On the Classification of Full Factors of Type III

16 Pages·0.18 MB·English

by  Dimitri Shlyakhtenko

ON THE CLASSIFICATION OF FULL FACTORS OF TYPE III.
DIMITRI SHLYAKHTENKO
2
0 ABSTRACT. We introducea new invariantS(M) for type IIIfactorsM with no almost-periodic
0 weights. We compute this invariant for certain free Araki-Woodsfactors. We show that Connes’
2 invariantτ cannotdistinguishallisomorphismclassesoffreeAraki-Woodsfactors. We show that

n thereexistsacontinuumofmutuallynon-isomorphicfreeAraki-Woodsfactors,eachwithoutalmost-
a
periodicweights.
J

1

]
A
1. INTRODUCTION.
O
ThenecessitytofindeffectiveinvariantstodistinguishfulltypeIIIfactorscomesfromtheprob-
.
h
lemofclassifyingtypeIIIfactorsnaturallyoccurringinfreeprobabilitytheoryofVoiculescu[22].
t
a These factors arise as free products of finite-dimensional or hyperfinite von Neumann algebras
m
[1, 14, 13, 6] and more generally from a second-quantization procedure involving the free Gauss-
[
ian functor (the so-called free Araki-Woods factors, [15]). The classification results so far all

1
relied on Connes’ S and Sd invariants [3, 4], and worked well for factors having almost periodic
v
7 weights (for example, in [15] a complete classification of free Araki-Woods factors for which the
0 freequasi-freestateisalmost-periodicwasgiven). However,notallfreeAraki-Woodsfactorshave
0
almostperiodicweights[17], andthequestionoftheircompleteclassificationremains open.
1
0 ThispaperisdevotedtotheexplorationoffreeAraki-Woodsfactors,havingnoalmost-periodic
2
statesorweights. Themostrefined existinginvariantforsuchfactorsisConnes’τ invariant[3,4],
0
/ whichmeasures the“degreeofcontinuity”ofthestates onM.
h
WeintroduceanewinvariantforafactorM, givenbytheintersectionoverthesetofallnormal
t
a
faithful states (or weights) φ on M of the collections of measures, absolutely continuous with
m
respect to the spectral measure of the modular operator of φ. This invariant is in spirit related to
:
v the Connes’ Sd invariant, where the intersection is taken over all almost-periodic weights on M.
i
X Amazingly,itturns outthatourinvariantcan becomputedforcertain free Araki-Woodsclasses.
r Using this invariant we are able to produce a pair of non-isomorphicfree Araki-Woods factors,
a
whichcannot bedistinguishedby theirτ invariant.
Acknowledgement. IwouldliketothankMSRIandtheorganizersoftheoperatoralgebrasprogram
fortheirhospitalityandfortheencouragingatmosphere. IamalsogratefultoProfessorsN.Brown,
T.Giordano,U. Haagerup, Y. Uedaand D.-V. Voiculescuformanyusefuldiscussions.
2. SOME EXAMPLES OF TOPOLOGIES ASSOCIATED TO UNITARY REPRESENTATIONS OF R
Thepurposeofthissectionistosetnotationandtoprovecertain resultsaboutunitaryrepresen-
tationofR, whichareneeded in therestofthepaper.
Date:1stFebruary2008.
ResearchsupportedbytheNationalScienceFoundation.
1
2 DIMITRISHLYAKHTENKO
2.1. Spectral measures of group representations. Let µ be a measure on a measure space X.
DenotebyC thecollectionofmeasures on X
µ
C = ν : µ(Y) = 0 ν(Y) = 0 forallmeasurableY X .
µ
{ ⇒ ⊂ }
In other words, C is the collection of all measures, which are absolutely continuous with respect
µ
to µ. We shall abuse notation and write C if T is a self-adjoint operator. By this we mean the
T
collectionofallmeasures, absolutelycontinuouswithrespect to thespectralmeasureofT onR.
