Higher spectral flow and an entire bivariant JLO cocycle

1 HIGHER SPECTRAL FLO W AND AN ENTIRE BIV ARIANT JLO COCYCLE MOULA Y-T AHAR BENAMEUR AND ALA N L. CAREY Abstract. F or a single Dirac op erator on a closed manifold the cocycle introduced b y Jaffe-Lesniewski- Osterwa lder [19 ] (abbreviated here to JLO), is a representat ive of Connes’ Chern c haracter map f rom the K-theory of the algebra of smooth functions on the manifold to its en tire cyclic cohomology . Give n a smo oth fibration of cl osed manifolds and a fami ly of generalized Dirac op erators along the fib ers, we define in this paper an asso ciated biv arian t JLO cocycle. W e then pro ve that, for an y ℓ ≥ 0, our biv ariant JLO cocycle is en tire when we endow smoo oth functions on the total manifold with the C ℓ +1 topology and f unctions on the base manifold with the C ℓ topology . As a b y-pro duct of our theorem, we deduce that the biv ariant JLO cocycle is entire for the F r´ ec het smo oth topologies. W e then prov e that our JLO biv ari an t co cycle computes the Chern character of the Dai-Zhang hi gher sp ectral flow. Contents Int ro duction 1 Statement of the main results 3 1. Preliminary results 4 1.1. Connections 4 1.2. Biv aria nt co chains 6 1.3. The heat semigroup and Duhamel 8 2. Multilinear functionals and identit ies 10 3. The biv ariant JLO co cycle is entire 12 3.1. Review of biv ariant entire cyc lic homology 12 3.2. Statement of the main theorem 13 4. Pro of of Theorem 3.5 14 4.1. Estimates 14 4.2. Last steps of the pr o of of the theor em 17 4.3. More general sup e r connections and transgres s ion 22 5. Compatibility with the higher sp ectral flow 25 5.1. Higher sp ectral flow 25 5.2. Second theorem and reduction to the third theor e m 26 5.3. Pro o f of the third theorem 28 References 31 Introduction Our ob jective in this pap er is to give a biv a riant ent ire Chern c haracter sufficiently general to encompass the index theo rem for families o f g eneralized Dira c opera tors. In other words we ma ke ex plicit the lo ng held view tha t the B ismut formalism may be incor po rated into noncommut ative geometry . F rom the authors’ po int o f view this question arose from a discussion at O b e rwolfac h (we thank Masoud Khalkha li a nd Alain Connes for comment s). A num ber of results in different alg ebraic and/or geometric situations hav e b een 1 Both authors ac kno wledge the financial supp ort of the Australian Researc h Council and of the PICS, Progr` es en Analyse G ´ eom´ etrique et Applications of the CNRS. 1 2 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 obtained previously fo r instance in [22, 25, 30, 18, 26]. (There is also the related question of the biv ar iant version of the Connes-Mos c ovici res idue cocyc le but we defer that to another place.) The first issue is to c ho ose a biv aria nt framework for this problem. In Meyer’s thesis [24] w e hav e found an appro priate for malism. Using Meyer’s idea s we define a biv ar iant JLO (Jaffe-Lesniewsk i-Osterwalder [19]) co cy cle that encompasses the lo cal index theor em for families [8, 9 ]. Recall that the ordinary J L O co cycle is a representativ e of Connes’ Chern c haracter map from the K - theory o f an algebra to its entire cyclic coho mology [14]. As explained in [10] it may be used to deduce the A tiyah-Singer lo ca l index theor em. Our biv aria nt JLO co cycle generalizes this r e s ult to the situation of families o f gener a lized Dirac op era tors asso ciated to a fibration. There are several in termediate results that are required to rea ch our ob jectiv e. O ur first main result is to prov e that our biv ar iant JLO co cycle is entire in the sense of our adaptation of the for ma lism of [24], when one consider s the C ℓ top ology on the a lgebra of smo o th functions on the base, and the C ℓ +1 top ology on the algebra of smo oth functions on the total space of the fibration. W e thank Ralf Me yer for his helpful comments on our approach to this result. T o describ e our further r e sults we need some notation. Throughout we shall consider a lo cally trivial fibration F → M π → B of clos ed manifolds endowed with smoo th metrics. F ollowing [8], w e fix an hermitian vector bundle E → M whose fibers are mo dules o ver the Clifford a lg ebra of the fib e rwise tangent bundle T v M = Ker( π ∗ ). While we for mu late our initial r esults in a general way our main in terest lies in the ca se of o dd dimensional fib e r s for E as this is not as well understo od as is the e ven dimensional situation. W e follow [9, 8] and introduce o n E a quasiconnectio n ∇ . W e cho ose a family of generalized Dirac ope r ators parametrise d smo othly by B denoted D , and different sup er connections as in [9]. They will b e fo r us given by A σ ( A ) := B σ + A, where A is a zer o-th or der fib erwise σ -pseudo differential op er ator with co efficients in differen tial fo rms of po sitive degree (in applications ≥ 2) on B . Here, the superco nnection B σ is defined by B σ = ∇ + σ D with σ being a Clifford v ariable as in Quillen’s work [2 7]. W e fo cus on A σ ( A ), B σ or on super connections obtained b y metr ic rescalings. In order to simplify the ex po sition o f our r e sults, w e r estrict o urselves to the case A (0) = B σ and briefly explain later ho w the pro ofs extend to the general c a se. Our main a pplication is to co nnect our biv ariant Chern character with the Dai-Zhang higher sp ectral flow. The method we use draws on some ideas in [17], [12] and [27] o n s up er connections and the JLO co cycle . W e sho w using [1 5] that the biv ariant JLO Cher n character in the case o f the Bism ut sup erconnec tion gives the Bism ut lo ca l fo r mula that Dai- Zhang o bserve computes higher s pe ctral flow. (The use of the Bismut sup e rconnection av oids the complicatio ns that ent er in to our biv ariant formula when ∇ 2 6 = 0.) Thu s our appro ach explains the Dai-Z ha ng results in terms of biv a riant cyclic cohomology . In the s ing le op erator case these r esults are known fro m Connes [13] and Getzler [1 7]. W e choose here to extend Getzler ’s metho d to dea l with families . In [26], similar issues a re discussed for in teresting nonco mm utative biv a riant situations, but only encompassing in the commutativ e case trivia l fibr a tions and fla t quasiconnectio ns ∇ . Our motiv ation for giving a deta ile d trea tmen t of the ge ne r al commutative cas e ha s to do with some a pplications in noncommut ative settings, see for instance [4], [5]. W e a re aware of a need to use our p oint of view and results in current w ork of colleague s A. Goro khovsky , J .- M. Lescur e and B.-L. W ang . Thus we hav e written the expo sition so that it will a da pt immedia tely to foliations and to sp ectr a l flo w fo r t wisted families. The c onceptual framework for our results is c a ptured by the following commutativ e squar e. JLO AND S PECTRAL FLOW FOR F AMILIES 3 K 1 ( M ) ❄ Ch SF( D, · ) ✲ JLO( D ) ✲ K 0 ( B ) ❄ ch H E 1 ( C ∞ ( M )) H even ( B , C ) W e digre s s to explain the notation in this diag ram. The left vertical arrow repr esents the entire Chern character, while the rig ht v ertical arrow represents the usual Chern c haracter with appropriate normaliza- tions. Our family of generalized Dirac op e rators pa rametrised by B defines an element [ D ] of K K 1 ( M , B ) and the top horizo ntal arrow, which is the higher sp e ctral flow map defined in [15] when the K 1 class of D is trivial, is still well defined in general as the Kasparov pro duct ∩ [ D ] b y [ D ]. Finally JLO( D ) is o ur en tire biv ariant Chern character which ta kes v alues in even de Rham cohomology of B . Statemen t of the main resul ts. The biv aria nt JLO co chain is defined by the sequence ( ψ n ) n of functionals which, for ( f 0 , · · · , f n ) ∈ C ∞ ( M ) n +1 , are g iven by the formula ψ n ( f 0 , · · · , f n ) := hh f 0 , [ B σ , f 1 ] , · · · , [ B σ , f n ] ii B σ ∈ Ω ∗ ( B ) , where the m ultilinea r functional on the right ha nd side is a generalized JLO functional whose precis e form is explained at the b eginning o f Se c tio n 2. O ur main results are then as follows. Theorem 3.5 . We assu me that the fib ers of our fibr ation ar e o dd dimensional manifold s. F or any ℓ ≥ 0 , ψ = ( ψ 2 n +1 ) n ≥ 0 is an ℓ -entir e bivariant c o cycle in t he sense made pr e cise in D efi n ition 3.3. This me ans that ψ is a b ounde d morphism fr om the entir e b ornolo gic al c ompletion of the universal differ ent ial algebr a of C ∞ ( M ) to the algebr a of s m o oth differ ent ial forms on B endowe d with the natur al b ornolo gy. Theorem 5 .5 . Assume t hat the index class of D in K 1 ( B ) is trivial so that the higher sp e ctr al flow SF( D, U ) is wel l define d, for any U ∈ GL N ( C ∞ ( M )) . Then the fol lowing r elation holds in t he even de Rha m c ohomolo gy of the b ase manifold B : 1 √ π < JLO( D ) , ch( U ) > = ch (SF( D , U )) . It is p oss ible to deduce from our computations a mo r e precise statement o n for ms rather than classes. W e see also as a co rollar y of our arguments, and of the main result o f [15], that JLO( D ) coincides with the top ologica l map H odd ( M ) → H even ( B ) given by ω 7− → Z M /B ω ∧ ˆ A ( T M | B ) . T o understand the structure of our expo sition we no w expand on the List of Conten ts. Section 1 gives the differential g eometric fra mework: f amilies of generaliz ed Dirac op er ators, connections a nd biv ar iant functionals. In Section 2 we intro duce our biv ariant JLO mult ilinear functionals and discuss the identities they s atisfy essent ially following [16] but a dapted to families. Our first o b jectiv e, to understand the entire prop erty for biv a riant JLO, b egins in Section 3. W e summarise Meyer’s p oint of view at the b eginning of this Section for the reader’s conv enience and then state our main theorem. Sec tio n 4 co ntains the pro of: the argument is a series of estimates that es tablish that our JLO is e nt ire in the sense o f Meyer (and hence ent ire). In Section 5 w e e stablish the commutativit y of the diagram a b ov e . A cknow le dgements. The a uthors are very grateful for a car eful reading by a referee who hig hlighted some confusions a nd g aps in the original exp os ition. W e have als o b enefitted from dis cussions with o ur co lle a gues, A. Connes, J. Cuntz, A. Gorokhovsky , J. Heitsch, M. Kha lk ahli, R. Meyer, D. Perrot, M. Puschnigg, A. Rennie, G. Y u, to whom we expr ess our appr eciation. This work was progressed while the first author was visiting the MSI in the Australian National Univ ersity , the sec o nd author was visiting the Lab oratoir e de Math´ ematiques et Applications in the Universit ´ e Paul V erlaine-Metz, and b oth author s were visiting the 4 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 Mathematisches F o rsch ung sinstitut Ober wolfac h a nd the Hausdor ff Institute for Mathematics . Both a utho r s are most grateful for the warm hospitalit y and g e ne r ous s uppo rt of their ho sts. 1. Preliminar y resul ts 1.1. Connections . W e b eg in by introducing some further nota tion a nd a genera l framework for our dis- cussion. As ab ov e w e denote b y T v M the fib erwise tang ent bundle T v M := Ker( π ∗ ) ⊂ T M . T he n we are assuming there is a Cliffo r d homomorphism of a lg ebra bundles c : C l ( T v M ⊗ C ) − → End( E ) w ith c ( ξ ) 2 = | ξ | 2 for ξ ∈ T v M , where C l ( T v M ⊗ C ) denotes the Clifford algebr a bundle a sso ciated with the hermitian bundle T v M ⊗ C , and End( E ) is the a lgebra bundle o f endomorphisms of E . W e as s ume as usual tha t E is endow ed with a Cliffor d connection ∇ E and consider the Dira c o p erator D asso ciated with this connectio n. Then D is a fib erwise first order differential ope r ator acting on s mo oth sections of E and ca n be regarded as a family of elliptic op erator s a long the fib er s smo o thly parametrized by the elements of the ba se manifold B , i.e. D = ( D b ) b ∈ B where D b : C ∞ ( M b , E | M b ) → C ∞ ( M b , E | M b ) is an es sentially self-adjoint g eneralized Dir a c op erato r. Notice that w e are using s e lf- a djoint op erator s while the authors of [8] use s kew-adjoin t op erato r s. Using the metr ic, we fix the horizontal distribution H = ( T v M ) ⊥ so that T M = H ⊕ T v M a nd for any b ∈ B , π ∗ ,m : H m → T π ( m ) B is a linear isomorphism . The dual vector bundle π ∗ T ∗ B can b e identified with the subbundle of T ∗ M consis ting of forms that v anish on vertical vectors. Using the splitting given by H , we deduce the existence of a re s triction pro jection  : T ∗ M → π ∗ T ∗ B . This pro jection