Hecke pairs and type III factors
Kroum Tzanev
May 13, 2004
Abstract
We show that for any λ ∈ [0, 1] there is a subgroup Gλ of Q o Q∗+ containing Γ = Z o 1, such that the von
Neumann algebra L (Gλ , Γ) associated to the Hecek pair (Gλ , Γ) is a type IIIλ factor.
1
Introduction
Let G be a discrete group and Γ ⊂ G a subgroup. We say that (G, Γ) is Hecke pair if the commensurability class of
Γ is fixed by the inner conjugations, i.e. if gΓg −1 is commensurable with Γ for any g ∈ G (definition 7). In this case
C (Γ\G/Γ), the space of finite supported functions on the set of double Γ-classes, has a natural algebra structure
and involution. This algebra is acting on l 2 (Γ\G) and the von Neumann algebra L (G, Γ) generated by this action
is the commutant of the right quasi-regular representation of G on l 2 (Γ\G). This quasi-regular representation is
isotypic iff L (G, Γ) is a factor.
In [Bo-C] the authors show that Q o Q∗+ , Z o 1 is a Hecke pair, and that L Q o Q∗+ , Z o 1 is the hyperfinite
factor of type III1 . This was a surprising result because in the case of single discrete group all the factors that we
can obtain are finite type I or II1 . After [Bo-C] the position of one discrete group inside another one is enough to
generate I∞ , II∞ and III1 factors. So the world of Hecke pairs appeared much richer then that of single discrete
groups.
A natural question was : Is this world rich enough to generate not only type III 1 factors, but also factors of any
type IIIλ ? Actually it is the case, as we show it in this paper.
For a sequence of prime numbers P, we note Z P −1 the localization ring of Z at P (page 6). Let G (P) =
∗
Z P −1 o Z P −1 + ⊂ Q o Q∗+ . We show that (G (P) , Z o 1) is a Hecke pair and that for any λ ∈ [0, 1] there is a
sequence (not unique) of prime numbers Pλ such that L (G (Pλ ) , Z o 1) is ITPFI2 factor of type IIIλ .
The idea of the construction is quite easy. First we remark that for any sequence of prime numbers P the von
n
o
−1
−1
Neumann algebra L (G (Pλ ) , Z o 1) is ITPFI2 factor with eigenvalue list p (1 + p) , (1 + p)
. Then we
p∈P
P
1
observe that itPSfrag
is a type
III factor if the size of P is not too small (i.e.
p∈P p = ∞). To obtain a IIIλ factor for
replacements
−1
λ ∈ ]0, 1[ we take a prime numbers that are “close” to all λ powers.
2δ j
2δ j+1
λ−j
λ−(j+1)
More precisely, we take all primes that are contained in
S j
−1
λ−j − δ j , λ−j + δ j for δ j = λ−j (log j) and we
use S and T invariants of Connes to show that the corresponding factor is the Power’s factor R λ . The III1 case is
done in [Bo-C] by taking for P1 the sequence of all primes. Actually we can take P1 = Pλ ∪ Pµ with
To obtain the III0 case we take λ and µ with
taking
log(λ)
log(µ)
to be a Liouville number.
log(λ)
log(µ)
log(λ)
log(µ)
∈
/ Q.
∈
/ Q and such that Pλ ∩ Pµ is big enough. This is done by
1
Remark
After I’ve finished this work, I discovered a paper of Boca and Zaharescu [B-Z] where similar results
for p-adic groups are proved using similar techniques. The main results in this paper can be deduced very easily
using the Schlihting completion described in [Tz2 ], so this paper won’t be published.
Acknowledgement
I want to thank Prof. Alain Connes, Prof. Georges Skandalis and Prof. Philippe Eyssidieux
for extremely useful discussions.
2
Preliminaries
2.1
ITPFI factors
Definitions
Let for k ∈ N, Mk = Mnk (C) be a finite type I factor with faithful normal state ϕk . For any such
ϕk there exist positive diagonal matrix ρk = diag (λk,1 , . . . , λk,nk ) with λk,1 ≥ · · · ≥ λk,nk > 0 such that tr (ρk ) = 1
and ϕk (a) = tr (ρk a) for a ∈ Mk .
