The Radial Masa in a Free Group Factor is Maximal Injective

0.3 MB·English

by  Jan Cameron

The radial masa in a free group fa
tor is
maximal inje
tive
8
0 Jan Cameron Junsheng Fang Mohan Ravi
handran
0
2 j
ameronmath.tamu.edu jfangmath.tamu.edu mohanr
isunix.unh.edu

t
c Stuart White
O
s.whitemaths.gla.a
.uk
1
2

]
A Abstra
t
O
The radial (or Lapla
ian) masa in a free group fa
tor is the abelian von Neumann
.
h algebra generated by the sum of the generators (of the free group) and their inverses.
at Themainresultofthispaperisthattheradialmasaisamaximalinje
tivevonNeumann
m subalgebra of a free group fa
tor. We establish this result by showing that sequen
es
[
entralising the radial masa have an orthogonality property in the ultrapower, using a

1 basisintrodu
edbyR dules
u. WBe⊂alsMoinvestigatIetensorprodu
tsofmaximalinje
tive
i i
v algebras. Given two in
lusions B of type von NeumannMalgebras in (cid:28)nite von
6 i i
0 Neumann algebBras⊗suB
h that ea
h is maximMal in⊗jeM
tive in , we show that the
1 2 1 2
9 tensor produ
t is maximal inje
tive in provided at least one of the
3 in
lusions satis(cid:28)es the asymptoti
orthogonality property we establish for the radial
.
0 masa. In parti
ular it follows that (cid:28)nite tensor produ
ts of generator and radial masas
1
will be maximal inje
tive in the
orresponding tensor produ
t of free group fa
tors.
8
0
:
v
i 1 Introdu
tion
X
r
In [14℄, Popa showed that the von Neumann algebra generated by a single generator of a
a
free group is a maximal inje
tive von Neumann subalgebra of the fa
tor generated by that
F K a ,...,a
K 1 K
free group. That is, if denotes the free group on generators , then the von
LF a
K 1
Neumann subalgebra of the free group fa
tor generated by is a maximal inje
tive
LF
K
von Neumann subalgebra of . This subalgebra is known as a generator masa (maximal
LF
K
abelianLsFubalgebra) in . There is anothKer (naat+uraal−ly1)o

