PRESSURE FOR AUTOMORPHISMS OF EXACT C*-ALGEBRAS AND A NONCOMMUTATIVE

VARUTIONAL PRINCIPLE

David Kerr

A thesis submitted in conformity with the requirements

for the degree of Doctor of Phiiosophy

Graduate Department of Mathematics

University of Toronto

@ Copyrzght by David Kerr 2001

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Pressure for Automorphisms of Exact C-algebras and a Noncommutat ive Variat ional Principle

Ph.D. thesis 2001

David Kerr

Gracluate Department of Mathematics, University of Toronto

Abstract

A notion of pressure with respect to a self-adjoint element is introduced for automor-

phisms of exact C*-algebras and a number of properties are established, including a gen-

eralization of a theorem of Pi. Brown for entropy asserting that the pressure remains the

sarne upon passing to the extension of an automorphism to the crossed product. A vari-

ationai inequality bounding the pressure below by the CNT and Sauvageot-Thouvenot

free energies is obtaïned in two stages via a locd state approximation entropy, which

is showvn to be an extension of M. Chodak nuclear entropy. We prove the variationai

principle for certain asymptoticaiiy Abelian automorphisms and introduce the class of

weakly AF C'-aigebras in order to descnbe a specid subcIass of these autornorphisms.

Contents

Introduction

1. Completely postive maps and nuclear and exact Cs-algebras

2. Noncommutative approximation pressure

Unitai e m c t C-algebras

Not-necessarily-unital exact Cs-aigebras

Unitai nuclear C*-algebras

3. Pressure and crossed products

4. Local state approximation entropy and variational inequalities

5. The variational principle

Zero entropy automorphisms FA automorphisms

6. Weakly AI' C*-algebras and traciaiiy FA automorphisms

7. Further examples of FA automorphisms

References

Introduction

Since its introduction fifty years ago in the work of Kolmogorov and Sinai, the notion

of entropy has p1ayed an important roIe in ergodic theory, as remarkably witnessed

in Ornstein's classification of Bernoulli shifts [Orn]. Entropy may be described as a

measure of the loss of information content of a system as it evolves over tirne or, ai-

ternatively. as a measure of the growth (in the structurai t e m of the systern type)

prodticed by the dynamics. In 1975 Connes and Stormer, motivated by quantum sta-

tistical mechanics and ergodic theory, initiated the theory of dynamical entropy for

operator algebras [CS]. This theory, among whose successes must be mentioned the

work of Connes, Narnhofer, and Thirring [CNT], has in the last [ive years b e n pushed

into a new phase of rapid growth with the seminai article [Vo3] of Voiculescu (see [Stol

[or a sumey). In this thesis we contribute to this development by establishing a theory

of pressure for automorphisms of exact C*-algebras and a variationai principle for a

special class of such automorphisms.

The classica1 variationai principle assefis that, given a compact metric space ,Y. a homeomorphism T : .Y - X, and a continuous function f : X - B. the pres-

sure P(T. f ) is equd to the supremum over ail T-invariant merisures p on X of che

free energies h,[T) + p( f ): where h, (T) is the Kolmogorov-Sinai entropy- Adapting

the definition of Bowen [Bow] for topological entropy, Rueiie introduced the notion

of topologicd pressure as a dynamical abstraction of the statisticd mechanicd notion

of pressure defined for an infinite lattice system as the logarithmic partition function

density under a thermodynamic limit. Rueiie proved the variationd principle under

certain restrictions on T in RU^], and then Walters established the result in NI gener-

di@ in FVal] motivated by the work of Dinaburg [Did, Goodman [Gm], and Goodwyn

[GwI][Gw2] that had completed the case f = O, when pressure reduces to topological

entropy. Rueiie's definition of pressure functions by sampling over the dynamicd iimits

of aii Enite subsytems, and thus does not refer to specific b i t e subsystems as in the

thermodynamic notion of entropy or pressure density for a Iattice system. Thus in

the statistical mechanical Framework the thermodynamic limit is reconcept uaIized as

a d ~ a m i c a l lirnit with space translations gnerating the sequence of subsystems. and the variational principle for trandation-invariant lat tice systems is subsumed into the

results of Ruelle and Walters.

In quantum statistical mechanics, as in classical statistical mechanics, pressure for a

Iattice system is defined as a density which is caiculated by taking the thermodynamic

Iimit over iïnite subsystems of averages of the logarithm of the partition function. We

Niil introduce a notion of pressure for automorphisms of exact Cs-algebras which, in

analogy to the classical situation above, captures the shift-invariant Iattice system of

quantum thermod~arnics as a special instance. The theory of C*-aigebras offers a

naturd mathematical extension of the topologica1 framework of classicd physics to the

quantum domain. In C'-aigebraic terms we may describe this extension as a passage

from the commutative to the noncommutative, and hence describe C8-aigebraic general-

izations of toplogical notions by tagging the letter with the qualifier "noncommutative"

(indeed the study of C"-aigebras is sometimes referred to as koncommutative topol-

ogf). The topological notion of dynamitai entropy has recently been subject to such a

generalization nrith Voiculescu's definition of entropy for automorphisms of nuclear C'-

dgebras [Vo31 and its recent extension by Brown to automorphisms of exact C*-algebras

[Brl]. The classes of nuclear and exact C'dgebras each admit a characterization in

terms of local approximation by finite-dimensionai C*-aigebras, and it is this idea of

Iocai approximation that forms the basis of Voiculescu's noncommutative topologicd

entropy? which uses the size of the appro-xhating finite-dimensionai C'-algebras as a

measure of the gowth produced by an automorphism. Neshveyev and St~rmer d e h e d

pressure in the nudear situation using Voiculescu's approximation approach PSj. Like- wise adopting Voiculescuts approacht our definition of noncommutative pressure in the

emct situation generalizes both the Neshveyev-Stormer pressure and the Voicuiescu- Brown entropy. As in the classicd commutative setting, for noncommutative dynamics

pressure wiii be defineci for a self-adjoint element of the aigebra. For quantum Iattice systems this element represents the energy contribution associated to one Iattice site, with the dynamics of space translation providing a mechanism to count out and thus sample over ail lattice sites in order to recapture the pressure density as a dynamicd

limit. After providing some background in Chapter 1 on completely positive maps and

exact C*-algebras, we d e h e pressure and establish several properties in Chapter 2. We

single out in Chapter 3 the result mhich, generalizing a theorem of Brown for entropy,

asserts that the pressure remains the same when paçsllig to the inner automorphism

implementing the ori,$nai action on the corresponding crossed product C-algebra.

in [NSI Neshveyev and Stormer established a variational principle for a class of

asymptoticdly Abelian autornorphisms of unital AF C*-aigebras. Our ultimate task,

carried out in Chapter 5, d l be to establish a "continuoils" generalization of this

result, which operates in a strong sense in the ~er~dimensional: ' domain of approxi-

mate finite-diniensionality. In formulating a noncommutative variational principle one immediately faces the problem that, compared to the ciassicai situation, C'-algebraic

ciynamics presents a much less definitive state of affairs for a theory of mensure-theoretic

entropy. Following Nestiveyev and Stormer, we will work with the entropy of Connes,

Narnhofer, and Thirring (henceforth abbreviated CNT) [CNT]. CNT's definition is

rnotivated by the physical idea of observing the systern via Abelian models. and thus

may be expected to yield a useful measure of dynamitai complexity for asymptotically

Abelian automorphisms. Indeed this apectation is borne out in part by Neshveyev

and Starmer's variatioiial principle and its extension in this thesis to automorphisms

(terrned "FA") which exhibit a certain type of asymptotic Abelianness that includes

the shift-invariant lattice systems of quantum statisticai mechanics as a paradigrnatic

example. FA automorphisms admit a simpler description in restricted circumstances in

terms of w e d y AF Cs-algebras. In Chapter 6 we introduce this class of C*-aigebras,

develop some of their properties? and use them to define the subclass of FA automor-

phisms called "tracially FA." Chapter 7 is devoted to a discussion of further examples

of FA automorphisms.

We separate out an initiai step toward the variationai principle in Chapter 4, es-

tablishing the variational inequality that asserts that the free energy in a given state

is bounded above by the pressure. This is the easier and evidently more clear-cut re-

lation benveen the pressure and free energies, and me show tbat it holds in complete

generality nith respect to the CNT entropy and Sauvageot-Thouvenot entropy [ST]

in two stages via the introduction of a third entropy whose associateci free energies

Function as an intermediate term. The definition of this "local state approximation"

entropy adopts the approximation framework of Voicuiescu's topologicai entropy and thus conceptuaiiy clarifies the variationai inequality relating pressure and the CNT

and Sauvageot-Thouvegeot free energies. Various properties of the local state a p

proxirnation entropy are estabiished in Chapter 4, in particuiar its reduction to the

KoIornogorov-Sinai entropy in the commutative case and its majorïzation of the CNT and Sauvageot-Thouvenot entropies.

Completely positive maps and nuclear and exact C*-algebras

CVe review in this section some fundamental aspects of the theories of completely pos-

itive maps and nuclear and exact C*-a1gebra.s. CornpieteIy positive maps play an

important role in the stntcttire theory for operacor algebras, notably in the formulation of local conditions which characterize the classes of nuclear and exact Ca-dgebras, as

nrill be described below. FVe Niii define a notion of dynmical pressure in Chapter

2 using (as does Brown for entropy in PrlI) the local characterization of aactness

(Tlieorem 1.5). so that compIetely positive maps will form the technical cornerstone

of the ensuing theory. The theory of C-algebraic exactness is expounded in detail in

FVü$2]. while [Pau] provides a comprehensive source for materid relating to completely

positive maps.

Given Cm-algebras A and B, a linear map 4 : A -+ B is said to be positive if

b(.4+) c B+: tvhere il+ and B+ denote the positive cones of A and B, respectively.

To define the notion of a completely positive map, we consider C'-tensor products

with the matrk algebras Mn(@). We note that the finite-dimensionality of lLI,(@) guarantees that the algebraic tensor product of a C*-algebra LI with &(@) is complete

in any crossnom: so that by the uniqueness of the C'-nom we obtain a unique Ca-

algebra tensor product D @ Mn (for a reference for Ca tensor products see [Tak]). We say that O : A + B is completely positive (sometimes abbreviated to c.p.)

if O 8 idn : A @ ?CI, - B 8 Mn is positive for aii n E PI. It is readily seen that

compositions and poinhvise limits of completely positive maps are agaîn completely

positive. Important examples of compIetely positive maps are *-homomorphisms and

projections of n o m 1 ont0 C*-subalgebras.

We will say that a linear map 6 : -4 -+ B is a contraction if 11qj(x)11 5 IIx(I for aiI

x E B and an isometry if Il$(x)c)ll = Ils[[ for ail x E B. If 4 : A -, B is a compIeteIy

positive isometry such that 4-' : $(A) + A is completely positive, then we Say that

4 is a complete order (c.o.) embedding. A complete order embedding which is surjective is termed a complete order (c.o.) isomorphism.

If A and B are unital C-algebras, then a h e a r map 4 is said to be unital if

&Il ) = 1. We Nil1 mainly be n r o r b g Mth unital completely positive (u.c.p.) rnaps and

unital complete order (u.c.0.) embeddings.

The significance of completely positive maps stems in large part from the charac-

terizacion dorded bÿ Stinespring's theorem, which is a dilation result asserting that

completely positive maps into B(3.C) are compressions of *-homomorphisrns.

Theorem 1.1. [Sti] If A is a ,unital C'-algebm and 1u : A - 'B(3C) is a unital

completely positive map, then there &ts a Hilbert space X containing 3C and a '- representation T : A - B ( X ) such that &(x) = qir(x)q for al1 z E At where q is the

pmjection of X ont0 X .

As an illustration of the consequences of Stinespring's theorem we record the foUowing

corollary, which says that a Ca-algebra is determined by its complete order structure.

Corollary 1.2. (see [BK, Cor. 4.1.41) A cornplete order isomorphism from one Cm- aigebm ta another is u "-isomorphism.

Xnother fundamental result concerning completety positive maps is the following

extension theorem of Arveson. to which we will frequently appeal.

Theorern 1.3. (Arveson's extension theorem) [Arv] If A is a unital Ce-algebm and

O is a contractive completely positive map /rom a self-adjoint unital subspace of A into

either B(X) or a finite-dimensional C*-algebm, then 4 admits a contmctiue completely

positive extension ta .A.

In the 1970's it was redized that the notion of nuclearity was fundamentai to the structure and classification theory of C-dgebras, especidy in view of Connes's work

(see [Con]) on the classification of amenable von Neumann aigebras, inside the class

of which appears the second dual of a Ca-aigebra il precisely when A is nuclear. El-

liott's IC-theoretic ~Iassication of approximately finite-dimensional (AE') C*-algebras

[Ell] initiated a program to classify the larger class of separable nuclear C*-algebras in

terms of K-theoretic and traciai data (for a m e n t report on the status of this project

see [E12)). We Say that a C*-aigebra is nuclear if it satisfies any one of the equivalent

conditions in the foUowing proposition.

Theorem 1.4. For a C*-algebm A the follo~uing are equaualent.

(i} Giuen any finite subset R of A and 15 > O them exists a finite-dimensional CL- algebm B and completely positive contmctions Q, : A - B and + : B + A such

that I)(îlr O o)(x) -211 < d jor eue7 x E Q, ie., the identity map on A Zs the

point-nom limit of completely positive contractions thmugh finite-dimensional

C* -algebras.

(ii) The algebmic tensor pmduct oiff A with any other C*-algebm admits a unique

C* -nom.

(iii) The second dual A'* is on amenable von Neumann algebm.

(iu) -4 zs amenable as a C*-algebra? Le.: every hounded derivation lmm A into a dual Banach A-module is inner.

E?smples of nuclear C*-algebras are type 1 C*-algebras (of which commutative and

finite-dimensional C*-algebras are specid instances)? AF aigebras (which in the sepa-

rable setting c m be characterized by strengthening condition (i) above so that the map

zb is required to be a '-homornorphisrn), and the Cuntz algebras O,. Nuclearity is pre-

served under taking inductive limits, quotients, tensor products, and crossed products

by amenable groups. See [La] for a survey of nuclearity and for references for the proofs

involveci in establishing the equivdences of Theorem 1.4.

The local condition (i) in Theorem 1.4 was used by Voiculescu in Po31 to define

a notion of dynamical entropy for nucIear Ca-algebras based on completely positive

approximation, which nriil be the basis of our approach to a theory of noncommutative

pressure. We wil l generaliy be working, however, within the broader class of exact

C*-algebras.

Definition 1.5 CVe say that a C-algebm A is exact ififfor any C8-algebm B and any

ideal J of B the sequence

s exact.

The class of exact C*-aigebras is strictly larger than the clacs of nuclear C'-&bras,

and, unlike nuclearity, the property of exactness passes to C*-subalgebras. An example

of a nonnuc1ea.r Exact C*-dgebra is the reduced gtoup C8-algebra of the fiee goup on two generators (see [WasS, Chap. 31). Exactness turns out to be equivalent to the

local approximation property termed nuclear embeddability, which is a weakening of

condition (i) in Theorem 1.4.

Definition 1.6 .4 C*-algebra A is said to be nuclearly embeddable if there e ~ s t s a

C*-algebm D and an injective '-homomorphism L : A - D such that for any finite

subset R of A and E > O the= exists a finite-dimensional C*-algebm B and completely

positive contractions rj : -4 - B and ~ : 3 -. D such that Il($ O +)(x) - L(x)[~ < E for

every x E i! (such a map L is said to be nuclear).

We note that for unital C*-algbras it suffices to consider unital compIetely positive

maps in the definition of nucleu embeddability.

Theorem 1.7. -4 C*-algebm is nuclearly embeddable if and only if it is ezact.

The fomard implication in Theorem 1.7 was proved in PVasl], while the much more

difficult converse wns established in Kir]. LVe note that for unital C*-algebras nuclear

embeddability can be defined using unitd completely positive maps. Using nuclear em-

beddability Brown carried over Voiculescu's approximation approach from the nuclear

setting CO define a notion of dynamicd entropy for exact C*-dgebras [Brl]. We will

develop our theory of dynamical pressure for exact C*-aigebras in the following sections

by cxtending Brown's definition of entropy.

For Our discussion of the variational principle we wiii find it useful to define the

notion of a 5veakly A F C-algebra by means of a finite-dimensionai approximation

property which is variation on those which characterize Ai?, nudear, and e..act C*- aigebras. These aigebras wili be introduced and some of their properties established in

Chapter 5.

Noncornmut at ive approximation pressure

Our first project wiii be to introduce a, notion of pressure for automorphisms of evact

C'-algebras which FoIIows the approximation approach of Voiculescu and extends both

the Brown-Voiculescu emct entropy Pr11 and the Neshveyev-Stormer nuclear pressure

[NS]. We Nill treat the unital case first: and then move to a generai definition which ivill be shonm CO be ZUI extension of the unital one (Proposition 2.17).

