Free products of finite dimensional and other von Neumann algebras with respect to non-tracial states

46 Pages·0.46 MB·English

by  Kenneth J. Dykema

FieldsInstituteCommunications
Volume00,0000
Free products of finite dimensional and other von
Neumann algebras with respect to non-tracial states.
6
9
9 Kenneth J. Dykema
1 Department of Mathematics
University of California
n
Berkeley CA 94720–3840
a
———-
J
present address:

3 Department of Mathematics and Computer Science
2 Odense Universitet, Campusvej 55
DK–5230 Odense M, Denmark

1 e-mail: [email protected]
v
3
0
Abstract. ThevonNeumannalgebrafreeproductofarbitaryfinitedimensionalvonNeumannalgebraswithrespect
0
toarbitraryfaithfulstates,atleastoneofwhichisnotatrace,isfoundtobeatypeIIIfactorpossiblydirectsuma
1
finitedimensionalalgebra. Thefreeproduct stateonthetypeIIIfactor iswhatwecallanextremal almostperiodic
0
6 state, and has centralizer isomorphic to L(F∞). This allows further classification the type III factor and provides
9 another construction of full type III1 factors having arbitrarySd invariant of Connes. The free products considered
inthispaperarenotlimitedtofreeproductsoffinitedimensionalalgebras,butcanbeofaquitegeneralform.
/
n
a
-
t
c Introduction.
n
u
In the context of Voiculescu’s theory of freeness, (see Voiculescu, Dykema and Nica [1992]), the free product
f
:
v of operator algebras is a basic operation, which is to freeness what the tensor product is to independence. Free
i
X products of several sorts of von Neumann algebras have been investigated, including free products of finite dimen-
r
a sional and hyperfinite von Neumann algebras with respect to traces, (Voiculsecu [1990], Dykema [1993(a)], Dykema
[1994(a)], Dykema [1993(b)]), certain amalgamated free products with respect to traces, (Popa [1993], Ra˘dulescu
[1994], Dykema [1995(a)]) and free products of various von Neumann algebras with respect to non–tracial states,
(R˘adulescu [preprint], Barnett [1995], Dykema [1994(b)]). In Ra˘dulescu [preprint], the free product of the diffuse
1 0
abelian von Neumann algebra and M (C) with respect to the state Tr 1+λ for 0 < λ < 1 was shown to
2 0 1+λλ ·
be a type III factor with core L(F ) B(H). In Barnett [1995]and Dy(cid:16)k(cid:16)ema [199(cid:17)4(b(cid:17))], some generalresults about
λ ∞
⊗
free products of vonNeumann algebraswith respectto non–tracialstates wereproved,but severalnaturalquestions
were left unanswered. For example, in the free product
1 0 1 0
( ,φ)=(M (C),Tr 1+λ ) (M (C),Tr 1+µ )
M 2 0 λ · ∗ 2 0 µ ·
(cid:18)(cid:18) 1+λ(cid:19) (cid:19) (cid:18)(cid:18) 1+µ(cid:19) (cid:19)
Supported byaNationalScienceFoundation Postdoctoral FellowshipandbyaFieldsInstitute Fellowship.
(cid:13)c0000AmericanMathematicalSociety
0000-0000/00$1.00+$.25perpage
1
2 K.J.Dykema
for 0<λ,µ<1, was by Dykema [1994(b)]known to be a factor only for certain values of λ and µ, and for other
M
values this remained unknown. One of the consequences of our main result is that such a free product is always a
factor, and the centralizer of the free product state φ is isomorphic to L(F ).
∞
In this paper, we investigate the free product of finite dimensional von Neumann algebras (and others) with
respect to arbitrary faithful states that are not traces. The main result can be summarized as follows:
Theorem 1 Let
( ,φ)=(A ,φ ) (A ,φ ) (1)
1 1 2 2
M ∗
be the von Neumann algebra free product of finite dimensional algebras with respect to faithful states, at least one of
which is non–tracial. Then
= D or = , (2)
0 0
M M ⊕ M M
where D is a finite dimensional algebra and where is a type III factor. The type I part, D, can be found from
0
M
def
knowledge of φ and φ , as described below. The restriction, φ = φ to the type III part, is an almost periodic
1 2 0 |M0
state (or functional) whose centralizer is isomorphic to the II –factor L(F ). The point spectrum of the modular
1 ∞
operator of φ , ∆ , is equal to the subgroup of R∗ generated by the union of the point spectra of ∆ and of ∆ .