urring masa in the free group
fa
tor K: the radial masa generated by i=1 i i , whi
h was shown to be maximal
abelian by Pytlik in [17℄. The main obje
tiPve of this paper is to show that this radial masa
LF
K
gives another example of an abelian maximal inje
tive von Neumann subalgebra of .
A
The radialmasa shares many properties withgeneratormasas. Dixmierde(cid:28)ned a masa
II M u ∈ M uAu∗ = A A
1
in a fa
tor to be singular if every unitary with lies in , and showed
in [5℄ that a generator masa is singular. Singularity of the radial masa was established by
R dules
u by means of an intri
ate
al
ulation in [18℄, whi
h provides a
entral ingredient
in this paper. An alternative
ombinatorial proof is given by Sin
lair and Smith in [20℄.
1
To establish singularity of the radial masa, R dules
u
omputed its Pukánszky invariant,
histori
ally the most su
essful invariant for singular masas (see [22℄ for a dis
ussion of this
invariant). The Pukánszky invariant of a generator masa is easily
omputed using the group-
subgroup methods of [21℄ and both the radialand generator masas have Pukánszky invariant
{∞} LF
K
. It is natural to ask whether these masas are
onjugate in . This seems unlikely,
but we have been unable to (cid:28)nd a proof. As noted in Proposition 3.1, it is easy to show that
LF
K
the radial masa is not inner
onjugate to a generator masa in .
LF
K
Question 1.1. Does there exist an automorphism of whi
h maps the radial masa onto
a generator masa?
The
riti
al ingredient in Popa's proof of the maximal inje
tivity of a generator masa
B LF
K
in a free group fa
tor is an asymptoti
orthogonality property at the Hilbert spa
e
x ,x (LF )ω
1 2 K
level. Lemma 2.1 of [14℄ shows that if are elements of an ultrapower with
EBω(x1) = EBω(x2) = 0 y1,y2 LFK EB(y1) = EB(y2) = 0
and are elements of with , then
x y y x L2((LF )ω)
1 1 2 2 K
is orthogonal to in the Hilbert spa
e . To do this, Popa (cid:28)rst shows
x ∈ LF a
K 1
that those elements whi
h approximately
ommute with the generator and are
B
orthogonal to must be essentially supported on the
olle
tion of words whi
h begin and
a a−1 y ,y
1 1 1 2
end with large powers of or . The asymptoti
orthogonality follows when are
F \{an : n ∈ Z}
group elements in K 1 and a density argument then gives the result for general
y y
1 2
and (see also [22, Se
tion 14.2℄).
Most of this paper is taken up with establishing an asymptoti
orthogonality
ondition
for the radial masa (Theorem 6.2). This is the key step in proving maximal inje
tivity.
ℓ2(F )
K
The natural orthonormal basis of does not behave well in
al
ulations involving the
radial masa, however to prove that the radial masa is singular R dules
u introdu
ed in [18℄
ℓ2(F )
K
a
olle
tion of ve
tors whi
h form a basis (albeit not quite an orthonormal basis) for
whi
h
anbethoughtofasspanningdouble
osets
omingfromtheradialmasa. In Se
tion3,
we set up notation for dis
ussing the radial masa and des
ribe the properties of R dules
u's
x
basis. In Se
tion 4, we establish the essential lo
ation of the support of an element in
LF
K
whi
h is both orthogonal to the radial masa and approximately
ommutes with it. We
do this in Lemma 4.3, the proof of whi
h is
ontained in the lemmas pre
eding it. Se
tion
5
ontains the se
ond step of [14, Lemma 2.1℄ for the radial masa. In this se
tion, we use
F
K
te
hniques from [20℄ to
ount
ertain words in whi
h we use in Se
tion 6 to establish
the asymptoti
orthogonality result from the results of Se
tion 4. All of the
al
ulations
LF ⊗ N N
K
des
ribed above are performed in , where is an arbitrary (cid:28)nite von Neumann
algebra. The only additional di(cid:30)
ulties this introdu
es are notational, and it enables us to
