On the Uniqueness of the Injective III1 Factor Preamble by Alain Connes

34 Pages·2016·0.28 MB·English

On the Uniqueness of the Injective III Factor
1
Uffe Haagerup
Received: August2,2016
CommunicatedbyJoachimCuntz
Abstract. We give a new proof of a theorem due to Alain Connes,
thataninjectivefactorN oftypeIII withseparablepredualandwith
1
trivialbicentralizerisisomorphictotheAraki–WoodstypeIII factor
1
R . This, combined with the author’s solution to the bicentralizer
∞
problemfor injective III factors providesa new proofof the theorem
1
that up to -isomorphism, there exists a unique injective factor of
∗
type III on a separable Hilbert space.
1
2010 Mathematics Subject Classification: 46L36
Preamble by Alain Connes
UffeHaagerupsolvedthehardestproblemoftheclassificationof
factors,namelythe uniquenessproblemforinjectivefactorsoftype
III . Thepresentpaper,takenfromhisunpublishednotes,presents
1
a direct proof of this uniqueness by showing that any injective fac-
toroftypeIII isaninfinitetensorproductoftypeIfactorssothat
1
the uniqueness follows from the Araki–Woods classification. The
proof is typical of Uffe’s genius, the attack is direct, and combines
his amazing controlof completely positive maps and his sheer ana-
lytical power, together with his solution to the bicentralizer prob-
lem. After his tragic death, Hiroshi Ando volunteered to type the
manuscript1. Some pages were missing from the notes, but eventu-
ally Cyril Houdayer and Reiji Tomatsu suggested a missing proof
of Lemma 3.4 and Theorem 3.1. We heartily thank Hiroshi, Cyril
and Reiji for making the manuscript available to the community.
We also thank SørenHaagerupfor giving permissionto publish his
father’s paper.
1Themanuscriptistyped byHiroshiAndo(ChibaUniversity)incooperation withCyril
Houdayer (Universit´e Paris-Sud), Toshihiko Masuda (Kyushu University), Reiji Tomatsu
(HokkaidoUniversity),YoshimichiUeda(KyushuUniversity)andWojciechSzymanski(Uni-
versityofSouthernDenmark).
1193
1194 Uffe Haagerup
Contents
1 Introduction 1194
2 Preliminaries 1196
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196
2.2 Connes–Woods’ characterizationof ITPFI factors . . . . . . . . 1197
2.3 Bicentralizers on type III factors . . . . . . . . . . . . . . . . . 1198
1
2.4 Almost unitary equivalence in Hilbert N-bimodules . . . . . . . 1199
3 Completely positive maps from m m-matrices into an in-
×
jective factor of type III 1200
1
4 Q-stable states on III factors 1208
1
5 Proof of Main Theorem 1213
1 Introduction
The problem, whether all injective factors of type III on a separable Hilbert
1
space are isomorphic, has been settled affirmatively. The proof of the unique-
nessofinjectiveIII factorsfallsintwoparts,namely(see 2.3forthedefinition
1
§
of the bicentralizer):
Theorem1.1([Con85]). LetM beaninjectivefactoroftypeIII onaseparable
1
Hilbert space, such that the bicentralizer B is trivial (i.e., B =C1) for some
ϕ ϕ
normal faithful state ϕ on M, then M is -isomorphic to the Araki–Woods
∗
factor R .
∞
Theorem 1.2([Haa87]). For anynormal faithful stateϕon an injective factor
M of type III on a separable Hilbert space, one has B =C1.
1 ϕ
In this paper we give an alternative proof of Theorem 1.1 above, based on
the technique of our simplified proof [Haa85] of Connes’ Theorem [Con76]
“injective hyperfinite” in the type II case2. The key steps in our proof of
1
⇒
Theorem 1.1 are listed below:
Step 1
By use of continuous crossed products, we prove that the identity map on an
injective factor N of type III has an approximate factorization
1
~>>R@@
Sλ~~~ @@@Tλ
~ @
~~ @
N //N
idN
2Typewriter’s note: Haagerup usedthis technique to give anew proof of the uniqueness
ofinjectivetypeIIIλ (0<λ<1)factor. Thisresulthasbeenpublishedin[Haa89].