For any subspace X of a C-algebra we denote by P f (,Y) the collection of fmitc subsets

of X. Lf A is a unital Cs-algebra, L : .A - D a unitd completely positive map, 9 f PJ(A) . and 6 > O. nTe denote by CPA(L,Q. b) the set of all triples B) where

B is a Finite-dimensional C*-dgebra and O : A - B and @ : B -, D are u.c.p. maps

such that I [ ( ~ J O d ) ( x ) - ~ ( x ) l l < S. Let -4 be a unital exact C'-algebra and û : A - A a u.c.p. map. Let D be a unital

injective Ca-algebra and L : .il - D a unitai complete order (henceforth abbreviated

u.c.0.) embedding. Since A is nudearly embeddable by Theorem 1.7 there is a unital

Ca-algebra E and an injective unital nucIear (see Definition 1.6) *-homomorphism

-, : A - E. and by the injectivity of D we cm extend L O I-L : $A} -* D to a u.c.p.

map T : E - D so that 'ï O -, is nuclear, from which it iç easily seen that L is nuclear

(see the proof of Proposition 2.2 below). Thus CPA(L, R, 6) is non-empty.

CVhenever the map 6 in question is char from the context we s h d abbreviate

Upzi di (R) to R:-' and Bi(h) to hg-'. For a finite-dimensional C-dgebra B ive

denote by Tr3 the trace which assumes the d u e I on each minimal projection.

Definition 2-1. If h E A,,7 Q E P f (A) , 6 > 0, and n E M we define

Proposition 2.2. I ~ L ' : A - Dl and L? : A -, D2 are U.C.O. embeddings into unita[

injective C*-algebm Di and D2, Q E Pf(-4), and 6 > O! then

Proof. It clearly suffices to establish the first inequality.

Since DL is injective we can extend the map L ~ O L ; ' : 4.4) -+ Dl to a U.C.P. map ï : Dr! - Dr- Let Q E P f (A) and S > O. Given n E N, if (&+, B) E CPA(Q, 8, Cl:-', 6) then every x E Cl,"-' satisfis

so that (9, ï O L!,. B ) E C P A ( L ~ , d , il,"-'. S). Thus

The reverse inequality foiIows by symmetry. Cl

Definition 2.3. In view of the above proposition and the fact that A always admits a

u.c.o, embedding into a unital injective Ca-algebm (consider, for instance, any faithfitl

unital representation), we can define the (unital exact approximation) pressure Pe(h) of B at h to be P o ( ~ , h) for any u.c.0. embeddz'ng L : A -+ D into a unital injective

Ca-algebra D. We may ais0 dmp L from Po(&, h, R, 6) and Po(i, h, R).

We NiU continue where necessq to use L to denote an arbitrary u.c.0. embedding into

a unital injective Ce-dgebra D, usuaiiy without mention. Occasiondy we will work &th faithful unitai representations, with which certain operations can be handled Nith

greater ease. We fl &O aiways assume that A is evact and 8 is a u.c.p. map unies stated ûtherwise.

Remark 2.4. If T : A + 23(w is a faithFul unita1 representation then Po(T, O, a, d) CO-

incides with the completely positive 6-rank rcp(r, R, 6) of Voicdescu-Brown [VoSIIBrlj,

and so Pe(0) is equai to the Voiculescu-Brown entropy ht(8). Note that, instead of

considering only faithful representations as does Brown in defuiing entropy p r l , Defn.

1-11: we have chosen to work with complete order embeddings into injective C'-algebras.

since Ive can absorb into the clehition arguments involving completely positive maps which would otherwïse have to be repeated in proofs which do not require the specific

structure of B(K) . Compare for instance Proposition 2.10 in [Brl] and Propostion

2.7 belom. As demonscrated by Proposition 2.2, it is the injectivity of B ( X ) which is

essential in Brown's definition.

In addition to the coincidence of Po(h) with (a suitable notion of) entropy when

h = 0. many other analogues of properties of topological pressure PVd] hoid in our

C'-exact approximation setting (see [NS: Prop. 2.41 for some of the correspondhg

properties in the nuclear case).

Proposition 2.5. Let 0 : A - A be a u.c.p. map and h,g E A,.

(il If g s h then Pdg) 5 Pa(h).

(ii) Po(h + cl) = Po(h) + c /or al1 c E R.

(iii) ht(8) + min spec(h) 5 Pe(h) 5 ht(8) + mrtuspec(h), whem min spec(h) and

m a spec(h) are the minimum and maximum elements, respectzvely, of the spec-

trum of h.

(iv) Eîther Po(h) = co for al1 h o r Pe(h) < CU for al1 h.

(v) if PB (-) is finite then 1 Po(h) - Pe(g) 1 5 Il h - gll . (vi) Pa (ch) 5 cPa (h) if c 2 1 and Ps(ch) > cPe(h) if c 5 1.

( v -4 IPa(h)l I Po(lhl).

(viii) Pe(h) = iP0t ( di(h)) for d l k E N.

(k) Pe(h + 0(g) - 9) = Pe(h)-

Proof. (i) Suppose g 5 h. If (@, $J, B) E CPA(L, Q,"-', G) then, since < Q(h,"-l)? by the PeierkBogoiiubav inequaiity [OP, Cor. 3-15] we have

from wbich (i) foiiows.

(ii) If (+? @, B) E CPA(L, R,"-', S) then

from which ive conclude (ii).

(iii) We infer from (i) and (ii) that

Po (h) 5 P e ( ( m a spec(h)) 1) = Po (O) + ma.. spec(h) = ht(9) + mxc spec(h)?

with the inequaiity Po(h) 2 ht(9) + min spec(h) foiiowing similarly.

(iv) From (iii) we see that Pe(h) = ca if and only if ht(9) = W.

(v) For any (4. Q, B) E CPA(L, RZ-'? 6) the Peierls-Bogoliubov inequality [OP? Cor.

3.151 yields

From which (v) foiiows.

(vi) If (9, @. B) E CPA(t, R,"-',6) then for c 5 1 we see by the convaity of the function x r zC on [O, cm) and the spectral mapping theorem that

from which the first inequality follows. The second inequality is established sirnilady

by using the concavity of the function x - zC for c 2 1. (vii) Using (i) we have Pe (f h) < Po (1 hl) and from (vi) Ive have - Pe(h) 5 Pe(-h),

from which (vü) foiiows.

(viii) For (4' +, B) E CPA(t, R$-', 6) we have

it folons that

1 kPo(h, R, O) = k iimsup - log Po,,(h. 0. J)

n-cc n

1 = Lim sup - log ~~k,,(hi- ' . fl0-l; 6) n-oc n

and. sirnilarly, kPo(h, R. S) 1 pek (hi-', RO-', 6). We thus conclude (viii). (Lu) Since 1::; Oi(h + O(g) - g) = hi-L+ 8k(g) - g. using (viii) and (v) we obtain

for al1 k E NI from which (ix) follows. O

The monotonicity of pressure, recorded in the following proposition, is a direct

consequence of the definition (cf. [Brl, Prop. 2.11). Note that C8-subalgebras of =act

C'-aigebras are exact, as is clear from Theorem 1.7.

Proposition 2.6. If A. is a 8-invariant C8-subalgebra of A and h .is a self-adjoint

elment of A. then POIAO (h) 5 Po(h).

Pmf. The proposition foliows by noting that the u.c.0. embedding L : A -+ D restricts

to a u.c.0. embedding tlAO : Ao - D and thato for Q E P f (Ao) and d > 0, (&Si, B) E

C'PL\('. ~;- l ,s) implies (4(.4,,$, B) E CPA(LIA, ,R~-~, 6). O

As is the case for the Voiculescu-Brown entropy prl, Prop. 2.101, we also have the

foilowing generaiiied version of monotonicity

Proposition 2.7. Let A and C be unital ezact C*-algebrus and 8 : A -+ A and

-/ : C 7 C u.c.p. maps. Suppose the= is a unital complete onler embedding p : C -t. A such that 8 o p = p o y. Then for any h E Cs, we have P,(h) 5 Po(p(h)).

Pmof. If L : A + D is a U.C.O. embedding then so is L O p, and so the propce

tion foiiows from the observation that if R E P f (C) and S > O then (4, $, B) E CPA(L. (p(!2})~-'~ 6) implies (4 o p, @, B) E CPA(t O p, nô-', 6). 0

Proposition 2.8 If (S)-X}~E:l iS Q net OJ sets in P f ( A ) such that the lznear span of

UxE., Uncl, Bn(RX) is dense in A then

Pe ( h ) = sup Po (h, il,\). ,€A

Proof. Given Q E P f (A ) and 8 > O there is by asumption a X E A and a r E N such

that if U:=, 8'(QA) = ( x l l . . . ? x,,) then for ail a E R there are s c d a s ca.k such that

If ive set C = m rr,a?c,.n: I c , ,~~ then it is easily verified that; any triple (4, +, 3) in CPA(c. (R~),""-'.C-~G) is also contained in CPA(~.fll- ' ,36), and since

nre infer t h

1 = limsup - log Povn(h, RA, c"S)

n-ca n

= Po(h, fil, ~ - ' 6 ) .

We conclude that Po(h) 5 sup,,,, Pe(h, nx). The opposite inequaiity follows immedi-

ateIy hom the defmition of Po(h). Cl

Proposition 2.9. Let BI : Al -t Al and Oz : A2 - - 4 be ,u.c.p. maps. II hl E (Ai),, and hi E (A2) , then

Proof. The inequality PoIeh((hLl h?)) 1 ma(Pe, (hl), Psl(h2)) is a consequence of nonotonicity (Proposition 2.6) relative to the embeddings ar H (al: 1) and as H (1, a l ) of Ar and 4, respectively, into AI $ A2.

To establish the reverse inequality, let L I : AI -+ %(KI) and 12 : A:, -r B(%) be

faithful representations, and note that 'B(3CL) 9 %(TL) is injective, so that Rie may

cornpuce the pressure of Bt S wing the embedding L I û L?. Suppose fil E AI? f12 E .A2: and b > O. For every n E N and i = 1.2 choose a tripIe (4i,,, +i,,,Bi,n) E

CP.4(ii. (R,):-', d) such chat

rve then have

Ranging over ail ni: II2, and b me conclude that

If -4 is a unital C*-algebra. 6 : A -+ A is a u.c.p. map, and Ai c -4.) C A:, C . . . is an increasing sequence of C*-subaigebras whose union is dense in A and each of which is

0-invariant. then we caiI B an inductive Iirnit u.c.p. endomorphisrn. This extends

the notion of inductive limit automorphism as defined in [Vol] for Xi' aigebras and in [Brl] for exact C*-dgebras.

Proposition 2.10. Let û be an inductive limit u.c.p. endomorphism of a unital ex-

act CW-algebm A with respect to a sequence {Ak}kErr . Let h E A,,, and suppose

that { h k ) k E r ~ is a seqaence of self-adjoint elements with hk r ilk for al1 k E N and

limk,, Ilhk - hl1 = O . Then

Proof. Given E > 0, choose R = {x~,. . , , x,) E P f (A) and 6 > O such that 5 5 E and

Po(r. h. R, 35) > Pe(h) - $ E , and choose T f W such that II is approximateiy contained

to within 6 in A, and Ilh- hkll < 6 for ail k 2 r. Next pick a i' E P f (A,) which contains

elements yi, . . . ,y, with Ilxi -yi( l < 6 for i = 1.. . . ? m. Now if (9, Jr. B) E CPA( t , ï, d) then for every i = lt .. . y m we have

from which we infer that Pe(c, h, r, 6) > P ~ ( L , h, R t 3 Q Thus, since P9(r, h, ï, 7) 3 PO(^! ht Rt 6) as is easily verified using .4rveson's extension theorern, it follows with an appeai to monotonicity (Proposition 2.6) and the proof of Proposition 2.5(v) that if

k 2 r then

On the other hand, if k 2 r then by rnonotonicity (Proposition 2.6) and Proposition 2.5(v) we have

PO!,, (hk) 5 P#(hk) 5 P0(h) f E:

and so we coiiclude that hmk,, PB,,, (hk) = Po(h). O

As observed by Brown in the context of entropy p r l ] , the above result readily extends

ta 1i.c.p. maps which respect a certain generalized inductive limit structure introduced

by Blackadar and IGchberg [BK]. ?Ve consider generalized inductive systems (Ak, u k ~ )

[BK, Defn. 2-1-11 in which each Ak is unitd and exact and each 0k.l is a unital complete order embedding. For each k E PiI Ive denote by a k the unital complete ordcr embedding

frorri =lk into A induced by the system. First we show that the generalized inductive

iimit Lim(..lk. cq.~) arising from such a system is e~act. - Proposition 2.11. If A = lim(&, akti) is a genemiized inductive limit of unital exact - C'-algebnrs urith connecting maps which are unitai complete onier embeddângs. then A is (unital and) exact.

Proof. Let ;r : -4 - 'B(3C) be a faithfid representation. Given R = {xi?. . . , rm) E

P f (A) and S > O there exists a k E N and a f = {yr,. . . y,) E P f (Ak) such that llx; - ûk(yi) 11 < S for every i = 1,. . , , m. Since -4 is exact and hence nucIearly embeddable by Theorem 1.7. the unitai complete order embedding st O al; is a nuclear map, and

so CPA(;r O ak, r. (5) is non-empty. Let (4 , $, B) E CPA(a 0 crk, ï, 6). By kveson's

extension theorem tve c m extend the map d o nii on ak(il) to a unitd completely positive rnap -( : A - B. For each i = 1,. . . ?rn we then have

Hence CPA(ir, il, 363, containing the triple (7, @, B), is non-empty. We conchde that

A iç nuclearly embeddable and hence exact.

Proposition 2.12. Let A = h ( A k , ak,[) 6e a genemlüed inductiue limit of unital

aact C*-algebras vith connectàng maps whàch are unital complete onier emlieddings,

and let 9 : A -. A be a u.c.p. map which leaues uk (Ak ) invariant for al1 k E H. Let

h E .As, and h k E (&)3a for euery k E N? and suppose that lirnk,, [(h - ak(hk)l( = 0- Then

= k-ai lim P 8 1 a k l i ) k l ( ~ k ( h k ) ) -

ProoJ The proof of Proposition 2.10 carries over with the obvious modifications. O

Next we show that the exact approximation pressure coincides with the classical

pressure in the commutative case. \.Ve use notation from [ÇVd].

Proposition 2.13. Let T : X -+ ,Y be a homeomorphism of a compact metric space,

and let B be the automorphism of C(X) defined &y d(f)(x) = ~ ( T x ) for al1 f E C ( X )

and z E .Y. If h E C(X. R) then

PmoJ By Proposition 4.15 ive have Po(h) 2 hh,(0) + o(h) for every 0-invariant state

o on C(X): where hh,(O) denotes the local state approximation entropy introduced

in Chapter 4. By the classical variationai principle [Wall, P(T. h ) is equai to the

supremum of hp(T) + p(h) over al1 T-invariant measuses p, where h,(T) denotes the

Kohogorov-Sinai entropy. By Proposition 4.13, any T-invariant state p on X satisfies

h,(T) = hh,(0)? where a is the state on C(X) defined by integration against p. W e

thus obtain the inequality P(T, h) 5 Pe(h).

To establish the reverse inequaiity, we identiS C ( X ) nrith its image under the

canonical embedding L into C(X)", which is commutative and hence injective. Let

R E Pft.4) and Ci > O. Let 'U be a finite open cover of ,Y such that if U E U and

xt y E U then 1 f (2) - f (y)l < Ci for al1 f E Q. Givea n E W let V = {VI,. . . : V,) be a

subcover of xzL T-'U which minimizes the sum

among di mch subcovers W. Note that if V E V and x,y E X then 1 f (z) - f (y)( < 6 for di f E Rt-L. Let be a partition of unity such that S U ~ ~ ( X ~ ) c 6 for each

i = 1: ... ,rn: and let X, = (xi,..-,xm) be a finite subset of X such that xi E for

each i = 1 ...., m. Define the u.c.p. maps b, : C ( X ) -+ C ( X n ) and In : C ( X n ) -+

C(X)" by f I- f lsn and g +. C K f l ( z & v i , respectively. If f E then For any

r E ;Y we have

and so (&, $,, C'(,Y,)) E C P A ( L , O$-'. 6). Furthermore.

rn

< 1 nup h(TJ4 i=L 'Et:

Thus Pe,,(h, R, cl) 5 P,(. h. U) and hence Pe(h, it. 6) 5 P(T, h: U ) . Since U could have been chosen to have arbitrarily small diameter we conclude that Pe(h) 5 P(T. f ) . 0

Proposition 2.14. Let O L : AL -t AL and fI2 : A2 + A2 be u.c.p. maps and suppose

hl E (AL) , , and h? E (A2)sa- Let B : Al @minA2 + Ar QminA2 be the ( u . c . ~ . ) eltension of the map 01 8 0:: : A i @ A? - AL 8 A? on the algeb~aic tensor pmduct. Then

and if we assume fiirrthennore that h l , h2 1 O then

Proof. instead of using arbitrary U.C.O. embeddings into injective C*-aigebras we work

with faithfid representations to more easiiy haadle the tensor product operation. Thus

we take f a i t h l representations TL : At - %(Ri) and - : A:! -t 'B(K2) and extend

TI 18 r 2 : -41 .A2 - 'B(3CL)g'B(%) 2 %(Ki @ fi) to a faithful representation ;r :

Al 'Pmin -42 + $(Xi '8 TC2)-

Let RL E P f(Al) , RI E P f(&), and Ji ,& > O? and set Ad = rnzc~{llxII : x E

f iL U Cl,). Suppose (ej7 tbj, Bj) € CPA(?, (fij):-', dj) for j = 1,2. By

qq 8 & : AL @ -42 - Br 8 B2 extends to a u.c.p. map 4 : AL @,in A? - Bi @ 82. Obserting that, for any x i E Al and 2 2 E A?,

IÏ {e~)Lsk5rank(Bi) and {e:)Lsllrank(B2) aïe masimai Sets of painvise orthogonal minimal spectral projections for d((hl @ l),"") and +((1@ h2)G-')t respectively! then

{ek 3 if) Llksrank( ),li15rank( B2) is a maximal set of pairwise orthogonal minimai spec- tral projections for +((hi 18 1 + 1 @ hz);-'). and so

It follows that

and since AI % A:! is dense in AL Bmin A2 an appeal to Proposition 2.8 yields

To estabIish the second inequality, note that AL embeds into Ai @,, A2 under the

map a1 H a @ 1, so that Pel (hl) 5 Pe(hI @ 1) by monotonicity (Proposition 2.6).