show that
ertain tensor produ
ts involving the radial masa are also maximal inje
tive, as
des
ribed below.
In [14, 4.5 (1)℄, Popa asked how maximal inje
tive von Neumann algebras behave under
B ⊂ M i = 1,2
i i
tensor produ
ts. That is, if ( ) are two in
lusions of von Neumann algebars
B M B ⊗ B
i i 1 2
with a maximal inje
tive von Neumann subalgebra of , must be a maximal
M ⊗ M
1 2
inje
tive von Neumann subalgebra of ? Various authors have subsequently worked
on this question. The (cid:28)rst progress was made by Ge and Kadison, who in [9℄ gave a positive
B = M
1 1
answer when is an inje
tive fa
tor. This result was subsequently improved by
Str til and Zsidó who removed the fa
tor assumption to show ([24, Theorem 6.7℄) that if
M M
1 2
is an inje
tive von Neumann algebra and is a von Neumann algebra with a separable
2
M ⊗B M ⊗M
1 2 1 2
predual, then is a maximal inje
tive von Neumann subalgebra of whenever
B M
2 2
is a maximal inje
tive von Neumann subalgebra of . In [19℄, Shen investigated the
tensor produ
t of
opies of a generator masa inside the a free group fa
tor, answering Popa's
question positively in this
ase. He was also able to obtain a positive result for an in(cid:28)nite
tensor produ
t of generator masas and so obtain an example of an abelian subalgebra whi
h
II
1
is maximalinje
tive in a M
Du(cid:27) fa
tor. In [19℄, it was ne
essary to develop an additional
orthogonality
ondition styled upon [14, Lemma 2.1℄ for tensor produ
ts of generator masas.
Re
ent progress was made by the se
ond author, who in [7℄ gave a positive answer to Popa's
Z(B ) M
1 1
question in the
ase that is atomi
and has a separable predual.
In Se
tion 2, we examine how asymptoti
orthogonality properties based on [14, Lemma
2.1℄implymaximalinje
tivity. UsingPopa'sintertwininglemmaandingredientsfrom[7,19℄,
we are able to give a te
hni
al improvement of the argument whi
h dedu
es the maximal
inje
tivity of the generator masa from the asymptoti
orthogonality
ondition. This argu-
ment doesn't require any further assumptions on the masa beyond singularity and is also
appli
able to the tensor produ
t of maximal inje
tive algebras, provided we
an establish a
slightly stronger asymptoti
orthogonality
ondition (the asymptoti
orthogonality property
of De(cid:28)nition 2.1) whi
h in all known
ases is established by exa
tly the same argument
used to establish the appropriate version of [14, Lemma 2.1℄. In parti
ular, it follows that if
A ⊂ M II
1
is a singluar masa of a fa
tor with the asymptoti
orthogonality property, su
h
B ⊂ N
as a generator masa or the radial masa in a free group fa
tor, and is a maximal
II B I A ⊗ B
1
inje
tive inje
tive von Neumann subalgebra of a fa
tor with of type , then is
M ⊗ N A
maximal inje
tive in (Theorem 2.8, taking to be a masa). This result strengthens
Shen's (cid:28)nite tensor produ
t results from [19℄ and enlarges the
lass of positive answers to
Popa's question.
The paper
on
ludes with Se
tion 7, whi
h
ontains some (cid:28)nal remarks. We show that
the additionalingredientsPopauses withasymptoti
orthogonalitytoshowageneratormasa
is maximal inje
tive, follow from maximal inje
tivity and Ozawa's solidity of the free group
fa
tors introdu
ed in [11℄. In this way, we see that these properties of the generator masas
are also satis(cid:28)ed by the radial masa. We end with a brief dis
ussion of maximal nu
learity
∗
of the generator and radial masas in the redu
ed group C -algebras of free groups.
A
knowledgement: The work in this paper originated during a visit of the fourth author
to the University of New Hampshire in April 2008. He would like to thank the fa
ulty
and students of the Department of Mathemati
