A noncommutative Atiyah-Patodi-Singer index theorem in KK-theory
aa r X i v : . [ m a t h . K T ] F e b A NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM INKK-THEORY
A. L. CAREY, J. PHILLIPS, AND A. RENNIE
Abstract
We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometrysetting framed in terms of Kasparov modules. We use a mapping cone construction to relate oddindex pairings to even index pairings with APS boundary conditions in the setting of KK -theory,generalising the commutative theory. We find that Cuntz-Kreiger systems provide a natural class ofexamples for our construction and the index pairings coming from APS boundary conditions yieldcomplete K -theoretic information about certain graph C ∗ -algebras. Contents
1. Introduction 12. Kasparov modules 43. K -Theory of the Mapping Cone Algebra and pairing with KK -theory 53.1. The mapping cone 53.2. The pairing in KK for the mapping cone 73.3. Dependence of the pairing on the choice of ( Y, P −
1) 84. APS Boundary Conditions and Kasparov Modules for the Mapping Cone 94.1. Domains 104.2. Elements in dom( ∂ t ⊗
1) are functions. 114.3. Self-adjointness of ˆ D away from the kernel 134.4. The adjoint on L ( R + ) ⊗ X and self-adjointness of ˆ D Introduction
This paper is about a noncommutative analogue of APS index theory. We will focus on one aspect ofgeneralising the APS theory. Namely we replace classical first order elliptic operators on a manifoldwith product metric near the boundary by a ‘cylinder’ of operators on a Kasparov module. We explainbelow how the classical theory provides an example of this more general framework. We also showin the last Section that there are many noncommutative examples as well. Our motivation is notsimply that we are trying to understand noncommutative manifolds with boundary but is derived from the fact that the construction in this paper can be applied to many index problems in semifinitenoncommutative geometry using [9] (which we plan to address elsewhere).To explain our point of view let us recast a simple special case, using the language of later Sections,the connection between spectral flow and APS boundary conditions discussed in [2]. Let X be aclosed Riemannian manifold, of odd dimension, and let D be a (self-adjoint) Dirac type operator on X . Then D determines an odd K -homology class [ D ] for the algebra C ( X ) and we may pair [ D ] withthe K -theory class of a unitary u ∈ M k ( C ( X )) to obtain the integerIndex( P k uP k ) = sf ( D k , u D k u ∗ ) . Here P k is the nonnegative spectral projection for D k := D ⊗ Id C k and the index of the ‘Toeplitzoperator’ P k uP k gives the spectral flow sf ( D k , u D k u ∗ ) from D k to u D k u ∗ .We may also attach a semi-infinite cylinder to X , and consider the manifold-with-boundary X × R + . If D acts on sections of some bundle S → X , then D determines a self-adjoint operator on the L -sectionsof S , H = L ( X, S ), with respect to an appropriate measure constructed from the Riemannian metricand bundle inner products. We defineˆ H = (cid:18) L ( R + , H ) L ( R + , H ) ⊕ Φ H (cid:19) , ˆ D = (cid:18) − ∂ t + D ∂ t + D (cid:19) , where Φ is the projection onto the kernel of D . It is necessary to single out the zero eigenvalue of D for special attention since it gives rise to ‘extended L -solutions’ which contribute to the index, [1].We let ˆ D act as zero on Φ H , and regard this subspace as being composed of values at infinity ofextended solutions (more on this in the text).We give ˆ D APS boundary conditions. That is, we take the domain of ∂ t + D to be { ξ ∈ L ( R + , H ) : ( ∂ t + D ) ξ ∈ L ( R + , H ) , P ξ (0) = 0 } where again P is the nonnegative spectral projection for D . The domain of − ∂ t + D is defined similarlyusing 1 − P in place of P . Then it can be shown, see for instance [1], that ˆ D is an unbounded self-adjoint operator and for any f ∈ C ∞ ( X × R + ) which is of compact support and equal to a constanton the boundary, the product f (1 + ˆ D ) − / is a compact operator on ˆ H .Such functions lie in the mapping cone algebra for the inclusion C ֒ → C ( X ). This is defined as M ( C , C ( X )) = { f : R + → C ( X ) : f (0) ∈ C X , f continuous and vanishes at ∞} . We have an exact sequence0 → C ( X ) ⊗ C ((0 , ∞ )) → M ( C , C ( X )) → C → K -theory. Since K ( C ) = 0, this sequence simplifies to0 → K ( C ( X )) → K ( M ( C , C ( X )) → K ( C ) → K ( C ( X )) → K ( M ( C , C ( X ))) → . A careful analysis, which we present in greater generality in this paper, shows that the map Z = K ( C ) → K ( C ( X )) takes n to the class of the trivial bundle of rank n on X , and so is injective.Thus we find that K ( C ( X )) ∼ = K ( M ( C , C ( X ))) , and the mapping cone algebra is providing a suspension of sorts. The relationship between the evenindex pairing for ˆ D and the odd index pairing for D is then as follows. Let e u be the projection over M ( C , C ( X )) determined by the unitary u over C ( X ), so that [ e u ] − [1] ∈ K ( M ( C , C ( X ))). ThenIndex( e u ( ∂ t + D ) e u ) − Index( ∂ t + D ) = h [ e u ] − [1] , [ ˆ D ] i = h [ u ] , [ D ] i = sf ( D , u D u ∗ ) . NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 3
The purpose of this paper is to present a noncommutative analogue of this picture. Our main result,Theorem 5.1, shows that the situation described above for the commutative case carries over to aclass of Kasparov modules for noncommutative algebras. We exploit a paper of Putnam [17] on theK-theory of mapping cone algebras to give an APS type construction for a Kasparov module withboundary conditions that implies an equality between even and odd indices. Not only will we find anew version of this index equality, but we will see that it allows us to use APS boundary conditions toobtain interesting index pairings, and consequences, that were previously unknown. For instance weshow that the complicated K -theory calculations of [14] can be given a simple functorial description.A description of the organisation and main results of the paper now follows. We begin in the nextSection with some preliminaries on Kasparov modules. In Section 3 we review [17], describing K ofmapping cone algebras, M ( F, A ) where F ⊂ A are certain C ∗ -algebras (replacing the pair C ⊂ C ( X ) inthe classical setting above). We make some basic computations related to these groups and associatedexact sequences.The application of APS boundary conditions for Kasparov modules is done in Section 4. We showthat certain odd Kasparov modules for algebras A, B with F a subalgebra of A , can be ‘suspended’to obtain even Kasparov modules for the algebras M ( F, A ) , B , using APS boundary conditions. Theproof is surprisingly complicated as there are substantial technical issues. Even self-adjointness of theabstract Dirac operator on the suspension with APS boundary conditions is not clear. We solve allof the difficulties using a careful construction in the noncommutative setting of a parametrix for ourabstract Dirac operators on the even Kasparov module.The main theorem (Theorem 5.1) shows that two index pairings – one from an odd Kasparov moduleand one from its even ‘suspension’ – with values in K ( B ) are equal. Replacing K ( C ) = Z with K ( B )gives us an analogue of the classical example above. The proof is quite difficult; solving differentialequations in Hilbert C ∗ -modules is a more complex issue than in Hilbert space.In Section 6 we explain one class of examples. There we calculate the K -groups of the mappingcone algebra M ( F, A ) for the inclusion of the fixed point algebra F of the gauge action on certaingraph C ∗ -algebras A . For these algebras, the application of Theorem 5.1 yields in Proposition 5.7 anisomorphism from K ( M ( F, A )) to K ( F ), which leads to a functorial description of the calculationsof K ( A ) , K ( A ) in [14].Readers familiar with [3] may be puzzled by the fact that we do not study the more general questionof boundary conditions parametrised by a Grassmanian. In fact we make, in our main theorem, anassumption that classicially corresponds to assuming that we can work with a fixed APS boundarycondition for all of the perturbed operators we study. We know that for classical index problemsit is often the case that a more general operator can be homotopied to one that preserves the APSboundary conditions. In the noncommutative context of this paper we have not studied this homotopyargument. The examples in Section 6 illustrate that for many cases our restricted analysis suffices andprovides complete information about the K -theory of the relevant algebras. Acknowledgements . We thank Rsyzard Nest for advice on Section 5, David Pask, Aidan Sims andIain Raeburn for enlightening conversations and Ian Putnam for bringing his work to the third author’sattention. The first and second named authors acknowledge the financial assistance of the AustralianResearch Council and the Natural Sciences and Engineering Research Council of Canada while thethird named author thanks Statens Naturvidenskabelige Forskningsr˚ad, Denmark. All authors aregrateful for the support of the Banff International Research Station where some of this research wasundertaken.
A. L. CAREY, J. PHILLIPS, AND A. RENNIE Kasparov modules
The Kasparov modules considered in this subsection are for C ∗ -algebras with trivial grading. Definition 2.1. An odd Kasparov A - B -module consists of a countably generated ungraded right B - C ∗ -module E , with φ : A → End B ( E ) a ∗ -homomorphism, together with P ∈ End B ( E ) suchthat a ( P − P ∗ ) , a ( P − P ) , [ P, a ] are all compact endomorphisms. Alternatively, for V = 2 P − , a ( V − V ∗ ) , a ( V − , [ V, a ] are all compact endomorphisms for all a ∈ A . One can modify P to ˜ P sothat ˜ P is self-adjoint; k ˜ P k≤ ; a ( P − ˜ P ) is compact for all a ∈ A and the other conditions for P holdwith ˜ P in place of P without changing the module E . If P has a spectral gap about (as happens inthe cases of interest here) then we may and do assume that ˜ P is in fact a projection without changingthe module, E . (Note that by 17.6 of [5] we may assume that P is a projection by changing to a newmodule in the same class as E . ) By [10], [Lemma 2, Section 7], the pair ( φ, P ) determines a KK ( A, B ) class, and every class hassuch a representative. The equivalence relation on pairs ( φ, P ) that give KK classes is generated byunitary equivalence ( φ, P ) ∼ ( U φU ∗ , U P U ∗ ) and homology: ( φ , P ) ∼ ( φ , P ) if P φ ( a ) − P φ ( a )is a compact endomorphism for all a ∈ A , see also [10, Section 7]. Later we will also require even , or graded , Kasparov modules. Definition 2.2. An even Kasparov A - B -module has, in addition to the data of the previousdefinition, a grading by a self-adjoint endomorphism Γ with Γ = 1 and φ ( a )Γ = Γ φ ( a ) , V Γ + Γ V = 0 . The next theorem presents a general result used in [15][Appendix] about the Kasparov product in theodd case.
Theorem 2.3.
Let ( Y, T ) be an odd Kasparov module for the C ∗ -algebras A, B . Then (assuming that T has a spectral gap around ) the Kasparov product of K ( A ) with the class of ( Y, T ) is representedby h [ u ] , [( Y, T )] i = [ker P uP ] − [coker P uP ] ∈ K ( B ) , where P is the non-negative spectral projection for the self-adjoint operator T . This pairing was studied in [15], as well as the relation to the semifinite local index formula in non-commutative geometry. It is also the starting point for this work. More detailed information aboutthe KK -theory version of this can be found in [9].In this paper we will employ unbounded representatives of KK -classes. The theory of unboundedoperators on C ∗ -modules that we require is all contained in Lance’s book, [12], [Chapters 9,10]. Wequote the following definitions (adapted to our situation). Definition 2.4.
Let Y be a right C ∗ - B -module. A densely defined unbounded operator D : dom D ⊂ Y → Y is a B -linear operator defined on a dense B -submodule dom D ⊂ Y . The operator D is closed if the graph G ( D ) = { ( x, D x ) : x ∈ dom D} is a closed submodule of Y ⊕ Y . If D : dom D ⊂ Y → Y is densely defined and unbounded, we define the domain of the adjoint of D to be the submodule:dom D ∗ := { y ∈ Y : ∃ z ∈ Y such that ∀ x ∈ dom D , hD x | y i R = h x | z i R } . Then for y ∈ dom D ∗ define D ∗ y = z . Given y ∈ dom D ∗ , the element z is unique, so D ∗ : dom D ∗ → Y , D ∗ y = z is well-defined, and moreover is closed. NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 5
Definition 2.5.
Let Y be a right C ∗ - B -module. A densely defined unbounded operator D : dom D ⊂ Y → Y is symmetric if for all x, y ∈ dom DhD x | y i R = h x |D y i R . A symmetric operator D is self-adjoint if dom D = dom D ∗ (so D is closed). A densely definedoperator D is regular if D is closed, D ∗ is densely defined, and (1 + D ∗ D ) has dense range. The extra requirement of regularity is necessary in the C ∗ -module context for the continuous functionalcalculus, and is not automatic, [12],[Chapter 9]. Definition 2.6. An odd unbounded Kasparov A - B -module consists of a countably generatedungraded right B - C ∗ -module E , with φ : A → End B ( E ) a ∗ -homomorphism, together with an un-bounded self-adjoint regular operator D : dom D ⊂ E → E such that [ D , a ] is bounded for all a in adense ∗ -subalgebra of A and a (1 + D ) − / is a compact endomorphism of E for all a ∈ A . An evenunbounded Kasparov A - B -module has, in addition to the previous data, a Z -grading with A evenand D odd, as in Definition 2.2. K -Theory of the Mapping Cone Algebra and pairing with KK -theory The mapping cone.
Let F ⊂ A be a C ∗ -subalgebra of a C ∗ -algebra A . Recall [17] that themapping cone algebra is M ( F, A ) = { f : [0 , → A : f is continuous , f (0) = 0 , f (1) ∈ F } . The algebra operations are pointwise addition and multiplication and the norm is the uniform (sup)norm. There is a natural exact sequence0 → C (0 , ⊗ A i → M ( F, A ) ev → F → . Here ev ( f ) = f (1) and i ( g ⊗ a ) = t → g ( t ) a . It is well known that when F is an ideal in the algebra A we have K ∗ ( M ( F, A )) ∼ = K ∗ ( A/F ).We will always be considering the situation where K ( F ) = 0, as is the case for graph C ∗ -algebras,though this is not strictly necessary. When K ( F ) = 0, the six term sequence in K -theory comingfrom this short exact sequence degenerates into(1) 0 → K ( A ) → K ( M ( F, A )) ev ∗ → K ( F ) j ∗ → K ( A ) → K ( M ( F, A )) → . We need to justify the notation j ∗ ; namely we need to display the map j which induces j ∗ . Lemma 3.1.
In the above exact sequence the map j ∗ : K ( F ) → K ( A ) is induced by minus theinclusion map j : F → A (up to Bott periodicity).Proof. The map we have denoted by j ∗ is actually a composite: j ∗ : K ( F ) ∂ → K ( C (0 , ⊗ A ) ∼ = → K ( A ) . The isomorphism here is the inverse of the Bott map
Bott : K ( A ) → K ( C (0 , ⊗ A ), where Bott ([ p ]) = [ e − πit ⊗ p + 1 ⊗ (1 − p )]. The boundary map ∂ is defined as follows, [8, p 113]. For[ p ] − [ q ] ∈ K ( F ), we choose representatives p, q over F , and then choose self-adjoint lifts x, y over M ( F, A ). Then e πix , e πiy are unitaries over C ( S ) ⊗ A which are equal to the identity modulo C (0 , ⊗ A . Then ∂ ([ p ] − [ q ]) = [ e πix ] − [ e πiy ] ∈ K ( C (0 , ⊗ A ) . A. L. CAREY, J. PHILLIPS, AND A. RENNIE
Now we choose the particular lifts over M ( F, A ) given by x ( t ) = tp and y ( t ) = tq (in fact these are t ⊗ j ( p ) and t ⊗ j ( q )). Both these elements are self-adjoint, vanish at t = 0 and at t = 1 are in F . Now (cid:2) e πix (cid:3) − (cid:2) e πiy (cid:3) = (cid:2) e πit ⊗ p (cid:3) − (cid:2) e πit ⊗ q (cid:3) = − Bott ([ p ] − [ q ]) ∈ K ( C (0 , ⊗ A ) . So modulo the isomorphism
Bott : K ( A ) → K ( C (0 , ⊗ A ), j ∗ ([ p ] − [ q ]) = − ([ j ( p )] − [ j ( q )]) . (cid:3) We now describe K ( M ( F, A )) [17]. Let V m ( F, A ) be the set of partial isometries v ∈ M m ( A ) suchthat v ∗ v, vv ∗ ∈ M m ( F ). Using the inclusion V m ֒ → V m +1 given by v → v ⊕ V ( F, A ) = ∪ m V m ( F, A ) . Our aim, following [17], is to define a map κ : V ( F, A ) → K ( M ( F, A )), and we proceed in steps.First, let v ∈ V ( F, A ) and define a self-adjoint unitary v via: v = (cid:18) − vv ∗ vv ∗ − v ∗ v (cid:19) , that is, v = 1 , v = v ∗ . So, v = p + − p − where p + = ( v + 1) and p − = (1 − v ) are the positiveand negative spectral projections for v . Then for t ∈ [0 ,
1] define v ( t ) = p + + e iπt p − so that we have a continuous path of unitaries from the identity ( t = 0) to v ( t = 1). Observe that v ( t ) is unitary for all t ∈ [0 , v ∈ C ([0 , ⊗ M m ( A ), v (0) = 1 and v (1) = v . Now define e v ( t ) = v ( t ) ev ( t ) ∗ , e = (cid:18) (cid:19) . Then e v ( t ) is a projection over the unitization ˜ M ( F, A ) of M ( F, A ) given by˜ M ( F, A ) = { f : [0 , → ˜ A : f is continuous , f (0) ∈ C , f (1) ∈ ˜ F } . Thus [ e v ] − [ e ] defines an element of K ( M ( F, A )). So with κ ( v ) = [ e v ] − [ e ] we find: Lemma 3.2. [17, Lemmas 2.2,2.4,2.5] κ ( v ⊕ w ) = κ ( v ) + κ ( w )
2) If v, w ∈ V m ( F, A ) and k v − w k < (200) − then κ ( v ) = κ ( w )
3) If v ∈ V m ( F, A ) , w , w ∈ U m ( F ) then w vw ∈ V m ( F, A ) , κ ( w ) = κ ( w ) = 0 , κ ( w vw ) = κ ( v ) .4) For v ∈ M m ( F ) a partial isometry, κ ( v ) = 0 , so, for p ∈ M m ( F ) a projection, κ ( p ) = 0 .5) The map κ : V ( F, A ) → K ( M ( F, A )) is onto.6) Generate an equivalence relation ∼ on V ( F, A ) by(i) v ∼ v ⊕ p for v ∈ V ( F, A ) , p ∈ M k ( F ) (ii) If v ( t ) , t ∈ [0 , is a continuous path in V ( F, A ) then v (0) ∼ v (1) .Then κ : V ( F, A ) / ∼→ K ( M ( F, A )) is a well-defined bijection. Hence we may realise K ( M ( F, A )) as equivalence classes of partial isometries in M m ( A ) whose sourceand range projections lie in M m ( F ). Observe that when K ( F ) = 0, K ( A ) embeds in K ( M ( F, A ))by regarding a unitary (possibly in a unitization of A ) as a partial isometry. We add the followinglemmas which we will need later. Lemma 3.3.