Since 1% h2 2 O it foilows from Proposition 2.5(i) that &(hl @ 1 + 1 @ h2) 2 Po, (hl) .

A pardel argument yields the inequality Pothl @ 1 + I 8 h2) 2 Ph (h?). LI

We round out this section by cornputing the pressure for a fundamental noncom-

mutative example, the tensor product shift system with one-body interaction iarnilios

from quantum statistical mechanim. In this case the unital exact approximation pres- sure agrees with the mean pressure, which is termed "thermod_vnamic free energy" by

Bratteli and Robinson (see Sections 6.2.3 and 6.2.4 of [BWlj, For ench k E Z let Nk be a copy of the matrilc dgebra !LI,(@), and consider the right shift automorphism

8 of the infinite tensor product BkEZ !Vk, wtiich is defined as the direct iimit of the

algebras Nk over the net of finite subsets K of L. with the natural embeddings

as the connecting '-homomorphisms. Let h be a self-adjoint element of L V ~ , which we

identify with its image in Nk. 1Ve thus obtain a madel of a trandation-invariant

quantirm-statistical-mechmicd systern over the lattice Z. This system may be inter-

preted phlicaliy as a lattice gas or spin mociel. Ln the lattice gas interpretation each

site is occupied by up to rn particles, which under t h e evolution may jump from site

eo site. In the spin mode1 interpretation. each lattice site is occupied by a L ~ e d particte

iicimitting m possible spin orientations which may change as the system evolves. The elemerit h represents the interaction energy contribution of one lattice site. so that

the rnean pressure may be computed as a dynamical approximation pressure using the

spatid symmetry 8. CVe have restricted h to be contained in the rnatrk algebra over

the zeroeth lattice site: which m e n s that there is no interaction between sites. Thus?

in the spin system interpretation, each particle interacts with an external field but not

with other paricles. The value of the pressure of h in this speciai case turns out to be

a simple funccion of its eigenvalues.

Proposition 2.15. The pressure O/ h in the aboue one-body tensorpmduct shift system

is given by

PB(h) = log(exl -i- . +. +- eAm):

where At,. . . , X, are the ezgenuahes (uith mdtzplzcity) of h a an elenaent of No.

Proof. Set A = BkSz Nk and let L : A A" be the naturd inclusion. Note that

A** is injective by [CE11 since A is AF and hence nuclear. Let R be the set {eijlij of standard matKu units for lVo, identifieci with its image in A. Then the lineu span of

Uka oki? is dense in A, s<j that by Proposition 2.8 we have Pe(h) = Pe(h,R). Let S > O

and n E N. Let B, be the C*-subalgebra of -4 generated by O:-', which is 8::; lVk iinder identification with its image in A. Let il>,, : B, -. A" be the inclusion map.

By Xrveson's extension theorem we c m extend the identity map on Bn to a projection

On : -4 - B,. Then (&, q,,, Bn) is an element of CPA(L, R:-', 4, anci? with XI , . . . ? X, denoting the eigenvaiues (with rnultiplicity) of h a s an element of iVo,

n- 1 n- 1 log T~B. e ~ ~ > - ( ~ ; - ' ) = log ( 'hl\ eek(h)) = log nLVk e6kV4

so that PO(t, h. fit 6) 5 log(ex1 + . + * + exm).

To establish the reverse inequaiity, let Co be the commutative C*-subalgebra of 1Vo generated by a maximal set P of pairwise orthogonai spectrai projections of h. and

for each A: E Z let CI; be the Ca-subalgebra @(CO) of A. Denote by C the &invariant commutative C*-subaigebra $)kES Ck of A. By monotonicity (Proposition 2.6) ive now

need only show that Poic(h) 2 log(exl + .* + + exm). Let p be the measure on the

spectnim X of C which induces the product state akEZuk on C given by

for each b E S and dl ;r. E Nk. Let T : X -+ X be the homeomorphism induced by BIc. Then. with S(.) denoting the von Neumann entropy of a state on a finite-dimensional

C*-aigebra, the Kolmogorov-Sinai entropy h,(T) is readily seen to be given by

1 hp(T) = -S(wo a - - - 8 = lim

n-a, n n-a ' n n S b ) k 0

= - 'li.,vo (eh log eh) + (eh log 'IlNo eh) eh 'li.,voeh

Thus. by the classicai variationai principle PVal],

= - nLv0 (eh h) + log Tr,v0 eh + nLv0 (ehh) TrLvo eh n,v0 eh

which completes the proof.

By replacing the u.c.p. maps in the unitai definition with c.p. contractions and working

with faithful representations instead of arbitrary complete order embeddings we can

obtain a definition of approximation pressure for c.p. contractions of (not necessarily

unital) exact C*-aigebras. If A is any C'-dgebra, 5r : A -+ 'B(3C) a faithful representa-

tion. R E Pf (A) , and S > O? we denote by CPA0(5r, O,&) the set of di triples (q, O, B) where B is a finite-dimensionai C*-algebra and a : A - B and $ : B - B ( X ) are c.p.

contractions such that 11 ( 9 O 9)(x) - L ( T ) [ I < S. For an exact (not necessarily unital)

Ca-algebra -4 and a c.p. contraction 8 : A - A ive can then replace CPA(n, R? S) with

C P R ( ~ . R. 6) iu the unital dehit ion to define the analogous quantities P&(r, h. R, d),

Pf(n, h, 9. B ) , Pi(n! h, R), and P ~ ( T , h). The proof of Proposition 2.2 readily adapts

to show that P;(ir, h) is independent of n with an appeai to a nonunitai version of

illsverson's extension theorem as in p r l , Prop. 2-31 (it is in order to be able to a p

ply this theorem that we work with representations and not arbitrary complete order embeddings as in the unital case). We can thus d e b e the (exact approximation) pressure P:(h) of 8 at h to be P ~ ( x , h) for any faithful representation n : A - %(m.

Ali of the results established above for the unital case that do not make cxplicit

reference to a unit carry over easiiy to Pi(h). E'urthermore, the two dehitions coincide

in the unital situation, as we now demonstrate- We d l require the foiiowing Iemrna.

Lernma 2.16. Let A and B be C*-algebnzs and $ : A -+ B a completely positive map. Let p be an element of A such that llpll 5 1 and - $(p)ll < E for a given E > 0- Then, for al1 x E At

Proof. We rnay assume that A. B , and u are unital, for if necessary ive may adjoin

iinits to -4 and B and consider the unital extension of iIr. which is completely positive

by [Brl. Lernma 1.61. Lie may ais0 assume that B is a subdgebra of 'B(3C) for some

Hilbert space 3C. By Stinespring'ç theorern (Theorern 1.1) there exists a Hiibert space

X containing 3C and a *-representation a : A - B ( X ) such that. nith q denotinp the

projection of X ont0 K. V ( r ) = qr(x)q

for al1 x E A. First observe that

and

so that by the triangle inequality Il$(pxp) - Sr(p)ll(x)Sr@) 11 < 2&11~11- O

LVe wouid like to thank G. Gong for pointing out why an equality such as the one in

the claim in the proof of the foiIowing proposition shouid hoid.

Proposition 2.17. If A i s unitd and ezcrct and 0 is u.c.p. then Pi(h) = Po(h).

Pmof. The inequality P:(h) < Pe(h) follows immediately from the definitions.

To establish the reverse inequality, let ir : A -+ %(!.J-C) be a Eaithful representation

and let r! be a finite subset of the unit b d of A containing 1, and suppose O 5 8 5 a. Let (O, t,b, B ) E CPAo(iil O:-', d), and set b = O(1). Let p be a spectral projection of

b such that bL := bp 1 ( 1 - &)p and b2 := b(1 - p ) < ( 1 - &)(1 - p). CVe claim that ~ ( 6 2 ) < fi. TO see this. suppose to the contrary that I1,1$1(62)[1 1 8. Since iIr is contractive we have li> bl + &b?)ll 5 1161 + &bal1 S 1- On the 0 t h hm4 II ( since w(b) >_ 1 - I I w O Q(1) - 111 > 1 - 6. the positivity of w yields

and since / /&v(h) > 6 this implies bl + &b2) 11 > 1. producing a contra- I I diction and thus establishing the claim.

Observe now t bat

Thus. with p denoting the support projection of bl , nre have

1 1

Set B' = pBp, and d e h e the u.c.p. map 4' : A -, Br by 4'(z) = b ~ ' ~ ( x ) b ~ ' , with

bi now being considered a s an eIement of Br. Let îCr/ : Br -+ B(!J€) be the u.c.p. map

and so estimating as do= Brown in the proof of Frl. Prop. 1.4] we obtain Il+'a$(x) -

m(pd(z)p)lI < 14(17&& whence

CVk therefore have (ci'! w'. B') f C P A ( T . ~ ! ~ - ' , 2 5 5 a ) ) . Also note that

= log ~ r ~ ~ @ ( ~ g - ~ ) ~ + 2nlthll (1 - Jq2

2 log . ~ ~ ~ ~ e ~ ( ~ ~ - ' ) ~ + 2nl[h(l & (1 - &)2

Nith the second Iast inequality FoUowing from Proposition 3.17 in [OP]- Tt foliows that

Pj(h! Q. 2 5 5 f l ) < e ( h , R : 6) +2llnll&, fiom which we condude that fe (h . fl) < Po ( F r , R) . Thus P: (h) 5 Po (h) . U

It is aIso the case that adjaining a unit to A does not aEect the d u e of the pressure (cf. prl, Lemma 1.71).

Proposition 2.18. Let A be exact, 8 : A -i A a c.p. contraction, and h f A,.

I/ is the unitization of A and é is the (unique) .unital extension of 8 to .i, then Pf (h) = Pi ( h ) .

Proof. FoUowing Brown's proof for entropy [Brl. Lemma 1-71, we first observe that the

inequality P:(h) 2 Pi(h) is a consequence of monotonicity (Proposition 2.6). To establish the reverse inequality we mite -4 as .4 x C and consider a given fi =

{(al ,XL), .. . .(a,.&)) E PI(.<) and 6 > O. Then R = {ai ,... ,G} E P f ( A ) . Let - ;i : -4 - 'S(W be a faithful representation and let ii : A - 'B(K) be its (faithful)

e~tension to ri. If (ci &. B ) E CPA(;r. Cl:-', 6 ) then by [Brl, Lemma 1.61 (4, &. 8) E

CPA(5 , fi,"-'. 5). where B is the unitization of B and 3 and ,$ are the unitai extensions

of ci and b! respectiveIy. It foUows that

I l A is n unital nuclear Cs-algebra we c m define pressure by using the local approxima-

tion characterization of nuclearity? as does Voiculescu to define entropy in [Vo3, Defn.

4.11. Thus. witli CPAmC((iL. R.6) defined as the collection of triples (@,& B) such that

B is a finite-dimensiond C"-algebras and 4 : A - B m d : B -+ A are u.c.p. maps

with I I (@oO) (x ) -211 < d for d x E Rt we set

P,lF(h. R. 6) = inf {Trs z(~:-') : (A ,+: B) E C P ~ "(A' 6 ) )

and define P p ( h , II, 6). P,""C(h, Q), and P r ( h ) by taking the appropriate suprema as

in the exact case. LVe then obtain the pressure of Neshveyev a d S tmmer [NS, Defn. 2.11

(up to a sign change). By adapting Voiculescu's Al? debition Po31 in a s i d m way we

can d e h e P k F ( h ) using the locd characterization for AF aigebras [Bra], and e N F ( h ) : p F ( h ) and ~ y ( h ) can be deûned analogously using local characterizations for strong

NF (see [BK]), NF (see Proposition 5-14): and quasidiagond exact C*-algebras (see

Proposition 5-16)? respectiveIy. Then

with each inequality applying to the appropriate domain of definition.

Proposition 2.19. Let A be a unital nuclear C*-algebmt 0 : A -+ A a u.c.p. map, and

h E A,,. If R E P f(A) and b > O then

Proof. The second statement is an irnmediate consequence of the first, which we estab-

lish by adapting Brown's argument for entropy prl, Prop. 1.41. LVe mny assume that

A is fnithfully and unitally represented R : A - '8(!K) on a Wilbert space N. Let iI E

P j(A) and 6 > O. IdentiFying il with its image under R, we have CPAnUc(A, Rô-'. 6) C

CPA(T ,Z~ , " - ' ~S ) For al1 n é PI. so that PY(h,Q,S) 2 Po(h,i2,G). For the reverse

inquürity. let E > O and rr E N ürid suppose (d, w, 3) E CPA(r? Q:-L. 6 ) . Given

(a.3.C) E C P A " " C ( A . $ ~ ~ - ~ , ~ ) we extend p to a u.c.p. map j : 'B(w C by .A.rveson's atension theorem. It is then readily checked that (4,D o p' O +. B ) E CPA"UC(A, iIô-L1 1d), from which we conclude that Pzy(h. R, 5) 5 Po,,(sr, h- R! 6) and hence P F ( h . Q. d) >_ Pe(h. 0.6). O

CHAPTER 3

Pressure and crossed products

For a C'-d~cunicd invariant such as entropy or pressure a problern which naturaiiy

arises is to calculate its value for an automorphism of a crossed product that acts via

conjugation by a. unitary implernenting the given action. Adapting a construction of

Sinclair 'and Smith [SS!, Brown proved in prll that if A is a unitd exact C*-algebra

and n is a group homomorphism frorn a discrete ilbelian group G into Aut(A), then

for any g E G the Voiculescu-Brown entropy of ctg coincides with that of the inner

~lutornorphism of the crossed product A x, G defmed by the iinitary implementing a,.

Following Brown's approach, we extend this result to pressure, restricting ounelves for

simplicity to the case of crossed products by a single automorphism. See p] for an

application of similar techniques to andogous probkrns in the setting of Cuntz-Kreiger-

type C*-dgebrs. See also [Stol for a brief survey of related results for entropy.

Suppose that A is a unital exact CW-algebra and û E Aut(A). The crossed product

C*-algebra A xo Z is constructed as follows. The idea is to encode the dynarnics C*- dgebraically by enlarging A with the adjunction of a iinitary mhich implements 8. First

we identiFy A with its image under a faithfui unitd representation on a Hilbert space

K. Let

r : A - $(12(Z.K)) 2 'B( f2(Z) @ !XI S 9 ( l 2 ( Z ) ) S $ ( K )

be the representation defined by

for aii x E A, 5 E L2(Z, N), and n E Z and Let X be the amphfieci left regular represen-

tation of Z on f2 (2:K) , that is,

for aii < E 3C and n, m E Z, where is the standard orthonormal basis of L2(2). Then for ail x E -4 and n E W we have r(On(x)c)) = X,R(X)X_,- FVe define the reduced

crossed product -4 x as the n o m cIosure of the linear span of {r(x)Xn : x E A: n E Z). We remark that this definition readïiy extends to actions by arbitrary discrete groups

(non-discrete groups can be handled with the introduction of hrther technicaiities; cf.

[PedI). X more generai construction is required to define the (fuli) crossed product, but

for arnenable groups the reduced and full crossed products coincide [Ped, Thm. 7.7.71,

and so there is no notational or terminologicai ambiguity in our conte-xt.

Proposition 3.1. If A is a unital emct Ca-algebm, O E Aut(A). h E ASaI and

u E -4 >a0 Z is the unitanj implementing 8: then

where h has been identified with its image under the canonical inclusion A - A >a0 Z.

Proof. In view of the canonicai inclusion A - A Xe Z, monotonicity (Proposition 2.6) yields the inequality Po(h) < PAd Jh).

To establish the reverse inequaliN we regard A XeZ as the subdgebra of 23(12(Z) @

N) by means of which it was defined prior to the proposition statement. Let i2 be a

set in P J ( A >a0 Z) of the form {n(xl)Xt,,. .. ,;r(xi)Xt,) with IIxj1l 5 1 for j = 1,. . . l ,

as in the proof of [Brl, Thm. 3.51. Note that the span of such sets is dense in A xo Z, and so by Proposition 3.8 we need only show that PAd u(id.,lxuZ: ht n) 5 Po(h). Set

0' = {T,, . . . . q} E P f (-4). The proof of [Brl, Lernma 3-41 eçtablishes the existence of

a finite set F E Z depending only on the distinct elements of {ti,. . . , tl} and S such

that

(4 Jt, B) E CPA(id.4, Clf, 6)

where d' is the u.c.p. map x - (l@4)((pFC3 l)(h)(pF@ l)) , Srf is a u.c.~. map, and MF

is the finite-dimensional C*-algebra P~(B(12(Z) ) )P~ , with p~ denoting the projection from L2(Z) ont0 span{Et : t E F). The same finite set F also works for Q:-' (with

(N),"-' replacing flf above), since

for alI n E M and F depends only on the distinct eIements of {tl,. . . , t r ) and 6. We

can assume that F is of the form {-m, -m + 1, .. . , -1,0,1,. . . , m - L,m) for some

positive integer m. Mthough we will not need the explicit form of 1V, we note that it

is given by Tf O (1 ,% O)? where TI is defined by

for al1 1: E 9 ( i 2 ( Z ) @ X). where t k ut is a u n i t q representation of L on %(!J-C)

implementing 0 (as nre may assume to e-xist) and mi is the multiplication operator in

8 ( L 2 ( Z ) ) by a function f E L2(L) with s u p ( f) c F and I[ f 112 = 1. chosen (as is possible

by the amenabilil of Z) so thrit I[(j + f ) ( t ) - 111 < 6 for -m 5 t 5 m. where is the function t t. f ( - t ) in 12(Z).

