Free Fisher Information for Non-Tracial States

12 Pages·0.13 MB·English

by  Dimitri Shlyakhtenko

FREE FISHER INFORMATION FOR NON-TRACIAL STATES
DIMITRISHLYAKHTENKO
1
0
0 ABSTRACT. WeextendVoiculescu’smicrostates-freedefinitionsoffreeFisherinformationandfreeentropy
2 tothenon-tracialframework.Weexplaintheconnectionbetweenthesequantitiesandfreeentropywithrespect
tocertaincompletelypositivemapsactingonthecoreofthenon-tracialnon-commutativeprobabilityspace.
n
WegiveaconditiononfreeFisherinformationofaninfinitefamilyofvariables,whichguaranteesfactoriality
a
ofthevonNeumannalgebratheygenerate.
J

7
1

]
A
1. INTRODUCTION.
O
Free entropy and free Fisher information were introduced by Voiculescu [13], [14], [16] in the context
.
h ofhisfreeprobability theory [17]asanalogs ofthecorresponding classical quantities. Thesequantities are
t
a usually considered in the framework of tracial non-commutative probability spaces; not surprisingly, the
m
most striking applications of free entropy theory were to tracial von Neumann algebras (see e.g. [15], [2],
[ [10]). Recently, however, itturned outthat sometypeIIIfactors associated withfree probability theory [5]

1 havecertainproperties incommonwiththeirtypeII1 cousins[9]. Thisgivesrisetoaspeculation thatthere
v isroomforfreeentropytoexistoutsideofthecontextoftracialnon-commutative probability spaces.
7
The goal of this paper is to initiate the development of free Fisher information, based on Voiculescu’s
3
microstates-free approach [16], in the non-tracial framework. The key idea is that all of the ingredients
1
1 going into the definition offree Fisher information in this case mustbehave covariantly withrespect to the
0 modular group [11] of the non-tracial state. The principal example of a family of variables for which free
1
Fisher information is non-trivial, and which belong to an algebra not having any traces, are semicircular
0
/ generators offreeAraki-Woodsfactors, takenwithfreequasi-free states[5].
h
We describe another route towards free Fisher information, which is based on first converting the non-
t
a
tracial von Neumann algebra into alarger algebra, the core (having an infinite trace), and then considering
m
free Fisher information relative to a certain completely positive map (in the spirit of [7]). We should point
:
v out that it isthis approach that is mostlikely to connect withthe microstates free entropy (as suggested by
i [3];seealso[8]),sinceitisatpresentunclearwhatamicrostatesapproach tofreeentropyinthenon-tracial
X
frameworkshould be.
r
a We finish the paper with a look at free Fisher information on von Neumann algebras that have traces.
Ourfirstresultisthatoncethealgebrahasatrace,thefreeFisherinformationisautomatically infinitewhen
computed with respect to a non-tracial state. It is likely that on any von Neumann algebra, free Fisher
information can be finite for only very special states (however, we do not have any results in this direction
in the non-tracial category). Another result of the present paper is a statement guaranteeing factoriality of
a tracial von Neumann algebra, once we know that it has an infinite generating family whose free Fisher
information isbounded inacertainway.
Acknowledgement. This work was supported by an NSFpostdoctoral fellowship. Theauthor is grateful to
MSRIandtheorganizers oftheoperator algebras programfortheirhospitality.
Date:February1,2008.
1
2 DIMITRISHLYAKHTENKO
2. FREE FISHER INFORMATION FOR ARBITRARY KMS STATES.
2.1. FreeBrownianmotioninthepresenceofamodulargroup. LetM beavonNeumannalgebra, φ :
M Cbeanormalfaithful state onM. Denotebyσφ themodular group ofφ. DenotebyL2(Msa,φ)
→ t ⊂
L2(M,φ)theclosureoftherealsubspace ofself-adjoint elementsMsa M.
⊂
LetX = X∗ M andletB M beasubalgebra. Assumethatσφ(B) B forallt R.
∈ ⊂ t ⊂ ∈
Consider the von Neumann algebra = Γ(L2(Msa,φ) L2(M,φ)), taken with the free quasi-
M ⊂
free state φ . Consider the element Y = s(X) (see [5] for definitions and notation). Then
M
∈ M
φ (YσφM(Y))= φ(Xσφ(X)),forallt R.
