I. J. Mah. Math. Sci.
(198) I-9
Vol. 6 No.
CLASSIFICATION OF INJECTIVE FACTORS:
THE WORK OF ALAIN CONNES
STEVE WRIGHT
Department of Mathematical Sciences
Oakland University
Rochester, Michigan 48063
(Received May 7, 1981)
ABSTRACT:
The fundamental results of A. Connes which determine a complete set
of isomorphism classes for most injectlve factors are discussed in detail.
After
some introductory remarks which lay the foundation for the subsequent discussion,
an historical survey of some of the principal lines of the investigation in the
classification of factors is presented, culminating in the Connes-Takesakl structure
theory of type III factors.
After a discussion of inJectlvity for finite factors,
the main result of the paper, the uniqueness of the injectlve
deduced, and the structure of
II.
II 1
factor, is
and type III injectlve factors is
then obtained
as corollaries of the main result.
1980 AMS (MOS) SUBJECT CLASSIFICATION:
PRIMARY
46L35, 46LI0;
SECONDARY- 46L50, 46L05
KEY WORI]S AND PHRASES:
and o
jiVe
yon
Nann o.%gebra, fi facr, dcre
p, f
tor
pro,
yon
N
gebra.
I.
INTRODUCTION.
From the beginning, one of the central problems of the theory of von Neumann
algebras has been their classification according to isomorphism type.
on the problem was initiated by
([I], [2], [3], [4]),
The attack
the" founding fathers Murray and von Neumann
who introduced the fundamental notion of types (I,
IIi, II,
and III), and who claimed early victories with the characterization of the hyperflnlte
S. WRIGHT
2
As the years
II I factor and a complete classification of the factors of type I.
passed, there was vigorous development of the theory, but no results appeared that
However,
were as definitive as these early advances.
the publication of the thesis [5] of Alain Connes.
[6], Connes
results of Tomita and Takesaki
in 1973 a new era dawned with
Basing his work on earlier
initiated a program for the classifica-
In [5],
tion of factors that can be termed as nothing short of revolutionary.
[7], [8], and [9],
he obtained a classification of factors of type III and auto-
morphisms of certain factors of type II
work of
[I0],
and
I
II
which culminated in the remarkable
in which appear the first major advances beyond the classical theory
in the classification of factors for the non-type I case.
The purpose of these
notes is to give a rather detailed discussion of the most important of these results.
We assume the reader has a familiarity with the theory of von Neumann algebras
on the level of
[II],
role in the sequel.
We recall some basic facts that play an important
say.
A factor is a von Neumann algebra with a trivial center, i.e.,
the only elements of the algebra which commute with every element are scalar
We will be concerned primarily with von Neumann algebras
multiples of the identity.
[12],
of finite type, and we will use the tracial characterization of this (see
Section
111-8).
on the factor
Recall that a trace
functional of norm
(ab)
I, i.e.,
N
a state on
V
(ba)
is a positive linear
which satisfies
N
b
a
N
(i.l)
A factor is said to be finite if it has a trace.
The trace on a finite factor is
uniquely determined among the states by (i.I), and it is automatically faithful
and ultraweakly continuous (Theorem 2.4.6 of
[13]).
These facts will be used
frequently.
We will also employ the standard representation of a finite factor
N
be the (canonical) trace on
Then the representation
,H,}
Naimark-Segal construction
corresponding to
ultraweakly continuous, and the cyclic vector
is cyclic for
L2(N, )
and the
(N)’
L2-nQrm of
x
N
is by definition
llxll 2
Let
in the Gelfandis faithful and
is separating for
denotes the commutant).
where
N
(N)
We denote
1/2
(x*x)
H
(i.e.,
by
is
CLASSIFICATION OF INJECTIVE FACTORS
N
called the standard representation of
,T(N)
L2(N,T),
H
in
products of C*-algebras and
act on Hilbert
N
acts @ta.ndardly in
H
(R)
I
H
A @ A2
1
the algebra
Form the algebraic tensor product
B(H 1
which can be viewed in a natural way as a *-subalgebra of
H
Tensor
C*-algebras, which for convenience we will assume
Ai,i=l,2 be
spaces H i,i=1,2
of bounded operators on
with its image
Neumann algebras w..ll play an important role in
yon
Let
what follows.
and when we identify
N
we will say that
3
A seminorm
2
p
A2
AI
on
(R)
H
2)
is called a
C*’-subcross seminorm if
p(x*x)
(i)
(ll)
p(a
I
p(x)
(R)
A1
x
A2,
llalll lla211
_<
a2)
2
is a C*-crossnorm if
p
a
A
i
(1), and (li) with -<
is a norm satisfying
replaced
If we take the supremum of all C*-subcross seminorms defined on
by
we will obtain a C*-crossnorm
A1 8 A
norm definable on
norm
acts on
A 1 8 A2
H
H
(R)
I
2
A
which we denote respectively by
(R)
1
maX
A
(R)
I
H
2)
A
and thus obtain a
information about tensor products is
M.1
and
N.1
isomorphisms, and let
A1
A2
mill
A2
the so-called
These algebras will in
are von Neumann
A2
and
relative to the weak operator
A
2
Neumann subalgebra of B(H (R) H2)
1
An excellent source of
an__d A2
1
[14].
We claim that
Ti,i
A
(R)
I
AI
yon
are finite factors,
denotes isomorphism).
respectively.
It can be shown that
2
A2
and
1
of
A (R) A
1
2
the (spatial) W*-tensor product
Suppose
H
(R)
A
and
2
If we further assume that
algebras, we may also take the closure of
B(H
2
-norm will yield C*-algebras
AI
maximal an__d minimal C*-tensor products o__f
topology on
HI
o[
v-norm and the
in the
general be distinct.
8 A
A 1 0 A2 by
and so each element of A @ A2 has a
I
is the smallest C*-crossnorm definable on
the operator norm
1
which by definition will be the lrgest C*-cross-
ohn-d ogerator
considered as a
Completing
v
A
Another crossnorm can be defined on
2
AI e A2
recalling that
(’-
1,2
i
i
M (R)
i
1,2,
M2"-= N 1
(R)
and
N
2
denote the canonical traces on
M
i
Let
N
i
i
i:Mi Ni
1,2
be
M.1 and N i
By uniqueness of the trace,
i
o
T
i
i
i
1,2
(I 2)
’4
S. WRIGHT
M
I (R) 2 is an isomorphism of
(I 0"2 o (I (R) 2 i (R) 2
Now
N (R) N
1
2
traces on
M
1
(R)
M
2
onto
N (R) N
I
N
onto
Since
(R)
i
[15], Lemma I, (see Lemma 9.2 infra)
Thus by
of
@ M
2
2
(R) N
1
(1.2),
and by
2
are the canonical
I (R) 2
and
respectively, they are faithful and normal.
M (R) M
1
2
and
1
I (R) 2
extends to an isomorphism
2
Let
Finally, we define the infinite tensor product of von Neumann algebras.
M
} be a sequence of von Neumann algebras, let
n
and let
n
n
C*-tensor product
be a normal state on
Mn
denote the infinite tensor product state on the infinite
(R)
n
M
of the
n
M’s
([13],
n
Section 1.23).
M
n n
determined by
elfand-Naimark-Segal representation of
Neumann algebra
n
(n Mn)"
is called the
denote the
Let
Then the
yon
tensor product of
(W*-)
M
n
}
relative to
2.
AN HISTORICAL PERSPECTIVE ON CONNES’ WORK
We will preface our discussion of [I0] by a rather brief overview of the main
lines of work which led up to it.
Because of the limitations of space and time,
we have concentrated on emphasizing only work which deals directly with the
classification of factors, and have reluctantly suppressed discussion of many other
interesting and important developments in the structure theory of von Neumann
algebras.
yon
Neumann algebras do, with the work
[I], [2], [3], and [4].
The classification of factors was
Our story begins, as all stories about
of Murray and von Neumann
the motivating problem for this work, and all other progress on the problem was
based on this pioneering effort.
In
fundamental notion of types (I, II I,
[I], Murray
II,
and
and von Neumann introduced the
III), and
obtained a complete classification of the factors of. type I:
H
type I, then there exists a Hilbert space
the algebra of all bounded operators on
H
such that
[2], they
in [i] and
M
if
M
is a factor of
is isomorphic to
B(H)
For our purposes, however, the most
important advance that Murray and von Neumann made was their famous characterization
of the hyperfinite II
factor.
Since this result will play a key role in the
sequel, we will describe it in detail.
CLASSIFICATION OF INJECTIVE FACTORS
M be a
Let
yon
Neumann algebra acting on a separable Hilbert space (from
now 0mtil the end of this section, all von
Neumann algebras will be assumed acting
on a separable Hilbert space without explicit mention).
{Mn}
finite if there is a sequence
is said to be
M
of flnite-dimensional
yon
If
weak operator topology.
M
{Xl,...,xn}
subfactor
C
of
of
M
for
there is a finite-dlmenslonal
such that
C
is
C of Murray-von Neumann):
>0
M and
Vl,...,vn E
M and
Neumann subalgebras
is a finite factor, hyperfiniteness of
equivalent to the following condition (condition
each finite subset
h_-
M relative to the
totally ordered by inclusion, whose union is dense in
M
of
5
llxi
-vill2 <
,
i
l,...,n
The following theorem will be used in a crucial way in our later discussion:
2. I.
II
I.