It should be mentioned that, following an idea of Mackey, C can be viewed as a kind of re-
µ
placement for the notion of the support of µ. Indeed, any ν C can be written as f µ for some
µ
∈ ·
function f, which is completely determined except on a set of µ-measure zero. One can form
intersectionsoftwofamiliesC andC ;thisoperationistoremindusofthenotionofintersection
µ ν
ofsets. Similarly,inclusionofC andC canbethoughtofastheinclusionofthesupportofµinto
µ ν
that of ν. We shall say that a collection of measures C is supported on a set Y if for all ν C,
thecomplementofY has measurezero. Notealso that ifµ is atomic,then theknowledgeof∈C is
µ
exactlyequivalentto theknowledgeoftheset ofatomsofµ.
We now recall some basic facts about representations and duality of locally compact abelian
groups (see e.g. [10]). Let σ : G U(H) be a -strongly continuous representation of G on
→ ∗
a Hilbert space H. Associated to σ it is spectral measure class C on the dual group Gˆ. The
σ
class C can be defined as the smallest collection of measures, so that (1) if µ C and µ′ is
σ σ
a.c. with respect to µ, then µ′ C and (2) for each ξ H, the measure obtained∈as the Fourier
σ
transform of the positive-defini∈tefunction g ξ,σ(g)∈ξ belongs to C . If H is separable, there
σ
is a measure µ in C , with the property that7→it gheneratesiC (i.e., C is the smallest collection of
σ σ σ
measures satisfying(1)and containingµ). We sometimesrefer tothisµ as “the”spectral measure
of σ. In particular, H can be decomposed as H = H dµ(χ), so that σ = σ , where
Rχ∈Gˆ χ Rχ∈Gˆ χ
σ (g) ξ = χ(g) ξ, ξ H .
χ µ
Ifth·egroupG·contai∈nsRas adensesubgroup,therepresentationσ can berestrictedtoR G.
Infact,allrepresentationsσ ofGariseasextensionsofrepresentationsofR,whicharecontin⊂uous
not just in the topology on R, but also in the restriction of the topology τ on G to R G.
G
⊂
The measure class C and the spectral measure µ of σ, when interpreted as objects on R Gˆ
σ
become exactly the measure class and the spectral measure of the restriction of σ to R, viebwe⊃d as
arepresentationofR(thisisevidentfromthedirectintegraldecompositionformulastatedabove).
Inparticular,arepresentationπ ofR extendstoarepresentationofGiffitsspectralmeasureC is
π
supportedonGˆ R.
Itiscustomary⊂tobchooseaparticularspectralmeasureofarepresentationofRonH,byfinding
on H anon-negativeoperator A, for which π(t) = Ait, and lettingµ be thespectral measureof A
(composed with some faithful state on B(H)). In particular, denoting by σ the extension of π to
G, wehaveC = C = C .
σ π ∆
2.2. Topologiesinducedbyunitaryororthogonalrepresentations ofR. Letµbeameasureon
thereal line,sothat µ( X) = µ(X). Denotebyπ theassociated (real orcomplex)representation
ofR onL2(R,µ)given−bythemap
π(t) = M ,
exp(2πitx)
ONTHECLASSIFICATIONOFFULLFACTORSOFTYPEIII. 3
whereM denotestheoperatorofmultiplicationbyg. Writeτ(µ)fortheweakesttopologymaking
g
the map t π(t) U(L2(R,µ)) continuous with respect to the strong operator topology on the
unitarygro7→up ofL2∈(R,µ). Ifµ is notsupportedon acyclicsubgroupofR, π is injectiveand τ(µ)
isaHausdorfftopology.
Proposition2.1. ThecompletionofRwithrespecttothetopologyτ(µ)isalocallycompactgroup
iffeitherτ(µ)istheusualtopologyonR,orµisatomic(inwhichcasethecompletioniscompact).
Proof. Denote by (G,τ) the completion of (R,τ(µ)). Then R G is an inclusion of locally
⊂
compactabeliangroups;byPontrjaginduality,thisinclusionisdualtotheinjectivedenseinclusion
Gˆ R.
B⊂ybthe structure theory for locally compact abelian groups [10], the connected component of
identityofGˆ musthavetheform R H, withR = Rn and H compactand connected.
∼
×
First note that H = e . Indeed, the image of H in R must be a connected compact subgroup
ofR, hencemustbethe{tr}ivialgroup. b
SbincetherearenocontinuousinjectivemapsfromRn intoRforn > 1,eithern = 1orn = 0. If
n = 1, all continuous injectivegroup homomorphismsfrom R to itself are surjective(their image
mustbeapath-connectedsubgroupofR). HenceinjectivityofGˆ R requires thatGˆ = R inthis
case, theinclusionbeahomeomorphismontoR and henceτ(µ) b⊂ethbeusualtopologyon R.