0 φ0 + φ1 φ2
Thus, in Connes’ classification of as type III , one can find λ and always 0<λ 1.
0 λ
M ≤
Note that in (1) when both φ and φ are traces, was found in Dykema [1993(b)]. In that case, was of
1 2
M M
a similar form, except that the non–(type I) part, , was then a II factor related to free groups, (see Ra˘dulescu
0 1
M
[1994] and Dykema [1994(a)]). In fact, the description in Dykema [1993(b)] of D is a special case of the description
that follows.
Let us nowdescribe the type I part,D, in(2). (A more precisediscussionis found in 7.) Givena faithful state,
§
ψ, on a finite dimensional von Neumann algebra, D = K M (C), we write
j=1 nj
L
K
(D,ψ)= M (C)
nj
Mj=1αj,1,...,αj,nj
to meanthatthe restrictionofψ to the jthsummandofD is givenbyadiagonaldensitymatrix withα ,... ,α
j,1 j,nj
down the diagonal. If
K1
(A ,φ )= M (C)
1 1 nj
Mj=1αj,1,···,αj,nj
K2
(A ,φ )= M (C)
2 2 mj
Mj=1βj,1,···,βj,mj
andif istheirfreeproductasin(1),todeterminewhether isafactorand,whenitisnot,tofindthetypeIpart
M M
of , we first “mate” every simple summand of A with every simple summand of A and examine the offspring, if
1 2
M
any. When the jth summand ofA , namely M (C) , is mated with the kth summand of A , namely M (C) ,
1 nj 2 mk
αj,1,···,αj,nj βk,1,···,βk,mk
then therecanbe offspringonlyif atleastoneofn andm is equalto 1,i.e. ifatleastone ofthe simple summands
j k
Free products 3
is just a copy of C in the corresponding A . In that case, say if n =1, then there is offspring if and only if
ι j
mk 1 1
< ,
β 1 α
k,i j,1
i=1 −
X
and then the offspring is
mj 1
M (C) where γ =β 1 (1 α ) .
γ1,m...k,γmk i k,i − − j,1 Xp=1βk,p!
Analogous offspring results if m = 1 instead of n = 1. A system of matrix units for the offspring is the meet, (cid:13)∧,
k j
defined in Dykema [1995(a)], of systems of matrix units for the simple summands that were mated, M (C) and
nj
M (C), (see 7). Thus the support projection of this offspring lies under the support projection of M (C) and
mk § nj
the supportprojection of M (C). The type I part of is equal to the direct sum of all offspring producedin this
mk M
way by all possible matings of the original simple summands. If none of the matings were successful in producing
offspring, then is a factor.
M
For example, we have
C C M (C)= M (C) (3a)
2 a 2
(cid:18)15 ⊕ 45(cid:19)∗ 32,31 M ⊕ 115,310
C C M (C)= (3b)
2 b
(cid:18)14 ⊕ 34(cid:19)∗ 32,31 M
M (C) M (C)= (3c)
2 2 c
∗ M
1,3 2,1
4 4 3 3
M (C) C M (C) C = M (C) C (3d)
2 2 d 2
(cid:18)10(1+1√8),10(1√+8√8) ⊕ 190(cid:19)∗(cid:18) 31,61 ⊕ 12(cid:19) M ⊕ 310,610 ⊕ 25
M (C) C M (C) C = C (3e)
2 2 e
(cid:18) 240,210 ⊕ 43(cid:19)∗(cid:18) 13,61 ⊕ 12(cid:19) M ⊕ 41
C C M (C) M (C) M (C) C = M (C) M (C) C, (3f)
3 2 2 f 3 2
(cid:18)411 ⊕ 4401(cid:19)∗(cid:18)116,116,18 ⊕ 410,490 ⊕ 81,81 ⊕ 14(cid:19) M ⊕ 6156,6156,3128 ⊕ 32258,32258 ⊕ 13674
where , , , , and are type III factors.