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Uniqueness of the Injective III Factor 1195
1
throughthehyperfinitefactorRoftypeII ,suchthat(S ) and(T ) are
1 λ λ∈Λ λ λ∈Λ
nets ofnormalunitalcompletely positive maps, andfor a fixednormalfaithful
state ϕ on N (chosen prior to S and T ), there exist normal fatihful states
λ λ
(ψ ) on R, such that for all t R and λ Λ,
λ λ∈Λ
∈ ∈
ϕ T =ψ , ψ S =ϕ,
λ λ λ λ
◦ ◦
σϕ T =T σψλ,
t ◦ λ λ◦ t
σψλ S =S σϕ,
t ◦ λ λ◦ t
and Sλ Tλ(x) x ϕ λ→∞0 for all x N, where y ϕ :=ϕ(y∗y)21 (y N).
k ◦ − k → ∈ k k ∈
Step 2
From Step 1, we deduce that a certain normal faithful state ϕ (Q-stable state
defined in 4) on an injective factor N of type III has the following property:
1
§
for any finite set of unitaries u ,...,u in N and for every γ,δ > 0, there
1 n
exists a finite-dimensional subfactor F of N such that
ϕ=ϕF ϕFc,
| ⊗ |
and such that there exist unitaries v ,...,v in F and a unital completely
1 n
positive map T: F N such that
→
ϕ T =ϕ,
◦
σϕ T T σϕ|F γ t, t R,
k t ◦ − ◦ t k≤ | | ∈
and
T(v ) u <δ, k =1,...,n.
k k ϕ
k − k
Step 3
WeprovethatifN,ϕ,F,u ,...,u ,v ,...,v areasinStep2,thenforeveryσ-
1 n 1 n
strongneighborhood of0inN,thereexistsafinitesetofoperatorsa ,...,a
1 p
V
in N such that
p p
(a) a∗a 1+ and a∗a 1,
i i ∈ V i i ≤
i=1 i=1
X X
p p
(b) ε a a∗ 1+ and ε a a∗ 1,
F,ϕ i i!∈ V F,ϕ i i!≤
i=1 i=1
X X
p
(c) a ξ ξ a 2 <δ′,
i ϕ ϕ i
k − k
i=1
X
p
(d) a u v a 2 <δ′, k =1,...,n.
k i k− k ikϕ
i=1
X
Documenta Mathematica 21 (2016)1193–1226
1196 Uffe Haagerup
Here ξ denotes the unique representing vector of ϕ in a natural cone. The
ϕ
above δ′ > 0 depends on γ and δ in Step 2, and δ′ is small when γ and δ
are small. Here, ε is the ϕ-invariant conditional expectation of N onto F.
F,ϕ
Moreover in (c), the standard Hilbert space H of N is regarded as a Hilbert
N-bimodule, by putting ηa:=Ja∗Jη (a N, η H).
∈ ∈
Assume now that the bicentralizer of any normalfaithful state on N is trivial.
Then by an averagingargument, we can exchange (b) by
p p
(b’) a a∗ 1+ and a a∗ 1.
i i ∈ V i i ≤
i=1 i=1
X X
Step 4
From(a),(b’),(c)and(d)above,wederivethatthereexistsaunitaryoperator
w N such that
∈
wξ ξ w <ε
ϕ ϕ
k − k
and
wu v w <ε, k =1,...,n,
k k ϕ
k − k
where ε is small when δ′ is small and is a small σ-strong neighborhood of 0
V
in N. The key partof Step4 is atheoremaboutgeneralHilbert N-bimodules,
which was proved in [Haa89].
Step 5
From Step 4, we get that for every finite set of unitaries u ,...,u N and
1 n
∈
every ε > 0, there exists a finite dimensional subfactor F (namely w∗Fw) of
1
N and n unitaries v′,...,v′ in F (namely w∗v w, k=1,...,n), such that
1 n 1 k
(i) v′ v <ε
k k− kkϕ
and
(ii) kϕ−ϕ|F1 ⊗ϕ|F1ck<ε.
The last inequality follows from the fact, that when w almost commutes with
ξ ,italmostcommuteswithϕtoo. Theproperties(i)and(ii)aboveshowthat
ϕ
ϕ satisfies the product condition of Connes–Woods [CW85] and thus N is an
ITPFI factor. But it is well-known that R is the only ITPFI factor of type
∞
III (cf. [AW68] and [Con73]).