If n = 0, Gˆ must be discrete. This corresponbds to the completionG being compact. Moreover,
it is not hard to see that µ must be supported on Gˆ R, since the representation π with spectral
measureµmustextend(by thedefinitionofτ(µ))to⊂thebcompletionG. Hence µis atomic.
Lemma 2.2. A sequence t ∞ converges to zero in τ(µ) iff µˆ(t ) 1, where µˆ is the Fourier
{ n}n=1 n →
transformof µ.
Proof. LetπbearepresentationofRassociatedtoµasabove,andletξ L2(R,µ)betheconstant
∈
function1. Then
µˆ(t) = π(t)ξ,ξ .
h i
By definition, t 0 in τ(µ) iff π(t ) 1 strongly. The vector state φ(T) = Tξ,ξ defines
n n
a faithful normal s→tate on the commutativ→e von Neumann algebra π(R)′′ B(L2(hR,µ))i. Hence
⊂
strongconvergenceofπ(t ) to1 isequivalentto
n
π(t ) 1 = φ((π(t ) 1)(π(t ) 1)∗))1/2 0.
n 2 n n
k − k − − →
In otherwords,π(t ) 1stronglyiff
n
→
φ(π(t )+π(t )∗) 1,
n n
→
i.e., µˆ(t ) 1. Since µˆ(t ) 1, thishappens iffµˆ(t ) 1.
n n n
ℜ → | | ≤ →
Theorem 2.3. There exists a continuum of non-atomic measures µ , λ I so that the topologies
λ
∈
τ(µ ) aremutuallynon-equivalent.
λ
Proof. Let c : n = 1,2,... be a sequence of real numbers, so that c 0, and c2 < + .
{ n } n ≥ P k ∞
Denotebyµ then-fold convolutionofdelta-measures
n
1 1 1 1
µ = ( δ + δ ) ( δ + δ ).
n 2 c1 2 −c1 ∗···∗ 2 cn 2 −cn
4 DIMITRISHLYAKHTENKO
Each µ isasymmetricprobabilitymeasure;and theFouriertransform ofµ is givenby
n n
n
µˆn(t) = Ycos(2πckt).
k=1
TheFouriertransform oftheweak limitµ of µ isgivenby thepointwiselimitofthis expression;
n
themeasureµisnon-atomic(seee.g. [8]).
Let c = 3−k!. Let 0 < λ 1 befixed. Let t = λ3n! = λc−1. We claimthat t 0 in τ(µ) iff
k ≤ n n n →
λ = 1. Recall thatt 0in τ(µ) iffµˆ(t ) 1.
n n
→ →
Ifλ < 1, µˆ(t ) cos(2πc t ) = cos(2πλ) < 1, sothat t doesnot convergeto 0in τ(µ).
n n n n
| | ≤ | | | |
Ifλ = 1,
n
µˆ(tn) = Ycos(2π3n!−k!) Ycos(2π3n!−k!) = Ycos(2π3n!−k!).
·
k=1 k>n k>n
Thereexistsω > 0 sothat for0 x ω,cos(2πx) 1 49x2. Hence
≤ ≤ ≥ −
µˆ(tn) Y(1 49 6n!−k!)
≥ − ·
k>n
as longas k > nand nis suchthat3n!−k! 3n!−(n+1)n! = 3−n·n! < ω.
≤
Sincefora [0,1/2],
∈
lim(1 49ap)p = 1
p→∞ −
uniformly, and the function p (1 49ap)p is increasing for any a < 1, given 1 > δ > 0, there
7→ −
existsap = p(δ) < + ,sothat
∞
(1 49ap) > (1 δ)1/p
− −
forall a [0,1/2]. Hence lettinga = 1,weget thatforanynso thatk! n! n n! > p,
∈ 6 − ≥ ·
µˆ(tn) Y(1 δ)1/(k!−n!).
≥ −
k>n
Hence
1
logµˆ(tn) Xlog(1 δ) .