If
([4],
THEOREM.
M
I
and
We will denote by
R
Let
M
both satisfy condition
C
Theorem XIV).
II
the hyperfinite
I
be fa,ztors of type
and
then
M
and
M2
are isomorphic.
factor, unique up to isomorphism by
1
Theorem 2. I.
Another important technique contained in the Murray-yon Neumann papers is the
so-called group-measure space construction of factors.
give the first examples of type
II 1
and
example of a factor of type III in [3].
construction right now:
II(R)
This technique was used to
factors in
[2],
and the first
We will not go into the details of this
it will emerge later as a special case of the Connes-
Takesaki crossed product construction.
The next major advance
in
the structure theory of von Neumann algebras
occured in 1949 with the publication of von
Neumann’s
reduction theory [16].
This paper introduced and used the concept of direct integral of Hilbert spaces
to decompose an arbitrary separably acting yon Nemann algebra into a direct
integral of factors, and thereby reduced the study of
yon
Neumann algebras, at
least in principle, to the study of factors.
This then was essentially the state of the art in the classification of von
Neumann algebras at the start of the 1950’s.
Most of the work in the area
consisted in a refining and strengthening of the tools bequeathed by Murray and
6
S. WRIGHT
von Neumann.
The theory at that time suffered from a dearth of examples of
nonlsoorphic factors, and consequently one of the sin lines of work was the
construction of such examples.
III factor in
],
3
Von Neumann had constructed one example of a type
but did not give another nonlsomorphic one.
In 1956, Pukansky
[17] constructed a pair of nonisomorphic type III factors using refinements of the
group-measure space construction, and there the matter lay until 1963 when J.T.
Schwartz [18] found a new invariant, his well-known Property
algebra
M
acting on
closed convex hull of
H has Property P, if for each
{U T U*: U a unitary operator
B(H)
T
M}
in
_P:
a
yon
Neumann
the weakly
has a nonempty inter-
section with
M’
hyperfinite II
and also to construct another type III factor different
1 factors,
Schwartz used this property to distinguish two nonlsorphlc,
from Pukansky’s examples [19].
P
property
He showed that all hyperflnlte factors have
and conjectured the converse.
This converse was one of the many
things established by Connes in [i0].
Since it had been a fairly difficult task to obtain such a small number of
examples of nonisomorphic factors, there gradually emerged a hope that a somewhat
complete classification of all factors just might be possible.
In 1967, that
hope began to fade when R.T. Powers [20] constructed the first uncountable family
of nonisoorphic factors.
Let
matrices.
RX
Powers factor
M
many copies of
We note that
R
1
RX2
if
I
and
+
is defined as the infinite tensor product of countably
is the hyperfinite
of type III for each
2
complex
defined by
relative to the product state
2
denote the 2 x 2
M2
and let
denote the state defined on
%
+-The
[0,I]
Let
X
(0,I)
II
1
factor.
(R)
n
n
R
where each
(0, I).
is
X
is a hyperflnite factor
and Powers showed that
are distinct elements of
n
RX
is not isoorphlc to
I
We will see this class of
factors again.
After
Powers’
examples appeared, the flood gates opened, and nonisomorphlc
7
CLASSIFICATION OF INJECTIVE FACTORS
In 1969, McDuff [21] exhibited an uncountable family
factors soon poured out.
factors, and in 1970, Sakai [22], [23] found uncountable families of
of II
and nonhyperfinite type III factors.
classi=fication of all factors.
II
This work dashed forever the hope of a complete
Attention began to focus on particular classes of
factors for which there did seem to be a possibility of a satisfactory classification
with results that were to shortly revolutionize the theory.
{M
Let
for each
in
T
Assume
n
Hn
that
M
n
Then the functional
acts on
x
H
n
is a state on
n
(R)
the product state
n
M
n
n
x
and let
defined on
M
n
by
n
m
<
m
n
be a fixed unit vector
I
of type
00
x
n
Form the infinite tensor product
n
M
M
be a sequence of finite factors, with
n
CTXn, Xn)
T
(R)
n
M
n
relative to
The resulting factor is an example of what Araki and
00
x
n
Woods [24] dubbed an ITPFI factor (for "infinite tensor product of finite type I’s").
In this paper, which appeared in 1968 and was motivated in part by Powers’ work,
these authors introduced an important new invariant for the ITPFI factors, the
asymtotic ratio set, and succeeded in obtaining an almost complete classification
of this class of factors.
Both for its historical importance and for its fore-
shadowing of even bigger things to come, we will briefly describe the asymptotic
ratio set and the Araki-Woods classification.
Let
of
[0, (R))
M
be a factor.
The asymptotic
r(M) o__f
rat___i_0_o se___t
M
is the subset
defined as follows:
{x
r(M)
U {x
[0, i]:
M is isomorphic to
M
(R)
R
(i,):
M is isomorphic to
M
(R)
R
x
/x
where
of
RX
r(M)
definition
is the Powers factor defined previously.
This is not the definition
first given by Araki and Woods for the ITPFI factors.
([24],
The first
Definition 3.2) was expressed in terms of a complicated limiting
procedure involving the ratios of the eigen-values of the density matrices of the
states occurring in the infinite tensor product decomposition of the ITPFI factors,
and Araki and Woods later showed that the definition given above was equivalent
to the original one
they deduced that
([24],
r=(M)
proof of Theorem 5.9).
If
M
is an
must have one of the following forms:
ITPFI
factor,
8
S. WRIGHT
so
o}
S
{1}
S
{0,xn
X
}
0,+I,_+2
n
{0,1}
So1
[0,)
s(R)
S
Araki and Woods showed that
corresponds to only one isomorphism class of
ITPFI factors, and it is of type III.
invariants
S
(0 i)
x
X
[0, I]
x
The
were also shown to come from only one isomorphism
class, the one determined by the Powers factor
R
Thus all isomorphism classes
X
of ITPFI factors, with the exception of those in the
determined.
R
We will denote this factor by
class, were completely
S01
The Araki-Woods classification scheme was the first significant
advance in the classification of factors beyond the initial results of Murray and
von
Neumann, and
it was to have a great influence on the young Alain
Connes.
The second major development which Connes would put to good use came from
Japan.
A problem of interest at the time concerned the commutant of a tensor
product of von Neumann algebras:
(M
(R)
M2)*
equal
i (R) M_?
if
M
are von Neumann algebras, does
M
2
and
In 1967, M. romita
[25],[26]
answered this question
affirmatively by a new and original analysis of the spatial relationship between
avon Neumann algebra and its commutant.
The exposition of [25] and [26] was
somewhat obscure, however, and in 1970, M. Takesaki published his seminal monograph
[6] which explained and extended Tomita’s earlier work.
be a von Neumann
M
Let
algebra with a vector which is both cyclic and separating for M. Takesaki associated
a
closed, densely defined, self-adjoint operator
which has two very useful properties
M
involutive isometry
J
with
The first is that
forms a one-parameter unitary group for which
@utomorphism roup of
A
A-itMit
and the second is that
A
M,
{A
o
the modular er,
it
M
t
the modular
M
and
M
.
induces a conjugate-linear,
of the underlying Hilbert space for which
This shows in particular that
(-,(R))}
JMJ
M
are anti-isomorphic, and is the key to
Tomita’s proof of the commutation theorem for tensor products.
In actuality, the
existence of a cyclic and separating vector is not necessary for the definition
CLASSIFICATION OF INJECTIVE FACTORS
of the modular operator, and in fact if
M
linear functional on
morphism group
(M)
where
{A t}
is any
faithful, normal, positive
A
then the modular operator
and the modular auto-
can be constructed in
corresponding to
{n,H}
9
is the representation of
H
relative to
arising from the GNS
M
The construction of the modular operator and the
construction induced by
verification of its main properties are quite technical, and for that reason we
will not go into the details.
We instead refer the reader to Rieffel and van Daele’s
excellent development of the Tomita-Takesaki theory [27], and, of course, also to
the original memoir [6].
The stage was now set, and in 1973
Connes’
thesis
[5] appeared.
This work
contained a classification scheme for factors of type III which was to have a
profound influence on all subsequent work in this area.
Taking his cue from the
Araki-Woods classification scheme and the Tomita-Takesaki theory, Connes introduced
the modular spectrum as an isomorphism invariant for type III von Neumann algebras.
M
Let
S(M)
be a separable acting type III von Neumann algebra.
operators
A
(see [29]) of the modular
corresponding to all faithful, normal, positive linear functionals
Connes and van Daele proved the remarkable fact that
M
on
is the intersection of the Arveson spectra
M
of
The modular spectrum
(0, -) and that
a subgroup of
invariant of
M
S(M)
S(M)\{0}
forms
is a closed set which is an isomorphism
Connes then divided the type III von Neumann algebras into
subtypes as follows:
M
is said to be
if
S(M)
{0,i}
of type IIIk
if
S(M)
{0,kn-n=0,_+l,_+2, ,...}
of type III
if
S(M)
[0, )
(i)
of type III
(ii)
(iii)
0
I
k
(0,I)’,
This is a direct generalization of the Araki-Woods classification:for a type III
ITPFI factor
M
M, r(M)
is of class
S(M)
and so
S01
M
is of type
III0;
lllk,
M
is of class
Sk
M
is of type
M
is of class
S.