Let v, w ∈ V m ( F, A ) have the same source projection, so v ∗ v = w ∗ w = p , say. Then [ v ⊕ w ∗ ] = [ v ] + [ w ∗ ] = [ v ] − [ w ] = [ vw ∗ ] . Remark If v = p we get a proof that − [ w ] = [ w ∗ ]. NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 7
Proof.
The homotopy is given by V θ = (cid:18) cos ( θ ) v + sin ( θ ) p cos( θ ) sin( θ )( w ∗ − vw ∗ )cos( θ ) sin( θ )( p − v ) cos ( θ ) w ∗ + sin ( θ ) vw ∗ (cid:19) , θ ∈ [0 , π/ . (cid:3) Lemma 3.4.
Suppose v ∗ v = p + q with p, q ∈ F projections, p ⊥ q . Then v = vp + vq , vv ∗ = vpv ∗ + vqv ∗ , vpv ∗ ⊥ vqv ∗ and if we assume that vpv ∗ ∈ F then [ v ] = [ vp ⊕ vq ] = [ vp ] + [ vq ] . Proof.
The first few statements are simple algebraic consequences of the hypothesis. The homotopyfrom v ∼ v ⊕ vp ⊕ vq is V θ = (cid:18) vp + vq cos ( θ ) vq sin( θ ) cos( θ ) vq sin( θ ) cos( θ ) vq sin ( θ ) (cid:19) , θ ∈ [0 , π/ . (cid:3) We will use the following equivalent definition of the mapping cone algebra, as it is more useful forour intended applications and agrees with the definition in the classical commutative case. We let M ( F, A ) = { f : R + → A : f continuous and vanishes at ∞ and f (0) ∈ F } . This way of defining the mapping cone algebra gives an isomorphic C ∗ -algebra and we will take thisas our definition from now on.3.2. The pairing in KK for the mapping cone. Using the Kasparov product, K ( M ( F, A ))pairs with KK ( M ( F, A ) , B ) for any C ∗ -algebra B . However, K ( M ( F, A )) also pairs with odd
A, B
Kasparov modules (
Y, V ) such that the left action by f ∈ F ⊂ A commutes with V . While all ourconstructions work for such A, B
Kasparov modules, we will restrict in the sequel to
A, F
Kasparovmodules. This will cause no loss of generality to those wishing to extend these results to the generalcase, but is the situation which arises naturally in examples.
Standing Assumptions (SA).
For the rest of this Section, let v ∈ A be an isometry with v ∗ v, vv ∗ ∈ F (the same will work for matrix algebras over A , F ). Let ( Y, V ) be an odd Kasparov module for
A, F such that the left action of f ∈ F ⊂ A commutes with V = 2 P − P is the non-negativespectral projection of V .To define the pairing we need a preliminary result. Lemma 3.5.
Let ( Y, V ) satisfy SA . The two projections vv ∗ P and vP v ∗ differ by a compact endo-morphism, and consequently P vP : v ∗ vP ( Y ) → vv ∗ P ( Y ) is Fredholm.Proof. It is a straightforward calculation that vP v ∗ = vv ∗ P + v [ P, v ∗ ] = vv ∗ P + 12 v [ V, v ∗ ]and, as [ V, v ∗ ] is compact, vv ∗ P and vP v ∗ differ by a compact endomorphism. One easily checks that P v ∗ P : vv ∗ P ( Y ) → v ∗ vP ( Y ) is a parametrix for P vP and the second statement follows. (cid:3)
Remarks.
In all the calculations we do here, if v ∈ M k ( A ) then we use P k := P ⊗ k in place of P :we will usually suppress this inflation notation in the interests of avoiding notation inflation. Definition 3.6.
For [ v ] ∈ K ( M ( F, A )) and ( Y, P − satisfying SA , define [ v ] × ( Y, V ) = Index(
P vP : v ∗ vP ( Y ) → vv ∗ P ( Y )) = [ker P vP ] − [coker P vP ] ∈ K ( F ) . A. L. CAREY, J. PHILLIPS, AND A. RENNIE
We make some general observations. • If v is unitary over A , we recover the usual Kasparov pairing between K ( A ) and KK ( A, F ), [9],[15, Appendix]. Thus the pairing depends only on the class of ( Y, P −
1) in KK ( A, F ) for v unitary. • In general the operator
P vP does not have closed range. However the operator ] P vP := (cid:18) P vP − P ) vP (cid:19) : (cid:18) v ∗ vP ( Y ) v ∗ vP ( Y ) (cid:19) → (cid:18) vv ∗ P ( Y ) vv ∗ (1 − P )( Y ) (cid:19) does have closed range, [7, Lemma 4.10], and the index is easily seen to beIndex( ] P vP ) = (cid:20)(cid:18)
P v ∗ (1 − P )( Y )(1 − P ) v ∗ P ( Y ) (cid:19)(cid:21) − (cid:20)(cid:18) (1 − P ) v ∗ P ( Y )(1 − P ) v ∗ P ( Y ) (cid:19)(cid:21) . The index of
P vP is in fact defined to be the index of any suitable ‘amplification’ like ] P vP , [7], andwe see that if the right F -module P v ∗ (1 − P )( Y ) is closed, then the ‘correction’ term (1 − P ) v ∗ P ( Y )arising from the amplification process cancels out. Since the K -theory class of the index does notin fact depend on the choice of amplification, we will ignore this subtlety from here on. That is, weassume without any loss of generality that the various Fredholm operators we consider satisfy thestronger condition of being regular in the sense of having a pseudoinverse [7][Definition 4.3]. Since wewill be concerned only with showing that certain indices coincide, this will not affect our conclusions. • The pairing depends only on the class of v in K ( M ( F, A )) with the module (
Y, V ) held fixed, inparticular it vanishes if v ∈ F . These statements follow in the same way as the analogous statementsfor unitaries, cf [15, Appendix]. • Since addition in the “Putnam picture” of K ( M ( F, A )) is by direct sum as is addition in the usualpicture of K ( A ) it is easy to see that the pairing is additive in the K ( M ( F, A )) variable with themodule (
Y, V ) held fixed. So with (
Y, V ) held fixed we have a well-defined group homomorphism: × ( Y, P −
1) : K ( M ( F, A )) → K ( A ) . Dependence of the pairing on the choice of ( Y, P − . The dependence on the Kasparovmodule ( Y, P −
1) is not straightforward. For instance, we require that P commute with the leftaction of F , and so homotopy invariance is necessarily broken. We now fix v ∈ V m ( F, A ) and showthat we can obtain an even
Kasparov module ( Y v , R v ) for ( A v , F ) := ( vv ∗ Avv ∗ , F ) so that the twoclasses [ v ] × ( Y, P −
1) and [1 A v ] × [( Y v , R v )] are equal in K ( A ), with the latter being a Kasparovproduct of genuine KK -classes.The purpose in doing this is to understand the homotopy invariance properties of Index( P vP ) bycharacterising it as a Kasparov product. In this subsection this is achieved by creating a ‘smaller’Kasparov module, which depends on v . In our main theorem, Theorem 5.1, we associate to anodd unbounded Kasparov module ( X, D ) a ‘larger’ even unbounded Kasparov module ( ˆ X, ˆ D ). Thislatter module is independent of v and allows us to characterise, for all [ v ] ∈ K ( M ( F, A )), the classIndex(
P vP ) as the Kasparov product [ v ] × [( ˆ X, ˆ D )]. Lemma 3.7.
With v, ( Y, P − as above, the pair ( Y v , R v ) := (cid:18)(cid:18) vv ∗ ( Y ) v ∗ v ( Y ) (cid:19) , (cid:18) R − R + (cid:19)(cid:19) where R − = ( P vP − (1 − P ) v ) and R + = R ∗− is an even ( vv ∗ Avv ∗ , F ) Kasparov module for the representation π ( a ) = (cid:18) a v ∗ av (cid:19) f or a ∈ vv ∗ Avv ∗ . NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 9
Proof.
First observe that vv ∗ Avv ∗ is always unital, with unit 1 A v = vv ∗ , and that π ( a ) leaves Y v invariant for a ∈ vv ∗ Avv ∗ . Next, R v is clearly self-adjoint and moreover, R − v ∗ v = vv ∗ R − . Takingadjoints we obtain R + vv ∗ = v ∗ vR + so that R v also leaves Y v invariant. Now since v and v ∗ commutewith P up to compacts we see that R − = (2 P − v ( mod compacts ) = v (2 P −
1) ( mod compacts ) and (2) R + = (2 P − v ∗ ( mod compacts ) = v ∗ (2 P −
1) ( mod compacts ) . (3)Hence, R v = (cid:18) vv ∗ v ∗ v (cid:19) = 1 Y v ( mod compacts ) . The compactness of commutators [ R v , π ( a )] can be reduced by (2) and (3) to the equations: a (2 P − v = (2 P − vv ∗ av and v ∗ avv ∗ (2 P −
1) = v ∗ (2 P − a ( mod compacts ) . This completes the proof using a = vv ∗ a = avv ∗ and [ P, a ] compact. (cid:3)
The following corollary is obvious once we note that π (1 A v ) = (cid:18) vv ∗ v ∗ v (cid:19) Corollary 3.8.
We have the equality in K ( F ) : [ v ] × ( Y, P −
1) = [1 A v ] × [( Y v , R v )] . Hence the pairing [ v ] × ( Y, P − depends only on [ v ] ∈ K ( M ( F, A )) and the class [( Y v , R v )] ∈ KK ( vv ∗ Avv ∗ , F ) . Remarks . In the Kasparov module ( Y v , R v ) there is a dependence on v . This result also shows thatwe can pair with any subprojection of vv ∗ in F instead of vv ∗ = 1 vv ∗ Avv ∗ . The Kasparov module( Y v , R v ) is formally reminiscent of the module obtained by a cap product of an odd module with aunitary. The remaining homotopy invariance is for homotopies of operators on Y v , or operators on Y commuting with vv ∗ .It should be clear by now that the mapping cone algebra provides a partial suspension, but mixes oddand even in a fascinating way. In the next section we relate the even index pairing for M ( F, A ) to theodd index pairing described here.4.
APS Boundary Conditions and Kasparov Modules for the Mapping Cone
In this Section we begin the substantially new material by constructing an even Kasparov modulefor the mapping cone algebra M ( F, A ) starting from an odd Kasparov F -module ( X, D ) for A. Inparticular we are assuming that D is self-adjoint and regular on X , has discrete spectrum and theeigenspaces are closed F -submodules of X which sum to X . Our even module ˆ X is initially definedto be the direct sum of two copies of the C ∗ -module: E = L ( R + ) ⊗ C X which is the completion ofthe algebraic tensor product in the tensor product C ∗ -module norm. That is, we take finite sums ofelementary tensors which can naturally be regarded as functions f : R + → X . The inner product onsuch f = P i f i ⊗ x i , g = P j g j ⊗ y j is defined to be h f | g i E = X i,j Z ∞ ¯ f i ( t ) g j ( t ) dt h x i | y j i X , where we have written h·|·i X for the inner product on X . Clearly the collection of all continuouscompactly supported functions from R + to X is naturally contained in the completion of this algebraic tensor product and for such functions f, g the inner product is given by: h f | g i E = Z ∞ h f ( t ) | g ( t ) i X dt. The corresponding norm is || f || E = ||h f | f i E || / . Remarks.
While many elements in the completion E can be realised as functions it may not betrue that all of E consists of X -valued functions. We also note that the Banach space L ( R + , X )of functions f defined by square-integrability of t
7→ k f ( t ) k is strictly contained in E . However, weshall show below that the domain of the operator ∂ t ⊗ E (free boundary conditions) consistsof X -valued functions which are square-integrable in the C ∗ -module sense above. We will define ouroperators using APS boundary conditions on the domains.4.1. Domains.
Let P be the spectral projection for D corresponding to the nonnegative axis and let T ± = ± ∂ t ⊗ ⊗ D (= ± ∂ t + D for brevity) with initial domain given bydom T ± = { f : R + → X D : f = n X i =1 f i ⊗ x i , f is smooth and compactly supported, x i ∈ X D , P ( f (0)) = 0 (+ case) , (1 − P )( f (0)) = 0 ( − case) } . By smooth we mean C ∞ , using one-sided derivatives at 0 ∈ R + . Then T ± : dom T ± ⊂ E → E . Theseare both densely defined, and so the operatorˆ D = (cid:18) T − T + (cid:19) is densely defined on E ⊕ E . An integration by parts (using the boundary conditions) shows that( T ± f | g ) E = ( f | T ∓ g ) E , f ∈ dom T ± , g ∈ dom T ∓ . Hence the adjoints are also densely defined, and so each of these operators is closable. This showsthat ˆ D is likewise closable, and symmetric.The subtlety noted above, namely that the module E does not necessarily consist of functions, forcesus to consider some seemingly circuitous arguments. Basically, to prove self-adjointness, we requireknowledge about domains, and we must prove various properties of these domains without the benefitof a function representation of all elements of E . However, we will prove below a function representationfor elements in the natural domain of ∂ t ⊗ , and therefore in the domains of the closures of T ± becauseif { f j } ⊂ dom T ± is a Cauchy sequence in the norm of E such that { T ± f j } is also Cauchy then as T ± is closable, the limit f of the sequence f j lies in the domain of the closure, and lim T ± f j = T ± f . Lemma 4.1.
For f ∈ dom T ± , the initial domain, we have:(1) h T ± f | T ± f i = h ( ∂ t ⊗ f | ( ∂ t ⊗ f i E + h (1 ⊗ D ) f | (1 ⊗ D ) f i E ∓ h f (0) |D ( f (0)) i X , and(2) ∓h f (0) |D ( f (0)) i X ≥ . Proof.
We do the case T + ; the proof for T − is the same. With a little computation it suffices to see: h ( ∂ t ⊗ f | (1 ⊗ D ) f i E + h (1 ⊗ D ) f | ( ∂ t ⊗ f i E = −h f (0) |D ( f (0)) i X NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 11 for f = P i f i ⊗ x i with f i compactly supported and f (0) ∈ ker P. Then, using integration by parts: h ( ∂ t ⊗ f | (1 ⊗ D ) f i E = X i,j Z ∞ ( ddt f i ( t ))( f j ( t )) dt · h x i |D x j i X = − X i,j (cid:26) f i (0) f j (0 + Z ∞ f i ( t ) ddt f j ( t ) dt (cid:27) h x i |D x j i X = −h X i f i (0) x i | X j f j (0) D x j i X − h X i f i ⊗ x i | X j ∂ t f j ⊗ D x j i E = −h f (0) |D ( f (0) i X − h f | ( ∂ t ⊗ D ) f i E . But, since D is self-adjoint and 1 ⊗ D commutes with ∂ t ⊗ h f | ( ∂ t ⊗ D ) f i E = h (1 ⊗ D ) f | ( ∂ t ⊗ f i E and item (1) follows. To see item (2), we have (1 − P )( f (0)) = f (0) where (1 − P ) = X ( −∞ , ( D ) sowe see that D restricted to the range of (1 − P ) is negative and therefore −h f (0) |D ( f (0) i X ≥ C ∗ -algebra. (cid:3) Corollary 4.2. If { f n } ⊆ dom ( T ± ) is a Cauchy sequence in the initial domain of T ± and { T ± ( f n ) } is also a Cauchy sequence in || · || E norm then both { ( ∂ t ⊗ f n ) } and { (1 ⊗ D )( f n ) } are also Cauchysequences in the || · || E norm. Therefore, the limit, f of { f n } in E which is in the domain of the closureof T ± , is also in the domain of the closures of both ( ∂ t ⊗ and (1 ⊗ D ) . Proof.
This follows from the lemma and the fact that if A = B + C are all positive elements in a C ∗ -algebra, then || A || ≥ || B || and || A || ≥ || C || . (cid:3) Lemma 4.3. (1) If g = P i f i ⊗ x i where the f i are smooth and compactly supported then h ( ∂ t ⊗ g | g i E = −h g (0) | g (0) i X − h g | ( ∂ t ⊗ g i E . (2) With g as above || g (0) || X ≤ || ( ∂ t ⊗ g || E · || g || E . Proof.
