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652 DAN VOICULESCU, More generally it is a consequence of the theory of the standard form. of von Neumann algebras that automorphismshave canonicalunitary im. plementationsin the standard form This implies that the perturbation. theoretic entropy is definedfor automorphismsof von Neumann algebras. without assumingthe existenceof invariant states, For the shift automorphism of the II factor of a free groupon genera. tors indexed by 7 the Connes Stormerentropy is either 0 or x but it is an. openquestionwhichof thesetwo is the actualvalue as I havelearnedfrom. Erling Stormer We proveherethat the perturbation theoreticentropyof. the free shift automorphismis, 2 Preliminaries Throughoutthis paper we will usedefinitionsand no. tations of 4 with someamendments This sectionis devotedto these. amendments, The definitionsof the perturbation theoreticentropy and of its vari. ant as well as the resultsof section3 of 4 are givenunderhyperfinite. nessconditions It is natural to expect that at a more advancedstage in. the study of the perturbation theoreticentropyhyperfiniteness. assumptions, shouldplay a role howeverin the definitionsand resultsof section3 of 4.

the hyperfinitenessassumptionsare not necessary We will thereforeremove. these unnecessaryassumptions We will usethis in the caseof the free shift. automorphism, As we mentionedin the introductionhere using the theory of the. standard form the perturbation theoreticentropy for automorphismsof. W algebras can be defined also in the absenceof invariant states In. more detail let M be a W algebra and let Q be a set of automorphisms. of M By 1 there is a standardform M T 7 P for M and this. standard form is uniqueup to unitary equivalence In particular for every. c Q there is a unitaryoperatoryu c on 7 whichimplements c i e. u o xu o c x for x G M By the uniqueness of the standardform. upto unitaryequivalence, it follows, thatHp u Q M and r u Q M. depend only on Q and M, Definition 2 1 If M is avon Neumannalgebraand Q a set of au. tomorphisms, the perturbation theoretic, entropyHp Q M andits variant. t rp Q M aredefined, tobeHp u Q M and respectively.

In the casewhereM is L X la thestandard formis onL2 X. with 7f f andP the positivefunctions, in L2 X I If C Tis the auto. ENTROPY OF DYNAMICAL SYSTEMS 653, morphisminducedby a transformationT of X for which is quasi inwariant. f T l k iT 1, In particular the perturbation theoretic. entropyH C T L X z is an, invariant of the transformationT with quasiinvariantmeasure Note that. this invariant doesnot dependon the choiceof within a given equivalence. Beforeclosingthis sectionlet uspointout that the assumption. that the Hilbert spaceswe considerare separableand that the yon Neumann. algebras have separablepredual will be in force throughout the present. paper The adaptation to the nonseparablecase being quite routine is. omitted here, 3 Bernoulli shifts We prove in this section that for Bernoulh shifts.

theperturbation theoretic, isproportional, to theKolmogorov. Sinai entropy We will use Sinai stheoremand the followinggeneralfact. Proposition 3 1 If u is a unitary operator normalizinga commutative. yonNeum n gebraA t e u A n u a, Proof The inequahty and the fact that the left handsideis not changed. whenn is replacedby n hasbeennotedearlierfor generalA Remark3 5. in 4 It will sufficeto provethe inequality whenn 0 This in turn. will followif givenB E A we find C E A suchthat k u C. SinceA is commutative, wem y defineC V0 k iu kBuk, ThereareX E N f C suchthatX TI asm ooand. Let Ym n 0 k lUkX u k WehaveY T I andY 7 1, fqo k n 1ukCu k. C 1 fiB Wehave, lim n uY u 1 Ym k u B, 654 DAN VOICULESCU.