a b c d e f
M M M M M M
Suppose that is not a factor, i.e. that some matings result in offspring, then one easily sees that there is a
M
dominant projection, p, which is by definition a minimal and central projection of A , for ι = 1 or ι = 2, such that
ι
all of the offspring are the progeny of the mating of p with simple summands from A where ι′ = ι. For example,
ι′ 6
in (3a), (3d) and (3f), the only dominant projection is 0 1 in the algebra on the left side of the , whereas in (3f),
⊕ ∗
both 0 1 on the left and 0 1 on the right side of are dominant projections. The dominant projection p A , if
ι
⊕ ⊕ ∗ ∈
it exists, must be the largest (with respect to φ ) of the projections in A that are both minimal and central.
ι ι
We now turn to a discussion of the type III factor, , appearing in Theorem 1. Connes [1973] defined a
0
M
state, φ, on a von Neumann algebra, , to be almost periodic if L2( ,φ) has a basis of eigenvectors for the
M M
4 K.J.Dykema
modular operator,∆ , of φ. In that case, Arveson–Connesspectraltheory (Arveson[1974],Connes [1973])provides
φ
a decomposition of as a direct sum of spectral subspaces. If, in addition, the centralizer, , of φ is a factor,
φ
M M
then, it is easily seen, the point spectrum of ∆ is a subgroup of R∗ and every spectral subspace is of the form
φ +
v or v∗ for an isometry v. In analogy with Connes’ results about full factors, Connes [1974], we call an
φ φ
M M
almost periodic state, φ, having centralizer that is a factor an extremal almost periodic state, and we define Sd(φ)
to be the point spectrum of ∆ . (At the end of 5, we show that term “extremal” is in some sense appropriate.)
φ
§
By Corollary 3.2.7 of Connes [1973], from the fact that is a factor we have also that Connes’ invariant S( )
φ
M M
is equal to the closure of Sd(φ). Hence in Theorem 1, φ is extremal almost periodic and Sd(φ ) is found from the
0 0
initial data, namely φ and φ . Moreover,regarding Connes’ classification of as type III , we have: is type
1 2 0 λ 0
M M
III for 0<λ<1 if Sd(φ )=λZ = λn n Z and otherwise Sd(φ ) is dense in R∗, and must be type III .
λ 0 { | ∈ } 0 + M0 1
For example,
in (3a), is a type III factor for λ=1/2;
a λ
M
in (3b), is a type III factor for λ=1/2;
b λ
M
in (3c), is a type III factor and the free product state, φ, has Sd(φ) equal to the (dense) subgroup of R∗
Mc 1 +
generated by 1/3 and 1/2;
in (3d), is a type III factor for λ=1/√2;
d λ
M
in (3e), is a type III factor for λ=1/2;
e λ
M
in (3f), is a type III factor and the free product state, φ, has Sd(φ) equal to the (dense) subgroup of R∗
Mf 1 +
generated by 1/2 and 1/9.
Extremal almost periodic states play a prominent role in the proof of Theorem 1. An example of the sort of
results proved and used in the course of this paper is the following theorem.
Theorem 2 Suppose for ι = 1,2 that A is a separable type III factor and φ is an extremal almost periodic
ι ι
state on A whose centralizer is isomorphic to the hyperfinite II factor, R, or to L(F ). Let
ι 1 ∞
( ,φ)=(A ,φ ) (A ,φ ).
1 1 2 2
M ∗
Then is a type III factor, φ is extremal almost periodic and Sd(φ) is the subgroup of R∗ generated by Sd(φ )
M + 1 ∪
Sd(φ ).
2
By taking inductive limits, Theorem 1 can be extended to more general algebras.
Theorem 3 Let
( ,φ)=(A ,φ ) (A ,φ ),
1 1 2 2
M ∗
where φ and φ are faithful states, at least one of which is not a trace, where A and A are nontrivial algebras,
1 2 1 2
each of which is one of the following:
(i) finite dimensional
(ii) B(H) for seperable, infinite dimensional Hilbert space H
(iii) a type III factor on which φ is extremal almost periodic and has centralizer isomorphic to the hyperfinite
ι
II factor, R, or to L(F );
1 ∞
Free products 5
(iv) a diffuse hyperfinite von Neumann algebra on which φ is a trace
ι
(v) an interpolated free group factor (of type II ), see Ra˘dulescu [1994], Dykema [1994(a)]
1
(vi) a possibly infinite direct sum of algebras from (i)-(v).