Item (1) is an integration by parts similar to the previous computation and item (2) followsfrom item (1) by the triangle and Cauchy-Schwarz inequalities. (cid:3)
Elements in dom( ∂ t ⊗ are functions.Definition 4.4. For each t ∈ R + , we define two shift operators S t and T t on L ( R + ) via: S t ( ξ )( s ) = ξ ( s + t ) and T t = S ∗ t . Clearly both have norm and S t T t = 1 and T t S t = 1 − E t where E t is theprojection, multiplication by X [0 ,t ] . Hence, S t ⊗ , T t ⊗ , and E t ⊗ are in L ( E ) and E t ⊗ convergesstrongly to E as t → ∞ . Lemma 4.5.
Let ∂ t ⊗ denote the closed operator on E with free boundary condition at . That is, ∂ t ⊗ is the closure of ∂ t ⊗ defined on the initial domain dom ′ ( ∂ t ⊗ consisting of finite sums ofelementary tensors f ⊗ x where f is smooth and compactly supported. Then, (1) S t leaves dom( ∂ t ⊗ invariant and commutes with ∂ t ⊗ . (2) If g ∈ dom ′ ( ∂ t ⊗ then for each t ∈ R + || g ( t ) || X ≤ || ( ∂ t ⊗ g || E || g || E . (3) If g ∈ dom( ∂ t ⊗ and { g n } is a sequence in dom ′ ( ∂ t ⊗ with g n → g in E and ( ∂ t ⊗ g n ) → ( ∂ t ⊗ g ) in E then there is a continuous function ˆ g : R + → X so that g n → ˆ g uniformly on R + . Moreover ˆ g ∈ C ( R + , X ) and depends only on g , not on the particular sequence { g n } . (4) If g ∈ dom( ∂ t ⊗ and ˆ g is the function defined in item (3) then for all elements h ∈ E whichare finite sums of elementary tensors of the form f ⊗ x where f is compactly supported and piecewisecontinuous we have: ( g | h ) E = Z ∞ h ˆ g ( t ) | h ( t ) i X dt. (5) If g ∈ dom( ∂ t ⊗ then: h g | g i E = lim M →∞ Z M h ˆ g ( t ) | ˆ g ( t ) i X dt := Z ∞ h ˆ g ( t ) | ˆ g ( t ) i X dt. Proof.
To see item (1), one easily checks that S t ⊗ ′ ( ∂ t ⊗
1) invariant and commutes with ∂ t ⊗ ∂ t ⊗ ′ ( ∂ t ⊗
1) and S t ⊗ || g ( t ) || X = || ( S t g )(0) || X ≤ || ( ∂ t ⊗ S t ( g ) || E || S t ( g ) || E = 2 || S t ( ∂ t ⊗ g ) || E || S t ( g ) || E ≤ || ( ∂ t ⊗ g ) || E || g || E . To see item (3), apply item (2) to the sequence { ( g n − g m )( t ) } to see that the sequence { g n ( t ) } in X is uniformly Cauchy for t ∈ R + . Since we can intertwine two such sequences converging to g , wesee that ˆ g is independent of the particular sequence. That ˆ g vanishes at ∞ follows immediately fromthe uniform convergence.To see item (4), let { g n } be a sequence satisfying the conditions of item (3). Then for h supported on[0 , M ] satisfying the conditions of item (4): h g | h i E = lim n →∞ h g n | h i E = lim n →∞ Z ∞ h g n ( t ) | h ( t ) i X dt = lim n →∞ Z M h g n ( t ) | h ( t ) i X dt = Z M h ˆ g ( t ) | h ( t ) i X dt = Z ∞ h ˆ g ( t ) | h ( t ) i X dt. To see item (5), fix
M > h g | E M ( g ) i E = lim n →∞ h g | E M ( g n ) i E = lim n →∞ Z ∞ h ˆ g ( t ) | E M ( g n )( t ) i X dt = lim n →∞ Z M h ˆ g ( t ) | g n ( t ) i X dt = Z M h ˆ g ( t ) | ˆ g ( t ) i X dt. Taking the limit as M → ∞ completes the proof. (cid:3) Corollary 4.6. (1) If g ∈ dom( ∂ t ⊗ then ( ∂ t ⊗ g ) is also given by a continuous X -valued functionas above. (2) If g ∈ dom( ∂ t ⊗ n for all n ≥ then ( ∂ t ⊗ n ( g ) is given by a continuous X -valuedfunction for all n. Proposition 4.7. (1) If g ∈ dom ( T ± ) the domain of the closure of T ± on its initial domain then g ∈ dom ( ∂ t ⊗ ∩ dom (1 ⊗ D ) . Moreover, g (0) is well-defined and P ( g (0)) = 0 in the T + case while NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 13 in the T − case, (1 − P )( g (0)) = 0 . Furthermore T ± g = ± ( ∂ t ⊗ g + (1 ⊗ D ) g. (2) If g ∈ dom ( T ± ) as above, then g (0) ∈ dom ( |D| / ) . Proof.
For the first item, by Corollary 4.2, g ∈ dom ( ∂ t ⊗ ∩ dom (1 ⊗ D ) . Then, by the previousLemma g (0) is defined. Since P is a bounded operator on X , P ( g (0)) = 0 in the T + case and(1 − P )( g (0)) = 0 in the T − case. To see item (2), we use part (2) of Lemma 4.1 to see that for f ∈ dom ( T ± ) we have: ∓h f (0) | D ( f (0)) i X = h|D| / ( f (0)) | |D| / ( f (0)) i X . If we apply this observation to f = g n − g m where { g n } is a Cauchy sequence in dom ( T ± ) we get theconclusion of item (2). (cid:3) Remark.
Note that evaluation at a point is continuous on dom( ∂ t ⊗
1) in the dom( ∂ t ⊗ not in the module norm.4.3. Self-adjointness of ˆ D away from the kernel. To show that ˆ D is self-adjoint we will followthe basic strategy of [1] and display a parametrix which is (almost) an exact inverse. Note that weassume that D has discrete spectrum with eigenvalues r k for k ∈ Z where the spectral projection of D corresponding to the eigenvalue r k is denoted by Φ k . We suppose that r k is increasing with k andif k > r k >
0, and conversely, so that the zero eigenvalue, if it exists, corresponds to the index k = 0. Moreover, the eigenspaces X k = Φ k ( X ) are F -bimodules which sum to X by hypothesis. Wenote that X = Φ ( X ) = ker D . We observe that if f is any real-valued function defined (at least) on { r k : k ∈ Z } , the spectrum of D ,then f ( D ) is the self-adjoint operator with domain: { x = X k x k ∈ X : X k f ( r k ) x k converges in X } , and is defined on this domain by f ( D ) x = P k f ( r k ) x k . The convergence condition on the domain isequivalent to P k | f ( r k ) | h x k | x k i X converges in F .We further note that if g : R + → X is continuous and compactly supported then for each k ∈ Z , thefunction g k := Φ k ◦ g : R + → X k is continuous with supp ( g k ) ⊆ supp ( g ) and g = P k g k converges in E . Furthermore, if g is smooth then so is each g k and ∂ t ( g k ) = ( ∂ t ( g )) k and by the previous sentence ∂ t ( g ) = ∂ t ( P k g k ) = P k ∂ t ( g k ) . As both ∂ t ⊗ ⊗ D leave the subspaces L ( R + ) ⊗ X k invariant, in order to construct parametrices Q + and Q − for T + and T − we can begin by considering homogeneous solutions f k to the equation T + ,k f k = ( ∂ t + r k ) f k = g k where g k is a smooth compactly supported function with values in X k for each k > . Setting f k ( t ) = Q + ,k ( g k )( t ) = Z t e − r k ( t − s ) g k ( s ) ds = Z ∞ H ( t − s ) e − r k ( t − s ) g k ( s ) ds, where H = X R + (the characteristic function of R + ) is the Heaviside function, we get a solutionsatisfying the boundary conditions, as the reader will readily confirm.Observe that for these homogeneous solutions our parametrix is given by a convolution operator f k ( t ) = Q + ,k ( g k )( t ) = ( G k ∗ g k )( t ) := L G k g k ( t ) . Here G k ( s ) = H ( s ) e − r k s ∈ L ( R ), and k G k k = 1 /r k . Since the operator norm of L G k on L ( R ) isbounded by k G k k , we have k Q + ,k k = k ( L G k ⊗ Φ k ) k End E ≤ k G k k ≤ /r k . For k < f k ( t ) = Q + ,k ( g k )( t ) = − Z ∞ t e − r k ( t − s ) g k ( s ) ds = − Z ∞−∞ X ( −∞ , ( t − s ) e − r k ( t − s ) g k ( s ) ds. The verification that T + f k = g k is again straightforward, and the solution is an L -function withvalues in Φ k ( X ) since it is given by the convolution of an L function and an L -function.Later when we have defined Q + , we will sum all the Q + ,k to obtain the parametrix Q + . At themoment we note that for a smooth compactly supported g we have:[ Q + (1 ⊗ ( P − Φ ))( g )] ( t ) := "X k> Q + ,k (1 ⊗ Φ k )( g ) ( t ) = "X k> Q + ,k g k ( t ) = X k> Z t e − r k ( t − s ) g k ( s ) ds. If we formally interchange the sum and the integral we get the equation:[ Q + (1 ⊗ ( P − Φ ))( g )] ( t )‘=’ Z t X k> e − r k ( t − s ) Φ k ( g )( s ) ds = Z t e −D ( t − s ) ( P − Φ )( g ( s )) ds. It is not hard to see that this convolution on the right actually converges to the expression on the leftin the norm of our module L ( R + ) ⊗ X. Similarly for the equation T − ,k f k = ( − ∂ t + r k ) f k = g k we have the solutions Q − ,k ( g k )( t ) = Z ∞ t e − r k ( s − t ) g k ( s ) ds = Z ∞−∞ X ( −∞ , ( t − s ) e r k ( t − s ) g k ( s ) ds, k > ,Q − ,k ( g k )( t ) = − Z t e r k ( t − s ) g k ( s ) ds = − Z ∞−∞ H ( t − s ) e r k ( t − s ) g k ( s ) ds, k < . Again this solution is given by a convolution, and in all cases k = 0 we get k Q ± ,k (1 ⊗ Φ k ) k ≤ / | r k | .We can get a similar operator convolution equation for P k< Q + ,k g k . Before proceeding we require a general lemma.
Lemma 4.8.
Let Y be a C ∗ - F -module and Y ⊆ Y a dense F -submodule. Let T : Y → Y be closableas a module mapping on Y , with closure T . Suppose there exists a bounded module mapping S on Y such that (1) S ( Y ) ⊂ Y , and (2) ST = Id Y and T S | Y = Id Y . Then S is one-to-one and T = S − : Image ( S ) → Y , dom T = Image ( S ) , S ◦ T = Id dom T , and T ◦ S = Id Y .Proof. This is essentially just a careful check of the definitions of the domains and closures in question.Let y ∈ dom ( T ) so there exists a sequence { y n } ⊂ Y converging to y and T y n → T y also. Now, since S is bounded, y n = ST y n → S ( T y ) and y n → y, so S ( T y ) = y and S ◦ T = Id dom ( T ) . This also shows dom( T ) ⊂ Image ( S ).On the other hand, let y = Sy ′ ∈ Image ( S ). Then y ′ = lim z n , where { z n } ⊂ Y , and so y = Sy ′ =lim Sz n . Since S : Y → Y , we see that { Sz n } ⊂ Y ⊂ dom ( T ), and so z n = T Sz n converges to y ′ ∈ Y . Hence y ∈ dom T and T y = y ′ . That is Image ( S ) ⊂ dom ( T ), and so they are equal. Finally, T Sy ′ = T y = y ′ , and as y ′ ∈ Y was arbitrary, T S = Id Y . Hence S is one-to-one, and T = S − . (cid:3) NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 15
Returning to the operators T ± and Q ± on the module E ⊖ (1 ⊗ Φ ) E , we have the following preliminaryresult. The proof is just a check of the hypotheses of the previous lemma. Corollary 4.9.
For k = 0 , let E k = L ( R + ) ⊗ X k and E k, ⊂ E k be the algebraic tensor product of C ∞ ( R + ) := { g ∈ C ∞ ( R + ) : g (0) = 0 and supp ( g ) is compact } with X k . That is, E k, = C ∞ ( R + ) ⊙ X k . Then T ± ,k , Q ± ,k map E k, to itself, and are mutual inversesthere. Hence dom ( T ± ,k ) = Image ( Q ± ,k ) , Q ± ,k is one-to-one, and the operators T ± ,k and Q ± ,k aremutually inverse (on appropriate subspaces). We extend this result by another application of Lemma 4.8:
Corollary 4.10.
Let the algebraic direct sum of the E k, with k = 0 be denoted E alg, := X alg,k =0 E k, = X alg,k =0 C ∞ ( R + ) ⊙ X k = C ∞ ( R + ) ⊙ X alg,k =0 X k Define Q ± on E alg, as the algebraic direct sum of the Q ± ,k , and similarly for T ± . Then Q ± extendsto an operator on the completion, E where it is bounded and one-to-one. Moreover, T ± = Q − ± :Image( Q ± ) → E so that Q ± ◦ T ± = Id dom T ± and T ± ◦ Q ± = Id E . We observe that E = E ⊕ ( L ( R + ) ⊗ X ) as an internal orthogonal direct sum. That is, E ⊥ = ( L ( R + ) ⊗ X ) . The adjoint on L ( R + ) ⊗ X and self-adjointness of ˆ D . On L ( R + ) ⊗ X the operator T + , becomes ∂ t ⊗ Id X with boundary conditions ξ (0) = 0 while T − , = − ∂ t ⊗ Id X with free boundaryconditions, and it is well-known that these two operators are mutual adjoints, cf [12, page 116]. Theparametrix Q + , for T + , is given by Q + , ( g )( t ) = Z t g ( t ) dt for g ∈ range ( T + , ) , while the parametrix Q − , for T − , is given by Q − , ( g )( t ) = − Z ∞ t g ( t ) dt for g ∈ range ( T − , ) . Of course, both Q + , and Q − , are unbounded operators and on L ( R + ) ⊗ X we have: T ± , Q ± , = Id range ( T ± , ) and Q ± , T ± , = Id dom ( T ± , ) . Letting Q ± denote the (closure of the) direct sum of all the Q ± ,k we get the parametrix for T ± . Proposition 4.11.
The adjoint of T ± : dom ( T ± ) → E is T ∓ . Moreover, T ± Q ± = Id range ( T ± ) and Q ± T ± = Id dom ( T ± ) . Proof.
In the following we write T ± for the closure of T ± . We write T ± = T ± (1 ⊗ Φ ) ⊕ T ± (1 E − (1 ⊗ Φ ))and observe from our last comments that ( T ± (1 ⊗ Φ )) ∗ = T ∓ (1 ⊗ Φ ) . Restricting to (1 E − (1 ⊗ Φ )) E = E we have Q ∗± = Q ∓ . To see this, recall that Q ± is bounded,and so it suffices to check on the dense submodule E alg, of Corollary 4.10. For ξ, η ∈ E alg, , there is ξ , η ∈ E alg, such that ξ = T ± ξ and η = T ∓ η ( ξ = Q ± ξ and similarly for η ). Then( Q ± ξ | η ) E = ( Q ± ( T ± ξ ) | T ∓ η ) E = ( ξ | T ∓ η ) E = ( T ± ξ | η ) E by symmetry= ( ξ | Q ∓ η ) E . Hence Q ∗± = Q ∓ on (1 E − (1 ⊗ Φ )) E = E . In order to deduce from this a similar relation for the T ± on E we need the following general considerations.For a densely defined module map T : E → E we have the relation between graphs G ( T ∗ ) = [ ν ( G ( T ))] ⊥ = ν [ G ( T ) ⊥ ] , where ν : E ⊕ E → E ⊕ E is the unitary given by ν ( x, y ) = ( y, − x ), [12, page 95]. Also for one-to-onemodule maps Q , G ( Q − ) = θ ( G ( Q )) where θ ( x, y ) = ( y, x ) and θν = − νθ . So restricting T ± to E wecalculate: G ( T ∗ + ) = [ ν ( G ( T + ))] ⊥ = [ ν ( G ( Q − ))] ⊥ = [ ν ( θ ( G ( Q + )))] ⊥ = − [ θ ( ν ( G ( Q + )))] ⊥ = − θ [ ν ( G ( Q + )) ⊥ ]= − θ [ G ( Q ∗ + )] = − θ [ G ( Q − )] = − [ G ( Q − − )] = − [ G ( T − )]= G ( T − ) . The same proof works for T − , and so T ∗± = T ∓ on all of E . (cid:3) The next step is to introduced the notion of extended solutions . In [1], the analogue of our mod-ule was introduced as a model of a (product) neighbourhood of the boundary for a manifold-with-boundary. Since the interest there, as here, was in the index of the operator on the whole manifold-with-boundary, it was necessary to modify the space of solutions considered to account for thosefunctions on the boundary which extended to interior solutions in a non-trivial way. Such functionsare not L on this product description of the boundary, but are bounded. Nevertheless they contributeto the index, and so we make a definition. Definition 4.12.