1
2 Preliminaries
2.1 Notation
We use M,N,... to denote von Neumann algebras and ξ,η,... to denote vec-
tors in a Hilbert space. Let M be a von Neumann algebra. (M) denotes the
U
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Uniqueness of the Injective III Factor 1197
1
unitary group of M. For a faithful normal state ϕ on M, we denote by ∆
ϕ
(resp. J ) the modular operator (resp. modular conjugation operator) asso-
ϕ
ciated with ϕ, and the modular automorphism group of ϕ is denoted by σϕ.
The norm x ϕ =ϕ(x∗x)12 defines the strong operatortopology (SOT) on the
k k
unit ball of M. The centralizer of ϕ is denoted by M .
ϕ
2.2 Connes–Woods’ characterization of ITPFI factors
RecallthatavonNeumannalgebraM withseparablepredualiscalledhyperfi-
niteifthereexistsanincreasingsequenceM M offinite-dimensional
1 2
*-subalgebras such that M = ( ∞ M )′′. ⊂A fact⊂or··M· is called an Araki–
n=1 n
Woods factor or an ITPFI (infinite tensor product of factors of type I) factor,
S
if it is isomorphic to the factor of the form
(M ,ϕ ),
i i
i∈I
O
where I is a countable infinite set and each M (resp. ϕ ) is a σ-finite type I
i i
factor (resp. a faithful normal state). Araki and Woods classified most ITPFI
factors:
Theorem 2.1 ([AW68]). There exists a unique ITPFI factor with separable
predual for each type I , II , II and III ,λ (0,1]. In particular, all ITPFI
∞ 1 ∞ λ
∈
factors of type III are isomorphic to
1
R := (M (C),Tr(ρ )),
∞ 3
·
n∈N
O
where ρ:= 1 diag(1,λ,µ) and 0<λ,µ satisfies logλ / Q.
1+λ+µ logµ ∈
It is clear that an ITPFI factor with separable predual is hyperfinite. The
convereseis also true for factors not of type III , but false in general. Namely,
0
Connes–Woods[CW85]characterizedhyperfinitefactorsoftypeIII withsepa-
0
rable predual which areisomorphic to ITPFI factors by the approximate tran-
sitivity of their flow of weights, while the existence of hyperfinite factors of
type III with separable predual which are not isomorphic to ITPFI factors
0
had been shown in [Con72]. Let N be a vonNeumann algebra,and let F be a
finite dimensionalsubfactor ofN with relative commutantFc :=F′ N in N.
∩
Then it is elementary to check, that the map
n n
x y x y , x F, y Fc (1 i n)
i i i i i i
⊗ 7→ ∈ ∈ ≤ ≤
i=1 i=1
X X
is an isomorphism of F Fc onto N. If ω is a normal state on F and ω is a
1 2
⊗
normal state on Fc, we let ω ω denote the corresponding state on N, i.e.,
1 2
⊗
(ω ω )(xy)=ω (x)ω (y), x F, y Fc.
1 2 1 2
⊗ ∈ ∈
Documenta Mathematica 21 (2016)1193–1226
1198 Uffe Haagerup
In our proof of
[N injective III and B =C1] N =R ,
1 ϕ ⇒ ∼ ∞
we shall need the following criterion for a factor to be ITPFI:
Proposition 2.2 ([CW85, Lemma 7.6]). Let N be a factor on a separable
Hilbert space. Then N is ITPFI if and only if N admits a normal faithful state
ϕ with the following property: for every finite set x ,...,x of operators in N,
1 n
for every ε > 0, and every strong* neighborhood of 0 in N, there exists a
V
finite dimensional subfactor F of N, such that
x F + , k =1,...,n
k
∈ V
and
ϕ ϕF ϕFc <ε.
k − | ⊗ | k
2.3 Bicentralizers on type III factors
1
In this subsection, we recall Connes’ bicentralizers. Let M be a σ-finite von
Neumann algebra, and let ϕ be a normal faithful state on M. We denote
by AC(ϕ) the set of all norm-bounded sequences (x )∞ in M such that
n n=1
lim ϕx x ϕ =0 holds.
n→∞ n n
k − k
Definition 2.3 (Connes). The bicentralizer of ϕ is the set B of all x M
ϕ
∈
such that lim xa a x =0 holds for all (a )∞ AC(ϕ).
