KK-theory and Spectral Flow in von Neumann Algebras

0.37 MB·English

by  Jens Kaad

KK
-THEORY AND SPECTRAL FLOW IN VON NEUMANN ALGEBRAS
7
0
0 J. KAAD, R. NEST, A. RENNIE
2

n Institute for Mathemati
al S
ien
es, University of Copenhagen
a
Universitetsparken 5, DK-2100 Copenhagen, Denmark
J

1
1 Abstra
t
J
We present a de(cid:28)nitionof spe
tral (cid:29)ow forany norm
losed ideal inany von Neumann algebra
] N N N/J
A
. Given a path of selfadjoint operators in whi
h are invertible in , the spe
tral (cid:29)ow
K (J)
O 0
produ
es a
lass in .
. ( ,H, ) (N,τ) [ ]
h Given a semi(cid:28)nite spe
tral triple A D relative to , we
onstru
t a
lass D ∈
t KK1(A, (N)) u u∗ u
a K . For a unitary ∈ A, the von Neumann spe
tral (cid:29)ow between D and D is
m [u]ˆA[ ]
equal to the Kasparov produ
t ⊗ D , and is simply related to the numeri
al spe
tral (cid:29)ow,
C
[ ∗
and a re(cid:28)ned -spe
tral (cid:29)ow.

1
v
Contents
6
2
3 1. Introdu
tion 2
1
0 2. K-theory-valued von Neumann Index Theory 2
7
0 3. Von Neumann Spe
tral Flow 6
/
h 3.1. Basi
De(cid:28)nitions and Properties 6
t
a
3.2. Von Neumann Spe
tral Triples and Spe
tral Flow 11
m
: 4. Kasparov Modules from Spe
tral Triples 13
v
i 4.1. Constru
tion of a Kasparov Module 13
X
K (A)
1
r 4.2. The Pairing with and Spe
tral Flow 14
a
C
∗
5. -Spe
tral Flow 16
5.1. Basi
De(cid:28)nitions 17
C
∗
5.2. The Relationship Between -Spe
tral Flow and von Neumann Spe
tral Flow 18
6. Numeri
al Spe
tral Flow 19
7. Appendix on Kasparov Produ
ts 23
K KK1
1
7.1. Produ
t between and 25
Referen
es 29
email: jenskaadhotmail.
om,rnestmath.ku.dk,renniemath.ku.dk.
1
2 J. KAAD,R. NEST, A. RENNIE
1. Introdu
tion
The theory of analyti
spe
tral (cid:29)ow has re
eived a great deal of attention in re
ent years, with
signi(cid:28)
ant progress being made by many authors, [2, 4, 8, 9, 10, 22, 23, 24, 27℄. The arti
le [27℄