s, and in parti
ular Don Hadwin, for their
hospitality during this visit. The authors would also like to thank Simon Wassermann for
∗ ∗
simulating
onversations regarding maximal nu
lear C -subalgebras of C -algebras.
2 Asymptoti
Orthogonality and Maximal Inje
tivity
We begin by making a te
hni
al de(cid:28)nition whi
h
an be used to dedu
e maximal inje
tivity
I
for von Neumann subalgebras of type .
A I II
1
De(cid:28)nition 2.1. Let be a type von Neumann subalgebra of a type von Neumann
M τ N
M
algebra with a (cid:28)xed faithful normal tra
e . Let be a (cid:28)nite von Neumann algebra
τ A ⊂ M
N
with a (cid:28)xed faithful normal tra
e . Say that has the asymptoti
orthogonality
3
N ω ∈ βN \ N
property after tensoring by if there is a non-prin
ipal ultra(cid:28)lter su
h that
x(1)y ⊥ y x(2) L2((M ⊗ N)ω) x(1),x(2) (A ⊗ C1)′∩(M ⊗ N)ω
1 2
in , whenever are elements of
E (x(i)) = 0 i = 1,2 y ,y ∈ M ⊗ N E (y ) = 0 i = 1,2
(A⊗N)ω 1 2 A⊗N i
with for , and with for . Say
A N = C1
that has the asymptoti
orthogonality property if it has this property when .
(M ⊗ N)ω
Note that the ultraprodu
t used in the de(cid:28)nition above is
onstru
ted with
τ ⊗τ
M N
respe
t to the produ
t tra
e .
In se
tion 2 of [14℄, Popa essentially shows that the generator masa inside a free group
fa
tor has the asymptoti
orthogonality property. The arguments given in [14℄ (see also [22,
Se
tion 14.2℄) also show that the generator masa has the asymptoti
orthogonality property
N
after tensoring by any without further work. As noted in [14℄, given any separable di(cid:27)use
I B 6= C1 B∗R
type von Neumann algebra , one
an form the free produ
t and the methods
B B ∗R B ⊂ B ∗R
of [14℄ show that is maximal inje
tive in . Indeed, these in
lusions have
N
the asymptoti
orthogonality property after tensoring by any .
In this se
tion we show how the maximal inje
tivity of a singular masa satisfying the
asymptoti
orthogonality property
an be established without further hypothesis (Corollary
2.3). We also examine tensor produ
ts, showing how the asymptoti
orthogonality property
N A⊗B
after tensoring by
an be used to show that
ertain tensor produ
ts of maximal
inje
tive subalgebras are again maximal inje
tive. In subsequent se
tions we shall show that
the radial masa in a free group fa
tor also has the asymptoti
orthogonality property after
N
tensoring by any . It will follow that (cid:28)nite tensor produ
ts of generator and radial masas
are maximal inje
tive. We need some te
hni
al observations, the (cid:28)rst of whi
h easily follows
from Popa's intertwining lemma.
L II
1
Lemma 2.2. Let be an inje
tive type von Neumann algebra equipped with a (cid:28)xed
A I L
faithful normal tra
e. Let be a type von Neumann subalgebra of . Then there exists a
u ∈ L′ ∩Lω EAω(u) = 0
unitary with .
L (L )
n
Proof. Sin
e isinje
tiveandsohyper(cid:28)nite we
an(cid:28)ndasequen
e of(cid:28)nitedimensional
L L ε > 0
subalgebras of whose union is weakly dense in . It su(cid:30)
es to show that for all
n ∈ N u ∈ L kE (u)k < ε u ∈ L′ ∩ L
and there exists a unitary with A 2 and n . However,
n ε > 0
if there exists some and for whi
h no su
h unitary
ould be found, then Popa's
4 ⇒ 1
intertwining lemma [16℄ (see also [3, Theorem F.12, ℄ for the exa
t statement we are
L′ ∩ L A L
using) shows that a
orner of n embeds into inside . This
an not happen, sin
e
L′ ∩L II A I
n 1
is ne
essarily type and is type (cid:28)n.
A II M
1
Corollary 2.3. Let be a singular masa in a von Neumann algebra with the asymp-
A
toti
orthogonality property. Then is maximal inje
tive.
L A ⊂ L ⊂ M p
Proof. Let be an inje
tive von Neumann algebra with . Let be the maximal
L Lp II p ∈ A p 6= 0
1