M t t ∈
Consider the algebra = (M,φ) ( ,φ ), and denote by φˆthe free product state on . Note that
M
N ∗ M N
σφˆ = σφ σφM. TheelementsX = X+√ǫY,ǫ 0formanaturalfreeBrownianmotion,whichbehaves
nitcelyutnd∗erttheactionofthemodǫulargroup. Inpa≥rticular, notethatforallǫ 0andt R,
≥ ∈
φˆ(X σφˆ(X )) = φ(Xσφ(X)) (1+ǫ).
ǫ t ǫ t ·
Furthermore, for each ǫ > 0, the distribution of X is that of a free Brownian motion at time ǫ, starting at
ǫ
X;thisisbecauseY isasemicircular variable, freefromX.
2.2. Conjugatevariables. LetB[X]denote that algebra generated byB and all translates σφ(X), t R.
t ∈
Assume that σφ(X) are algebraically free over B, i.e., satisfy no algebraic relations modulo B. Denote
{ t }
by∂ :B[X] thederivation givenby:
X
→ N
φ φˆ
1. ∂ (σ (X)) = σ (Y)
X t t
2. ∂ (b) = 0,b B.
X
∈
Noticethattherangeof∂ actuallyliesinthesubspaceB[X]YB[X] . Notealsothatsince∂ (σφ(X))
is self-adjoint, wehave thXat for P B[X], ∂ (P∗) = ∂ (P)∗, i.e.,⊂∂Nis a -derivation. ObseXrvetfinally
X X X
∈ ∗
that∂ iscovariantwithrespecttothemodulargroupsσφ andσφˆ:
X t t
φ φˆ
∂ (σ (P)) = σ (∂ (P)), P B[X].
X t t X ∈
Definetheconjugatevariable J (X : B) L2(B[X],φ)tobesuchavectorξ that
φ
∈
(2.1) ξ,P = Y,∂ (P) , P B[X],
h iL2(B[X],φ) h X iL2(N,φˆ) ∀ ∈
if a vector ξ satisfying such properties exists. Formally, this means that ξ = ∂∗ (Y), where ∂ :
X X
L2(B[X],φ) L2( ,φˆ)isviewedasadenselydefinedoperator.
→ N
Itisclear, becauseofthedensity ofB[X]inL2(B[X],φ),thatξ isunique,ifitexists.
Itisconvenient totalkaboutJφ(X : B)eveninthecasethat σt(X) t∈R arenotalgebraically freeover
{ }
B (such isthe case, forexample, when φisatrace, and hence σφ(X) = X for all t). Inthis case, one can
t
view ∂ asamulti-valued map, the setofvalues given bythe results ofapplication of thedefinition of∂
X X
inallpossible ways;thedefinitionofJ isthenthat(2.1)isvalidforallvaluesof∂ .
φ X
NotethatJ (X : B)dependsonmorethanjustthejointdistributionofX andB withrespecttothestate
φ
φ;itdepends onthejointdistribution ofthefamilyB σφ(X) : t R).
∪{ t ∈
We continue to denote by σφ the extension of σφ to the Hilbert space L2(M,φ) (this is precisely the
t t
one-parameter group ofunitaries ∆it,where∆ isthemodularoperator). Inparticular, ifφisatrace, then
φ φ
thedefinitionofJ is(uptoamultiple)precisely thatoftheconjugate variableofVoiculescu [16].
φ
Lemma2.1. Assumethatξ = J (X : B)exists. Thenξ L2(Msa,φ),andσφ(J (X :B)) = J (σ (X) :
φ ∈ t φ φ t
B).
FREEFISHERINFORMATIONFORNON-TRACIALSTATES 3
Proof. Wenotethat,because∂ isa -derivation,
X
∗
P∗,ξ = Y,∂ (P∗) = ∂ (P),Y = ξ,P .
X X
h i h i h i h i
From this it follows that ξ is in the domain of the S operator of Tomita theory, and moreover that Sξ = ξ.
Henceξ L2(Msa).
∈
Onealsohas
φ
σ (ξ),P = Y,∂ (P)
h t i h X i
φ
= σ (Y),∂ (P) ,
h t σtφ(X) i
sincethejointdistributions ofB[X]and{σtφˆ(Y)}t∈R isthesameasB[X]and{σsφˆ+t(Y)}t∈R,foranys. It
followsthatσφ(J (X : B))= J (σφ(X) : B).
t φ φ t
Lemma2.2. LetP,Q B[X],andassumethatξ = J (X)existsandisinM. Then
φ
∈
φ(PξQ) = φˆ(PY∂ (Q))+φˆ(∂ (P)YQ).