ontains a mu
h more detailed review of other aspe
ts of spe
tral (cid:29)ow.
K
Here we take a slightlydi(cid:27)erent ta
k, repla
ingnumeri
almeasures ofspe
tral(cid:29)ow by -theory
valued measures, as in [18, 27℄. The advantages of this approa
h are the great generality in
whi
h it
an be de(cid:28)ned, and its
ompatibility with the various numeri
al notions.
This
ompatabilityyields
onstraintson thepossible valuesof spe
tral (cid:29)ow, whi
h, forexample,
in the semi(cid:28)nite setting of [22, 23℄, is a priori any real number. Our des
ription of spe
tral
K
(cid:29)ow allows one to fa
tor through a -theory group, and so
onstrain the possible values of the
K
spe
tral (cid:29)ow. The more re(cid:28)ned we
an be about the target -theory group, the more re(cid:28)ned
our information.
J
We de(cid:28)ne von Neumann spe
tral (cid:29)ow for any norm
losed ideal in any von Neumann algebra
N N N/J
. Given a path of selfadjoint operators in whi
h are invertible in , we obtain a
lass in
K (J) K (J)
0 0
. In order tobe able towork insu
h a general
ontext, we need todevelop a -valued
N,J
index theory for any su
h pair . Su
h an index theory is developed in Se
tion 2, and then
in Se
tion 3 we de(cid:28)ne and study the von Neumann spe
tral (cid:29)ow. We then follow the approa
h
of [22, 23℄, de(cid:28)ning spe
tral (cid:29)ow in terms of relative indi
es of proje
tions.
A
losely related idea whi
h we introdu
e is a von Neumann spe
tral triple, modelled on the
N J
de(cid:28)nition of semi(cid:28)nite spe
tral triples, but valid for any von Neumann algebra and ideal .
We show that su
h a triple de(cid:28)nes a Kasparov
lass, and relate the spe
tral (cid:29)ow to Kasparov
produ
ts.
KK
In parti
ular, every semi(cid:28)nite spe
tral triple represents a -
lass, just as ordinary spe
tral
K
triples represent -homology
lasses. This extends the observed relation in [18, 19℄.
K K ( )
0
In Se
tion 5 we dis
uss the
onsequen
es of re(cid:28)ning our target -theory group to B , where
J σ
B ⊂ is a -unital subalgebra. We show that this
an always be done for a von Neumann
C
∗
spe
tral triple, and so we
an de(cid:28)ne a -spe
tral (cid:29)ow. We relate this spe
tral (cid:29)ow to our
previously de(cid:28)ned von Neumann spe
tral (cid:29)ow.
C
∗
Se
tion 6 relates both von Neumann and spe
tral (cid:29)ow for a semi(cid:28)nite spe
tral triple to the
numeri
al spe
tral (cid:29)ow obtained from a tra
e.
KK
The Appendix summarisessomeresultsfrom -theorythatwe require, andproves anexpli