M
is of type III
k
(0,i);
I0
S. WRIGHT
Connes then proceeded to prove a fundamental structure theorem for the factors of
type
k
lllk,
[0,i)
in terms of discrete crossed products of von Neumann
The case of type III
algebras.
was not treated, but Connes was not alone in this
In the same year as Connes’ thesis appeared, Takesaki also offered [28],
work.
which solved the III
case by the introduction of continuous crossed products,
and which also developed a duality theory for crossed products that was to be very
We now proceed to describe the structure theorems of Connes and
influential.
Takesaki.
G
Let
be a locally compact abelian group,
of
continuous action
o__n
M
a (x)
g
is *-strongly continuous.
g
we call
L (G;H)
M
H
H-
valued,
of
M
h l(x)(h);
(g-lh);
((g)) (h)
W*(M,)
Th__e crosse__d
product
B(L2(G;H))
generated by
discrete action
n- 8
n
of
L
M}
x
G,
L
G
of
on
2(G;H);
2(G;H).
is the von Neumann subalgebra of
M b_
M
h
G,
g,h
{ (x)
W*(M,@)
the crossed product
of
M
x
-
G
square integrable
and
as follows
(h)
the mapping
denote Haar measure on
Let
We define the representations
((x))
M
x
M
denote the Hilbert space of all
G
Aut(M)
into
R (as will be the case in the sequel),
G
If
acts on a Hilbert space
L2(G;H)
functions on
2
such that for each
a one-parameter action on
Suppose
and let
M
G
of
is a homomorphism
the group of all *-automorphisms of
A
a von Neumann algebra.
M
by
n--0, +I, +2,...
8
and
{k(g)
g 6 G }
Aut(M)
If 8
is the crossed product of
M
by the
W*(M,8)
We will often refer to
as a
discrete crossed product or a discrete decomposition.
The oldest, and in many ways the most important, example of a crossed product
is the classical group-measure space construction of Murray and von
(X,)
be a
o-
bijective mapping
of
X
onto
X
an automorphism of
X
Let
and call a
finite measure space with positive measure
T
Neumann.
if
T
and
T-I
are
CLASSIFICATION OF INJECTIVE FACTORS
T
measurable and
T
automorphism
-
maps sets of
X
of
s
H
Let
L2(X,)
L(X,)
X
on
f 6 L (X,).
f- f O T
T
-
denote the Hilhert space of all
B(H)
maximal abelian von Neumann subalgebra of
s
(X,),T)
T
automorphism
U
X
of
g- allmost all
for
T
and
x
X
of
2
and
s
is a *-automorphism of
T
W*(L
such that
then
T
and
W*(L(X,),TI
T2
(X,),T)
is ergodic, it is a factor.
T
Two
are weakly equivalent if there exists an
{T I n (x)
Z}
n
(UT
2nu
(x)
Z}
n
In a famous paper which generalized the work of Araki
and Woods, Krieger [30] proved that if
X
in the
The von Neumann algebra given by the group-measure
is hyperfinite, and if
automorphisms
L (X,)
and thereby forms a
H
space construction is simply the discrete crossed product
W*(L
of
T
An
square integrable functions
acts by pointwise multiplication on
this von Neumann algebra.
g- measure zero.
measure zero to sets of
induces an algebra automorphism
natural way
Ii
T
W*(L(X,),T2
is isomorphic to
are weakly equivalent.
are ergodic automorphisms of
T2
and
if and only if
For this reason, Connes calls a discrete
crossed product of an abelian von Neumann algebra by an ergodic automorphism of
the algebra a
Krieer factor,
and so therefore will we.
(Incidentally, Connes
[24])
in the introduction to
cites the work of Krieger (along with Araki-Woods
[5] as being one of the primary motivations for developing his classification
scheme for type III factors).
We are now in a position to state what may appropriately be called the first,
second, and third fundamental structure theorems for type III factors.
2.2.
(Takesaki, [28]).
THEOREM.
exists avon Neumann
on
trace
o s
(i)
(ii)
N
M
algebra
N
of type
Let
II.,
and a one-parameter action
t
e
-t
is isomorphic to
W*(N,).
M
be a factor of type III.
There
a semifinite, faithful, normal
s
t
on
N
such that
12
S. WRIGHT
(Connes, [5]).
THEOREM.
2.3.
There exists a factor
N
moe
(i)
(ii)
M
II,
of type
normal, semifinite trace
N
of
Let
M
be a factor of type
an automorphism
II,
N
onto
1
and a faithful
such that
(N,8)
(i)
satisfying
and any two such pairs
Aut(N2)
quotient map of
p(n
such that
2
81-I)
onto the quotient
gives rise to a crossed product
(N2,82)
(NI,8 I)
factors if and only if they correspond to the same
N
(0,i).
W*(N,8).
is isomorphic to
factor of type
of
k 6
=h m
Furthermore, any pair
,
N
of
8
lllk,
k
give isomorphic
and there is an isomorphism
p(82) ,where
of Aut(N2) by
p
is the canonical
its subgroup of inner
automorphisms.
exists a von
8
N
of
(Connes, [5]).
THEOREM.
2.4.
Neumann algebra
N
Let
of type
M
II
which is ergodic on the center of
trace
on
N
There
0.
with a diffuse center, an automorphism
N
and a faithful, normal, semifinite
such that
k < 1
for some
(i)
be a factor of type III
k0T(x
(8(x))
0
for all positive elements
x
of
N
(li)
M
Any pair
III
is isomorphic to
(N,8)
W*(N,8).
satisfying the above conditions gives rise to a factor of type
o
These theorems in principle reduce the study of factors of type III to the
study of von Neumann algebras of type
II
and their automorphisms, and Connes lost
no time in beginning such a study.
If
M
II.
is a
[4],
of Murray and von Neumann
product of a
II
1
H
factor acting on a separable Hilbert space
factor
N
and
Theorem IX) allows one to write
B(H)
M
II
as the tensor
Thus the study of automorphisms of
factors can be effectively reduced to the study of automorphisms of
Now the hyperfinite
a theorem
II
II.
factors.
factor is in many ways the simplest of all the
II
1
factors, and hence any program which aspires to a classification of automorphisms
of
II
1
factors must first of all handle the case of the hyperfinite
II
1
CLASSIFICATION OF INJECTIVE FACTORS
R
factor
In
[9],
this is precisely what Connes did.
13
In this deep and penetrat-
ing study, he determined all outer conjugacy classes of automorphisms of
in particular showed that the quotient of
R
and
Aut(R) by its subgroup of inner auto-
morphisms is a simple group with only countably many conjugacy classes.
previous comments, all automorphisms of the
II
factor
R
NB (H)
0,I
By our
can now
be determined.
The factor
is also of interest for another reason.
R0,1
of Murray-vonNeumann asked whether all hyperfinite
to
R
0 ,I
?
II
An old question
factors are isomorphic
This question arose naturally from their work on the hyperfinite
factor, and for
II
a long time was viewed as one of the most important open questions
in the theory of von
Neumann algebras.
In [i0] it received an affirmative answer.
The key to this answer lies in the concept of injectivity, an idea introduced by
Tomiyama in [31] and exploited by him and others in the study of tensor products
of
yon
Neumann algebras (the terminology is due to Effros and Lance
[14]).
The
great achievement of [10] was to identify injectivity and hyperfiniteness for
separably acting factors, first in the
finally for the type
and 2.4.
III
II
1
case, then for the
II
case, and
case using the fundamental structure Theorems 2.2, 2.3,
It is to a detailed discussion of these ideas that we now turn.
A few words are in order concerning the organization of the remainder of the
paper.
The main theorem occurs in Section 8, and asserts that all injective
factors are isomorphic.
II
In Sectio 3, we define injectivity for von Neumann
algebras and give a tracial characterization of injectivity for finite factors.
Sections 4 and 5 are concerned with
establishing
a certain type of finite dimensional
approximation property for standardly acting, injective, finite factors which plays
a central role in the proof of the main theorem.
discreteness for injective finite factors.
Section 6 discusses semi-
Automorphisms of factors are briefly
treated in Section 7, and several important results of Connes on automorphisms of
II
factors are stated for use in the proof of the main theorem.
commences with the proof of the
Section 8
main theorem, Section 9 provides some necessary
lemmas on embeddings in ultraproducts, and proof of the main theorem is completed
14
S. WRIGHT
in Section I0.
The eleventh and final section uses the results of Sections 2 and
8, together with some results from [7] and [8], to obtain the classification of injective factors of type
3.
III
and
II(R)
k
[0,i)
INJECTIVE VON NEUMANN ALGEBRAS AND THE HYPERTRACE.