entral proje
tion in so that is type . Note that . If , use Lemma 2.2
u (Lp)′ ∩ (Lp)ω EAω(u) = 0 v ∈ Lp
to (cid:28)nd a unitary in with . Choosing some non-zero
E (v) = 0 uv ⊥ vu
A
with , the asymptoti
orthogonality property gives . On the other hand
uv = vu uv = 0 v = 0 p = 0 L I
, so and , giving a
ontradi
tion. Hen
e and is a (cid:28)nite type
A L
von Neumann algebra. By [10℄, is regular in (see also [19, Lemma 2.3℄, where this fa
t
A M L = A

an be found expli
itly), so by singularity of in it follows that , as required.
4
The proof of our se
ond te
hni
al observation, whi
h allows us to handle tensor produ
ts,
is
ontained in [19℄ modulo a theorem from [24℄. We in
lude the details for
ompleteness.
A ⊂ M B ⊂ N
Lemma 2.4. Let and be two in
lusions of (cid:28)nite von Neumann algebras
A′ ∩M ⊂ A A I B
with separable preduals. Suppose that and is type and suppose that is a
N A ⊗ B ⊂ L ⊂ M ⊗ N
maximal inje
tive von Neumann subalgebra of . Let be an inje
tive
intermediate subalgebra. Then
E (L) = A ⊗ B.
A⊗N
A I (A′∩M)′∩M = A A I
n
Proof. Sin
e is(cid:28)niteandtype , itfollowsthat . Indeed if isoftype ,
A Z⊗M Z
n
wMe=
anM˜ex⊗prMess as (A′ ∩Mfo)r =som(Ze′a∩beM˜lia)n⊗Ca1nd
onstru
t a
orrrespAon′d∩inMg f⊂a
tAorisation
n
Z . ThenM˜ (A′∩M)′∩M = (Zso⊗tChe1)
′o∩nd(Mit˜io⊗nMtha)t= Z ⊗M = Aimplies
n n
that is a masa in . Hen
e . The
general
ase follows by taking a dire
t sum.
x ∈ L E (x) k·k co2{uxu∗ : u ∈ U((A′ ∩
Given , A⊗N is the unique element of minimal 2 in
M) ⊗ Z(N))} ((A′ ∩M) ⊗ Z(N))′ ∩ (M ⊗ N) = A ⊗ N (A′ ∩ M) ⊗ Z(N) ⊂
as . Sin
e
A ⊗ B ⊂ L E (L) ⊂ L x ∈ L E (x) ∈/ A ⊗ B
A⊗N A⊗N
, it follows that . In parti
ular, if has ,
then we have
A ⊗ B ( (A ⊗B,E (x))′′ ⊆ A ⊗ N,
A⊗N
(A ⊗ B,E (x))′′ L
A ⊗ N
where is a subalgebra of and hen
e inje
tive. This
ontradi
ts [24,
A⊗B A⊗N A
Theorem6.7℄, whi
hshows that ismaximalinje
tivein ,whenever isinje
tive
B N
and is maximal inje
tive in a von Neumann algebra with separable predual.
These ingredients enable us to gain
ontrol over
ertain intermediate inje
tive subalge-
bras.
M,N A
Lemma 2.5. Let be (cid:28)nite von Neumann algebras with separable preduals. Let be
I M A′ ∩M ⊆ A
a type von Neumann subalgebra of with and the asymptoti
orthogonality
N B I N
property after tensoring by . Let be a type von Neumann subalgebra of whi
h is
N L
a maximal inje
tive von Neumann subalgebra of . If is an intermediate inje
tive von
A ⊗ B M ⊗ N L I
Neumann algebra between and , then is ne
essarily type .
p L Lp II
1
Proof. Let be the maximal
entral proje
tion of su
h that is type . Note that
p ∈ L′∩(M ⊗N) ⊂ A ⊗ B p 6= 0 u ∈ (Lp)′∩(Lp)ω
. If , then Lemma 2.2 gives us a unitary ,
E (u) = 0 (Lp)ω
((A⊗B)p)ω
with , where the ultrapower is
onstru
ted from the produ
t tra
e on
M ⊗ N (A ⊗ B)p ( Lp v ∈ Lp E (v) = 0
A⊗B
. Sin
e , there is some non-zero with . Regarding
u ∈ (M ⊗ N)ω v ∈ M ⊗ N
and , we have
E (v) = E E (v) = E E (v) = 0,
A⊗N A⊗B A⊗N A⊗N A⊗B
E (v) ∈ A ⊗ B u = (u )
A⊗N n
where the (cid:28)rst equality uses Lemma 2.4 to see that . By writing
u ∈ L E (u ) = 0 n E (u ) = 0 n
n A⊗B n A⊗N n
for some with for all , it follows that for ea
h just
E (u) = 0 A ⊂ M
(A⊗N)ω
as above and so . As has asymptoti
orthogonality property after
N uv ⊥ vu u ∈ (Lp)′ ∩(Lp)ω v ∈ Lp u v
tensoring by , we have . Sin
e and , and
ommute.
uv = 0 v = 0 p = 0
Therefore and hen
e , a
ontradi
tion. Therefore .
5
A
With the notation of the previous lemma, if we in addition assume that is a singular
B
masawiththeasymptoti
orthogonalitypropertyand isamasawhi
hismaximalinje
tive,
A ⊗ B M ⊗ N L
we
an then dedu
e the maximal inje
tivity of in as follows. Suppose
A ⊗ B M ⊗ N L
is an inje
tive von Neumann algebra between and . By Lemma 2.5, is a
I B M
(cid:28)nite type von Neumann algebra. Sin
e is singular in (by virtue of being maximal
A ⊗ B M ⊗ N
inje
tive, see [12, Lemma 3.6℄ for example), is singular in by [23, Corollary
I
2.4℄. On the other hand, every masa in a (cid:28)nite type von Neumann algebra is regular by
L = A ⊗ B
[10℄ (see also [19, Lemma 2.3℄). Ergo .
I A B
Returning to the more general situation of type algebras and , the argument of the
previous paragraph breaks down as there need not be a non-trivial normalising unitary of
A ⊗ B L P M (C) ⊕ C1 Q = M (C)
2 3
in . Indeed, take to be , naturally embedded inside ,
P′ ∩ Q ⊆ P P Q P
then but every normalising unitary of in lies in . To
ir
umvent this
P ⊂ Q
di(cid:30)
ulty, we need to use groupoid normalisers. Re
all that if is an in
lusion of von
P′∩Q ⊆ P v ∈ Q
Neumann algebras with , then a partial isometry is a groupoid normaliser
P vPv∗ ⊆ P v∗Pv ⊆ P GN (P)
Q
of if and . We write for the
olle
tion of all groupoid
A M
normalisers. Re
all too that if is a singular masa in a (cid:28)nite von Neumann algebra , then
GN (A) ⊂ A
M
. To use the argument of the pre
eding paragraph in a groupoid normaliser