Let ( X, D ) be an unbounded odd Kasparov A − F -module. Let E = L ( R + ) ⊗ X be the M ( F, A ) − F -module defined above. As seen in Lemma 4.5, any element in the domain of theoperator ∂ t ⊗ (free boundary conditions) is given by a uniformly continuous X -valued function g which vanishes at ∞ and the integral h g | g i E = R ∞ h g ( t ) | g ( t ) i X dt converges in F + . We enlarge E to aspace ˆ E consisting of formal sums, f = g + x where g ∈ E and x ∈ X . For g ∈ dom( ∂ t ⊗ , the element f = g + x is naturally a function on R + where f ( t ) = g ( t ) + x and lim t →∞ f ( t ) = x ∈ X . We callsuch an f an extended L -function and we may regard f as a function f : R + → X with a limit: lim t →∞ f ( t ) := f ( ∞ ) such that f − f ( ∞ ) is in L ( R + ) ⊗ X and f ( ∞ ) ∈ X , that is, D f ( ∞ ) = 0 . Notewe reserve the terms extended L -function and extended solution to the case where f ( ∞ ) = 0 . So, we have a new module ˆ E = { f = g + x | g ∈ E and x ∈ X } . We let F act on the left and rightof this extra copy of X by its natural action. The F -valued inner product on ˆ E is given by: h f + x | h + y i = h f ( t ) | h ( t ) i E + h x | y i X . The left action of M ( F, A ) on the extra component X is naturally defined to be zero since M ( F.A )consists of functions which vanish at ∞ . However, when we extend the left action to the unitizationof M ( F, A ) the added identity will of course act as the identity on the extra copy of X . While D naturally acts as zero on this extra copy of X , functions f ( D ) act as multiplication by f (0) so thatin particular, P acts as the identity operator on this copy of X and the operator, ∂ t naturally extendshere as the zero operator.We now modify our earlier definition of ˆ X to include ˆ E only in the second component. Hence, bydefinition: ˆ X = (cid:18) E ˆ E (cid:19) . NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 17
For the first component any solution (i.e. element of the kernel of T + ) necessarily vanishes on theboundary, and classically cannot contribute to the index and the same situation persists in this non-commutative setting.We extend the action of T − to a map: ˆ E → E via T − ( f + x ) = T − ( f ) . Similarly we extend the actionof T + to a map: E → ˆ E via T + ( f ) = T + ( f ) + 0 and we extend the definitions of the actions of Q + and Q − . In order to emphasize the extension of T − we use the somewhat clumsy notation:ˆ D = (cid:18) T − ⊕ T + (cid:19) . The addition of the zero map does not affect the adjointness properties proved above, and so( T − ⊕ ∗ = T + and T ∗ + = T − ⊕ . Thus ˆ D is self-adjoint. We summarise this lengthy discussion. Proposition 4.13.
Let X be a right C ∗ - F -module, and D : dom D ⊂ X → X be a self-adjoint regularoperator with discrete spectrum. Then the operator ˆ D = (cid:18) − ∂ t ⊗ ⊗ D ) ⊕ ∂ t ⊗ ⊗ D (cid:19) defined on (cid:18) E ˆ E (cid:19) satisfying APS boundary conditions as above is self-adjoint and regular on ˆ X = ( E ⊕ ˆ E ) T . Proof.
It remains only to show that ˆ D is regular, namely (1 + ˆ D ) has dense range. We begin withˆ D restricted to ( E ⊕ E ) T . We restrict ourselves further to the invariant subspace ( E ⊕ E ) T . To thisend let R = Q + Q − . This is a bounded, positive endomorphism on E which is injective and has denserange (both Q + , Q − are injective with dense range, and are mutual adjoints by Proposition 4.11).Hence the (unbounded) densely defined operator R − = ( Q + Q − ) − = Q − − Q − = T − T + on E is aone-to-one positive operator which is onto. As the operator R +1 is bounded, positive and (boundedly)invertible, it is surjective. Thus on dom ( T − T + ) consider the operator( R + 1) R − = 1 + R − = 1 + T − T + . This is the composition of two surjective operators and so is surjective (on E ). Similar commentsapply to 1 + T + T − (on E ). Thus (1 + ˆ D ) restricted to (its domain in) ( E ⊕ E ) T maps onto ( E ⊕ E ) T . Next, inside E , we have E ⊥ = L ( R + ) ⊗ X and ˆ D on ( E ⊥ ⊕ E ⊥ ) T is just (cid:18) − ∂ t ∂ t (cid:19) ⊗ X . Asregularity is automatic on ( L ( R + ) ⊕ L ( R + )) T , we have regularity on all of ( E ⊕ E ) T . Now, on X ֒ → ˆ E , ˆ D is defined as zero, so (1 + ˆ D ) | X = 1 X , which is surjective. Putting the pieces together,1 + ˆ D is surjective on ˆ X . (cid:3) For use in the next proposition, we consider a more explicit discussion of regularity. So we considerthe equation (cid:18) T − T +
00 1 + T + T − (cid:19) (cid:18) f f (cid:19) = (cid:18) − ∂ t + D
00 1 − ∂ t + D (cid:19) (cid:18) f f (cid:19) = (cid:18) g g (cid:19) . Here we initially suppose each of ( g , g ) T is in C ∞ ( R + ) ⊙ P alg X k . With the exception of the extrakernel term, such pairs are dense in ˆ X. We need to find f = ( f , f ) T in the domain of ˆ D satisfyingthis equation. In solving this equation we may therefore assume that all terms are homogeneous, meaning that the general solution is built from functions that map R + to a single eigenspace for D ,corresponding to the eigenvalue r k . Thus the equation we must solve, for given ( g , g ) T ∈ ˆ X , is (cid:18) − ∂ t + r k
00 1 − ∂ t + r k (cid:19) (cid:18) f f (cid:19) = (cid:18) g g (cid:19) . The boundary conditions are r k ≥ (cid:26) f (0) = 0(( − ∂ t + r k ) f )(0) = 0 , r k < (cid:26) f (0) = 0(( ∂ t + r k ) f )(0) = 0We use the notation b r k := (1 + r k ) / as this term appears so often. The solution for f is f ( t ) = (2 b r k ) − (cid:18)Z ∞ t e b r k ( t − w ) g ( w ) dw + Z t e − b r k ( t − w ) g ( w ) dw (cid:19) + Ae − b r k t , where for r k ≥ , A = − b r k Z ∞ e − w b r k g ( w ) dw, and for r k < , A = 12 b r k b r k + r k b r k − r k Z ∞ e − w b r k g ( w ) dw. Observe that in terms of the Heaviside function H : f ( t ) = 12 b r k (cid:18)Z ∞−∞ H ⊥ ( t − w ) e b r k ( t − w ) g ( w ) dw + Z ∞−∞ H ( t − w ) e b r k ( t − w ) g ( w ) dw + ( −h e − b r k · , g ( · ) i e − b r k t r k ≥ b r k + r k b r k − r k h e − b r k · , g ( · ) i e − b r k t r k < ! . The point of this observation is that it displays the integral as a convolution by an L -function, plus arank one operator, namely a multiple of the projection onto span { e − b r k t } . Thus f is an L -function.For f the situation is analogous. We have f ( t ) = (2 b r k ) − (cid:18)Z ∞ t e b r k ( t − w ) g ( w ) dw + Z t e − b r k ( t − w ) g ( w ) dw (cid:19) + Be − b r k t , where for r k < , B = 12 b r k Z ∞ e − w b r k g ( w ) dw, and for r k ≥ , B = 12 b r k b r k − r k b r k + r k Z ∞ e − w b r k g ( w ) dw. Now we consider elements of ˆ X which only have a nonzero component in X . For such elements(0 , x ) T we have (1 − ∂ t + D ) x = (1 − x = x, so we have surjectivity for such elements. Now write a general g = ( g , g + x ) T ∈ ˆ X as g = (cid:18) g g + 0 (cid:19) + (cid:18)
00 + x (cid:19) . Then the above solutions show that for any g in a dense subspace of ˆ X , we can find f ∈ dom ˆ D with(1 + ˆ D ) f = g . Hence, we have a second proof that ˆ D is regular which we now exploit.In the next result APS boundary conditions mean that ˆ D is defined on those ξ = ( ξ ⊕ ξ ) T in ( E ⊕ ˆ E ) T such that ˆ D ξ ∈ ˆ X, P ξ (0) = 0 , (1 − P ) ξ (0) = 0 . This is all well defined thanks to Lemma 4.5.
NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 19
Proposition 4.14.
Let ( X, D ) be an ungraded unbounded Kasparov module for C ∗ -algebras A, F with F ⊂ A a subalgebra satisfying A · F = A . Suppose that D also commutes with the left action of F ⊂ A ,and that D has discrete spectrum. Then there is an unbounded graded Kasparov module ( ˆ X, ˆ D ) = (cid:18)(cid:18) E ˆ E (cid:19) , (cid:18) T − T + (cid:19)(cid:19) = L ( R + ) ⊗ X \ L ( R + ) ⊗ X ! , (cid:18) − ∂ t + D ∂ t + D (cid:19)! (with APS boundary conditions) for the mapping cone algebra M ( F, A ) .Proof. The most important observation is that the left action of M ( F, A ) on ˆ X preserves the APSboundary condition, and therefore the domain of ˆ D because for every f ∈ M ( F, A ), f (0) ∈ F andhence commutes with the spectral projections defining the boundary conditions. We note that tosee that the action of M ( F, A ) on ˆ X is by bounded module maps requires the strong boundednessproperty of all adjointable mappings [12] Proposition 1.2. We let A ⊂ A be the ∗ -subalgebra of A such that for all a ∈ A , [ D , a ] is bounded (on X ) and a (1 + D ) − / is a compact endomorphism of X . We define the algebra M ( F, A ) = { f : R + → A : f (0) ∈ F and f ∈ C ∞ ( R + ) and [ ˆ D , f ] is bounded } . We observe that the *-algebra of finite sums: { X i f i ⊗ a i : f i ∈ C ∞ ( R + ) and f i (0) = 0 if a i F } is dense in M ( F, A ) and is a *-subalgebra of M ( F, A ).By Proposition 4.13, the operator ˆ D is regular and self-adjoint, so we may employ the continuousfunctional calculus [12], to prove that f (1 + ˆ D ) − / is a compact endomorphism. It suffices to showthat f (1 + ˆ D ) − is compact. To see this, observe that f (1 + ˆ D ) − / is compact if and only if f (1 + ˆ D ) − f ∗ = f (1 + ˆ D ) − / (1 + ˆ D ) − / f ∗ is compact and this follows if f (1 + ˆ D ) − is compact. The latter follows by observing that from oursecond proof of Proposition 4.13 we have that each diagonal entry of f (1 + ˆ D ) − (cid:18) (1 ⊗ Φ k ) 00 (1 ⊗ Φ k ) (cid:19) := f (1 + ˆ D ) − ((1 ⊗ Φ k ) ⊗ )can be expressed as a finite sum of terms of the form f ( L g k ⊗ Φ k )+ f ( R k ⊗ Φ k ) where L g k is convolutionby an L -function and R k is a rank one operator. We consider a single elementary tensor in the abovesubalgebra of M ( F, A ): f = h ⊗ a , where a = a · b , where b ∈ F and a ∈ A . For such an elementarytensor the diagonal entry is ( h · L g k + h · R k ) ⊗ a · b Φ k . Since g k is in L , the product h · L g k isa compact operator on L ( R + ), and of course hR k is compact. Since b (1 + D ) − / is a compactendomorphism on X , it is straightforward to check that b Φ k is a compact endomorphism. So as End F ( L ( R + ) ⊗ X ) = End C ( L ( R + )) ⊗ End F ( X ), [18][Corollary 3.38], the endomorphism B k := f (( L g k + R k ) ⊗ Φ k ) = (1 ⊗ a )( h ( L g k + R k ) ⊗ b Φ k ) = (1 ⊗ a ) C k is compact: indeed each C k is compact on L ( R + ) ⊗ X k . The importance of this description is that f (1 + ˆ D ) − = (1 ⊗ a )( ⊕ k C k ) is a direct sum of compacts on ⊕ k ( L ( R + ) ⊗ X k ) times the boundedoperator (1 ⊗ a ).The operator norm of L g k on L ( R + ) is bounded by the L -norm of g k , and so k L g k k op ≤ k g k k = (1 + r k ) − / . The norm of the rank one operator R k on L ( R + ) is given by Cauchy-Schwarz as k R k k op ≤ (2(1 + r k )) − . (This inequality is unaffected by multiplication by ( b r k + | r k | ) / ( b r k − | r k | ), so can be applied to both r k < r k ≥ k C k k op ≤ k h k op k L g k k op k b k op + k h k op k R k k op k b k op ≤ k h k op k b k op ((1 + r k ) − / + (2(1 + r k )) − ) . Since 1 + r k → ∞ as | k | → ∞ , the sequence of compact endomorphisms { (1 ⊗ a ) P N − N C k } convergesin norm to f (1 + ˆ D ) − , which is therefore compact. Since an arbitrary f ∈ M ( F, A ) is the normlimit of finite sums P f j ⊗ a j we see that f (1 + ˆ D ) − is compact for general f in the mapping conealgebra.We can now show that we do indeed obtain a Kasparov module. First V = ˆ D (1 + ˆ D ) − / is self-adjoint. Also f (1 − V ) = f (1 + ˆ D ) − is a compact endomorphism for f ∈ M ( F, A ). Since V clearlyanticommutes with the grading operator Γ = (cid:18) − (cid:19) , we need only show that [ V, f ] is compactfor all f ∈ M ( F, A ). For f a sum of elementary tensors (using smooth functions), we may write thiscommutator as [ V, f ] = [ ˆ D , f ](1 + ˆ D ) − / + ˆ D [(1 + ˆ D ) − / , f ]Now for an elementary tensor f ⊗ a , we get [ ˆ D , f ⊗ a ] = ∂f ⊗ a + f ⊗ [ D , a ] and so the first term inthe above equation is compact. In the proof of Proposition 2.4 of [6] we have the formula:ˆ D [(1 + ˆ D ) − / , f ]= 1 π Z ∞ λ − / { ˆ D (1 + ˆ D + λ ) − / n (1 + ˆ D + λ ) − / [ f, ˆ D ](1 + ˆ D + λ ) − / o ˆ D (1 + ˆ D + λ ) − / + ˆ D (1 + ˆ D + λ ) − [ f, ˆ D ](1 + ˆ D + λ ) − } dλ. where the integral converges in operator norm and we have grouped the terms in the integrand so thatthey are clearly compact by the discussion above. It follows that [ V, f ] is a compact endomorphismfor f a sum of elementary tensors. Since these are norm dense in M ( F, A ) and V is bounded, [ V, f ]is compact for all f ∈ M ( F, A ). So we have an even Kasparov module for ( M ( F, A ) , F ) with anunbounded representative for ( M ( F, A ) , F ). (cid:3) Remark . It should be noted that in this context, discreteness of the spectrum of D does NOT implythat (1 + D ) − / is a compact endomorphism. We are assuming that we have a Kasparov module, sothat for all a ∈ A a (1+ D ) − / is a compact endomorphism, but these two compactness conditions arenot equivalent unless A is unital. Kasparov modules corresponding to infinite graphs provide examplesof this phenomenon, [15].5. Equality of the index pairings from the Kasparov modules.
We formulate our main theorem in this Section demonstrating how even and odd Kasparov modulesgive equal index pairings.We recall that given a partial isometry v ∈ A with range and source projections in F (observe thisincludes unitaries in A ), we defined v = (cid:18) − vv ∗ vv ∗ − v ∗ v (cid:19) . This is a self-adjoint unitary in
NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 21 M ( ˜ A ), and hence there exists a norm continuous path of self-adjoint unitaries in M ( ˜ A ) from v tothe identity. We choose the path v ( t ) = 12 ( e i tan − ( t ) ( v − ) + ( v + 1 )) , so that v (0) = v and v ( ∞ ) = 1 . Now define a projection e v ( t ) over ˜ M ( F, A ) by e v ( t ) = v ( t ) (cid:18) (cid:19) v ( t ) ∗ = (cid:18) − t vv ∗ − it t v it t v ∗ t v ∗ v (cid:19) , where we have used some elementary trigonometry to simplify the expressions. It is important toobserve that this is a finite sum of elementary tensors P f j ⊗ a j with f j smooth and square integrableor f j − f j ( ∞ ) smooth and square integrable. As such it maps ( ˆ E ⊕ ˆ E ) T to itself and leaves ( E ⊕ E ) T invariant.The difference of classes [ e v ( t )] − (cid:20)(cid:18) (cid:19)(cid:21) lies in K ( M ( F, A )): see Lemma 3.2 and the discussion preceding it, as well as [17]. Let e = (cid:18) (cid:19) ,a constant function, then the index pairing of [ v ] ∈ K ( M ( F, A )) with [( ˆ X, ˆ D )] is h [ e v ] − [ e ] , [( ˆ X , ˆ D )] i := Index( e v ( ˆ D ⊗ ) e v ) − Index( e ( ˆ D ⊗ ) e ) ∈ K ( F ) . Remarks.