≥ − k! n!
k>n −
Sincelog(1 δ) < 0,and
−
1 1 1 1 1
X X = X X < e,
k! n! ≤ k! k! k! · 1 1 ≤ k!
k<n − k<n − n k<n − n k<n
weget that
logµˆ(t ) elog(1 δ).
n
≥ −
Hence
µˆ(t ) (1 δ)e
n
≥ −
foranynso that(1)3−n·n! < ω and (2)n n! > p(δ). Itfollowsthatµˆ(t ) 1as n .
n
· → → ∞
ONTHECLASSIFICATIONOFFULLFACTORSOFTYPEIII. 5
Nowfix 0 < λ < 1 andset
µ (X) = µ(λ X)
λ
·
foranyBorel setX R. Then
⊂
µˆ (t) = µˆ(t/λ).
λ
It follows that for any 0 < ν λ, the sequence ν3n! is convergent to 0 in τ(µ ) iff ν = λ. It
λ
≤
followsthat τ(µ ) : 0 < λ 1 are pairwisenon-equivalent.
λ
{ ≤ }
Theauthoris indebtedtoU. Haagerup forcommunicatingtohimthefollowingexample:
Theorem 2.4. There exists a measure µ, so that µ as well as its arbitrary convolution powers
µ µ (any number of times) are singular with respect to Lebesgue measure, but τ(µ) is the
∗ ···∗
usualtopologyon theadditivegroupofreal numbers.
Proof. Let µ beas intheproofofTheorem 2.3,withc = 3−n. Then
n
t
µˆ(t) = Ycos(2π ).
3k
k≥1
Wefirstclaimthatτ(µ)istheusualtopologyontherealline. Assumethatt ,butµˆ(t ) 1.
n n
→ ∞ →
ChooseN so that for all n > N, t > 9. Choosek so that c = t 3−k > 1 but t 3−k−1 = c/3 1.
n n n
≤
Then
µˆ(t) cos(2πt 3−k) cos(2πt 3−k−1) cos(2πt 3−k−2) = cos(2πc) cos(2πc/3) cos(2πc/9),
n n n
≤ · · · ·
where, 1 < c 3. Let
≤
f(c) = cos(2πc) cos(2πc/3) cos(2πc/9).
· ·
It is not hard to see that f(c) is strictly less than 1 on the interval 1 < c 3. It follows that
≤
µˆ(t) < 1 whenevert > 9. Contradiction.
All convolution powers of µ are singular with respect to Lebesgue measure (see the discussion
of Taylor-Johnson measures [8] for examples of similar measures µ but satisfying even stronger
propertiesthan whatweneed here).
2.3. Bimoduledecompositionsofcrossedproducts. Let(M,φ)beavonNeumannalgebra,φa
faithful normal state, and let G be a locally compact abelian group. Assume that α is an action of
G onM, which leavesφ invariant. Then thecrossedproduct vonNeumannalgebra C = M ⋊ G
α
contains a canonical copy A of the group algebra L(G) = L∞(Gˆ); moreover, the weight φ gives
∼
rise to a normal faithful conditional expectation E : C A. Let ψˆ be a normal faithful weight
→
on A, and let ψ = ψˆ E. This is a normal faithful weight on C. Moreover, L2(C,ψ) is an
◦
A,A-bimoduleinanatural way.
Fix an isomorphism(A,ψˆ) = L∞(Gˆ,ν ). Denote by ℓ2 the Hilbert space with basis e ,e ,...
∼ G 1 2
and by ℓ ℓ2 the subspaces spanned by e ,... ,e . Given a measure η on X X whose
n 1 n
⊂ ×
projectionsontothethecoordinatedirectionsonX X arebothequivalenttoν ,andamultiplicity
G
functionn : X X N ,let ×
× → ∪{∞}
H(η,n) = L2(X X,η,n)
×
6 DIMITRISHLYAKHTENKO
be the space of square-integrable functions from X X ℓ2, so that f(x,y) ℓ for
n(x,y)
× → ∈
all x,y X (where for convenience we set ℓ = ℓ2). Endow H(η,n) with an A,A-bimodule
∞
∈
structureby letting
(f h g)(x,y) = f(x)h(x,y)g(y), f,g A, h H.
· · ∈ ∈
In fact [2, 11, 5] any bimoduleover A can be represented in this way. It is easily seen that if η′ is
another measure on X X, projecting onto µ, and η′ is mutually absolutely continuous with η,
×
thenH(η,n) = H(η′,n) asbimodulesoverA.
