On the Noncommutative Residue for Projective Pseudodifferential Operators

On the Noncomm utativ e Residue for Pro jectiv e Pseudo differen tial Op erators J¨ org Seiler and Alexander Strohmaier Abstract. A well kno wn result on pseudo differen tial operators states that the noncomm utative residue (W odzicki residue) of a pseudodifferen tial pro jec- tion v anishes. This statemen t is non-local and im plies the regularit y of the eta inv ariant at zero of Dirac t ype operators. W e pro ve that in a filtered algebra the v alue of a pro jection under an y r esidual trace depends only on the princi- pal part of the pro jection. This general, purely algebraic statemen t applied to the a lgebra of pro jective pseudodifferent i al op er ators implies that the noncom- mut ative r esi due factors to a map from the twisted K -theory of the co-sphere bundle. W e use argument s from t wisted K -theory to show that this map v an- ishes in o dd dimensions, thus sho wing that the nonco mmutativ e r esidue o f a pro jective pseudo differen tial p r o jection v anishes. This also give s a v ery direct proof i n t he classical setting. 1. In tro duction The algebra of sym b ols of classical pseudodiffer e n tial op erators ΨDO cl ( X, E ) on a closed manifold X acting on sections of a vector bundle E can b e defined a s the quotient of ΨDO cl ( X, E ) by the ideal of smo othing op erato rs. Since pseudo differ- ent ia l operato rs are smo oth off the diagonal the symbol algebra is loc a lized on the diagonal and it therefor e ca n als o b e defined lo cally , using the pro duct expa nsion formula and the change of c har ts formula for pseudo differential op e rators. That the lo cal heat k er nel co efficien ts a nd the index of elliptic pseudo differential op e rators are lo cally computable relies on the fact that the index and asymptotic spectral prop erties of pseudo differential o pera tors dep end o nly on their class in the s y m b ol algebra. Note that the pr incipal sy mbo l of a pseudo differential op erator is a section of the bundle of endomorphisms of π ∗ E , where π : T ∗ X → X is the canonical pro jection. 2010 Mathematics Subject Classific ation. Primary 58J42; Secondary 47G30, 19L50. Key wor ds and phr ases. Non commutativ e resi due, pr o jectiv e pseudodifferential operators, Azuma ja bundle, twisted K- theory . 1 2 J ¨ OR G SEILE R AND ALEXANDE R STROHMAIER The bundle of endomorphisms of a co mplex her mitian vector bundle is a bun- dle of simple matrix alg ebras with ∗ -structure. How ever, not all bundles of simple matrix algebra s with ∗ -structure, so-ca lle d Azumay a bundles, are isomorphic to e n- domorphism bundles of hermitian vector bundles. The obstructio n is the so- called Dixmier-Douady class in H 3 ( X, Z ). Given an Azumay a bundle A it is po ssible to construct algebra s of symbo ls whos e pr incipal s ym b ols takes v alues in the space of sections o f the Azumay a bundle π ∗ A (see for example [ MMS05 ] a nd the disc us sion in [ MMS06 ]). F o llo wing [ MMS05 ] we re fer to such a symbo l alg ebra a s the alge- bra of s y m b ols of pro jectiv e pseudo differen tial op erato r s. F o r such symbol algebra s one ca n define an index and Mathai, Melrose and Singe r [ MMS06 ] proved an index formula for pro jective pseudo different ia l op erato r s, analog o us to the Atiy ah- Singer index formula. The top ological index in this case is a map from t wis ted K -theor y to R . It has also been sho wn in [ MMS06 ] that any oriented manifold admits a pro jectiv e Dirac ope rator even if the manifold do es not admit a spin structure. In this case its index may fa il to b e an integer. Another imp ortant quantit y that dep ends o nly on the clas s of the symbol of a pseu- do differen tial op erator is the so-called W o dzic ki residue or noncommutativ e r esidue. Up to a factor it is the unique trace o n the algebr a of pseudo differential op erators. The