n→∞k n− n kϕ n n=1 ∈
Since B is avonNeumannsubalgebraofM [Haa87,Proposition1.3],itholds
ϕ
that lim xa a x ♯ =0.
n→∞k n− n kϕ
It was conjectured by Connes that for all factors of type III with separable
1
predual, the bicentralizer B of any normal faithful state ϕ on M is trivial,
ϕ
i.e., B =C1 holds. This is still an open problem. We will need the following
ϕ
result on type III factors, known as the Connes–Størmer transitivity:
1
Theorem 2.4 ([CS78]). Let M be a type III factor with separable predual.
1
Then for every faithful normal states ϕ,ψ on M and ε > 0, there exists a
unitary u (M) such that uϕu∗ ψ <ε holds.
∈U k − k
Connes showed that by the Connes-Størmer transitivity, for a type III factor
1
M with separable predual, the triviality of B for one fixed faithful normal
ϕ
state ϕ on M implies the triviality of B for every faithful normal state ψ
ψ
(see [Haa87,Corollary1.5]forthe proof). He alsoshowedthatthe trivialityof
the bicentralizer is equivalent to the following property (the proof is given in
[Haa87, Proposition 1.3 (2)]):
Proposition 2.5 (Connes). Let M be a von Neumann algebra with a normal
faithful state ϕ. Then B =C1 holds, if and only if the following condition is
ϕ
satisfied: for every a M and δ >0,
∈
conv u∗au;u (M), uϕ ϕu δ C1= ,
{ ∈U k − k≤ }∩ 6 ∅
where conv is the closure of the convex hull in the σ-weak topology.
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Uniqueness of the Injective III Factor 1199
1
We will use the following variant of Proposition 2.5.
Proposition 2.6. Let M beatypeIII factor with separable predual, andlet ϕ
1
be a normal faithful state on M whose modular automorphism group σϕ leaves
a finite-dimensional subfactor F globally invariant. Let ε : M F be the
F,ϕ
→
normal faithful ϕ-preserving conditional expectation. Assume that B = C1.
ϕ
Then for every δ >0 and a M, we have
∈
ε (a) conv u∗au; u (Fc), uξ ξ u δ . (1)
F,ϕ ϕ ϕ
∈ { ∈U k − k≤ }
Here, ξ is the representing vector of ϕ in the natural cone.
ϕ
Proof. Theproofisessentiallythe sameasProposition2.5,soweonlyindicate
theoutline. NotethatbyAraki-Powers-Størmerinequality,foreveryu (M)
∈U
one has:
ξ uξ u∗ 2 ϕ uϕu∗ ξ uξ u∗ ξ +uξ u∗ .
ϕ ϕ ϕ ϕ ϕ ϕ
k − k ≤k − k≤k − k·k k
Thereforeintheargumentsbelow,wemayreplacethecondition“ uξ ξ u
ϕ ϕ
k − k≤
δ” in Proposition 2.6 with the condition “ uϕ ϕu δ”, as we take δ > 0
k − k ≤
to be arbitrarily small. As was pointed out in [Haa87, Remark 1.4], it follows
from the proof of Proposition 2.5 that the condition B =C1 is equivalent to
ϕ
the next condition that for all a M and δ >0,
∈
ϕ(a)1 conv u∗au;u (M), uξ ξ u <δ . (2)
ϕ ϕ
∈ { ∈U k − k }
δ>0
\
Let a∈M. Since M ∼=F ⊗Fc with ϕ=ϕ|F ⊗ϕ|Fc, we may now apply (2) to
Fc(∼=M) and ϕ|Fc to obtain
εF,ϕ(a)=idF ϕFc(a) conv u∗au; u (Fc), uξϕ ξϕu δ .
⊗ | ∈ { ∈U k − k≤ }
Note that we usedthe factthat ϕu uϕ = ψu uψ ,where ψ :=ϕFc and
k − k k − k |
u (Fc)thankstotheexistenceofanormalfaithfulϕ-preservingconditional
∈U
expectation from M onto Fc.