Let {e,.t),,t be the standard matriv units in 'B ( i2 (Z ) ) . so that

for di s, t , p E Z. By [Brl, Lemma 3-11, a(h) = Et,== etat @ 0- ' (h) , where the conver-

gence is in the strong operator topology. 1 n-1 Suppose now that n 3 7m + 1 and (4: 9, B ) E CPil(idli, (R ? 6). CVe will show

that the absolute difference

is bounded by an expression independent of n by splitting the estimation into three

parts. Two of these estimates wiii follow readily hom the PeierlsBogoiiubov inequality

[OP. Cor. 3-15], nrhile the other reduces to a tensor product cdculation reminiscent of

the proof of Propostion 2.14. Since

ive have

= I J c ~ - ~ k-rn xk t=-m PF.L,LPF ~3 (4 0 O-*) (h) +

< xm-l k=-rn zk l=-m I I ~ F e t . t w B ( $ 0 O-*) (h)(l+

by [OP. Cor. 3.151. Ncyt observe that if B and 7 are maximal sets of painvise or- n-rn-L thogonal spectral projections for EL-, p ~ e ~ , ~ p ~ and (4 O d-"(h), respec-

tively, then @ @ y is a ma..rnai set of paim-se orthogonal spectral projections for n-rn-L

E k P F ~ L J P F ( 4 0 Ok)(h)? and thus

Since the above holds for any (rb,., B ) E CPA(idA, il, S), it foIlows that

Thus, letting n Vary while m remains iï.xed, we infer that

We conclude t hat PAd ,(idAxos, ~ ( h ) , n) 5 &(idAr h). cornpteting the proof of the

proposition. 0

Local state approximation entropy and variat ional inequalit ies

In this chapter we introduce a notion of C*-dgebraic entropy with respect to an invari- ant state tvhich adopts the approximation frametvork of Voiculescu's definition [VO~].

but exercises the entropy of the induced local state instead of the logarithm of the rank of the local algebra. This -local state approximation7 entropy rviii be used as a concep

tually illuminative intermediq in the proof of a variational inequality (Corollary 4-16)

with respect to the CNT and Sauvageot-Thouvenot entropies, which dl be needed to

establish the variational principIe in Chapter 5.

The framework of the dehition will be similx to that of the definition of pressure

in Chapter 2. Let -4 be a unital exact C*-aigebra, 8 : il - A a u.c.p. map, and a

a O-invariant state on -4. Let D be an injective Ca-algebra and L : A - D a u.c.o.

embedding. The notation P J ( A ) and CPA(t. R. S) d l continue to have the meaning

given in Chapter 2. Denote by E(a, L ) the set of ail States w on D which extend the

state u O L-' on ~ ( t l ) . We recail that the von Neumann entropy S(w) of state on a

hite-dimensionai Ca-aigebra B is defined as - Cr=, X i log A, where Xi,. . . . A, are the

eigenvalues. with multipiicity. of the density rnatrk for r j with respect to Dg, with

x log x defined CO be zero at x = 0.

Definition 4.1. If w is a date on D, R E P f (A), and 6 > O , we define the completely positive 6-entropy

of R with respect to (t,w), and for w E E(a, L) we define the dymmical entropies

1 hhu(8! L, w, R, 6) = limsup -cpe(L, w, Rg-', 6)

n-oo n

hh,(8, L, u) = sup hh,(dl 6, w, R) REP~ (br)

hh, (B ,~)= sup hh,(O,~,u). & ~ ( D , u , L )

Proposition 4.2. If L I : -4 -, Dl and : A - 0 3 are u.c.0. embeddings into

injective C*-algebras Dl and D2 then

Proof. Since Dl is injective we can extend the map L I O 11' : L ~ ( A ) - Dl to a U.C.P.

msp T : D2 - Dl. Let wi E ~ ( O L I ) and define w- E ~ ( u : L ~ ) by ( ~ 2 = ( J I o Y . Let

0 E P f (A) and b > O. If (9,.1$~ B) E C P A ( L ~ , R:-'? 4 then for ail x E R:-' we have

so that (b, T o ~. B ) E CPA(LI , C2on-l. 4. Since S(wl O ( T o d ~ ) ) = S(u2o1C>), we conclude

that

cpe(ti:q:R:-'. 6) 5 C ~ ~ ( L ~ , U ? , no"-', 6).

Therfore hhu(Bo L L , Y I ) 5 hh,(O. t2, q), and taking the supremum over ( J I E C(a, L l )

yields hh,(B. L I ) 5 hh,(O, L - ) . The reverse inequality foiiows by symmetry. D

Definition 4.3. In view of the above proposition and the fact that A always admits

a .u.c.o. embedding into a n injective C'-algebm (conside . for instance, its universal

representation), ure can define the local state approximation entropy hh,(8) to be

hh,(O, L ) for any U.C.O. embedding L : A -+ D into a n injective C'-algebm D.

.As when referring to pressure, we wiii often tacitly assume that L represents an ubitrary

u.c.0. embedding in to rui injective C*-algebra D.

Rernark 4.4. If A is nuclear, we can dispense with state extensions and d e h e h h y ( 6 ) by replacing the logarithrn of the rank of the local dgebra in VoicuIescu's topological

definition [Vo3] with the entropy of the induced locd state. Tii other words, with

CPPUc(A, $2, S) defined as the collection of tripIes (4, $J, B) such that B is a hi te-

dimensional Cs-algebras and cb : A - B and $J : B -+ A are u.c.p. maps with Il(.$ O

@ ) ( x ) - rll < b for al1 x E Rt we set

cpe(Q: 6) = inf {S(w 0 $) : (4, $, B) E CPAnUC(A, fi, 8))

and clefine h h y ( 0 , R, 4, hh:uC(dt 0)' and h h y ( d ) by takÏng the appropnate suprema

as in the exact case. CVe then obtain the entropy introduced under the notation ht,(O)

hy Choda in [Ch]. We can also adapt Voiculescu's SLF dehition [Vo3, Defn. 2-11 in a

similar way to definc hh:F(d) using the local characterization for AF algebras [Bra],

aiid hhzxF(0) can be defined analogously using the locd characterization for strong NF algebras PIC]. Then

with cadi inequality applying to the appropriate domain of definition.

Proposition 4.5. If k E M then

Proof. The definition of cpe(i,u, R, d;) as an infimum over C P A ( L ? R? 6) permits us to adapt the seconcl half of the proof of Proposition 1.3 in Po31. The inequality

hh,(dk) 5 khh,(B) follows from the inclusion

for di n E M. for this implies, given w E E(a, L ) , that

, n-l 1

To establish the reverse inequaiiw note that

for al1 n E M. so that, for w E E(q L ) ,

1 h h , ( 6 * , ~ ~ ~ ~ f l ~ - ~ ~ ~ ) = ümsup n-cc - ~ X ( L , W n ; - 6 j ( ~ ) ; d )

k L i j -1

L'W, U 6"(fl).6) n-cc n

j=O

Proposition 4.6. (Concavity) If xb, Aiq is a conver combination of O-inuo~+zd

states oi on A then

C & h h , (8 ) 5 h h y A,,, (6) -

Pmof. Let L : A - D be an embedding into an injective Ca-algebra. Set o = x f=l &ci- For each i = l.. . . . k let ui E E(oi, L). Then the state defined by C &yi lies in C(u. L ) ~

and 1 1 - C ri log epe (L, di, s) 5 - epe ( L : C Aiiii , O:-', 6) n n

by the concavity of S(+) on state spaces of Enite-dimensional Cm-nlgebrns. Thereiore

and hence C Xi&, (O, L : w ~ ) 5 hhx (8, L. C &yi)* Taking the suprernum successive~y for each i = 1, . . . , k over wi E E(q, L ) yields

We next show that if 0 is an automorphism of a nuclear dgebra d and r is a

&invariant tracid state on A then h k ( 0 ) coincides with the approximation entropy