it
KK
formula for
ertain odd pairings in -theory, whi
h plays a key role throughout the paper.
A
knowledgements It is a pleasure to thank Alan Carey and John Phillips for many helpful

onversations about spe
tral (cid:29)ow.
2. K-theory-valued von Neumann Index Theory
N H
Throughout this se
tion, we let be a von-Neumann algebra a
ting on a Hilbert spa
e and
J N π : N N/J
let be a norm
losed ideal in . Let → be the quotient map.
KK
-THEORY AND SPECTRAL FLOW IN VON NEUMANN ALGEBRAS 3
S : H H
In all the following, we will distinguish between the kernel of an operator →
alled
ker(S) N(S) (H)
and the proje
tion onto the kernel
alled ∈ L . Likewise we have the image of
S (S) (S) R(S) (H) S
, Im and the proje
tion onto the norm
losure of Im , denoted ∈ L . If is in
N N(S) R(S) N
then and are in also.
p,q (H) (p) (q) p q (H)
Foranytwoproje
tions ∈ L wedenotetheproje
tionontoIm ∩Im by ∩ ∈ L .
p,q N S N Sp = pS Sq = qS Sp q = p qS
′
If ∈ and ∈ , then and thus ∩ ∩ . It follows easily that
p q N = N
′′
∩ ∈ .
S N
We re
all some fa
ts about the polar de
omposition of an operator. Let ∈ . The partial
u N S S S
isometry ∈ from the polar de
omposition of is
alled the phase of . The phase of
has the following properties
u S = S S = u S
∗ ∗ ∗
| | | |
uu = R(u) = R(S) u u = R(u ) = R(S )
∗ ∗ ∗ ∗
1 uu = N(u ) = N(S ) 1 u u = N(u) = N(S)
∗ ∗ ∗ ∗
− −
K C
∗
See [12, Appendi
e III℄ for more details. Sin
e -theory is well-de(cid:28)ned for non-separable -
K
algebras, we
an ask what the generalised index map in -theory gives us for an invertible in
N/J
the `Calkin algebra' .
[π(S)] K (N/J) K (N/J) π(S)
1 1
LemmSa 2.M1. (LNet) ∈ n N be a
lass in represented by the unitary ,
n
where ∈ for some ∈ . Then
∂[π(S)] = [N(S)] [N(S )] K (J),
∗ 0
− ∈
∂ : K (N/J) K (J) K
1 0
where → is the boundary map in -theory. See for instan
e [3, De(cid:28)nition
8.3.1
℄.
M (N) = M (C) N
n n
Proof. The algebra ⊗ is a von-Neumann algebraa
ting on the Hilbert spa
e
n H S M (N) u M (N) S u
⊕i=1 so
an be polar de
omposed in n . Let ∈ n be the phase of . Now, is
π(S)
a lift of sin
e
π(S) = π(u S ) = π(u)π(S S)1/2 = π(u)
∗
| |
And we
on
lude from the de(cid:28)nition of the boundary map, [3, 14, 26℄, that
∂[π(S)] = [1 u u] [1 uu ] = [N(S)] [N(S )]
∗ ∗ ∗
− − − −
(cid:3)
as
laimed.
S
The generi
situation where the index of an operator is relevant for appli
ations is when
S : H H
1 2
→ . Even to de(cid:28)ne `odd' index pairings one requires su
h operators. Thus one must
N qNp p,q N