To explain this notation we review even index theory. On (cid:18) E ˆ E (cid:19) , ˆ D = (cid:18) T − T + (cid:19) while the grading operator Γ = (cid:18) − (cid:19) . That is ˆ D is odd while the action of M ( F, A ) is even ,i.e., diagonal. Then, on (cid:18)
E ⊗ C ˆ E ⊗ C (cid:19) we have: ˆ D ⊗ = (cid:18) ˆ D
00 ˆ D (cid:19) and Γ ⊗ = (cid:18) Γ 00 Γ (cid:19) while e v = (cid:18) f gh k (cid:19) ∈ M ( M ( F, A )) acts as (cid:18) f ⊗ g ⊗ h ⊗ k ⊗ (cid:19) . Let ([
E ⊕ ˆ E ] ⊕ [ E ⊕ ˆ E ]) T ∼ = ([ E ⊕E ] ⊕ [ ˆ E ⊕ ˆ E ]) T bethe obvious unitary equivalence. Under this equivalence ˆ D ⊗ becomes (cid:18) T − ⊗ T + ⊗ (cid:19) , while e v = (cid:18) f gh k (cid:19) ∈ M ( M ( F, A )) acts as (cid:18) e v e v (cid:19) . Also, Index( e v ( ˆ D ⊗ ) e v ) really means the indexof the lower corner operator of (cid:18) e v e v (cid:19) ( ˆ D ⊗ ) (cid:18) e v e v (cid:19) = (cid:18) e v ( T − ⊗ ) e v e v ( T + ⊗ ) e v (cid:19) : e v (cid:18) T + T + (cid:19) e v : as a mapping e v (cid:18) EE (cid:19) → e v (cid:18) ˆ E ˆ E (cid:19) . That is we must compute both:ker( e v ( T + ⊗ ) e v ) ⊆ e v (cid:18) EE (cid:19) and ker( e v ( T − ⊗ ) e v ) ⊆ e v (cid:18) ˆ E ˆ E (cid:19) ⊆ (cid:18) ˆ EE (cid:19) . Similarly, Index( e ( ˆ D ⊗ ) e ) means the index of the lower corner operator: e (cid:18) T + T + (cid:19) e , that is, T + as a mapping from E → ˆ E , which we will write as Index( ˆ D ). With this reminder, and the conventionthat if T is an operator on the module Y , we write T k for T ⊗ k on the module Y ⊗ C k , we now stateour key result. Theorem 5.1.
Let ( X, D ) be an ungraded unbounded Kasparov module for the (pre-) C ∗ -algebras A ⊂
A, F with F ⊂ A a subalgebra satisfying A · F = A . Suppose that D also commutes with the leftaction of F ⊂ A , and that D has discrete spectrum. Let ( ˆ X, ˆ D ) be the unbounded Kasparov M ( F, A ) , F module of Proposition 4.14. Then for any unitary u ∈ M k ( A ) such that P k and (Φ ) k both commutewith u D k u ∗ and u ∗ D k u we have the following equality of index pairings with values in K ( F ) : h [ u ] , [( X, D )] i := Index( P k u ∗ P k ) = Index( e u ( ˆ D k ⊗ ) e u ) − Index( ˆ D k )=: h [ e u ] − (cid:20)(cid:18) (cid:19)(cid:21) , [( ˆ X , ˆ D )] i ∈ K ( F ) . Moreover, if v is a partial isometry, v ∈ M k ( A ) , with vv ∗ , v ∗ v ∈ M k ( F ) and such that P k and (Φ ) k both commute with v D k v ∗ and v ∗ D k v we have h [ e v ] − (cid:20)(cid:18) (cid:19)(cid:21) , [( ˆ X , ˆ D )] i = − Index(
P vP : v ∗ vP ( X ) → vv ∗ P ( X )) ∈ K ( F )= Index( P v ∗ P : vv ∗ P ( X ) → v ∗ vP ( X )) ∈ K ( F ) . (4) Remarks. (1) In the last statement we really are taking a Kasparov product when we consider K ( M ( F, A )) × KK ( M ( F, A ) , F ) → K ( F ) . Hence the index is well-defined, depends only on the class of [ e v ] − [1] = [ v ] and the class of the ‘APSKasparov module’.(2) We note that our hypothesis that P and Φ commute with v ∗ D v is equivalent to P and Φ commuting with v ∗ dv since P, Φ commute with D and with v ∗ v . Thus P, Φ commute with allfunctions of v ∗ D v , and in particular with each spectral projection v ∗ Φ k v . Similarly, the first set ofcommutation relations imply that D and all of D ’s spectral projections commute with v ∗ P v and v ∗ Φ v .(3) Whether every class [ v ] ∈ K ( M ( F, A )) possesses a representative satisfying the hypotheses of thetheorem is unknown to us in general. Just as with the issues of regularity, it may be that one canalways homotopy v and/or ( X, D ) so that the hypotheses are satisfied. We leave this issue for futurework, noting that for the applications we have in mind the hypotheses are satisfied.(4) With regards to the regularity of P vP (in the sense of having a pseudoinverse [7, Definition 4.3,]), weobserve that since P commutes with v ∗ P v, the operator
P vP is regular as an operator from v ∗ vP ( X ) to vv ∗ P ( X ), where the pseudoinverse of P vP is provided by
P v ∗ P . That is, ( P vP )( P v ∗ P )( P vP ) =
P vP and (
P v ∗ P )( P vP )( P v ∗ P ) = P v ∗ P. Thus our hypotheses guarantee the regularity of
P vP , and theindependence of the index of
P vP on which regular ‘amplification’ we take gives some evidence thatthe hypotheses may be relaxed.The proof of Theorem 5.1 will occupy the rest of the Section.5.1.
Preliminaries.
As is usual for an index calculation such as this, we will assume without loss ofgenerality (by replacing A by M k ( A ) if necessary) that the partial isometry v lies in A . To begin theproof it is helpful to write e v as an orthogonal sum of subprojections in L ( ˆ X ⊕ ˆ X ) which, of course,commute with e v : e v = t t vv ∗ − it t v it t v ∗ t v ∗ v ! + (cid:18) − vv ∗
00 0 (cid:19) := b e v + e v . (5)Note that to prove the Theorem it suffices to demonstrate the equality in Equation (4), and that iswhat we shall do. Using the decomposition of e v into orthogonal subprojections in (5) an elementarycalculation now gives: NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 23
Lemma 5.2. (1)
Let ξ = (cid:18) ξ ξ (cid:19) ∈ (cid:18) EE (cid:19) . Then ξ ∈ e v (cid:18) EE (cid:19) if and only if v ∗ vξ = ξ and vv ∗ ξ = − itvξ . In this case by Equation (5) we get an orthogonal decomposition: (cid:18) ξ ξ (cid:19) = e v (cid:18) ξ ξ (cid:19) = b e v (cid:18) ξ ξ (cid:19) + e v (cid:18) ξ ξ (cid:19) = (cid:18) η ξ (cid:19) + (cid:18) ζ (cid:19) , (6) where η = vv ∗ ξ = − itvξ and ζ = (1 − vv ∗ ) ξ ; and both (cid:18) η ξ (cid:19) and (cid:18) ζ (cid:19) lie in e v (cid:18) EE (cid:19) . (2) The same statement (mutatis mutandis) holds for ξ = (cid:18) ξ ξ (cid:19) ∈ (cid:18) ˆ E ˆ E (cid:19) In order to solve the differential equations to find the index in the Theorem we need the commutationrelations recorded in the following lemma.
Lemma 5.3.
The operators v ∗ D v , v ∗ v D and v ∗ dv preserve the subspaces of v ∗ v ( X ) (intersected withthe appropriate domains where necessary) given by v ∗ QvP ( X ) , v ∗ Qv (1 − P )( X ) , where Q is any ofthe projections P, P − Φ , − P, − P + Φ , Φ .Proof. In the remarks after the statement of Theorem 5.1, we noted that all spectral projections of v ∗ v D commute with the projections v ∗ Qv with Q . As v ∗ v D also commutes with P and 1 − P , v ∗ v D preserves these subspaces. Likewise, v ∗ D v commutes with v ∗ Q ′ v for any spectral projection Q ′ of D ,and by the hypotheses on v , v ∗ D v commutes with P and so 1 − P . Thus v ∗ D v preserves all thesesubspaces. The result for v ∗ dv = v ∗ D v − v ∗ v D follows immediately. (cid:3) Simplifying the equations.
The main consequence of Lemma 5.2 is that we can consider twoorthogonal subspaces of solutions separately and this greatly reduces the complexity of our task. Inthis subsection we will cover the T + case: ker( e v ( T + ⊗ ) e v ).We observe that ( ∂ t + D ) ⊗ commutes with the projection (cid:18) − vv ∗
00 0 (cid:19) (which is ≤ e v ). Thuswith Q + the parametrix for T + = ∂ t + D constructed earlier we have (cid:18) (1 − vv ∗ ) 00 0 (cid:19) ( Q + ⊗ ) (cid:18) (1 − vv ∗ ) 00 0 (cid:19) e v (( ∂ t + D ) ⊗ ) e v (cid:18) − vv ∗
00 0 (cid:19) = (cid:18) (1 − vv ∗ ) 00 0 (cid:19) ( Q + ⊗ )(( ∂ t + D ) ⊗ ) (cid:18) − vv ∗
00 0 (cid:19) = (cid:18) (1 − vv ∗ ) 00 0 (cid:19) ( Id ⊗ ) (cid:18) − vv ∗
00 0 (cid:19) = (cid:18) − vv ∗
00 0 (cid:19)
Thus the kernel is { } on this subspace, and so we need only calculate the kernel on the range of b e v . Using the notation da := [ D , a ] and recalling that vv ∗ and v ∗ v commute with D , so that v ∗ vdv ∗ = dv ∗ and vv ∗ dv = dv we now obtain: b e v [( ∂ t + D ) ⊗ ] b e v = t (1+ t ) vv ∗ + t t vv ∗ ( ∂ t + D ) + t (1+ t ) vdv ∗ it (1+ t ) v + − it t v ( ∂ t + D ) + − it (1+ t ) dv i (1+ t ) v ∗ + it t v ∗ ( ∂ t + D ) + it (1+ t ) dv ∗ − t (1+ t ) v ∗ v + t v ∗ v ( ∂ t + D ) + t (1+ t ) v ∗ dv ! = 11 + t (cid:18) t vv ∗ ( ∂ t + D ) − itv ( ∂ t + D ) itv ∗ ( ∂ t + D ) v ∗ v ( ∂ t + D ) (cid:19) + 1(1 + t ) (cid:18) tvv ∗ + t vdv ∗ it v − it dviv ∗ + itdv ∗ − tv ∗ v + t v ∗ dv (cid:19) . Using this formula, we obtain b e v (cid:18) ( ∂ t + D ) 00 ( ∂ t + D ) (cid:19) b e v (cid:18) ξ ξ (cid:19) = − itv ( ∂ t + D ) ξ − it t vξ − it t dvξ ( ∂ t + D ) ξ + t t ξ + t t v ∗ dvξ ! . Since this vector is also in the range of b e v we check that the first coordinate is − itv times the secondcoordinate as required by Lemma 5.2. We may rewrite the second coordinate in the preceding equation: ρ ( t ) = ( ∂ t + D ) ξ + t t ξ + v ∗ dvξ − v ∗ dv t ξ using ξ = v ∗ v ( ξ ) , and 1 − / (1 + t ) = t / (1 + t ) as: ρ ( t ) = (cid:18) √ t ∂ t ◦ p t + v ∗ v D + t v ∗ dv t (cid:19) ξ =: ( ˜ D v + V ) ξ where ˜ D v = (cid:16) √ t ∂ t ◦ √ t + v ∗ v D (cid:17) and V = t t ⊗ ( v ∗ dv ) := V ⊗ ( v ∗ dv ) . So in order to computethe kernel of b e v [( ∂ t + D ) ⊗ ] b e v acting on the range of b e v , it suffices to compute the kernel of ˜ D v + V acting on vectors ξ ∈ dom( ˜ D ) satisfying v ∗ v ( ξ ) = ξ and tξ ∈ L ( R + ) ⊗ X . In the T + case only, such vectors are precisely those ξ in dom( ˜ D ) which lie in L ( R + , (1 + t ) dt ) ⊗ v ∗ v ( X ). We make theimportant observation that ˜ D v is naturally a densely defined closed operator on L ( R + , (1 + t ) dt ) ⊗ v ∗ v ( X ) completely analogous to the operator T + = ∂ t + D of Section 4 which acts on L ( R + ) ⊗ X. Now we consider boundary values. For the equation e v (( ∂ t + D ) ⊗ ) e v ξ = 0 we want to impose theboundary condition e v (0)( P ⊗ ) e v (0) ξ (0) = 0 where P is the non-negative spectral projection for D .This projection is e v (0) (cid:18) P P (cid:19) e v (0) = (cid:18) (1 − vv ∗ ) P v ∗ vP (cid:19) . Observe that our boundary projection is also the non-negative spectral projection of e v (0)( D⊗ ) e v (0).As noted above, the only solution which lies in the range of e v ( P ⊗ ) e v = (cid:18) (1 − vv ∗ ) P
00 0 (cid:19) is thezero solution, for which this condition is automatically satisfied. Hence, we need not concern ourselvesany further with this subcase.5.3.
Solutions, integral kernels and parametrices.
In the following we make some notationalsimplifications.
We replace v ∗ v D by D , and similarly for other operators, since everything commuteswith v ∗ v and we will always be working on the subspace v ∗ v ( X ). In the notation of the previoussubsection we aim to find the solutions of ( ˜ D v + V ) ρ = 0 on L ( R + , (1 + t ) dt ) ⊗ v ∗ v ( X ).We will break our space up into orthogonal pieces preserved by ˜ D v + V . We first split our space as theimage of 1 ⊗ P and 1 ⊗ (1 − P ). On the image of 1 ⊗ P we define a two parameter family of boundedoperators which will be the integral kernel of a local left inverse for ˜ D v + V on this space. The reasonfor our notation ˜ D v + V is that we regard V as a (time dependent) perturbation, and we will define ourintegral kernels using a variant of the Dyson expansion for time dependent Hamiltonians, [19, X.12].So for t ≥ s ≥ P v ∗ v ( X ) by U ( t, s ) = e − ( t − s ) P D + ∞ X n =1 ( − n Z ts Z t s · · · Z t n − s e − ( t − t ) P D V ( t ) e − ( t − t ) P D · · · V ( t n ) e − ( t n − s ) P D dt , where we write: dt n · · · dt dt = dt , and where P D really means D restricted to P v ∗ v ( X ) . NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 25
Lemma 5.4.
For all t ≥ s ≥ the integrals and the infinite sum defining U ( t, s ) converge absolutelyin the operator norm on the space P v ∗ v ( X ) . For all t ≥ s ≥ we have k U ( t, s ) k ≤ k e − ( t − s ) P D k e ( t − s ) k v ∗ dv k . Moreover U ( t, s ) satisfies the differential equations ddt U ( t, s ) = − ( D + V ( t )) U ( t, s ) and dds U ( t, s ) = U ( t, s )( D + V ( s )) . Proof.
To see the convergence and the norm inequality, we use the crude estimate k V ( t ) k ≤ k v ∗ dv k together with the equalities: k e − ( t k − t k +1 ) P D k = k e − P D k ( t k − t k +1 ) and Z ts Z t s · · · Z t n − s dt n · · · dt = ( t − s ) n /n ! , to obtain the inequality: k U ( t, s ) k ≤ k e − ( t − s ) P D k ∞ X n − ( t − s ) n k v ∗ dv k n n ! = k e − ( t − s ) P D k e ( t − s ) k v ∗ dv k . Differentiating formally yields the two differential equations but to see that the difference quotientsconverge in operator norm to the formal derivative takes a little effort. For example, using the meanvalue theorem and the functional calculus for unbounded self-adjoint operators, one shows that for any f ∈ C (2) ( R + ) which satisfies x | f ′′ ( x ) | ≤ C for all x ∈ R + we have: ddt ( f (( at + b ) D )) = a D f ′ (( at + b ) D )when ( at + b ) > f ( x ) = e − x for( t − s ) > ddt e − ( t − s ) D = −D e − ( t − s ) D , and dds e − ( t − s ) D = D e − ( t − s ) D . As for differentiating the integral terms, formally one uses a product rule which technically is invalid asone term is unbounded; however, by using the product rule trick of adding in a term and subtracting itout, one shows the formal calculation works. Since the original series and the series for the derivativesconverge uniformly and absolutely, we are done. (cid:3)
Using these results we now construct a (local) left inverse for ( ˜ D v + V )(1 ⊗ P ). We define for any t ≥ ρ ∈ ( L ( R + , (1 + t ) dt ) ⊗ P v ∗ v ( X )),( ˜ Qρ )( t ) := 1 √ t Z t U ( t, s ) p s ρ ( s ) ds. Observe that ( ˜ Qρ )(0) = 0, and is differentiable. First we need an elementary operator-theoretic lemma. Lemma 5.5.
Let T be a closed densely defined operator on a Banach space B and let S ⊆ dom( T ) bea dense subspace of dom( T ) in the domain norm. Let A : dom( T ) → B be a bounded operator in the dom( T ) norm, and let Q be a densely defined closable linear operator whose domain contains T ( S ) and such that QT = 1 S + A | S . Then, range( T ) ⊆ dom( Q ) and QT = 1 dom( T ) + A. Proof.
Let
T x ∈ range( T ) , so there exists a sequence { x n } in S with x n → x and T x n → T x.
Butthen, the fact that lim n T x n = T x and lim n Q ( T x n ) = lim n ( x n + A ( x n )) = x + A ( x ) implies that T x ∈ dom( Q ) and Q ( T x ) = x + A ( x ) . (cid:3) Lemma 5.6.
The equation ( ˜ D v + V ) ρ = 0 has no nonzero solutions in ( L ( R + , (1 + t ) dt ) ⊗ P v ∗ v ( X )) . Proof.