extends to exterior powers and we obtain, in the obvious notation,  : C ∞ ( M , E ⊗ Λ T ∗ M ) → C ∞ ( M , E ⊗ Λ π ∗ T ∗ B ) . Notice that C ∞ ( M , E ⊗ Λ π ∗ T ∗ B ) is a mo dule o ver the algebra of differential forms on the ba se manifold B . More pr ecisely , this module str ucture is obta ined using the pull-back map asso ciated with the pro jection π . So if we denote by ξ ω the action of a differential form ω on B , on a smo oth section ξ of E ⊗ Λ π ∗ T ∗ B , then ( ξ ω )( m ) := ξ ( m ) ∧ ω ( π ( m )) , or ξ ω := ξ ∧ π ∗ ω . Here ∧ is denoting the usual action of differential forms on the exterior algebra . F or any m ∈ Z , w e denote by Ψ m ( M | B , E ) the s pa ce of (1 -step p olyho mo geneous) class ical pseudo dif- ferential op er ators of o rder m , a cting along the fib ers of π : M → B , see [2]. The lo cal co efficients of such op erator s are thus smo oth in the base v ar iables a nd the s pa ce Ψ m ( M | B ; E ) is a mo dule ov er the alge- bra C ∞ ( B ) of smo oth functions on B . W e shall also need the space of such o pe rators with co efficients in differential forms on the base. More precisely , we s e t ψ m ( M | B , E ; Λ ∗ B ) = Ψ m ( M | B , E ) b ⊗ C ∞ ( B ) Ω ∗ ( B ) . According to [2], the space Ψ m ( M | B , E ; Λ ∗ B ) can b e endow ed with a co mplete smo oth top o lo gy as a pro jectiv e tens o r pro duct o f suc h top olog ical spaces. An element o f ψ m ( M | B , E ; Λ ∗ B ) is th us equiv aria nt for the action of the alg ebra Ω ∗ ( B ) of smo oth differential for ms on B . As a cons equence we can co mp o s e an element P of ψ m ( M | B , E ; Λ k B ) with an element Q of ψ m ′ ( M | B , E ; Λ h B ) to get an element QP of ψ m + m ′ ( M | B , E ; Λ k + h B ). W e shall denote b y ψ ∞ ( M | B , E ; Λ ∗ B ) the alg e bra o btained in this way , i.e. ψ ∞ ( M | B , E ; Λ ∗ B ) := [ m ∈ Z ψ m ( M | B , E ; Λ ∗ B ) . As we sha ll see, the order of the pseudo diff erential op erato rs will not b e in volv e d in the gradings used in the seq ue l. In particular, for us , ψ ∞ ( M | B , E ; Λ ∗ B ) is Z 2 -graded by the pa rity of the degr ee of the forms on B . W e also denote by ψ −∞ ( M | B , E ; Λ ∗ B ) the ideal of fib erwise smo o thing op er ators with coefficie nt s in differential forms on B , i.e. ψ −∞ ( M | B , E ; Λ ∗ B ) := \ m ∈ Z ψ m ( M | B , E ; Λ ∗ B ) . JLO AND S PECTRAL FLOW FOR F AMILIES 5 Int ro duce the fib er pro duct G = M × B M := { ( m, m ′ ) ∈ M × M , π ( m ) = π ( m ′ ) } , which is a smo o th group oid. By the fib er wise Schw ar tz theorem applied to fib erwise smo o thing op era tors, ψ −∞ ( M | B , E ; Λ ∗ B ) can and will b e iden tified with the conv olutio n a lgebra of smo oth sections o f the bundle Hom( E ) ⊗ Λ ∗ T ∗ B ov er G whose fiber is (Hom( E ) ⊗ Λ ∗ T ∗ B ) m,m ′ := Hom( E m ′ , E m ) ⊗ Λ ∗ T ∗ π ( m )= π ( m ′ ) B . Definition 1.1. We define t he op er ator ∇ by ∇ :=  ◦ ∇ E : C ∞ ( M , E ⊗ Λ π ∗ T ∗ B ) → C ∞ ( M , E ⊗ Λ T ∗ M ) → C ∞ ( M , E ⊗ Λ π ∗ T ∗ B ) , Then ∇ is c al le d a quasi-c onne ction. Remark 1.2. In Bismut’s viewp oint ∇ is a c onne ction on an infinite dimensional F r´ echet bund le on B . Lemma 1.3 . The qu asi-c onne ction ∇ incr e ases the de gr e e of the forms by one and satisfies t he L eibniz rule ∇ ( ξ ω ) = ( − 1) ∂ ξ ξ d B ω + ( ∇ ξ ) ω , wher e ∂ ξ is the form de gr e e of the se ction ξ . Mor e over, the cu rvatur e op er ator ∇ 2 of ∇ is a fib erwise first or der differ ential op er ator with c o efficients in 2 -forms on the b ase manifold B . Pr o of. This argument is ana logous to one introduced in [5]. As ∇ E is a connection and choo sing the mo dule action on the r ight, we have: ∇ E ( ξ ω ) = ( ∇ E ξ ) ∧ π ∗ ω + ( − 1) ∂ ξ ξ ∧ dπ ∗ ω = ( ∇ E ξ ) ω + ( − 1) ∂ ξ ξ ∧ π ∗ d B ω = ( ∇ E ξ ) ω + ( − 1) ∂ ξ ξ d B ω . Therefore, applying  , w e o btain ∇ ( ξ ω ) =  ( ∇ E ξ ) ω + ( − 1) ∂ ξ ξ d B ω = ( ∇ ξ ) ω + ( − 1 ) ∂ ξ ξ d B ω . Therefore, we hav e by the clas sical computation ∇ 2 ( ξ ω ) = ( ∇ 2 ξ ) ω , i.e. ∇ 2 is Ω ∗ ( B ) linear, and hence it is a fib e rwise differential op er ator with co efficients in 2-forms on the base. B y reducing to lo cal cordina tes on the ba se manifold B , one computes ∇ 2 using the lo cal expression of ∇ E . It is then easy to chec k that ∇ 2 is indeed a (fiberwise) firs t o rder differential op era to r, see also [5] for more details.  W e shall say that an Ω ∗ ( B )-linea r map on C ∞ ( M ; E ⊗ π ∗ Λ ∗ B ) has degree k ∈ Z , if it increases the form degree of the sections by k . Hence, such a map sends C ∞ ( M ; E ⊗ π ∗ Λ h B ) into C ∞ ( M ; E ⊗ π ∗ Λ h + k B ). W e then set ∂ T = k and deno te by C k the filtra tion obtained in this wa y , that is C k is the space of such maps with degree ≤ k , which are Ω ∗ ( B )-linea r ope r ators. Lemma 1.4. L et T b e an Ω ∗ ( B ) -line ar map of de gr e e k , acting on C ∞ ( M ; E ⊗ π ∗ Λ ∗ B ) . Then t he gr ade d c ommutator ∂ ( T ) = [ ∇ , T ] := ∇ ◦ T − ( − 1) k T ◦ ∇ , is an Ω ∗ ( B ) -line ar map which b elongs to C k +1 . Pr o of. Fix T ∈ C k . F or ξ ∈ C ∞ ( M ; E ⊗ π ∗ Λ h B ) and ω ∈ Ω ∗ ( B ), w e ca n wr ite ∂ ( T )( ξ ω ) = ∇ (( T ξ ) ω ) − ( − 1) k T ( ∇ ξ ) ω − (1) k + h ( T ξ ) d B ω = ∇ ( T ξ ) ω + ( − 1 ) h + k ( T ξ ) d B ω − ( − 1) k T ( ∇ ξ ) ω − (1) k + h ( T ξ ) d B ω = ( ∂ ( T ) ξ ) ω.  Prop ositi on 1.5. The derivation ∂ : C ∗ → C ∗ pr eserves the subsp ac e ψ −∞ ( M | B , E ; Λ ∗ B ) . Mor e pr e cisely, the derivation ∂ pr eserves e ach ψ h ( M | B , E ; Λ ∗ B ) for h ∈ Z . 6 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 Pr o of. A clas s ical computation shows that, in lo ca l co or dinates ov er a small op en set V of M , the quasi- connection ∇ has the ex pr ession ∇ = d ν ⊗ I N + ω , with ω ∈ M N ( C ∞ ( V , π ∗ ( T ∗ B ))) , where we have used a v ector bundle iso morphism E | V → V × C N and d ν denotes the transverse de Rham deriv ative (in the direction π ∗ (Λ T ∗ B )) g iven with o bvious notations by d ν =  ◦ d. T aking co mm utators with the 0-th or der term ω clearly prese rves ψ h ( V | B , E ; Λ ∗ B ), since ω belo ng s to ψ 0 ( V | B , E ; Λ 1 B ). Using a trivialization ( x ; b ) = ( x 1 , · · · , x p ; b 1 , · · · , b q ) : V − → R p × R q . of the fibration V → U and the op en set U ⊂ B , o ne finds tha t there exists A ∈ M q,p ( C ∞ c ( R n )) suc h tha t d ν ( f ) = q X i =1 [ ∂ f ∂ b i + p X j =1 A ij ∂ f ∂ x j ] db i . The ex pr ession P q i =1 db i P p j =1 A ij ∂ ∂ x j is an element o f ψ 1 ( V | B ; Λ 1 B ), a sca lar op era to r. Therefore , when tensored with the identit y of C N it has a diago nal matrix as fib erwise principal s ymbol. Such a diagonal matrix graded comm utes w ith the principal symbol of an y e lement of ψ h ( V | B , E ; Λ ∗ B ). So , the commu- tator with this op erato r prese rves the order of the pseudo differential oper a tors. It th us remains to com- pute the co mmut ator of d B = P q i =1 db i ∂ ∂ b i with an element of ψ h ( V | B , E ; Λ ∗ B ). Since the elements P o f ψ h ( M | B , E ; Λ ∗ B ) ar e Ω ∗ ( B )-linea r, w e can re strict to P ∈ ψ h ( M | B , E ) and by pseudolo cality , we can even assume that P ∈ ψ h ( V | B , V × C N ) is g iven in the lo cal co ordinates ( x, b ) by P ( f )( x ; b ) = 1 (2 π ) p/ 2 Z V b × R p a ( x ; b ; ξ ) e i ( x − x ′ ) ξ f ( x ′ ; b ) dx ′ dξ , where f ∈ C ∞ c ( V , C N ) and a is the lo cal total symbol of the classical fib er wise pseudodiffer e ntial op erator P , an N × N matrix. A simple computation shows tha t the commutator [ d B ⊗ I N , P ] is given b y the same lo cal formula but with a replaced by the ma trix d B ( a ). Since this latter matrix is a classica l sy mbol o f or der h , the pro of is co mplete.  1.2. Biv arian t co c hains. As b efor e we ar e considering a smo oth lo cally trivial fibration π : M → B o f closed ma nifolds, together with the fib er wise ge ne r alized Dirac op e r ator D acting on the smoo th sections of the Clifford bundle E ov er M . The dimensio n o f the fib ers is denoted by p and the dimension of the base is p ′ . Recall that Ω ∗ ( B , E ) is the space of smo oth sections over M o f the bundle E ⊗ π ∗ Λ ∗ T ∗ B . This is a graded mo dule ov er the g r aded Grassmann algebr a of differential for ms o n the base manifold B . W e defined in Section 1.1 the quasi- connection ∇ a sso ciated with a connectio n o n E and the choice of a ho rizontal distribution H . Our biv aria nt ( n, k )-cochains are linear maps f : B n = A ⊗ n → L k where L = ⊕ k L k is a gra ded v ector space (or a graded mo dule ov er some algebra ) or a graded algebr a that will o ften b e endowed with a ‘connection’. Mor e precis ely , we are int erested in the graded spa ces L = Ω ∗ ( B ; E ) , L = Ω ∗ ( B ) , L −∞ = ψ −∞ ( M | B , E ; Λ ∗ B ) a nd L = L ( E ) . Here ψ −∞ ( M | B , E ; Λ ∗ B ) is the algebra of fib erw is e smo othing op er ators with co efficien ts in for ms. The space E is the Hilb ert mo dule over the C ∗ -algebra C (Λ ∗ T ∗ B ), o f contin uous sectio ns of Λ ∗ T ∗ B ov er B , which is the completion o f the s mo oth sections ov er M of the bundle E ⊗ π ∗ Λ ∗ T ∗ B . (F or background on Hilber t C ∗ mo dules see [21].) W e c ho ose connections on these g raded spaces (except for the las t o ne ) ∇ , d B = de Rham different ial and [ ∇ , · ] resp ectively . Note that we us e the conv en tion that all the commutators are gra ded ones. The space Ho m( B , L ) will then b e bi-graded and we shall use the total gr ading for the commutators. F or any P ∈ ψ h ( M | B , E ; Λ ∗ B ) and any b ∈ B , the o p er ator P b belo ngs to ψ h ( M b , E | M b ) ⊗ Λ ∗ ( T ∗ b B ). Ther efore, for h ≤ 0, P b extends to a Λ ∗ ( T ∗ b B )-line a r bo unded o per ator of the Hilbert space L 2 ( M b , E | M b ) ⊗ Λ ∗ ( T ∗ b B ), hence an element of B ( L 2 ( M b , E | M b )) ⊗ Λ ∗ ( T ∗ b B ) . JLO AND S PECTRAL FLOW FOR F AMILIES 7 Next, using a basis of the finite dimensio nal vector space Λ ∗ ( T ∗ b B )), we define a Λ ∗ ( T ∗ b B )-v alued graded trace τ : L 1 ( L 2 ( M b , E | M b )) ⊗ Λ ∗ ( T ∗ b B ) . − → Λ ∗ ( T ∗ b B ) , where L 1 ( L 2 ( M b , E | M b )) is the usual ideal of trace class op era to rs on the Hilbe r t space L 2 ( M b , E | M b ). The expres sion “graded tra ce” means that τ v anishes on gr aded commutators, the gra ding being pro duced by the de g ree of the for ms in Λ ∗ ( T ∗ b B ). The class ical theory o f pseudo differential o pe r ators shows that ψ h ( M b , E | M b ) ⊂ L 1 ( L 2 ( M b , E | M b )), for h < − p . Putting these traces together , we inherit a gr aded tr ace τ : L h = ψ h ( M | B , E ; Λ ∗ B ) − → Ω ∗ ( B ) for any h < − p. W e shall for simplicit y restrict τ to fiberwise smo othing op era to rs a nd o nly consider τ : L −∞ = ψ −∞ ( M | B , E ; Λ ∗ B ) − → Ω ∗ ( B ) . Lemma 1.6. The gr ade d tr ac e τ is close d, i.e. it satisfies t he r elation τ ◦ ∂ = d B ◦ τ . Pr o of. ( cf [5]). Fix a P ∈ ψ −∞ ( M , E ⊗ Λ ∗ ( T ∗ B )). Using the mo dule structure, we can assume tha t P ∈ ψ −∞ ( M , E ). Notice that each s uch P has a smo oth Schw ar tz kernel k P . In lo ca l co or dinates, ∇ = d ν + M ω with d ν =  ◦ d the de Rham der iv ative in the direction π ∗ (Λ T ∗ B ) and M ω a z ero-th order differential op erator with coefficie nts in 1- forms o n B . Clearly , the trace of the comm utator [ M ω , P ] is the in tegral of the trace of a co mmut ator and hence is trivial. Using compactly supp or ted smo oth cut-off functions, we can assume that the smo oth kernel k P is supp orted within a trivia l op en set diffeomo rphic to U × U ′ × W wher e U and U ′ are trivializing op en sets in the typical fiber manifold F a nd W is a trivia lizing op en set in the base B . The o p erator d ν is giv en in the lo c a l coo rdinates ( x 1 , · · · , x p ; b 1 , · · · , b q ) of U × W , by d ν = d B + q X i =1 db i p X j =1 A ij ∂ ∂ x j with A ∈ M q,p ( C ∞ c ( U × W )) and d B = q X i =1 db i ∂ ∂ b i . W e observe that τ ([ d B , P ])( b ) = Z M b ( d B k P )( b ; x, x ) dx = ( d B ◦ τ )( P )( b ) . It thus r emains to show tha t τ ([ ∂ ∂ x j , P ]) = 0. But since P is fib er w is e smoo thing and ∂ ∂ x j is a fib erwise first order differen tial op erator , the pro of is complete.  W e follo w [27] a nd introduce an extr a Clifford v ariable σ of degr ee 1 a nd central in the gra ded sense (it graded commutes with all op erator s). Hence we replace the algebra L −∞ by L −∞ [ σ ]. W e ass ume that the fiber s of our fibration are o dd dimensiona l, s o p is o d d. Then we extend the graded closed trace τ so that ∂ : L −∞ [ σ ] → L −∞ [ σ ] and τ σ : L −∞ [ σ ] − → L = Ω ∗ ( B ) , by setting τ σ ( T + σ S ) := τ ( S ) . W e may a lso co nsider the algebra ψ ∞ ( M , E ; Λ ∗ B )[ σ ] in the seq uel. The total degree of an element A in o ne of thes e extensions then takes in to account σ and will b e denoted | A | . Lemma 1. 