ht,(O) of Choda [Ch], also denoted above by hhrC(0). in establishing this fact we dl use Choda's notation. so that

~~~(06) = inf {S(w O ~) : (Q, Ilr , B ) E CPAnUC(~, R!d)}

1 ht,(O, R: 6 ) = limsup - ~ c ~ ( S l ~ - ' : b ) ~

n - m n

ht,(O, R) = sup ht4O.Q; 6): 5>0

Proposition 4.7. Let .A be a unital nuclear C'-algebra, 8 : -4 - A a u.c.p. map, and

o a O-invariant state on A. Let L : -4 -r A*' be the canonical embedding of A into its

second dual (which is injective by [C'El] since A is nuclear) and. identifijng A with its

image under L , let 8 be the normal eztension of o to A". Then

Proof. First note that i l R E P f ( A ) and S > O then

so that hh,(O,~,ü,R,d) 5 ht,(B?Q?S). Consequently hh,(8,t,ü) <ht,(O).

To establish the reverse inequaiity let Q be a finite subset of the unit b d of A and let b > O. Let (&,î1i,B) E CPA(L ,R ,~ ) . Let E > O. We -di show the existence

of an n x n matriv aIgebra 1 W m ( @ ) and unital completely positive maps a : A*' - la(@) and 6 : Mn(@) - A such that (+,P O a O $,B) E CPPUC(A, R,3Q and I(a O /3 O a O +)(y) - (5 O Ilr)(y)[ c 3~ for ail x in the unit b d of B. If E is chosen

suiEcient1y smaii then this will imply, by the continuity of the von Neumann entropy

on the state space of a lînitwiimensional C*-aigebra [OP, Prop. 1.81, that

so that we di be able to conclude that

from which the desireci inequality wiU foiiow.

So Iet I' be the image under $ of a h i t e subset of B which approximately contains

the unit baü o f 3 to rvithin E ! which Ive may assume to be Iess than 6. ive Niil folIow an

argument from the proof of [CE?, Thrn. 3-11. Let S(fl:&u) be the set of d l restrictions

eo ci of compositions 3 A*= A Mn (C) - A

Cor some compIeteiy positive maps û and 0 with u(1) = 1 and some n x n matriv

algebra Mn(@) which satisfy II(P O a)(x) - X I I , < c for al1 x f i', where I] - IIa is the

semi-norm on -4" defineci by llrlld = i?(x*x)k We note that $(f!,&o) is a convex

subset of the collection of di Iinear maps from il to itself. for il

arc compositions of compIetely positive maps Nith ui(l) = 1 then, for any X f [O! 11,

defining the complecely positive maps a : Ag* - ri&,,+,, (C) and @ : Mn,,,, (@) -. A bv

respectiveIy. Ive obtain a(1) = 1 and ,8 O a = ,\(Pi O q) + (1 -A)(,& O a?), and, for aU X E r.

New since A"* is injective and hence semidiscrete by [CEII, the arguments of Choi

and Effros [CE2: pp. 75-76] show that idA4-- Les in the point-a-weak domire of the

collection of maps fiom A" to itseif which factorîze through a matrix algebra via O-

weakly continuous unitai comp1eteIy positive maps. 'CVe cm thus h d a net

of compositions of a-we&y continuous unital compIetely positive maps su& that I[(Pxo

aA)(z) - ~115 < c for d 3: f ï for each X and (Px o a x ) ( x ) :) z in the point-o-weak

topolom for al1 x E -4. For each X the map 8 : !CI,, (A*') -, L(lC.I,,(@), A") given by

for all r = [x,]] E Mn,(A") and b = [bijl E A&,(@) is an order isomorpliism nrith respect to the corie L(M,,, (C). .AN*)% of completely positive maps by [CE. Lemma 2.11. Since hi,,, (.A) is weakly dense in M,,, (ilv*). we caii apply I<aplansky's density theorem

to obtain a bounded net {xx,), of self-adjoint eIements in iVRx(A) which converges

strongly to (Q-' (3,)) t . Then { x $ ) , is a net of positive elements converging a-weakly

to @ - ' ( a ) . and so ( Q ( X & ) ) ~ is a net of completely positive maps from iCI,,(C) to

-4 which converges in the point-c-weak topology to BA. Setting O,,, = Q(x&) we may

assume. by passing to a subnet if necessary, that I [ ( , & o a x ) ( x ) - xljo < E for every

x E i' ancl al1 p. CVe tliiis see that ida4 lies in the point-a-weak closure of J ( R . 6, a). But on -4 the a-weak topology restricts to the weak topology, and since 3 ( R , 6. a) is a

conva subset of the coiiection of al1 continuous Linear maps from A to A its point-weak

and point-nom closures coincide by [DS. Cor. VI.1.51. So we can find a composition

of o-weakly continuous completely positive maps such that a ( 1 ) = 1,11(/3oa)(~)-x11~ < E for every x E r. and I](P O a ) ( x ) -xi[ < 6 for al1 x E R. CVe may also assume that

[ [ ( P 0 a ) ( l ) - 111 < E. and so [[/3([ = ([P(1)([ > 1 - E , so that upon replacing ,O by

/3/IIPII we obtain a unital completely positive map for which it is easily checked that

Il($'oa)(x)-xlle < 2~ for every r E r and ~ [ ( ~ o a ) ( x ) - x 1 ~ < 25 for aii x E R (since E 5 S).

An application of the triangle inequality then yields (poao$, d, B) E CPAnUC(A, R. 367.

.&O notice that if x E ï then by the Cauchy-Schwarz inequality

so that if y is in the unit bail of B then

and so the maps a and p satisfy our requirements, compIeting the proof. O

Coroilary 4.8. If A is unitai and nuclear, B E Aut(A), and a a 6-invariant state on

A: then

hh,(O) = htr(6).

Proof. The inequaiity hh,(O) 2 ht,(O) is m immediate consequence of Proposition

4.7. The reverse inequality foiiows by appljlng the observation in the first paragraph

of the proof of Proposition 4.7 to an arbitrary c.0.e. L of A uito an injective C'-algebra

to conclude that hho(O.~,u,R,d) i) ht,(O,Q,d) for any w f rE(q~) , R E Pf(A)? and

6 > O. O

Cornbining Proposition 4.7 and Coroiiary 4.8 yields the foiionring.

Corollary 4.9. Let -4 be a unitai nuclear C-algebra, O E Aut(A), and o a 8-invariant state on A. If L : A -. .4** is the canonical embedding and ü is the normal extension

of o to =I'* under the identification of A with its image under L, then

F'e next show that hh,(O) rnajorizes the Sauvageot-Thouvenot entropy h,(6), wfiose

definitiori we briefly recaii from [ST]. Sauvageot and Thouvenot's entropy, which is a k h

to that of Connes. Narnofer, and Tliirring [CNT] (with which it coincides in the nuclear

setting [ST. Prop. 4-11 and which it rnajorizes in general), exercises decompositions of

the given state into convex combinations of other states defined via a coupling of the

djnamics with a commutative dynamical system.

Let A be a unitai (not necessarily exact) C*-dgebra, 8 an automorphism of -4. and o a O-inwiant state on A. Let C be a unital commutative C-algebra, c an

automorphism of C, and p a faithful &invariant state on C. A stationary coupling of the Ca-dynamical systern (A, 8, a) with the commutative Ca-dynamical system (C! c, p) is a 19 B <-invariant state X on A @ C such that XIA = a and XIc = p. IF P is a &te

partition of C into projections then for each p E P we d e h e the state gP on A by

for aii x E A. Then cr decomposes as the convex combînation '&,p(p)a,. The

mutual and conditional entropies of X with respect to u and P are defined by

respectiveI> where SI.. -) denotes the quantum relative entropy of kak i [Aral and

Hp(P) is the classical entropy of the partition P. Setting P- = Vk=l C-~IP. we define

the txo quantities

h(P. A) = H,,(P 1 P-) - HA(P [ A @ P-):

and

h(P:X) = Hp(P [ P-) - Hx(P 1 A),

where H P ( . 1 -) denotes the classical conditional entropy and LI,!(- 1 -) denotes the con-

ditional entropy as defined in Section 2 of [ST], here with respect to the stationary

couphg (&O denoted for notational simpticitJ. and consistency with [ST] by A) of (A8 C, dl.%: A) with (C1 <! p ) defined by composing X with id,\@S, where S : C@ C - C acts by restricting functions to the diagonal. The Sauvageot-Thouvenot entropy hL(O) is then defined as the suprenium of h(P. A) over stationary couplings X and al1

partitions P of C into projections, or. equivdently, as demonstrated in Lemma 3.2 of [ST]. as the stipremum of hf(P, A) over the same set of X and P.

We wïll find it conveluerit to invoke the following quivalent mems of formulating

h(P. A). as do Sauvageot and Thouvenot in Proposition 3.3 of [ST]. The validity of this

formula remains impücit in [ST], aithough it foUows readily from the observations on the fourth line of p. 416.

Lemma 4-10, If X is a stationary coupling of (A, u) with the unital commutative

systena (C, ç, p) and P Zs a finite partition of C into pmjections then

PmoJ As described above and in the paragraph preceding Lemma 2.2 in [STJ, the

stationary coupting X dehes a stationary coupling (denoted &O by A) of ( A @ C: 8 @

q: A) with (C7c,p) via the map from C @ C to C which restricts functians to the

diagonal. We may &O similady d e h e a stationaxy coupling (again denoted by A) of

(A@C$~~C,~GI~@F, , ! ) with(.~,p) usingthesamemapfiomC@Cto C. Foresch

n 2 1 we then have by Lemma 2.2 of [ST]

and since

t his leads inductively to

Noting now that

by another application of Lemma 2.2 of [ST], we obtain

n

H A ( V ? P I A ) = H x ( P I A ) + ( n + l ) H * ( P I A @ P-). k= 1

Dividing by n and taking the iimit as n tends to infini@ yields

& ( P / A @ P-) = n-m lim L H ~ ( n \j ~ L P ~ A ) . h=L

1 n

K J P I P-) = n-œ lira -H,(V n CCP) kL

fiom the classical theory, we conclude that

1 n

= n-m iim -EA(A, n V SP) - I;= 1

Proposition 4.11. If 6' E -4uttA) and r~ is a $-invariant state on A, then

2 ha(0).

Proof. Let L : A - D be a U.C.O. etnbedding i~ito an injective Cg-algebra D. Suppose X

is a stationary coupling of ( - 4 . 0 , ~ ) with (C.q), nith the restriction of X to C denoted

by p. which we c m assume to be faithful. h t e n d the state X O L-I on r(A) to a state - X on D B, C. Suppose P is a finite partition of projections in C. For each n E M and

p E V~Z; s k ~ . let up be the state on d defined by z - p(p)-LX(s@p) and wp the state

on D defined by y I- 8 p ) . Note that i+ extends the state op o L-' on ((A). Let w be the state on D given by the convex combination CpEP p(p)uP (which is equal

such t hat 1

hh,(O, L, w1 Z2' 4) = lim sup -S (w 0 s(n,4,n) . n-ca 71

Set A = P f ( A ) x R,o. For each n E Ny { @ a , n ~ & , n ) a E ~ is a net converging pointWise in

n o m to L, so that {w o ?ban, o &a,n)aEA converges weak" to a and, for d p E v::: c",

{w, O O fia.n)aEA converges meak" to up. The weak* lower semicontinuity of the

relative entropy S(-: -) and the weak* compactnes of the state space of A then yields

an a0 = (Roy qo. JO) E A such that, for al1 n E R and p E VZ: ?PI

by the monotonicity of S(-: -)? we therefore have

n-L

~ ( P , x ) = n-cu lim -=A(V n ?PA) k a

1 5 limsup -S(w O n-c~ n

= hh,(8, L ~ U , Qo, 60).

T&ng the supremum over d l stationary coupiings X and finite partitions P. we ohtain the proposition. O

To show that the local Stace approximation entropy agrees Mth the Kolmogorov-

Sinai entropy in the commutative case ive need the folloning lemma. For an open cover

U of a topalogical space X denote by 6(U) the set of d l x E X which are contained in

oniy one member of U.

Lemma 4.12. Let p be a measure on a compact Hausdorfl space X . if U = {UL.. . . . Um) is a finite open cover of XI then for e v e q E > O there is a open re-

finement V = {VI,. . . Vm) of U such that there are closed sets Hi c for i = 1,. . . . m

S U C ~ that Ukl Ki c B(V) and p(X \ UE1 Hi) < E .

PmoJ Let U = {UL,. . . , Um) be a finite open cover of X and E > O. Set 6 = UL. Let

Gl C Cil be a closed set such that p(VI \ Gl) < + and set = U2 fl (X \ GI). We continue inductively for k = 3, . . . , m so that at the kth stage we choose a closed set

Gk-1 C Vk-1 sudi that p(Y-1 \ Gk-1) < 5 and set = Ukn (X \ $:Gj). Put Kr = Gl and, for i = 2,. . . , m7 Hi = Gi\(&u- - .uK-~). Then Uzl Hi C G(V),

and since Gi, . . . , G, are pairwise disjoint so are Hi,. . . ,Hm. Eùrthermore, for each

i = l , . . . ,m we have

as required.

Proposition 4.13. Let T : X + .Y be a homeomorphism of a compact metric space

and IL a T-inuan'ant measure on X. Let il be the automorphisna of C(X) defined bg

û ( f ) ( r ) = f ( T x ) for al1 j E C(X) and x E ,Y, and let a denote the date on C(X) @en by integrution against p. Then

where h,(T) is the Kolmogorov-Sznai entrupy of T .

Pmof. By Corollary 4.16 below the Local state appro?rimation entropy is bounded

beloiv by the CNT entropy, which agrees with the Kolmogorov-Sinai entropy in the

comniutative case (see the ciiscussioa followirig CotoIlary 10.14 in [OP]). Hence h,(T) 5 hh,(h-).

To establish the reverse inequaiitv: Let 1 : C(X) 3 C'(;Y)" be the canonicd em-

bedding. Note that since C(,Y)" is commutative it is injective. Suppose w E e(a, L),

fl E P f ( C ( X ) ) and S > O. Let U = {UL,. , . .Ur) be an open cover of X such that

if and x.y E Ui for some i = 1, ..., r then l f ( x ) - f(y)I 5 b for di f E il. Let Cl

be the Borel partition {Ui \ U;:: Uj : 1 1 ,i < r ) reiining U. Fiu n E N. Note that

if U E V:,: T-'ZL and x . y , ~ U then If(=) - f(y)l 5 6 for ail f E RI-'. Let E > O

be s m d enough so that if O 5 a, b 5 1 and [a - bl < E then la log a - b logb[ < r-n.

By Lemma 4.12 there is a refinement V = (K, . . . V,) of K ! T-'U nich that there are painvise disjoint closed sets Hi C Vi for i = 1,. . . , rn such that UEl Hi c 6 ( U )

and p(X \ UL"=,J < E. Let Z = {xi,. . . , ;Y,) be a partition of unity such that

s.upp(,yi) C K for each i = 1, ... ,m, and let X, = (xi, ..., xm) be a Enite subset of

,Y such that xi E Hi for each i = 1: -. . ,m. Dehe the u.c.p. 4, : C(X) -c C(Xn) and A : C(XJ - C(-Y)** by f I+ f lx, and g - Clsism g(xi)( t O xi), respectively. Then for d f E R:-' and L E x we have

mhence (h,, iy,, C(-Y,,)) E CPA(L, R:-'. d).

Now for each i = 1.. . . . m, we have

since p(X \ UL,i,,) < .E and does not intersect Hi for j = 1, ... , m, j # i. Thus, - - since m < r-". our choice of E yields

Setting K = UEl Hi ive have

for ai1 Q E c ! T% Mso note that the partition {Q U K : Q E vY'Z; T-iQ} r e h e s

{Hi : a 5 i 5 m) since each Q E ri: T-'9 intersects at most one of Hl:. ..' Hm. Thus

The above two estimates combine to produce

from which we hfer that

Dividing by n anci taking the limit supremum yields hh,(Of~,w.fl,s) 5 H,(Y.T). Taking the supremum over '211 6 > 0, Z! E P f (A) , and w E E(u, L) , we conclude that

hh&O) = hh,(O. L) 5 h,(T). O

We next establish a variational inequaiity bounding the free energy in a given state

by the approximation pressure. First ive r e c d the principle of minimum free energy for

iinite systems. which says that on a finite-dimensional C*-dgebra the Gibbs state is the

unique state that minimizes (or mavimizes in our case. since we have favoured the sign

convention from ergodic t heory rat her than that from physics) the Gee energy a t a k e d

energy (i.e.. self-adjoint element). This non-dynamical variationai principle typicaiiy

forms the local basis upon which any (dynamitai) variationai principle is established

(see Lemma 5.7 and [NS. Sect. 31). The Gibbs state relative to a self-adjoint element

h in a finite-dimensional C*-algebra B is defined to be the state on B with the density

rnacrix eh(TrBeh)-' Mth respect to the trace TrB.

Lemma 4.14. (see [BR-. Prop. 62-22]) Let A be a finite-dinaensional C'-algebm and

h a seif-adjoint element of A. Then the Gibbs state is the unique state on B maxirnizing

the functional

u - S(w) f w(h)

on the date space of B, and the mazimum uafue is log ~ ~ e ~ .

Proposition 4.15. If h E il,, and a is a O-invariant state on At then

Proof. Let .r; : A - B ( q be a faithful representation of A- Then Pe(h) = Pe(hlr) and

hh,(0) = hh,(0,r).

Suppose w E E(q.r). Let R be a set in Pf(A) containing h, 6 > 0, and n E N. If (4: +. B) E CPA(?r. flg-', 6) then by Lemma 4.14

2 S(w 0 Ilr) + n(w O ir)(h) - n6

= S(w O dt) + no(h) - nd.

and so Po(h. r ) > hh,(9. T. w ) + a(h). Taking the supremum over iii E f(u? xj yields Pe(h, T ) 2 hh,(8, T) + a@). thus

establishing the proposition. O

[a view of Proposition 4.16 and Proposition 4.11 and the fact that the Sauvageot- Thoirvenot entropy majorizes the CNT entropy, we immediately obtain the folionring

corollary generdizing a result of Dykema, who showcd that the Voicuiescu-Brown en-

tropy mnjorizes the CNT entrapy in the exact case [Dyk. Prop. 91.

Corollary 4.16. 1'0 E ..lut(A) and a is a B-znvar+ant state on A, then the varia-

tianal s'nequality of the previous proposition alsa holds when hh,(O) is ~ p l a c e d by the

Sauuageot-Thouuenot or CrVT entmpy.

CHAPTER 5

The variational principle

The principal god of this chapter is to establish a variational principle which ex-

tends that of Neshveyev and Stormer [NS? Thm. 3-31 beyond the "zero-climensional"

situation of approximate finitedimensionality. Our "continuous" version of Neshveyev

and Stormer's result demands the same type of Iocal eventud Abelianness, but relues

the assumption of finite-dimensional approxirnability, which is only required to hold within a weak operator closure. As Rie NiII propose at the end of the chapter. the proof

demonstrates that the C*-dgebraic notion of quasidiagondity provides the most a p p r s

priate framework for Starmer and Neshveyev's argument and our elaboration thereof.

As a warm-up, we will first establiçh the variationai principle in the specid case of

u.c.p. maps with zero Voiculescu-Brown entropy.

Proposition 5.1. Let .A be a unital ezact Cg-algebm, 8 : A -+ A a .u.c.p. rnap, and

h E A,,. Suppose that ht(8) = O . Then

Po(h) = sup [h,(B) + a(h)] = sup a(h), u O

where h,(B) zs the CNT? Sawuageot-Thouuenot, o r Local state approximation entropy

and the supremum is taken over al6 B-invariant dates a.

Pmof. First note that for any &invariant state o we have h,(8) 5 ht(8) by Corollary

4.16 and hence h,(B) = O. Thus by Proposition 4.15 and Coroiiary 4.16 it suffices to

show that Pe(h) 5 sup, u(h). Let il € P f(A) and 6 > O. To establish the desired inequality we wiil find a

B-invariant state a on A such that

Since ht(B) = O we can find a triple (cj,, &, 3,) E CPA(L, O:-', 6) for each n E N such

that I

lim -logrank(B,) = 0. n-ca n

We can then find an increasing sequence {nk)kEpl of positive integers such that