X X
Proof. Recall that φ(and φˆ)satisfy the KMScondition: for all a,b M (or ), there exists a(unique)
∈ ∈ N
function f(z),analytic onthestrip z :0 < z < 1 ,andsothat(writingσ foreitherσφ orσφˆ)
{ ℑ } t t t
φ(aσ (b)) = f(t),
t
φ(σ (b)a) = f(t+i), t R.
t
∈
FixP,Q B[X]andletf beasabove,sothat
∈
φ
φ(σ (P)ξQ) = f(t+i),
t
φ
φ(ξQσ (P)) = f(t).
t
Then
φ
f(t) = ξ,Qσ (P)
h t i
φ
= Y,∂ (Qσ (P))
h X t i
φ φˆ
= Y,∂ (Q)σ (P) + Y,Qσ (∂ (P)) ,
h X t i h t X i
where inthe last step weused the fact that ∂ intertwines σφ and σφˆ. Using the KMS-condition for φˆ, we
X t t
thenget
f(t+i) = φˆ(σφˆ(P)Y∂ (Q))+φˆ(σφˆ(∂ (P))YQ)
t X t X
= φˆ(σφ(P)Y∂ (Q))+φˆ(∂ (σφ(P))YQ).
t X X t
Sincef(t+i) = φ(σφ(P)ξQ),weget,settingt = 0,that
t
φ(PξQ) = φˆ(PY∂ (Q))+φˆ(∂ (P)YQ),
X X
asclaimed.
4 DIMITRISHLYAKHTENKO
2.3. Conjugate variables as free Brownian gradients. As pointed out above, X + √εY is a natural
free Brownian motion, which is covariant with respect to the appropriate modular groups. The following
proposition showsthatJ (X : B)playstheroleofthefreeBrowniangradient ofX.
φ
Proposition 2.3. Assumethatξ = J (X : B)existsandbelongstoM L2(M,φ). LetP(Z ,...,Z ),be
φ 1 n
⊂
φ
anynon-commutative polynomial innvariables Z ,...,Z ,withcoefficients from B. WriteX = σ (X),
1 n t t
Y = σφˆ(Y),ξ = σφ(ξ),t R.
tThentforalltt ,..t.,t ∈R,wehave
1 n
∈
1
φˆ(P(X +√εY ,... ,X +√εY )) = φ(P(X +εξ ,... ,X +εξ )
t1 t1 tn tn 2 t1 t1 tn tn
+O(ε2).
Proof. Wemayassume,bylinearity,thatP isamonomial,i.e.,P(Z ,...,Z ) = b Z b b Z b ,for
1 n 0 1 1 n−1 n n
···
b B. Inthiscase,wehave
j
∈
φˆ(b (X +√εY )b b ) = φ(P(X ,... ,X ))+O(ε2)
0 t1 t1 1··· n t1 tn
+ε φˆ(b X b Y b X b Y
0 t1··· l tl+1 l+1 tl+2··· k tk+1 ·
Xk<l
b X X b )
· k+1 tk+2··· tn n
= φ(P(X ,... ,X ))+O(ε2)
t1 tn
1
+ ε φˆ(b X X b Y ∂ (b X X b ))
2 0 t1··· tl l tl+1 Xtl+1 l+1 tl+2··· tn n
Xl
1
+ ε φˆ(∂ (b X X b )Y b X X b )
2 Xtk+1 0 t1··· tk k tk k+1 tk+2··· tn n
Xk
= φ(P(X ,... ,X ))+O(ε2)
t1 tn
1
+ φ(b X X b εξ b X X b ),
2 0 t1··· tk k tk k+1 tk+2··· tn n
Xk
thelastequalitybyLemma2.2. ThisimpliesthestatementoftheLemma.
2.4. Examplesofconjugatevariables.
2.4.1. Tracial case. We have seen before that if φ is a trace, then the definition of J (X : B) coincides
φ
with the definition of conjugate variables given by Voiculescu, up to a constant (which has to do with the
factthatwechooseY sothat Y = X ,andnot1). Inparticular,
k kL2(φˆ) k kL2(φ)
1
J (X :B)= J(X :B) , ifφisatrace.