onsider operators not in a von Neumann algebra , but in a skew-
orner where ∈
are proje
tions. This situationwas (cid:28)rst
onsidered in [11℄ for semi(cid:28)nite von Neumann algebras.
The following de(cid:28)nition generalises the semi(cid:28)nite notion of Fredholm.
p,q N S qNp (q p)
De(cid:28)nition 2.2. Let be proje
tions in . Then ∈ is a - -Fredholm operator if
T,R pNq
there exists a ∈ su
h that
π(TS) = π(p) π(SR) = π(q)
and
π(T) = π(TSR) = π(R) R = T T
Sin
e , we
an
hoose . The operator is
alled a parametrix
S
for .
4 J. KAAD,R. NEST, A. RENNIE
S qNp
Remark 2.3. Suppose we have an operator ∈ . Then
N(S) p = N(S)p N(S ) q = N(S )q
∗ ∗
∩ and ∩
This follows immediately sin
e
(1 p)N(S) = (1 p) = N(S)(1 p) pN(S) = N(S)p
− − − ⇒
N(S) p = N(S)p N(S ) q
∗
so ∩ . Similar
omments apply to the proje
tions and .
S qNp u N S u qNp
Lemma 2.4. Let ∈ and let ∈ be the phase of . Then ∈ and we have the
identities
p u u = N(S) (1 p) = N(S) p
∗
− − − ∩
q uu = N(S ) (1 q) = N(S ) q.
∗ ∗ ∗
− − − ∩
S qNp (q p) π(u u) = π(p) π(uu ) = π(q)
∗ ∗
Furthermore if ∈ is a - -Fredholm operator then and .
u (q p) N(S) p,N(S ) q J
∗
In parti
ular is - -Fredholm and ∩ ∩ ∈
u qNp (1 p)H (S) = (u) (u) = (S) qH
Proof. First, is in sin
e − ⊆ Ker Ker and Im Im ⊆ . Next,
(1 p)N(S) = (1 p) N(S) (1 p) = N(S) N(S)(1 p) = N(S)p = N(S) p
− − so − − − − ∩ by
N(S ) q
∗
Remark 2.3. The statement
on
erning and is proved in the same way.
S qNp (q p) T pNq
Now, suppose that ∈ is a - -Fredholm operator with parametrix ∈ . Then
S S pNp (p p) TT pNp π(S S)
∗ ∗ ∗
∈ is a - -Fredholmoperator with parametrix ∈ . This means that
C π(p)N/Jπ(p) π(SS ) C
∗ ∗ ∗
is invertible in the -algebra . Similarly is invertible in the -algebra
π(q)N/Jπ(q) u qNp S qNp π(S)π(S S) 1/2
∗ −
Clearly, then the phase ∈ of ∈ is a lift of ∈
π(q)N/Jπ(p)
. This allows us to dedu
e the identities
π(u u) = π(S S) 1/2π(S S)π(S S) 1/2 = π(p)
∗ ∗ − ∗ ∗ −
and
π(uu ) = π(S)π(S S) 1π(S ) = π(q)
∗ ∗ − ∗
(cid:3)
as desired.
The result allows us to make the following de(cid:28)nition.
S qNp (q p) (q p) S
De(cid:28)nition 2.5. Let ∈ be a - -Fredholm operator. We de(cid:28)ne the - -index of as
the
lass
(S) = [N(S) p] [N(S ) q]
(q ) ∗
Ind -p ∩ − ∩
K (J)
0
in .
S qNp (q p) u qNp S
Let ∈ be a - -Fredholm operator and let ∈ be the phase of . The triple
(p,q,u) K [S] := [p,q,u] K (N,N/J)
0
is a relative -
y
le and thus de(cid:28)nes the
lass ∈ in relative
K K K (N,N/J) K K (J)
0 0
-theory. The relative -theory is related to the -theory of the ideal
through the ex
ision map
: K (J) K (N,N/J)
0 0
Ex
→
4.3.7 4.3.8
as de(cid:28)ned in [14, De(cid:28)nition ℄. The ex
ision map is an isomorphism, [14, Theorem ℄.
(q p) S
In the next theorem we will see that the - -index of is simply the inverse of the ex
ision
[S] K (N,N/J) (q p)
0
map applied to the
lass ∈ . Many properties of the - -index will thus follow
immediately, and we will state the ones we need as
orollaries.
KK
-THEORY AND SPECTRAL FLOW IN VON NEUMANN ALGEBRAS 5
S qNp (q p) u qNp S
Theorem 2.6. Let ∈ be a - -Fredholm operator and let ∈ be the phase of .
Then the identity
1[S] = (S)
− q p
Ex
Ind -
K (J)
0
is valid in
[S] K (N,N/J)
0
Proof. We
an express the
lass ∈ as a sum of
lasses
[S] = [p,q,u] = [p u u,q uu ,0]+[u u,uu ,u]
∗ ∗ ∗ ∗
− −
K (u u,uu ,u)
∗ ∗
The relative -
y
le is degenerate so a
tually
[S] = [p u u,q uu ,0]
∗ ∗
− −
p u u = N(S) p q uu = N(S ) q J
∗ ∗ ∗
The proje
tions − ∩ and − ∩ are in by Lemma 2.4, so
1[S] = [p u u] [q uu ] = (S)
− ∗ ∗ q p
Ex
− − − Ind -
(cid:3)
as desired.
S qNp S qNp (q p)
0 1
Corollary 2.7. Let ∈ and ∈ be - -Fredholm operators. Suppose that there
(q p) S S
0 1
is a norm-
ontinuous path of - -Fredholm operators
onne
ting and . Then
(S ) = (S )
(q ) 0 (q ) 1
Ind -p Ind -p
t S qNp S S
t 0 1
Proof. Let 7→ ∈ be the norm-
ontinuous path
onne
ting and . The norm-
t π(S )π(S S ) 1/2 π(q)N/Jπ(p) π(p)N/Jπ(p)