ϕ
−it
is the group of modular automorphisms of ϕk .
The one parameter group σ t k (a) = ρit
k aρk
Nl
Nl
We note by liml→∞ k=0 Mk the C*-inductive limit of k=0 Mk = Mn0 ...nl (C) and by ϕ = liml→∞ ⊗lk=0 ϕk
the faithful state determined by
ϕ (a0 ⊗ · · · ⊗ al ⊗ 1 ⊗ 1 ⊗ · · · ) = ϕ0 (a0 ) · · · ϕl (al ) .
Let
N∞
k=0
(Mk , ϕk ) be the von Neumann algebra generated by liml→∞
ciated to ϕ.
This von Neumann algebra
N∞
k=0
Nl
k=0
Mk in the KMS representation asso-
(Mk , ϕk ) is a factor called ITPFI (Infinite Tensor Product of Factors of Type
I). When Mk = M2 (C) (resp. Mk ⊂ MN (C) for some N > 0) we specify this by saying that it is ITPFI2 (resp.
ITPFIb with “b” like “bonded”). Actually M is ITPFIb iff M is ITPFI2 but not all ITPFI factors are ITPFI2
[Gi-S].
The ITPFI construction is due to von Neumann, however the detailed study of this factors was done 25 years
later by H. Araki and F.J. Woods [Ar-W] motivated by the Power’s construction of ITPFI 2 factors Rλ of type IIIλ
for λ ∈ ]0, 1[.
Invariants
The construction of M =
N∞
k=0
(Mk , ϕk ) depends only on the eigenvalue list {λk,l ; 0 ≤ k, 1 ≤ l ≤ nk }
but different eigenvalue lists can produce the same factor. Given an ITPFI factor M , H. Araki and F.J. Woods
consider two invariants r∞ (M ) ∈ R+ and ρ (M ) ∈ R∗+ defined from {λk,l }. This invariants are then generalized
by Krieger for factors constructed from ergodic dynamic systems and then by A. Connes for general factors. The
Connes’ invariants S (M ) ∈ R+ and T (M ) ∈ R are such that for an ITPFI factor M we have the equalities
S (M ) = r∞ (M ) and T (M ) = 2π/ log (ρ (M )).
In this paper we use r∞ (M ) and T (M ) that we describe now.
Given {λk,l ; 0 ≤ k, 1 ≤ l ≤ nk } let Xk = {1, . . . , nk } and µk be the probability measure such that µk (l) = λk,l .
Q∞
Q∞
Let X = k=1 Xk be the product probability space with the probability measure µ = k=1 µk . And for any index
Q
Q
subset I ⊂ N let X (I) = k∈I Xk with the probability measure µ = k∈I µk .
The asymptotic ratio set r∞ (M ) is the set of all x ∈ R+ such that there exist mutually disjoint finite index
2
sets Iα ⊂ N and for any Iα two disjoint subsets Kα1 ,Kα2 ⊂ X (Iα ) with a set bijection φα : Kα1 → Kα2 such that :
X
µ Kα1
α
µ (φα (a)) lim max1 x −
n→∞ a∈Kα
µ (a) =
∞
=
0
∗
H. Araki and F.J. Woods showed that r∞ (M ) is an invariant of M and that r∞
(M ) = r∞ (M ) \ {0} is a
subgroup of R∗+ or the empty set. When r∞ (M ) 6= {0, 1} this invariant determine completely M as follows : if
r∞ (M ) = ∅ then M ' Mn (C) with n2 = dim (M ) < ∞ ; if r∞ (M ) = 0 then M ' B l2 , the I∞ factor ; if
r∞ (M ) = 1 then M = R, the II1 hyperfinite factor ; if r∞ (M ) = {λn ; n ∈ Z} ∪ {0} for λ ∈ ]0, 1[ then M = Rλ ,
the Power’s factor ; if r∞ (M ) = R+ then M = R∞ , the Araki-Woods III1 factor. When r∞ (M ) = {0, 1} there
are two cases : either M = R0,1 = R ⊗ I∞ is the II∞ hyperfinite factor or M is type III0 .