W o dzicki re s idue app eared first as a residue of a ze ta function measur ing sp ec- tral asy mmetry ([ APS76, W o84 ]). W o dzicki show ed that the r egularity of the η -function of a Dirac- t yp e op erator at zero – a necessar y ingredient to define the η -inv aria nt – follows as a sp ecial case fr o m the v anishing of the W o dzic ki residue on pse udodifferential pro jections (as r e marked by Br ¨ uning and Lesch [ BL99 ] the regular ity of the η -function at z ero for a n y Dirac type op erator and the v anishing o f the W o dzicki residue o n ps e udodifferential pro jections are a ctually e quiv a len t). The regular ity of the η -function was proved by At iyah, Pato di and Singer in [ APS76 ] in the case when X is o dd dimensional and later by Gilkey ([ Gi81 ]) in the general case using K -theore tic arguments. Note that where a s the W o dzicki res idue can b e lo cally co mputed its v anishing o n pseudo differential pr o jections is no t a lo cal phe- nomenon. Gilkey [ Gi79 ] c onstructed a ps eudodifferential pro jection w ho se res idue density is non-v anishing but integrates to zer o . In our pap er we s how that the W o dzicki residue can a lso be defined for pr o jective pseudo differen tial op e rators (this has alrea dy b een observed in [ MMS06 ]) and show that it v anishes o n pro jections in cas e the dimensio n of the manifold is o dd. Our pro of is ba sed o n the Ler a y- Hirsch theo r em in twisted K -theory and a purely algebraic result on ‘r esidue-traces’ in filter ed r ing s. If L is a filtered ring then we call a linear functional τ : L → C a r esidue tr ace if τ ( L − N ) = { 0 } for N lar ge enough. W e pr o ve that the v alue of τ on pro jections de- pends only on their class in L (0) := L/L − 1 . Thus, if the map K 0 alg ( L ) → K 0 alg ( L (0) ) is surjective the map τ des cends to a map from the algebra ic K -theory of L (0) to NONCOMMUT A TIVE RESIDUE FOR PROJECTIVE ψ DO 3 C . This result ca n b e a pplied to the W o dzicki residue showing that it descends to a map from twisted K -theor y K 0 ( S ∗ X , π ∗ A ) to C . W e then use the Leray-Hirsch theorem to show that this map actua lly v anishes. W e r e duce the pro blem to p ositive sp ectral pr o jections o f gener alized Dirac op erators for which it is known [ BG92 ] that the residue density v anishes . 2. Con volution bundles and Azuma ja bundles Pseudo differential o pera tors on a smo oth closed Riema nnia n manifold X a cting on sections of a vector bundle E ca n be under stoo d as co- normal distributional sections in the vector bundle E ⊠ E ∗ 1 ov er the spa ce X × X , by identifying the o pera tors with their distributional kernel. The bundle E ⊠ E ∗ has the following structure that allows to conv olve kernels of in tegr al o pera tors: a n y element in the fibre ov er ( x, y ) may b e mult iplied by a n elemen t in the fibr e ov er ( y , z ) to give an element in the fibre ov er ( x, z ). Moreover, this multiplication satisfies natural conditions such as a sso ciativit y . In order to define pro jective pseudo differen tial op erators it is conv enient to for malize this structur e , as we shall do in this section. 2.1. Con v ol ution bundles. Le t U denote an op en ne ig h b orho o d of the diag o- nal ∆( X ) in X × X which is sy mmetr ic under the reflection map s : ( x, y ) 7→ ( y , x ). Let p ik : X × X × X → X × X b e defined by p ik ( x 1 , x 2 , x 3 ) = ( x i , x k ) and set e U := p − 1 12 ( U ) ∩ p − 1 23 ( U ) ∩ p − 1 13 ( U ). Denote by e p ik the res tr iction of the map p ik to e U . Definition 2.1 . L et π : F → U b e a lo c al ly trivial ve ctor bund le with typic al fibr e Mat( k ) , the c omplex k × k - matric es. We c al l F a conv o lution bundle if ther e exists a homomorphism of ve ctor bund les m : e p ∗ 12 F ⊗ e p ∗ 23 F → F su ch that the fol lowing c onditions ar e satisfie d: (i) The fol lowing diagr am is c ommutative: e p ∗ 12 F ⊗ e p ∗ 23 F   m / / F   e U e p 13 / / U (ii) m is asso ciative, i.e., whenever f ij b elong to the fibr e F ( x i ,x j ) then m ( m ( f 12 ⊗ f 23 ) ⊗ f 34 ) = m ( f 12 ⊗ m ( f 23 ⊗ f 34 )) . 