7. The map τ σ is a gr ade d tr ac e on the Cliffor d ex t ension L −∞ [ σ ] , with values in the gr ade d algebr a Ω ∗ ( B ) . Mor e over, it satisfies the re lation τ σ ◦ ∂ + d B ◦ τ σ = 0 . Pr o of. Let A = T + σ S and A ′ = T ′ + σ S ′ be elemen ts of L −∞ [ σ ] with deg rees k and k ′ resp ectively . This means that T and T ′ hav e re s p e ctively degr ees k and k ′ while S and S ′ hav e re s p e ctively degrees ( k − 1 ) and ( k ′ − 1). W e then co mpute τ σ ( AA ′ ) = ( − 1) k τ ( T S ′ ) + τ ( S T ′ ) = ( − 1) k ( − 1) k ( k ′ − 1) τ ( S ′ T ) + ( − 1) ( k − 1) k ′ τ ( T ′ S ) = ( − 1) kk ′ h τ ( S ′ T ) + ( − 1 ) k ′ τ ( T ′ S ) i = ( − 1) kk ′ τ σ ( A ′ A ) . In the same w ay we have τ σ ([ ∇ , T ] − σ [ ∇ , S ]) = − τ ([ ∇ , S ]) = − d B τ ( S ) = − d B τ σ ( T + σ S ) . 8 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018  1.3. The heat semigrou p and Duhamel. The ope r ator ∇ is used to ass o ciate with the generalized Dirac op erator D , different sup erconnections [9]. They will b e for us giv en b y A σ ( A ) := B σ + A, where A is a zero-th order fib e rwise pseudo differential op era to r with co efficients in differential forms of po sitive degree (in applications ≥ 2) on B . Here, the superco nnection B σ is defined by B σ = ∇ + σ D . W e are mainly interested in the s up er connection B σ or in the Bismut s upe r connection together with its metr ic rescalings . In order to simplify the exp ositio n of our results, w e restrict our selves to the ca se A (0) = B σ and shall briefly explain later ho w the pr o ofs e x tend to the general case. W e have B 2 σ = D 2 + X where X = ∇ 2 − σ [ ∇ , D ]. Note that the oper ator X is a fiber wis e differential op erator of order one with co e fficient s in differen tial fo rms of pos itive degree ≤ 2. Definition 1.8. F ol lowing [8] we wil l use t he notation e − u B 2 σ to denote the semigr oup (that is, the solution to t he he at e quation) given by the fol lowing finite p erturb ative sum of str ong inte gr als e − u B 2 σ = X m ≥ 0 ( − u ) m Z ∆( m ) e − uv 0 D 2 X e − uv 1 D 2 · · · X e − uv m D 2 dv 1 · · · dv m . wher e ∆( m ) = { ( u 0 , · · · , u m ) ∈ R m +1 , P u j = 1 } is the m - simplex. Since the bas e manifold B is finite dimensional, the ab ov e sum is finite. Note also that class ic al results show that the op erator D is a self-adjoint reg ular op erato r on the Hilb ert C ( B , Λ ∗ B )-mo du le E , see for instance [29] o r [7]. Hence the heat op erato r e − tD 2 can be viewed as an adjointable (bounded) op era to r on E , so that it belong s to the algebra L ( E ) of all such op erato r s. Mor eov er, since the o p erator e − tD 2 is a smo othing op erator , the heat op e r ator e − u B 2 σ asso ciated with the supe rconnection B σ defined ab ov e, is also a smo othing op era tor but with co efficients in differ ent ial for ms of the base B . As a consequence, for a ny u > 0, the fiber wise gra de d trace τ σ ( e − u B 2 σ ) makes sense as a differen tial fo r m on the base ma nifo ld B . Lemma 1. 9. (Duhamel principle) F or any element A of the algebr a ψ ∞ ( M | B , E ; Λ ∗ B )[ σ ] , the fol l owing e quality hold s in L ( E ) [ A, e − B 2 σ ] = − Z 1 0 e − s B 2 σ [ A, B 2 σ ] e − (1 − s ) B 2 σ ds. Pr o of. This lemma ca n b e proved following [8]. W e sketc h the arg umen t. F o llowing [2 8] p. 263-264 the Duhamel formula is known to b e sa tisfied by the family D 2 using just the fact that for ξ 0 in our Hilb ert space of L 2 sections ξ t = e − tD 2 ξ 0 solves the heat equa tion. If ξ 0 ∈ E then using the Sc hw ar tz kernel for e − tD 2 we see that ξ t ∈ E from which Duhamel follows in E . Now if w e repla ce e − s B 2 σ and e − (1 − s ) B 2 σ by the finite expansion sums, w e obtain e − s B 2 σ [ A, B 2 σ ] e − (1 − s ) B 2 σ = X m,m ′ ≥ 0 ( − 1) m + m ′ s m (1 − s ) m ′ Z ∆( m ) × ∆( m ′ ) e − v 0 sD 2 X e − v 1 sD 2 · · · X e − v m sD 2 [ A, D 2 + X ] e − w 0 (1 − s ) D 2 X e − w 1 (1 − s ) D 2 · · · X e − w m ′ (1 − s ) D 2 dv 1 · · · dv m dw 1 · · · dw m ′ . Now if w e in tegrate ov er (0 , 1), ma ke a suitable c hange of v a riables using m X j =0 sv j + m ′ X i =0 (1 − s ) w i = 1 , and apply the Duhamel principle for D 2 we obtain the result.  JLO AND S PECTRAL FLOW FOR F AMILIES 9 It is worth p ointing o ut that the op er ator [ A, e − B 2 σ ] is a fib erwise smo othing op erator with co efficients in Ω ∗ ( B ). Mo reov er, for any s ∈ (0 , 1) the o pe rator e − s B 2 σ [ A, B 2 σ ] e − (1 − s ) B 2 σ is a lso fib erwis e s mo othing. It is then straightforw ard to chec k, using the dominated conv er gence theore m, that the following holds τ σ ([ A, e − B 2 σ ]) = − Z 1 0 τ σ ( e − s B 2 σ [ A, B 2 σ ] e − (1 − s ) B 2 σ ) ds. Lemma 1 .10. Consider for any u ∈ [0 , 1 ] , a sup er c onne ction A u , given by A u := B σ + u A , wher e A is a (o dd for the gr ading) zer o-t h or der fib erwise pseudo differ ential op er ator with c o efficients in p ositive de gr e e differ ential forms on B . Then we have d du e − A 2 u = − Z 1 0 e − s A 2 u [ A, A u ] e − (1 − s ) A 2 u ds in t he str ong op er ator top olo gy of L ( E ) . Pr o of. W e set for any small real num ber h 6 = 0, Y u ( h ) := [ A, A u ] + hA 2 . Then from the definition o f e − A 2 v we deduce that 1 h h e − A 2 u + h − e − A 2 u i = X k ≥ 1 h k − 1 Z ∆( k ) e − u 0 A 2 u Y u ( h ) e − u 1 A 2 u · · · Y u ( h ) e − u k A 2 u du 1 · · · du k . Both sides are well defined as op era tors on E as they are fiber wise smo o thing op era to rs. Applying b oth sides to elemen ts of E we end the pro of b y letting h → 0.  W e shall also nee d the following lemma. Lemma 1.11 . F or any A 0 , · · · , A n in t he algebr a ψ ∞ ( M | B , E ; Λ ∗ B )[ σ ] we have , τ σ  [ σ D , A 0 e − u 0 B 2 σ A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ ]  = 0 for u j > 0 . Pr o of. By definition e − u j B 2 σ is given by the p erturbative s um e − u j B 2 σ = X m ≥ 0 ( − u j ) m Z ∆( m ) e − u j v 0 D 2 X e − u j v 1 D 2 · · · X e − u j v m D 2 dv 1 · · · dv m . F or u 0 > 0, w e know tha t the oper ator σ DA 0 e − u 0 B σ / 2 is fib erwise smo othing with co e fficient s in differen tial forms and tha t its deg r ee is | A 0 | + 1. Therefore, the graded tra cial prop erty of τ σ shows that τ σ h ( D A 0 e − u 0 B 2 σ / 2 )( e − u 0 B 2 σ / 2 A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ ) i = ( − 1) ( | A 0 | +1) P n j =1 | A j | τ σ h ( e − u 0 B 2 σ / 2 A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ σ D )( A 0 e − u 0 B 2 σ / 2 ) i Now as befor e, the oper a tor A 0 e − u 0 B 2 σ / 2 , as w ell a s e − u 0 B 2 σ / 2 A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ σ D are fiber wise smo othing ope r ators with coe fficie nt s in differen tial fo r ms. Therefore, τ σ h ( e − u 0 B 2 σ / 2 A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ σ D )( A 0 e − u 0 B 2 σ / 2 ) i = ( − 1) | A 0 | (1+ P n j =1 | A j | ) τ σ h ( A 0 e − u 0 B 2 σ / 2 )( e − u 0 B 2 σ / 2 A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ σ D ) i . Hence we hav e τ σ h σ D A 0 e − u 0 B 2 σ A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ i = ( − 1) P n j =0 | A j | τ σ h A 0 e − u 0 B 2 σ A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ σ D i .  10 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 2. Mul tilinear functionals and identities In this Section w e record some useful identities satisfied by the multilinear functionals that enter into the biv ariant JLO co cycle. W e shall extensively use ideas developed in the seminal pap er [1 6] by E . Getzler and A. Szenes. Let A i ∈ ψ ∞ ( M | B , E ; Λ ∗ B )[ σ ] and with ∆( n ) b eing a s b efore, the n -simplex, we define m ultilinear functionals [16, 30] hh A 0 , · · · , A n ii B σ := Z ∆( n ) τ σ ( A 0 e − u 0 B 2 σ A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ ) du 1 · · · du n ∈ Ω ∗ ( B ) . and < A 0 , · · · , A n > := Z ∆( n ) τ σ ( A 0 e − u 0 D 2 · · · A n e − u n D 2 ) du 1 · · · du n ∈ Ω ∗ ( B ) . The following lemma is stated in the case o f flat co nnec tions in [30] and is a stra ightforw ard extensio n of [16][Lemma 2.2]. W e give the pro of for completeness and beca use it will b e use d in the sequel. Lemma 2.1. L et A 0 , · · · , A n ∈ ψ ∞ ( M | B , E ; Λ ∗ B )[ σ ] and let ǫ i = ( | A 0 | + · · · + | A i − 1 | )( | A i | + · · · + | A n | ) . • F or 1 ≤ i ≤ n , hh A 0 , · · · , A n ii = ( − 1) ǫ i hh A i , · · · , A n , A 0 , · · · , A i − 1 ii ; • P n i =0 hh A 0 , · · · , A i , 1 , A i +1 , · · · , A n ii = hh A 0 , · · · , A n ii ; • hh [ B σ , A 0 ] , A 1 , · · · , A n ii + P n i =1 ( − 1) | A 0 | + ··· + | A i − 1 | hh A 0 , · · · , [ B σ , A i ] , · · · , A n ii + d B hh A 0 , · · · , A n ii = 0; • F or 0 ≤ i < n , hh A 0 , · · · , A i − 1 A i , A n ii − hh A 0 , · · · , A i A i +1 , A n ii =  A 0 , · · · , [ B 2 σ , A i ] , A n  ; and for i = n , hh A 0 , · · · , A n − 1 A n ii − ( − 1) ( | A 0 | + ··· + | A n − 1 | ) | A n | hh A n A 0 , A 1 , · · · , A n − 1 ii =  A 0 , · · · , A n − 1 , [ B 2 σ , A n ]  . • L et ( B σ,s = B σ + sA ) s b e a 1 -p ar ameter famil y of sup erc onne ctions asso ciate d with σ D as in L emma 1.10, t hen we have d ds hh A 0 , · · · , A n ii B σ,s + n X i =0 hh A 0 , · · · , A i , [ B σ,s , A ] , A i +1 , · · · , A n ii B σ,s = 0 . Pr o of. The first re la tion is a consequenc e o f the fact that τ σ is a g raded trace. Indeed, B 2 σ is homoge ne o us of even total degree. The second rela tion is also clea r, see [16] for more details. Let us chec k the third r elation. F rom the p erturba tive finite sum whic h defines e − u B 2 σ we hav e seen in Lemma 1.9 that the Duhamel pr inciple holds, hence w e obtain the following Bianc hi iden tit y [ B σ , e − u B 2 σ ] = − Z 1 0 e − us B 2 σ [ B σ , B 2 σ ] e − u (1 − s ) B 2 σ ds = 0 . Therefore, the left hand side o f the third r e la tion coincides with Z ∆( n ) τ σ  [ B σ , A 0 e − u 0 B 2 σ A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ ]  du 1 · · · du n + d B hh A 0 , · · · , A n ii . On the o ther hand, b y Lemma 1.11 τ σ ([ σ D , A 0 e − u 0 B 2 σ A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ ]) = 0 . Hence, the left hand side of the third relation coincides with Z ∆( n ) h τ σ  [ ∇ , A 0 e − u 0 B 2 σ A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ ]  + d B  τ σ ( A 0 e − u 0 B 2 σ A 1 e − u 1 B 2 σ · · · A n e − u n B 2 σ ) i du 1 · · · du n . Lemma 1.7 then co mpletes the pr o of of the third item. The fourth relatio n is again a consequence of Lemma 1.9 and is a stra ightforw ard gener alization of the similar relation prov ed for a single o p erator in [16]. JLO AND S PECTRAL FLOW FOR F AMILIES 11 Now, notice that d ds hh A 0 , · · · , A n ii B σ,s = n X i =0 Z ∆( n ) τ σ ( A 0 e − u 0 B 2 σ,s · · · A i de − u i B 2 σ,s ds A i +1 e − u i +1 B 2 σ,s · · · A n e − u n B 2 σ,s ) du 1 · · · du n . But, Duhamel’s formula 1.10 shows that de − B 2 σ,s ds + Z 1 0 e − u B 2 σ,s [ B σ,s , A ] e − (1 − u ) B 2 σ,s du = 0 . The pro of is thus complete.  Definition 2. 2. The bivariant JLO c o chain is define d by the se quen c e ( ψ n ) n given for ( f 0 , · · · , f n ) ∈ C ∞ ( M ) n +1 by the formula ψ n ( f 0 , · · · , f n ) := hh f 0 , [ B σ , f 1 ] , · · · , [ B σ , f n ] ii B σ . One deduce s from Lemma 4.3 that for any C 1 function f on the close d manifold M , the commutator [ B σ , f ] is a bounded ope r ator with v a lues in horizontal 1-for ms. F or simplicity , w e work with smo oth functions and smo oth forms, although the constructions work obviously with less reg ularity , and leav e it to the in ter ested reader to transp ose the s tatements for more restrictive reg ularity co nditions. W e r ecall that Ω A deno tes the universal differen tial graded algebra o f a topo logical algebra A , that is , Ω n ( A ) = A + ⊗ A ⊗ n for n ≥ 1 (where A + means there is a unit a djoined to A ). Also r e call the op e rators b, B on Ω A defined b y: b ( a 0 da 1 · · · da n ) = n − 1 X j =0 ( − 1) j a 0 da 1 · · · d ( a j a j +1 ) da j +2 · · · da n + ( − 1) n a n a 0 da 1 · · · da n − 1 , B ( h a 0 i da 1 da 2 · · · da n ) = n X j =0 ( − 1) nj da j · · · da n d h a 0 i da 1 · · · da j − 1 . resp ectively . Her e h a 0 i means either a 0 or 1, where 1 deno tes the a dditional unit. Notice tha t the letter B is already used fo r the base manifold B but this should not cause any confusion. Lemma 2.3. The se quenc e ψ = ( ψ n ) n ≥ 0 is a chain map of o dd de gr e e. Mor e pr e cisely, it is a chain map fr om the u niversal differ ential algebr a of C ∞ ( M ) t o the gr ade d Gr assmann algebr a Ω ∗ ( B ) . Inde e d, its c omp onent s whose form de gr e e have t he p arity of n ar e trivial, and it satisfies the c o cycle r elation ψ ◦ ( b + B ) + d B ◦ ψ = 0 . Remark 2. 