Corollary 3 2 f o is an automorphism, of an abelianyonNeumannalge. braA then rp n A n rp A, With the notationsof Theorem4 1 of 4 we havethe followingtheo. Theorem 3 3 Thereis a universal, 2 1 18 suchthat, p UT L X h T whenever. T isa Bernoulli, Proof Combining, 4 1of 4 ndthefactthatHp p 3Hp. 2 h T UT L X lSh T, Let T d T2 be BernouUis fts by Sin s theoremh T h T2.

imp esthat T is a factorof T2 d hencein viewof Retook 3 5 of 4 we. v L X S V L X, Hencethereis increasing, function 0 0 suchthat 2 t. if T is a Bernou i shift, If T is a Bernou ishift thenTn n 0 is so a Bernoul shift of. entropynh T henceby Proposition 3 1 we have, nitp UT L X. Sinceh T maybe any numberin 0 ee we inferck nt nck t for all. t e 0 ee It followsthat ck qt qq t for all positiverationalnumbers. q Combiningthis with the fact that bis increasingwe easily get that. 4 The free shift Let G be a freegroupon generators. gn n6 Z Let, L G be the yonNeumannalgebra A G whereA is the left regulax. representationof G Further let a be the free shift automorphism i e. a e Aut L G is suchthat a gn gn, ENTROPY OF DYNAMICAL SYSTEMS 655.

Proposition 4 1 Let c be the freeshift automorphism. on L G Then, Proof L G is in standard, formon 2 G, andtheunitaryimplementing. a is u B 2 G whichactsonthecanonical, basisof 2 G, G as the freeshift automorphism. of the groupG We have, Further let v e A g0 be a unitaryoperatorwith v I and suchthat. the trace of the spectralmeasureof v is Haar measureon the groupof n th. roots of unity We have, te L a kZo v v I uv, If is thecanonical. trace vector, forL G then, P ttVP2 ttVP, andit is easilyseenthat uv uv and satisfythe assumptions.

Proposition2 3 in 4 It followsthat, He u L G log n 1. Sinceu is arbitrary this provesthe proposition Q E D. 5 Non singular transformations In this section we present some. facts concerningHe for generalnon singulartransformationswhich may. lead to the constructionof exampleswithout invariant measurewhere the. perturbation theoreticentropyis finite and non zero We also prove that. non atomic dissipativetransformationshave infinite He. We begin with a generalizationof the upper boundpart of Theorem. Proposition 5 1 Let T be a non singularinvertible transformationof. the probabilitymeasurespace X y p suchthat the Radon Nikodym. derivative, d is a simple, onlyfinitelymanyvalues, 656 DAN VOICULESCU. isaconstant, that for everyfinite partition q we have. p O q oP q l, for a e x in X Then we have, p UT L X 2C 6Alog 2. whereUT U aT, Proof Let B L X andlet fl be the corresponding.

partition In, viewof the assumption, onthe Radon Nikodym derivative. wemayassume, afterenlarging, B that d B andde T, Ifp q and 0 let. x xl 1 q p I V TJ 3 x, Let further, It is easily seenthat. if a 1 0 1 Ourassumptions, implylim v o p wN C, 0 Foreachn NIwedefine. projections, P A andRN where, is multiphcation, by thecharacteristic.

functionof w X whileR is the, orthogonalprojection. ontothe subspaceof L2 X consistingof function, constantsontheatomsofT V V T. If a 1 0 1 wehave, ENTROPY OF DYNAMICAL SYSTEMS 657. Note that the assumptions, dlao T dttoT 1 B, implythat if a 1 O 1 then. U R vU RN I, A E 2t 3 1, k N l t N l, We haveP v C e T I and Rt T Q whereQ is the projectiononto.

TJ measurable, Thisimplies, Yn2 I andZ vTQ, UTYNU Yn N 1 UrPke C E 2tql. k N l t N l, N 1 Pk le C, k N l N 2t 3, P 2t 3 t N l. 658 DAN VOICULESCU, P Ie C 2t 3 inN i 1, Pk e C E 2t 3 N i. e C E 2t 3 N I, sothatdenoting, P v l P2 l e C Ac wehave. N P v l UTY VU Y v N P 2N l, It follows that, N 1 P V iKNP N i.