Fix
M > E M be the orthogonal projection of L ( R + , (1 + t ) dt ) onto the subspace L ([0 , M ] , (1 + t ) dt ) . Then E M ⊗ L ( R + , (1 + t ) dt ) ⊗ P v ∗ v ( X ))onto the subspace ( L ([0 , M ] , (1 + t ) dt ) ⊗ P v ∗ v ( X )) . Now, we see that ˜ Q defines a linear operatoron the dense subspace of ( L ([0 , M ] , (1 + t ) dt ) ⊗ P v ∗ v ( X )) consisting of continuous functions, callit ˜ Q M . This operator has a densely defined adjoint defined on the same subspace, ˜ Q M , given by theformula: ( ˜ Q M ρ )( t ) := 1 √ t Z Mt U ( s, t ) ∗ p s ρ ( s ) ds. Thus, ˜ Q M is not only densely defined, but also closable on ( L ([0 , M ] , (1 + t ) dt ) ⊗ P v ∗ v ( X )) . The smooth functions ρ in the domain of ( ˜ D v + V )(1 ⊗ P ) form a domain-dense subspace and( E M ⊗ D v + V )( ρ ) = ( ˜ D v + V )( E M ⊗ ρ ) ∈ dom( ˜ Q M ) . Let ρ M = ( E M ⊗ ρ ) , fix t ∈ [0 , M ] and calculate:( ˜ Q M ( ˜ D v + V ) ρ M )( t ) = ( ˜ Q M ( ˜ D v + V ) ρ )( t ) = ( ˜ Q ( ˜ D v + V ) ρ )( t )= 1 √ t Z t U ( t, s ) (cid:16) ∂ s ( p s ρ ( s )) + p s ( D + V ( s )) ρ ( s ) (cid:17) ds = 1 √ t Z t ∂ s ( U ( t, s ) p s ρ ( s )) ds − √ t Z t ( ∂ s U ( t, s )) p s ρ ( s ) ds + 1 √ t Z t U ( t, s ) p s ( D + V ( s )) ρ ( s ) ds = ρ ( t ) − √ t U ( t, ρ (0) = ρ ( t ) = ρ M ( t ) , As ρ (0) = P ( ρ (0)) = 0 the previous lemma implies that ( ˜ D v + V )( E M ⊗ P ) is injective and ( ˜ D v + V ) ρ = 0has no nonzero local solutions on [0 , M ] for any M > . Hence, ( ˜ D v + V ) ρ = 0 has no nonzero globalsolutions in ( L ( R + , (1 + t ) dt ) ⊗ P v ∗ v ( X )). (cid:3) Next we split the range of 1 ⊗ (1 − P ) into two pieces, namely1 ⊗ (1 − P ) = 1 ⊗ v ∗ (1 − P + Φ ) v (1 − P ) ⊕ ⊗ v ∗ ( P − Φ ) v (1 − P ) . Lemma 5.7.
The equation ( ˜ D v + V ) ρ = 0 has no nonzero solutions in the subspace L ( R + , (1 + t ) dt ) ⊗ v ∗ (1 − P + Φ ) v (1 − P )( X ) . Proof.
Suppose we did have a solution ρ ∈ L ( R + , (1 + t ) dt ) ⊗ v ∗ (1 − P + Φ ) v (1 − P )( X ). We write ρ ( t ) = √ t σ ( t ), where σ is now an (ordinary) L function with values in v ∗ (1 − P + Φ ) v (1 − P )( X ).A brief calculation shows that11 + t ddt h σ ( t ) | σ ( t ) i X = 1 √ t ddt p t h ρ ( t ) | ρ ( t ) i X = h− ( D + V ( t )) ρ ( t ) | ρ ( t ) i X + h ρ ( t ) | − ( D + V ( t )) ρ ( t ) i X . Since v ∗ D v is non-positive and D strictly negative on v ∗ (1 − P + Φ ) v (1 − P )( X ), we have the estimate D + V ( t ) = (1 + t ) − ( t v ∗ D v + D ) < − c / (1 + t ) , where c > < c < | r − | where r − is the first negative eigenvalue of D on this subspace. Thus11 + t ddt h σ ( t ) | σ ( t ) i X ≥ c t h ρ ( t ) | ρ ( t ) i X . NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 27
Multiplying by 1 + t and integrating from 0 to s gives (this is an integral of a continuous functioninto the positive cone of the C ∗ -algebra F ) Z s ddt h σ ( t ) | σ ( t ) i X dt = h σ ( s ) | σ ( s ) i X − h σ (0) | σ (0) i X ≥ c Z s h ρ ( t ) | ρ ( t ) i X dt. The right hand side is a nondecreasing function of s , and if ρ is nonzero, this function is eventuallypositive. Hence h σ ( s ) | σ ( s ) i X is a continuous non-decreasing function of s in F + , and so can not beintegrable as can be seen by evaluating on a state of F. Hence σ is not an element of L and there are nononzero solutions ρ of ( ˜ D v + V )( ρ ) = 0 in the space L ( R + , (1+ t ) dt ) ⊗ v ∗ (1 − P + Φ ) v (1 − P )( X ). (cid:3) Finally, we come to the subspace L ( R + , (1 + t ) dt ) ⊗ v ∗ ( P − Φ ) v (1 − P )( X ). On this subspace wewill define a parametrix which is a right inverse, but is not a left inverse, instead providing solutionsto our equation. Thus, for t ≥ s ≥ H ( t, s ) on the space v ∗ ( P − Φ ) v (1 − P )( X )by: e − ( t − s ) v ∗ D v + ∞ X n =1 Z ts Z t s · · · Z t n − s ((1 + t ) · · · (1 + t n )) − e − ( t − t ) v ∗ D v v ∗ dv · · · v ∗ dv e − ( t n − s ) v ∗ D v dt . where v ∗ D v means v ∗ D v restricted to the subspace v ∗ ( P − Φ ) v (1 − P )( X ) . Lemma 5.8.
For all t ≥ s ≥ the integrals and the infinite sum defining H ( t, s ) converge absolutelyin norm. For t ≥ s ≥ , H ( t, s ) is an endomorphism of the module v ∗ ( P − Φ ) v (1 − P )( X ) with norm k H ( t, s ) k ≤ k e − ( t − s ) v ∗ D v k e tan − ( t ) k v ∗ dv k ≤ e − ( t − s ) r e tan − ( t ) k v ∗ dv k , where r is the smallest positive eigenvalue of v ∗ D v on this subspace. The family of endomorphisms H ( t, s ) satisfies the differential equations ddt H ( t, s ) = − ( D + V ( t )) H ( t, s ) , dds H ( t, s ) = H ( t, s )( D + V ( s )) . Proof.
Except for the final estimate the proof of this is similar to the proof of Lemma 5.4. Now, thenorm of H ( t, s ) (on v ∗ ( P − Φ ) v (1 − P )( X )) can be estimated as follows: k H ( t, s ) k ≤ k e − ( t − s ) v ∗ D v k ∞ X n =1 k v ∗ dv k n Z t Z t · · · Z t n − ((1 + t ) · · · (1 + t n )) − dt ! = k e − ( t − s ) v ∗ D v k ∞ X n =1 k v ∗ dv k n n ! (tan − ( t )) n ! = k e − ( t − s ) v ∗ D v k e tan − ( t ) k v ∗ dv k ≤ e − ( t − s ) r e tan − ( t ) k v ∗ dv k , where r is the smallest positive eigenvalue of v ∗ D v on the subspace. (cid:3) We now define a local parametrix on the space L ( R + , (1 + t ) dt ) ⊗ v ∗ ( P − Φ ) v (1 − P )( X ) . Let ρ begiven by a continuous function in L ( R + , (1 + t ) dt ) ⊗ v ∗ ( P − Φ ) v (1 − P )( X ) and let t ≥ . Define( ˜ Rρ )( t ) := 1 √ t Z t H ( t, s ) p s ρ ( s ) ds. As in the proof of Lemma 5.6 ˜ R defines a closable linear mapping locally on [0 , M ] on it’s initial densedomain of continuous functions. We note that ˜ R ( ρ ) is differentiable. Lemma 5.9.
For every vector x in the subspace v ∗ ( P − Φ ) v (1 − P )( X ) there exists a unique element ρ ∈ L ( R + , (1 + t ) dt ) ⊗ v ∗ ( P − Φ ) v (1 − P )( X ) with ρ (0) = x and ( ˜ D v + V ) ρ = 0 . Moreover, theseare the only solutions in the space L ( R + , (1 + t ) dt ) ⊗ v ∗ ( P − Φ ) v (1 − P )( X ) . Proof.
As in the proof of Lemma 5.6 we work locally with t in the interval [0 , M ], however, we suppressthe local notations ρ M , etc. Take ρ a continuous function in L ( R + , (1+ t ) dt ) ⊗ v ∗ ( P − Φ ) v (1 − P )( X )with values in dom( D ) and compute using the differential equations from Lemma 5.8.1 √ t ∂ t p t ( ˜ Rρ )( t ) = 1 √ t ∂ t (cid:18)Z t H ( t, s ) p s ρ ( s ) ds (cid:19) = ρ ( t ) − ( D + V ( t ))( ˜ Rρ )( t ) , Thus ( ˜ D v + V ( t ))( ˜ Rρ )( t ) = ρ ( t ) and ˜ R is injective. The injectivity is first proved locally on [0 , M ] byusing Lemma 5.5 which easily implies global injectivity. On the other hand if ρ is smooth and lies inthe domain of ˜ D v + V then ( ˜ D v + V )( ρ ) is continuous and so locally we get:( ˜ R ( ˜ D v + V ) ρ )( t ) = 1 √ t Z t H ( t, s ) (cid:16) ∂ s ( p s ρ ( s )) + p s ( D + V ( s )) ρ ( s ) (cid:17) ds = 1 √ t Z t ∂ s ( H ( t, s ) p s ρ ( s )) ds − √ t Z t ( ∂ s H ( t, s )) p s ρ ( s ) ds + 1 √ t Z t H ( t, s ) p s ( D + V ( s )) ρ ( s ) ds = ρ ( t ) − √ t H ( t, ρ (0) , where we have again used the differential equations from Lemma 5.8. Applying Lemma 5.5 we obtainthis equation for all ρ ∈ dom( ˜ D v + V ) . By the estimate on k H ( t, k in the previous lemma, thefunction √ t H ( t, ρ (0) is in L ( R + , (1 + t ) dt ) ⊗ v ∗ ( P − Φ ) v (1 − P )( X ) , and so if ρ is in the kernelof ˜ D v + V we have locally and hence globally:(7) ρ ( t ) = (1 + t ) − / H ( t, ρ (0) . Conversely, with x = ρ (0) ∈ v ∗ ( P − Φ ) v (1 − P )( X ), Eq. (7) defines a solution as ˜ R is injective. (cid:3) Putting together Lemmas 5.6, 5.7, 5.9, we have the following preliminary result.
Corollary 5.10.
The kernel of ˜ D v + V on L ( R + , (1 + t ) dt ) ⊗ v ∗ v ( X ) is isomorphic to the right F -module v ∗ ( P − Φ ) v (1 − P )( X ) . Consequently ker( e v (( ∂ t + D ) ⊗ ) e v ) = ker( b e v (( ∂ t + D ) ⊗ ) b e v ) ∼ = ker( ˜ D v + V ) ∼ = v ∗ ( P − Φ ) v (1 − P )( X ) . Thus we have part of the Index of ( e v ( ˆ D ⊗ ) e v ) . To complete the calculation, we compute the kernelof the adjoint operator e v ( − ∂ t + D ) e v . We follow an essentially similar path, but must take a littlemore care with the extended L -space ˆ E . The kernel of the adjoint.
As explained above, we must compute the kernel of the operator e v (cid:18) − ∂ t + D − ∂ t + D (cid:19) e v as a map from e v (cid:18) ˆ E ˆ E (cid:19) to e v (cid:18) EE (cid:19) . Recall that M ( F, A ) acts as zero on the constant X -valued functions but the added unit element acts as the identity. Thus for a pairof constant functions (cid:18) x x (cid:19) ∈ (cid:18) ˆ E ˆ E (cid:19) we have e v (cid:18) x x (cid:19) = (cid:18) x (cid:19) . Hence e v (cid:18) ˆ E ˆ E (cid:19) ⊆ (cid:18) ˆ EE (cid:19) . For ξ ∈ e v (cid:18) ˆ E ˆ E (cid:19) to be in the domain of e v (( − ∂ t + D ) ⊗ ) e v we impose the boundary condition: (cid:18) (1 − vv ∗ )(1 − P ) 00 v ∗ v (1 − P ) (cid:19) ξ (0) = 0 . NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 29
For the constant function (cid:18) x (cid:19) ∈ e v (cid:18) ˆ E ˆ E (cid:19) to be in the domain this means that x must satisfy(1 − vv ∗ )(1 − P )( x ) = 0 . However, this is automatic as x ∈ X so that (1 − P )( x ) = 0 . Thus thedomain of e v (( − ∂ t + D ) ⊗ ) e v extended to the constant X -valued functions ( X ⊕ X ) T is ( X ⊕ T . Of course, the extended operator e v (( − ∂ t + D ) ⊗ ) e v is identically 0 here. It is important to notethat: dom( e v (( − ∂ t + D ) ⊗ ) e v ) ⊆ e v ( ˆ E ⊕ E ) T . As before we use the orthogonal decomposition of e v to enable separate analysis of the two subspaces: b e v (cid:18) ˆ E ˆ E (cid:19) → b e v (cid:18) EE (cid:19) and e v (cid:18) ˆ E ˆ E (cid:19) → e v (cid:18) EE (cid:19) . Now, e v (cid:18) ˆ E ˆ E (cid:19) = e v (cid:18) EE (cid:19) ⊕ (cid:18) (1 − vv ∗ )( X )0 (cid:19) . As in the case of e v (( ∂ t + D ) ⊗ ) e v we have e v (( − ∂ t + D ) ⊗ ) e v is one-to-one on e v ( E ⊕ E ) T and so thekernel there is 0 . Since e v (( − ∂ t + D ) ⊗ ) e v is identically 0 on e v ((1 − vv ∗ )( X ) ⊕ T ∼ = (1 − vv ∗ )( X ) , we have the following result. Proposition 5.11.
The kernel of e v (( − ∂ t + D ) ⊗ ) e v restricted to e v (cid:18) ˆ E ˆ E (cid:19) is isomorphic to theright F -module (1 − vv ∗ )( X ) . These solutions are a rather trivial type of extended solution to the adjoint equation. Next: b e v (cid:18) ( − ∂ t + D ) 00 ( − ∂ t + D ) (cid:19) b e v (cid:18) ξ ξ (cid:19) = − itv ( − ∂ t + D ) ξ + it t vξ − it t dvξ ( − ∂ t + D ) ξ − t t ξ + t t v ∗ dvξ ! . That is, any vector ( ρ , ρ ) T in the range of b e v [( ∂ t + D ) ⊗ ] b e v satisfies ρ ( t ) = − itv ( ρ )( t ) and asbefore, after simplifying, ρ ( t ) = (cid:18) − √ t ∂ t ◦ p t + v ∗ v D + t v ∗ dv t (cid:19) ξ =: ( b D v + V ) ξ where b D v = (cid:18) − √ t ∂ t ◦ p t + v ∗ v D (cid:19) and V = t t ⊗ ( v ∗ dv ) := V ⊗ ( v ∗ dv ) . So in order to compute the kernel of b e v [( − ∂ t + D ) ⊗ ] b e v acting on the range of b e v , it suffices tocompute the kernel of b D v + V acting on vectors ξ ∈ E satisfying v ∗ v ( ξ ) = ξ and − itv ( ξ ) ∈ ˆ E . As opposed to the T + case, such vectors ξ need only lie in the larger space L ( R + ) ⊗ v ∗ v ( X ) , while ξ ( t ) = − itv ( ξ ( t )) may have a nonzero limit at ∞ in X subject to the boundary conditions P ( ξ (0)) = ξ (0) . Again we split L ( R + ) ⊗ v ∗ v ( X ) into the range of 1 ⊗ P and 1 ⊗ (1 − P ). On the image of 1 ⊗ (1 − P )we define a two parameter family of bounded operators which will be the integral kernel of a localparametrix for b D v + V on this space. Thus with D standing for (1 − P ) D and for t ≥ s ≥
0, define anoperator on (1 − P ) v ∗ v ( X ) by W ( t, s ) = e ( t − s ) D + ∞ X n =1 ( − n Z ts Z t s · · · Z t n − s e ( t − t ) D V ( t ) e ( t − t ) D V ( t ) · · · V ( t n ) e ( t n − s ) D dt . Lemma 5.12.
For all t ≥ s ≥ the integrals and the infinite sum defining W ( t, s ) converge absolutelyin norm. For all t ≥ s ≥ we have (in the operator norm for endomorphisms of v ∗ v ( X ) ) k W ( t, s ) k ≤ k e ( t − s ) D k e ( t − s ) k v ∗ dv k . Moreover W ( t, s ) satisfies the differential equations ddt W ( t, s ) = ( D + V ( t )) W ( t, s ) , dds W ( t, s ) = − W ( t, s )( D + V ( s )) . Proof.
This is very similar to the proof of Lemma 5.4 so we omit the details. (cid:3)
Using these results we construct a local parametrix for ( b D v + V )(1 ⊗ (1 − P )) . For ρ a continuousfunction in L ( R + ) ⊗ (1 − P ) v ∗ v ( X ) define( b Qρ )( t ) := − (1 + t ) − / Z t W ( t, s ) p s ρ ( s ) ds. Observe that ( b Qρ )(0) = 0, and is differentiable, and so if ρ has range in dom( D ) then b Q ( ρ ) is locallyin the domain of b D v + V . As in the proof of Lemma 5.6, b Q defines a closable linear mapping locallyon [0 , M ] on its initial dense domain of continuous functions. All our calculations below are local asin Lemma 5.6. Lemma 5.13.
In the space L ( R + ) ⊗ (1 − P ) v ∗ v ( X ) the equation ( b D v + V ) ρ = 0 has no nonzerosolutions and therefore it has no nonzero solutions in the subspace L ( R + , (1 + t ) dt ) ⊗ (1 − P ) v ∗ v ( X ) . Proof.