Then
= D or = ,
0 0
M M ⊕ M M
with and D as described in Theorem 1, (and more explicitly below).
0
M
In Theorem3, one finds the finite dimensional part, D, of , if it exists, by mating every matrix algebradirect
M
summand of A with every matrix algebrasummand of A , as described above,and D is the sum of the offspring of
1 2
these matings.
Theorem 4 Let
( ,φ)= * (A ,φ ),
ι ι
M ι∈I
where I is finite or countably infinite, I 2, each φ is a faithful state on A , at least one φ is not a trace and
ι ι ι
| | ≥
each A is of the form described in Theorem 3. Then
ι
= D or = ,
0 0
M M ⊕ M M
with and D as described in Theorem 1, (and more explicitly below).
0
M
In this case, one finds the finite dimensional part, D, of , if it exists, by, for every choice of a matrix algebra
M
direct summand, say M (C) from each A , mating them all together, and letting D be the sum of the offspring
nι ι
αι,1,...,αι,nι
resulting from all such matings. A mating of one summand from each A results in offspring only if all but one of
ι
the n equal 1, say n =1 if ι I ι′ , and then offspring results if and only if
ι ι
∈ \{ }
1 n′ι 1
>
(1 α ) α
ι∈I\{ι′} − ι,1 Xj=1 ι′,j
and the offspring is P
n
ι′ 1
M (C) where γ =α 1 (1 α ) .
n i ι,i ι,1
γ1,..ι.′,γnι′ ′  −(cid:18)ι∈XI\{ι′} − (cid:19)Xp=1αι′,p
We also have, as in Theorem 1, that is a type III factor, φ = φ is extremal almost periodic and Sd(φ ) is
M0 0 |M0 0
equal to the subgroup of R∗ generated by the union over ι I of the point spectra of ∆ .
+ ∈ φι
For example,
1/2 ∗ 3/4
1/4 3/16
(B(H),Tr( 1/8 )) (B(H),Tr( 3/64 ))= g (4g)
 .. · ∗  .. · M
. .
   
1/2
1/4
(B(H),Tr( 1/8 )) C M2(C)= h M2(C) (4h)
(cid:18)  .. · ⊕9/10(cid:19)∗ 3,1 M ⊕ 7,7
. 4 4 20 60
 
M (C) = , (4k)
2 k
n≥∗11+1n√2,1+n√n√22 M
6 K.J.Dykema
where , and are type III factors and
g h k
M M M
in (4g), is a type III factor for λ=1/2;
g λ
M
in (4h), is a type III factor and the free product state restricted to , call it φ , has Sd(φ ) equal to the
h 1 h 0 0
M M
subgroup of R∗ generated by 1/2 and 1/3;
+
in (4k), is a type III factor and the free product state, φ, has Sd(φ) equal to the subgroup of R∗ generated
Mk 1 +
by √n2 n 1 .
{ | ≥ }
The proofs of the above theorems involve what is loosely speaking a “type III version” of the following fact
from group theory. Let F = a,b be the free group with free generators a and b. Let π : F Z be the group
2 2
h i →
homomorphism such that π(a) = 0 and π(b) = 1. Then kerπ ∼= F∞ and kerπ is freely generated by (bnab−n)n∈Z.
By “type III version” we mean that trying to find the centralizer of certain free product states is like trying to find
kerπ above, where instead of b we have a nonunitary isometry. Of course, there are some technically more involved
aspects,butthe basicideaisaseasyasthis. We planto writeupthe proofofaspecialcase,namely M (C) M (C)
2 2
∗
with respect to arbitrary faithful states, in Dykema [SNU]. This proof will contain most of the essential ideas of
the proof of the general case, but should more transparent, and we would recommend it to those interested in an
introduction to the techniques.
Many results of this paper are proved using the technique of free etymology. This technique consists of proving
the freeness of certain algebras by looking at words in those algebras, and investigating the “roots” of those words.