∼
Choosevectorsξ ,ξ ,... L2(M,φ), ξ = 1, ξ 1,ameasureµonGˆ andamultiplic-
1 2 1 2 i 2
∈ k k k k ≤
ityfunctionn : Gˆ N,so that:
→
α (ξ ) α (ξ ) i,j g,h G
g i h j
⊥ ∀ ∀ ∈
L2(M,φ) = span α (ξ ) : g G,i = 1,2,...
g i
{ ∈ }
ξ ,α (ξ ) = µˆ (g),
i g j i
h i
where µi = µ n−1(i) andˆdenotes the Fourier transform. Let (X,µ) = (Gˆ,νG), where νG is the
| ·
Haarmeasureon G. Let
η(x,y) = µ(x y), n(x,y) = n(x y)
− −
beameasureandamultiplicityfunctiononGˆ Gˆ,andletH = H(η,n)(notethattheprojections
×
of η onto the coordinate directions are precisely µ ν = µ(Gˆ) ν ). We claim that H(η,n) =
G G ∼
∗ ·
L2(C,ψ) as bimodules. To see this, one can verify that the map pξip p(x)p(y)χn−1({i}) for a
7→
projection p L∞(Gˆ) with ψˆ(p) < + , is a bimodule isometry from the linear span of pξ p
i
∈ ∞ ∈
L2(C) toH(η,µ).
Notethatthemeasureµ(whichisthe“spectralmeasure”oftherepresentationofGonL2(M))is
uniquelydetermineduptoabsolutecontinuitybytheL(G),L(G)-bimodulestructureofL2(C,ψ).
3. FULL TYPE III FACTORS.
Assume that M is a full factor, so that its group of inner automorphisms Inn(M) is a closed
subgroup of the group of all automorphisms Aut(M), endowed with the u-topology [9, 4]. Let
Out(M) = Aut(M)/Inn(M), with the quotient topology. Denote by π the quotient map from
Aut(M) to Out(M), and by δ the composition π σφ (which is independent of t by Connes’
◦ t
Radon-Nikodymtypetheorem[3]).
Assume that the action of R on M by t σφ extends, for some φ, to an action of a locally
compact completion G of R (by Proposition7→2.1tabove, this means that either G is just R, or G
is compact, and φ is almost-periodic). In this case, call φ a G-state (or weight) on M. Call the
crossed product
M ⋊ G
σφ
theG-coreofM. Itisknown[3,4]thattheG-coreofM isindependentofthechoiceoftheG-the
stateφ(havingtheproperty thatitsmodulargroupextendstoG).
Definition 3.1. Let M befull. Then define:
(a) τ(M) to be the weakest topology on R making the map δ : R Out(M) continuous (this
→
invariant was introduced by Connes in [4]). Thus t 0 in τ(M) iff there exists a sequence of
n
→
ONTHECLASSIFICATIONOFFULLFACTORSOFTYPEIII. 7
faithfulnormalsemifiniteweightsφ onM, forwhichσφn idin theu-topology.
n tn →
(b) Denote by τ the weakest topology on R making the modular group σφ of a normal faithful
φ t
semifinite weight φ on M continuous in the u-topology on Aut(M). Then define τ(M) to be the
weakest topology among all τ . In other words, t t in τ(M) iff σφ id for some normal
φ n → tn−t →
faithfulsemifiniteweightφ onM;
(c)TheS invariantto betheintersection
S(M) = \ C (∆φ)⊗n.
Ln
φf.n.stateonM
Wesimilarlydefine
W (M) = \ C (∆φ)⊗n.
Ln
φaf.n.strictlysemifiniteweightonM
Notethat W (M) S(M).
⊂
Part (c) of the definition is equivalent to the Sd invariant of Connes [4] if the intersection were
tobetaken overall faithfulnormalalmost-periodicweights.
Note that since we are dealing only with states in the definition of S(M), the delta measure at
1 isalways inS(M).
Notethat τ(M) isnotweaker thanτ(M).
Proposition3.2. W (M) is containedin C ,theclassof theHaarmeasureon R = R.