Let ρ be a smooth function in the domain of ( b D v + V )(1 − P ):( b Q ( b D v + V ) ρ )( t ) = − √ t Z t W ( t, s ) (cid:16) − ∂ s ( p s ρ ( s )) + p s ( D + V ( s )) ρ ( s ) (cid:17) ds = 1 √ t Z t ∂ s ( W ( t, s ) p s ρ ( s )) ds − √ t Z t ( ∂ s W ( t, s )) p s ρ ( s ) ds − √ t Z t W ( t, s ) p s ( D + V ( s )) ρ ( s ) ds = ρ ( t ) − (1 + t ) − / W ( t, ρ (0) = ρ ( t ) , where, as ρ has values in the range of (1 − P ), we have ρ (0) = 0 . Arguing as in the proof of Lemma5.6 this implies that ( b D v + V )(1 − P ) is injective on its whole domain. Hence, ( b D v + V ) ρ = 0 has nononzero solutions in L ( R + ) ⊗ (1 − P ) v ∗ v ( X ). (cid:3) Next we split the range of 1 ⊗ P into three pieces, namely1 ⊗ P = [1 ⊗ v ∗ ( P − Φ ) vP ] ⊕ [1 ⊗ v ∗ (1 − P ) vP ] ⊕ [1 ⊗ v ∗ Φ vP ] . Lemma 5.14.
In the subspace L ( R + ) ⊗ v ∗ ( P − Φ ) vP v ∗ v ( X ) the equation ( b D v + V ) ρ = 0 has nononzero solutions and therefore has no nonzero solutions in L ( R + , (1 + t ) dt ) ⊗ v ∗ ( P − Φ ) vP v ∗ v ( X ) . Proof.
First, suppose we have a solution ρ with ρ ( t ) ∈ v ∗ ( P − Φ ) vP v ∗ v ( X ) for all t ≥
0, and ρ ∈ L ( R + ) ⊗ v ∗ ( P − Φ ) vP v ∗ v ( X ). Then − itv ( ρ ( t )) ∈ ( P − Φ ) vP v ∗ v ( X ) and so if this has a limitat ∞ in Φ ( X ), the limit must be 0 . That is, − itv ( ρ ) ∈ L ( R + ) ⊗ ( P − Φ ) vP v ∗ v ( X ) and so oursolution ρ actually lies in the smaller space: L ( R + , (1 + t ) dt ) ⊗ v ∗ ( P − Φ ) vP v ∗ v ( X ) . NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 31
Arguing as in Lemma 5.7 write ρ ( t ) = (1 + t ) − / σ ( t ), where σ is now an (ordinary) L function withvalues in v ∗ ( P − Φ ) vP v ∗ v ( X ):(1 + t ) − ddt h σ ( t ) | σ ( t ) i X ddt p t h ρ ( t ) | ρ ( t ) i X = h ( D + V ( t )) ρ ( t ) | ρ ( t ) i X + h ρ ( t ) | ( D + V ( t )) ρ ( t ) i X . Since v ∗ D v is strictly positive and D is non-negative on v ∗ ( P − Φ ) vP v ∗ v ( X ), we have the estimate D + V ( t ) = (1 + t ) − ( t v ∗ D v + D ) > r t / (1 + t ) , where r is the first positive eigenvalue of v ∗ D v on this subspace and therefore11 + t ddt h σ ( t ) | σ ( t ) i X ≥ r t t h ρ ( t ) | ρ ( t ) i X . Multiplying by 1 + t and integrating from 0 to s gives Z s ddt h σ ( t ) | σ ( t ) i X dt = h σ ( s ) | σ ( s ) i X − h σ (0) | σ (0) i X ≥ r Z s t h ρ ( t ) | ρ ( t ) i X dt. The right hand side is a nondecreasing function of s , and if ρ is nonzero, this function is eventuallypositive. Thus arguing further as in Lemma 5.7 there are no nonzero solutions ρ of ( b D v + V )( ρ ) = 0in L ( R + , (1 + t ) dt ) ⊗ v ∗ ( P − Φ ) vP v ∗ v ( X ) , and hence none in L ( R + ) ⊗ v ∗ ( P − Φ ) vP v ∗ v ( X ) . (cid:3) Next, we come to the subspace L ( R + ) ⊗ v ∗ (1 − P ) vP v ∗ v ( X ). On this subspace we will define alocal parametrix which is a right inverse, but is not a left inverse, instead providing solutions to ourequation. So for t ≥ s ≥ G ( t, s ) (on the module v ∗ (1 − P ) vP v ∗ v ( X )) by e ( t − s ) v ∗ D v + ∞ X n =1 ( − n Z ts Z t s · · · Z t n − s ((1 + t ) · · · (1 + t n )) − e ( t − t ) v ∗ D v v ∗ dv · · · v ∗ dv e ( t n − s ) v ∗ D v dt . Lemma 5.15.
For all t ≥ s ≥ the integrals and the infinite sum defining G ( t, s ) converge absolutelyin norm. For t ≥ s ≥ , G ( t, s ) is a bounded endomorphism of the module v ∗ (1 − P ) vP v ∗ v ( X ) withnorm bounded by k G ( t, s ) k ≤ k e ( t − s ) v ∗ D v k e ( t − s ) k v ∗ dv k ≤ e ( t − s ) r − e ( t − s ) k v ∗ dv k , where r − is the largest negative eigenvalue of D . The family of endomorphisms G ( t, s ) satisfies thedifferential equations ddt G ( t, s ) = ( D + V ( t )) G ( t, s ) , dds G ( t, s ) = − G ( t, s )( D + V ( s )) . Proof.
The proof of this is very similar to the proof of Lemma 5.8. (cid:3)
Now define a local parametrix on continuous functions ρ by( b Rρ )( t ) := (1 + t ) − / Z t G ( t, s ) p s ρ ( s ) ds, ρ ∈ L ( R + ) ⊗ v ∗ (1 − P ) vP v ∗ v ( X ) . As in the proof of Lemma 5.6 b R defines a closable linear mapping locally on [0 , M ] on the initial densedomain of continuous functions. We note that b R ( ρ ) is differentiable. Lemma 5.16.
For every vector x in the space v ∗ (1 − P ) vP v ∗ v ( X ) there exists a unique element ρ ∈ L ( R + ) ⊗ v ∗ (1 − P ) vP v ∗ v ( X ) with ρ (0) = x and ( b D v + V ) ρ = 0 . Moreover, these are theonly solutions in the subspace L ( R + ) ⊗ v ∗ (1 − P ) vP v ∗ v ( X ) . In fact these solutions ρ clearly lie in L ( R + , (1 + t ) dt ) ⊗ v ∗ (1 − P ) vP v ∗ v ( X ) and satisfy lim t →∞ − itv ( ρ ( t )) = 0 . Proof.
We work locally as in the proofs of Lemmas 5.6 and 5.9. Take ρ a continuous function in L ( R + ) ⊗ v ∗ (1 − P ) vP v ∗ v ( X ) with values in dom( D ) and compute − √ t ∂ t p t ( b Rρ )( t ) = 1 √ t ∂ t (cid:18)Z t G ( t, s ) p s ρ ( s ) ds (cid:19) = ρ ( t ) − ( D + V ( t ))( b Rρ )( t ) , where we have used the computations from Lemma 5.15. Thus ( b D v + V ( t ))( b Rρ )( t ) = ρ ( t ) and b R isinjective. The injectivity is first proved locally on [0 , M ] by using Lemma 5.5 which easily impliesglobal injectivity. On the other hand if ρ is smooth and lies in the domain of b D v + V then ( b D v + V )( ρ )is continuous and so locally we get:( b R ( b D v + V ) ρ )( t ) = − √ t Z t G ( t, s ) (cid:16) − ∂ s ( p s ρ ( s )) + p s ( D + V ( s )) ρ ( s ) (cid:17) ds = 1 √ t Z t ∂ s ( G ( t, s ) p s ρ ( s )) ds − √ t Z t ( ∂ s G ( t, s )) p s ρ ( s ) ds − √ t Z t G ( t, s ) p s ( D + V ( s )) ρ ( s ) ds = ρ ( t ) − (1 + t ) − / G ( t, ρ (0) , where we have again used the derivative computations from Lemma 5.15. Applying Lemma 5.5 weget this formula for all ρ ∈ dom( b D v + V ) . Now if ρ is in the kernel of b D v + V we have locally and hence globally(8) ρ ( t ) = (1 + t ) − / G ( t, ρ (0) , and this lies in L ( R + ) ⊗ v ∗ (1 − P ) vP v ∗ v ( X ) by the estimate: k G ( t, k ≤ e tr − e tan − ( t ) k v ∗ dv k where r − is the largest negative eigenvalue of D on the subspace. Conversely, given any vector ρ (0) ∈ v ∗ (1 − P ) vP v ∗ v ( X ), Equation (8) defines a solution since b R is injective. (cid:3) Finally we need to consider the subspace L ( R + ) ⊗ v ∗ Φ vP ( X ). This subspace gives rise to extendedsolutions. That is, the solutions we seek here are the second components ξ of a solution ξ = ( ξ , ξ ) T in e v ( ˆ E ⊕ ˆ E ) T ⊆ ( ˆ E ⊕ E ) T to the equation e v ( − ∂ t + D ) e v ξ = 0, where ξ ∈ ˆ E satisfies ξ = − ivtξ .Hence, a true extended solution (one where ξ / ∈ E ) comes from those ξ which behave like (1 + t ) − / as t → ∞ . With this reminder, we have Lemma 5.17.
For every vector x ∈ v ∗ (Φ ) vP ( X ) there exists a unique solution to the equation ( b D v + V ) ρ = 0 in the space L ( R + ) ⊗ v ∗ (Φ ) vP ( X ) with ρ (0) = x. Moreover, every solution in thisspace is of the form ρ ( t ) = (1 + t ) − / e − (tan − ( t )) v ∗ dv ρ (0) and (1) lim t →∞ − itv ( ρ ( t )) = − ive − π/ v ∗ dv ( ρ (0)) ∈ Φ ( X ) and(2) t (cid:16) − ivtρ ( t ) + ive − ( π/ v ∗ dv ( ρ (0) (cid:17) is in L ( R + ) ⊗ v ∗ (Φ ) vP ( X ) . Proof.
We define a local parametrix for ρ a continuous function in L ( R + ) ⊗ v ∗ (Φ ) vP ( X ) by( b Eρ )( t ) := − √ t e − (tan − ( t )) v ∗ dv Z t e (tan − ( s )) v ∗ dv p s ρ ( s ) ds. NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 33
We observe that ( b Eρ ) is differentiable and satisfies ( b Eρ )(0) = 0 . To show that this a parametrix, firstuse v ∗ dv = v ∗ D v − v ∗ v D to rewrite b D v + V = − √ t ∂ t p t + v ∗ D v −
11 + t v ∗ dv. As v ∗ D v acts as zero on v ∗ Φ v ( X ), this reduces on v ∗ Φ v ( X ) to b D v + V = − √ t ∂ t p t −
11 + t v ∗ dv. Applying − √ t ∂ t √ t to b E ( ρ ) and using the product rule gives − √ t ∂ t p t ( b Eρ )( t ) = 1 √ t ∂ t (cid:18) e − (tan − ( t )) v ∗ dv Z t e (tan − ( s )) v ∗ dv p s ρ ( s ) ds. (cid:19) = ρ ( t ) − v ∗ dv t ( b Eρ )( t ) . Thus ( b D v + V )( b Eρ ) = ρ locally for continuous functions As in previous cases b E is locally a closableoperator and so by Lemma 5.5 we get that ( b D v + V )( b Eρ ) = ρ locally for all ρ in the domain of b E .Hence, b E is globally injective. Integration by parts for smooth ρ in the domain gives( b E ( b D v + V ) ρ )( t ) = ρ ( t ) − (1 + t ) − / e − tan − ( t ) v ∗ dv ρ (0) . Applying Lemma 5.5, we get this equation for all ρ ∈ dom( b D v + V ) . Hence if ρ ∈ ker( b D v + V ) we have ρ ( t ) = (1 + t ) − / e − tan − ( t ) v ∗ dv ρ (0) . On the other hand if x ∈ v ∗ (Φ ) vP ( X ) and we define ρ by this equation with ρ (0) = x then we havea solution of ( b D v + V )( ρ ) = 0 in the space ρ ∈ L ( R + ) ⊗ v ∗ (Φ ) vP ( X ) . Since for each t ≥ ρ ( t ) ∈ v ∗ Φ v ( X ), we also have − itv ( ρ ( t )) ∈ Φ v ( X ) ⊆ Φ ( X ) , and therefore:lim t →∞ − itv ( ρ ( t )) = − ive − π/ v ∗ dv ( ρ (0)) ∈ Φ ( X ) . It is an exercise to check that t (cid:0) − ivtρ ( t ) + ive − ( π/ v ∗ dv ( ρ (0) (cid:1) is in L ( R + ) ⊗ v ∗ (Φ ) vP ( X ) . (cid:3) Putting together Proposition 5.11 and Lemmas 5.13, 5.14, 5.16, 5.17, we have the following.
Corollary 5.18.
The kernel of ( b D v + V ) on L ( R + ) ⊗ v ∗ v ( X ) is isomorphic to the right F -module ker( b D v + V ) ∼ = [ v ∗ (1 − P ) vP ( X )] ⊕ [ v ∗ Φ vP ( X )] , where the first summand consists of ordinary solutions in L ( R + , (1+ t ) dt ) ⊗ v ∗ v ( X ) , while the secondsummand consists of extended solutions whose second component is in L ( R + ) ⊗ v ∗ v ( X ) . Consequently,taking into account the (trivial) extended solutions of Proposition 5.11, (1 − vv ∗ )Φ ( X ) we have thefull kernel ker( e v (( − ∂ t + D ) ⊗ ) e v ) ∼ = [ v ∗ (1 − P ) vP ( X )] ⊕ [ v ∗ Φ vP ( X )] ⊕ [(1 − vv ∗ )Φ ( X )] . Completing the proof of Theorem 5.1.
Consider the pairing of (cid:18) (cid:19) with ∂ t + D .Examining our earlier parametrix computations shows that ∂ t + D with boundary condition P has nokernel, while − ∂ t + D with boundary condition 1 − P has extended solutions: the constant functionswith value in X . The projection onto these extended solutions is Φ and Index( ∂ t + D ) = − [ X ] . Since the mapping cone algebra is nonunital, we can not just pair with the class of e v , but must pair with [ e v ] − [ (cid:18) (cid:19) ]. We have computed the pairing of ( ˆ X, ˆ D ) with both these terms, and so wehave the following intermediate result: Proposition 5.19.
The pairing of [ e v ] − [ (cid:18) (cid:19) ] with ( ˆ X, ˆ D ) is given by Index( e v ( ∂ t + D ) e v ) − Index( ∂ t + D ) = Index( e v ( ∂ t + D ) e v ) + [ X ]= [ v ∗ ( P − Φ ) v (1 − P )( X )] − [ v ∗ (1 − P ) vP ( X )] − [ v ∗ Φ vP ( X )] − [(1 − vv ∗ )( X )] + [ X ]= [ v ∗ P v (1 − P )( X )] − [ v ∗ Φ v (1 − P )( X )] − [ v ∗ (1 − P ) vP ( X )] − [ v ∗ Φ vP ( X )] + [ vv ∗ ( X ) ]= [ v ∗ P v (1 − P )( X )] − [ v ∗ (1 − P ) vP ( X )] − [ v ∗ Φ v ( X )] + [ vv ∗ ( X )]= [ v ∗ P v (1 − P )( X )] − [ v ∗ (1 − P ) vP ( X )] . The last line follows because w = v ∗ Φ is a partial isometry with ww ∗ = v ∗ Φ v and w ∗ w = vv ∗ Φ , showing that the modules defined by these projections are isomorphic.Now we can finalise the proof of the Theorem by computing the index of P vP : v ∗ vP ( X ) → vv ∗ P ( X ) , where P is the non-negative spectral projection for D . The kernel of P vP is given by the set { ξ ∈ v ∗ vP ( X ) : vξ ∈ vv ∗ (1 − P )( X ) = (1 − P ) v ( X ) } = P v ∗ (1 − P ) v ( X ) , while the cokernel is given by { ξ ∈ vv ∗ P ( X ) = P v ( X ) : ξ = vη, η ∈ v ∗ v (1 − P )( X ) } = (1 − P ) v ∗ P v ( X ) . Thus Index(
P vP ) = [
P v ∗ (1 − P ) v ( X )] − [(1 − P ) v ∗ P v ( X )] ∈ K ( F ) . Hence Index(
P vP : v ∗ vP ( X ) → vv ∗ P ( X )) = − (Index( e v ( ∂ t + D ) e v ) − Index( ∂ t + D )) , and the proof of Theorem 5.1 is complete.*********************************** Remark
When [ D , v ∗ dv ] = 0, enormous simplifications occur in the preceeding analysis. In this caseone can verify that for the equation ˜ D v + V in v ∗ v E , a solution of ρ = ( ˜ D v + V ) ξ vanishing at zero isgiven by ξ ( t ) = e v ∗ dv tan − ( t ) √ t Z t e − v ∗ D v ( t − s ) p s e − v ∗ dv tan − ( s ) ρ ( s ) ds, and we require ρ ∈ v ∗ ( P − Φ ) v E . This formula can be obtained by performing the sums and integralsin the definition of our more general parametrix. Similar comments apply to the other cases.In the next section we apply Theorem 5.1 to graph algebras and the Kasparov module constructedfrom the gauge action in [15]. We will see that in this case we can always assume that v ∗ dv commuteswith D , so that we are in the simplest situation described above. NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 35 Applications to certain Cuntz-Krieger systems
For a detailed introduction to Cuntz-Krieger systems as graph algebras see [20]. A directed graph E = ( E , E , r, s ) consists of countable sets E of vertices and E of edges, and maps r, s : E → E identifying the range and source of each edge. We will always assume that the graph is locally-finite which means that each vertex emits at most finitely many edges and each vertex receives atmost finitely many edges. We write E n for the set of paths µ = µ µ · · · µ n of length | µ | := n ; that is,sequences of edges µ i such that r ( µ i ) = s ( µ i +1 ) for 1 ≤ i < n . The maps r, s extend to E ∗ := S n ≥ E n in an obvious way. A sink is a vertex v ∈ E with s − ( v ) = ∅ , a source is a vertex w ∈ E with r − ( w ) = ∅ however we will always assume there are no sources .A Cuntz-Krieger E -family in a C ∗ -algebra B consists of mutually orthogonal projections { p v : v ∈ E } and partial isometries { S e : e ∈ E } satisfying the Cuntz-Krieger relations S ∗ e S e = p r ( e ) for e ∈ E and p v = X { e : s ( e )= v } S e S ∗ e whenever v is not a sink.There is a universal C ∗ -algebra C ∗ ( E ) generated by a non-zero Cuntz-Krieger E -family { S e , p v } [11,Theorem 1.2] . A product S µ := S µ S µ . . . S µ n is non-zero precisely when µ = µ µ · · · µ n is a path in E n . The Cuntz-Krieger relations imply that words in { S e , S ∗ f } collapse to products of the form S µ S ∗ ν for µ, ν ∈ E ∗ satisfying r ( µ ) = r ( ν ) and we have(9) C ∗ ( E ) = span { S µ S ∗ ν : µ, ν ∈ E ∗ and r ( µ ) = r ( ν ) } . There is a canonical gauge action of T on A := C ∗ ( E ) determined on the generators via: γ z ( p v ) = p v and γ z ( S e ) = zS e . Because T is compact, averaging over γ with respect to normalised Haar measuregives a faithful expectation Φ from A onto the fixed-point algebra F = A γ :Φ( a ) := 12 π Z T γ z ( a ) dθ for a ∈ C ∗ ( E ) , z = e iθ . As described in [15], right multiplication by F makes A into a right (pre-Hilbert) F -module with innerproduct: ( a | b ) R := Φ( a ∗ b ) . Then X denotes the Hilbert F -module completion of A in the norm k a k X := k ( a | a ) R k F = k Φ( a ∗ a ) k F . For each k ∈ Z , the projection Φ k onto the k -th spectral subspace of the gauge action is defined byΦ k ( x ) = 12 π Z T z − k γ z ( x ) dθ, z = e iθ , x ∈ X. The generator of the gauge action on X , D = P k ∈ Z k Φ k , is determined on the generators of A = C ∗ ( E )by the formula D ( S α S ∗ β ) = ( | α | − | β | ) S α S ∗ β . The following result is proved in [15].