For each k E M let (JI, be the Gibbs state on Bnk relative to h, that is, the state

tvith density operator ( ~ r ~ ~ , e ~ ~ k ( ~ ~ ~ - ' ~ ) - ~ e ~ ~ t ( ~ ~ ~ ~ ~ ) with respect to TrBnk. Then by Lernma -4.14

iog~re,,, = S(q) + u*($., (h8k-L)). nk-1

Set C T ~ = '- Ci=-, m d let cr be a weak' lïmit point of the sequence {rk}k.rr- n k

By passing to a subsequence if necessary Ive may assume that limk,, uk(x) = u(x) for

di x r -4. Note that a is B-invariant. for if x E A then given 6 > O there is an k E N aich that lok(z) - CT(Z)~ c ++ Iok(B(x)) -=(8(1))1 < fe, and $ 1 ~ 1 < &. whence

Since O <_ S(wk) 5: log ranb(Bn,) for d l k E W we have

and so we conclude that Po(h) 5 sup, o(h). O

5.Vë point out that Proposition 5.1 appiies to the situation in which A is finite-

dimensional. That the Voiculestni-Brown entropy ht(0) is zero in this case for any

O E Aut(A) is easiiy deduced fiom the fact that if s : A -t B(3C) is a faithful *-

representation then for any R f l'/(A) and ii > O the triple (idA, sr, A) is an element

of CPA(sr. RI 6). We also note that if 0 is an inductive Iimit u.c.p. endomorphism with

respect to a sequence AL c A2 c - - - of C*-subalgebras of A with dense union and 8Iti, has zero Voiculescu-Brown entropy for every n E W, then by [Brl, Prop. 3.141 0 has

zero Voicuiescu-Brown entropy. Thus, in particular, inductive limit automorphisms of

.AE' aIgebras in the sense of VoicuIescu Fol] fa11 within the scope of Proposition 5.1.

Before stating the definition of the terms &FA aiid FA, ive introduce a couple of

pieces of terminology and notation. If ,Y and Y are subsets of a C*-aigebra and d > O

then ive say that Y approximately contains X to within 6 if for every x E X there

is a y E Y such that [lx - y11 < b. If A is a unital C8-algebra, 0 E Aut(A), and C c A is a C*aubnlgbra, then for m E M we denote by the C*-subaigebra of A generated

by U" 0". and we denote by Colm the C*-subalgebra of A generated by UkEz 0%-

Definition 5.2. Let -4 be a unitalC*-algebra and 8 an automorphism of A. We say that

a vnital C8-subalgebra C c -4 is &FA if @C commutes with C for k # O and for euery

R f P f(C) and b > O there is a faithful unital representation îr : Ce." - 'B(X) , an

alrtornorphism of a(C'+p)" which extends îro û O T - ' I , ( ~ ~ . ~ ) , and a jhite-dimensional

Cm-subolgebra N c îr(C)Ir approzimately containing îr(R) to vithin b srtch that there

ezists a nonnal conditional eqectation of r ( ~ ' . ~ ) " onto (No.")" which maps îr(C)"

ont0 N and commutes with 8.

Definition 5.3. Let A be a unital C'-algebm and û an automorphism of A. W e say

that 0 is FA if for evenj R E P f ( A ) and b > O there m'sts a unital Ca-subalgebm

C C .4 approximately containing R to within 6 and an r E N such that Ce?"-' k Bk-EI4 for al1 k 2 r .

If 6 is FA then -4 must be quasidiagonal, as we show in the next proposition. W e recdi

that a separable set X of operators in B(!J-C) is said to be quasidiagonal if there is an increasing sequence {p,), of finite-rank projections in %(X) such that p, -, 1 strongly

and llpnx - q n l l -. O for di x E X. A Cm-algebra A is saîd to be quasidiagonal i i there is a faithful representation î r : A - 8(3C) such that ïr(A) is a quasidiagonal

set of operators. CVe wii i use the foiiowing characterization of quasidiagonality due to

Voiculescu. (For a survey of quasidiagonality see [Br2]).

Proposition 5.4. [VOS] A C*-algebm A 2s quasidiagonal if and odg if for euery

il E P f ( A j and 6 > O there is a bite-dimensionai C*-algebnt B and a contmdive

conrpletely positiue map 4 : A + B such that I I ~ ~ ( x ) I I > 11x11 - 8 for al1 x E Q and

I[Q(q) - d(x)4(y)II < 6 for al1 x: y E R.

Proposition 5.5. If a unital C*-algebra A adnaits an FA automorphism then A i s

quasidiagonal and exact.

Proof. Let 9 be an FA automorphism of -4, and let fl E P f ( A ) and 6 > O. With a

view to verifying the conditions characterizhg quasidiagonalil in Proposition 5.4 we

c m assume that the elements of R have norm less than one. Since 8 is FA R E P f (A) is approximately contained to within d in a Cinite subset fi of a unital Ca-subdgebra

C C -4 such that C0vk is @-FA for some k E N. Let a : c0vk -+ 'B(9-C) be a faithful

unital representation such that z(R) is approximately contained to within S in a finite-

dimensional C-subalgebra B c IB(3C). Let E be a conditional expectation from IB(!H)

onto B. By ,Ameson's extension theorem (Theorem 1.3) E O a extends to a unital

compIetely positive map d : A -, B. ifx, y E R then choosing 5, r j E fi with 112-511 < 6. Ily - YI1 C S. and 11?(1? Ili11 5 1. we have

'and

and so we conclude that A is quasidiagonal by Proposition 5.4.

To see that A is enct: let L : A + B(K) be a faithful unital representation, and Iet R E PJ(A) and S > O. By the definition of FA there is a unital Cs-subalgebra

C C il with a finite subset RI c C approximately containing fl to within 8 aod a

unitai representation ;r : C 4 IB(X), such that 'B(3C) admits a finite-dimeosional C'- aigebra 1V which approximately contains ir(R1) to within 6. Let E : $(X) + N +Cl be a conditional expectation, as guaranteed to exist by Aweson's extension theorem

(Theorem 1.3). Again using h e s o n ' s extension theorem we c m extend Ecsr to a u.c.p-

map 4 : A - N +Cl and L 0 ~ r - l : n(C) -+ %(X) to a u.c.p. map $ : IB(3C) --, $ ( X ) . Then, denoting by CY the inclusion N + Cl - B(W, it is readily checked that

for al1 x f Q. We conclude that A is nuclearly embeddabIe and hence evact by Theorem 1.7. 0

Our aim is to estabhh a -ational principle for FA automorphisms of unital evact

C*-dgebras. Proposition 5.5 hints that quasidiagonality is the appropriate fiarnework

for approaching this problem, and indeed the discussion at the end of the chapter will bear this out, but in the meantime we state and prove the theorem working with our

usual emct definition of pressure. Following the approach of Neshveyev and Stormer

we first treat the situation of local one-iterate commutativity (Lemmas 5.?), then re-

duce the local eventually commutative scenario to this case by considering powers of 0 (Lemma 5.8), and finally pass from the local (i.e., with respect to finite subsets O € the

algebra) CO the global to complete the proof. ,As d l become clear in the proofs Ieading

to the variational principle, the FA property means that the dynarnics c m be locdy

approximated in norm by roots of tensor product shifts in a weak cperator closure.

The crucial proper l of tensor product shift systems, to which ive appeaI i ~ i Lemma

5.7, is that the CNT entropy is a lower semicontinuous function of the state space of

shift-invariant maximal Abelian subalgebras.

CVe first recaU the definition of the CNT entropy for an automorphism 8 of a unital

C*-aigebra -4 relative to a O-invariant state a [CNT]. Let C be a finite-dimensiond

commutative C'dgebra and P : A -. C a unital positive Linear map (note that P is automaticdy completely positive by [Tale, Cor. 3.51). If p l , .. . ?pr are the minimal

projectioils of C then there are states 0 1 ~ . . . .ar on A such thac P ( x ) = ~i (x)pi for al1 x E .A: so that

r

which espresses o as a convex sums of states. With S(-l -) denoting the relative entropy

of Ar& [Aral, ive set r

and define the entropy defect

Nom suppose that Bi,. . . , B, are finite-dimensional Cv-algebras and r k : Bk - A u.c.p. maps for i = 1,. . . , n. An Abelian mode1 for (A, u, (-/k)E=L) is a quadruple

(. (C)r=lt P, j ~ ) where C is a finite-dimensional commutative C8-algebra with measure

p, P r A - C ïs a unitai positive iinear map such that p O P = a, and Ci, -. . ,Cn are C8-subalgebras of C which generate C. Denoting by Ek : C -+ CI, the pinvariant

conditional evpectation. we define the entropy of (C, (C)r!l, P:p) to be

We then define H,(-il,.. . , -1,) to be the supremum of the entropies over all Abelian

models for (A. o. (T~):!~). We neyt incorporate the dynamics by defining, for every

finite-dimensional C*-algebra B and u.c.p. map 7 : B + A,

with the existence of the limit following from by [CNT, Props. III.6.b and III.6.d]. Finally the CNT entropy h,(B) is defined to be the supremum of h,o(-y) over d l

pairs (B.-t). The iollowing lemma captures a property of the CNT entropy which we will need

for the proof of Lemma 5.7.

Lemma 5.6. Let -4 be a unital C8-algebm and 0 : A + A a u.c.p. map. Let D be

a unital C*-algebra. q : D - D a u.c.p. mapl and 9 : A -+ D a surjective unital

completely positive map which intertwines 6 and q, i.e., 9 o B = q o 4. Suppose there

ezists a unital completely positive map a : D - =I such that 9 o a = idD, as is the

case for instance if .-I is separable and nuclear and 9 is a *-homomorphzsm (by the

Choi-Effms [.ifling theorem), or if CP is a conditional qectation onto a C*-subalgebra,

in which case we can take a to be the inclusion. Then for any q-invariant state c on

D we have h,(q) 5 h,,+(B).

Proof. Let B be a finitedimensional C*-algebra and 7 : B - D a u.c.p. map.

Given n E W, let (C. (Ck)FEL: P,p) be an AbeIian model for (D: o, ($ o y);::). Then

(C. (Ck)rel, Po@. p) is an Abelian model for (A, mQ, ( 8 h o a o y ) ~ ! ~ ) , since Po@ : A -r C is a positive linear map and p a (P o a) = u o ch. Since 9 o gk = 7k o cP and 9 o a = i dD

we also have k @ ~ e ~ o a ~ ~ = ~ 0 7

for k = O?.. . , n - 1, and so the entropies of the two Abelian models coincide. W e

conclude that H,(y, q o y, . . . , 9n-I a ?) Kc,+ (a o y, 8 o a o 7: . . . : en-' o a o y), so that

h , d c 0 5 L + . e ( a 0 7) and hence h d d < hm+(e)-

S

Lemma 5.7. Let A be a unital ezact C*-algebnz? O E Aut(A): and h E Asa. Let R be an

eiement of P f (A) containing h and let 6 > O? and let C be a unital 8-FA C*-subalgebm

of A containing R such that CO." = A. Then the= &sts a O-inuariant state CT on A such that

Pe(h. R. 26) 5 hm($) + r ( h ) .

PmoJ By the definition of O-F-4 there is a faithful representation ;r : A - 'B(K), an

automorphism 8 of ~ ( 4 ) " which extends noO~n-~I,(~e.,), and a finite-dimensionai C*-

subaigebra 8' c n(C)" approximately containing n(R) to within S such that there exists

a normal conditionai evpectation E of r(A)" ont0 (NRm)" which maps n(C)" ont0 !Y and cornmutes with 8. CVe identik A with its image under îr. For each k E M let Ek be

a conditionai expectation frorn n(Co*")" ont0 Nëtk which factors through E. Now for

each 6 E N if p is the unit of of ( N ~ ~ ) ' ' and u is a state on (1 - p)Xf(l - p ) then it is

easily seen that x I- Ek(x) + ~ ( ( 1 - p)x(l - p))(l - p ) defines a conditionai expectation

frorn T ( C ~ + ~ ) " ont0 (N + while if q is the unit of ((IV + ~ 1 ) ~ ~ ~ ) " and v is a 8- invariant normal state on ( 1 -q)il"(l -q) and then x - E(x) +v((l -q)x(l -q))(l - q )

defines a normal conditional expectation of A'' onto ((N + CI)^.")" which cornmutes

with 8. CVe rnay thus assume that N is a unitai C-subdgebra of il" by replacing it

with N + Cl if necessary.

Let T be the right shift automorphism on the C'-algebra Na'. Since B ~ ( c " ) corn-

mutes with C" for k # O. by [Takl. Prop. 4.71 there is a *-homomorphism @ : NaZ -+ &'

such that (bINa~ = BkEF(PIN) for every Bnite subset F C Z, and cP intertwines T with 8. that is. <POT = go@. For each n E L let Bn be the Enite-dimensiond C*-aigebra

- Ne*" = @(1~8[~J" '~1) , and set B = Nevm = @(N@z). Note that for ail n E N we have

E[B, = En, for if x E B, then E(x) = E(E,(x)) = E,(x). Following the proof of WS, Lemma 3.31 we next show the &stence of a ëlB-invariant

state w on B such that

1 iim sup - log nB,, eEn(hi-L) 5 hW (gl B) + (u O E) (h) .

n-a: ri

For n E Pi let <,, be the Gibbs state on B, with respect to E,(~Ô-'), that is, the state

with density operator -

eEn(ht-L 1) l e h ( h ~ - ' l

reiative to Tr8,. Identifying ( i ~ @ [ ~ ~ - ~ ] ) @ " with N@" we define on N@= the T-invariant

state v, = (G ~ < h l ~ ~ p . , . - q ) ~ " + Then h,,(Tn) = S(S;r 09l ,~ep.~- i l ) = S(<n)- Define the T-invariant state Y, = un o T ~ . Using the concavity, adclitivity, and covariance

of the CNT entropy we have

By Lemma 4.14 we have

Since cach En factors through E and E commutes with ê ive have

where (Eo(h)) is regarded in the k t mo expressions as an element of 8 anci from the

second expression ontvard as an eIement in the zeroeth tensor product factor in Ma'. We thus now have

Let 2 be a ive& limit point of the sequence Since each c ~ , is T-invariant, so

is 2. Let LU be a commutative mbalgebra of N of the same rank wkch contains E(h).

Since each En factors through E: ICI@= lies in the centralizer of each un, and so by

Corollary mI.8 and Theorem W.4 of [CW] we see that hW, (T) = bn(TIM@z). But this Iast term coïncides with the KohogorovSinai entropy of the underlying shift (see

[OP? Chap. which is an upper semicontinuous function of the space of dynamicaliy

invariant states by [DGS, Prop, 16-81. Since kPZ aIso centralizes r;i we condude by

this upper semicontinuity and Lemma 5.6 that

N e ~ t we show that ker4 c kerW so that ive may induce a state on B. It suffices by the density of U,,, i~@[~,"l in NB' to show that, for al1 m < n, ker QIIvob.nl C

k e ~ G l , ~ a [ ~ . , ~ for aii m < n. If x E ker41iva[m.nl then T ( x ) E k e ~ @ l ~ ~ ~ ( ~ + k , ~ + k ~ C

ker91,vap.pl for k = -m. -m + 1.. . . , p - n and p 2 m. so that

from which we conclude that G(r) = O. We may therefore define a state LJ on B via

the equation

y(@(.)) = S(3)

for al1 z E Np". Since h , ( 8 [ ~ ) and h;(T) are determineci by the Kolmogorov-Sinai entropies of the repective restrictions of # and T to @ ( 1 ~ 1 @ ~ ) and i P Z (see C o r o l l q

VIII.8 and Theorem VIL4 of [CNTI) ive must have ~ ( 8 1 ~ ) = k . (T) , and so we obtain

the inequality (*) as desired.

Let u be the state on A given by w o Ela4. Since 80 E O #-' = E nre have

that is. u is &invariant. Now for each n E NI letting L, : 3, + 'B(9-C) be the inclusion

we have (En, L,, B,) E CP~(;r,fl~-',28)~ for if z E O:-' then writing a: = for s o m e y ~ R a n d O I k ' n - l w e c a n h d a z ~ B o = N s u c h t h a t IIz-yll Cdwhence

Appealing to (*) we therefore get

1 Pe(h, R, 26) < limîup - log ?1.~,,8"(~~-~) < h(#l~) + ~(h). n* n

Thus to complete the proof we need only show that hW(&) 5 h,(e), which we now

proceed to do.

For each n E N we identiiy B, with a C*-subaigebra of a full matci.. algebra :\frank(Bn,(@) of the same rank and let -,, : ?Ltrmk(&)(@) -) B be the composition of

a norm-one projection of ibfmnk(&)(C) ont0 B, and the inclusion of B, into B. By

[CNT: Thm. V.21 ive have

and JO it suffices CO show that h,(O) 3 h, , j I , ( - tn) for every n E N. So let n E N, and let K : B, + -4'' be the incliision. By [CE? Lemma 3.11 the map O : L~&,~(~, , ) (A") +

L(LLI,,~~(B,,) (@), -4") @en by

for al1 x = [xij] E ~ ~ ~ ~ ( ~ , , ) ( r l " ) and b = [bij] f i\frrinkIBn)(@) is an order isomorphism with respect to the cone (C)! A " ) ~ of completely positive maps. Since

~ L [ r ~ ~ ( ~ ~ ) ( - i l ) is strongly dense in 1b&,,,~(&)(-4~') K a p l a n s w density theorem can be

appliecl to obtain a bounded net ( x ~ ) ~ of self-adjoint elements in it.Irmk(Bn)(A) which 1

converges strongly to (€FL(~ O y,)) 2. Then (G}x is a net of positive eIements con-

verging u-weakly to O-'(rl O 7,): and thus setting px = O(<) ive obtain a net { p x ) x of

completely positive maps from lbfmnk(Bnl (C) to A which converges point-a-weakly to

K O 7,. Since E is normal E O px converges point-u-weakly to E O K O 7, = -~n, and so

limA ( 1 E o ph - afn(lc = O . Then by [CNT, Th. VI.31 we have

and since the second half of the proof of Lemma 5.6 yields for each X

we obtain

as desired.

For the r a t of this subsection expressions of the form hi-' and fia-' will refer to

sums (resp. unions) of iterates under the automorphkm 8.

Lemma 5.8. Let A be a unilal exact C*-algebm, 8 E Aut(A), and h E A,,. Let R be

an element of P f (-4) containing h and C a Cs-subalgebm of A containing R. Suppose that there exists an r E W such that 0'15' cornmutes with C for al1 k 2 r . Then

Proof. The proof of WS. Lemma 3-41 works with minor modifications in this generdized

setting, the argument running as foilows. Let b > O m d choose an s E N sucb that

for aii r E R. Let rr E M and set rn = 151. Yote chat (c'~')""*" iS Os+'-invariant.

Pick a tripIe

such that

Using Amson's extension thmrem we extend 6, to a u.c.p. map a, from A to LV*. Let B, be the direct s u m of s copies of Nm, Dehe the unital cornpletely positive maps

Om : -4 - Bm and zlt, : B, - A by

for aii x E A and - d

for di (yl,. . . ,ys) f Bm. Now if z E 0:-l then at most r - 1 members of the set

b. d(z), . . . ; es-l(x)) are not contained in K=; @+') (Q:-~), md so we have

Next observe that for each i = O.. . . . s - 1 the sets Ji = [i, i + n - 11 and J = u ~ = ~ ~ ~ ( s + r ) , j ( s + r ) + SI have ~ymmetrk difference of cardinality nt most mr + 2s. and t hus

by the Peierls-Bogoliubov inequality [OP, Cor. 3.151. Hence

Recaiiing the dependence of m on n we thus have

1 r - - ha-^, QT'. 6) + ;r;([hll, s + r

and since this holds for a.ii sufticientIy Iarge s we obtain the desired inequality. Ci

Theorem 5.9. If -4 is a unital nuclear C*-algebru, 0 is an FA automorphism of il,

and h E A,,, then

where the supremum is taken ouer ail 0-inuanant dates o.

Proof. It suffices to establish the inequality Po(h) 5 sup, [hO(0)+a(h)], since the reverse ineqiiality follows from Corollary 4.16. Let r! be an element of P f (A) containing h and

let S be a positive number l e s than 1. To obtain the desired inequality we need only show the existence of a 0-invariant state a on -4 such that