φ · X 2
k kL2(φ)
2.4.2. Freequasi-free states. Letµ be apositive finite Borelmeasure on R,and let R be thereal Hilbert
space L2(R,R,µ) of µ -square-integrable real-valued functions. Denote by U the rHepresentation of R on
t
R,givenby
H
(U f)(x) = e2πitxf(x), x,t R.
t
∈
Lethdenotethevector1 R,andconsider
∈ H
M = Γ( R,Ut)′′, φ= φU, X = s(h) M
H ∈
(see[5]fordefinitionsandnotation).
FREEFISHERINFORMATIONFORNON-TRACIALSTATES 5
ThenX = J (X : C). Indeed, setX = σφ(X) = s(U h);thenwehave
φ t t t
φ(X X X ) = φ(XX )φ(X X ) φ(X X )
· t1··· tn tk t1··· tk−1 · tk+1··· tn
Xk
= φˆ(Yσφˆ(Y))φ(X X ) φ(X X )
tk t1 ··· tk−1 · tk+1··· tn
Xk
= φˆ(YX X YX X )
t1··· tk−1 tk+1··· tn
Xk
= φˆ(Y∂ (X X )),
X t1 ··· tn
sothatX satisfiesthedefiningproperty ofJ (X :C),andhenceJ (X :C)existsandequalsX.
φ φ
2.5. FreeFisherinformation. Following[16],wedefinethefreeFisherinformation Φ∗(X : B)tobe
φ
Φ∗(X :B)= J (X : B) 2 X −2
φ k φ k2·k k2
(the extra factor X −2 comes from the fact that ∂ (X) does not have unit norm in our definition). For
k k2 X
severalvariables, weset
Φ∗(X ,...,X ) = Φ∗(X : W∗(σφ(X ),... ,Xˆ ,... ,σφ (X ) :t ,... ,t R))
φ 1 n φ i t1 1 i tn n 1 n ∈
X
(hereXˆ meansthatX isomitted).
i i
3. FREE FISHER INFORMATION RELATIVE TO THE CORE.
Recall[11]thatif(M,φ)isasabove, itscoreisdefinedtobethevonNeumannalgebra crossed product
P = M ⋊ R. Thereisacanonical inclusion M P,andP isdensely spannedbyelementsoftheform
σφ
⊂
mU , t R,
t
∈
wherem M,andU satisfyU mU∗ = σφ(m). TheelementsU : t Rgenerateacopyofthegroupvon
Neumann∈algebraL(Rt) P;thtematp t t ∈
⊂
Eφ : mU φ(m)U , m M,t R
t t
7→ ∈ ∈
extendstoanormalconditional expectation fromP ontoL(R).
ForX M self-adjoint, definethecompletely positivemapη :L(R) L(R)by
X
∈ →
η (g) = Eφ(XgX), g L(R).
X
∈
Identify L(R)withL∞(R)viaFouriertransform. Foreacht R,set
∈
η(t) = X,σφ(X) = Eφ(XU X).
h t i t
Thenη (f)= ηˆ f,iff L∞(R)= L(R);hereηˆdenotesFouriertransform.
DefiXneonP an∗L(R)-va∈luedinner∼product
a,b L(R) = Eφ(a∗b), a,b P.
h i ∈
DenotebyL2(P,Eφ)theL(R)-HilbertbimodulearisingfromthecompletionofP withrespecttothenorm
inducedbythisinnerproduct. Notethattherestrictionof , L(R) toM P isvaluedinthecomplexfield,
h· ·i ⊂
andcoincides withtheinnerproduct a,b = φ(a∗b)onL2(M).
Denoteby , theL(R)-valuedhinneriproductonP P (algebraic tensorproduct) givenby
η
h· ·i ⊗
a b,a′ b′ = Eφ(b∗η(Eφ(a∗a′)b′)), a,a′,b,b′ P.
η
h ⊗ ⊗ i ∈
Denoteby1 1thevector1 1 P P.
η
⊗ ⊗ ∈ ⊗
6 DIMITRISHLYAKHTENKO
Letδ :B[X] L(R) P P begivenby
X
· → ⊗
δ (X) = 1 1, δ (B L(R)) = 0
X η X
⊗ ·
andthefactthatδ isaderivation.