4. The ab ove lemma is state d un der the assumption that the fib ers ar e o dd dimensional. When these fib ers ar e even dimensional an analo gous r esult hold s. Pr o of. The third relation in Lemma 2.1 applied to A 0 = f 0 and A i = [ B σ , f i ] for i ≥ 1 gives hh [ B σ , f 0 ] , · · · , [ B σ , f n ] ii − n X i =1 ( − 1) i  f 0 , [ B σ , f 1 ] , · · · , [ B 2 σ , f i ] , · · · , [ B σ , f n ]  + d B hh f 0 , [ B σ , f 1 ] , · · · , [ B σ , f n ] ii = 0 . On the o ther hand computing ( B ψ n +1 )( f 0 , · · · , f n ), w e find, using the second item of Lemma 2.1: ( B ψ n +1 )( f 0 , · · · , f n ) = hh [ B σ , f 0 ] , · · · , [ B σ , f n ] ii . Using the la st item of L e mma 2 .1, we finally deduce that ( bψ n − 1 )( f 0 , · · · , f n ) + n X i =1 ( − 1) i  f 0 , [ B σ , f 1 ] , · · · , [ B 2 σ , f i ] , · · · , [ B σ , f n ]  = 0 . Therefore, w e obtain the desired r esult B ψ n +1 + bψ n − 1 + d B ψ n = 0 .  12 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 3. The biv ariant JLO cocycle is entire W e are using cy c lic homology for b or nological a lgebras due to R. Meyer [2 4] and will need so me prelimi- naries which we now describ e. 3.1. Review of biv arian t en tire cyclic homology. F or the convenience of the reader, w e summarise in this subsection what w e need ab out biv a riant entire homology . The reader is encourage d to consult [24] for more details , esp ecially for the definitions and prope r ties of bo rnolog ies. See also [11] for the basic (non- trivial) concepts of bor nologica l functional a nalysis. Our task her e is to a dapt this fo rmalism to the fa milies situation. The idea of using a b or nology in the study of en tire c y clic cohomology is due to Connes [14] (see pages 370-3 71). Given a loca lly convex top olog ic a l algebra A , it is pro p o sed there to use the b ounded subsets on A to define ent ire cyclic cohomology . Meyer develops this idea using bo rnolog ic al functional analy sis in a form that is appropriate for this pap er in [24]. Here A , A 1 , A 2 etc will denote complete lo cally conv ex top olo gical algebra s. W e will use a family of different bo rnologie s o n A j denoted g enerically b y S ( A j ). An algebra A equipp ed with a particular bo rnolog y will b e denoted ( A, S ( A )). Denote by Ω A the universal differ ent ial graded a lgebra of A . F ollowing [24], Section 3, w e intro duce the follo wing notions. Definition 3.1. (i) F or S ∈ S ( A ) ( dS ) ∞ denotes the union over n of elements ds 1 ds 2 . . . ds n ∈ Ω n ( A ) wher e s 1 , . . . , s n ar e fr om S . L et, as b efor e, h S i = S ∪ { 1 } wher e 1 denotes an additional unit and not the identity of A (we use A + to denote the adjunction of this unit to A ) and then define h S i ( dS ) ∞ = S ( dS ) ∞ ∪ ( dS ) ∞ ∪ S ⊂ Ω A, S ( dS ) ev = h S i ( dS ) ∞ ∩ Ω ev A, h S i ( dS ) odd = h S i ( dS ) ∞ ∩ Ω odd A. (ii) The notation h a 0 i da 1 . . . da n ∈ Ω n A me ans either a 0 da 1 . . . da n or da 1 . . . da n dep ending on c ont ex t. (iii) S an is the b ornolo gy on Ω A gener ate d by h S i ( dS ) ∞ for al l S ∈ S ( A ) and Ω an A denotes the c ompletion of Ω A in the b ornolo gy S an . Equivalently S an is gener ate d by the u nion over n of the sets {h s 0 i ds 1 ds 2 . . . ds n | s j ∈ S, S ∈ S ( A ) } . Remark 3.2. If A is F r´ ech et, then A is alr e ady c omplete in the b ornolo gy given by t aking the b ounde d sets in t he F r´ ec het t op olo gy, se e [2 6] for instanc e. In this paper A will alwa ys b e o ne o f the F r´ echet algebras C ∞ ( M ) or C ∞ ( B ) w he r e F → M → B is a fibratio n of compact s mo oth manifolds. How ever they will b e equipp ed with bo rnolog ies defined by the subsets bounded in certa in fa milies of norms. If ( V , S ( V )) and ( W, S ( W )) are complete lo ca lly conv e x b o rnolog ic a l space s then bounded linear maps ℓ : V → W are linear ma ps with the prop erty that ℓ ( S ) ∈ S ( W ) whenever S ∈ S ( V ). This notion extends to m ultilinear maps as well. More ov er bo unded linear maps ℓ : Ω an A → W a re in bijection with b ounded line a r maps o n Ω A equipp e d with the b ornolo g y S an . These in turn ar e in bijection with linear maps ℓ : Ω A → W satisfying ℓ ( h S i ( dS ) ∞ ) ∈ S ( W ), for any S ∈ S ( V ). W e now explain some key results. W e denote by n ! S an the bo rnology on Ω A generated by the union ov er n and S ∈ S ( A ) of the s ets n ! h S i h dS i ( dS ) 2 n which are defined to be { n ! h s 0 i ds 1 ds 2 . . . ds 2 n , | s j ∈ S } ∪ { n ! h s 0 i ds 1 ds 2 . . . ds 2 n +1 | s j ∈ S } . Let C ( A ) b e the algebra Ω A completed in the b ornolo gy n ! S an . If we equip Ω( A ) with the Ho chsc hild bo undary b and then with Connes’ op er ator B s atisfying the usual relatio ns b 2 = 0 = B 2 = B b + bB then we define a bicomplex. The pair ( b, B ) extend to b ounded maps on C ( A ) and ( C ( A ) , b + B ) is a Z 2 -graded complex of complete b o rnologic a l vector spa ces called Connes’ entire complex. An imp or ta nt fac t is that Meyer shows that his analytic cyclic co homology o f A is the same as Connes’ ent ire cyc lic cohomo lo gy of A . The idea of the pro of is to consider the dual complex C ( A ) ′ of b ounded linear maps C ( A ) → C . These a re just b ounded linea r maps (Ω A, n ! S an ) → C . The b ounded linear JLO AND S PECTRAL FLOW FOR F AMILIES 13 functionals on (Ω A, n ! S an ) are those linear ma ps Ω A → C that remain b ounded on all sets o f the for m n ! h S ih dS i ( dS ) 2 n . Ident ifying Ω A ∼ = P ∞ n =0 Ω n A and Ω n A ∼ = A + ˆ ⊗ A ˆ ⊗ n , C ( A ) ′ bec omes the space of families ( φ n ) n ∈ Z + of n + 1 - linear maps φ n : A + × A n → C satisfying the en tire growth condition | φ n ( h a 0 i , a 1 , . . . , a n ) | ≤ const ( S ) / [ n/ 2]! for all h a 0 i ∈ h S i , a 1 . . . , a n ∈ S and for all S ∈ S ( A ). Here [ n/ 2 ] := k if n = 2 k or n = 2 k + 1 a nd const ( S ) is a constant dep ending o n S but not on n . The b ounda ry on C ( A ) is comp ositio n with B + b . This motiv a tes us to use the b orno logical approach of Mey er in the con text o f Co nnes ( b, B ) bicomplex. The p oint of view of [24] is to define the biv aria nt cyclic cohomology of a pair A 1 , A 2 to b e the homolog y of the complex of b ounded linear maps from Ω an ( A 1 ) to Ω an ( A 2 ). In this pap er we replace the analytic bo rnolog y of C ∞ ( M ) by the equiv alent entire b orno lo gy but do no t work with the universal graded algebra Ω( C ∞ ( B )) and instead co nsider the smo o th exterior algebra Ω ∗ ( B ) (that is smo oth sectio ns of the exterio r bundle) asso ciated with the smo oth manifold B . W e will equip this smo oth algebra with v a rious b o r nologies which we give in the next subsection. The reas on for doing this is that we wish to w ork with sup er connections. 3.2. Statement of the main theorem. The techniques use d here a re inspired by [6]. W e denote by C ℓ ( M ), for any ℓ ≥ 0, the algebra of complex v a lued functions on the smo oth manifold M which are of class C ℓ . The algebr a C ℓ ( M ) can b e endow ed with a Banach space topo lo gy as usua l. This is achiev e d for instance by using lo cal co ordinates and a pa rtition of unit y subo rdinated with a (finite) ope n cov er . Using lo cal orthono rmal fr ames extended to vector fields ov er M using this par tition, we can define this to po logy using a finite set X of vector fields ov er M . More precise ly , the s emi-norms p q ( f ) := sup X j ∈X k X 1 ◦ · · · ◦ X q ( f ) k ∞ , 0 ≤ q ≤ ℓ, induce a Banach spa ce top o logy on C ℓ ( M ). F o r simplicity , we have omitted the finite set X from the notation. W e shall denote b y Σ ℓ the b ornolo gy on C ℓ ( M ), and also its restriction to C ∞ ( M ), which is given b y the bounded se ts of the norm max 0 ≤ q ≤ ℓ p q . W e also introduce for any vector field Y on B , the notation d Y to denote the op erato r i Y ◦ d B . The b ornolog ical a lgebra (Ω ∗ ( B ) , Σ ℓ ) of smo o th differential fo rms on B is endow ed similarly with the b orno lo gy giv en b y the b ounded sets o f the usual C ℓ top ology on forms. Recall that this latter is asso ciated with the s emi-norms obtained on Ω k ( B ) b y using as for C ℓ ( M ) a finite s et Y of vector fields on B and b y considering semi-norms p r ( ω ) := sup Y j ∈Y , k Z j k≤ 1 1 2 kr k ( d Y 1 ◦ · · · d Y r ( i Z 1 ◦ · · · ◦ i Z k ω )) k ∞ , 0 ≤ r ≤ ℓ. Our go a l is to prove tha t the biv ariant JLO co chain constructed in the for mal s pirit of Q uillen’s seminal pap er [27], is a b ounded cyclic co cycle from the entire completion o f the universal differential algebra asso ciated with the underlying b orno logical algebra ( C ∞ ( M ) , Σ ℓ +1 ) on the one hand and the b orno logical algebra (Ω ∗ ( B ) , Σ ℓ ) of smoo th differe nt ial forms o n B endowed with the Σ ℓ bo rnolog y on the o ther hand. Again, we only consider smo o th for ms and the restriction of Σ ℓ to them. As a corollary we sha ll obtain an entire biv ariant cyclic co cycle, following Connes [14]. T o shorten the statements of our results we need some further notation. W rite n ! S ℓ +1 an for the en tire b o rnolog y on Ω C ∞ ( M ) arising from the Σ ℓ +1 bo rnolog y on C ∞ ( M ). Definition 3.3. • A morphism ϕ fr om Ω C ∞ ( M ) to Ω ∗ ( B ) is an ℓ -entir e bivariant c o chain if it is b ounde d when Ω C ∞ ( M ) is endowe d with t he entir e b ornolo gy n ! S ℓ +1 an and Ω ∗ ( B ) with the b ornolo gy Σ ℓ . An ℓ -ent ir e bivariant c o cycle is an ℓ -ent ir e bivariant c o chain which satisfies ϕ ◦ ( b + B ) + d B ◦ ϕ = 0 . • A morphism ϕ fr om Ω C ∞ ( M ) to Ω ∗ ( B ) is c al le d an entir e bivariant c o chain her e if it is b ounde d when Ω C ∞ ( M ) is endowe d with the entir e b ornolo gy n ! S ∞ an and Ω ∗ ( B ) with the b ornolo gy Σ ∞ . Remark 3.4. A morphism ϕ is a biva riant entir e c o chain if and only if it is a bivariant ℓ -entir e c o chain, for al l ℓ ≥ 0 . W e are now ready to state our first theorem. Recall that the fib e r s of our fibration are odd dimensional. There is a similar statemen t in the even cas e. 14 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 Theorem 3.5. F or any inte ger ℓ ≥ 0 , the bivariant J LO c o chain ψ is an ℓ -entir e biva riant c o cycle. Corollary 3 .6. The bivariant JLO c o chain is ent ir e with r esp e ct to the first variable for the F r´ ec het C ∞ - top olo gy of C ∞ ( M ) and C ∞ ( B ) . Pr o of. If S is a b ounded set for the F r´ echet C ∞ top ology of C ∞ ( M ), then fo r an y ℓ ≥ 0, S is bounded in the C ℓ +1 top ology . Therefore, if A is a subset o f h S i ( dS ) ∞ then a pplying Theorem 3.5, its image under the morphism defined b y ψ will b e cont ained in some set h S ′ i ( dS ′ ) ∞ for a bounded set S ′ in the C ℓ top ology o f Ω ∗ ( B ). Since this is true for an y ℓ ≥ 0, we deduce that ψ ( A ) is bo unded for the C ∞ top ology .  Corollary 3.7. F or any U ∈ GL N ( C ∞ ( M )) and for any ℓ ≥ 0 , the fol lowing series of differ ential forms on B c onver ges in the C ℓ -top olo gy to a close d differ ent ial form whose c ohomolo gy class is denote d < JLO( D ) , U > : X k ≥ 0 ( − 1) k k !  U − 1 , [ B σ , U ] , · · · , [ B σ , U − 1 ] , [ B σ , U ]  2 k +1 . Pr o of. This coro llary is the precise re phrasing of the fo llowing fact. Since the biv a r iant JLO co chain is entire and clos ed, it pairs with Connes’ [13] entire cyclic cycles of C ∞ ( M ) to yield a closed differential form on B , and direct inspectio n of the pair ing of [1 3] gives precisely the one in the statement of the cor ollary .  Recall that a sequence ( φ n ) n ≥ 0 of co chains φ n : C ∞ ( M ) ⊗ n +1 → C is entire in the sense of Connes’ definition [1 4] for the C s norm k · k s if and only if for a ny bounded set S in ( C ∞ ( M ) , k · k s ) there exists a constant C ( S ) such that | φ n ( f , · · · , f n ) | ≤ C ( S ) / [ n/ 2]! , for any f i ∈ S a nd a ny n ≥ 0 . This allo ws us to define in the same way en tire co cycles for the F r´ echet topo logy . Another consequence of Theorem 3.5 is the following: Corollary 3.8. L et C b e a close d de Rham curr ent on the b ase manifold B of de gr e e N ∈ { 0 , · · · , dim( B ) } . Then the fol lowing se quenc e ψ C = ( Z C ψ n ) n − N ∈ 2 Z +1 , is an entir e cyclic c o cycle on t he algebr a C ∞ ( M ) . Mor e over, the fol lowing series c onver ges in C to the p airing of the Chern char acter of U with the c omp osition of JLO( D ) with C : X k ≥ 0 ( − 1) k k ! Z C  U − 1 , [ B σ , U ] , · · · , [ B σ , U − 1 ] , [ B σ , U ]  2 k +1 = Z C h J LO ( D ) , U i . Pr o of. Co mputing ( b + B ) ψ C we find bψ C n − 1 + B ψ C n +1 = < C, bψ n − 1 + B ψ n +1 > = − < C , d B ψ n > = 0 . The last equality is true since C is clo sed. It remains to chec k the entire pr op erty . Notice that the closed cur- rent C defines a cy c lic co cycle on C ∞ ( B ) which yie lds a C ∞ -contin uous g raded trace on Ω ∗ ( B ). Comp osing this trace with the JLO biv ariant co cycle, w e conclude using Corollary 3 .7.  