ontinuous path 7→ t t∗ t − ∈ , where the inverse is in ,
t v qN/Jp (p,q,v ) K t [0,1]
t t
lifts to a path 7→ ∈ su
h that are relative -
y
les for all ∈ , [14,
4.3.13 u qNp u qNp S S
0 1 0 1
Lemma ℄. If ∈ and ∈ are the phases of and respe
tively, then
π(u ) = π(v ) π(u ) = π(v )
0 0 1 1
and so we have the identity
[S ] = [p,q,u ] = [p,q,v ] = [p,q,v ] = [p,q,u ] = [S ]
0 0 0 1 1 1
K (N,N/J)
0
in . It thus follows immediately by Theorem 2.6 that
(S ) = 1[S ] = 1[S ] = (S )
(q p) 0 − 0 − 1 (q p) 1
Ind - Ex
Ex
Ind -
(cid:3)
as desired.
S qNp (q p) T rNq (r q)
Corollary 2.8. Let ∈ be a - -Fredholm operator and let ∈ be an - -
TS (r p)
Fredholm operator. Then is an - -Fredholm operator and
(T)+ (S) = (TS)
(r q) (q p) (r p)
Ind - Ind - Ind -
v rNq u qNp w rNp T S TS
Proof. Let ∈ , ∈ and ∈ be the phases of , and respe
tively. From
the
al
ulation
π(w) = π(TS)π(S T TS) 1/2
∗ ∗ −
= π(T)π(SS T T) 1/2π(S)
∗ ∗ −
= π(T)π(T T) 1/2π(S)π(S S) 1/2
∗ − ∗ −
= π(vu)
[p,r,vu] = [p,r,w] K (N,N/J)
0
we dedu
e the identity in .
6 J. KAAD,R. NEST, A. RENNIE
[T] [S] K (N,N/J)
0
Summing the
lasses and in we get
[T]+[S] = [q,r,v]+[p,q,u] = [p,r,vu] = [p,r,w] = [TS]
K (N,N/J)
0
where the se
ond equality follows from the relations in .
This allows us to
on
lude that
(T)+ (S) = 1[T]+ 1[S] = 1[TS] = (TS)
(r q) (q p) − − − (r p)
Ind - Ind - Ex
Ex
Ex
Ind -
(cid:3)
as desired.
N
Let be a semi(cid:28)nite von Neumann algebra equipped with a (cid:28)xed normal, semi(cid:28)nite, faithful
τ τ
N
tra
e . Let K be the -
ompa
t operators as de(cid:28)ned in De(cid:28)nitionτ6.1:.KA(ll pr)oje
tRions in
N 0 N
K have (cid:28)nite tra
e by Theorem 6.5. Applying the homomorphism ∗ K → from
Theorem 6.4 to the main theorems of this se
tion we obtain some of the important results from
Breuer-Fredholm theory, [2, 5, 6, 8, 9, 10, 11, 22, 23, 24℄.
3. Von Neumann Spe
tral Flow
3.1. Basi
De(cid:28)nitionsand Properties. HavingsetuptheappropriateindextheoryforFred-
pNq
holm operators in skew-
orners , [11℄, we now analyse spe
tral (cid:29)ow. This is asso
iated with
p q
odd index pairings, and so self-adjoint operators. Spe
ialising our de(cid:28)nition of - -Fredholm to
p = q = 1
the
ase we have the following.
T N J π(T) N/J
De(cid:28)nition 3.1. An operator ∈ is said to be -Fredholm if is invertible in . The
J J
spa
e of -Fredholm operators is denoted by F. The spa
e of selfadjoint -Fredholm operators
sa
is denoted by F
χ : R R [0, )
Let → be the indi
ator fun
tion for the interval ∞ de(cid:28)ned by
1 t [0, )
χ(t) = ∈ ∞
0 t ( ,0)
(cid:26) ∈ −∞
4.3
The following lemma was (cid:28)rst proved in [2, Lemma ℄ in the semi(cid:28)nite
ontext. In fa
t it
makes sense and is true in the more general
ontext
onsidered here. We quote the statement
and proof for
ompleteness.
T π χ(T) = χ π(T)
sa
Lemma 3.2. Let ∈ F then .
(cid:0) (cid:1) (cid:0) (cid:1)
χ π(T) 0 / π(T) ε > 0
Proof. Note that makes sense sin
e ∈ Sp thus we
an (cid:28)nd an su
h that
[ ε,ε] π(T)
fthe:iRntervRal − (cid:0)is in
lu(cid:1)ded in the resolvent set of(cid:0) .(cid:1)Now de(cid:28)ne the
ontinuous fun
tions
1
→
0 t ( , ε]
∈ −∞ −
f (t) = ε 1t+1 t [ ε,0]
1 −
 ∈ −
1 t [0, )