X
Theorem3.1. Let (M,φ) be as above, and let P be its core. Let i : L2(M,φ) L2(P,Eφ) be the
→
extension oftheinclusion ofM P. Thenζ = i(J (X :B))satisfies
φ
⊂
(3.1) hζ,QiL(R) = h1⊗η 1,δX(Q)iηX
forallQ B[X] L(R). Conversely, ifthereexistsavectorζ L2(P,Eφ),sothat(3.1)issatisfied, then
∈ ∨ ∈
J (X :B)existsandζ = i(J (X :B)).
φ φ
Proof. Assume first that J (X : B) exists. Set ζ = i(J (X : B)). We must verify that (3.1) holds.
φ φ
By linearity, and the fact that L(R)BL(R) BL(R), it is sufficient to consider the case when Q =
⊂
b Us1X b Us2 X b Usn, with b B and X = σφ(X). Then Q = P Ur, where r = s , and
0 t1 1 ··· tn n j ∈ t t · j
P = b0Xt′1b′1···b′n,withb′j = σsφj−1 ◦···σsφ1(bj),t′j = sj−1+···+s1+tj. Notethatforx,y,Px′,y′ ∈ P,
x y,x′ Uy′ = x y,x′U y ,and x y,Ur(x′ y′)Us = U−rx yU−s . Usingthis,we
η η
h ⊗ ⊗ i h ⊗ ⊗ i h ⊗ ⊗ i h ⊗ i
get
ζ,Q L(R) = ζ,P L(R)g
h i h i
= ζ,P Ur
L2(M,φ)
h i ·
= φˆ(Y∂ (P)) Ur
X
·
= φ(b′0Xt′1···Xt′jb′j)φ(b′j+1Xt′j+2···Xt′nb′n)·φˆ(YYtj)Ur
Xj
= φ(b′0Xt′1···Xt′jb′j)φ(b′j+1Xt′j+2···Xt′nb′n)·φ(XUt′jXU−t′j)Ur
Xj
= φ(b′0Xt′1···Xt′jb′j)φ(b′j+1Xt′j+2···Xt′nb′n)·Eφ(XUt′jXU−t′jUr)
Xj
= φ(b′0Xt′1···Xt′jb′j)φ(b′j+1Xt′j+2···Xt′nb′n)·ηX(Ut′j)Ur−t′j
Xj
= ηX ◦Eφ(b′0Xt′1···Xt′jb′jUt′j)·Eφ(Ur−t′jb′j+1Xt′j+2···Xt′nb′n)
Xj
= h1⊗η 1,b′0Xt′1···Xt′jb′jUt′j ⊗Ur−tjb′j+1Xt′j+2···Xt′nb′niη
Xj
= h1⊗η 1,b0Us1Xt1···XtjbjUsjUt′j ⊗Ur−tjUsj+1bj+1Xtj+2···XtnbnUsnU−riη
Xj
= 1 1,b Us1X X b Usj Usj+1b X X b Usn
h ⊗η 0 t1··· tj j ⊗ j+1 tj+2··· tn n iη
Xj
= 1 1,δ (Q) .
η X η
h ⊗ i
Conversely, assume that ζ satisfying (3.1) exists. Since the argument above is reversible, it is sufficient
to prove that ζ is in the image of i : L2(M) L2(P,Eφ). Let θ be the dual action of R on P, given
t
→
by θ (U ) = exp(2πist), θ (m) = m, m M. It is sufficient to prove that θ (ζ) = ζ, since i(L2(M))
t s t t
∈
consists precisely ofthose vectors, whichareleft fixedbyθ. Itissufficient toprove thatθ (Eφ(ζmU )) =
s t
FREEFISHERINFORMATIONFORNON-TRACIALSTATES 7
exp(2πist)ifm M. Sinceζ isassumedtobeintheclosureofB[X] L(R),itissufficienttocheckthis
∈ ∨
form B[X]. Butthenby(3.1),
∈
Eφ(ζmUt) = ζ,mUt L(R)
h i
= 1 1,δ (mU)
η X η
h ⊗ i
= 1 1,δ (m) U
η X η
h ⊗ i
M U ,
t
∈ ·
whichgivesthedesiredresult,sinceθ actstrivially onM.
s
Note that (3.1) means that ζ is equal to J(X : B R,η) in the notation of [7]. (This is strictly speaking
incorrect, sincethesettingof[7]presumestheexist∨enceofafinitetraceonB[X] L(R);however,itisnot
∨
hardtocheck thatthearguments in[7]gothrough also inthecase ofasemifinite trace, which existsinour
case).