Actually there are no other possibilities for r∞ (M ). In particular if λ, µ ∈ r∞ (M ) with log λ/ log µ ∈
/ Q then
r∞ (M ) = R+ and M = R∞ .
The definition of T (M ) is based on the Tomita-Takasaki theory but using Connes’ noncommutative version of
the Radon-Nikodým theorem it can be simplified and in the case of ITPFI factors we have (corollary 1.3.9 in [Co]) :
!
∞
X
X
1+iτ λk,l .
1−
τ ∈ T (M ) ⇔
l
k=1
c∗ is the orthogonal
The invariant T (M ) is a subgroup of R and if M is not type I or III0 then T (M ) ⊂ R =R
+
∗
group of r∞
(M ) ⊂ R∗+ .
Some lemmas
Lemma 1 If λ ∈ r∞ (M ) and 2π/ log λ ∈ T (M ) for λ ∈ ]0, 1[ then M = Rλ .
⊥
∗
(M ) ⊂ (2πZ/ log λ) = {λn ; n ∈ Z} and as λ ∈ r∞ (M ) ⇒ {λn ; n ∈ Z} ⊂
Proof. if 2π/ log λ ∈ T (M ) then r∞
r∞ (M ) we obtain that r∞ (M ) = {λn ; n ∈ Z} ∪ {0}.
In this paper we consider only special kind of ITPFI2 factors. A general ITPFI2 factor M has the form
⊗nj
N
−1
with nj ∈ N∗ ∪ {∞}, ϕj (a) = tr ρj a where ρj = (1 + λj ) diag (1, λj ) and λj ∈ R∗+ .
M = j M2 (C) , ϕj
It is completely determined by the sequence of pairs (nj , λj ) and we write M = M (nj , λj ). If nj ≡ 1 we use the
simpler notation M = M (λj ). Of course M (nj , λj ) = M nj , λ−1
.
j
The factors that appear here are such that nj ∈ N∗ and λj → 0.
Lemma 2 If M = M (nj , λj ) with nj ∈ N∗ and λj → 0 then :
• M is type I ⇔
P
• M is type III ⇔
nj λj < ∞.
P
nj λj = ∞.
Proof. For a general ITPFI factor M with eigenvalue list {λk,l } we have the following criteria (lemma 2.14 of
P
[Ar-W]) : M is type I iff
|1 − λk,1 | < ∞. In our case as λj → 0 we have (for j big enough) that |1 − λk,1 | =
P
P
λk,2 = λj / (1 + λj ) ∼ λj where n1 + · · · + nj ≤ k < n1 + · · · + nj+1 . So
|1 − λk,1 | < ∞ ⇔
nj λj < ∞.
By the same result (lemma 2.14 of [Ar-W]) : If it exists δ > 0 such that λk,1 > δ for all k (and this is true in our
n
o
n
o
P
P
2
2
case) then M is type III iff
λk,l inf |λk,1 /λk,l − 1| , 1 = ∞. In our case as
λk,l inf |λk,1 /λk,l − 1| , 1 =
n
n
2 o
2 o
P
−1
−1
∼ 1 we obtain that M is type III iff
and (1 + λj ) inf λ−1
nj λj (1 + λj ) inf λ−1
j − 1 ,1
j − 1 ,1
P
nj λj = ∞.
3
Some informations about the factors that appear in this paper can be summarized in the following table :
type of M (nj , λj ) with λj → 0
type III ⇔
P
subtype
r∞
T
III0
subgroup of R
IIIλ
{0, 1}
{0} ∪ {λ }n∈Z
2πZ
log λ
III1
R
1
n j λj = ∞
n
Let λ ∈ ]0, 1[ and {lj },{nj } be two eventually increasing sequences of positive integers converging to infinity.
−1
Lemma 3 2π (log λ) ∈ T M nj , λlj .
Proof. This is a direct consequence of corollary 1.3.9 de [Co].
P
Lemma 4 The factor M nj , λlj is type III iff
nj λlj = ∞.
Proof. Direct consequence of Lemma 2 for λlj → 0.
Lemma 5 If
P
nj λj = ∞ then M nj , λj is the hyperfinite factor of type IIIλ .