∈ ∞
f : R R
2
and → by 
0 t ( ,0]
∈ −∞
f (t) = ε 1t t [0,ε]
2 −
 ∈
1 t [ε, )

∈ ∞

KK
-THEORY AND SPECTRAL FLOW IN VON NEUMANN ALGEBRAS 7
f = χ = f (π(T)) f χ f (T)
1 2 1 2
So on Sp while ≥ ≥ on Sp . Thus
χ π(T) = f π(T) = π f (T) π χ(T) π f (T) = f π(T) = χ π(T)
1 1 2 2
≥ ≥
χ(cid:0) π(T(cid:1)) = π(cid:0) χ(T)(cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
yielding as desired.
(cid:0) (cid:1) t (cid:0) B (cid:1) t χ π(B )
t sa t
Lemma 3.3. Let 7→ be a norm
ontinuous path in F . Then 7→ is a norm
C N/J
∗

ontinuous path in the -algebra . (cid:0) (cid:1)
C
∗
To prove this lemma we need a general result from the theory of -algebras. The result is
probably well-known to the experts, but as we
ould not (cid:28)nd a referen
e, we in
lude a proof.
A C U R A
∗ sa
Lemma 3.4. Let be a -algebra and let be an open subset of . Denote by the real
A
subspa
e of selfadjoint elements with the indu
ed topology from . Then the set
a A (a) U
sa
{ ∈ |Sp ⊆ }
A
sa
is open in
a A (a) U ( ,Uc) : C [0, [
sa
Proof. Let ∈ with Sp ⊆ . The fun
tion dist · → ∞ de(cid:28)ned by
(λ,Uc) = inf λ µ µ Uc
dist {| − || ∈ }
λ C (a)
for all ∈ is
ontinuous. It attains thus its minimumon the
ompa
t set Sp . Furthermore
λ (a) (λ,Uc) > 0 λ / Uc = Uc
for ∈ Sp we have dist be
ause ∈ , so the minimumis stri
tly positive.
Set (a),Uc inf λ µ λ (a),µ Uc
ε = dist Sp = {| − || ∈ Sp ∈ } > 0
2 2
(cid:0) (cid:1)
b A b a < ε λ (b)
Now take ∈ sa with k − k 2 and suppose for
ontradi
tion that there exists a ∈ Sp
with
B (λ) (a) =
ε
∩Sp ∅
B (λ) ε > 0 λ µ B (λ) µ / (a)
ε ε/4
Here denotes the ball of radius and
enter . Let ∈ . Then ∈ Sp
and 3ε
(µ a) 1 1 = sup µ α 1 α (a) 1 = µ, (a)
− − − −
k − k {| − | | ∈ Sp } dist Sp ≥ 4
(cid:0) (cid:1)
Furthermore ε ε
(λ b) (µ a) λ µ + a b < + (µ a) 1 1
− −
k − − − k ≤ | − | k − k 4 2 ≤ k − k
λ b 17.3
So − is a
tually invertible whi
h is a
ontradi
tion, see [17, Proposition ℄. Hen
e for
λ (b)
∈ Sp we
annot have
B (λ) (a) =
ε
∩Sp ∅
ε (b) U (b) U
Bb e
aAuse of thebwaay t<heε/2was
hosen we
on
lude that Sp ⊆ . Thus Sp ⊆ for an(cid:3)y
sa
∈ with k − k
t [0,1] ε > 0 [ ε,ε]
0
Proof. of Lemma 3.3. Let ∈ . Choose an su
h that the interval − is in
luded
π(B )
in the resolvent set of t0 . Now
π(B ) ( , ε) (ε, )
Sp t0 ⊆ −∞ − ∪ ∞
t π(B ) δ > 0
(cid:0) (cid:1) t
By Lemma 3.4 and the
ontinuity of 7→ there is a su
h that
π(B ) ( , ε) (ε, )
t
Sp ⊆ −∞ − ∪ ∞
(cid:0) (cid:1)
8 J. KAAD,R. NEST, A. RENNIE
t (t δ,t +δ) [0,1] t (t δ,t +δ) [0,1]
0 0 0 0
for all ∈ − ∩ . So for all ∈ − ∩ we have the identity
χ π(B ) = f π(B )
t t
f (cid:0) (cid:1) (cid:0) (cid:1) f
1
where issome(cid:28)xed
ontinuous fun
tion(forinstan
e the fun
tion fromtheproof ofLemma
3.2). But the fun