4. Proof o f Theorem 3.5 The pr o of is long and will b e split into many s ubpa rts. 4.1. Estim ates. The pro o f of Theorem 3.5 rests on establis hing some estimates on our biv ariant JLO functional. W e collect the preliminary facts in this s ubsection. W e denote by d v f the different ial of f in the fiber wise direction and by d H f the differential of f in the hor izontal direction defined by the hor izontal distribution H . W e choos e the metric on M so that H and the fib erwise bundle T v M are o rthogona l. Recall that if c is the fib erwise Clifford r epresentation then for f ∈ C ∞ ( M ) we hav e [ D , f ] = c ( d v f ). Note also that [ ∇ , f ] = d H f ∧ · JLO AND S PECTRAL FLOW FOR F AMILIES 15 F or a vector field Y on B , we de no te by ˜ Y the horizontal v ector field on M satisfying π ∗ ˜ Y = Y . Rec a ll that ∂ denotes the graded (with resp ect to the degree of the forms ) commu tator asso cia ted with the quasi- connection ∇ , a (exterior) g r aded deriv ation of the algebra ψ ∞ ( M | B , E ; Λ ∗ B ) of fib erwise pseudo differential op erator s with co efficients in horizontal differen tial forms. W e denote, for any horizontal (i.e. H v alued) vector field Z o n M , by ∇ Z the comp os ition i Z ◦ ∇ wher e i Z is co ntraction by Z . As usual, for P ∈ ψ h ( M | B , E ; Λ k B ) we a ls o deno te by ∂ Z ( P ) the e le ment [ ∇ Z , P ] o f ψ h ( M | B , E ; Λ k B ). If P ∈ ψ h ( M | B , E ) with h ≤ 0, then w e set k P k := sup b ∈ B k P b k , where the norm k P b k is the op erator norm on the L 2 sections. In genera l, if P ∈ ψ h ( M | B , E ; Λ k B ) for h ≤ 0 and k ≥ 0 we define the unifor m norm of P by the same ex pr ession, except that now k P b k is obtained b y taking the s upremum ov er k -multiv ec to rs Z ∈ Λ k ( T b B ) of norm ≤ 1, of the o p er ator norms k i Z P b k . W e use the op erato r s (1 + D 2 b ) s/ 2 to define the So b olev pre - Hilber t H s top ology on C ∞ ( M b , E | M b ). W e may extend the definition given above for op era tors of zeroth or negative o rder to op erator s o f pos itive order α . The H s -norm of an op er ator A b of o rder α is defined a s the nor m o f A b : H s → H s + α and thes e no r ms are compar a ble for all s . By tak ing the supremum of o ver b ∈ B as a b ove we obtain comparable norms for different c hoices of s for op erator s of order α . In the dis c ussion b elow w e will for co nv enience use thes e s -norms in terchangeably without comment. Lemma 4.1. F or any q ≥ 0 , sup Y 1 , ··· , Y r ∈Y k ( I + D 2 ) − 1 / 2 [ ∂ ˜ Y 1 · · · ∂ ˜ Y r ]( D 2 )( I + D 2 ) − 1 / 2 k = α r ( D 2 ) < + ∞ , and sup Y 1 , ··· , Y r ∈Y k ( I + D 2 ) − 1 / 2 [ ∂ ˜ Y 1 · · · ∂ ˜ Y r ]( D ) k = α r ( D ) < + ∞ wher e t he Y j ’s ar e ve ct or fields on B . Pr o of. The same pr o of works for b oth o p e r ators and we only give the pr o of for D 2 . W e firs t p oint out that the o pe r ator [ ∂ Y 1 · · · ∂ Y q ]( D 2 ) is a second order fib erw is e differential op erator, with smo o th co efficients. Therefore, the oper ator ( I + D 2 ) − 1 / 2 [ ∂ ˜ Y 1 · · · ∂ ˜ Y q ]( D 2 )( I + D 2 ) − 1 / 2 , is a zero-th o rder fiber wise pseudo differential op era to r who s e norm is finite. Moreover, a s B is compact, by a partition of unity argument w e may ass ume that we are g iven a lo ca l orthono rmal basis ( ∂ 1 , · · · , ∂ b ) of the tang ent bundle to B ov er an op en set U ⊂ B . Then, we ca n replac e the op er ators ∂ Y j by opera tors of the form ˜ ∂ j := ∂ j + ω ( ∂ j ), where ω is a matrix of differen tial 1 -forms. Now, the finite family of oper ators ( I + D 2 ) − 1 / 2 [ ˜ ∂ j 1 · · · ˜ ∂ j q ]( D 2 )( I + D 2 ) − 1 / 2 , for 1 ≤ j 1 , · · · , j q ≤ b , is uniformly bounded o ver U . Since the vector fields Y 1 , · · · Y q belo ng to the finite family Y , the pro o f is thus co mplete.  Lemma 4.2. F or any r ≥ 0 , sup k Z 1 k≤ 1 , k Z 2 k≤ 1 ,Y 1 , ··· , Y r ∈Y k ( I + D 2 ) − 1 / 2 [ ∂ ˜ Y 1 · · · ∂ ˜ Y r ]( i ˜ Z 1 ∧ ˜ Z 2 ∇ 2 ) k = β r ( ∇ , D ) < + ∞ , wher e Z 1 , Z 2 , Y 1 , · · · , Y r ar e ve ctor fields on B that we view thr ough their unique horizontal lifts. Pr o of. The pro of follo ws the sa me lines as the previous lemma. More precisely , using the compactness of B w e can reduce to lo cal co ordinates. But as can be c heck ed in these lo ca l co ordinates, since the oper ator i Z 1 ∧ Z 2 ∇ 2 is a smo oth family of differential op erator s o f order 1, the smo oth family o f zero- th order op erato rs ( I + D 2 ) − 1 / 2 [ ∂ Y 1 · · · ∂ Y q ]( i Z 1 ∧ Z 2 ∇ 2 ) is unifor mly bo unded.  Recall that for f ∈ C ∞ ( M ), k f k s := max 0 ≤ j ≤ s p j ( f ) . 16 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 Lemma 4.3. F or any s ≥ 0 , ther e ex ists a c onstant C s ≥ 0 such that k [ ∂ ˜ Y s ◦ · · · ◦ ∂ ˜ Y 1 ]( f ) k ≤ C s k f k s and k [ ∂ ˜ Y s ◦ · · · ◦ ∂ ˜ Y 1 ]([ D , f ]) k ≤ C s k f k s +1 , for any f ∈ C ∞ ( M ) and any ve ctor fields Y j fr om the finite family Y . Pr o of. F or any j ≤ s , we hav e: [ ∂ ˜ Y j ◦ · · · ◦ ∂ ˜ Y 1 ]( f ) = [ ˜ Y j ◦ · · · ◦ ˜ Y 1 ]( f ) . The RHS mea ns the multiplication op erato r b y the function [ ˜ Y j ◦ · · · ◦ ˜ Y 1 ]( f ). Since M is compact, there obviously exists a constant C s > 0 only depending o n the distribution H such that k [ ˜ Y j ◦ · · · ◦ ˜ Y 1 ]( f ) k ≤ C s p j ( f ) , for any j ≤ s. Using the fact that [ D , f ] is Clifford multiplication by d f w e ca n expand in loca l co or dinates to easily prov e in a similar fashion the seco nd estimate.  Lemma 4.4 . Fi x any ǫ ∈ ]0 , 1 / 2] , t hen for fib erwise pseudo differ ential op er ators ( A j ) 0 ≤ j ≤ N and ( B j ) 0 ≤ j ≤ N with A j ∈ ψ 0 ( M | B , E ) and B j ∈ ψ 2 ( M | B , E ) for any j , we have: k < A 0 , B 0 , · · · , A N , B N > k ≤  π 2 ǫ  N +1 k τ ( e − (1 − ǫ ) D 2 ) k N ! × Π N j =0 k A j kk ( I + D 2 ) − 1 / 2 B j ( I + D 2 ) − 1 / 2 k . Pr o of. By inspection we have < A 0 , B 0 , · · · , A N , B N > = < A 0 ( I + D 2 ) 1 / 2 , ( I + D 2 ) − 1 / 2 B 0 , · · · , A N ( I + D 2 ) 1 / 2 , ( I + D 2 ) − 1 / 2 B N > . Therefore, using H¨ older’s ineq ua lity fib e rwise, we obtain (writing du = du 1 · · · du N , dv = dv 0 dv 1 · · · dv N ): k < A 0 , B 0 , · · · , A N , B N > k ≤ Z ∆(2 N +1) Π N j =0 k A j ( I + D 2 ) 1 / 2 e − u j D 2 k 1 /u j k ( I + D 2 ) − 1 / 2 B j e − v j D 2 k 1 /v j du dv ≤ Π N j =0 k A j kk ( I + D 2 ) − 1 / 2 B j ( I + D 2 ) − 1 / 2 k Z ∆(2 N +1) Π N j =0 k ( I + D 2 ) 1 / 2 e − u j D 2 k 1 /u j k ( I + D 2 ) 1 / 2 e − v j D 2 k 1 /v j du dv ≤ Π N j =0 k A j kk ( I + D 2 ) − 1 / 2 B j ( I + D 2 ) − 1 / 2 k Z ∆(2 N +1) Π N j =0 k ( I + D 2 ) 1 / 2 e − u j ǫD 2 kk ( I + D 2 ) 1 / 2 e − v j ǫD 2 kk τ ( e − (1 − ǫ ) D 2 ) u j τ ( e − (1 − ǫ ) D 2 ) v j k du dv ≤ k e − (1 − ǫ ) D 2 k 1 Π N j =0 k A j kk ( I + D 2 ) − 1 / 2 B j ( I + D 2 ) − 1 / 2 k Z ∆(2 N +1) Π N j =0 k ( I + D 2 ) 1 / 2 e − u j ǫD 2 kk ( I + D 2 ) 1 / 2 e − v j ǫD 2 k du dv . But, for an y α > 0, we hav e by the sp ectral theorem in L ( E ) (see for instance [7]), k ( I + D 2 ) 1 / 2 e − αD 2 k ≤ e α − 1 / 2 √ 2 α . Therefore, Π N j =0 k ( I + D 2 ) 1 / 2 e − u j ǫD 2 kk ( I + D 2 ) 1 / 2 e − v j ǫD 2 k ≤ e ǫ − 1 / 2 (2 ǫ ) N +1 × Π N j =0 ( u j v j ) − 1 / 2 . Now we complete the pro of by computing the following integral: Z ∆(2 N +1) Π N j =0 ( u j v j ) − 1 / 2 du dv = π N +1 N ! .  JLO AND S PECTRAL FLOW FOR F AMILIES 17 W e s hall a lso need the intermediate estimate co rresp onding to p + 1 ent ries of second order fib erwise pseudo differential op era tors in < · · · > n + p +1 . In fact a similar method of pro of establishes our next result. Lemma 4.5. F or any ǫ ∈ ]0 , 1 / 2] , for any A 0 , · · · , A N ∈ ψ 0 ( M | B , E ) and any B j 0 , · · · , B j p ∈ ψ 2 ( M | B , E ) with p < N and 0 ≤ j 0 < · · · < j p ≤ N , the fol lowing estimate holds    A 0 , · · · , A j 0 , B j 0 , A j 0 +1 , · · · , A j 1 , B j 1 , · · · , A j p , B j p , A j p +1 , · · · , A N    ≤  π 2 ǫ  p +1 k τ ( e − (1 − ǫ ) D 2 ) k N ! × Π N i =0 k A i k Π p i =0 k ( I + D 2 ) − 1 / 2 B j i ( I + D 2 ) − 1 / 2 k . Pr o of. W e apply again the method of pro of of Lemma 4.4 and use the equality Z u 0 + ··· + u N + v j 0 + ··· + v j p =1 du 1 · · · du N dv j 0 · · · dv j p √ u j 0 · · · u j p v j 0 · · · v j p = π p +1 p ! N ! . More precisely , we hav e k τ ( A 0 e − u 0 D 2 · · · A j 0 e − u j 0 D 2 B j 0 e − v j 0 D 2 A j 0 +1 e − u j 0 +1 D 2 · · · A j p e − u j p D 2 B j p e − v j p D 2 A j p +1 e − u j p +1 D 2 · · · A N e − u N D 2 ) k ≤ Π N i =0 k A i k Π p i =0 k ( I + D 2 ) − 1 / 2 B j i ( I + D 2 ) − 1 / 2 kk e − (1 − ǫ ) D 2 k 1 Π p i =0 k ( I + D 2 ) 1 / 2 e − u j i ǫD 2 kk ( I + D 2 ) 1 / 2 e − v j i ǫD 2 k . Next w e apply the sp ectra l theorem in E to estimate k ( I + D 2 ) 1 / 2 e − u j i D 2 kk ( I + D 2 ) 1 / 2 e − v j i D 2 k ≤ e − 1 / 2+ ǫu j i p 2 u j i ǫ e − 1 / 2+ ǫv j i p 2 v j i ǫ ≤ 1 p 2 u j i ǫ p 2 v j i ǫ . The rest of the pro of is straightforward.  4.2. Last steps of the pro o f o f the theorem. Recall fro m [24] that the universal different ial graded algebra Ω C ∞ ( M ) is endow ed with the entire b or nology Σ ℓ +1 generated by the sets n ! h S i ( dS ) ∞ where S describ es the b o unded subse ts o f C ∞ ( M ) for the C ℓ +1 top ology recalled in the b eginning of s ubsection 3.1. Recall that Ω ∗ ( B ) is similarly endow ed with the b or nology given by the b ounded sets for the C ℓ top ology on smoo th forms. In o rder to estimate the semi-norms o f ψ N ( f 0 , · · · , f N ), we nee d to expand in to its homoge ne o us co mp o - nent s. W e denote by J the subset of { 0 , 1 } 3 given by J = { (1 , 0 , 0); (0 , 1 , 0); (0 , 0 , 1) } F or α ∈ J we denote by α ( j ) the j -th comp onent of α , j = 1 , 2 , 3. So o nly one of the integers α ( j ) is non trivial a nd equals 1 . W e shall set b α ( j ) in a given expression to mean tha t when α ( j ) = 1 , we take into account b but when α ( j ) = 0 then we simply erase b from the ex pression. F or instance ( a 0 , · · · , a k , b α ( j ) , a k +1 , · · · , a n ) , equals the ( n + 2)-tuple ( a 0 , · · · , a k , b , a k +1 , · · · , a n ) when α ( j ) = 1 and the ( n + 1)-tuple ( a 0 , · · · , a n ) when α ( j ) = 0. F or α ∈ J , we set X α ( b ) := [ ∇ , b ] α (1) ( σ [ ∇ , D ]) α (2) ∇ 2 α (3) . F or any m ≥ 0, n = ( n 0 , · · · n m ) ∈ N m +1 and α = ( α 1 , · · · , α m ) ∈ J m , w e define an P m j =0 n j + P m i =1 α (1) i co chain φ m α,n with v alues in m + P m i =1 α (3) i differential forms on B , by the formula φ m α,n ( f 0 , · · · , f n 0 , g α (1) 1 1 , f n 0 +1 , · · · , f n 0 + n 1 , · · · , g α (1) m m , f n 0 + n 1 + ··· + n m − 1 +1 , · · · , f n 0 + ··· + n m ) := < f 0 , σ [ D, f 1 ] , · · · , σ [ D , f n 0 ] , X α 1 ( g 1 ) , σ [ D, f n 0 +1 ] , · · · , σ [ D , f n 0 + n 1 ] , · · · , X α m ( g m ) , σ [ D, f n 0 + ··· n m − 1 +1 ] , · · · , σ [ D , f n 0 + ··· n m ] > . 18 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 Lemma 4.6. The c o chains ψ N of t he JLO c o cycle c an b e exp ande d as a fi nite algebr aic sum over m ≥ 0 , n = ( n 0 , · · · n m ) ∈ N m +1 and α = ( α 1 , · · · , α m ) ∈ J m of t he bihomo gene ous c o chains φ m α,n . Mor e over the numb er of such φ m α,n is b oun de d by (dim B ) N +1 2 dim B . Pr o of. W e fir st replace in ψ N , ea ch factor e − u j B 2 σ by its definition, a finite p erturbative sum, and by using a s traightforw ard change of v aria bles, we eas ily deduce that ψ N ( f 0 , · · · , f N ) is a finite signed sum ov er ( m 0 , · · · , m N ) ∈ N N +1 of the terms * f 0 ; m 0 times z }| { X , · · · , X ; [ B σ , f 1 ]; m 1 times z }| { X , · · · , X ; · · · ; [ B σ , f N ]; m N times z }| { X , · · · , X + Now, repla cing X by its v a lue ∇ 2 − σ [ ∇ , D ] a nd [ B σ , f j ] by its v alue [ ∇ , f j ] − σ [ D , f j ], it is ea s y to rewrite each such term as a finite signed sum o f a ppropria te φ m α,n ’s. Next we s ee that P N j = o m j is bounded by the dimension dim B of the base b ecause we cannot ha ve mo re than dim B differen tial forms in a ny term. If we set λ N = ♯ { ( m 0 , m 1 , . . . , m N ) | X j m j ≤ dim B } and | m | = m 0 + . . . + m N then necessarily w e ha ve 2 | m | × λ N ≤ (dim B ) N +1 2 dim B as r equired.  