Tobe morespecific,suppose wewantto showthatafamily (A ) ofsubalgebrasofAis free in(A,φ), andeachA
ι ι∈I ι
is defined in terms ofsome other subalgebras(B ) that we know are free. Since we work so muchwith words,let
j j∈J
us use a specialnotation, which we have used before in Dykema [1993(b)], Dykema [1994(a)]and Dykema [1995(a)].
Definition5 SupposeAisanalgebra,φ:A Cislinear. ForB Aanysubalgebra,wedefineBo(pronounced
→ ⊆
B–bubble) to be Bo = B kerφ. Suppose X A, are subsets, (ι I). The set of reduced words in (X ) is
ι ι ι∈I
∩ ⊆ ∈
denoted Λo((X ) ), and is defined to be the set of a a a such that n 1, a X and ι = ι = ι . If
ι ι∈I 1 2··· n ≥ j ∈ ιj 1 6 2 6 ··· n
I =2, we may call them alternating products.
| |
Now a free etymologyproof ofthe freeness of (A ) wouldgo as follows: we need only showφ(x)=0 for every
ι ι∈I
x Λo((Ao) ); after doing some work, one shows that x is equal to, for example if A is a vonNeumann algebra,a
∈ ι ι∈I
strong–operatorlimit of sums of reduced words in (Bo) , and hence one concludes that φ(x)=0.
j j∈J
In 8, using a result of Barnett [1995], we show in many cases that in Theorems 1, 3 and 4 is a full factor.
0
§ M
Thus, for these factors, Connes’ invariant for full factors, Sd( ), is equal to Sd(φ ), and thereby, for instance by
0 0
M
taking free products as in example (4k), we obtain for an arbitrary countable dense subgroup, Γ, of R∗, a free
+
product factor that is a full type III factor, has Γ as its Sd invariant and has extremal almost periodic state with
1
centralizer isomorphic to L(F ). See Corollary 4.4 of Connes [1974] and Golodets and Nessonov [1987] for other
∞
constructions of full III factors with arbitrary Sd invariant.
1
Acknowledgements. Most of this work was done while I was at the Fields Institute during the special year in
Operator Algebras, 1994/95. I would like to thank George Elliott for the organizing that marvelous year.
Free products 7
Contents:
1. Free Subcomplementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.7
§
2. Freeness with amalgamation over a subalgebra . . . . . . . . . . . . . . . . . . . . . . . . . p.11
§
3. Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.11
§
4. Extremal almost periodic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.13
§
5. Some technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.16
§
6. Free products of factors with respect to extremal almost periodic states. . . . . . . . . . . . . . . p.19
§
7. Free products of finite dimensional algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . p.31
§
8. Fullness of free product factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.44
§
9. Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.45
§
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.46
1. Free Subcomplementation.
InDykema[1993(b)],thefreeproductofarbitraryfinitedimensionalorAFDvonNeumannalgebraswithrespect
to faithful traces was determined, and found to be an interpolated free group factor (R˘adulescu [1994], Dykema
[1994(a)]) possibly direct sum a finite dimensional algebra. The free product of finite dimensional algebras was
determinedusingthetechniquethatisnowcalledfree etymology, andpassingtoAFDalgebraswaspermittedbythe
use of the notionof a standard embedding ofinterpolatedfree groupfactors. The definitionof a standardembedding
of interpolated free group factors, L(F ) ֒ L(F ) is a generalization of the sort of embedding L(F ) ֒ L(F ),
s t n m
→ →
n < m or n = m = , obtained by mapping the n free generators of F to n (not all) of the m free generators of
n
∞
F .
m
The characterization of standard embeddings in Proposition 1.2 is pleasing to the intuition and will be used in
this paper.
Definition 1.1 Suppose ( ,φ) is a von Neumann algebra with normal state, and let 1 A be a von
M ∈ ⊆ M
Neumannsubalgebra. WesaythatAisfreely complementedin( ,φ)ifthere isavonNeumannsubalgebraB
M ⊆M
suchthat is generatedby A B andA and B are free in ( ,φ). To be more specific, we cansaythat A is freely
M ∪ M
complemented by B.