Proof. 2π (log λ)
−1
∈ T M nj , λ j
as shown in Lemma 3. Following Lemma 1 it is enough to show that
λ ∈ ρ∞ (M ). To obtain this result we use the definition of r∞ (M ).
−1
For j ∈ N, we put Xj = {0, 1} the two point space with the probability measure µj given by µj (0) = 1 + λj
−1
P
and µj (1) = λj 1 + λj
. For any k ∈ N we note r (k) the smallest integer such that k < j≤r(k) nj . In other
words, r : N N is the increasing function such that #r −1 (j) = nj for all j ∈ N. Given a subset J of N we set
Q
Q
X (J) = j∈J Xr(j) with the product measure µJ = j∈J µr(j) . To show that λ ∈ r∞ (M ), it’s enough to show
the existence of a sequence {Jα } of finite disjoint subsets of N, and subsets Kα0 and Kα1 of X (Jα ) with bijections
'
φ : Kα0 → Kα1 , such that :
X
and
µJα Kα0 = ∞
µJα (φ (x)) lim max0 λ −
= 0.
α→∞ x∈Kα
µJα (x) Let j0 be such that {nj }j≥j0 is increasing. Let {Iα } be the sequence of two element set k, k + nr(k) for r (k) even
2
and r (k) ≥ j0 . These sets are disjoints because r k + nr(k) = r (k) + 1 is odd. As X k, k + nr(k) = {0, 1}
with the product measure µr(k) × µr(k)+1 , let Kα1 and Kα2 be the one point sets : Kα1 = {(1, 0)} and Kα2 = {(0, 1)}
, with the evident isomorphism φα : Kα1 → Kα2 . So we have :
X
µ Kα1
=
X
k, j0 ≤r(k)=2j
=
X
µ2j × µ2j+1 (1, 0) =
n2j λ2j 1 + λ2j
2j≥j0
The last equality hold because
P
j≥j0
−1
X
2j≥j0
1 + λ2j+1
n2j µ2j (1) × µ2j+1 (0) =
−1
>
1 X
n2j λ2j = ∞.
4
2j≥j0
nj λj = ∞ and n2j λ2j > n2j λ2j + n2j−1 λ2j−1 λ/2.
We can conclude by observing that
µ (φα (a)) max1 λ −
µ (a) a∈Kα
=
=
µ Kα1 λ −
=
µ (Kα0 ) µ (0) µ2j+1 (1) = 0.
λ − 2j
µ2j (1) µ2j+1 (0) 4
i
S h
Lemma 6 Let {δ j } be a sequence of positive real numbers and {π k } be a real sequence in j λ−lj − δ j , λ−lj + δ j
i
h
P
such that nj = # λ−lj − δ j , λ−lj + δ j ∩ {π k }. If δ j = o λ−lj and
nj δ 2j λ3lj < ∞, then M (π k ) ' M nj , λlj .
P
Proof. As in the preceding Lemma, for k ∈ N we note r (k) the smallest integer such that k < j≤r(k) nj . As the
o
n
is obtained by repeating nj times λlj , we have M nj , λlj = M λlr(k) . Using lemma 2.13
sequence λlr(k)
k∈N
of [Ar-W], to show the isomorphism M (π k ) ' M nj , λlj it is enough to show that
X
k≥k0
r
πk
−
1 + πk
s
1
1 + λlr(k)
!2

r

+
1
−
1 + πk
s
2
λlr(k) 
< ∞,
1 + λlr(k)
where k0 is such that for j > r (k0 ) one has λ−lj + δ j < λ−lj+1 − δ j+1 . Using the mean value theorem for
q
q
x
1
f (x) = 1+x
and g (x) = 1+x
, as π k − λ−lr(k) < δ r(k) and δ r(k) = o λ−lr(k) , we obtain the existence of a
3
constant C such that f (π k ) − f λ−lr(k) < Cδ r(k) λ2lr(k) and g (π k ) − g λ−lr(k) < Cδ r(k) λ 2 lr(k) . Finally
X
k≥k0
2.2
r
πk
−
1 + πk
s
1
1 + λlr(k)
!2

+
r
1
−
1 + πk
s
2
X
λlr(k) 
2
<
2C
nj δ 2j λ3lj < ∞.