tion
t f π(B )
t
7→
(cid:3)
(cid:0) (cid:1)
is
learly
ontinuous and the lemma is thereby proved.
K (J)
0
With these tools at hand we
an now de(cid:28)ne spe
tral (cid:29)ow as a
lass in .
t B
t sa
De(cid:28)nition 3.5 (Spe
tral (cid:29)ow). Let 7→ be a norm
ontinuous path in F . By Lemma 3.3
the path
t π χ(B ) = χ π(B )
t t
7→
0 = t < t < ... < t = 1
(cid:0) 0 (cid:1)1 (cid:0) n(cid:1)
is norm
ontinuous. Find a partition su
h that
π χ(B ) π χ(B ) < 1/2 t,s [t ,t ]
t s i 1 i
k − k for all ∈ −
p = χ(B ) (cid:0) (cid:1) (cid:0) (cid:1) B
Set i ti . We now de(cid:28)ne the spe
tral (cid:29)ow of the path { t} to be
n
B = (1 p ) p (1 p ) p K (J)
t i i 1 i 1 i 0
sf{ } − ∩ − − − − ∩ ∈
i=1
X(cid:2) (cid:3) (cid:2) (cid:3)
This de(cid:28)nition raises several questions whi
h we will answer in the following lemmas.
p p p Np (p p ) i 1,...,n
i i 1 i i 1 i i 1
(1) Are the elements − ∈ − - − -Fredholm operators for all ∈ { } ?
(2) Is the spe
tral (cid:29)ow independent of the partition
hosen ?
B
t
(3) Is the spe
tral (cid:29)ow invariant under homotopies of the path { } ?
B C B C J
t t t t
(4) Is the spe
tral (cid:29)ow of { } equal to the spe
tral (cid:29)ow of { } if − ∈ for all
t [0,1]
∈ ?
p,q N π(p) π(q) < 1
Lemma 3.6. Suppose that ∈ are two proje
tions su
h that k − k . Then
qp qNp (q p) (1 q) p J
∈ is a - -Fredholm operator. Thus, by Lemma 2.4, we have − ∩ ∈ and
(1 p) q J
− ∩ ∈ .
Proof. The inequality
π(pqp) π(p) π(p) π(q) < 1
k − k ≤ k − k
π(pqp) π(pNp) T pNp π(Tpqp) =
shows that is invertiblein , so there is an operator ∈ su
h that
π(p)
. Likewise the inequality
π(qpq) π(q) π(q) π(q) < 1
k − k ≤ k − k
π(qpq) π(qNq) R qNq π(qpqR) =
sπh(oqw) s that is iqnpvertib(leqipn) , so there is an operator ∈ su
h that (cid:3)
. It follows that is a - -Fredholm operator.
B 0 = t < t < ... < t = 1
t sa 0 1 n
Corollary 3.7. For a path { } in F and a partition su
h that
π χ(B ) π χ(B ) < 1/2 t,s [t ,t ]
t s i 1 i
k − k for all ∈ −
(cid:0) (cid:1) (cid:0) (cid:1)
KK
-THEORY AND SPECTRAL FLOW IN VON NEUMANN ALGEBRAS 9
i 1,...,n (p p )
i i 1
for all ∈ { } we
an express the spe
tral (cid:29)ow of the path as the sum of - − -indi
es
n
B = (p p )
sf{ t} Ind(pi-pi−1) i i−1
i=1
X
p = χ(B ) i 0,...,n
where i ti for all ∈ { }. Thus by Theorem 2.8 we a
tually have
B = (p ...p ) = [N(p ...p ) p ] [N(p ...p ) p ]
sf{ t} Ind(pn-p0) n 0 n 0 ∩ 0 − 0 n ∩ n
p,q,r N
Lemma 3.8. Suppose that are three proje
tions in with
π(p) π(q) < 1/2 , π(q) π(r) < 1/2 π(r) π(p) < 1/2
k − k k − k and k − k
then
(rq)+ (qp) = (rp)
(r q) (q p) (r p)
Ind - Ind - Ind -
Thus the spe
tral (cid:29)ow is independent of the partition
hosen -it doesn't
hange if a (cid:28)ner one is