Proposition 1.2 Suppose π :L(F )֒ L(F ) is an injective, normal, trace–preserving –homomorphism. Then
s t
→ ∗
π is a standard embedding if and only if π(L(F )) is freely complemented in L(F ) by an algebra isomorphic to
s t
L(F ) C or simply to L(F ), some 1<r .
r r
⊕ ≤∞
Proof Let A=π(L(F )) be freely complemented in L(F ) by B =L(F ) C or =L(F ). Then by 4.4 and 1.2
s t ∼ r ⊕ ∼ r
of Dykema [1993(b)], π is a standard embedding. Since for any two standard embeddings π : L(F ) ֒ L(F ) and
1 s t
→
π :L(F )֒ L(F ),thereisanautomorphismαofL(F )suchthatπ =π α,(thisiseasilyseenfromthedefinition
2 s t t 1 2
→ ◦
of a standard embedding Dykema [1993(b)]), and since for anys and t we can find B =L(F ) C and a trace on it
r
⊕
such that L(F ) B =L(F ), it follows that every standard embedding is of this form. (cid:3)
s ∗ ∼ t
8 K.J.Dykema
The above characterization of standard embeddings allows one to easily prove the following useful facts about
standard embeddings, some of which first appeared in Dykema [1993(b)].
Proposition 1.3
(i) The composition of standard embeddings is a standard embedding.
(ii) Suppose A = L(F ), 1 < t and π : A A is a standard embedding (n 1). Consider the
n tn n ≤ ∞ n n → n+1 ≥
inductive limit von Neumann algebra (taken with respect to the unique traces on A ), A=lim(A ,π ) with
n n n
−→
the associated inclusions ψ : A ֒ A. Then A = L(F ) where t = lim t and each ψ is a standard
n n t n→∞ n n
→
embedding.
We now turn to an extension of the notion of a freely complemented subalgebra.
Definition 1.4 Suppose ( ,φ) is a von Neumann algebra with normal state, and let 1 A be a von
M ∈ ⊆ M
Neumann subalgebra. Let A denote the centralizer of φ . We say that A is freely subcomplemented in ( ,φ)
φ |A M
if, for some index set I, there is for each ι I a self–adjoint projection h A and a von Neumann subalgebra
ι φ
∈ ∈
h B h h such that
ι ι ι ι
∈ ⊆ M
(i) =W∗(A B )
M ∪ ι∈I ι
(ii) Bι and hιMI\{Sι}hι are free in (hιMhι,φ(hι)−1φ|hιMhι), where MI\{ι0} d=efW∗(A∪ ι∈I\{ι0}Bι).
Tobemoreprecise,wewillsaythatAisfreelysubcomplementedin( ,φ)by(Bι)ι∈I andwithSassociatedprojections
M
(h ) .
ι ι∈I
The condition (ii)’ of the following lemma will be the most useful one for proving properties of free subcomple-
mentation.
Lemma 1.5 Suppose ( ,φ) , A, h and B are as in Definition 1.4, except without assuming (ii) holds and
ι ι
M
suppose also that the GNS representation of associated to φ is faithful. Let Ω be the set of reduced words
M
a a a Λo(A, Bo) such that whenever 2 j n 1, a A, and a ,a Bo for some ι I, then
1 2··· n ∈ ι∈I ι ≤ ≤ − j ∈ j−1 j+1 ∈ ι ∈
aj (hιAhι)o. ThenSspan(Ω 1 ) is a s.o.–dense –subalgebra of . Consider the condition
∈ ∪{ } ∗ M
(ii)’ φ(z)=0 z Ω A.
∀ ∈ \
Then (ii)’ = (ii). Moreover, (ii)’ implies the existence of a s.o.–continuous projection of norm 1, E : A,
⇒ M →
such that E(z)=0 z Ω A. Conversely, if each h has central support in A equal to 1 then (ii)= (ii)’.
ι φ
∀ ∈ \ ⇒
Proof One easily sees that span(Ω 1 )is the –algebrageneratedby A B , hence is s.o.–dense in .
∪{ } ∗ ∪ ι∈I ι M
For F ⊆ I, let ΩF be the set of all words z ∈ Ω such that whenever z has a leStter coming from Bιo, ι ∈ I, then
ι F. Let us show that (ii)’ implies the existence of the projection E : A. We will realize this projection
∈ M →
by compressing to L2(A,φ) in the Hilbert space L2( ,φ). The defining mapping L2( ,φ) will be denoted
M M → M
x xˆ and we let ξ d=efˆ1.