1 + λlr(k)
j≥r(k )
0
Hecke pairs
General framework
Let G be a group with two subgroups Γ1 and Γ2 . We say that Γ1 and Γ2 are commen-
surable (and we denote this by Γ1 ∼ Γ2 ) if the indices [Γ1 : Γ1 ∩ Γ2 ] and [Γ2 : Γ1 ∩ Γ2 ] are finite.
Definition 7 A subgroup Γ of a group G is almost normal (and we write Γ E G) if Γ ∼ gΓg −1 for any g ∈ G.
In other words Γ is almost normal in G if any double Γ-class contains finitely many right Γ-classes.
If Γ E G we say that (G, Γ) is a Hecke pair.
Hecke algebras were introduced in noncommutative geometry by J.-B. Bost and A. Connes in their work on
spontaneous symmetry breaking of a dynamic system associated to the Hecke pair Q o Q∗+ , Z o 1 [Bo-C]. In
that paper they show the relationship between this dynamic system and the distribution of prime numbers.
All the constructions in this part are based on [Bo-C] (see also [Tz2 ]).
For a discrete set X we denote by C (X) the space of complex valued functions with finite support on X. So
for Γ E G the space C (Γ\G/Γ) is naturally identified with the subspace of Γ-invariant functions in C (G/Γ) (or in
C (Γ\G)). In the space C (Γ\G/Γ) there are a natural product and involution given by:
(f1 ∗ f2 ) (g) =
X
g1 ∈hΓ\Gi
f1 gg1−1 f2 (g1 ) =
X
g2 ∈hG/Γi
f1 (g2 ) f2 g2−1 g
for g ∈ G,
f ∗ (g) = f (g −1 ),
where the functions of C (Γ\G/Γ) are identified with Γ-biinvariant functions on G and g 1 ∈ hΓ\Gi (resp. g2 ∈
hG/Γi) means that g1 (resp. g2 ) runs over a set of representatives of left (resp. right) Γ-sets. In this manner one
obtains an involutive algebra C (G, Γ) called the Hecke algebra of (G, Γ). When Γ C G one has the isomorphism
C (G, Γ) ' C (G/Γ), where C (G/Γ) is the group algebra of G/Γ.
There are a natural representation λ of C (G, Γ) in B l2 (Γ\G) given by the formula:
[λ (f ) ξ] (g) =
X
g1 ∈hΓ\Gi
f gg1−1 ξ (g1 )
for f ∈ C (G, Γ) , ξ ∈ C (Γ\G) and g ∈ Γ\G .
5
00
The von Neumann algebra of (G, Γ) is the bi-commutant λ (C (G, Γ)) and we denote it by L (G, Γ).
As far as I know, there is no general criteria (as ICC for discrete groups) that allows one to see for which (G, Γ)
the von Neumann algebra L (G, Γ) is a factor.
Before considering particular Hecke pairs we need to remember some general results.
For a Hecke pair (G, Γ) we note ϕΓ the vector state corresponding to the characteristic function of the Γ-classe
Qn
Q
Q∞
of identity, ξ Γ ∈ l2 (Γ\G). Given a sequence of pairs (Gj ⊃ Γj ) we note res Gj = lim j=0 Gj × j=n+1 Γj .
Proposition 8 If (Gj , Γj ) is a sequence of Hecke pairs then (
N
L (Gj , Γj ) , ϕΓj .
Proof. (
Q
res
Gj ,
Q
Q
res
Gj ,
Q
→
Γj ) is a Hecke pair and L (
Q
res
Gj ,
Q
Γj ) =
Q∞
Qn
Γj ) is a Hecke pair because for (g, γ) ∈ j=0 Gj × j=n+1 Γj we have


n
∞
∞
∞
Y
Y
Y
Y
−1
−1


Γj g ×
Γj (g, γ) = g
Γj .