2.4 Almost unitary equivalence in Hilbert N-bimodules
We recall a result about almost unitary equivalence in Hilbert bimodules es-
tablished in [Haa89] which is a generalizationof [Haa85, Theorem 4.2]. Let N
be a von Neumann algbera, and H be a normalHilbert N-bimodule, i.e., H is
a Hilbert space on which there are defined left and right actions by elements
from N:
(x,ξ) xξ, (x,ξ) ξx, x N, ξ H
7→ 7→ ∈ ∈
such that the above maps N H H are bilinear and
× →
(xξ)y =x(ξy), x,y N, ξ H.
∈ ∈
Moreover, x L , where L ξ := xξ (ξ H) is a normal unital -
x x
7→ ∈ ∗
homomorphism, and x R , where R ξ := ξx (ξ H) is a normal unital
x x
7→ ∈
-antihomomorphism.
∗
Documenta Mathematica 21 (2016)1193–1226
1200 Uffe Haagerup
Definition 2.7. Let N be a von Neumann algebra, let (N,H) be a normal
HilbertN-bimodule,andletδ R . Twon-tuples(ξ ,...,ξ )and(η ,...,η )
+ 1 n 1 n
∈
of unit vectors in H are called δ-related, if there exists a family (a ) of
i i∈I
operators in N, such that
a∗a = a a∗ =1
i i i i
i∈I i∈I
X X
and
a ξ η a 2 <δ, k =1,...,n.
i k k i
k − k
i∈I
X
We will use the followingresultwhichrelates the δ-relatednessto approximate
unitary equivalence in Hilbert N-bimodules:
Theorem 2.8 ([Haa89, Theorem 2.3]). For every n N and ε > 0, there
∈
exists a δ = δ(n,ε) > 0, such that for all von Neumann algebra N and δ-
related n-tuples (ξ ,...,ξ ) and (η ,...,η ) of unit vectors in a normal Hilbert
1 n 1 n
N-bimodule H, there exists a unitary u (N) such that
∈U
uξ η u <ε, k =1,...,n.
k k
k − k
Remark 2.9. As canbe seeninthe proofof[Haa89, Theorem2.3],in orderto
showthattheconclusionofTheorem2.8holds,itsufficestoshowthefollowing:
for every σ-strong neighborhood of 0 in N, there exist a ,...,a N such
1 p
V ∈
that
p
a ξ η a 2 <δ, k =1,...,n (3)
i k k i
k − k
i=1
X
p p
a∗a 1, a a∗ 1 (4)
i i ≤ i i ≤
i=1 i=1
X X
p p
a∗a 1+ , a a∗ 1+ . (5)
i i ∈ V i i ∈ V
i=1 i=1
X X
Thisisbecausewecanobtaintheconclusionsof[Haa89,Lemma2.5]outof(3),
(4) and (5), which is enoughto proveTheorem2.8. We will use this variantin
the proof of Lemma 5.6.
3 Completely positive maps from m m-matrices into an injective
×
factor of type III
1
The main result of this section is:
Theorem 3.1. Let N be an injective factor of type III with separable predual,
1
and let ϕ be a faithful normal state on N. Then for every finite set u ,...,u
1 n
of unitaries in N, and every ε,δ > 0, there exists m N, a unital completely
∈
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Uniqueness of the Injective III Factor 1201
1
positive map T: M (C) N, and n unitaries v ,...,v in M (C), such that
m 1 n m
→
ψ =ϕ T is a normal faithful state on M (C), and
m
◦
σϕ T T σψ δ t, t R,
k t ◦ − ◦ t k≤ | | ∈
T(v ) u <ε, k =1,...,n.
k k ϕ
k − k
In the following we let M =N ⋊ R be the crossedproduct of N by σϕ with
σϕ
generators π (x) (x N) and λ(s) (s R). We identify π (x) with x N.
σϕ σϕ
∈ ∈ ∈
Let a be the (unbounded) self-adjoint operator for which λ(s)=exp(isa) (s
∈
R). For f L1(R), we define the Fourier transform fˆby
∈
1 ∞
fˆ(s)= e−istf(t)dt, s R.