λ +
b
Proof. Let φ be any G-weight on M. Denote by µ the spectral measure of infinitesimal generator
∆ of the unitary group σφ acting on (M,φ). Denote by d the function x x on R . On
φ t 7→ +
B(L2(Rˆ)), considerthenormalfaithfulweightψ = Tr(d ). Then themodulargroupofψ is given
·
byconjugationwithdit,andhenceψ isaweightonB(L2(R)). Inparticular,themodulargroupof
ψ, actingon L2(R), is theleft regularrepresentation ofR.b
Consider the nbormal faithful weight θ = φ ψ on M B(L2(R)) = M. Then the spectral
∼
measure class ∆ d is the Lebesgue measure⊗λ on R, sin⊗ce the lebft regular representation of R
φ
is absorbing for te⊗nsor products. Hence W (M) can be no bigger than the class of the Lebesgue
measure.
AssumethattheG-core ofM is afactor. Notethat sincetheG-corehas asemifinitenormal trace,
it is a full iff its compressionby a finite projection is non-Γ. In particular, if the G core is a factor
and isfull,it hasno non-trivialcentral sequences.
Theorem 3.3. Assumethat for someG-state φ on a factor M the G-core is a full factor. Then M
isfull.
Proof. Let C = M ⋊ G be the G-core of M. Assumefor contradiction that M is not full. Then
σ
by [4, Corollary 3.6, Proposition 2.8] there exists a sequence of unitaries m M, φ(m ) = 0,
n n
∈
and so that [m ,m] 0 -strongly for all m M and [m ,φ] 0 in norm on M for any
n n ∗
→ ∗ ∈ →
φ M . View m C M. We claim that m is a non-trivial central sequence in C. Let
∗ n n
∈ ∈ ⊃ { }
E : C L(G) C be the canonical conditional expectation. Then E(m ) = φ(m ) = 0,
n n
→ ⊂
so that m is not asymptotically scalar. For any m M, [m ,m] 0 -strongly. Denote by
n n
∈ → ∗
U L(G) C, g G, the implementing unitaries. Then by Connes’ results [4, Theorem
g
∈ ⊂ ∈
8 DIMITRISHLYAKHTENKO
2.9(3)], for any t R G, [m ,U ] 0 -strongly. Hence [m ,u] 0 -strongly for any
n g n
u W∗(M,U : t∈ R)⊂= C. Hence m→is a∗n asymptotically central seq→uence∗in C. Since C is
t n
∈ ∈
assumedtobea fullfactor, and isoftypeII ,wehavearrivedat acontradiction.
∞
Theorem 3.4. Assumethatfor someG-stateφ on M, theG-core of M is a fullfactor. Then if for
some H not necessarily locally compact containing R as a dense subgroup, there is an H-weight
ψ onM, onemusthavethatH G.
⊂
Proof. Let C = M ⋊ G. Let L(G) C be the canonical copy of the group algebra of G;
σφ
⊂
for g G, denote by w L(G) the implementing unitary. We write E for the canonical
g L(G)
∈ ∈
conditionalexpectationfromC ontoL(G).
Assume that H G, so that the topology defined by the inclusion R H is not stronger than
the topology defin6⊂ed by the inclusion R G. Hence there exists a se⊂quence t R, so that
n
⊂ ∈
σψ id, butσφ does notconverge. Letu = [φ : ψ] M C. It followsthat
tn → tn t t ∈ ⊂
Ad = σψ.
utwt|C t
In particular, [u w ,x] 0 -strongly for all x M C. Note moreover that u w com-
tn tn → ∗ ∈ ⊂ t t
mutes with u w (since they form a one-parameter subgroup of U(C)). Hence for s fixed,
s s
[u w ,u w ] = 0, and since u w asymptotically commutes with M C, it follows that
tn tn s s tn tn ⊂
also [u w ,w ] 0 -strongly. It followsthat u w is a central sequence in M. Hence, by the
tn tn s → ∗ tn tn
assumption that C is a full factor, and by the fact that C is type II , we find that λ u w 1
-stronglyfor somescalars λ T. Butthen ∞ n tn tn →
n
∗ ∈
E (λ u w ) 1
L(G) n tn tn →
-strongly. Sinceu M C,E (u ) C,sothatλ′ w 1 -stronglyforsomesequence
∗λ′ C, λ 1. Htnen∈ce λ′⊂π(t ) L(G1) -tnstro∈ngly, where πn istnth→e lef∗t regular representation of G
n ∈ | | ≤ n n → ∗
(sincetherepresentation ofGon L2(C,Tr) isa multipleofitsleftregularrepresentation).