ontext, we need the groupoid normaliser analogue of Kadison's fa
t that masas in (cid:28)nite
I
type von Neumann algebars are regular.
P ⊂ Q I P′∩Q ⊆
Lemma 2.6. Let be an in
lusion of (cid:28)nite type von Neumann algebras with
P GN (P)′′ = Q
Q
. Then .
Z(Q) ⊆ Z(P) Q I n ∈ N
n
Proof. Sin
e ,we
anassumethat istype forsome . By[13,Theorem
P Q B Q = A ⊗ M
n
3.3℄,
ontains a masa of , say . By [10, Theorem 3.19℄, we
an write ,
A Q B = A⊗D D M
n n n
where is the
entre of and , where are the diagonal matri
es of . An
p⊗e P p ∈ A
i,j
easy
al
ulation shows that is a groupoid normaliser of for every proje
tion
(e )n M
i,j i,j=1 n
and the are the standard matrix units of . These elements evidentally generate
Q
.
In [6℄, the se
ond author introdu
ed the notation of
omplete singularity of an in
lusion
P ⊆ Q GN (P) ⊂ P
Q
, whi
h implies that . Sin
e it was shown in [6, Proposition 3.2℄ that a
maximal inje
tive von Neuamnn subalgebra is
ompletely singular, the next lemma follows
immediately. Alternatively, one
an establish the lemma dire
tly from Connes'
hara
teri-
P v
sation of amenability in [4℄ by showing that if is an inje
tive von Neumann algebra and
P (P ∪{v})′′
is a groupoid normaliser of , then is also inje
tive.
P
Lemma 2.7. Let be a maximal inje
tive von Neumann subalgebra of a von Neumann
Q GN (P) ⊂ P
Q
algebra . Then .
We are now in a position to prove the main result of this se
tion.
M,N A
Theorem 2.8. Let be (cid:28)nite von Neumann algebras with separable preduals. Let be a
I M GN (A) ⊂ A
M
a type von Neumann subalgebra of with and the asymptoti
orthogonality
N B I N
property after tensoring by . Let be a type von Neumann subalgebra of whi
h is
N A ⊗ B
a maximal inje
tive von Neumann subalgebra of . Then is maximal inje
tive in
M ⊗ N
.
6
L A ⊗ B ( L ⊆ M ⊗ N
Proof. Let be an inje
tive von Neumann algebra with . By Lemma
L I A ⊗ B ⊂ L I
2.5, is type and so is an in
lusion of (cid:28)nite type von Neumann algebras
(A ⊗ B)′ ∩L ⊂ A ⊗ B v A ⊗ B
with . By Lemma 2.6, there is a groupoid normaliser of in
L v ∈/ A ⊗ B GN (A) ⊂ A GN (B) ⊂ B
M N
with . On the other hand by hypothesis and by
Lemma 2.7. Corollary 5.6 of [8℄ shows that
GN (A ⊗ B)′′ = GN (A)′′ ⊗ GN (B)′′ = A ⊗ B,
M⊗N M N
A ⊗ B M ⊗ N
and this gives a
ontradi
tion. Hen
e is maximal inje
tive in .
B = N = C
Taking in the previous theorem immediately yields the next
orollary, whi
h
we
ould also establish dire
tly in exa
tly the same way as Corollary 2.3.
A ⊂ M I
Corollary 2.9. Let be an in
lusion of a type von Neumann algebra inside a
GN (A) ⊂ A A
M
(cid:28)nite von Neumann algebra su
h that and has the asymptoti
orthogonality
A M
property. Then is maximal inje
tive in .
Finally we note that the asymptoti
orthogonality property does not pass to tensor
A ⊗ B ⊂ M ⊗ N A B II
1
produ
ts. Indeed, no in
lusion with and masas in type fa
tors
an
u A′ ∩Mω
have the asymptoti
orthogonality property. To see this, take a unitary in with
EAω(u) = 0 v ∈ N EB(v) = 0 (u⊗1)
by Lemma 2.2 and some non-zero with . Then lies in
(A ⊗ B)′∩(M ⊗N)ω E (u⊗1) = 0 1⊗v ∈ M ⊗ N E (1⊗v) = 0
(A⊗B)ω A⊗B
with and with ,
uv = vu
but .
3 The radial masa in a free group fa
tor
K ≥ 2 F K a ...,a
K 1 K
For , let denote the free group on generators. Write for the generators
F F LF
K K K
of . We regard as a subset of the free group fa
tor . The unique faithful normal
τ LF kxk = τ(x∗x)1/2 LF
tra
ial state on K indu
es a pre-Hilbert spa
e norm 2 on K and
LF ℓ2(F )
K K