This fact has many consequences for the conjugate variables J (X : B), coming from the properties of
φ
J(X : B L(R),η). Note in particular that if X is free from B with amalgamation over D B with
respecttos∨omeconditional expectation E : B D,andE isφ-preserving, thenX isfreefrom⊂B L(R)
withamalgamation overD L(R)(see[12],[6→]). Werecordthisas ∨
∨
Theorem3.2. AssumethatE : B Disaφ-preserving conditional expectation. IfX isfreefromB over
→
D,then
J (X : B)= J (X : D).
φ φ
Inasimilarway,onecangeneralizetoJ (X :B)allthepropertiesoftheconjugatevariableJ(X :B,η)
φ
provedin[7].
Reformulating gives the following properties of Φ , which we list for reader’s convenience, since they
φ
areneededintherestofthepaper:
Theorem3.3. Letφbeanormalfaithful stateonM,B M beglobally fixedbythemodular group(i.e.,
σφ(B) = B forallt),andX M. Then: ⊂
(at)Φ∗(λX ,... ,λX : B)=i ∈λ−2Φ∗(X ,... ,X :B)forallλ R 0
φ 1 n φ 1 n ∈ \{ }
(b)IfB A M andAisglobally fixedbyσφ,thenΦ∗(X ,... ,X :A) Φ∗(X ,... ,X :B).
⊂ ⊂ φ 1 n ≥ 1 n
(c) If C M is globally fixed by σφ, and W∗(X ,... ,X ) and B are free with amalgamation over C
1 n
(with resp⊂ect to the unique φ-preserving conditional expectation from M onto C), then Φ∗(X ,...,X :
φ 1 n
B C)= Φ∗(X ,...,X :B).
∨ φ 1 n
(d) If Y M are self-adjoint, D B, D C subalgebras of M, which are globally fixed by σφ, and
i
∈ ⊂ ⊂
B[X ,... ,X ] is free from C[X ,... ,X ] over D (with respect to the unique φ-preserving conditional
1 n 1 n
expectation from M onto D), then Φ∗(X ,... ,X ,Y ,... ,Y : B C) = Φ∗(X ,... ,X : B) +
φ 1 n 1 m ∨ φ 1 n
Φ∗(Y ,... ,Y : C).
φ 1 n
(e)Φ∗(X ,... ,X ,Y ,... ,Y :B) Φ∗(X ,... ,X :B)+Φ∗(Y ,... ,Y : B).
φ 1 n 1 n ≥ φ 1 n φ 1 m
(f)Φ∗(X ,... ,X : B) φ( X∗X )2 n2. Equalityholdsiff σφ(X ),...,σφ (X ) : t ,...,t R
φ 1 n · i i ≥ { t1 1 tn n 1 n ∈ }
have the same distribution aPs the semicircular family κs(σφ(X )),...,κs(σφ (X )) : t ,...,t R
{ t1 1 tn n 1 n ∈ }
withrespecttothefreequasi-free state, κ> 0.
Wemention that all of the statements in sections 3 and 4 of [7] remain valid for Φ∗; we leave details to
φ
thereader.
8 DIMITRISHLYAKHTENKO
One can also define and study free entropy χ∗(X ,...,X ) by setting Xǫ = X + Y to be the free
φ 1 n i i i
Brownianmotiondescribed inthebeginning ofthepaper, andletting
1 ∞ n
χ∗(X ,...,X )= Φ∗(Xǫ,... ,Xǫ) dt.
φ 1 n 2 Z (cid:18)1+t − φ 1 n (cid:19)
0
Theproperties ofχ∗( ,η)onceagaingeneralize toχ∗ (comparesection8of[7]).
··· φ
4. STATES ON A II1 FACTOR.
4.1. Φ∗ vs. Φ∗. Thefollowingtheoremissomewhatsurprising,sinceitshowsthatΦ∗ isidenticallyinfinite
φ τ φ
for most states φ on a II factor (the analogy with classical Fisher information would instead suggest that
1
φ Φ∗ wouldhavesomeniceconvexityproperties). This,ontheotherhand,goeswellwiththe“degener-
7→ φ
ateconvexity”propertyofthemicrostatesfreeentropyχ[15](whichisreflectedinthatitisidentically
−∞
ongenerators ofanyvonNeumannalgebrawhichmorethanoneunitaltrace).