hosen.
Proof. We want to prove that
(rq)+ (qp) (rp) = 0
(r q) (q p) (r p)
Ind - Ind - −Ind -
By Theorem 2.8 this amounts to show that
(rqpr) = 0
(r r)
Ind -
Verify the inequality
π(rqpr) π(r) π(qp) π(r)
k − k ≤ k − k
π(qp) π(q) + π(q) π(r)
≤ k − k k − k
π(p) π(q) + π(q) π(r)
≤ k − k k − k
< 1
t [0,1]
Let ∈ , then
π (1 t)rqpr+tr π(r) = (1 t) π(rqpr) π(r) < (1 t)
k − − k − k − k −
π (1 t)rqp(cid:0)r + tr (cid:1) π(rNr) t [0,1]
thus − is invertible in for all ∈ . This means that the path
t (1 t)rqpr+tr (r r) rqpr r
7→ (cid:0)−
on(cid:1)sists entirely of - -Fredholm operators and it
onne
ts with .
To (cid:28)nish the proof we simply refer to Theorem 2.7 whi
h gives
0 = (r) = (rqpr)
(r r) (r r)
Ind - Ind -
(cid:3)
as desired.
B C J
t t
Lemma 3.9. [2, 23℄ Let { } and { } be two paths of selfadjoint -Fredholm operators. Let
H : [0,1] [0,1] B C
sa t t
× → F be a homotopy
onne
ting { } and { } leaving the endpoints (cid:28)xed.
H H(t,0) = B H(t,1) = C t [0,1] H(0,s) = B
t t 0
Thatis isnorm-
ontinuouswith , for all ∈ and ,
H(1,s) = B s [0,1] B = C B = C B = C
1 0 0 1 1 t t
for all ∈ . In parti
ular and . Then sf{ } sf{ }.
10 J. KAAD,R. NEST, A. RENNIE
ζ : [0,1] [0,1] N/J
Proof. The map × → de(cid:28)ned by
ζ(t,s) = π χ H(t,s)
(cid:16) (cid:17)
(cid:0) (cid:1)
is
ontinuous and thus uniformly
ontinuous, so we
an
hoose a grid
0 = t < t ... < t = 1 , 0 = s < s ... < s = 1
0 1 n 0 1 n
[0,1] [0,1] (t,s),(u,v) [t ,t ] [s ,s ] ζ(t,s) ζ(u,v) < 1
of × su
h that for any ∈ i−1 i × j−1 j we have k − k 2
i,j 1,...,n
where ∈ { } are (cid:28)xed.
i,j 1,...,n
Now look at the spe
tral (cid:29)ow along the borders of the squares. That is, for ∈ { }
J
there are eight paths of selfadjoint -Fredholm operators. For instan
e we have
u H (1 u)t +ut ,s
i 1 i j 1
7→ − − −
(cid:0) (cid:1)
as one of them. The spe
tral (cid:29)ow of this path will be denoted by
(t ,s ),(t ,s )
H i 1 j 1 i j 1
sf − − −
(cid:0) (cid:1)
Likewise for the spe
tral (cid:29)ow of the other paths. Applying Lemma 3.8 and the de(cid:28)nition of
spe
tral (cid:29)ow gives
(t ,s ),(t ,s ) + (t ,s ),(t ,s )
H i 1 j 1 i j 1 H i j 1 i j
sf − − − sf −
+ (t ,s ),(t ,s ) + (t ,s ),(t ,s ) = 0
(cid:0) H i j i 1 (cid:1)j (cid:0)H i 1 j i (cid:1)1 j 1
sf − sf − − −
(cid:0) (cid:1) (cid:0) (cid:1)
Furthermore
sf (t ,s ),(t ,s ) = sf (t ,s ),(t ,s )
H i 1 j 1 i j 1 H i j 1 i 1 j 1
− − − − − − −
(cid:3)
And an easy
ombinato(cid:0)rial argument yields(cid:1)the result(cid:0). (cid:1)
p,q N p q < 1
Remark 3.10. Suppose that ∈ are two proje
tions with k − k , then
(p) (q) = 0 = (q) (p)
Ker ∩Im Ker ∩Im
J
so the -index of the proje
tions
(pq) = [(1 p) q] [(1 q) p] = 0
(pq)
Ind - − ∩ − − ∩
1 p+pqp N
To see this we start by dedu
ing that − is invertible in from the inequality
p pqp p q < 1
k − k ≤ k − k
x (q) (p)
If now is in Ker ∩Im we immediately have
(1 p+pqp)x = 0
−
1 p + pqp x = 0 (q) (p) = 0
but − was invertible so . Therefore Ker ∩ Im . To prove that
(p) (q) = 0 p q
Ker ∩Im simply inter
hange and .
B C J B
t t t
Lemma 3.11. Let { } and { } be two paths of self adjoint -Fredholm operators with −
C J t [0,1]
t
∈ for all ∈ and
(p q ) = (q p ) = 0
Ind(p0-q0) 0 0 Ind(q1-p1) 1 1
p = χ(B ) p = χ(B ) q = χ(C ) q = χ(C ) B = C
0 0 1 1 0 0 1 1 t t
where , , and . Then sf{ } sf{ }. The

ondition (3.11) is true if for instan
e
χ(B ) χ(C ) < 1 χ(C ) χ(B ) < 1
0 0 1 1
k − k and k − k