W e have chosen to ex pa nd the JLO co cycle as a finite comb ination of bihomo geneous co c hains with r esp ect to the cochain grading and the form gr a ding. By doing so, our fo rmulae are explicit enoug h to b e paired with closed currents on the base. Prop ositi on 4.7. Set N = n 0 + · · · + n m + m , then for N lar ge, the bihomo gene ous c o chain φ m α,n c an b e estimate d as fol lows: p r  φ m α,n ( f 0 , · · · , f N )  ≤ C ′ ( S ) N +1 N ! , 0 ≤ r ≤ ℓ and f j ∈ S. wher e C ′ ( S ) is some c onstant which only dep ends on the b ounde d set S for the C ℓ +1 top olo gy on C ∞ ( M ) . Pr o of. F or s implicit y , w e denote b y i Y either contraction by the vector field Y over B , or b y its ho rizontal lift on M or o n ψ ∞ ( M | B , E ; Λ ∗ B ) and ψ ∞ ( M | B , E ; Λ ∗ B )[ σ ]. Then for fib er wise smo othing o p er ators T ∈ ψ −∞ ( M | B , E ; Λ ∗ B )[ σ ], we have ( i Y ◦ τ σ )( T ) = − ( τ σ ◦ i Y )( T ) and ( d B ◦ τ σ )( T ) + ( τ σ ◦ ∂ )( T ) = 0 . Therefore, if d Y := i Y ◦ d B is the deriv ative in the direction Y in B , then ( d Y ◦ τ σ )( T ) − ( τ σ ◦ ∂ Y )( T ) = 0 W e need to estimate for 0 ≤ s ≤ ℓ , the se mi- norms p s ( φ m α,n ( f 0 , · · · , f N )) where N = n 0 + · · · + n m + m o dd , for given f 0 , · · · , f N in a bo unded se t S for the C ℓ +1 top ology . So , we ass ume that there exists a constant C ≥ 0 such that k f j k t ≤ C for any 0 ≤ t ≤ ℓ + 1 and for 0 ≤ j ≤ N . F or the c o nv enience of the r e ader, w e firs t ex plain the pr o of for m = 0 th us g iving a guide to the general case. Step I: m = 0 W e b e gin by estimating the p s semi-norms of functions on B . They are given b y φ 0 N ( f 0 , · · · , f N ) = h f 0 , σ [ D, f 1 ] , · · · , σ [ D , f N ] i . JLO AND S PECTRAL FLOW FOR F AMILIES 19 Let k · k α denote the supremum over B of the fib erwis e α -Schatten nor m of a compact op erator o n L 2 sections. Using H¨ older’s inequa lity for each b ∈ B and taking the suprem um over B , we hav e: k φ 0 N ( f 0 , · · · , f N ) k ≤ Z ∆( N ) k τ  f 0 e − u 0 D 2 [ D , f 1 ] e − u 1 D 2 · · · [ D , f N ] e − u N D 2  k du 1 · · · du N ≤ Z ∆( N ) k f 0 kk e − u 0 D 2 k 1 /u 0 k [ D , f 1 ] kk e − u 1 D 2 k 1 /u 1 · · · k [ D , f N ] kk e − u N D 2 k 1 /u N ≤ k e − D 2 k 1 N ! × k f 0 k Π N j =1 k [ D b , f j ] k . Using Lemma 4 .3, we deduce k φ 0 N ( f 0 , · · · , f N ) k ≤ C N 0 k e − D 2 k 1 N ! × k f 0 kk f 1 k 1 · · · k f N k 1 ≤ k e − D 2 k 1 C N 0 C N +1 N ! . In the same w ay , let Y 1 , · · · , Y s be vector fields on B taken from the finite collection Y . Using Lemma 1.7, we can write d Y 1 · · · d Y s φ 0 N ( f 0 , · · · , f N ) = Z ∆( N ) τ  [ ∂ Y 1 · · · ∂ Y s ]( f 0 e − u 0 D 2 [ D , f 1 ] e − u 1 D 2 · · · [ D , f N ] e − u N D 2 )  du 1 · · · du N . Note that the op er ators ∂ Y j 1 · · · ∂ Y j k ( f ) and ∂ Y j 1 · · · ∂ Y j k [ D , f ] a re zero-th o rder differen tial op erator s on M and b y Lemma 4 .3, w e can estimate k ∂ Y j 1 · · · ∂ Y j k ( f ) k ≤ C k k f k k and k ∂ Y j 1 · · · ∂ Y j k [ D , f ] k ≤ C k k f k k +1 . Therefore, one can apply the H¨ older inequality exactly , as in the case s = 0 treated ab ove, and deduce the required estimates for all the terms that in volve no de r iv atives of the fib erwise smo othing ope r ators e − u j D 2 . Thu s, w e may co ncentrate on terms of the form τ  A 0 [ ∂ Y k 0 1 · · · ∂ Y k 0 β 0 ] e − u 0 D 2 A 1 [ ∂ Y k 1 1 · · · ∂ Y k 1 β 1 ] e − u 1 D 2 · · · A N [ ∂ Y k N 1 · · · ∂ Y k N β N ] e − u N D 2 ]  , where A j is a zero-th order pseudo differential op erator , 0 ≤ β l ≤ s and Σ β l is at mo st s a nd is pr escrib ed b y the n um ber of deriv a tives applied to g e t the op era tors A j out of the op er ators f 0 and [ D, f j ]. Now, apply again Duhamel’s form ula: ∂ Y e − uD 2 = − u Z 1 0 e − utD 2 ∂ Y ( D 2 ) e − u (1 − t ) D 2 dt = − Z u 0 e − tD 2 ∂ Y ( D 2 ) e − ( u − t ) D 2 dt. F or instance, Z ∆( N ) τ ( A 0 ∂ Y e − u 0 D 2 A 1 e − u 1 D 2 · · · A N e − u N D 2 ) du 1 · · · du N = Z ∆( N +1) τ ( A 0 e − v 0 D 2 ∂ Y ( D 2 ) e − v 1 D 2 A 1 e − v 2 D 2 · · · A N e − v N +1 D 2 ) dv 1 · · · dv N +1 =  A 0 , ∂ Y ( D 2 ) , σ A 1 , · · · , σ A N  N +1 20 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 So, the norm o f this ter m can b e estimated using Lemma 4.5 and Lemma 4.1. More pec is ely , we get for any ǫ ∈ ]0 , 1 / 2 ] (one can tak e here ǫ = 1 / 2 for simplicit y) k  A 0 , ∂ Y ( D 2 ) , A 1 , · · · , A N  N +1 k ≤ π k e − (1 − ǫ ) D 2 k 1 ǫN ! k ( I + D 2 ) − 1 / 2 ∂ Y ( D 2 )( I + D 2 ) − 1 / 2 k Π N i =0 k A i k ≤ π α 1 ( D 2 ) k τ ( e − (1 − ǫ ) D 2 ) k ǫN ! Π N i =0 k A i k The o ther, more complicated, terms are dealt with in a similar way using Lemma 4.5 a nd the metho d of Step II I where we sho w how to esta blish the b o und: k τ σ  A 0 [ ∂ Y k 0 1 · · · ∂ Y k 0 β 0 ] e − u 0 D 2 A 1 [ ∂ Y k 1 1 · · · ∂ Y k 1 β 1 ] e − u 1 D 2 · · · A N [ ∂ Y k N 1 · · · ∂ Y k N β N ] e − u N D 2 ]  k ≤ C ( ǫ ) N ! Π N i =0 k A i k . Again, the oper ators A j are here der iv atives of f 0 or [ D , f j ]’s. Step II: ge ner al m The general case inv olves differential forms o n the base manifold B obtained from co mm utators o f functions with ∇ , commutators of D with ∇ a nd also with the curv ature ∇ 2 . The latter presents some difficulties. The commutators of functions with ∇ are easy to handle, as they give zero -th order differential op er ators and can be estimated using the H¨ older inequa lity again and Lemma 4.3. On the other ha nd, co mm utators of D with ∇ and terms inv olving ∇ 2 int ro duce a dditional deriv a tives in the fib erwise direction, as these op erator s ar e first order fib er wise differential o p erator s with co efficients in differential forms of degree 1 for the firs t and degr ee 2 for the second. So, w e cannot apply direc tly the a rgument of Lemma 4.5 and we need to give ca reful estimates for such terms. The w orst situation arise s when the e nt ries B 0 , · · · , B N in the expression < B 0 , · · · , B N > are co mp o sed of ‘to o many’ fib e rwise pseudo differential op er ators of p ositive orders. By this we mean that, in addition to the B j ’s, we have the maximum n umber of commutators [ ∇ , D ] or [ ∇ 2 , D ] a nd also the maxim um num b er of directional der iv atives of the heat kernel e − u j D 2 . The latter int ro duce op era tors of or de r 2. F or tunately , a nd this seems to b e a crucial p oint here, the ba se manifold is finite dimensional and we ar e taking a t most ℓ direc tional deriv atives. Hence the num b er of entries inv o lv ing [ ∇ , D ] or [ ∇ 2 , D ] is limited by the dimension and the n um be r of deriv atives is also bounded by ℓ . Denote by k the degr ee of the differential form φ m α,n ( f 0 , · · · , f n 0 , g α (1) 1 1 , · · · , g α (1) m m , · · · , f P m i =0 n i ) . So, k = m + P m i =1 α (3) i . W e fix vector fields Z 1 , · · · , Z k on the ba se manifold B with nor ms ≤ 1 and thus need to estimate, for s ≤ ℓ , the s seminorm of the function i Z 1 · · · i Z k φ m α,n ( · · · ). This reduce s to the computation of the s seminorm o f ter ms < A 0 , · · · , A N + q + q ′ > where N entries A j are zero-th o rder, q entries ar e first order and q ′ ent ries are second or der, and where as explained ab ov e, w e can assume that N is as larg e as allow ed, while q and q ′ are b ounded by sup( ℓ, dim B ). Indeed, for N ≥ q + q ′ , we obtain the desir ed es timate as follows. First to illustrate the idea s assume that the o rder is as follows: < A 0 , B 1 , A 1 , · · · , B q , A q , C 1 , A q +1 , · · · , C q ′ , A q + q ′ , A q + q ′ +1 , · · · , A N > where the A j ’ are zero - th o rder, the B j ’s ar e first or der and the C j ’ are second or der. By using the H¨ older inequality , we reduce to the issue of estimating the expression k e − uD 2 E e − vD 2 A k 1 / ( u + v ) ≤ k e − uD 2 E ( I + D 2 ) − 1 / 2 k 1 /u k ( I + D 2 ) 1 / 2 e − vD 2 A k 1 /v . where E is at most second order. This gives for any ǫ ∈ ]0 , 1 / 2] k e − uD 2 E e − vD 2 A k 1 / ( u + v ) ≤ k e − uǫD 2 ( I + D 2 ) 1 / 2 kk ( I + D 2 ) − 1 / 2 E ( I + D 2 ) − 1 / 2 k × k ( I + D 2 ) 1 / 2 e − vǫ D 2 A k sup b τ ( e − (1 − ǫ ) D 2 b ) u + v . JLO AND S PECTRAL FLOW FOR F AMILIES 21 Now, again we hav e by the sp ectral theor em k e − uǫD 2 ( I + D 2 ) 1 / 2 k ≤ e ǫ − 1 / 2 √ 2 uǫ , and the estimate go es exactly as for the previo us simpler cases. W e thus obtain the existence of a constant C ( ǫ ) ≥ 0 such that k < A 0 , B 1 , A 1 , · · · , B q , A q , C 1 , A q +1 , · · · , C q ′ , A q + q ′ , A q + q ′ +1 , · · · , A N > k ≤ C ( ǫ ) N ! Π N i =0 k A i k Π q i =1 k ( I + D 2 ) − 1 / 2 B i k Π q ′ i =1 k ( I + D 2 ) − 1 / 2 C i ( I + D 2 ) − 1 / 2 k . Step II I: the worst case T o indicate how to handle general terms we now consider < A 0 , A 1 , · · · , A k , A k +1 , · · · , A r , A r +1 , · · · , A N > where A 1 , · · · , A k are of or de r t w o, the next r − k are of order one and the last N − r + 1 a re o f o rder zero. W e may assume in this ex pression that N is chosen so that k + r ≤ ( N − 1) / 2 = N ′ . The in tegrand of the JLO t yp e functional in this insta nce ma y be written as k Y j =0 [( I + D 2 ) j / 2 A j ( I + D 2 ) − j / 2 − 1 ( I + D 2 ) 1 / 2 e − u j D 2 ] × r Y j = k +1 [( I + D 2 ) ( k +1) / 2 A j ( I + D 2 ) − ( k +1) / 2 − 1 / 2 ( I + D 2 ) 1 / 2 e − u j D 2 ] × r + k Y i =0 ( I + D 2 ) ( k +1 − i ) / 2 A r + i +1 ( I + D 2 ) − ( k +1 − i ) / 2 ( I + D 2 ) 1 / 2 e − u r + i + i D 2 × A r +2+ k e − u r + k + 2 D 2 · · · A N e − u N D 2 Now we may integrate o ver the simplex and e s timate the norm of the resulting e xpression and we find that it is bo unded by N Y j =0 k A j ( I + D 2 ) − α j / 2 kk τ ( e − (1 − ǫ ) D 2 k Z ∆( N ) r + k +1 Y j =0 (2 ǫu j ) − 1 / 2 du 1 · · · du N = k τ ( e − (1 − ǫ ) D 2 k (2 ǫ ) − ( r + k ) / 2+1 N Y j =0 k A j ( I + D 2 ) − α j / 2 k Z ∆( N ) ( u 0 · · · u r + k +1 ) − 1 / 2 du 1 · · · du N = k τ ( e − (1 − ǫ ) D 2 k (2 ǫ ) − ( r + k ) / 2+1 N Y j =0 k A j ( I + D 2 ) − α j / 2 k π ( r + k +2) / 2 Γ( r + k 2 + 1)( N − ( r + k ))! β (( r + k ) / 2+1 , N − ( r + k +1)) where w e ha ve used Z ∆( N ) ( u 0 · · · u ℓ ) − 1 / 2 du 1 · · · du N = π ( ℓ +1) / 2 Γ( ℓ 2 + 1)Γ( N − ℓ ) β (( ℓ + 1) / 2 , N − ℓ ) . W e c a n assume, to simplify the ev a luation of the b eta function w.l.o.g., that r + k is ev en say 2 γ . The n, using the expression for the b e ta function in ter ms of gamma functions, the pr evious expression is b ounded by  π 2 ǫ  γ +1 k τ ( e − (1 − ǫ ) D 2 ) k × N Y j =0 k A j (1 + D 2 ) − α j / 2 k γ ! ( N − ( α + 1))! 22 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 Now γ ≤ N ′ , and N − ( γ + 1 ) − γ ≥ N ′ so that we can estimate the r atios of gamma functions a nd b ound the preceding expression b y  π 2 ǫ  γ +1 k τ ( e − (1 − ǫ ) D 2 ) k ( N − 1)!( N − γ )( N − γ − 1 ) × N Y j =0 k A j (1 + D 2 ) − α j / 2 k This estimate ob viously suffices to deduce the allowed estimate.  Now, in order to deduce the pro of of Theore m 3.5, we p oint out that ψ N is a sum of at most (dim B ) N +1 × 2 dim B comp onents φ m α,n . Therefore , using the Stirling estimate, it is easy to deduce the ex is tence o f a c o nstant C ( S ) dep ending only on the bo unded set S of ( C ∞ ( M ) , Σ ℓ +1 ) such that p r ( ψ N ( f 0 , · · · , f N )) ≤ C ( S ) [ N / 2]! , 0 ≤ r ≤ ℓ. 4.3. More general sup erconnections and transgressio n . W e show in this s ubsection ho w to extend Theorem 3.5 to more genera l sup erconnections as so ciated with the o dd op era tor σ D a nd prov e that the entire biv ariant cyclic homolog y class do es not dep end on certain choices made in the course o f the argument. Since the tec hniques are classical, w e sha ll be br ief. More precisely , we consider sup erconnectio ns A given a s A := B σ + A where A is an odd element of Ψ 0 ( M | B , E ; Λ ∗ B )[ σ ] , whose differential form deg rees a re p ositive. Recall that B σ = σ D + ∇ so that A = σ D + ∇ + A . Given such a superco nnection A , w e can wr ite A 2 = D 2 + X ′ where X ′ = ∇ 2 + A 2 + [ ∇ , σ D + A ] has o nly p ositive degree forms and is therefor e nilpotent. W e define the heat kernel e − A 2 of the sup er c o nnection A b y the usua l finite Duhamel expa nsion where we simply replace the opera tor X by X ′ : e − A 2 := X m ≥ 0 Z ∆( m ) e − v 0 D 2 X ′ e − v 1 D 2 · · · X ′ e − v m D 2 dv 1 · · · dv m . where ∆( m ) = { ( u 0 , · · · , u m ) ∈ R m +1 , P u j = 1 } is again the m -simplex. As with the sup erco nnection B σ which cor r esp onds to A = 0, w e define for any o dd integer n : ψ n ( f 0 , · · · , f n ) := hh f 0 , [ A , f 1 ] , · · · , [ A , f n ] ii A Prop ositi on 4 . 