(g, γ)
Γj ∼
j=0
j=0
j=0
j=n+1
Q
Nn 2
Qn
n
It is obvious that under the isomorphism l 2
j=1 l (Γj \Gj ) the vector ξ
j=0 Γj \
j=0 Gj '
Nn
spond to j=1 ξ Γj . And so we have the isomorphism
 

 

n
n
n
n
O
O
Y
Y
C ∗ 
C ∗ (Gj , Γj ) ,
ϕ Γj  .
Gj ,
Γj  , ϕ nj=0 Γj  ' 
j=0
j=1
j=0
We can conclude by observing that
 

∞
∞
n
Y
Y
Y
C ∗ 
Γj  , ϕ
Γj ,
Gj ×
j=0
j=n+1
∞
j=0
Γj
j=0


n
j=0
Γj
corre-
j=1

 ' C ∗ 
n
Y
j=0
Gj ,
n
Y
j=0

Γj  , ϕ
n
j=0
Γj

.
Given a Hecke pair (G, Γ) there is a unique locally compact group G containing a compact open subgroup Γ
such that L (G, Γ) ' L G, Γ ' pΓ L G pΓ where pΓ is the projection corresponding to the characteristic function
χΓ ∈ Cc G of Γ (see [Tz2 ],[Tz1 ]). These are spacial isomorphisms based on l 2 (Γ\G) ' l2 Γ\G ' pL2 G so we
have that (L (G, Γ) , ϕΓ ) ' pΓ L G pΓ , ϕΓ . The pair G, Γ is called the Schlihting completion of (G, Γ).
Hecke pairs associated to prime numbers Let us consider now a particular examples of Hecke pairs. Let
−1 ∗
αk
α1
P be a sequence of prime numbers. Let Z P −1 = m
=
n ∈ Q ; n = p1 . . . pk , pj ∈ P, αj ∈ N , Z P
+
−1 α1
−1 ∗
αk
∗
∗
p1 . . . pk ∈ Q+ ; pj ∈ P, αj ∈ Z and G (P) = Z P
oZ P
⊂ Q o Q+ .
+
Theorem 9 Let P = {pj }j≥1 be a sequence of prime numbers. Then (G (P) , Z o 1) is a Hecke pair.
P −1
If
pj = ∞ then L (G (P) , Z o 1) is type III and
L (G (P) , Z o 1) ' M (pj ) .
Proof. Let Qp be the field of p-adic rationals and Zp be the ring of p-adic integers. Then Zp is compact
−1 ∗
Q
Q
Q
.
open subgroup of Qp o pZ and so
Zpj is compact-open subgroup of res Qpj o pZj =
res Qpj o Z P
+
Q
Q
Z
Actually as G (P) can be embedded as dense subgroup of res Qpj o pj [Se] in such a way that G (P) ∩ Zpj =
Q
Q
Z
Zpj is the Schlihting completion of (G (P) , Z o 1) (proposition 4.1, [Tz2 ]). So
Zo1, the pair
res Qpj o pj ,
Q
Q
Z
L (G (P) , Z o 1) ' L
Zpj and using Proposition 8 we obtain
res Qpj o pj ,
O
L (G (P) , Z o 1) '
L Qpj o pZj , Zpj , ϕZp .
j
6
−1 −2
Let ρp ∈ B l2 (N) be the diagonal operator ρp = p−1
, p , . . . and ψ p the corresponding faithful
p diag 1, p
normal state given by ψ p (a) = tr ρp a . Bost and Connes show in [Bo-C] that L Qpj o pZj , Zpj , ϕZp
'
j
2
B l (N) , ψ p .
. As in Lemma 2 we use lemma
So L (G (P) , Z o 1) is an ITPFI factor with eigenvalue list (p − 1) p−l
k
k,l≥1
P
P −1
2.14 of [Ar-W] to found that L (G (P) , Z o 1) is type III iff
|1 − pk / (pk − 1)| = ∞ ⇔
pk = ∞. Let π 2 be
Q
the projection on F2 = (xj ) ∈ l2 (N) ; j ≥ 2 ⇒ xj = 0 . Then as 1 − ψ p (π 2 ) = p−2 we have that ψ pj (π 2 ) > 0
N
N
B l2 (N) , ψ pj '
and so π = ⊗π 2 ∈
B l2 (N) , ψ pj is a nonzero projection in a type III factor. So
hN i
N
−1
π
B l2 (N) , ψ pj π '
B (F2 ) , ψ pj (π) ψ pj ' M (pj ).