Choose now φ a function on G, supported in a compact neighborhood of identity and so that
φ = 1. Then λ π(t ) φ φ in L2. In particular, it means that the support X of φ and its
2 n n
k k · →
translate X + t cannot be disjoint once n is sufficiently large. It follows that t X X for
n n
∈ −
sufficientlylargen. It followsthatt 0in G. Contradiction.
n
→
Corollary3.5. If the G-core of M is full, then τ(M) = τ , the weakest topology making the
G
inclusionR G continuous.
⊂
Wealso have:
Proposition3.6. If M hasa G-stateandG H is aproperinclusionoflocallycompactabelian
⊂
groups,then theH-coreof M is nota fullfactor.
Proof. Let φ be a G-state on M, and denote by C the H-core M ⋊ H. Assume that C is a
σφ
factor. By assumption, there exists a sequence t R, t 0 in the topology of G, but t
n n n
∈ →
not convergent in H. Denote by w C, h H, the implementing unitaries for the H action
h
∈ ∈
on M. Then Ad (x) x -strongly for all x M C; moreover, Ad (w) = w for all
wtn → ∗ ∈ ⊂ wtn
w W∗(w : h H) = L(H). Hence w form a central sequence. Arguing exactly as in the
las∈t paragrahph of∈the proof of Theorem 3.4tnwe find that for no sequence of scalars λ T does
n
∈
ONTHECLASSIFICATIONOFFULLFACTORSOFTYPEIII. 9
λ w 1 -strongly in the group algebra L(H). Hence w is a non-trivial central sequence in
n tn → ∗ tn
M.
Choose p L(H) C a projection of finite trace. Then [p,w ] = 0 and hence pw p is a
∈ ⊂ tn tn
central sequence in pCp, which has a finite trace. Thus pCp has property Γ. Hence by Connes’
results[4], pCp and henceC isnotfull.
4. CROSSED PRODUCTS, FREE ENTROPY DIMENSION AND THE S INVARIANT.
The main result of this section is a computation of the W and S invariants of some type III
factors M, forwhichthecorehas asequenceofgenerators withlargefree entropydimension. We
first recall somepreliminaries.
4.1. Free entropy dimension for infinite families of generators. It is useful for us to consider
Voiculescu’sfreeentropy dimensioninthecontextofan infinitefamilyofgenerators x ,x ,...in
1 2
a von Neumann algebra M. We point out the necessary modifications of Voiculescu’s approach
([20, 21]; see also [18], where a similar modification was necessary). We freely use the notations
of[19, 20, 21].
Let x , 1 i < + and y , 1 i < + be in M. Fix a free ultrafilter ω and an element κ in
i i
≤ ∞ ≤ ∞
theStone-Chech compactificationof(0,1],notlyingin thisinterval. Define
χω(x ,... ,x : y ,y ,... ;m,ε) = liminfχω(x ,... ,x : y ,y ,... ,y ;m,ε)
1 p 1 2 1 p 1 2 q
q→∞
(note that the liminf is actually a limit in this definition). Define χω(x ,... ,x : y ,y ,...) in
1 p 1 2
exactlythesameway as in [20], but usingχdefined above. Onestillhastheproperty
χω(x ,... ,x : y ,y ,...) χω(x ,... ,x : z ,... ,z )
1 p 1 2 1 p 1 l
≤
forall z ,... ,z W∗(x ,... ,x ,y ,y ,...).
1 l 1 p 1 2
∈
Define
δ0 (x ,x ,... ,x : y ,y ,...) =
ω,κ 1 2 p 1 2
χω(x +√εS ,x +√εS ,... ,x +√εS : S ,... ,S ,y ,y ,...)
1 1 2 2 p p 1 p 1 2
p lim ,
−ε→κ logε
whereS ,... ,S are afree semicircularfamily,freefrom x y . Nowdefine
1 p i i j j
{ } ∪{ }
δ(x ,x ,...) = liminfδ0 (x ,... ,x : x ,x ,...).
