consider two unitary operators U and U ∞ H = ⊕n=1 Hn , ∞

∞

n=1

n=1

U = ⊕ Un and U = ⊕ zn Un . Then hω (αU ) = hω (αU ) for any αU - and αU -invariant state ω on A(H). The same holds for the restrictions of the automorphisms to the even CARalgebra A(H)e . Proof. Consider the unitary operator V = ⊕∞ n=1 zn 1. For each n ∈ N choose an increasing sequence {Hnk }∞ of ﬁnite-dimensional subspaces of Hn such k=1 that ∪k Hnk is dense in Hn . Set Kn = H1n ⊕ . . . ⊕ Hnn . Then Kn is ﬁnite-dimensional, Kn ⊂ Kn+1 , and ∪n Kn is dense in H. Since k V Kn = Kn and αU = αV ◦ αU = αU ◦ αV , we have αU (A(Kn )) = k αU (A(Kn ) for any k ∈ Z, whence hω (A(Kn ); αU ) = hω (A(Kn ); αU ). As {A(Kn )}n is an increasing sequence with dense union in A(H), it follows that hω (αU ) = hω (αU ). Similarly, using the algebras A(Kn )e instead of A(Kn ), we get the result for the even CAR-algebra.

Lemma 13.4.4. Let X1 , X2 be measurable subsets of T such that µ(X1 ) = µ(X2 ) > 0. Then there exist a countable measurable partition X1 = ∞ n=1 Yn ∞ of X1 and a sequence of numbers {zn }∞ n=1 ⊂ T such that X2 = n=1 zn Yn modulo a set of measure zero. Proof. It suﬃces to prove that there exists a subset Y ⊂ X1 of positive measure and z ∈ T such that zY ⊂ X2 . Then an application of Zorn’s lemma ﬁnishes the proof. Denoting by 1X1 and 1X2 the characteristic functions of the sets X1 and X2 we have 1X1 (w)1X2 (zw)dµ(w)dµ(z) = µ(X1 )µ(X2 ) > 0. T×T

Hence there exists z such that 1X1 (w)1X2 (zw)dµ(w) > 0, T

that is, zX1 ∩ X2 has positive measure. Now we can extend Proposition 13.3.1 to arbitrary unitaries.

13.4 Dynamical Entropy

241

Proposition 13.4.5. For ε > 0 and C, 0 < C < 1, choose δ > 0 as in Proposition 13.3.1. Let U be a unitary with absolutely continuous spectral measure and a direct integral decomposition ⊕ ⊕ H= Hz dµ(z), U = z dµ(z). T

T

Set X = {z ∈ T | Hz = 0}. Let A, 0 ≤ A ≤ 1 be an operator commuting with U such that Sp A ⊂ (λ0 − δ, λ0 + δ) for some λ0 ∈ (0, C). Then hωA (αU ) ≥ µ(X)(η (λ0 ) + η (1 − λ0 ) − ε). The same inequality is true for the entropy of the restriction of αU to A(H)e . −1 Proof. Let {nk }∞ k=1 ⊂ N be a sequence such that µ(X) = k nk . By Lemma 13.4.4 there exist a measurable partition X = ∞ k,m=1 Xkm and a set {zkm }∞ k,m=1 ⊂ T such that for all k ∈ N 2 3 1 1 = exp 2πi 0, zkm Xkm nk m ⊕ modulo sets of measure zero. Let Hkm = Xkm Hz dµ(z) be the spectral subspace of U corresponding to the set Xkm . Set Hk = ⊕m Hkm , and deﬁne a unitary operator Uk on Hk by Uk = ⊕ zkm U |Hkm . m

By construction the multiplicity function of Uk is positive on exp (2πi[0, n−1 k ]), so that Uknk has Lebesgue spectrum. By Lemma 13.4.3 and Proposition 13.3.1 we get 1 hω |A(Hk )e (αU nk |A(Hk )e ) k nk A 1 (η (λ0 ) + η (1 − λ0 ) − ε). ≥ nk

hωA |A(Hk )e (αU |A(Hk )e ) =

By (13.12) for any n ∈ N we get hωA (αU ) ≥ ≥

n

hωA |A(Hk )e (αU |A(Hk )e ) k=1 n k=1

1 nk

(η (λ0 ) + η (1 − λ0 ) − ε).