8. Given a su p er c onne ction A asso ciate d with σ D as ab ove, the c o chains ( ψ n ) n form, for al l ℓ ≥ 0 , an ℓ -entir e bivariant c o chain JLO( A ) fr om the u n iversal gr ade d algebr a of C ∞ ( M ) to the gr ade d algebr a Ω ∗ ( B ) of differ ent ial forms on B , in the sense of Definition 3.3 and henc e J L O( A ) is also entir e. Pr o of. Notice first that the alge br aic relations prov ed in Le mma 2.1 (all of them b esides the last one) are still v alid with A repla c ing B σ . Hence, the collection ( ψ n ) n is a gain a biv a riant cyclic co cycle by exactly the same pr o of as for Lemma 2.3. The pro of of boundedness is a rephrasing of the pro o f of 3.5. The only difference is that w e hav e to deal with new terms involving A 2 + [ B σ , A ]. The term A 2 + [ ∇ , A ] is a zero-th order fib erwis e pseudo differential op e rator with coefficients in positive degr e e forms a nd causes no tr ouble. W e only hav e to explain how to estimate terms in volving − σ [ D , A ] = − σ P k> 0 [ D , A [ k ] ], where A [ k ] is the comp onent of A which incr eases the form degree by k . But this is done using the follo wing mo dification of Lemma 4.2 and whic h is pr ov ed in the same wa y by reducing to lo cal co or dinates: sup k Z 1 k≤ 1 , ··· , k Z k k≤ 1 ,Y 1 , ··· , Y q ∈Y k ( I + D 2 ) − 1 / 2 [ ∂ Y 1 · · · ∂ Y q ]( i Z 1 ∧··· Z k [ D , A [ k ] ]) k = β ′ q < + ∞ , W e omit the pro of here. Then the rest of the pro of is tedious but is exactly a r ephrasing of the pro of giv en in the previous s ubsection.  Remark 4.9. We show in The or em 4.12 that the ℓ -entir e c ohomolo gy class of JLO( A ) c oincides with t he ℓ -entir e c ohomolo gy class of JLO( B σ ) . JLO AND S PECTRAL FLOW FOR F AMILIES 23 W e now pro ceed to prove the main res ult of this subsection, namely the transgr e ssion formula for our JLO ent ire biv ariant co cycle. W e follow the method ado pted in [16]. Set, for any sup erconnection A ass o ciated with σ D as ab ov e, and with V a homog eneous fib erwis e pse udo diff erential op erato r with co efficients in differential forms on the base B : Ch( A , V )( f 0 , · · · , f n ) := n X i =0 ( − 1) i | V | hh f 0 , [ A , f 1 ] , · · · , [ A , f i ] , V , [ A , f i +1 ] , · · · , [ A , f n ] ii A , and α ∗ ( A , V )( f 0 , · · · , f n ) := n X i =1 ( − 1) ( i − 1)( | V | +1) hh f 0 , [ A , f 1 ] , · · · , [ V , f i ] , · · · , [ A , f n ] ii A . Lemma 4.10 . • Assume that the pseudo differ ential or der of V is ≤ 1 , then Ch( A , V ) is an ℓ -entir e bivariant c o chain, for any ℓ ≥ 0 . • Assu me that the pseudo differ ential or der of V is ≤ 0 , then α ∗ ( A , V ) is an ℓ -ent ir e bivari ant c o chain, for any ℓ ≥ 0 . Pr o of. W e o nly give the pro of for A = B σ and leave the g eneral case a s an exercise. Let us no w prov e for instance the first item, the se c ond b e ing easier since V is bo unded. Applying the definition of e − u B 2 σ in the expression hh f 0 , [ B σ , f 1 ] , · · · , [ B σ , f i ] , V , [ B σ , f i +1 ] , · · · , [ B σ , f n ] ii B σ , we reduce to the estimates of terms o f the form < f 0 , X , · · · , X , [ B σ , f 1 ] , X , · · · , X , · · · , [ B σ , f i ] , X , · · · , X ; V ; X , · · · , X, [ B σ , f i +1 ] , X , · · · , X , · · · , [ B σ , f n ] , X , · · · , X > The p o int is to apply the argument of Steps I I and I I I in the pro o f o f Theorem 3.5, b y simply a dding one op erator of orde r 1 in the en tr ies. Recall that this can b e done as lo ng as the num b er o f ope r ators of order 1 or 2 is no t too big with re sp ect to the num ber o f oper ators of order 0 . But, no tice that V only app ears once, the first o rder op era tor X has co efficients in differential forms of p os itive deg r ee only and hence cannot app ear more than the dimension of B times. Therefore, since w e only need the estimates for n large, the same pro o f w orks and we obtain the r equired estimates ex actly as in Steps II and I I I of the pro of of Theorem 3.5. Notice that we hav e to estimate the sum of n + 1 terms o f the form hh f 0 , [ B σ , f 1 ] , · · · , [ B σ , f i ] , V , [ B σ , f i +1 ] , · · · , [ B σ , f n ] ii B σ , but since the estimate involv es 1 /n !, we get the allow ed estimate of the kind C ( S ) [ n/ 2]! .  Prop ositi on 4.11. The fol lowing identity hold s ( d B + ( − 1) | V | ( b + B ) ) Ch( B σ , V ) + Ch( B σ , [ B σ , V ]) + ( − 1) | V | α ∗ ( B σ , V ) = 0 . Pr o of. F or 0 ≤ i ≤ n , w e a pply the third r elation of Lemma 2.1 to the op era tors A 0 = f 0 , A j = [ B σ , f j ] for 1 ≤ j ≤ i, A i +1 = V and A j = [ B σ , f j − 1 ] for j ≥ i + 2 . So, for i = 0 for insta nce, this means that we apply that relation to A 0 = f 0 , A 1 = V and A j = [ B σ , f j − 1 ] for j ≥ 2. F or any i this gives us a relation θ i + d B θ ′ i = 0 where ( − 1) i | V | θ i = X i 1 + X i 2 + X i 3 = 0 where X i 1 = ( − 1) i | V | hh [ B σ , f 0 ] , · · · , [ B σ , f i ] , V , [ B σ , f i +1 ] , · · · , [ B σ , f n ] ii X i 2 = X 1 ≤ j ≤ i ( − 1) i | V | + j − 1  f 0 , [ B σ , f 1 ] , · · · , [ B 2 σ , f j ] , · · · , [ B σ , f i ] , V , [ B σ , f i +1 ] , · · · , [ B σ , f n ]  + n X j = i +1 ( − 1) j − 1+( i +1) | V |  f 0 , [ B σ , f 1 ] , · · · , [ B σ , f i ] , V , [ B σ , f i +1 ] , · · · , [ B 2 σ , f j ] , · · · , [ B σ , f n ]  and X i 3 = ( − 1) i ( | V | +1) hh f 0 , [ B σ , f 1 ] , · · · , [ B σ , f i ] , [ B σ , V ] , [ B σ , f i +1 ] , · · · , [ B σ , f n ] ii . 24 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 Finally , θ ′ i is giv en b y θ ′ i = hh f 0 , [ B σ , f 1 ] , · · · , [ B σ , f i ] , V , [ B σ , f i +1 ] , · · · , [ B σ , f n ] ii . Thu s the expression P n i =0 ( − 1) i | V | ( θ i + d B θ ′ i ) = 0 allows us to write n X i =0 X i 1 + n X i =0 X i 2 + n X i =0 X i 3 + d B Ch( B σ , V ) = 0 . Now we hav e, by insp ectio n, the relation n X i =0 X i 3 = Ch( B σ , [ B σ , V ])( f 0 , · · · , f n ) . Similarly , we leav e it to the reader to directly compute P n i =0 X i 2 . One finds n X i =0 X i 2 = ( − 1) | V | [ b Ch( B σ , V ) + α ∗ ( B σ , V )]( f 0 , · · · , f n ) . Next, using the s econd relation of Lemma 2.1, we see that B Ch( B σ , V )( f 0 , · · · , f n ) = n X i =0 ( − 1) ( i +1) | V | hh [ B σ , f 0 ] , · · · , [ B σ , f i ] , V , [ B σ , f i +1 ] , · · · , [ B σ , f n ] ii . The conclusion follo ws immediately .  W e are no w in p os ition to prov e Theorem 4. 12. L et A b e, as b efor e, the su p er c onne ction A := B σ + A . Then the JLO ℓ -entir e c o cycle JLO( A ) asso ciate d with the sup er c onne ction B σ + A is c ohomolo guous to the J LO ℓ -entir e c o cycle JLO( B σ ) asso ciate d with the sup er c onne ction B σ = ∇ + σ D . Pr o of. Let B σ,s := B σ + sA b e the smo o th linear path of super connections ass o ciated with σ D . Then w e can write, using the fifth rela tion of Lemma 2.1: d ds hh f 0 , [ B σ,s , f 1 ] , · · · , [ B σ,s , f n ] ii B σ,s = − n X i =0 hh f 0 , [ B σ,s , f 1 ] , · · · , [ B σ,s , f i ] , [ B σ,s , A ] , [ B σ,s , f i +1 ] , · · · , [ B σ,s , f n ] ii B σ,s + n X i =1 hh f 0 , [ B σ,s , f 1 ] , · · · [ B σ,s , f i − 1 ] , [ A, f i ] , · · · , [ B σ,s , f n ] ii B σ,s Notice that | A | = 1 while | [ B σ,s , A ] | = 2. Hence we o btain d ds hh f 0 , [ B σ,s , f 1 ] , · · · , [ B σ,s , f n ] ii B σ,s = − Ch( B σ,s , [ B σ,s , A ])( f 0 , · · · , f n ) + α ∗ ( B σ,s , A )( f 0 , · · · , f n ) . But w e kno w from P rop osition 4.11 tha t − Ch( B σ,s , [ B σ,s , A ]) + α ∗ ( B σ,s , A ) = [ d B − ( b + B ) ] Ch( B σ,s , A ) , which completes the pro of since Ch( B σ,s , A ) is an even ℓ -e ntire cochain.  JLO AND S PECTRAL FLOW FOR F AMILIES 25 5. Comp a tibility with the higher spectral flow 5.1. Highe r sp e ctral flo w. An a pplication of o ur ent ire JLO co cycle comes from its r elation with the higher sp ectr a l flow. Using our pr evious r e sults we expla in in this Sectio n how to prove the equality betw een the Chern character of the higher sp ectra l flow and the corresp onding JL O pairing , which is well defined in the F r´ echet topolo gies. Higher sp e ctral flow, introduced in [15] (see also [22]) for a family of fiberwis e self-adjoint elliptic op erator s D = ( D b ) b ∈ B , is only well defined under the assumption that the K 1 class defined by the family is tr ivial. W e assume this from now o n. Our pr o of that the Chern character o f hig her sp ectral flow coincides with the pair ing with our entire biv ar iant JLO co cycle is a gener a lization o f Getzler ’s pro of in the case of a single o p er ator [17]. Recall that the fib erwise generalized Dirac o p erator D defines a class [ D ] in the Ka sparov gr oup K K 1 ( M , B ) [20], and hence using the Kasparov pro duct, a ho momophism K 1 ( M ) → K 0 ( B ) which assigns to U ∈ K 1 ( M ) the clas s U ∩ [ D ]. This Kasparov pro du ct is an index map which is describ ed b elow us ing either families of T o eplitz op erator s or the notio n of higher sp ectral flow. Denoting by E the entire cyclic homolo g y w e prov e in the present Section c ommut ativity o f the following diagram: K 1 ( M ) ❄ Ch SF( D, · ) ✲ JLO( D ) ✲ K 0 ( B ) ❄ ch H E 1 ( C ∞ ( M )) H even ( B , C ) Thu s the ℓ -entire c y clic cohomolo gy class JLO( D ) is precise ly the biv ar iant Che r n-Connes character of [ D ]. Com bining our re s ult with the main result of [1 5], w e deduce that JLO( D ) co incides up to the (obviously b ounded) Ho chsc hild-K ostant-Rosenberg-Connes (HKRC) map for M , with the top ologica l map H odd ( M ) → H even ( B ) giv en up to consta nt by ω 7− → Z M /B ω ∧ ˆ A ( T M | B ) . By using the r esults of [23], we k now that the K 1 index of D = ( D b ) b ∈ B is zero if and only if there exists a (s mo oth) spec tr al section P for D , tha t is, a s mo oth fa mily of self-adjoint fib erwise ps eudo differential pro jections P = ( P b ) b ∈ B acting on the L 2 -sections such that fo r s ome smo oth non-neg ative function  on B , P b 1 ]  ( b ) , + ∞ ) ( D b ) = P b and P b 1 ] −∞ , −  ( b )[ ( D b ) = 0 , ∀ b ∈ B . The following result is taken from [23]. Prop ositi on 5.1. [23] L et D = ( D b ) b ∈ B b e as b efor e the fib erwise gener alize d D ir ac op er ator along the smo oth fibr ation π : M → B and assume that the index of D in K 1 ( B ) is trivial. Given a sp e ctr al se ction P for D , ther e ex ists a self-adjoint fib erwise zer o-th or der pseudo differ ential op er ator A ∈ Ψ 0 ( M | B ; E ) su ch that for any b ∈ B the op er ator D b + A b is invertible and P b c oincides with 1 [0 , + ∞ ) ( D b + A b ) . W e as s ume from now on that the op erator A is chosen as in the previous pr op osition and thus asso c iated with a fix ed s p e c tral section P for D . So P = 1 [0 , + ∞ ) ( D + A ) and D + A is a zer o-th o r der p er turbation of D and is a family of in vertible op era tors. Prop ositi on 5.2. F or any U ∈ GL N ( C ∞ ( M )) , t he op er ator P U P := ( P ⊗ 1 N ) ◦ U ◦ ( P ⊗ 1 N ) , acting fi b erwise on the image of L 2 ( M b , E ) ⊗ C N under t he pr oje ction P b ⊗ 1 N , is a smo oth family of F r e dholm op er ators whose index class in K 0 ( B ) is denote d Ind( T U ) . Then, Ind( T U ) do es not dep end on the choic e of the sp e ctr al se ction P and only dep ends on the K 1 class of U . 26 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 Pr o of. Compare with [15]. F o r any fixe d b ∈ B , the us ual pro of for a single T o eplitz op erato r on the o dd dimensional closed manifold M b shows that P b U b P b = ( P b ⊗ 1 N ) ◦ U | M b ◦ ( P b ⊗ 1 N ) is a F redholm op era tor in the Hilbert space P b ( L 2 ( M b , E | M b )) N . Hence, P U P is a smo oth family of F redholm op erator s on the image of P . W e then know that the homotopy cla s s of this family defines a class in K 0 ( B ), see [1]. T o explicitly define this cla s s a s the At iyah-Singer index o f a fib erwise elliptic op era tor, we follow [3] and define the ze r o-th order fiberwise elliptic pse udo differential op era to r T U,P by T U,P := I − P + P U P . The index Ind ( P U P ) is, by definition, the Atiy a h-Singer index c lass of T U,P in K 0 ( B ) [2]. If we c ho ose another spectra l section P ′ then the usual computation shows that the op era tor T U,P ′ is a p e rturbation of T U,P , therefore the index cla ss is unc hanged. A homotopy class o f in vertibles U t yields a homotopy cla ss of principal symbols of the fiberwis e opera tor T U,P and hence the index is unc ha nged.  