and furthermore. by appealing to Proposition 2.5, the observation that P ~ ( L , h, Q', 26) 2 P e ( ~ . h , R, 5) if il' approximately contains il to nrithin S. and the fact that O is FA. vie rnay assume that R is contained in a C*-subdgebra C c il such that, for some r f NI ~ a . k - r is @-FA foc al1 k 2 r. By Lemma 5.8 we ca~i find an s E N such that

and r(s + r)-L[lhll < S. By Lemma 5.7 there exiçts s Os+'-invariant state G on ( c ~ . ~ ) ~ ' + ~ * ~ such t hat

By [NS, Lemma 3.51 W extends to a d'+-invariant state w on A such that

Define the &invariant state t~ = & rzcL u O dk- Using the addivia afünity. and covariance [CNT, VII.51 of the CNT entropy we have

Noting that

we obtain

completing the proof. O

We obtain the foUowing tnro coroUaries generalizing Corollaries 3.6 and 3.7. respec- tively. of PSI.

Corollary 5.10. If -4 is a unital nuclear C-algebm and 0 is an E4 automorphism of

-4 then the pressure is a conuex function of h.

Proo/. The function h t, h,(B) + a(h) is affine for each &invariant state a on A, and

thus if h l? h:! E .1,, and X E [O, 11 then by Theorem 5.9

where the suprema are taken over all &invariant states.

Corollary 5.11. If Ai is a unital nuclear CD-algebm and Oi an i?4 automorphism of

Ai for i = 1.2 then

ht(e, 8 e,) = q e L ) + ht(e2).

Proof. The CNT is superadditive [SV, Lemrna 3-41 and the Voiculescu-Brown entropy

is subadditive p r l , Prop. 2-11 on tensor products, and so for any &invariant states ci

for i = 1.2 we have

wïth the middle inequality foIiowing Crom Proposition 4.16. We now appeaI to the

theorem, mhich in this case asserts that ht(&) is the supremum of h,(Bi) over aI1 &invariant states for each i = 1,2. Dl

It is not known if the Voicdescu-Brown entropy is additive on tensor products in

general (there is a counterexmple, however, for the CWT entropy [NST]). Another

important open problem asks whether the the Voiculescu-Brown entropy decreases in

quotients. We can answer this in the &mative in the nuclear setting for quotients of

FA automorphisms which are again FA using Lemma 5.6 and Theorem 5.9. We can

obtain a more easily verifiable set of hypotheses by specializing the context of Theorem

5.9 according to the description in the fotiowing chapter, where this restricted case is

recorded as Proposition 6.13.

To conclude this chapter we argue that the conciusion of Theorern 5.9 by can be strengthened by substituting a suitabie quasidiagond-exct definition of pressure for

the exact definition. For the definition of qusiçidiagonality see the discussion prior to Proposition 5.4. We first note the folIowing r m l t which was established by Dadariat

for faithful essential representations P a ] and then extended to generd representations

by Brown [Br31.

Proposition 5.12. [Da][Br3] If A 2s an exact C - a l g e 6 m and n : rl - 8(X) is a

sepamble representation. then P ( A ) is a quasidiagonal set of operators if and only if

for e v e q R E P f ( A ) and d > O there i.s a finite-dimensional Cn-subalgebra B c B(K) which appmzimately contains ;r(R) to within 6.

Wit h t his result we can estabbsh a 1 0 4 approximation characterization of quasidiag-

onal exact Cn-algebras which is idedIy suited to our situation. We wiii present the

unital version, keeping within the general Framework of the chapter.

Proposition 5.13. A unital sepamble C - a l g e b m il is quasidiagonal and exact if and

only if for any unital complete order embedding L : A -+ D into a unital injective

C*-algebru D and any R E Pf(A) and S > O t h e z &ts a finite-dimensional C- algebm B and unital completely positive maps # : A - B and $ : B -. D such that

II(@ O o)(x) - L(x)]~ < S for ail x E R and I[P(q) - #(x)g5(y)ll < S for al1 x, y E R.

Proof. (+=) That A is quasidiagond foUows immediateiy from Proposition 5.4: and A is evidently nuclearly embeddable (Defmition L.6) and hence exact by Theorem 1.7.

(=+) Let D be a unital injective C-algebra and L : A -+ D a unitai compIete order

embedding. Let R E P f (A) and 6 > O. Set K = mau{ilxll : x E fi) + 1. Since A is quasidiagonal there is by definition a faithful representation T : A + 'B(3C) such that a(d) is a quasidiagonal set of operators. We may assume that 9C is separable by

[BrL Lemma 15-31. By Proposition 5.12 there is a finite-dimensionai C'-subalgebra

B c 8(3C) whicù approximateiy contains 40) to tvithin &5. We may assume that B is a unital subdgebrrt by replacing ic wit h B +Cl if necessary. Let P be an idempotent

~1.c.p. map from BI#) onto B, By the injectivitg of D we can extend the map L O ?r-'

on a(--!) to a u.ç.p. map -/ : 'E(3-C) - D. If x. y E Tt then dioosing 5, y' E B with

115 - ii(r)ll < -&6 and Ilfi - r(g)ll < $6 we have

and

Thiis taking O = P o sr and = y 1~ we obtain the result. O

Thiis if A is a unital C-algebra wbich is quasidiagond and e x c t and h f A,,: then,

as we hinted in the ciiscussion prior to Proposition 2.N, for a given unitd compIete

order embeddiig t : A -. D into a unita1 injective C'-algebra D and any 0 E P f ( A ) and 8 > O we can defiiie cPAQ~(L,~,~) as the collection of triples (c$:$! B) such that

B is a finite-dimensional C'-aIgebra and 4 : A - B and @ : B - A are u,c.p. maps

with I[(d O rb)(x) - X I [ < 6 for al1 r f R and Il$(q) - q5(x)qj(y)[l < d for aU r , y E fl, and thcn set

We can then d e h e pBE(~, h_fl:6), P:'(L, h, Cl): and P ? ~ ( L ? h) by taking the appropri-

ate suprema as in the Eyact and nuclear cases. The prwf of Proposition 2.2 shows that

ai i of these quantities are independent of tt so that we rnay defme the qunsidiagond-

exact pressure ~ : ~ ( h ) as P:~(L. h) for ûny L.

Now suppose that A is a separable unitd quasidiagond evact idgebra and the

hypotheses of Lemma 5.7 hold. CVe daim that the proof Lemma 5.7 shows in fact that

wbere C = mxc{llxll : x E S?) + 1 and x : A - !B(!H) is the faithhil representation o b

tained from the definition of &FA. Indeed suppose that B is a unitd finitedimensionai

C-subalgebra of B(X) which approximately contains x(~:-~) to nrithin &b and P is an idempotent u.c.p. map from %(X) onto B. Let rc : B - 'B(3C) be the inclusion.

Noting that C bounds the norm of elements of for every n E N. for a y x. y E S? choosing 2.c E 3 Nith with 111 -x(x)ll < $6 and Ili - x(y)II < $6 we then have

and

= CS.

Thus ( P a T, K? B) E C P A Q ~ ( ~ T : !2:-', C4)- In view of this construction, we notv see by

inspecting the proof of Lemma 5.7 that (r) holds. To see that the quasidiagonal-euact

andogue of Lemma 5.8 also holds it suffices to observe that a direct sum of u.c.p. maps which are approximatsly multiplicative to nrithin 6 on a bite set is itseif approximately

multiplicative to tvithin 6 on the same set, as such a direct sum construction provides

the basis for estabMing the desired inequality. Theorem 5.9 then goes through easiiy

if we replace 4 ( h ) with ~ ; ~ ( h ) .

In Theorem 5.9 we may &O substitute the NF definition of pressure for the emct

definition. A separable Cm-algebra A is said to be an NF algebra if it is isomorphic to

an inductive liniit of a gneralized inductive system of finite-dimensionai C*-algebras

Mth completely positive contractive connecting maps [BK]. Condition (iü) in the foiIow-

ing proposition yields a locd characterization for NF algebras that enables us to d e h e

a notion of NF approximation pressure in anaiogy to the exact and quasidiagond-exact

cases.

Proposition 5.14. [BK] For a separable C*-algebm A the following are equiualent.

(i) A is an IVF algebra.

(ii) A is nuclear and quasidiagonal.

(iii) Giuen any finite subset Z! of -4 and 6 > O the= ezists a finite-dimensional Ca- algebra B and completely positive contractions 4 : il -, B and $ : B -+ il such

that Il(zhod)(z) -211 < b for a11 2 E 12 and IlO(xy) -r$(z)~$(y)ll < 6 for al1 x. y E R.

If -4 is unital then the maps in condition (iii) may be taken to be unitd. in the

nonseparable case we may take condition (iii) as the definition of an NF algebra. Notice

that imposing the the assumption of quasidiagonaiity in both the nuclear and exact

cases simply changes the usual local approximation conditions by iurther requiring that

the u.c.p. maps into the finite-dimensional dgebra be approximately multiplicative to

within 6 on R. So if A is an NF algebra then for R f P f (A) and 5 > O we c m define

C P A ~ ~ ( A . R. 5) as the collection of triples (4, $, 5) such that B is a finite-diniensional

Cg-algebras and Q : A - B and + : B - A are u.c.p. maps with II(+ O 4)(x) - 211 < 6 for al1 3: E R and Ilci(xy) - O(x)o(y)ll < 5 for a . x, ÿ E S2, and then set

and define ptT(h, R, 6). PtF(h, R). and pBF(h) by taking the appropriate suprema as

in the nuclear, euact, and quasidiagonal-e-act cases.

Now suppose that A is a unitai NF algebra and the hypotheses of Lemma 5.7 hold.

Similariy to the case of quasidiagonal eyact C*-idgebras above, we argue that the proof

Lemma 5.7 shows that

where C = ma..{llx[l : x E S I ) + 1. Suppose that sr : A -, B(K ) is a faithful unital

representation. IdentiFying A with its image under îtr suppose that B is a unital fuiite-

dimenrional Ce-subdgebra of 23(w wwhich approxhately contains 0:-L to within &6

and P is an idempotent u.c.p. map fiom 'B(2-C) ont0 B. Let rc : B -t $(X) be the inclusion. Since A is uuclear by Proposition 5-14, there is a finite-dimensional C*- algebra N and u.c.p. maps # : A - N and $ : N -i A such that Il($ O C#J)(z) - zll < $6 for aii x E Q. By Arveson's extension theorem we c m extend C#J to a u.c.p. map

d : 'B(K) - N. Since C bounds the noms of elements of R G - ~ _ for any x, y E R choosing 2. t j E B with 112 - n(x) I I < &S and III - n(y) 11 < &S we then have

and

= CS.

Tbus ( P O 7. I$ O 4 O K, C ) E C P A " ~ ( A . R,"-'. Cd). Inspecting the proof of Lemrna 5.1

we nom see by virtue of this construction that (**) holds. Lemma 5.8 aiso holds for

the sanie remon as in the quasidiagonal-exact case, and so Theorem 5.9 remains valid

upon replacing Pe(h) with PtF(h). In fact. the quasidiagonal-exact and N F approximation pressures agree in the nu-

clear caset a s can be seen by adapting the proof of Proposition 2.19.

CHAPTER 6

Weakly AF C*-algebras and tracially FA automorphisms

In order to simplify the verification of the hypotheses of Theorem 5.9 for certain exam-

ples we can try to specialize the definition of FA automorphisms by disentangling to a

greater d e p e the topological component (norm appro-uimation by finite-dimensionai

aIgebras) frorn the mesure-theoretic component (existence of suitabIe conditionai ex-

pectations), If WC assume the existence of a dj.namicaily-invariant traciai state? for

instance, then a theorem of Takesaki [Taid, Prop. 2.361 wiU take care of the condi-

tional expectations. To handle the finite-dimensional approximation separately we are

naturaiiy lead to the concept of a weakly AF C*-aigebra.

Definition 6.1. -4 separable C*-algebra A is said to be weakly AF B/ for evey

representation ;r : .4 -* 'B(2-C) and evey fl E P /(A) and S > O there is a finite-

dimensional Cs-subalgebm of 7r(.4)" which appmximatelg contains T(R) to within 6.

It is not surprising that weakly AF C-algebras appear in an initiai search for a "contin-

uous" noncommutative variationai principle, since their definition juxtaposes topoiogi-

cal (norm approximation by finitedimensional algebras) and mesure-theoretic (within

a weak operator closure) notions. as does a postulated equality between pressure and

the supremum of the free energies. Before specializing the definition of FA automor-

phisms to a notion of YracialIy FA" using weakly AF Ca-algebras. we wilI spend some

tirne estabiishing several properties of the latter.

Proposition 6.2. A C*-algebm A is ueakly AP if und only if Jar every $2 E P f (A ) and

5 > O there is a jnite-dimensional Cs-subalgebra of A*' which approximately contaiw

R to vrithin B.

Pmoj To check whether a (7-algebra is weakly AF it is enough to verify the finite-

dimensional approximation condition for nondegenerate repraentations, since for a

degenerate representation we may always cut-dom to a nondegeneratc one by a prw jection in the weak operator closure of A which acts as a unit on the image of A. The proposition then foflows from [Ped, Thm. 3-7-71, which asserts that any nonde- generate representation sr : A + %(X) evtends to a n-weakly continuous surjective

Remark 6.3. Proposition 6.2 shows that foc separable C'-aigebras we need only

consider separable repreçentations in Definition 6.11 since s von Neumann dgebra with

separable precluai is *-isomorphic to a von Neumann dgebra acting on a separable

Hilbert space.

A separable C*-algebra A is said to be strongly quasidiagond if for every sepa- rable representation ir : .4 + 'B(w the set T(A) is quasidiagond.

Proposition 6.4. A sepamble weakly AF C*-algebra is nvclear and stnmgly quasidi-

agonal.

Pro06 Let A be a weakly AF C*-dgebra. and let {Rk)kErr be an increasing sequence

of finite subsets of A nith dense union. For each k E N let Bk be a finite-dimensionai

C'-algebra of A*" which approxirnately contins flk to within i, and let pk : A - B be a nom-one projection. iis guaranteed ta exist by Arveson-s ~xtension theorem. Let

KC : Br, -, AInk (63) be a unitai embedding into a mat~ix aigebra and 7 : iVnk(C) - Bk a conditional expectation. Now if x E A and E > O then there is an k E W such that

112 - rkll < E . and so

Thus ive obtain a sequence

of compositions of unitai completely positive maps such that (tk O yk o iq o pk j (x ) = pk(x ) + x in nom. We may nonr apply the argument of Choi and Eifros on p. 72 of

[CE21 to conchde that the identity map on A is the point-norm closure of maps of the

fom # O A - Mn(C) - A!

that isl A is nuclear.

To show that A is strongly quasidiagonal, let n : A -r B(9-C) be a separabLe represen-

tation. Then for every fl E P f (A) and 6 > O there is a finite-dîmensiond C*-subalgebra

of x(A) which approximately contains T(Q) to within 6. This implies by [Br3, Thm.

1.21 that a(A) is a quasidiagonal set of operators. We thus conclude that A is strongly

quasidiagonal. O

Proposition 6.5. Finite-dimensionaI Cs-algebm and sepamble comrnutatiue C*- algeb.ms are weakly -4 F.

Proof. Every finite-dimensional C*-algebra is isornorphic to a direct sum of rnatris

alqebras and thus clearly weakly M. Consider a separable commutative C*-algebra At which by the Gelfand-Naimark

theoreni niay be assumecl to be Co(X) for some locdy compact Hausdorff space X. if R E P f (A) and 8 > O then we c m find a finite Borel partition {PL, . . . P,) of X such

that, for every f E R and k = 1.. ..' n, Jj(x) - f(y)f < d for ail x, y E ,Yk. Now under

the sup norm the Borel functions over X form a C'-aigebra B ( X ) containing A as a

C*-subalgcbra. .and any separable representation of A &ends to a representation of

B ( X ) since the former is unitarily equivdent to a direct s u m of representations given by

multiplication on L 7 X . p ) for some Borel measure p. It thus rernains to observe that

if C is the finite-dimensional C'-subalgebra of B(X) generated by the characteristic

functions of Pl.. . . P,,. then for the extension ;r to B(.Y) of any separable representation

of A we have that a(A)" = r (B(X) ) and x(C) approximately contains R to within S.

0

Proposition 6.6. Tensor pmducts of uieakly AF Ca-algebm a7e weaklp AF.

Proof. Let =Ir and -42 be weakly LW C*-algebras. Note that by Proposition 5.10 weakly

AF C*-aigebras are nuclear: and so there is a unique C*-tensor product Ai 8 A2. CVe

may assume that Al and -42 are unital, since unitizing wil i not change the double

commutant under a representation. Let a : Ai @ A2 -t 'B(3f) be a representation, and

let S I E P f (A1 8 A2) and 6 > O. Since the set of the eiementary tensors spans a dense

subset of Al i&, -42, there is finite set {CjEJx, @ yij}iEr of finite iinear combinations

of elementary tensors which approximately contains Q to within $6. Identifying AI and A2 with their canonical copies in AI 8 A?, we can h d finitedimensionai C*-

subalgebras Bi E ir(Al)" and B.L E 7r(A2)" which approximately contain

and {ir(y)ij}isrjsJ, respectively, to within &. Since r(Ai) and n(A2) cornmute so do i;(Al)" and 7r(A2)": and so Br and B2 generate a finite-dimensionai C*-subdgebra

B C R(A~)"A(&)" Z n(AI @ A2)lr. Since two ekmentary teusors are ciose in norm by

a factor of 2 whenever their components are factorwise close in norm (as can be s h o m

by applying the triangle inequaiity as in the proof of the mntinuiw of muItiplication

in a C*-algebra). it is easily seen that B apptoximately contains ( x j E J ~ i j 8 g i j I i E I to

within &dt and hence contains R to within 6 by the triangle inequdty. We conclude

that -41 #a .-12 is weakIy AF. O

Proposition 6.7. Inductive lirnits of wenkiy AF C - a l g e b n t s are uienlcly AF.

PmoJ Suppose that A is the inductive limit of the inductive system (A,,&,). Let

O, : An 1 A be the compatible maps. Let rr : A - 'B(K) be a representation, and

Iet R E P f (A) and b. For some n E PI we c m h d an R, E Pf(A,) such that

&(Cl,) apptoximately contains R to within id. Since A, is weakiy .AF there is a finite

dimensionai C'-subalgebra B C ( l ia &)(An)" C n(A)" which spproxhately contains

(noc+J(R,) to within id, ruid it foIlows by the triangle inequality t hat B approximately

contains R to within 6. We conchde that A is weakly M. 0

Proposition 6.8. Hereditanj C'-subalgebras of sepamble ureakly d P C ' - a l g e b m are

uieakly -4 F.

PmoJ Let A be a separable weakiy Al' C'dgebra md C a hereditary Cv-subalgebra

of A. Let R E P j (C) and d. E > O. Since the postive elements in A with n o m less than