2.3
Distribution of prime numbers
Given a real number x, we denote by π (x) the number of primes not exceeding x. It was shown almost simultaneously by J. Hadamard and C. J. de La Vallé-Poussin (1896) that π (x) ∼ logx x . In 1908 E. Landau establish the
R
√
R x du
1−δ
formula π (x) = Li (x) + O xe−c log x where Li (x) = limδ→0+ 0 + 1+δ log
u and c is the Euler constant.
Let π (a, b) be the number of primes in the interval [a, b]. We will need this result in the following form.
Theorem 10 (E. Landau, 1908) If {xn } and {δ n } are two sequences of real numbers such that xn → ∞ and
δ n = o (xn ) then
π (xn − δ n , xn + δ n ) =
Z
xn +δ n
xn −δ n
√
du
+ O xn e−c log xn
log u
For constructing a type III0 factor we need the existence of particular Liouville number.
n
o
−1
Lemma 11 For ξ ∈ R let Lξ = l ∈ N , l > 1, d (lξ, Z) < (log l)
. There exist ξ ∈ R\Q such that
X
l∈Lξ
1
= ∞.
l log l
Proof. We will construct by recurrence two increasing sequences (dk )k∈N and (sk )k∈N such that for the Liouville
P∞
number ξ = k=1 10−sk we will have :
X
1
>k
l log l
l∈Lξ ∩[2,dk ]
Let d0 = 2 and s0 = 1. Let dk+1 > 10sk and such that
Pdk+1
1
l=10sk l log l
> 1. We choose rk ∈ N such that 10rk > dk+1 .
Let sk+1 = sk + 2rk .
P∞
−1
Let ξ = k=1 10−sk . For l ∈ [10sk , dk+1 ] ⊂ [10sk , 10rk ] we have d (lξ, Z) < 10−rk < l−1 < (log l) . So using
P
Pk
P dm
1
1
that Lξ ∩ [10sk , dk ] = N ∩ [10sk , dk ] we obtain l∈Lξ ∩[2,dk ] l log
m=1
l=10sm−1 l log l > k.
l >
Remark 12 The set of ξ ∈ R such that
R 1 P
P∞
2
1
n=2 n(log n)2 < ∞.
l∈Lξ l log l dξ =
0
3
P
1
l∈Lξ l log l
= ∞ has measure zero. This is clear from the equality
Main result
Now we can proof the main result of this paper :
Theorem 13 For all λ ∈ [0, 1] there is an infinite set Pλ of primes such that the von Neumann algebra of the
Hecke pair (G (Pλ ) , Z o 1) is a factor of type IIIλ .
7
Proof. It follows from theorem 9 that it’s enough to construct for any λ ∈ [0, 1] a sequence of primes P λ = {pj }
such that the von Neumann algebra M (pj ) is a factor of type IIIλ .
S −1
The case λ ∈ ]0, 1[. Let δ j = λ−j (log j)
and Pλ the set of primes contained in j λ−j − δ j , λ−j + δ j . One
puts nj = # λ−j − δ j , λ−j + δ j ∩ Pλ , the number of primes contained in λ−j − δ j , λ−j + δ j . Using the prime
numbers theorem we have
nj =
Z
λ−j +δ j
λ−j −δ j
√
du
+ O λ−j e−c j|log λ|
log u
√
δ
δ
so, as δ j = o λ−j and λ−j e−c j|log λ| = o jj , we obtain nj ∼ K jj with K = −2/ log λ. As a consequence
P
−3
nj δ 2j λ3j ∼ Kj −1 (log j) and so
nj δ 2j λ3j < ∞. Applying Lemma 6 we found that M (pj ) = M nj , λj . Now,
P
K
as nj λj ∼ j log
nj λj = ∞, we can apply Lemma 5 to obtain that M (pj ) = M nj , λj is the hyperfinite
j ⇒
factor of type IIIλ .