A von Neumann algebra
N
E
jective if there is a projection
E _> 0
follows easily that
H
acting on a Hilbert space
B(H)
of
onto
N
is said to be in-
E
with
It
and by a well-known theorem of Tomiyama
([31]),
E
is a conditional expectation, i.e.,
E(axb)
aE(x)b
Suppose now that
N
B(H)
x
B(H),
onto
x
B(H)
is an injective finite factor acting on
N
denote a tracial state on
norm 1 of
N
a,b
N
o E
and set
Then
E
where
is a state on
B(H)
Let
H
is a projection of
such that for all
N
a
(xa)
(E(xa))
(E(x)a)
(aE(x))
(E(ax))
(ax)
A state
(3.1)
B(H) satisfying
of
N
Conversely, suppose
hypertrace
(3. i) is called a hypertrace of
N
is a finite factor acting standardly on
We will eventually show that this forces
N
H
with
to be injective,
L2(N,T)
but before doing this we need to recall some facts about the Hilbert space
of Segal
on
H
An element
affiliated with
operator
(E)
set
([32]).
T
,
(x*x) <
o
contained in
of
N
%1N_
is a
closed, densely defined operator
o
(E a Borel subset of (0, + )) satisfies
d(k)
and the
2(N,)
[32].
A positive
is called .integrable when its spectral measure
d(k) <
and we
2(N,) consists of all integrable x satisfying
1/2
norm of 2(N,) is given by
x
N is
(x*x)
as a dense submanifold, and for
If we restrict the hypertrace
and so
L2(N,)
in the sense of Definition 2.1 of
affiliated with
(XE(T))
(T)
N
x
to
N
a
N
a tracial state on
(a*a)
a
N
1/2
results
The Schwartz inequality for positive linear functionals therefore
CLASSIFICATION OF INJECTIVE FACTORS
A
yields for each
15
B(H)
1/2
1/2
1/2
(A*A) (a*a)
(A*A) (a*a)
A ll(a*a) 1/2
V a N
l(aA) -<
A
Thus each
L 2(N,)
B(H)
E(A)
in
(a,E(A)*)62(N,)
A >_ 0
If
Claim:
A
determines a bounded linear functional
and hence a unique
(aE(A))
(3.2)
A
then
(a*aT*T)
L2(N,)
is positive on
(aT*Ta*)
on
such that
V a
(aA)
(aA)
a
(3.3)
N
N
Let
T
N
a
B(H)
Then
is a hypertrace)
(aT* (aT*) *) >_ 0
This verifies the claim.
E(A)
We can now show that
N
N
for
A >_ 0
E(A)
Since
it suffices by the double commutant theorem to show that
Let
a, b
b
a11 2
112 _< 1 Then
l(b*E(A)a)l
by [32],
l(ab*E(A))
l(ab*A)
lA(ab*)l
N
I(E(A) a,
blHl
A(a*a) A(bb*)
_<
N
operator on
H
with
IIE(A) II-<
We conclude that
to deduce from
(3.3) that
A
0
by (3.2)
it follows that
E(A)
extends to a bounded
IIAII
N
E(A)
by (3.3)
l al1 /2 llbll
[2(N,)
H
is dense in
extends to a
Corollary 11.2
since
[[A[[ (a a*) 1/4 (b b*) 1/4
IIAII
Since
E(A)
[2 (N,)
H--
bounded operator on
is affiliated with
for all
E(A)
A
A
B(H)
It is now straightforward
is a projection of norm
B(H)
of
onto
N.
The following proposition now obtains:
PROPOSITION.
3.1.
Let
N
be a finite factor.
if
N
4o
THE METHOD OF DAY FOR INJECTIVE FINITE FACTORS
N
is injective if and only
admits a hypertrace in its standard representation.
Let
N
be avon Neumann algebra,
a linear functional on
N
u
a unitary
S. WRIGHT
16
o ad
We denote by
N
in
(uxu*)
x-+
H
N
x
be an injective finite factor acting standardly in
N
Let
4.1. PROPOSITION.
the linear functional
u
Ul,...,Un}
Then for each finite set
there exists a normal state
II@-
aduj)ll
( o
@
< e
on
B(H)
j
l,...,n
N
of unitaries in
and
0,
e >
such that
(4.1)
The proof uses a separation argument first employed by Day [33] in
PROOF.
the context of countable amenable discrete groups.
We consider the Banach space
direct sum of
(, o ad
(B(H),)
n
( o ad u I)
Now there exist
(B(H) ,)
Consider the set
}
B(H),
C
norm closure of
0 (
C
f
on
such that
> 0
and
n
B(H)
Then there is a norm-continuous linear functional
C
0
Re f(
formed by taking the Banach space
( o ad u )) :@
n
Ul),...,@
To verify (4.1), it suffices to show that
Suppose
n
copies of the predual of
n
(,
C
(B(H),)
( o ad
B(H)
x
n
x
V
Un ))
f(g)
such that
B(H),
n
Z
gi(xi)
g
for
(gl
’gn
Hence
n
Re Z
j=l
((xj) -(ujxjuj*))
By Proposition 3.1,
Theorem
on
for
B(H)
2.14]),
Since
But
is the
N
V
has a hypertrace
o(B(H)*,B(H))
(4.2) holds for each
%(xj)
>_ a
%(ujxjuj*)
j
(4.2)
B(H),
%
By the bipolar theorem
limit of a net
(@)aA
we hence conclude that
@a
n
We conclude that
O
([34,
of normal states
(4.2) holds
CQED
5.
AN ANALOG OF
F#LNER’S CONDITION FOR INJECTIVE FACTORS.
As mentioned in the previous section, Day used the technique of Proposition
4.1 to show that when
G
is a
countable, amenable, discrete group, there exists
CLASSIFICATION OF INJECTIVE FACTORS
a sequence
F
llgFn
0
Fnlll
g
for all
G
S
n
#
for all
A
#
Let
V
0
A c G
cardfnality of
N
of finite subsets of
n
given a
G
such that
and
n
(Sn A Sn
where
such that
Finer [35] had
Somewhat earlier
stronger result by finding a sequence
S
gl(G)
of normalized positive functions in
n
17
A
and
G
cj
denotes synnetric difference.
be an injective finite factor acting standardly in
Flnr’s
proposition gives an analog of
role in the sequel.
N
result for
H
The next
and will play a crucial
Before giving its statement and proof, we need a rather
technical lemma concerning certain approximations for positive Hilbert-Schmidt
operators.
For its proof, consult [I0], Theorem 1.22 or [37], Section 2.5.
LEMMA.
Let
H
be a Hilbert space.
(a,+)
function of the interval
Schmidt operators on
H
exists
g
[[HS
where
> 0
n
]IX a (hi)
Z
j=l
5.1.
in
H
X a (h
PROPOSITION.
Let
E
a finite-rank projection
N
I)[]2
HS <
and
B
j
I,
n
Then there
3nell) a
(hl)l12HS
be an injective finite factor acting standardly
Xl,
B(H)
II[E,xj]IIHS < glIEIIHS
A
be positive Hilbert-
# 0 and
For each finite subset
(For operators
hl,...,hn
denotes the Hilbert-Schmidt norm.
Xa(hl)
such that
a > 0
denote the characteristic
such that
{lhj -h1{{HS < {IhI{{HS
for some
a
Let
0
a
X
Let
xn
such that
j
we denote
} c N and
E # 0
and
l,...,n
AB-BA
by
[A,B].)
> 0
there exists
18
S. WRIGHT
By Proposition 4.1, for any set
PROOF.
we can find a normal state
{{- ( o
Let
For
has density matrix
u.3
H
functional on
I
B(H)
x
we have
u.3(x) (ujxu.*)3 Tr(oujxuj*) Tr((uj*guj)x)
o ad
o ad
i.e.,
Tr(o)
B(H)
canonical trace on
N
(5.2)
l,...,n
j
p >_ 0
Tr(p-)
of unitaries of
such that
denote the density matrix of
0
Tr
Thus
< s
3
(-)
where
u.){{
ad
B(H)
of
{ul,...,Un}
and since the norm of a normal
uj*ouj
is the trace norm of its density matrix, we conclude from
(5.2)
that
lluj*puj -911Tr <
For
I,
j
n
Schmidt operator.
s
set
uj*huj
hj
llhj hllHS -< llhj
<
h211Tr
2
Ilhmllrr
Thus by the lemma, there exists
[[Xa(hj)
Let
E
a
-X
(h)
a
(h)[[ HS
X
a
9
1/2
is a positive Hilbert-
inequality [36],
lluj*ouj 911Tr
1[91ITr
a > 0
(hi)
such that
X (h) # 0
1/2)1/2[IX a (h)[[ HS
+ 1) s
< (3(n
Then
h
l,...,n
j
I[h[IHS
where
Powers-Strmer
By (5.3) and the
2
(5.3)
l,...,n
j
u.*Eu,
3
3
and
and
(5.4)
,n
j
Tr E<
Thus
E
has finite rank,
and by (5.4)
1/21/2
uj*Euj]IHS < (3(n + I) )IIEIIHS
Claim: I[[E,uj]IIHS
liE- uj*Euj[IHS
lIE-
basis for
H
lIE
and let
uj*Euj I]2HS
uj * E, uj
Aj
IIAjII2HS
Z
k
(5.5)
l,...,n
Let
e
Then
Z (AjAj*e,e)
([E,uj] [E,uj]*uje,uje=)
}
be an orthonormal
19
CLASSIFICATION OF INJECTIVE FACTORS
ll[E,uj]ll2HS
j
n
i,
(5.1) now follows by the claim, (5.5), the arbitrariness of
N
that
6.
s
and the fact
is the span of its unitary group.