7→
Lemma 1.6 If is a von Neumann algebra with normal state φ and if h,k are self–adjoint projections
φ
P ∈ P
in the centralizer of φ such that φ(hk)=0, then φ(hak)=0 a .
∀ ∈P
Free products 9
Proof 0=φ(hk)=φ(khk).The left supportofkhis the supportofkhk whichis k k (1 h). Since khk 0
− ∧ − ≥
and φ(khk)=0 we have φ(k k (1 h))=0. Thus
− ∧ −
φ(hak)=φ(khak)=φ((k k (1 h))ha(k k (1 h)))=0.
− ∧ − − ∧ −
Hence Lemma 1.6 is proved. (cid:3)
Continuing with the proof of Lemma 1.5, using Lemma 1.6 we see for ι ,ι I that
1 2
∈
h Ah kerφ if φ(h h )=0
ι1 ι2 ⊆ ι1 ι2
(5)
h Ah =(h Ah ) kerφ+Ch h if φ(h h )=0
ι1 ι2 ι1 ι2 ∩ ι1 ι2 ι1 ι2 6
Using(ii)’,equation(5)andthatspan(Ω 1 )isdensein ,wefindthatL2( ,φ)canbecharacterizedasfollows.
∪{ } M M
o o
Let α I, H d=efL2(A,φ ) Cˆ1, for ι I let H d=efL2(B ,φ ) Chˆ , and let
6∈ α |A ⊖ ∈ ι ι |Bι ⊖ ι
o o o
Hd=efCξ H H H .
⊕ j1 ⊗ j2 ⊗···⊗ jn
n≥1
M
jk∈I∪{α}
j16=j26=···6=jn
Then L2( ,φ) is the closed subspace of H that is spanned by ξ together with all vectors
M
o o
ζ ζ H H , j = =j
1⊗···⊗ n ∈ j1 ⊗···⊗ jn 1 6 ···6 n
such that
if 2 k n 1, j =α, (hence j ,j I), then ζ aˆ a h Ah kerφ
≤ ≤ − k k−1 k+1 ∈ k ∈{ | ∈ jk−1 jk+1 ∩ }
if n 2 and j =α, (hence j I), then ζ aˆ a Ah kerφ
≥ 1 2 ∈ 1 ∈{ | ∈ j2 ∩ }
if n 2 and j =α, (hence j I), then ζ aˆ a h A kerφ .
≥ n n−1 ∈ n ∈{ | ∈ jn−1 ∩ }
o
LetP be the self–adjointprojectionofL2( ,φ) ontoL2(A,φ )=Cξ H L2( ,φ). Let E :B(L2( ,φ))
A M |A ⊕ α ⊆ M M →
B(L2(A,φ )) be defined by E(x)=P xP . Then E is a projection of norm 1 and is s.o–continuous. We easily see
|A A A
that
E(a)=a a A
∀ ∈
E(z)=0 z Ω A.
∀ ∈ \
Thus, by s.o.–continuity of E, we see that E projects onto A.
M
Now using the projectionE, we willshow (ii)’= (ii). Letι I. We willshowthat B andh h arefree
ι ι I\{ι} ι
⇒ ∈ M
in h h . Clearly span(Ω 1 ) is a dense –subalgebra of . Using the projection E : A, we see
ι ι I\{ι} I\{ι}
M ∪{ } ∗ M M→
that each element of (h h )o is the s.o.–limit of a bounded net in span(h (Ω A)h (h Ah )o), hence it
ι I\{ι} ι ι I\{ι} ι ι ι
M \ ∪
will suffice to show φ(y)=0 whenever
y Λo(h (Ω A)h (h Ah )o,Bo).
∈ ι I\{ι}\ ι∪ ι ι ι
But by regrouping we see that y (Ω A) (h Ah )o, hence φ(y)=0 by (ii)’.