√2π ∈
Z−∞
In the sequel, von Neumann algebra-valuedintegrals are understood to be the
σ-weaksense. Let(θsϕ)s∈R be the dualactionofσϕ onM. By [Haa79-2], there
exists a normal faithful semifinite operator-valued weight P: M N (N
+ + +
→
is the extended positive part of N) given by
b b
∞
P(x)= θϕ(x)ds, x M . (6)
s ∈ +
Z−∞
Following [CT77], if we put
c
m:=span x M ;sup θϕ(x)dt < ,
∈ + t ∞
(cid:26) c>0(cid:13)Z−c (cid:13) (cid:27)
(cid:13) (cid:13)
then the formula (6) for x m makes s(cid:13)ense and P(x(cid:13)) N. Moreover, m
∈ (cid:13) (cid:13) ∈ ∋
x P(x) N defines a positive linear map.
Fo7→r all x ∈ m, the σ-weak integral c θϕ(x)dt is σ-strongly convergent as
∈ −c t
c . The range of P is contained in π (N), because π (N) is the fixed
σϕ σϕ
→ ∞ R
point algebra in M under the dual action.
Lemma 3.2. Let t x(t) be a σ-strongly* continuous function from R to N
7→
such that t x(t) is in L1(R) L∞(R). Put
7→k k ∩
∞
x:= λ(t)x(t)dt M.
∈
Z−∞
Then x∗x m, and
∈
∞
P(x∗x)=2π x(t)∗x(t)dt.
Z−∞
Proof. Note first, that
x∗x= x(s)∗λ(t s)x(t)dsdt
R2 −
ZZ
= x(s)∗λ(t)x(s+t)dsdt.
R2
ZZ
Documenta Mathematica 21 (2016)1193–1226
1202 Uffe Haagerup
Put f (s)=e−s2/(4n) (s R), and
n
∈
g (t)= 1 ∞ f (s)e−itsds= n 21 e−nt2 (t R).
n n
2π π ∈
Z−∞ (cid:16) (cid:17)
Using that θϕ(y) = y (s R,y N), θϕ(λ(t)) = e−istλ(t) (s,t R)3 and the
s ∈ ∈ s ∈
Fubini Theorem, we have for every ψ M ,
∗
∈
∞
ψ, θϕ(x∗x)f (u)du =
h u n i
Z−∞
∞ ∞ ∞
= e−ituf (u)ψ(x(s)∗λ(t)x(s+t))dsdtdu
n
Z−∞Z−∞Z−∞
∞ ∞
= g (t) 2πψ(x(s)∗λ(t)x(s+t))ds dt.
n
Z−∞ (cid:18)Z−∞ (cid:19)
Since t ∞ ψ(x(s)∗λ(t)x(s+t))ds is in C (R) and g n→∞ δ (weak∗ in
7→ −∞ 0 n → 0
C (R)∗), we have
0
R
∞ ∞
lim ψ, θϕ(x∗x)f (u)du = ψ,2π x(s)∗x(s)ds .
n→∞h Z−∞ u n i h Z−∞ i
Sinceψ M isarbitrary,θϕ(x∗x) 0andf 1uniformlyoncompactsets,
∈ ∗ u ≥ n ր
it follows that
∞ ∞
lim θϕ(x∗x)f (u)du=2π x(s)∗x(s)ds (σ-strongly).
u n
n→∞
Z−∞ Z−∞
∞
Therefore x∗x m, and P(x∗x)=2π x(t)∗x(t)dt.
∈
Z−∞
Lemma 3.3. Let a be the (unbounded) self-adjoint operator affiliated with M
for which exp(ita) = λ(t) (t R) holds. Let α > 0, and let e be the spectral
α
∈
projection of the operator a corresponding to the interval [0,α]. Then for each
x N, one has e xe m and
α α
∈ ∈
∞ 1 cosαt
P(e xe )= σϕ(x) − dt, x N. (7)
α α t πt2 ∈
Z−∞
Proof. It is sufficient to consider the case x 0, so we can assume that x =
≥
y∗y (y N). For f L1(R) L∞(R) C(R), we have
∈ ∈ ∩ ∩
1 ∞ 1 ∞
yfˆ(a)= yλ( t)f(t)dt= λ( t)σϕ(y)f(t)dt
√2π − √2π − t
Z−∞ Z−∞
1 ∞
= λ(t)σϕ (y)f( t)dt.
√2π −t −
Z−∞
3Typewriter’s note: Haagerup used the convention θsϕ(λ(t)) = eistλ(t). However, since
thenegativesignconventioniswidelyaccepted, wedecidedtochangethedefinition.
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