QED
SEMIDISCRETENESS AND PURIFICATION OF FACTOR STATES.
A
Let
c
conjugate Hilbert space with
For
x E B(H)
EH
denote by
Ac
Let
Let
{x
c
x
(A)"
A
Let
:A
@
(a
(R) b
&
where
Ac
A }
A
A
c
c
c)
((a)
A
(@,H,)
,%
we say that
admits
N
A
(R).
mn
x
denote the
H
cc
H
onto
(x)
c
c
B(H c)
corresponding to
is separating for
@
(b))H
A
a
b
A
and
and
Ac
c
the existence of which
always extends to a state on
Ac
Ac
A
the spatial tensor product on
Ac
(R)
max
(R)
H
H
c
_a purification.
be a factor acting on
[I] asserts that the homomorphism
N
is an isomorphism.
c
Assume for simplicity that in the Gelfand-
is the modular operator corresponding to
extends to a state on
Let
such that
is a C*-subalgebra of
follows from Tomita-Takesaki theory.
if
B(Hc)
denote the algebraic tensor product of
by
H
a conjugate-linear isometry of
the element in
be a factor state on
Naimark-Segal construction
define
c
x
Let
H
be a C*-algebra acting on a Hilbert space
An early result of Murray and von Neumann
H
N @
defined on
is semidiscrete
N’
:a
by
(R)
a’
aa’
(in the sense of Effros and Lance,
N
if this isomorphism extends to an isomorphism of
min
N’
[14])
i.e., if
n
n
E
a
j=l
J
(R) b
E
j HH
j=l
a.b H
J j
V
a
l’’’’’an
bl"’’’bn
N
N’
Connes characterizes semidiscreteness in terms of purification of states as follows
([37]
Section
6.1.
PROPOSITION.
N
and only if
Now, let
trace.
2.8):
Let
N
be a factor on
H
Then
N
is semidiscrete if
has a faithful normal state which admits a purification.
N
be an injective finite factor, with
is normal and faithful.
We claim that
denoting the canonical
admits a purification.
To
S. WRIGHT
20
(N) I, vizo.,
on
trace vector
lla(1)ll H
Thus,
(S’S)
A
(a
I
(R) b
([37],
(b*a)
2.7),
Section
N @ Nc
we have the following inequality:
n
n
I(
c
a (R) b
((a) l,(b)l) H
But by Proposition 5.1 and
is isometric, since
lla*(1)ll
whence for all
c)
a*(1)
S:a(1)
(aa*)
(a*a)
1/2
induced by the involution and the canonical
S
see this notice first that the map
I(
b.*a.)l
]
]
%
j=l
a.b.*)l
j
%
j=l
n
-<
llj=IY, a.j bjCllHC, V aj,bj
(R)
c
N @ N
is bounded on
It follows that
relative to the spatial tensor product
N
norm, and therefore has an extension to
E N.
%in
Nc
We can now deduce from
Proposition 6.1:
6.2.
7.
PROPOSITION.
Injective finite factors are semidiscrete.
AUTOMORPHISMS OF FACTORS.
Let
M
(by automorphism, we will always mean
M
automorphism group of
Aut(M)
We set
be a von Neumann algebra.
*-automorphism)
inner automorphisms of
Int(M)
form
adu: x
u*xu
M
i.e., the set of all automorphisms of the
x
M
u a unitary operator in
M.
This is a
normal subgroup of Aut(M)
Out(M)
p :Aut (M)
Aut(M)/Int(M)
canonical quotient map.
Out (M)
The weak topology on
predual
M,
of
M
Aut(M)
c
Let
f
8()
for the action
automorphisms converges weakly to
We set Int M
is the topology of point-norm
-o o
closure of
@
(8-I)
8
i.e., a net
of
Aut(M) if and only if
I1:o, vq
Int(M)
convergence in the
M,.
in the topology just described.
be a fixed linear functional on
M
and let
a
M
The linear
CLASSIFICATION OF INJECTIVE FACTORS
af
functionals
M
of
if
{an
for all
(
M,
An automorphism
of
n
M
is
_c
M
Aut(M)
is a normal subgroup of
Int(M)
M,
Ct(M)
then
a
(xn)
[u, Xn ]u*
any unitary
x
ll[u,xn ]il
=
lira
n
[u*,xn *]
M
u
M,
For
(*)
6
II[u
M,
y
[U,Xn](y)
0
x
n
n
Xn]*ll
]II
and setting
(7 I)
M,
0
(7.1),
it suffices to show that for
{x }
n
(
and any
M,
have
(7 2)
0
[n’ u*u]
](y)
We must
to show that
Thus to verify
ll[u
[u,xn ]u*
x
One verifies by direct computation that for
[UXnU*,
both tend to
be a centralizing sequence.
any centralizing sequence
lim
n
Let
llanII, llan*ll
8(xn)
Now
0 *-strongly.
n
and
{xn}
and let
it suffices by
n
[U,Xn]*
M
0 *-strongly.
n
ad u (Int(M)
lira
But
{x }CM
*-strongly, for tverv, centralizing sequence
is a bounded sequence in
Let
n
A bounded sequence
We notice first that
PROOF.
a
0
fa
group of all centrally trivial autoorphisms of
LEMMA.
show that
6
for all
O(Xn) -Xn
Ct(M)
(*)
af
f(xa)
af:x
We set
Ct(M)
7. io
[a,f]
We set
centrally trivial if
n
are defined respectively by
ll[Xn,]ll-O
is centralizing if
{x }
M
on
x 6 M
f(ax)
fa:x
fa
and
21
y
(u*yu)
M
(7.3)
M.
YXnU)
(yux
n
u(yux u*
n
u(yux u*
YXn)
n
ux u’y) + u(UXnU*Y
yxn)
n
[UXnU*,U]
u(UXnU*Y YXn)
[Xn,U] (u*yu) + u(UXnU*Y- yxn)
(y) +
Now,
ux u*y
n
yx
n
ux u*y
n
(ux u*yu*
n
yu*ux
n
yu*UXnU*)U
by (7.3)
(7.4)
22
S. WRIGHT
[ux u* yu*]u
n
Therefore,
uq)
u2q) ([UXnU*’yu*])
YXn)
(UXnU*Y
2
-u
[yu*, UXnU*]
q)
u*,-u2q)]
[ux
n
-[x
n’
(yu*)
u@u] (u’y)
(7.5)
by (7 3)
Thus by (7.4) and (7.5)
[x ou] (u*yu)
[u,xn ]q)(y)
n
Ixn
uqu] (u’y)
{x
(7.2) now follows from (7.6) and the fact that
LEMMA.
7.2.
0
Then for each
M
Let
-nt
M
p()p(O)
M,
For
denote the seminorm
Aut(M)
n n
n
-1 c
V
n
IlO(u)
ull
lie(u)
ull
-i
< 2
u
and
II@
o
o
-lll
< 2
M
ad u
n
Let
nV
llO(Vn) Vnll
W
< 2
Un+lUn*
-I
<
IIO(Vn*)Vn
so that
2-n
-ill
o
Vn
-2n
Let
WnWn
e
.
< 2
V
n
of
V
n
in
so we must show that
-i
2-n
Aut(M)
Then
+
2
-n
such that
M
be a unitary in
u
n
and therefore
adu
n
Since
-n
00 -I
Then
n
u
ad
with
lim ad(u
n)_
n
converges *-strongly to a unitary in M
that
II@,
by
-n
o o (e -1-1)
-I
1/2
there is a neighborhood
n
be a decreasing basis of heighborhoods of
W
Let
W W
such that for all unitaries
(x*x)
x
is centrally trivial, for each positive integer
in
QED
is centralizing.
n
be a von Neumann algebra with separable predual.
Ct(M)
p(O)p()
PROOF.
(7.6)
3
2
-n
u
n
such
*(U_)n
CLASSIFICATION OF INJECTIVE FACTORS
since
W
ad u
n
23
Thus
n
u
llO(Vn*)Vnn
<3"2
u
n
-n
and
llS(Un+l*)Un+
ll8(Un*)O(Vn*)VnUn e(Un*)nll
8(Un*)Unll
< 3-2
-n
Also because
(Vn+ i
< 2
Vn+l
o
and
I!
o (ad 0 (u -I ))
n
we obtain
-
0
JJUn+l*O(Un+l) u*O(u)ll
This shows that
ad u
Un*0(Un)
-n
(0= -10-1 )11 <
< 3
2
2-n
-n
converges *-strongly to a unitary
M
u
such that
-I00-I
QED
structure of
Int(M),
Int(M)
The relation between
and Ct(M)
By Lemma 7.2, one always has
M
has a strong bearing on the
_c
p(Ct(M))
commutant of
pC[ M)o
On the other hand, Connes proves the following remarkable theorem:
([ 9]
THEOREM
7.3.