Letting n → ∞ we obtain the required estimate. The same proof works for the restriction of the automorphism to the even part of the algebra.

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13 Bogoliubov Automorphisms

We are now ready to proof a half of Theorem 13.4.1. Proof of the lower bound for the entropy in Theorem 13.4.1. Since hωA (αU ) ≥ hωA |A(Ha ) (αU |A(Ha ) ) by (13.12), to prove the result it suﬃces to consider unitaries with absolutely continuous spectral measure. Assuming that U is such a unitary, let us ﬁrst show that if hωA (αU ) < ∞ then Az has pure point spectrum for almost all z ∈ T. Fix δ0 ∈ (0, 1/2) and take ε ∈ (0, η(δ0 )). Let δ be as in the formulation of Proposition 13.4.5 with C = 1 − δ0 . For any Borel subset X of R, let 1X (A) be the spectral projection of A corresponding to X. Then ⊕ 1X (A) = 1X (Az )dµ(z). T

Deﬁne a measurable function ϕX on T, 1, if 1X (Az ) = 0, ϕX (z) = 0, otherwise. By Proposition 13.4.5 if X is a Borel subset of (λ0 − δ, λ0 + δ) for some λ0 ∈ (δ0 , 1 − δ0 ) then hωA (αU |A(1X (A)H)e ) ≥ (η (λ0 ) + η (1 − λ0 ) − ε) ϕX (z)dµ(z) T

≥ η(1 − δ0 )

T

ϕX (z)dµ(z),

(13.13)

where we have used that η (λ0 ) + η (1 − λ0 ) ≥ η (δ0 ) + η (1 − δ0 ) in the second inequality. Let t0 = δ0 < t1 < . . . < tm = 1 − δ0 , tk − tk−1 < δ. Then, similarly to the proof of Proposition 13.4.5, using (13.12) we obtain from (13.13) the inequality m hωA (αU ) ≥ η(1 − δ0 ) ϕ(tk−1 ,tk ] (z)dµ(z). T k=1

If we let max(tk − tk−1 ) → 0 then the integrand in the expression above converges to inﬁnity at every point z ∈ T such that (δ0 , 1 − δ0 ) ∩ Sp Az is inﬁnite. Therefore if hωA (αU ) < ∞, then the intersection (δ0 , 1 − δ0 ) ∩ Sp Az is ﬁnite for almost all z ∈ T. Since δ0 is arbitrary, it follows that Az has pure point spectrum for almost all z ∈ T. Thus to get the lower bound for the entropy we may assume that Az has pure point spectrum for almost all z ∈ T, since otherwise both the entropy and the integral in the formulation of the theorem are inﬁnite. But then, see C.6, we have N H = ⊕ L2 (Xn , dµ), n=1

where Xn is a measurable subset of T, N ∈ N ∪ {∞}, and U and A act on L2 (Xn , dµ) as multiplications by functions z and λn (z), respectively. We must prove that

13.4 Dynamical Entropy

hωA (αU ) ≥

N n=1

243

(η (λn (z)) + η (1 − λn (z)))dµ(z).

Xn

Using (13.12) once again, we see that it suﬃces to estimate hωA (αU |A(H)e ) assuming N = 1. As in the proof above, ﬁxing δ0 > 0, ε > 0 and choosing t0 = δ0 < t1 < . . . < tm = 1 − δ0 , we get that hωA (αU |A(H)e ) ≥

m k=1

{z∈X1 | tk−1

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