ompleting in this norm yields the Hilbert spa
e . Therefore, we
an regard
LF ℓ2(F ) ℓ2(F )
K K K
as a subset of as well as an von Neumann algebra a
ting on .
g ∈ F |g| g n ≥ 0 w ∈ LF
K n K
Given , write for the length of . For , let be the sum of all
n F
K
words of length in . The re
urren
e relations
w +(2K −1)w n > 1
n+1 n−1
w w = w w = w +2Kw n = 1
1 n n 1  2 0
(3.1)
w n = 0
1

w  LF A
1 K
show that generates an abelian von Neumann subalgebra of , whi
h we denote by .
LF LF
K K
This is the radial or Lapla
ian subalgebra of whi
h was shown to be masa in by
L2(A) A ℓ2(F ) (w /kw k )∞
Pytlik, [17℄. Write for the
losure of in K and note that n n 2 n=0 forms
L2(A)
an orthonormal basis for .
As promised in the introdu
tion, we give an easy argument whi
h shows that the radial
LF
K
and generator masas are not inner
onjugate in .
u ∈ LF uAu∗ = {a }′′
K 1
Proposition 3.1. There is no unitary with .
7
Φ LF LF
K K
Proof. Let be the automorphism of indu
ed by the automorphism of whi
h
a a Φ(w ) = w
1 2 1 1
inter
hanges the generators and and (cid:28)xes the other generators. Note that
Φ u Φ(u)AΦ(u)∗ =
so that (cid:28)xes the radial masa pointwise. If su
h a unitary existed, then
Φ(uAu∗) = Φ({a }′′) = {a }′′ θ = ad uΦ(u)∗ LF
1 2 K
and then gives an inner automorphism of
θ({a }′′) = {a }′′
2 1
with and this
ontradi
ts [15, Corollary 4.3℄.
LF ℓ2(F )
K 2
To show that the radial masa is singular in , R dules
u de
omposed into a
A−A
dire
t sum of orthogonal -bimodules, [18℄. Before setting out this de
omposition, we
l ≥ 0 W ℓ2(F )
l K
need some preliminary notation. For , let denote the subspa
e of spanned
l F q ℓ2(F ) W
K l K l
by the words of length in . Let be the orthogonal proje
tion from onto .
ξ ∈ M r,s ≥ 0
l
Given a ve
tor , and integers , de(cid:28)ne
q (w ξw )
r+s+l r s
ξ = .
r,s (2K −1)(r+s)/2 (3.2)
(2K −1)(r+s)/2
R dules
u's de(cid:28)nition does not have the s
aling fa
tor on the denominator,
whi
h we introdu
e simplify our subsequent
al
ulations. Indeed, this s
aling fa
tor ensures
kξ k = kξk r,s ≥ 0 ξ 7→ ξ ℓ2(F )⊖L2(A) →
that r,s 2 2, for all and so r,s extends to an isometry K
ℓ2(F )⊖L2(A) ξ = 0 r < 0 s < 0
K r,s
. De(cid:28)ne when either or .
(ξi)∞ ℓ2(F )
i=1 K
R dules
u
onstru
ts a sequen
e of orthonormal ve
tors in with the fol-
lowing properties.
ξi W l(i) ≥ 1
l(i)
1. Ea
h lies in , for some .
2 2
Span(AξiA) ℓ2(F ) ⊕Span(AξiA) =
K
2. The subspa
es are pairwise orthogonal in and
ℓ2(F )⊖L2(A)
K
.
2
i l(i) > 1 Span(AξiA) (ξi )
r,s r,s≥0
3. For those with , the spa
es have orthonormal bases .
i,j > 0 T ξi ξj
i,j r,s r,s
4. For ea
h , the map whi
h sends to extends to a bounded invertible
k2 k2
Span(AξiA) Span(AξjA)
operator from onto . Furthermore, there exists a
onstant
C K i j
0
(whi
h depends on but not or ) su
h that
kT k, T−1 ≤ C , i,j > 0.
i,j i,j 0 (3.3)
ξi (cid:13) (cid:13) (ξi )∗ = ξi
We
an also
hoose the to be self-adjoi(cid:13)nt. It(cid:13)then follows that r,s s,r. Note that the
{ξi : i ≥ 1,r,s ≥ 0} ℓ2(F )⊖L2(A)
r,s K
set does not give an orthonormal basis for , due to the
i l(i) = 1
presen
e of some indi
es with . However, the last two fa
ts above show that this
ℓ2(F )⊖L2(A) 2
K
set is at least a basis for and the -norm it indu
es is equivalent to the norm
k·k
2. We shall subsequently use this in a tensor produ
t setting.
H η ∈ (ℓ2(F )⊗H)⊖(L2(A)⊗H)
K
Proposition 3.2. Let be a Hilbert spa
e and let . Then
(αi ) H η = ξi ⊗αi .
there exist ve
tors r,s r,s≥0,i>0 in su
h that r,s r,s Furthermore
i≥1
X
r≥0,s≥0
C−1kηk ≤ kαi k2 ≤ C kηk ,
0 2 r,s 2 0 2
i≥1
X
r≥0,s≥0
C > 1
0
where is the
onstant appearing in equation (3.3).
8
A
riti
al
omponent of R dules
u's
omputations in [18℄ are re
urren
e relations analo-
ξi w
gous to (3.1) showing how the r,s behave under multipli
ation with 1. We will need these
relations subsequently so we re
all them here. Again it is slightly more
onvenient for our
w˜ = w /(2K−1)1/2 i ≥ 1
1 1
purposes to use a suitable normalisation, so we de(cid:28)ne . Then, for
r,s ≥ 0 σ(i) = ±1
and , there exist values of so that
ξi +ξi r ≥ 1
r+1,s r−1,s
w˜ ξi = ξi r = 0, l(i) ≥ 2
1 r,s  1,s (3.4)
ξi + σ(i) ξi r = 0, l(i) = 1
1,s 2K−1 0,s−1