R
predual, and let
Theorem 2.2. I)
Let
M
be a factor with separable
denote the hyperfinite factor of type
II
The following
l
are equivalent:
(a)
M
(b)
p(Int M)
(c)
Int M
Moreover, if
is isomorphic to
_
(a)
M(R) R
is nonabelian.
Ct(M)
holds, then
p(Ct(M))
commutant of
p(Int M)
We now list several theorems which allow us to manipulate
for a factor
N
of type II
1
consult the indicated references
Let
N
has pro__p_ert_y
be a finite factor,
Int N
and Ct(N)
Regretfully we must omit all proofs; for details,
(all results are due to Connes)o
the canonical trace on
of Murray and von Neumann
([4])
N
Recall that
if for each finite subset
N
24
S. WRIGHT
[u,xj]112 <
N
of
Xl,...,xn
g
j
predual.
there is a unitary
([7],
Corollary 3.8)
([10],
Let
H
ardly on
N
has property
N
Since
7.6.
and
F
N’
H
([i0],
0 and
factor with separable
factor acting stand-
C*(N,N’)
H
the C*-
C*(N,N’)
Int N
Theorem 3.1)
is irreducible.
II
for
Let
(0)
K
N
factors.
be a II
e
if and only if
The next theorem
factor acting
extends to an automorphism
such that
([10],
N’
Identity on
N’
THEOREM.
(Ni)
F
be a II
N
C*(N,N’)
H
Int N
IN
7.7.
Let
if and only if
Then
Aut (C* (N,N’))
8.
(u)
Then
acts standardly on
THEOREM
standardly on
(R)
be a II
has property
Theorem 2.1)
gives a complete characterization of
I
N
Let
denote the compact operators on
K
algebra generated by
N
N
if and only if
THEOREM.
7.5.
Aut
with
Then
Int(N) # Int N
’i
M
u
n
i,
THEOREM.
7.4.
> 0
and
Let
Corollary 4.4)
NI,N2
be II
1 factors.
Then for
1,2
i
2 Ct(NI (R) N2)
if and only if
UNIQUENESS OF THE INJECTIVE
111
i
Ct(N i
i
1,2.
FACTOR.
In this section, we begin the proof of the main theorem.
The proof will end
in Section I0.
8.1
([I0],
THEOREM.
Theorem 5.1)
All injective factors of type II
acting on a separable Hilbert space are isomorphic.
PROOF.
Let
N
denote an injective II
space, and acting standardly on
to the hyperfinite
lllfactor
L2(N,)
We will show that
N
is isomorphic
R.
N is semidiscrete, ioeo, the mapping D:N e N’
n
n
7. a.b. is isometric as a mapping from B(H (R) H)
rl: 7.. a.1 (R) b.
1 1
1
By Proposition 6o2,
given by
H
factor acting on a separable Hilbert
1
B(H)
B(H)
CLASSIFICATION OF INJECTIVE FACTORS
25
Since
n
C*(N,N’)
7, a
norm closure of
b
a
i i
i
extends to an isomorphism of
it follows that
N
b
N (R)
N’
N’
i
Z+
n
C*(N,N’).
onto
rain
We claim that
sided ideals.
N
N’
and
N
Since
is a finite
no nonzero maximal ideals.
N’
([13],
factor, by [38], Theorem 6.2
N
Since every two-sided ideal of
maximal one, we conclude that
isomorphic to
are simple, i.e., they contain no closed, two-
N
is simple.
N
Since
N
contains
is contained in a
is conjugate-linearly
N’
Proposition 2.9o2), it follows that
is also
simple.
By a theorem of Takesaki ([39]
N (R) N’ is simple so that
we conclude that
min
C*(N,N’)
Hence if
is simple.
operators on
H
I 6 C*(N,N’)
and
C*(N,N’)
K
C*(N,N’) U
then
H
denotes the (closed, two-sided) ideal of compact
K
K
(0)
is either
Since
C*(N,N’) K # C*(N,N’)
is infinite-dimensional,
(0)
C*(N,N’)
or
so
Thus by Theorems 7.4 and 7.5,
Int(N) # Int N
Let
o
denote the flip automorphism of
F
o
F (x
(R)
y)
y
We want to show that
o
To do this, let
M
let
8
Aut(M)
n
7.
(8.1)
(R) x
N (R) N
N
x,y
Int (N (R) N)
F
be a standardly acting semidiscrete
Then for all a
a
n
M, b
factor, and
n
aibi[
E
7,
a.1
(R)
b.lll
(M is semidiscrete)
8(a i)
(R)
8(ai)
bill
bil
n
7.
This shows that
8
II
M’
,b n
.
n
aiM-
i.e.,
[M’-
since 8 (R) id extends to
M’
an automorphism of M
m.n
(semidiscreteness again)
extends to an automorphism
identity on
M’
of
C*(M,M’)
Thus by Theorem 7.6, 8
We have hence shown that
Aut M
a
Int Mo
such that
Int M.
26
S. WRIGHT
Now by a theorem of Effros and Lance ([14], Proposition 5.6)
N
semidiscreteness from
N (R) N
Since
N (R) N
and so
N
o
F
II
6 Int
has separable predual.
OF(8
(N
(R)
We hence conclude that Aut (N (R) N)
N)
N (R) N
and therefore
p(Ct(N (R) N)) c commutant of
By Lemma 7.2,
whence by Theorem 7.7,
p(8
Since
factor.
acts on a separable Hilbert space, so does
p(Int(N (R) N))
inherits
and standard facts about tensor products imply that
is a standardly acting
Int (N (R) N),
N (R) N
(R)
l)P(OF)
(R)
I)o F
-I
p(l
(R)
p(oF)p(8
(R) 8
8)
(R)
V 8
i)
6 Ct(N)
we get
p(oF(8
(R)
I)o F
I)
p(8
(R)
I)
whence
8 (R) 8 -I
Now a theorem of R. Kallman
M
2
are
Int(M
yon
(R)
([40],
Neumann algebras,
M2)
(8.2),
In t(N (R) N)
Corollary 1.14)
Aut(Mi)
i
I
if and only if either
Int(N)
(8.2)
1,2
i
Int(Ml)
M
asserts that if
or
then
S
2
and
81 (R) 82
Int()
Thus by
Recalling Lemma 7.1, we have thus shown that
Ct(N)
(8.3)
Int(N)
We conclude from (8.1) and (8.3) that
Int(N) c Int N
Ct(N)
N
so by Theorem 7.3,
is isomorphic to
N
(R)
R
This completes the first major step of the proof of Theorem
to fully exploit this isomorphism, we must relate
all, we are trying to show that
by embedding
N
N
in the ultraproduct
and
R
R
N
8.1.
In order
R
more closely to
are in fact the same)
free ultrafilter.
(after
This is done
We take up
the details in the next section and it is there that Proposition 5.1 plays a
crucial role.
9.
EMBEDDINGS OF
Let
Z+
N
IN ULTRAPRODUCTSo
denote the positive integers with the discrete topology.
Let
CLASSIFICATION OF INJECTIVE FACTORS
C
g
(Z+)
b
g
of
Z+
n
Points
Z+
Z+\Z+
Z+\Z+
Suppose
n
}
k
and let
the
Nk’S
({
if
{x
kernel of
lim
k-
NNk
p. 451)
Denote by
k(Xk
c
is a
Fn
Nk’S
of
as follows:
gi occuring
9.1o
R
into
Fn
m
If
gl,...,gn
as the sum of the absolute values of all exponents
m
LEMMAo
in
Let
n)
(Ul,...
if
m
u
Let
m
Fk
set of all
F
n
m
by
of unitaries defines a unitary
u
ul,o.o,
u(m)
m
F
of
n
Un
n
be unitaries in
Q
representation
is the element formed by replacing
L,...,
i
i
there is a finite-dimensional factor
such that
N
k
Each n -tuple u
each
is a closed, two-sided
corresponding __t
denote the free group on the generators
appearing in a reduced presentation of
n
Z
the ultraproduct formed by countably many copies of the factor
M
we define the
of F
0 }
Neumann algebra with finite normalized trace
yon
We proceed to construct an embedding of
-<
the C*-direct sum of
iNk
Let
@iNk
called the ultraproduct of the
length
a.
On @
@iNk:(x*x)
00
Let
Then for a sequence
a
n
Z+
be a free ultrafilter on
([41],
u
Mg
be a sequence of factors with finite normalized traces
()kZ+
Then
M.
n
we define the trace
Z
ideal in
tL= maximal ideal space
correspond to omomorphisms not of this form.
a
n
:(xk)
Let
Mg
with
corresponds to
we write lima
Now, let
Z+
of
correspond to the homomorphism of evalutaion at
and free ultrafilters
a
We identify the
denote the C*-algebra of bounded sequences.
Stone-Cech compactification
27
N
For each
and unitary operators
> 0
Vl,..Vn
Q
28
S. WRIGHT
l(u(m))
where
(v(m))l
-(Ul""’ n-U)
u
m E F
<
_(Vl,... n-V)
v
k
and
is the normalized trace on
q
Q
PROOF.