The case λ = 0. Let ξ ∈ R\Q be a number like in Lemma 11 and λ, µ ∈ ]0, 1[ such that log λ/ log µ = ξ. Let
−1
Lξ = {lj } and {mj } such that |lj ξ − mj | < (log lj ) . So we have for any j ∈ N, log λ−lj − log µ−mj <
−1
−1
|log v| (log lj ) ⇒ λ−lj − µ−mj < Cλ−lj (log lj ) for some constant C.
i
S h
−1
Let δ j = λ−lj (log lj )
and P0 the set of primes contained in j λ−lj − δ lj , λ−lj + δ lj . One puts nj =
h
i
# λ−lj − δ j , λ−lj + δ j ∩ P0 . Using the prime numbers theorem we have
nj =
Z
λ−lj +δ j
λ−lj −δ j
√
du
+ O λ−lj e−c lj |log λ|
log u
√
so, as δ j = o λ−lj and λ−lj e−c lj |log λ| = o δ j lj−1 , we obtain nj ∼ Kδ j lj−1 with K = −2/ log λ. As a
P
−3
consequence nj δ 2j λ3lj ∼ Klj−1 (log lj )
and so
nj δ 2j λ3lj < ∞. Applying Lemma 6 we found that M (pj ) =
P
lj
lj
K
M nj , λlj . Now, as nj λlj ∼ lj log
⇒
n
λ
=
∞,
we
can
apply
Lemma
4
to
obtain
that
M
(p
)
=
M
n
,
λ
j
j
j
lj
is a type III factor. Let Q be the sequence obtained by repeating nj times the number mj for any j ∈ N. Using
that λ−lj − µ−mj < Cδ j and applying once more the Lemma 6 for the sequence Q, we found that M nj , λlj =
M (nj , µmj ). Finally from Lemma 3 it follows that 2π (log λ)
consequence, using that
log(λ)
log(µ)
−1
∈ T (M (pj )) and 2π (log µ)
∈
/ Q, we obtain that M (pj ) is a type III0 factor.
−1
∈ T (M (pj )). As a
The case λ = 1. This case is done in [Bo-C], using [Bl], by showing that the von Neumann algebra of Q o Q∗+ , Z o 1
is a type III1 factor. So by taking P1 the set of all primes we obtain this last case. This result is also an easy
consequence of the case λ ∈ ]0, 1[ : as P1 ⊃ Pλ for any λ ∈ ]0, 1[ we have that r∞ (M (P1 )) ⊃ r∞ (M (Pλ )) = {0, λn }
⇒ r∞ (M (P1 )) = R.
References
[Ar-W] H. Araki and J. Woods, A classification of factors, Publ. Res. Inst. Math. Sci. Kyoto Univ., Ser.A 4 (1968),
51-130
[Bl]
B. Blackadar, The Regular Representation of Restricted Direct Product Groups, J. Funct. Anal. 25 (1977),
267-274
[B-Z] F. Boca and A. Zaharescu, Factors of type III and the distribution of prime numbers, Proc. London Math.
Soc. (3) 80 (2000), no. 1, 145-178.
[Bo-C] J.-B. Bost and A. Connes, Hecke algebras, type III factors and phase transitions with spontaneous symmetry
breaking in number theory, IHES, 1995
8
[Co] A. Connes, Une classification des facteurs de type III, Ann. Sci. Ecole. Norm. Sup. (4) 6 (1973), 133-252
[Gi-S] T. Giordano and G. Skandalis, Krieger Factors Isomorphic to Their Tensor Square and Pure Point Spectrum
Flows, J. Funct. Anal. 64 No.2 (1985), 209-226
[Se]
J. P. Serre, Cours d’arithmétique, PUF, 1970
[Tz1 ] K. Tzanev, C*-algèbres de Hecke et K-théorie, PhD thesis, Université Paris 7 - Jussieu, 2000
[Tz2 ] K. Tzanev, Hecke C*-algebras and amenability,J. Operator Theory 50 (2003), no. 1, 169–178
Kroum Tzanev, Laboratoire Emile Picard, Université Paul Sabatier, 118 route de Narbonne,
F-31062 Toulouse Cedex 4, France. Email : [email protected]
9