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