ι ι
∈ \ ∪
Assumethateachh hascentralsupportinA equalto1andletusshow(ii)= (ii)’. Takeanyz =a a a
ι φ 1 2 n
⇒ ··· ∈
Ω A, where F is a finite subset of I. It will suffice to show φ(z) = 0, and we will proceed by induction on F ,
F
\ | |
10 K.J.Dykema
proving the F = 1 case and the inductive step simultaneously. Let ι F be such that z has a letter from Bo.
| | ∈ ι
Now we may assume n 3 and a ,a A. By hypothesis, there are partial isometries v ,... ,v A such that
1 n 0 k φ
≥ ∈ ∈
v∗v h 1 j k and k v v∗ = 1. Now φ(z) = k φ(v v∗ zv v∗ ) = k φ(v∗zv ), so we may
j j ≤ ι ∀ ≤ ≤ j=1 j j j1,j2=0 j1 j1 j2 j2 j=0 j j
assume without loss of generaPlity that a1 ∈ hιA and an ∈ APhι. Regrouping in z by singlPing out all letters from Bιo
and groupingall other strings of neighboring letters together,we see that z Λo(h Ω h ,Bo) and if a letter of z
∈ ι F\{ι} ι ι
is in h Ω h A then it is in (h Ah )o, unless it is the first or the last letter, in which case it is in h Ah . Using
ι F\{ι} ι ι ι ι ι
∩
if necessary h Ah =(h Ah )o+Ch to expand the first and/or the last letter, we see that z is a linear combination
ι ι ι ι ι
of up to four elements of
Λo((h Ω h A) (h Ah )o,Bo).
ι F\{ι} ι\ ∪ ι ι ι
By inductive hypothesis in the case F >1 and by the fact that Ω Ao if F =1, we have that
F\{ι}
| | ⊆ | |
z spanΛo((h h )o,Bo),
∈ ιMI\{ι} ι ι
hence by (ii), φ(z)=0. (cid:3)
Nowweshowthatthecompositionof(certain)freelysubcomplementedembeddingsisafreelysubcomplemented
embedding.
Corollary 1.7 Suppose
1 A A (6)
1 2
∈ ⊆ ⊆···⊆M
is a finite or infinite chain of inclusions of von Neumann algebras, φ is a normal state on and
M
A if the chain (6) is finite of length n
n
=
M ( n≥1An if the chain (6) is infinite
Suppose for all j that A is freely subcoSmplemented in (A ,φ ) by (B ) with associated projections (h )
j j+1 |Aj+1 ι ι∈Ij ι ι∈Ij
such that each h has central support in (A ) equal to 1 and is Murray–von Neumann equivalent in (A ) to a
ι j φ j φ
projection in A . Then A is freely subcomplemented in by (algebras isomorphic to) (B ) , with I = I .
1 1 M ι ι∈I j j
S
The above result is a special case of the next one.
Corollary 1.8 Let J be a set and a well–ordering on J. Suppose j J A is a von Neumann algebra
j
≤ ∀ ∈
with normal state φ and suppose there are compatible, φ –preserving inclusions A A j j . Let ( ,φ)=
j j j1 ⊆ j2 ∀ 1 ≤ 2 M
lim (A ,φ ) be the inductive limit von Neumann algebra. Let j J denote the smallest element and suppose
j∈J j j 0 ∈
−→
j J j that d=efW∗( 1 A ) is freely subcomplemented in W∗( A ) (with respect to the
∀ ∈ \{ 0} M(cid:8)j { }∪ j1(cid:8)j j1 M(cid:8)j ∪ j
restriction of φ to W∗(M(cid:8)j ∪Aj)) bSy (Bι)ι∈Ij with associated projections (hι)ι∈Ij, such that each hι has central
support in (A ) equal to 1 and is Murray–von Neumann equivalent to a projection in A . Then A is freely
j φ j0 j0
subcomplemented in ( ,φ). by (algebras isomorphic to) (B ) where I = I .
M ι ι∈I j∈J j
S
Proof Let (h ) be the projections associated to (B ) . We may suppose h (A ) ι I. Now it will
ι ι∈Ij ι ι∈Ij ι ∈ j0 φ ∀ ∈
sufficetoshow(ii)’ofLemma1.5,(withA=A here). Letz Ω A,whereF I isanyfinitesubsetandletj J
j0 ∈ F\ ⊆ ∈
be the largest element of J such that F I is nonempty. We show φ(z)=0 by transfinite induction on j. If j =j
j 0
∩
then Ω A and there is nothing to prove. Otherwise, regrouping in z by grouping letters from all Bo for ι I
F ⊆ ι 6∈ j