or equd to 1 form an approxirnate unit ive c m find a positive a E C Nith Ilal 5 1 such

that [lx - axaIl < 8 for all x E Q. Since A is weakiy AF there is a finitedimensional

Ce-subalgebra 5 C rl** rvhich approxhnately contains R to Nithin S and contains an element b such that Ilb - al1 < E. We may rissume that B is a unitai C'-subalgebra by

taking B -k Cl.+.. Notv if y f B then

and so by [Ch , Thm. 5.31 ive can take E to have b e n chosen smaii enough so that

there is a unitary u E A** such that I[l..p- - uli < 6 and u(6Bb)u' c a P a . We c m

furthemore assume that E is s m d enough so that (2 + E ) E I I ~ [ I c 6 for any y E 3 which Lies nrithin 6 in n o m from an eIement of $2. Thus if a: f f2 then Iettirg y be an eIement

in B Nith I[x -y [ < S we have

Hence the finite-dimensionai C*-algebra u(bBb)u* approiomately contains R to within

56. We &O have aAla C ( a h ) * * C C**, since (aila)" identifies with the weak

operator closure of a P a in A" and in the weak operator topology lima x, = x implies

lim, axas = axa. Thus by Proposition 6.2 ive conclude that C is weakiy AF. 0

A C*-aigebra is said to be approximately homogeneous or AH if it is isomorphic

to an inductive limit of algebras of the form p{Mn(@) @ C ( X ) ) p where X is a sepwable

locdly compact Hausdorff space and p is a projection in &.I,(@) @ C(X).

Proposition 6.9. A H C*-algebnrs are weakly AF.

Proof. Combine Propositions 6.5, 6.6: 6.7? and 6.8. O

Proposition 6.10. Quotients of weakly AF C - a l g e b m are weakly AF.

Pmof. Siippose A is a weakiy AF C*-dgebra and 3 a closed ideai of A. CVe canonicaily

identify (A/J)** with A"p for some central projection p of A*'. Let 52 E P f (AI J )

and d > 0. and let fi be an set in P f ( A ) whose image under the quotient map is il, Since A is tveakly AF there exiçts a finite-dimensional C*-subdgebra B C A" which

approsimately contains 6 to nrithin 5. Shen Bp is a finite-dimensional C'-subalgebra

of ( A I J ) " which approximately contains to wichin 5. Hence A / J is weakly AF by

Proposition 6.2. O

Definition 6.11. Let A be a unital C*-algebm and 0 a n avtomorphism of A. W e

say that a representation IF : A -r 'B(K) Ls t r a c i d y O-FA if ir a 0 O ~ - ' l , ( ~ ) extends

to a n automorphism ë of ;r(A)" and ir(A)'* admits a normal 8-invariant trace. W e

say that O is tracially FA if for euerg Q E P f (A) and 6 > O the= &sts a unital

C*-subalgebm C E -4 appmzimately containing !J t o v i th in 6 and an r E Pi such that,

for al1 k 2 r , 0%' commutes with C and ( c ' * ~ - ' ) ' ~ ~ ~ i s a weakly AF C*-subalgebm

admitting a faithful .unital tmcially &FA representation.

Remark 6.12. We note that if A admits a O-invariant faithhil tracial state r then

any O-invariant C*-subalgebra C c A admits a faithful traciaiiy 8-FA representation,

since we may take the GNS representation n of the restriction of T to C (by [BRI, Cor.

2.3.171 TO O O n-Ll,(c) extends to an automorphism Bof a ( ~ ) " under which the normal

extension of r o ~ - ~ l , ( ~ ) is invariant). In this case the topologicai (finit~dirnensional

approximation) and measure-theoretic (existence of a 8-invariant trace) conditions are compIetely and conveniently separated,

Proposition 6.13. A tracially FA automorphism of a unital CT-algebm is FA.

Prnof. It suffices to show that a w e d y AF C*-subalgebra C c A with Ckvffi adrnitting

a tracially 8-FA representation satisfies the appro-uimation component of the definition of 19-FA. So let T : CO-" + $(K) be a faithful tracidly &FA representation, 8 an

aiitomorphism of x(Co,")" e~tending n~Box-~I,(~), and r a faithful normal #-invariant

trace on ;i(CRm)" whose restriction to semifinite von Neumann subalgebras is again

semifinite. Ive identiFy Co," with its image under ir. Since C is wenkly Al?, for

every R E P f (C) and S > O we can finci a finite-dimensional C*-subalgebra of C" which appro.uimately contains R to w i t h i 6. Next note that by [ T a , Prop. 3-36]

every von Neumann siibaigebra M of (c~,w)'' is the image of a unique T-preserving

normal conditional expectation E. Furthemore, if the subalgebra 1Ci is &invariant

then 80 E o #-' : (Ce+x)" - hl is a normal conditional expectation, a d it is normal

and T-preserving since E and 8 are, so that by the uniqueness of E we rnust have

g o E o 8-L = E: that is. E cornmutes with 8. O

Exarnple 6.14. If A is a unital weakiy Ai? C"-dgebra with a faithful tracial state

then it is readily seen that the shift on the infinite tensor product ABZ is tracidy FA.

Proposition 6.15. If a separable unital C-algebra A admits a tmcî'ally FA automor-

phism then A is nuclear and strongly quasidiagonal.

Proof. The proposition foliows easily from the fact that any $2 E P f ( A ) is approxi-

mately contained to within any given 6 > O in a weakly AF C*-algebra, which is nuclear

and strongly quasidiagonal by Proposition 6.4. O.

We record in the follonring proposition the observation that the property of being

tracially FA passes to quotients wîth a dynarnically invariant tracial state.

Proposition 6.16. Let A be a unital CT-algebm, 6 a tracially FA automorphism of A, 3 a closed two-sided ideal in A, and 7 an automorphism of A I J such that the quotient

rnap n : A - A I J intertuines 0 and 7, i.e., n O 8 = y O n. Suppose that A I J admits a

faithfîll y-inuarïant tmcial &te. Then 7 is tracially FA.

Pmof. Tt suffices to note that images of weakiy FA C*-algebras under *-homomorphisms

are we&y FA by Proposition 6.101 and that the quotient map preserves commutativity.

O

Proposition 6.17. CVith the hypotheses of Pmposition 6.16 and the additional as-

mmption that A is separable zue have ht(-f) 5 ht(8),

Pmof. Since by Lemma 5.6 the CNT entropy does not increase when passing to quo-

tients of unitai separable nuclear C*-algebras and every -(-invariant state c on A/J gives rise to a 8-invariant state a O 7;- on A we have by Theorem 5.9

where the siiprema are taken over ai1 dynamically invariant states. O

In view of Remark 6.12 it wouId usefu1 to determine conditions which guarantee the existence of a dynarnically invariant faithful tracial state. The foiiowing proposition

addresses such a situation. W e say that a Cg-algebra A with an automorphism 0 is

19-simple if A admits no non-trivial proper 0-invariant ideals.

Proposition 6.18. Let A be a unital stabty finite exact C*-algebru and 8 an automor- phism of -4, If A is 8-simple then A ahnits a faithjid 9-huariant tracial state.

Pmof. By [Bq and M A admits a tracid state r. Let i be any weak* limit point of the

sequence {! r o B ~ ) . , ~ of traciai aates. It is easily verified that i is 8-invariant.

By [Tai& Lemma 9.61 and the tracid property of 7. the set Jr = {a: A : ~ ( x ' x ) = 0) is a closecl ideal in A? and it is O-invariant. for if 3: E J then ?(B(x)*d(x)) = ?(B(z*x)) = ~(x'x) = O. Thus J = A by the 8-invariance of i? and so ? is faithful. O

Example 6.19. Let T : ,Y - X be a minimal homeomorphism of a compact metric

space and 8 is the automorphism of C(X) defîned by O( f ) (x) = f (Tx) for al1 f E C(X) and x E X. Then 8 is traciaüy FA. since from the correspondence between open subsets of X and ide& of C(X) we infer that T is minimal if and onIy if C(X) is 8-simple, and so we can apped to Proposition 6.18. More generaiiy, it is easily seen

that any topoiogicai dynamical system admitting a dynamically invanant measure with

full support gives rise to a tracially FA automorphism at the C'dgebra level. Note

that C(X) is nuclear, so that Theorem 5.9 applies.

Further examples of FA automorphisms

We show in the follonring proposition that the FA situation includes the dynarnical

systems considered by Neshveyev and Starmer. IF A is a unital C'-aigebra and B an automorphism, then the dynamical system (A , 0) is said to be asymptotically Abelian with locality if there is a dense *-subalgebra B of A such that every pair

r,g E B generates a finite-dimensional C-aigebra and satisfies, for some p = p(x, y),

[f?"(z)? = O for ail Ikl 2 p PS, Defn. 3.11, As Neshveyev and Stormer note: since finite-dimensionai Ca-algebras are singIy generated it follows by an inductive argument

that nny finite subset of B generates a finitedimensionai C8-algebra, and so A must

be AF.

Proposition 7.1. Let A be a unital C*-algebm and B an automorphism of A. If the

dynamical system (A, O) is asymptotically Abelian uith locality then B is FA.

Proof. Let B be a dense '-siibaigebra of .4 witnessing the condition of asymptotic

AbeIianess with Iocaiity. Given R E P f ( A ) and d > O choose a finite subset 6 of B approximateiy containhg R to within 8. Let C be the C'-subaigebra of A generated

by 6. For some r E M we must have [Bk(r),yj = O for di =,y E 6 and 1k1 2 r, since such an r exists for every such pair x. y. Now for al1 k 2 r the C*-subalgebra

c6.k-r c A is generated by fi;-r-'. and so 8 k ~ e . k - r cornmutes nrith and is finite-dimensionai. To show that is Br-FA for every k >_ r it remains to check

the appro.uimation component of the definition. The automorphism 0 extends to an

automorphism of .P by the discussion in PVas, 5-11, and we can simply take N in

Definition 5.2 to be itse- since the identity map on ( (c '*~- ' )~?~)" fuüills the

requirement for the 8-invariant nomai conditional expectation. O

Exarnples 7.2. As indicated in Section 11 of [Stol, we have the foiiowing examples

of dynamicai systems which are asymptaticaily Abelian with locaiity: (i) shifts on the

idn i t e tensor product A@' for an AF dgebra A, (ii) the shift defined on the Jones

projections in the Temperley-Lieb dgebra, and (iü) the canonicd shift arising fiom a

finiteindex subfactor inclusion.

The cIass of FA automorphisrns is cIosed under tensor products and inductive limits,

as ive show in the folIowing trvo propositions.

Proposition 7.3. Let A, be a unital C'-algebm and di an FA automorphism for i = 1: 2. men the autornorphism 81 O & of Al Bmi, -A2 is FA.

Proof. Let R E P f (Al 18 =in) and b > O. To verify the requirements of the definition

of an F.4 automorphism for B i 8 & we may assume that R is of the form fiL @ R2 for

some fir f P f (-Al) and T?:! f P f (A?) since the linear span of such sets is dense in .Ai @,,in il2 (see the proof of Proposition 6.6). Then for each i = 1.3 there is a un i t d

C8-nibalgebra Ci C Ai ilpproximately containing Ri to Nithin S and an ri E W such that

is @-FA for d l k >_ ri. Then CL @,in C2 approximateiy contains !2 = R i @ il2 to within 28 (as noted in the proof of Proposition 6.6), so ive need only check now

that (cr ~~)&@w-- ' = ~ 0 ~ ~ k - r 3 min - CO"^-' is &-FA for dl k 2 mm(rl, r2) . But for i = 1.2 the CW-subdgebra c'I.~-' is ciearly 85F.4 for d k >_ mau(rl, r?), and the

tensor product of @-FA Ca-subdgebras is dk-~ .4 , since representations of the factors

natirra1Iy give rise to a representation of the spatial tensor prodiict. the double com-

mutant of tlie latter factors into the product OP the double commutants. and the tensor

prodrict of finite-tlirnensiona1 C'-subdgebras is Bnit~dimensional (and appraximately

contains an eIcrnentary tensor to within 26 i l factomise approdmate containment holds

to within d), whiie the nonnaiity of tlie product of two normal completely positive mnps

(conditional expectations in o u case) on the double commutant is assured by [Tafi, Prop. CV.5.13]. O

Proposition 7.4. Let 8 be an inductive tirnit automorphism of a unital C8-algebm

Urith respect to a sequence Ai c Aq c A3 c . . . O/ Cf-subdgebm (see [Bd? Defn. 2.13/) sach that 61.4, is FA !or ail k E N. Then 0 is FA.

Pr00 ff This follows immediately Erom the fact that if Q f P f ( A ) and 8 > O then, for

some k E E, R is approximately contained in Ak to within 6. O

CVe next =amine certain tensor product automorphisms whiçh combine an action

on a l o c d y compact Hausdorff space r with another C-algebra autornorphism. The C'-algebra ~der l~ving the latter automorphism wiii be assumed to be weakiy AF and ta admit a f ~ t W dynamicalIy-invariant trace, while the local eventual Abelianess in the definition of an FA automorphism wiil be taken care of by requiring that the action

on T be %anderin$ in a certain sense, so that every compact subset gets sent out

to inhity under the dynamics. Because we require our automorphisms to be unitd,

we wiii unitize the tensor product, in which case the dynamics admit a convenient

alternative description as the restriction of a larger tensor product with the first factor

trivially extending the action on l? to a cornpactification by the addition of fived points.

Exarnple 7.5. Let T : X -t X be a horneomorphism of a compact metric space, and

let r be the complement of the cIosure of the set of non-wandering points. FVe recall

that a point r E -Y is said to be non-wandering if for every open set U containing

x the set {n E % : F ( U ) n Ii # 0) is unbounded, and note that the closure of the

set of non-ivandering points. and hence ais0 i'. is invariant under T. Suppose that ï admits a T-invariant a-finite Borel measure p. For instance, we can take a translation

on Rn. inclucle a compactifying hed point c at infinity (so that t' = {c ) ) , and take

p to be Lebesgue measure. Next we introduce a weakiy .4F Cm-algebra D with an autoinorphism -1 and a faithful -pinvariant tracial state T. Consider the Ca-subalgebra

of C'(,Y. C ) 2 C(X) 8 C, where + denotes unitkation. Let 6 be the automorphism of A

given by restricting rl@ -f to A. where q is the automorphism of C(X) to which T gives

rise. Then 8 is FA, which can be seen as foUorvs. Suppose R E P f (A) and 6 > O. Let

Y c i? be a compact subset such that I[ f (x) - f (y)[[ < 6 for di f E R, x E X \ Y, and

y E ,Y\ i'. By compactness there euists an r E N such that T ~ ( Y ) U Y = 0 for al1 k 2 r!

and a finite Borel partition '9 of Y such that if P E tP and f E 52 then I l f (x) - f (y)ll < 6 for di 2: y E P. For each k E M the C*-aigebra c ( IP~ ' , D) S c(P$-') 8 D of functions

from the finite partition 'PO-' = T v T-'T v - - - v T-kf 'P to D naturaiiy identifies

with a C*-subalgebra of the C*-dgebra B(LY. D) of bounded Borel functions from X to D: which is itself contained in the envdoping Borel C*-algebra B(A) of A. Aisol by

Proposition 6.6 C(T!-')@ D is weakly AF. Set C = C(P, D). It is easily checked that C approximately contains R to wïthin 8. Denoting by LJ the state that p d e h e s on Co(r), we have by [BRl. Cor. 2.3.171 that the extension a of u@ 7 to A gives rise to a, faithful

representation ?r : A + B(X) such that A O 0 o n-'lA extends to an automorphisrn 8 of ir(A)" and o O 5r-'lri extends to a normal &invariant semifinite trace 5. Identify A with its image under X. Now ëkC cornmutes with C for dl k 2 r, and so to show that

c@.~-' S C(P;+~-') @ D is 0-FA for ali k 2 r it remains to estabiish the e-xïstence of

the appropriate conditional expectations. But for every k 1 r the restriction of a to

( c ~ . ~ - ~ ) " , and hence to each of its Cf-subaigebras, is f i t e , and so normal conditional

expectations from ((CO.k-r)Br.m)" ont0 C*-subalgebras of ((Cs~k-')dr~m)" for any rn E Z exist by [Tak2, Prop. 2.361. AIso, if .N C (Co*-')" then the restrictioo of o to (H ' -~) ' ' is semifinite. since its restriction to is finite for every m E Z. Thus by [Tai$. Prop. 2.361 there b a unique norma! conditional =pectation ont0 ( L V " ~ ) ' ' ~ and by

uniqueness it must commute with ë since (x ' .~) ' ' is B-kvarimt (ë o E o PL is again a

normal conditional e'cpectatioo).

The exarnples of 7.5 actudiy have zero Voiculescu-Brom entropy, as c m be readi1y

verified. WC can generalize these exarnples beyond the zero entropy situation to more

generai quasi-locd algebrs over a configuration space on which a %andering? home-

ornorphisrn is given. For the general definition of a quasi-IocaI dgebra over a directed

index set with orthogondity relation see pR1. Defn. 2.6.31. We wiiI consider quasi-Iocai

algebras A defined as the norm dosure of the union of a net (.4n)IiEu(s) of C8-algebras

indexed by the collection U ( X ) of open subsets of a locally compact metric space X Rith compact dosure such that (i) if hi c Ai then A,\, c An,? (ii) if Ai n A2 = 0 then -Aii, and =I,\, commute, and (iü) the Ca-algbras Ax share an identity.

Let T : .Y - X be a homeomorphism of a locdy compact metric space X and p

a T-invariant a-Enite mesure on X. Suppose that T admits no non-wandering points

(see Example 7.5). AS in Example 7.5 Ive cau think of the special case: of relevance to

continuous quantum statisticai mechanics (see PRZ]), where X is Rw and T is the shift

1: F- x + r by a nonzero vector r E IRV, Nith j i taken to be Lebesgue measure. h o t h e r

special case is given by shifts, and in particular tensor product shifts (see Exmple

7.2(i) and [BR21), over Zw. Let A be a quasi-local C*-algebra Nith respect to a net {A4,\),,nl(s) and 8 an automorphism of A that is T-covariantl that isl 8AiL = AT^^ for

ail A E U(X). We suppose that there is a diected subset h of U(X) such that =ILL is contained in a weakly IIF Cg-subalgebra of A and admïts a faithfd tracid state for

ati A E A. More generdy! we c m relax the assumption that for 11 E A the dgebra

il,, be weakiy AF and simply require that it be locdy weakIy AF, in the sense that

any h i t e subset of A,\ is approximately contained to Nithin any gïven b > O within a

weakly ..iF C-subalgebra of Aa. Then 8 is an FA automorphism, since any x E A can

be nom-approrcimated by an eIement XA f Act for some A E A andr a s in Example 7.5:

the compactnes of impIies that, for some r E N, ~ ~ i i u .& = 0 for all k 2 r, so that

AI\ is #-FA.

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