We first strengthen Proposition 5.1 as follows:
N
Xl,...,xn
such that for
s > 0
and
I,...,
j
If[E,
xj]ll
given
E # 0
there exists a finite-rank projection
on
H
n
HS
< s
IIEIIHS
,
(x .E,E)
l(xj)
{Ul,..., Um}
To see this, we first claim that there exist unitaries
and
6 > 0
one has
such that for any state
l(xj) -@(xj)l
finite subset
o.
<
I
j
of unitaries of
N
on
N
]II -<
For suppose not.
n
6 > 0
and
ll[uj,
with
6
N
I,
j
m
Then for each
there is a state
o.,6
on
N
such that
Io.,6
(ii)
If we partially order the set of
o.1
c o-
2
and
(i),
for some
(o.,6)’s by
setting
o.,6
N
But since
(xj)
(xj)l <
e,
j
I
I,..., n
j
(o.2,62)
}
if
o.,6 +
of
By
re(weak*),
n
This verifies the claim.
Now use Proposition 5.1 to find a finite rank projection
i
_<
n
such that
lo, 5
contradicting (ii)
(o.i,61)
then
is unitarily invariant, and so
we may find
{I,...,
j E
{o., 6 } is a net and so by weak *-compactness
{o,6} weak *-accumulates at a state of N
61 62
the state space of
>_
(xj) -(xj)
I,...,
m
E # 0 such that for
CLASSIFICATION OF INJECTIVE FACTORS
29
E]{IHS < e {{E{{HS
6
[l[uj, E][[HS < [IE[[HS
ll[x i,
(iii)
(iv)
We assert that the state
E)HS
(xE,
(x)
llEll
ll[uj,
satisfies
u.
]II-<
2HS
6
I,...,
j
(uj
3
x
defined by
Tr (ExE)
Tr E
,
(x)
N
on
Tr
uj
This is so because
n
Euj
rr
xu.. *
E
E
Tr (Eu.xu.* Eu.u.*)
Tr E
Tr( (uj *Euj x (uj *Euj
Tr E
(xEj
E. HS
3
E.
Tr E
3
u.*Eu.
3
3
Therefore,
]*(x)
I(xE
*u.3 (x)
-< (I(x(E-
IIxlIIIEII
Since
[uj,
l[[uj,
%][[ -<
.
+
Ej), E)HS
EjlIHsIIEII2S
liE
-I
HS
211EIIHS
(%-%u.) (xuj)
3
6
i,..., n
(xj
(xjE, E)H S
,,"Z’’2HS
(IIEIIHS + IIEjlIHS)
by (iv)
% ](x)
j
IIEII2S
I(xEj,E-Ej)HS "IIEII2S
EIHs (xEj ,Ej )HS
x E N
as asserted.
< e
j
we conclude that
Thus, by the previous claim,
n
30
S. WRIGHT
which together with (iii) is what we are seeking.
s/2k,
Now, letting 6
E
we may hence find a nonzero finite-rank projection
such that
II[E,uj]IIHS < 611EIIHS
(u(m) E,E) HS
l(u(m))
Let
u.Eu.*33
E.j
Let
v.’
w.u.
J
v.’E
3
v.
J
factor
F
m
(9.1)
k
and, by [i0], Lemma 1.4, there are
E and
w.E.w.*3
3 3
with
Ev.’
3
Ev.’
llv.’
J
Then
J
v.
IIIHS.. < 3611EIIHS....
llwj.
u
j HS
The normalized trace
q
(xE,E) HS
(x)
Since for any unitaries
B(H)
llaujb-
Q
on
q
J
n
is given by
we have
g
avj
HS
HS
we conclude by (9.2) and induction on the length of the word
flu(m)
By
(9.2),
E
(9 2)
Q
x
IIEIIHS2
a,b
IIEII HS
<
is a unitary operator in the finite-dimensional
3
EB(H)E
Q
s,
Then
3 J
v.’E
3
Let
i<
2
B(H)
w.3
,n
..IIEj- EIIHS,. < 611EIIHS
then
unitary operators
J
v’(m)llHS
commutes with all v
3
-< (length
so that
of
m
that
m)-IIEIIHS
(v’(m))E
(9.3)
v(m)
for each word
m
Thus by (9.3) and the Schwartz inequality for the Hilbert-Schmidt norm,
I(u(m)E,E)Hs (v(m)E,E)HsI
By (9.1) and the definition of
q
<
2
slIEII HS
F
m
k
QED
this is what we seek.
We are now ready to construct an embedding of
N
into
construction depends on the following extensiorr-lemma of
R
The
Pearcy and Ringrose
CLASSIFICATION OF INJECTIVE FACTORS
([15],
31
(In what
Lemma I), which we record for the convenience of the reader.
follows,-WOT denotes closure in the weak operator topology)o
LEMMA.
9.2.
be avon Neumann algebra,
Mo
Let
Mo
positive linear functional on
A
-WOT
A
i =1,2
Suppose that
and that
2
M.
A2
onto
of
*-isomorphism
LEMMA.
9.3.
Let
T
i
onto M
M
N
F=
Let
on A
such that
M.
can be extended to a
Then
2
o
i
on M
there exists a trace-preserving
Z+\Z+
R
into
R
acts separably, it
N
Since
contains a strongly dense sequence, and so we may find a sequence
N
unitary operators generating
F
_c F
k
k
o(vk(m))l
l(u(m))
where
v
k
e
R
unitary in
to
F
k
k
(v
v
A
let
(x)
<
F
m
unitaries
(9.4)
k
For each
I, I,...
(vj k) k
(v(m))
lim
k-o
Y.X v(m)
%X u(m)
If
m
Ek
m m
m
u
0
(u(m)*u(m’))
n
(u(m))
}
For
v
m
x
be the
v.
let
Then since each
Z+
o(vk(m))
*-algebra generated by
Z+
j
sufficiently large, we get, with
k
T
Let
k
k
represented by
for all
of
such that
R
in
Vk
k
choose for each
R
}
and with length <_ k
gl’’’’’ gk
Using Lemma 9.1 and the hyperfiniteness of
u
n
Finally, let
as avon Neumann algebra.
be the set of all words involving only
k
{gn
denote the free group on countably many generators
denote the normalized trace on
0
a *-subalgebra of
is an algebraic *-isomorphism from
with
2
For each
*-monomorphism of
PROOF.
o
A.
and
a faithful, normal,
i
m
F
belongs
(Vl,..., vj,...
(9.5)
F
EX u(m)
m
then
0
whence by (9.5)
%%mXm
and so
EX v(m)
m
0
T0,
Thus
(v(m)*v(m’))
0
is well-defined.
By construction
is a
32
S. WRIGHT
A
*-homomorphism of
R
into
which preserves the trace, and is hence monomorphic.
We now apply Lemma 9.2 to
N
to obtain the desired *-monomorphism of
into
QED
R
N (R) N
of
(x
(R)
(IN
For each
IR) v
Z+
(b)
For each
zv)v Z+
(R)
(x)
(N
into
(a)
PROOF.
I
For any
LEMMA.
9.4.
R)
(x (R)
( (Y)n)n
is represented by the constant sequence
N)
(R)
(1N
(R)
is represented by a sequence of the form
y)
R
I
be the unique isomorphism of
n (
Z+..
N
the isomorphism of
where
N
y
there is trace-preserving *-monomorphism
Z+\Z+
such that
(x
x ( N
Zv
Let
R)
(R)
e (
e
into
*-monomorphism
(R)
R)
such that
x (R) y
r (x)r
N (R) N
(N
(R)
2
(y)
and
R)
N. Since
(R)
min
of
(N
(R)
c
2(N)
R)
Murray-von Neumann
I
I(N)’
Sec. 6
e
(N)
so if
(
(R)
(y)
and
such that
(y))
(R)
c
2
Z+,
n
n
R
in
y
N
(b)
(a)
satisfies
o)
o
N
((x
(R)
y))
i
(N)
2
N @ N
(R)
(b)
and
so
2
rl-isomorphism of
preserves the trace.
trace on
(
(R)
o
R
Then
[(x
)
lim T(x)T (
O
n-+(o
(R)
(y)n)
is thus a
N @ N
-WOT
(N @ N)
Since
defines a
are commuting subfactors
denotes the
"q
(I(R) 2
suffices by Lemma 9.2 to show that
trace on
and
we have
well-defined *-monomorphism on
Let
IN
(y)
(a)
satisfying
are *-isomorphisms, and so
2
N
*-isomorphism on
R)
is well-defined and can be extended as claimed.
we must show that
I
(R)
extends to a trace-preserving
With this definition, it is clear that
Now
2
is a representing sequence for
Z+
We claim that
(N
(N
into
the isomorphism of Lemma 9.3, and
N
x
N
(y)n)]
N (R) N
it
CLASSIFICATION OF INJECTIVE FACTORS
T(x) lira o (n(y))
n
33
(x)
O e
n-0
(x)(y)
LEMMA.
9.5.
R
Zl,...,Zn
PROOF.