Definition 5.3. [1 5] Assume that [0 , 1] ∋ t 7→ D t := ( D t,b ) b ∈ B is a sm o oth p ath of fib erwise el liptic pseudo- differ ential op er ators such that the index class of the endp oints, D 0 and D 1 in K 1 ( B ) , ar e trivial. Cho ose sp e ctra l se ctions P 0 , P 1 for D 0 , D 1 r esp e ctively and fix a sp e ctr al se ction Q = ( Q t ) t ∈ [0 , 1] for the total family viewe d as a fib erwise op er ator over B × [0 , 1] . Then the sp e ctr al flow of the p ath ( D t ) t ∈ [0 , 1] with r esp e ct to P 0 and P 1 is the class in K 0 ( B ) define d by: SF( D ; P 0 , P 1 ) := [ P 1 − Q 1 ] − [ P 0 − Q 0 ] . It is easy to check that SF( D ; P 0 , P 1 ) do es not depend o n the choice of the global sp ectral section Q . In this pap er w e ar e ma inly interested in the affine path D t := D + tU − 1 [ D , U ] where D is a family of ge neralized Dirac op erators ov e r B who se index clas s in K 1 ( B ) is triv ial, and U is a given element o f GL N ( C ∞ ( M )). In this c a se the endpoints are conjugate and we consider the spectra l flow with res pe ct to the sp ectral sections P 0 = P and P 1 = U − 1 P U , where P is a fix e d sp e c tral section for D . It turns out that the spectr a l flow do e s not dep end on P either a nd is an inv ar iant of the principal symbol of D and of the homotopy cla s s of U . W e denote it SF( D, U ). Indeed, Dai a nd Z ha ng prov ed the following Prop ositi on 5.4. [1 5] . We have in K 0 ( B ) , Ind( T U ) = − SF( D , U ) . 5.2. Second theorem and reduction to the third theorem. The pairing of ℓ -entire biv a riant cyclic homology with en tire cyclic ho mo logy with r esp ect to Σ ℓ reads in o ur case as follo ws: < JLO( D ) , U > := X n ≥ 0 ( − 1) n n !  U − 1 , [ B σ , U ] , · · · , [ B σ , U − 1 ] , [ B σ , U ]  B σ , 2 n +1 Theorem 5.5. Assu me t hat the index class of D in K 1 ( B ) is trivial. Then for any U ∈ GL N ( C ∞ ( M )) , the fol lo wing r elation holds in the even de Rham c ohomol o gy of t he b ase m anifold B : 1 √ π < JLO( D ) , U > = ch(SF( D , U )) = − ch(Ind ( T U )) . wher e ch is the u sual Chern char acter on the manifold B . The las t rela tion is clear from the previous pro p o sition and the fact that the Chern c haracter only dep ends on the K -theory cla ss. The pro of of this theorem is long a nd w e split it in to several lemmas and prop ositions. Lemma 5. 6 . L et A b e a sup er c onne ction asso ciate d with t he op er ator σD as in the pr evious se ctions. Define the affine p ath of sup er c onne ctions ( A t ) 0 ≤ t ≤ 1 given by A t := A + tU − 1 [ A , U ] . Then (i) the differ ent ial form R 1 0 τ σ ( U − 1 [ A , U ] e − A 2 t ) dt is a close d form on B , (ii) the c ohomolo gy class of R 1 0 τ σ ( U − 1 [ A , U ] e − A 2 t ) dt do es not dep end on the choic e of sup er c onn e ction A . JLO AND S PECTRAL FLOW FOR F AMILIES 27 Pr o of. (1) Let ˜ A := dt ∂ ∂ t + A t be the asso cia ted super connection for the fibration M × [0 , 1] → B × [0 , 1]. Therefore, the differen tial form τ σ ( e − A 2 ) is clos ed in B × [0 , 1]. A straightforw ard computation using the fact that τ σ ( e − A 2 t ) is itself closed in B , pr ov es the following relation d B  τ σ ( ˙ A t e − A 2 t )  = d dt τ σ ( e − A 2 t ) , Hence, d B Z 1 0 τ σ ( ˙ A t e − A 2 t ) dt = τ σ ( e − A 2 1 ) − τ σ ( e − A 2 0 ) = τ σ ( U − 1 e − A 2 0 U ) − τ σ ( e − A 2 0 ) = 0 . The last equa lity is deduced fr om the rela tion σ U = U σ and the g raded tracial pro p e rty of the functiona l τ . (2) Assume that w e ar e given another sup erconnection B ′ asso ciated with σ D . Cons ider the corres po nding affine path B ′ t := B ′ + tU − 1 [ B ′ , U ] a s b efor e, and the smo oth family A t,s = B t + s ( B ′ t − B t ) o f super connections asso ciated with σD , where s als o runs over [0 , 1]. W e then set D := A t,s + dt ∂ ∂ t + ds ∂ ∂ s . Clearly , D is a sup erconnec tio n asso ciated with σ D but for the smo oth fibration M × [0 , 1] 2 → B × [0 , 1] 2 . Therefore, the differen tial form τ σ ( e − D 2 ) is closed in B × [0 , 1] 2 . Using this fact and that d B τ σ ( e − A 2 t,s ) = 0 , we obtain the relation d B τ σ ( e − A 2 t,s ∧ K 2 t,s ) = dt ∧ ds  ∂ ∂ s τ σ ( ∂ A t,s ∂ t e − A 2 t,s ) − ∂ ∂ t τ σ ( ∂ A t,s ∂ s e − A 2 t,s )  , where K t,s = dt ∧ ∂ A t,s ∂ t + ds ∧ ∂ A t,s ∂ s . Now we ca n compute Z 1 0 τ σ ( ∂ A t, 1 ∂ t e − A 2 t, 1 ) dt − Z 1 0 τ σ ( ∂ A t, 0 ∂ t e − A 2 t, 0 ) dt = Z 1 0 ∂ ∂ s  Z 1 0 τ σ ( ∂ A t,s ∂ t e − A 2 t,s ) dt  ds = Z [0 , 1] 2 ∂ ∂ s τ σ ( ∂ A t,s ∂ t e − A 2 t,s ) dt ∧ ds = Z [0 , 1] 2 ∂ ∂ t τ σ ( ∂ A t,s ∂ s e − A 2 t,s ) dt ∧ ds + d B Z [0 , 1] 2 τ σ ( e − A 2 t,s ∧ K 2 t,s ) = Z 1 0  τ σ ( ∂ A 1 ,s ∂ s e − A 2 1 ,s ) − τ σ ( ∂ A 0 ,s ∂ s e − A 2 0 ,s )  ds + d B Z [0 , 1] 2 τ σ ( e − A 2 t,s ∧ K 2 t,s ) . Notice that A 1 ,s = U − 1 A 0 ,s U a nd ∂ A 1 ,s ∂ s = B ′ 1 − B 1 = U − 1 ∂ A 0 ,s ∂ s U. Therefore, the proo f is complete.  The ne x t prop osition is an ea sy rephrasing of a r esult of Dai and Zha ng: Prop ositi on 5.7. [1 5] L et A b e the Bismut sup er c onne ction asso ciate d with σ D , then the c ohomolo gy class of t he differ ent ial form − 1 π 1 / 2 R 1 0 τ σ ( ˙ A t e − A 2 t ) dt c oincides with the Chern char acter of the sp e ctr al flow, i.e. ch (SF( D , U )) = − 1 π 1 / 2  Z 1 0 τ σ ( ˙ A t e − A 2 t ) dt  . 28 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 The pro of of this prop os ition r elies on the go od b ehaviour of the asymptotics of the rescaled Bismut sup e rconnection. T o sum up, in or de r to pr ov e Theor em 5.5, we are r educed to pr oving the following auxiliary result. Theorem 5. 8. When A = B σ and as differ ential forms on the b ase m anifold B , we have t he fol lowi ng e quality: Z 1 0 τ σ ( ˙ A t e − A 2 t ) dt = 1 2 X k ≥ 0 ( − 1) k k !(  U − 1 , [ A , U ] , [ A , U − 1 ] , · · · , [ A , U ]  2 k +1 , A −hh U, [ A , U − 1 ] , [ A , U ] , · · · , [ A , U ] , [ A , U − 1 ] ii 2 k +1 , A ) + exact forms on t he b ase . Note that the additio nal exact fo rms can b e g iven explicitly . 5.3. Pro of of the third theorem . As ex plained before , the pro of of this theorem follows the lines of [17] and w e split the argument into a n umber of steps. First we double up our Hilber t space and replace U by V :=  0 iU − 1 − iU 0  , so that V 2 = I , and let ˜ B =  B σ 0 0 − B σ  . Consider the op erator A a sso ciated with the fibr ation M × [0 , 1] × [0 , + ∞ ) → B × [0 , 1] × [0 , + ∞ ) , and giv en b y A := ˜ B t,x +  d 0 0 d  where ˜ B t,x := ˜ B t + xV and ˜ B t = ˜ B − tV [ ˜ B , V ] . W e us e gra ded commutators so that for insta nce [ ˜ B , V ] = ˜ B V + V ˜ B . The differential d is the de Rham differential on [0 , 1] × [0 , + ∞ ). Moreov er , we extend τ σ to a super trace τ s given b y τ s ( A ) := τ σ ( A 11 ) + τ σ ( A 22 ) . It is then straightforw ard to c heck, using the Bianchi identit y satisfied b y B σ , that d B τ s ( e − A 2 ) = 0. Com- puting the square of A one finds using for instance the relation V [ ˜ B , V ] V = [ ˜ B , V ]: A 2 = Y t,x + dxV − dtV [ ˜ B , V ] where Y t,x = ( ˜ B t ) 2 + x (1 − 2 t )[ ˜ B , V ] + x 2 . Recall then that the differential form τ s ( e − A 2 ) is automatically closed as a differential form on the manifold with bounda ry B × [0 , 1] × [0 , + ∞ ). Lemma 5.9. L et R ( x 0 ) denote the r e ctangle [0 , 1] × [0 , x 0 ] , then in Ω ∗ ( B ) we have: Z ∂ R ( x 0 ) τ s ( e − A 2 ) ∈ d B Ω ∗ ( B ) . Pr o of. W e hav e by a direct computation Z ∂ R ( x 0 ) τ s ( e − A 2 ) = Z ∂ R ( x 0 ) ( dxτ s ( V e − Y t,x ) − dtτ s ( V [ ˜ B , V ] e − Y t,x )) . Hence the differen tial form R ∂ R ( x 0 ) τ s ( e − A 2 ) on the bas e ma y be written as Z ∂ R ( x 0 ) τ s ( e − A 2 ) = Z ∂ R ( x 0 ) dxω t,x − dtα t,x , where ω t,x and α t,x are smoo th families of differential forms on B . W e thus have Z ∂ R ( x 0 ) τ s ( e − A 2 ) = Z 1 0 [ α t,x 0 − α t, 0 ] dt − Z x 0 0 [ ω 1 ,x − ω 0 ,x ] dx = Z 1 0 Z x 0 0 [ ∂ α ∂ x − ∂ ω ∂ t ] dtdx. JLO AND S PECTRAL FLOW FOR F AMILIES 29 The closedness of τ s ( e − A 2 ) implies in pa rticular that the compo nent that contains dt ∧ dx , say β t,x dt ∧ dx satisfies d B β + ∂ α ∂ x − ∂ ω ∂ t = 0 . But this is pr ecisely what we need to complete the pro o f.  W e denote for x > 0 by γ x the path [0 , 1] × { x } oriented in the direction of increasing t ∈ [0 , 1 ]. W e als o consider the path Γ x t = { t } × [0 , x ] for t ∈ [0 , 1] and x > 0, oriented in the direction of increasing y ∈ [0 , x ]. Lemma 5.10 . We have the fol lowing e quality of the c orr esp onding even forms Z γ 0 τ s ( e − A 2 ) = 2 Z 1 0 τ σ ( ˙ B t e − B 2 t ) dt. Pr o of. W e hav e Y t, 0 = ˜ B 2 t =  ( B σ + tU − 1 [ B σ , U ]) 2 0 0 ( B σ + tU [ B σ , U − 1 ]) 2  Hence we obtain Z γ 0 τ s ( e − A 2 ) = − Z 1 0 τ s ( V [ ˜ B , V ] e − Y t, 0 ) dt = Z 1 0 τ σ (( U − 1 [ B σ , U ]) e − ( B σ + tU − 1 [ B σ ,U ]) 2 ) dt − Z 1 0 τ σ ( U [ B σ , U − 1 ] e − ( B σ + tU [ B σ ,U − 1 ]) 2 ) dt = 2 Z 1 0 τ σ ( U − 1 [ B σ , U ] e − ( B σ + tU − 1 [ B σ ,U ]) 2 ) dt where the last step is obaine d by t → 1 − t in the second term. Notice that only the even forms ar e relev ant for us.  Lemma 5.11 . We have lim x → + ∞ " Z Γ x 1 τ s ( e − A 2 ) + Z Γ x 0 τ s ( e − A 2 ) # = 0 . Mor e over up t o forms that ar e exact on B , lim x → + ∞ " Z Γ x 0 τ s ( e − A 2 ) # = 1 2 ( < JLO( D ) , U > + < JLO( D ) , U − 1 > ) . Pr o of. Only the term − τ s ( V e − Y t,x ) contributes to the integrals R Γ x t . W e thus need to compare Y 1 ,x with Y 0 ,x . But notice that Y 1 ,x = ˜ B 2 1 − x [ ˜ B , V ] + x 2 = V ˜ B 2 V − x [ ˜ B , V ] + x 2 . On the o ther hand, V [ ˜ B , V ] V = [ ˜ B , V ] so that Y 1 ,x = V h ˜ B 2 − x [ ˜ B , V ] + x 2 i V = V Y 0 , − x V . Hence, τ s ( V e − Y 1 ,x ) = τ s ( V 2 e − Y 0 , − x V ) = τ s ( e − Y 0 , − x V ) = τ s ( V e − Y 0 , − x ) . Therefore w e obtain Z Γ x 1 τ s ( e − A 2 ) = Z 0 − x τ s ( V e − Y 0 ,y ) dy . Now, since Y 0 ,x = ˜ B 2 + x [ ˜ B , V ] + x 2 and using Duhamel we know that Z R τ s ( V e Y 0 ,x ) dx = X k ≥ 0 DD V , [ ˜ B , V ] , · · · , [ ˜ B , V ] EE ˜ B Z R x k e − x 2 dx, a series which conv erges in the F r´ echet to p o logy of Ω ∗ ( B ) b ecause R R x k e − x 2 = Γ( k +1 2 ) while the JLO brack et introduces a fac to r of 1 /k ! (see the pro of of the next lemma for details). Next, c omputing, hh V , [ ˜ B , V ] , · · · , [ ˜ B , V ] ii ˜ B in ter ms o f the mult ilinear functional corresp onding to B σ , s hows that it is triv ial 30 M-T. BENAMEUR AND A. L. CAREY NO VEMBER 19, 2018 when the num be r of co mmut ators [ ˜ B , V ] is e ven. Mor eov er, when k = 2 ℓ + 1 is o dd clear ly the integral R R x k e − x 2 dx v anishes, and so Z R τ s ( V e − Y 0 ,x ) dx = 0 or equiv alently lim x → + ∞ Z Γ x 1 τ s ( e − A 2 ) = lim x → + ∞ Z Γ x 0 τ s ( e − A 2 ) . If w e in teg rate ov er (0 , + ∞ ) rather than R in the previous computation, then w e obtain Z + ∞ 0 τ s ( V e − Y 0 ,x ) dx = − X ℓ ≥ 0 hh V , [ ˜ B , V ] , · · · , [ ˜ B , V ] ii 2 ℓ +1 , ˜ B Z ∞ 0 x 2 ℓ +1 e − x 2 dx = − 1 / 2 X ℓ ≥ 0 ℓ ! hh V , [ ˜ B , V ] , · · · , [ ˜ B , V ] ii 2 ℓ +1 , ˜ B . Hence V e − u 0 ˜ B 2 [ ˜ B , V ] e − u 1 ˜ B 2 · · · [ ˜ B , V ] e − u 2 ℓ +1 ˜ B 2 = ( − 1) ℓ +1 U − 1 e − u 0 B 2 σ [ B σ , U ] e − u 1 B 2 σ · · · [ B σ , U ] e − u 2 ℓ +1 B 2 σ 0 0 − U e − u 0 B 2 σ [ B σ , U − 1 ] e − u 1 B 2 σ · · · [ B σ , U − 1 ] e − u 2 ℓ +1 B 2 σ ! So that, using the fact that the differential forms inv o lved are even, ( − 1) ℓ +1 hh V , [ ˜ B , V ] , · · · , [ ˜ B , V ] ii 2 ℓ +1 = hh U − 1 , [ B σ , U ] , [ B σ , U − 1 ] , · · · , [ B σ , U − 1 ] , [ B σ , U ] ii 2 ℓ +1 − hh U, [ B σ , U − 1 ] , [ B σ , U ] , · · · , [ B σ , U ] , [ B σ , U − 1 ] ii 2 ℓ +1 .  Now we r emark that the last line of the proo f can be simplified, up to the addition of exac t forms on the base, using the co cycle prop er t y of JLO( D ) lim x → + ∞ Z Γ x 0 τ s ( e − A 2 ) = X k ≥ 0 ( − 1) k +1 k ! hh U − 1 , [ B σ , U ] , · · · , [ B σ , U − 1 ] , [ B σ , U ] ii 2 k +1 , B σ . T o end the pro of of Theorem 5.8, we ar e reduced to the following Lemma 5.12 . In the F r´ ech et top olo gy of Ω ∗ ( B ) , we have : lim x 0 → + ∞ R γ x 0 τ s ( e − A 2 ) = 0 . Pr o of. Recall that Z γ x 0 τ s ( e − A 2 ) = − Z 1 0 τ s ( V [ ˜ B , V ] e − Y t,x 0 ) dt. Moreov er, an applica tion of our main theorem 3.5 shows that the following Duhamel expansion is co nv er gent in the F r´ echet top o logy of Ω ∗ ( B ), with s um precisely τ s ( V [ ˜ B , V ] e − Y t,x 0 ) e − x 2 0 X k ≥ 0 x k 0 (1 − 2 t ) k Z ∆( k ) τ s ( V [ ˜ B , V ] e − u 0 ˜ B 2 t [ ˜ B , V ] e − u 1 ˜ B 2 t · · · [ ˜ B , V ] e − u k ˜ B 2 t ) du 1 · · · du k . Repro ducing the estimates of the s emi-norms ( p q ) q ≥ 0 of the expression Z ∆( k ) τ s ( V [ ˜ B , V ] e − u 0 ˜ B 2 t [ ˜ B , V ] e − u 1 ˜ B 2 t · · · [ ˜ B , V ] e − u k ˜ B 2 t ) du 1 · · · du k we see that w e can find co nstants C q depe nding on the unitary U such that p q ( Z ∆( k ) τ s ( V [ ˜ B , V ] e − u 0 ˜ B 2 t [ ˜ B , V ] e − u 1 ˜ B 2 t · · · [ ˜ B , V ] e − u k ˜ B 2 t ) du 1 · · · du k ) ≤ C k q /k ! . JLO AND S PECTRAL FLOW FOR F AMILIES 31 Finally notice that R 1 0 | 1 − 2 t | k dt = 2 / ( k + 1) . 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