Theorem4.1. Let M be a tracial von Neumann algebra, φ a faithful normal state on M, B M a
subalgebra sothat σφ(B) = B for allt,andX = X∗ M. ThenifJ (X : B)exists, the modula⊂r group
t ∈ φ
ofφmustfixX.
Proof. Let d M be a positive element, so that φ(x) = τ(dx), where τ is a normal faithful trace on M,
∈
and d is an unbounded operator on L2(M,τ), affiliated to M. The modular group of φ is then given by
σφ(x) = ditxd−it,x M. DenotingbyX theelementσφ(X),wethenget
t ∈ t t
X = X = d−itX dit, t R.
0 t
∈
Consider
φ(X2) = φ(J (X :B) X )= φ(J (X : B)d−itX dit).
0 φ · 0 φ t
Letb andb betwoelementsinthedomainof∂ ,sothatb = b∗. Thenweget,writingY = σφˆ(Y):
1 2 X 1 2 t t
φ(J (X : B)b X b ) = φˆ(Y b Y b )
φ 1 t 2 0 1 t 2
+φˆ(Y ∂ (b )X b )+φˆ(Y b X ∂ (b ))
0 X 1 t 2 0 1 t X 2
= φ(b )φ(b )φˆ(Y Y )
1 2 0 t
+φˆ(Y ∂ (b )X b )+φˆ([∂ (b∗)X b∗Y ]∗)
0 X 1 t 2 X 2 t 1 0
= φ(b )φ(b∗)φˆ(Y Y )
1 1 0 t
+φˆ(Y ∂ (b )X b∗)+φˆ([∂ (b )X b∗Y ]∗).
0 X 1 t 1 X 1 t 1 0
Now,forallm,n M,wehave
∈
φˆ(Y mY n)= φ(m)φ(n) = φˆ(mY nY ),
0 0 0 0
sothat
φˆ(Y ∂ (b )X b∗)+φˆ([∂ (b )X b∗Y ]∗) = φˆ(Y ∂ (b )X b∗)+φˆ([Y ∂ (b )X b∗]∗)
0 X 1 t 1 X 1 t 1 0 0 X 1 t 1 0 X 1 t 1
= φˆ(Y ∂ (b )X b∗)+φˆ(Y [∂ (b )X b∗])
0 X 1 t 1 0 X 1 t 1
R.
∈
Itfollowsthat
φ(J (X :B)b X b∗) = φˆ(Y b Y b∗).
ℑ φ 1 t 1 ℑ 0 1 t 1
FREEFISHERINFORMATIONFORNON-TRACIALSTATES 9
Nowfixt Randchoosea inthedomainof∂ , a 1,sothat
n X n
∈ k k ≤
a dit, a∗ d−it strongly.
n n
→ →
Onecanchoosea ,forexample, tobeelementsofthealgebraB[X]. Then
n
0 = φ(X2) = φ(J (X :B) X )
ℑ 0 ℑ φ · 0
= φ(J (X :B)d−itX dit)
φ t
ℑ
= lim φ(J (X :B)a X a∗)
φ n t n
n→∞ℑ
= lim φˆ(Y a Y a∗)
0 n t n
n→∞ℑ
= lim (φ(a )φ(a∗)φˆ(Y Y )
n n 0 t
n→∞ℑ
= lim φ(a )φ(a∗) φˆ(Y Y )
n n 0 t
n→∞ ℑ
= φ(dit)φ(d−it) φˆ(Y Y ).
0 t
ℑ
Sinceφˆ(Y Y )= φ(X X ),fortsufficientlyclosetozero(sothatφ(dit)= 0),wegetthat
0 t 0 t
6
φ(XX ) R.
t
∈
Thus
0 = τ(dXX ) τ(d(XX )∗)
t t
−
= τ(dXditXd−it ditXd−itXd)
−
= τ((dX Xd)ditXd−it)
−
= τ([d,X]ditXd−it).
Differentiating thisint,andnotingthat(d/dt) (ditXd−it)= i[d,X]gives
t=0
iτ([d,X]2)= 0.