Proposition 6.1.
Let A be the graph C ∗ -algebra of a directed graph with no sources. Then ( X, D ) isan odd unbounded Kasparov A - F -module. The operator D has discrete spectrum, and commutes withleft multiplication by F ⊂ A . Set V = D (1 + D ) − / . Then ( X, V ) defines a class in KK ( A, F ) . We are going to investigate relations in K ( M ( F, A )). As graph algebras are generated by partialisometries in A with range and source in F , so K ( M ( F, A )) contains a lot of information about A and the underlying graph. The main result of Section 5 will give us more information. Proposition 6.2.
Let A be the graph C ∗ -algebra of a locally finite directed graph. Let α = α α · · · α | α | be a path in the graph, and S α the corresponding partial isometry in A . If µ is also a path let P µ = S µ S ∗ µ .Then in K ( M ( F, A )) we have the relations [ S α P µ ] = | α |− X j =1 [ S α j S α j +1 S α j +2 · · · S α n P µ S ∗ α n · · · S ∗ α j +2 S ∗ α j +1 ] + [ S α | α | P µ ] , [ S α S ∗ β ] = [ S α ] − [ S β ] , α, β paths . Proof.
This proceeds by induction on | α | . If | α | = 0 then [ S α ] = [ p r ( α ) ] = 0 and if | α | = 1, there isnothing to prove. So suppose the relation is true for all α with | α | < n . Let α be a path with | α | = n and write α = αα n where | α | = n −
1. Then[ S α P µ ] = [ S α S α n P µ ] = [ S α S α n P µ S ∗ α n S α n P µ ] = [ S α S α n P µ S ∗ α n ] + [ S α n P µ ] by Lemma 3.3= | α |− X [ S α j S α j +1 S α j +2 · · · S α n P µ S ∗ α n · · · S ∗ α j +2 S ∗ α j +1 ] + [ S α | α |− S α n P µ S ∗ α n ] + [ S α n P µ ] , the last line following by induction. The application of Lemma 3.3 requires( S α S α n P µ S ∗ α n ) ∗ ( S α S α n P µ S ∗ α n ) = S α n P µ S ∗ α n = ( S α n P µ )( S α n P µ ) ∗ . The second relation follows from Lemma 3.3 also, since S ∗ α S α = p r ( α ) = S ∗ β S β . (cid:3) Lemma 6.3.
Let A be the graph C ∗ -algebra of a locally finite directed graph E with no sources. Thenfor all edges e ∈ E , the class [ S e ] ∈ K ( M ( F, A )) is not zero. Similarly if r ( e ) = s ( α ) then [ S e P α ] = 0 .Proof. The assumptions on the graph ensure the existence of the Kasparov module ( X, D ) constructedfrom the gauge action. The pairing h [ S e P α ] , [( X, D )] i is given by [ S e P α S ∗ e Φ ] = [ S e P α S ∗ e ] ∈ K ( F ),where Φ is the kernel projection of D , whose range is the trivial F -module F . This class is nonzerosince F is an AF algebra, and so satisfies cancellation. (cid:3) Remark . The hypothesis of ‘no sources’ was introduced so that we could use the nonzero indexpairing to infer nonvanishing of the class [ S e P α ]. This restriction may be loosened provided we useother ways of deducing the nonvanishing. For instance, if the class [ P α ] − [ S e P α S ∗ e ] = ev ∗ ([ S e P α ]) = 0in K ( F ), then the class [ S e P α ] cannot be zero. On the other hand, if [ P α ] = [ S e P α S ∗ e ] in K ( F ), thensince F is AF, there exists a partial isometry v ∈ F such that S e P α S ∗ e = vv ∗ and P α = v ∗ v . Then u = 1 − P α + v ∗ S e P α is a unitary, and so defines a class in K ( A ). Since the map K ( A ) → K ( M ( F, A ))is an injection, and takes [ u ] to [ S e P α ], we would know that [ S e P α ] = 0 if we knew that [ u ] = 0. Corollary 6.4.
Let A be the graph C ∗ -algebra of a locally finite connected directed graph with nosources. Two nonzero classes [ S e P α ] , [ S f P α ] , with e, f edges in the graph and α an arbitrary path, areequal if and only if r ( e ) = r ( f ) . Two nonzero classes [ S e ] , [ S f ] , r ( e ) a sink, are equal, [ S e ] = [ S f ] , ifand only if r ( e ) = r ( f ) .Proof. Suppose that r ( e ) = r ( f ), and that [ S e P α ] = 0 (otherwise there is nothing to prove). Then as S e P α S ∗ f ∈ F we have 0 = [ S e P α S ∗ f ] = [ S e P α ] − [ S f P α ] , by Lemma 3.3. Conversely, if r ( e ) = r ( f ) at least one of these classes is zero.For the second statement we observe that if r ( e ) = r ( f ) then S e S ∗ f is nonzero, and then [ S e ] =[ S e S ∗ f ] + [ S f ] = [ S f ] by Lemma 3.3. If r ( e ) = r ( f ), we suppose [ S e ] = [ S f ], for a contradiction, and NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 37 compute the index pairing with the Kasparov module ( X, D ) constructed from the gauge action. Thepairing is given by h [ S e ] , [( X, D )] i = − [ S e S ∗ e ] = − [ S f S ∗ f ] = h [ S f ] , [( X, D )] i . Hence the class of S e S ∗ e in K ( F ) ( F is the fixed point algebra) coincides with the class of S f S ∗ f .Since F is an AF algebra, there exists a partial isometry v ∈ span { S µ S ∗ ν : | µ | = | ν |} such that S e S ∗ e = vS f S ∗ f v ∗ . Thus p r ( e ) = S ∗ e vS f S ∗ f v ∗ S e = X j c j c k S ∗ e S µ j S ∗ ν j S f S ∗ f S ν k S ∗ µ k S e . Here the paths µ j start from s ( e ) and end at some vertex v j , while the corresponding path ν j startsfrom s ( f ) and ends at the same vertex v j . Moreover there is at least one path µ j with S ∗ e S µ j = 0 so µ j = eµ j · · · µ j k , where | µ j | = k . However, r ( e ) is a sink, so any such path is of the form µ j = e . Thisforces the length of the corresponding ν j to be 1, and ν j = f . The only way the product S µ j S ∗ ν j = S e S ∗ f can now be non-zero is if r ( e ) = r ( f ), contradicting our assumption. (cid:3) Corollary 6.5.
Let A be the graph C ∗ -algebra of a locally finite connected directed graph with nosources. Then if two partial isometries of the form [ S e ] , [ S f ] satisfy [ S e ] = [ S f ] ∈ K ( M ( F, A )) thenthere exists a partial isometry ρ in F such that ρS e = S f and ρ ∗ ρS e = S e = ρ ∗ S f .Proof. The required partial isometry ρ is S f S ∗ e . The remaining statements are immediate. (cid:3) Lemma 6.6.
Let E be a row-finite directed graph. Then the group K ( M ( C ∗ ( E ) γ , C ∗ ( E ))) is generatedby the classes [ S e P α ] , where e is an edge and α is a path.Proof. Let [ v ] ∈ K ( M ( C ∗ ( E ) γ , C ∗ ( E ))) and consider ev ∗ [ v ] = [ v ∗ v ] − [ vv ∗ ] ∈ K ( C ∗ ( E ) γ ) . Now K ( C ∗ ( E ) γ ) is generated by the classes [ p µ ], p µ = S µ S ∗ µ , where µ ∈ E ∗ is a path, [14]. As C ∗ ( E ) γ is an AF algebra, there are partial isometries W, Z over C ∗ ( E ) γ such that(10) W ∗ W = v ∗ v, W W ∗ = X j p µ j , ZZ ∗ = vv ∗ , Z ∗ Z = X k p ν k , and [ v ] = [ Z ∗ vW ∗ ]. The latter follows because Z, W are partial isometries over F and so representzero, while [ Z ∗ vW ∗ ] = [ Z ∗ ]+[ v ]+[ W ∗ ]. In Equation (10) the sums are necessarily orthogonal, and maybe in a matrix algebra over C ∗ ( E ) γ , and some zeroes (place-holders to make the matrix dimensionsequal) may have been omitted from the sums. Observe that ev ∗ [ Z ∗ vW ∗ ] = P k [ p ν k ] − P j [ p µ j ]. Byconsidering p ν k Z ∗ vW ∗ p µ j we may suppose without loss of generality that we have only one summandso that W W ∗ = p µ and Z ∗ Z = p ν . Then ev ∗ [ Z ∗ vW ∗ S µ S ∗ ν ] = [ p ν ] − [ p ν ] = 0 . Hence [ v ] = [ Z ∗ vW ∗ ] = [ S ν S ∗ µ ] modulo the image of i ∗ , and Lemma 6.2 completes the proof for[ v ] Image( i ∗ ). Observe that S ν S ∗ µ = 0 (and so r ( µ ) = r ( ν )) is a consequence.In the case ev ∗ [ v ] = 0, so that [ v ] ∈ Image( i ∗ ) we observe that there is a partial isometry X over C ∗ ( E ) γ such that X ∗ X = v ∗ v and XX ∗ = vv ∗ so that 1 − v ∗ v + X ∗ v is unitary. Then, again since allpartial isometries are over F ,[ v ] = [ W X ∗ vW ∗ ] = [ W X ∗ ZZ ∗ vW ∗ ] = [ W X ∗ ZS ν S ∗ µ ] = i ∗ [1 − p µ + W X ∗ ZS ν S ∗ µ ]gives a unitary representative of v . Since i ∗ [1 − p µ + W X ∗ ZS ν S ∗ µ ] = [ S ν S ∗ µ ], Lemma 6.2 completes theproof. (cid:3) The structure of K ( M ( F, A )) is even simpler.
Lemma 6.7. If E is a row-finite directed graph, A = C ∗ ( E ) and F = C ∗ ( E ) γ , then K ( M ( F, A )) = 0 .Proof.
The exact sequence 0 → A ⊗ C (0 , → M ( F, A ) → F → K ( F ) = 0 yields(11) 0 → K ( A ) → K ( M ( F, A )) ev ∗ → K ( F ) → K ( A ) → K ( M ( F, A )) → . By Lemma 3.1, the map K ( F ) → K ( A ) is induced (up to sign and Bott periodicity) by inclusion j : F → A . This map is surjective on K by [14][Lemma 4.2.2], and so K ( M ( F, A )) = 0. (cid:3)
In [14], the K -theory of a graph algebra C ∗ ( E ), where E has no sources or sinks, was computed asthe kernel ( K ) and cokernel ( K ) of the map given by the vertex matrix on Z E (there are subtletieswhen sinks are involved). The proof of this result involves the dual of the gauge action and thePimsner-Voiculescu exact sequence for crossed products. In Equation (11), we see the K -theory againexpressed as the kernel and cokernel of a map, but this time it arises with no serious effort. Thedifference of course is that the groups K ( M ( F, A )) and K ( F ) are in general harder to compute.While the map ev ∗ : K ( M ( F, A )) → K ( F ) is neither one-to-one nor onto in general, we can deducethat the two groups K ( M ( F, A )) and K ( F ) are in fact isomorphic in a wide range of examples. Welet ( ˆ X, ˆ D ) be the APS Kasparov module arising from the Kasparov module ( X, D ). Proposition 6.8.
Let A be the graph C ∗ -algebra of a locally finite connected directed graph with nosources and no sinks. Then the map Index ˆ D : K ( M ( F, A )) → K ( F ) given by the Kasparov productwith the Kasparov module of the gauge action is an isomorphism.Proof. First the index map is a well-defined homomorphism, [10]. We begin by showing that the indexmap is one-to-one. So suppose that we have edges e, g and paths α, β in our graph (with no range asink), and suppose that Index ˆ D ([ S e P α ]) = Index ˆ D ([ S g P β ]). A simple computation using Theorem 5.1yields Index ˆ D ([ S e P α ]) = [ S e P α S ∗ e ] = [ S g P β S ∗ g ] = Index ˆ D ([ S g P β ]) . As F is an AF algebra, we can find a partial isometry v in F such that S e P α S ∗ e = vS g P β S ∗ g v ∗ . Then setting w = P α S ∗ e vS g P β = 0 we have P α = ww ∗ = wP β w ∗ and P β = w ∗ w = w ∗ P α w. We will use Lemma 3.3 below and need to check that some partial isometries have the same sourceprojections. First observe that ( S e P α wP β ) ∗ ( S e P α wP β ) = P β = w ∗ w , so[ S e P α ] = [ S e P α wP β w ∗ ] = [ S e P α wP β ] + [ w ∗ ] = [ S e P α wP β ] = [ S e P α S ∗ e vS g P β ] , the second last last equality following since w is a partial isometry in F . Now since ( S g P β )( S g P β ) ∗ = S g P β S ∗ g and ( S e P α S ∗ e v ) ∗ ( S e P α S ∗ e v ) = S g P β S ∗ g , we can apply Lemma 3.3 again to find[ S e P α ] = [ S e P α S ∗ e vS g P β ] = [ S e P α S ∗ e v ] + [ S g P β ] = [ S g P β ] . Thus Index ˆ D is one-to-one. Now supposing that our graph has no sinks, every class in K ( F ) is asum of classes [ p µ ] = [ S µ S ∗ µ ], where µ is a path in the graph of length at least one. For a given µ = µ · · · µ | µ | , define µ = µ · · · µ | µ | . Then it is straightforward to check thatIndex ˆ D ([ S µ S ∗ µ ]) = [ p µ ] . Hence the index map is onto and we are done. (cid:3)
NONCOMMUTATIVE ATIYAH-PATODI-SINGER INDEX THEOREM IN KK-THEORY 39
Observe that this does not mean that the K -theory of the graph algebra is zero! The evaluation mapand the index map are very different. For the Cuntz algebra O n , n ≥
2, for example, the fixed pointalgebra has K -theory K ( F ) ∼ = Z [1 /n ] and so we have ev ∗ ([ S µ ]) = [1] − [ S µ S ∗ µ ] ∼ − n | µ | = ( n | µ | −
1) 1 n | µ | , with ker( ev ∗ ) ∼ = K ( O n ) = 0 and coker( ev ∗ ) ∼ = K ( O n ) = Z n − . The index map gives usIndex ˆ D ([ S µ ]) = | µ |− X j =0 [ S µ S ∗ µ Φ j ] . This equality follows from Theorem 5.1, and to determine the right hand side more explicitly, set µ = µ j +1 · · · µ | µ | and define the partial isometry W = S µ S ∗ µ Φ . Then W W ∗ = S µ S ∗ µ Φ j and W ∗ W = S µ S ∗ µ Φ . Thus in K ( F ) we haveIndex ˆ D ([ S µ ]) = | µ |− X j =0 [ S µ S ∗ µ Φ j ] = | µ |− X j =0 [ S µ S ∗ µ Φ ] = | µ |− X j =0 [ S µ S ∗ µ ] ∼ | µ |− X j =0 n − ( | µ |− j ) = n | µ | − n − ! n | µ | . The evaluation map and the mapping cone exact sequence gives us K ( M ( O γn , O n )) ∼ = ( n − Z [1 /n ](those polynomials all of whose coefficients have a factor of n −
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