Let
(R)
X(I N
R
such that
zj)X*ll 2 <
(R)
(N
R)
(R)
there is a unitary
IIxj
D
Now
(R)
v(l
N
(Xv)v
(v)
Z+
(N
(R)
(R)
I)
(R)
(R)
xj)
lira
(9.6)
n
be the isomorphism of Lemma 9.4.
o
Aut(N
F
(R)
N)
Int(N
is in
xj)v*II 2 <
(v)S(l
R)
As we
(R)
N)
so
such that
(R)
I
j
(9.7)
n
L2-norm.
xj)(v)*ll 2 <
(9.7)
Thus by
(9.8)
n
j
be a representing sequence of unitary operators in
(the injectivity of
(zj(U))D Z+
let
N
1
j
preserves the trace and hence the
lls(xj
Let
N (R) N
v
Then there exist
0
>
N (R) R
X
:N (R) N
Let
N
saw in Section 8, the flip automorphism
be a sequence in
here!)
is invoked
R
(I
such that
(R) z.
()
N (R) R
for
For each
j
represents
Then by (9.8),
IIxj
(R)
xu (I N
1R
(R)
zj
(U))xu,ii2 <
Since the evaluations at points in
from (9.9) that
I0.
N
QED
Xl,...,xn
and a unitary
llxj
(I
V x, y
y)
(R)
preserves the trace.
i.e.,
X
(x
((y))
Xu
and
z
(u)
I
Z+
,...,Zn
are weak *-dense in
(u)
(9.9)
n
j
Mg
it follows
Z+
satisfy (9.6) for suitable
THE PROOF OF THEOREM 8.1 COMPLETED.
Recall from Section 2 that a finite factor
{Xl,...,xn }
for each finite subset
dimensional subfactor
IIxj vjll 2 <
s
j
C
of
n
M
and
of
Vl,
M
and
o,Vn
M
satisfies condition
s > 0
in
In order to show that
C
if
there exists a finite-
C
such that
N
is isomorphic to
R
34
S. WRIGHT
it suffices by Theorem 2.1 to verify condition
C
for
N
This is what we
now proceed to do.
N (R) R
N
We saw that
R
Now
in Section 8.
M (R) M (R)
2
2
where
2 x 2 matrices and the infinite tensor product is taken relative to the
M
2
(R)n’ where
product state
n
each
n
R
follows that
is the normalized trace on
2
It
where the infinite tensor product is taken
R (R) R (R)
relative to the infinite tensor product of the canonical trace on
pp. 205-206)
M
N
Hence we may identify
with
N
k
an increasing sequence
k
N (R) R (R) R (R)
contains
of subfactors such that
N (R) R(R)
N
It follows
N (R) R (R) R (R)
from the definition of the infinite tensor product that
R (see [13],
(R)
R
k times
N
The relative commutant
N (R) R
Thus
in
c
N
R
N (R) R (R) R (R)
N
of
k
k
N
(R)
such that
c
Also
k
U
N
k
for all
N
is
k
k
is
is isomorphic to
N}
L2-dense,
is an increasing sequence
and
N
k
k
N (R) R (R)
N
of subfactors of
(ii)
all
N
M
k
U M
k
(iii)
(R)
R
N
k times
By the foregoing discussion, we may choose an increasing sequence
(i)
R
R (R) R (R)
k
is
M
k
with the following properties:
for all
k
L2-dense
in
The relative commutant
N
<
in
N
is isomorphic to
By (ii), choose
k
and
of
M
k
R, for
k
(iv)
N
Now, let
Mk (R)
<
for all
N
Xl’’’’’Xn
k
x
such that
j
j
2
By (i) and (iii), there are isomorphisms
(io. i)
CLASSIFICATION OF INJECTIVE FACTORS
I:
0
81
So we may extend
B
(R)
N
02:
<
R
M (R)
k
to an isomorphism of
2
35
< L2-norms.
N (R) R
which
preserves the trace (see the introduction), and hence the
M
identifying
with its image under the isomorphism
k
we may assume by (iv) that each
of
N-- M
X
M (R) R
(R)
M
R
R
N
(R)
zj)X*II 2
(R)
3
a 6 M
k
R
in some factorization
R
Zl’’’’’Zn
R
<
I
j
and a unitary
(10.2)
n
is hyperfinite, there is a finite-dimensional subfactor
Q
(R)
Q
R
of
and
such that
llzj qjll 2 <
X(
m.
(R)
such that
ql,...,qn
Then
has the form
3
By Lemma 9.5, there exist
N
X(I
Since
x.’
l(a)
a
Thus, by
Q)X*
I,
j
(10.3)
n
is a finite-dimensional subfactor of
N
M (R) R
and by
(10.1)- (10.3),
llxj
X(l
C
This verifies condition
II.
(R)
M
qj)X*ll 2 <
for
n
j
N
QED
CLASSIFICATION OF INJECTIVE FACTORS OF TYPE
II(R), lllk, k
[0,I).
With Theorem 8.1 and the fundamental structure Theorems 2.2, 2.3, and 2.4
now at our disposal, the precise structure of injective factors of type
III,
X E
(0,I]
H
THEOREM
II
factors of type
PROOF.
H
acting II
R
is the hyperfinite II
0
factor
R (R) B(H)
is a separable Hilbert space.
Ii.i.
space
and
can now be deduced in a fairly straightforward fashion.
Recall from Section 2 that
where
II
Let
By
Theorem 7.4).
are isomorphic to
M
[4],
factor
([I0],
be an injective
R
II
All separably acting injective
0,I
Theorem IX, we can write
N
M
-
factor acting on a separable Hilbert
M
N (R) B(H)
for a separably
can thus be viewed as the algebra of -X- matrices
S. WRIGHT
36
N
with entries in
of
M
N
we see that
M
E M
(xij)
defines a projection of norm
Xll
N
By Theorem 8.1,
is injective.
M
onto
R
is isomorphic to
whence
QED
}
is an increasing family of injective von
follows by [14]
Proposition 5 7 that the weak closure of
M
H)
(R)
R0, 1
is isomorphic to
If
B(H
and composing this with a projection of
N
onto
F
The map
Neumann algebras, it
U sM
is injective.
Since every finite-dimensional algebra is injective, we conclude that all
Applying Theorem II.i, we deduce
hyperfinite von Neumann algebras are injective.
11.2.
All separably acting, hyperfinite,
COROLLARY.
II
factors are
isomorphic.
11.3.
([I0],
THEOREM.
acting injective factors of type
PROOF.
M
Let
space with discrete decomposition
to show that
N
[28],
factor acting on a separable Hilbert
[28], Theorem 8.1,
A
on
(R)
N
=
g (R)
that
t(x)’)}
(E(x),)
defines a projection
A
(R)
M
E
of norm
N
is injective.
of
N,)
By
A (R) N
it is straightforward to verify
,
M
x
(R)
is precisely
M
on
W*(A
M
such that
the fixed point algebra of
is an invarient mean on
we conclude that
We want
obtained via Theorem 2.3.
W*(N,e)
p. 303, we can find an abelian von Neumann algebra
and a one-parameter action
Hence, if
III
Rk
is injective.
By Theorem 2.2 and
A
All separably
are isomorphic to Powers factor
lllk
be an injective
(0, I).
E
Let
Theorem 7.7).
H
A
onto
Thus by
(R)
N
[14],
Since
M
is injective,
Proposition 5.6,
N
is
injective.
N
We conclude by Theorem 11.1 that
Corollary 6 and
[5], Th6orme 4.4.1,
Rk
is isomorphic to
is the only III
of whose discrete decomposition is isomorphic to
to
Rk
11.4.
R0,
k
But by [9]
R0, 1
factor the
Hence
M
II
part
is isomorphic
QED
THEOREM
([i0],
Theorem 7.5).
If
M
is a separably acting injective
CLASSIFICATION OF INJECTIVE FACTORS
factor of type
III0,
Let
PROOF.
then
W*(N,8)
M
is a Krieger factor.
be a discrete decomposition of
As in the proof of Theorem 11.3,
a factor.
37
N
is injective;
M
as in Theorem 2.4.
however, it is not in general
On the other hand, it is a direct integral of factors, and each factor
appearing in its direct integral decomposition (except possibly for a set of
measure 0) is injective by
isomorphic to
[I0],
Proposition 6.5 and of type
II
and hence
But these are precisely the hypotheses of [8], Theorem II.I,
R0,1
which asserts that under these conditions
M
is a Krieger factor
QED
It follows from [30] that injective III 0 factors are classified up to
isomorphism by ergodic non-transitive flows.
can consult [30]
[I0],
For further details, the reader
and [37].
The only known injective factor of type III
and Woods (see Section 2).
1
is the factor
R
of Araki
It is not known whether this is the only possible
one.
ACKNOWLEDGEMENTS.
I would like to thank Sze-kai Tsui for several stimulating
and instructive conversations about the material contained herein.
I would also
like to thank Jim Agler, John B. Conway, and Andrew Lenard for very active
participation in a seminar given during the summer of 1978 at Indiana University,
during which many inaccuracies in an early version of this work were detected
and corrected.
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