1 2 ω,κ 1 p p+1 p+2
p→∞
For a finite family x ,... ,x this is exactly Voiculescu’s definition of free entropy dimension. In
1 n
general, δ(x ,x ,...) liminf δ0 (x ,... ,x ). Moreover,
1 2 ≤ p→∞ ω,κ 1 p
0 δ(x ,x ,...)
1 2
≤
iffW∗(x ,x ,...) can beembeddedintoRω, theultrapowerofthehyperfiniteII factor.
1 2 1
10 DIMITRISHLYAKHTENKO
Ifx ,... ,x arefreeformx ,x ,...,thenδ0 (x ,... ,x : x ,...) = δ0 (x ,... ,x ).
1 p p+1 p+2 ω,k 1 p p+1 ω,κ 1 p
In particular, if the families x ,... , x ,... , y ,y ,... are free, and y ,y ,... ′′ is em-
1 p 1 2 1 2
{ } { } { } { }
beddable,we getby [21]that
δ(x ,y ,x ,y ,...) = lim δ0 (x ,...x ,y ,... ,y : y ,y ,...)
1 1 2 2 ω,k 1 p 1 p 1 2
p→∞
= lim δ0 (x ,... ,x )+δ(y ,y ,...)
ω,k 1 p 1 2
p→∞
Xδ(xk).
≥
k
Definition 4.1. Let M beaII vonNeumann algebra. Denoteby
1
δ(M) = sup δ(x ,x ,...)
1 2
x1,x2,...∈M
where the supremum is taken over all self-adjoint families (finite or infinite) x ,x ,... of gener-
1 2
ators of M. If N is type II , we write δ(N) for the supremum over all finite-trace projections
∞
p N ofδ(pNp).
∈
Remark 4.2. It is quite likely that δ(M) ,0,1,+ if M is type II . Note also that
∞
∈ {−∞ ∞}
δ(M) δ(M B(H))forall M.
≤ ⊗
Whileδ(M) is clearly an invariantofM, ourinabilityto provethat thenumberδ(x ,x ,...) is
1 2
independent of the choice of generators x ,x ,... [19, 20, 21] results in the inability to compute
1 2
the exact value of δ for infinite-dimensional von Neumann algebras. However, as we pointed out
above, if M = L(F ) N, with N Rω and n = 1,2,... or + , we have δ(M) n and
n
∗ ⊂ ∞ ≥
δ(M B(H)) n (in fact, = + by [7]). Furthermore, as Voiculescu provedin [20], δ(R) = 1
⊗ ≥ ∞
if R is the hyperfinite II (or II ) factor; more generally, δ(M) 1 if M has property Γ or has a
1 ∞
≤
Cartansubalgebra(sincethesepropertiesareinheritedbycompressionsofavonNeumannalgebra,
thesestatementsare validforM typeII ortypeII ).
1 ∞
The following theorem essentially follows from the results of [20]; we sketch the necessary
modifications of the proof coming from the fact that we may be dealing with infinite families of
generators.
Theorem 4.3. Let M be a II or II von Neumann algebra. Let L∞(X,µ) = A M be a
1 ∞ ∼
⊂
diffuse abelian subalgebra, so that Tr is semifinite. View L2(M) as an A,A bimodule, and
M A
|
identifyit with H(η,n) for some measure η on X X. Assumethat η is disjointfrom µ µ, i.e.,
× ×
X X = Y Y , so thatη(Y ) = 0 and(µ µ)Y = 0. Then δ(M) 1.
1 2 1 2
× ∪ × ≤
Proof. (Sketch). We first reduce to the case that M is type II . Given t (0,+ ), let p A be
1
∈ ∞ ∈
a finite projection, correspondingto thecharacteristic function of someset Y X, µ(Y) < + .
⊂ ∞
Then view pMp as a bimodule over pAp. It is not hard to see that pMp can be identified with
H(η′,n′), with η′ absolutely continuous with respect to η , n′ = n . If η is
Y×Y⊂X×X Y×Y⊂X×X
| |
disjoint from µ µ, then η′ is disjoint from µ′ = µ . If the statement of the theorem can be
Y
× |
proved for pMp and pAp pMp, we would have that δ(qMq) 1 for all q M of finite trace
⊂ ≤ ∈
(since qMq depends up to isomorphism only on the trace of q). Hence by definition we get that
δ(M) 1.
≤