and ξi +ξi s ≥ 1
r,s+1 r,s−1
ξi w˜ = ξi s = 0, l(i) ≥ 2
r,s 1  r,1 (3.5)
ξi + σ(i) ξi s = 0, l(i) = 1.
r,1 2K−1 r−1,0
ξi ξi 
Re
alling that −1,s and r,−1 arede(cid:28)ned to be zero, it follows that the se
ond
ases of the
relations above are identi
al to the (cid:28)rst
ases.
A′ ∩ (LF )ω
K
4 Lo
ating the support of elements of
In this se
tion we establish the (cid:28)rst half of the te
hni
al estimates required to show that the
LF
K
radial masa has the asymptoti
orthogonality property in . Our obje
tive, realised in
A′∩(LF )ω Aω
K
Lemma 4.3, is to
ontrol the support of elements of whi
h are orthogonal to .
We perform these
al
ulations after tensoring by an arbitrarily (cid:28)nite von Neumann algebra
N
as our obje
tive is the asymptoti
orthogonality property after tensoring by an arbitrary
N i
. For ea
h , de(cid:28)ne
1, l(i) ≥ 2
λ(i) = .
(1− σ(i) , l(i) = 1
2K−1
N x ∈ ℓ2(F )⊗L2(N)
K
Lemma 4.1. Let be a (cid:28)nite von Neumann algebra and suppose that
x ⊥ L2(A ⊗ N) kxk = 1
has and 2 . Write
x = ξi ⊗αi ,
r,s r,s
r,s≥0, i≥1
X
ℓ2(F )⊗L2(N) αi ∈ L2(N) s ≥ 1
K r,s
with
onvergen
e in , for some . Then, for ea
h
1/2 1/2
kαi +λ(i)αi +···+λ(i)αi +λ(i)αi k2 − kαi k2
r−s,0 r−s+2,0 r+s−2,0 r+s,0 2 r,s 2
(cid:12)(cid:16)Xr≥s (cid:17) (cid:16)Xr≥s (cid:17) (cid:12)
(cid:12) i≥1 i≥1 (cid:12)
(cid:12) (cid:12)
≤ 3s−1C kx(w˜ ⊗1)−(w˜ ⊗1)xk ,
0 1 1 2 (4.1)
C
0
where is the
onstant of Proposition 3.2.
9
ε = kx(w˜ ⊗1)−(w˜ ⊗1)xk αi = 0 r < 0 s < 0
Proof. Write 1 1 2 and de(cid:28)ne r,s whenever or .
ξi r < 0 s < 0
Re
alling the
onvention that r,s is zero whenever or , the re
urren
e relations
(3.4) and (3.5) show that
ξi +ξi −ξi −ξi , l(i) ≥ 2, r,s ≥ 0
r+1,s r−1,s r,s+1 r,s−1
ξi +ξi −ξi −ξi , l(i) = 1, r,s ≥ 1
 r+1,s r−1,s r,s+1 r,s−1
[w˜ ,ξi ] = ξi + σ(i) ξi −ξi −ξi , l(i) = 1, r = 0, s > 0
1 r,s  1,s 2K−1 0,s−1 1,s+1 1,s−1

ξi +ξi −ξi − σ(i) ξi , l(i) = 1, r > 0, s > 0
r+1,1 r−1,1 r,1 2K−1 r−1,0
ξi −ξi , l(1) = 1, r = s = 0.
 1,0 0,1





Colle
ting terms together, the relations above yield
[w˜ ⊗1,x ] = ξi ⊗(αi +αi −αi −αi )
1 n r,s r−1,s r+1,s r,s−1 r,s+1
l(i)≥2, r,s≥0
X
+ ξi ⊗(αi +αi −αi −αi )
r,s r−1,s r+1,s r,s−1 r,s+1
l(i)=1, r,s≥1
X
+ ξi ⊗(λ(i)αi +αi −αi )
r,0 r+1,0 r−1,0 r,1
l(i)=1, r≥1
X
+ ξi ⊗(αi −αi −λ(i)αi )
0,s 1,s 0,s−1 0,s+1
l(i)=1, s≥1
X
+ λ(i)ξi ⊗(αi −αi ),
0,0 1,0 0,1
l(i)=1
X
ℓ2(F )⊗L2(N)
K
with
onvergen
e in . Proposition 3.2 then gives the estimate
C2ε2 ≥ kαi +αi −αi −αi k2
0 r−1,s r+1,s r,s−1 r,s+1 2 (4.2)
l(i)≥2, r,s≥0
X
+ kαi +αi −αi −αi k2
r−1,s r+1,s r,s−1 r,s+1 2
l(i)=1, r,s≥1
X
+ kλ(i)αi +αi −αi k2
r+1,0 r−1,0 r,1 2
l(i)=1, r≥1
X
+ kαi −αi −λ(i)αi k2
1,s 0,s−1 0,s+1 2
l(i)=1, s≥1
X
+ kλ(i)(αi −αi )k2,
1,0 0,1 2
l(i)=1
X
C
0
where is the
onstant of Proposition 3.2.
s = 0
Considering the terms with from the (cid:28)rst line of right hand side of (4.2) and those
terms from the third line gives
kαi +αi −αi k2 + kλ(i)αi +αi −αi k2 ≤ C2ε2,
r−1,0 r+1,0 r,1 2 r+1,0 r−1,0 r,1 2 0
r≥1 r≥1
X X
l(i)≥2 l(i)=1
10