Since[d,X]isanti-self-adjoint, thisimplies thatτ([d,X]2) = 0,sothat [d,X] = 0,because τ isfaithful.
| |
φ
Thismeansthatσ (X) = X forallt.
t
Corollary 4.2. Suppose that X ,...,X are self-adjoint generators of a II factor M. Letφ be a normal
1 n 1
faithful stateonM,anddenotebyτ theuniquefaithful normaltraceonM. ThenΦ∗(X ,...,X ) < +
φ 1 n ∞
impliesthat:
(1)Φ∗(X ,...,X ) < and
τ 1 n
∞
(2)φisamultipleofthetraceτ onM.
Proof. Clearly, the second statement implies the first. Toget the second statement, write φ() = τ(d) and
· ·
apply the theorem to conclude that [d,X ] = 0. Since X ,...,X generate M, d must be in the center
i 1 n
of M, which must consist of multiples of identity, since M is a factor. But then d is a scalar multiple of
identity, sothatφandτ areproportional.
4.2. Factoriality. Voiculescu showed [15] that for his microstates entropy χ the following implication
holds:
χ(X ,...,X ) > W∗(X ,...,X )isafactor.
1 n 1 n
−∞ ⇒
Infact,theconclusionisstronger: notonlyisthecenterofW∗(X ,...,X )istrivial,butsoisitsasymptotic
1 n
center. Unfortunately, wedon’tknow ifthesameimplication holds forthenon-microstates freeentropy χ∗
10 DIMITRISHLYAKHTENKO
introduced byVoiculescu in[16],orevenunder thestronger assumption thatΦ∗(X ,...,X )isfinite. We
1 n
proveaweakerversionoftheassertion aboveforΦ∗ = Φ∗. Wefirstneedatechnical lemma:
τ
Lemma4.3. Let φ be a normal faithful state on M. Let X M be self-adjoint and B M be a subal-
gebra, sothat σφ(B) = B for allt. Assume that p B is a∈self-adjoint projection, φ(p)⊂= α, and so that
t ∈
φ
σ (p) = pforallp. Assumethat [X,p] < δ. Then
t k k2
α2(1 α)2
Φ∗(X : B)> 4 − .
φ δ2
Proof. Let(A,τ)beacopyofL(F ),freefromB[X]. SinceΦ∗(X : B) = Φ∗(X : B A),andsincethe
2 φ φ ∨
centralizerof σφ(B) Aisafactor[1],wecanfindaprojectionq B A,whichisfixedbythemodular
{ t }t∨ ∈ ∨
group, andsothat [X,q] < δ,andτ(q) = β = m/nisrational andclosetoα. Wemaymoreoverfinda
2
k k
familyofmatrixunitse B A,1 i,j n,fixedbythemodulargroup, andsothat
ij
∈ ∨ ≤ ≤
e∗ = e , e e = δ e
ij ji ij kl jk il
m
1
τ(e ) = , q = e .
ii ii
n
Xi=1
Denote by C the algebra generated in B ∨Aby{eij}1≤i,j≤n. Note that C ∼= Mn×n, the algebra of n×n
matrices. Therestriction ofφ τ toC istheusualmatrixtrace. Then
∗
Φ∗(X :B)= Φ∗ (X :B A) Φ∗ (X : C).
φ φ∗τ φ∗τ
∨ ≥
WriteX = e Xe . Thentheinequality [X,q] < δ impliesthat
ij 1i j1 2
k k
δ > xq qx
2
k − k
= qxq+(1 q)xq qxq qx(1 q)
2
k − − − − k
= (1 q)xq qx(1 q)
2
k − − − k
= √2 qx(1 q) ,
2
·k − k
since(1 q)xq andqx(1 q)areorthogonal. Hence
− −
qX(1 q) < δ/√2.
2
k − k
Itfollowsthat
φ(X∗X )+ φ(X∗X ) < δ2.
ij ij ij ij
1≤i≤mX,m<j≤n m<i≤Xn,1≤i≤n
Denotebyφ′ thestaten(φ τ)(e e )one W∗(X,C)e . Then
11 11 11 11
∗ ·
Φ∗ ( X ) Φ∗ ( X )
φ′ { ij} ≥ φ′ { ij}
Xi,j
1
> 2m(n m)
− n(δ2/2m(n m))
−
(2m(n m))2
= −
nδ2
β2(1 β)2
= n34 − .
δ2
Arguingexactlyasin[4,Proposition 4.1],wegetthat
1 β2(1 β)2
Φ∗φ∗τ(X : C)= n3Φ∗φ′({Xij}) > 4 δ−2 .