with IlK vii 1, ZN N 1R2N iLNR2N i, whereIlL vii 1. Let D v Z vY vZ v We haveDN t5 7 1, DN Q 3 and, D v B 0 We have. I D v Ur 17o, IUTDNU DNI, 3N logrank, ENTROPY OF DYNAMICAL SYSTEMS 659. where 5 is a constantindependentof N, ofR2 v l2 v 1. V T2 V l whichmeetco2 v e, C Acer Sincethemeasureof such.

an atomis at leaste c AcN 4N s, it follows, theirtotalnumberdoes. not exceede 4 v s e c cN 4, Thisin turn implies, lim D STll o 12C 6A log 2 12e. ReplacingD v by Q 3 DlvQ 3 gives, kL UrIQ 3 L2X I BIQ 3 L2X. 12C 6Alog2, thisto B E f L X suchthatB D B andQ B T l. we easily get, kL UrIB 12 6A log 2, and hence the desiredconclusion Q E D.

Proposition 5 2 Let Tj be transformations, of probabilityLebesgue. surespaces, Xj Ej PJ J 1 2 suchthatT1isnon singular. wfiileT is, me ure preserving and ergodie Then we have. rp UTlxT2 L X, Proof Note that it will sufficeto prove that if T2 is a Bernoulli shift with. i p UTi xX2 I log n, xT2 L X1 1, The generalcasewill then followusingProposition3 1 and Sinai stheorem.

Indeed if for an integerM we haveMh T2 log n then by Sinai stheo. Mhasafactor, isomorphic, toaBernoulli, shiftwithweights. Moreover by Proposition 3 1, xx2 Urfxrf x xx2, 660 DAN VOICULESCU. wherewe usedProposition3 3 of 4 for the last inequality Hencefor. M oo we get the desiredresult, AssumenowT2 is the Bernoullishift In the proofof the lowerbound. in Theorem4 1 4 usingLemma4 2 4 weprovedtheexistence of a unitary. elementV A whereA L X2 suchthat there is a unit vector. L2 X2 sothatif Wj VJUT2 1, j n 1 wehavethatthevectors. Wj Wj rn O l jk n 1 l k m, are pairwiseorthogonal, Let L2 X1 be a unitvectorandlet r Thenthevectors.

UT Wj UT Wj rllm O I j n 1 l k m, are pairwiseorthogonal By Proposition. 2 3 in 4 we have, H Vrxr2 L Xx 1log n, Corollary 5 3 Let T be a non singulartransformationot a non atomic. probabilitymeasure space, X y p lit isdissipative, thenHp Ur L X. Proof Indeed T T1 x T2 with T T2 like in the precedingproposition. and with h T2 oo Q E D, Remark 5 4 It is an openproblemwhetherthereexistsa nonsingular. T without invariant measure such that, 0 Hp UT L X.

Propositions5 1 and 5 2 may providea way towardsfindingsucha T It is. howevernot clearwhetherthe conditionsin Proposition5 1 can be satisfied. by a transformationfor which there is no equivalentinvariant measure. Also checkingthe assumptionon the lim supof the informationfunction. appearing in Proposition5 1 for a given T seemsto representa serious. difficulty Perhapsthe resultsin 3 and 2 mayprovidesomeinspiration. on how to deal with this question, ENTROPY OF DYNAMICAL SYSTEMS 661. REFERENCES, 1 U Haagerup The standard form of yon Neumann algebras Math Scand 37. 1975 271 283, 2 U Krengel IYansfovraationswithout finite invariant measure have finite strong. generators in SpringerLectureNotes in Math 160 1970 133 157. 3 W Parry An ergodic theorem of information theory without invariant measure. ProceedingsLondonMath Soc XII 52 1963 605 612, 4 D Voiculescu Entropy of dynamical systems and perturbations of operators Er. godic Theory and Dynamical Systems to appear, Department of Mathematics.

University of CMifornia, Berkeley CA 94720, Received June 12 1991. We begin with a generalization of the upper bound part of Theorem 4 1 in 4 Proposition 5 1 Let T be a non singular invertible transformation of the probability measure space X y p such that the Radon Nikodym derivative d22 T is a simple function i e takes only finitely many values