2 (iii) Ther e is an atlas {O α } of U t o gether with lo c al trivializations φ α : π − 1 O α → O α × Mat( k ) , 1 E ⊠ E ∗ denotes the external tensor pro duct of E and its dual bundle E ∗ , i. e., the fibre o ver a poi nt ( x, y ) is E x ⊗ E ∗ y . 2 Here we implicitely assume that ( x 1 , x 2 , x 3 ), ( x 1 , x 3 , x 4 ), and ( x 1 , x 2 , x 4 ) belong to e U . 4 J ¨ OR G SEILE R AND ALEXANDE R STROHMAIER such that φ α ( m ( f 12 ⊗ f 23 )) = φ α ( f 12 ) · φ α ( f 23 ) whenever f ij ∈ F ( x i ,x j ) with ( x 1 , x 2 , x 3 ) ∈ e p − 1 12 ( O α ) ∩ e p − 1 23 ( O α ) . Definition 2 .2 . A ∗ - structure on F is a c onjugate line ar map ∗ : F → F of ve ctor bund les such that F   ∗ / / F   U s / / U c ommutes, such that ( m ( f ⊗ g )) ∗ = m ( g ∗ ⊗ f ∗ ) , and such that the ab ove lo c al trivializatio ns additional ly satisfy ∀ f ∈ π − 1 ( O α ∩ s ( O α )) : φ α ( f ∗ ) = φ α ( f ) ∗ , wher e the star on the right hand side denotes the hermitian c onjugation of matric es. We wil l r efer to a c onvolution bund le with ∗ -stru ctur e as a ∗ - con volution bundle . Note that E ⊠ E ∗ is a par ticula r example for a ∗ - con volution bundle; in this case we can choos e U = X × X . The restric tion of a ∗ -conv olution bundle F to the diagonal in X × X is a bundle A of finite dimensiona l simple C ∗ -algebra s. F ollowing the literature we refer to such bundles of ma trix alge bras as Azuma ja bundles. As shown in [ MMS05 ] an y Azuma ja bundle A on X gives rise to a conv o lution bundle near the diago nal in the fo llowing way , using an atlas of lo cal trivializa tions with resp ect to a go o d cov er 3 { U α } of X : T he transition functions σ αβ are smo oth functions on U αβ = U α ∩ U β with v a lues in the automorphisms of Mat( k ). Since all automor phims are inner we ca n choo se lo cal functions ϕ αβ : U αβ → S U ( k ) that implemen t σ αβ , i.e., σ αβ ( x )( A ) = ϕ αβ ( x ) Aϕ − 1 αβ ( x ). In general, the functions ϕ αβ may violate the co-cyc le conditio n and therefor e are not the transitio n functions of a vector bundle. The co cycle condition for the σ αβ together with the condition that the ϕ αβ are chosen in S U ( k ) show that any ϕ αβ ϕ β γ ϕ γ α m ust b e a constant function on U α ∩ U β ∩ U γ , equal to an k -th ro ot o f unity times the identit y matrix . 4 Then we obta in a conv olution bundle F with typical fibre Mat( k ) on a neighbor ho o d of the diagonal b y choo sing the trans ition functions φ αβ ( x, y )( A ) = ϕ αβ ( x ) A ϕ αβ ( y ) − 1 , A ∈ Ma t( k ) , on U αβ × U αβ . There are also other poss ible ex tensions of A , cf. [ MMS0 6 ], and Prop osition 2 .4, b elow. 3 A co ver i s go od if finite intersect ions of elemen ts therein are either empt y or con tractible. 4 On differen t triple inte r sections, the r esulting unit-ro ot can be different . Thi s i nduces a torsion elemen t in H 3 ( X, Z ), the D i xmier-Douady class. NONCOMMUT A TIVE RESIDUE FOR PROJECTIVE ψ DO 5 Remark 2.3 . In the se quel it wil l b e o c c asional ly c onvenient to cho ose an atlas for F c onsisting of set s O α := U α × U α , wher e { U α } is a go o d c over of X ; t he c orr esp ondi ng t rivial isations we shal l denote by φ α ( so we u se the same notation as in Definition 2.1.(iii) ab ove, but p ossibly have change d t he atlas ) . 2.2. T ransition functions . In the previous section w e hav e seen how an Azu- ma ja bundle leads to a conv olution bundle by cho osing ce rtain transition func- tions. Let us no w hav e a clos e r lo ok to the tra nsition functions of an ar bitr ary ∗ - conv olution bundle. Fix an atlas as explained in Remark 2.3 and let φ αβ : O α ∩O β =: O αβ − → GL (Mat( k )) b e the transition functions defined by φ β ◦ φ − 1 α  ( x, y ) , A  =  ( x, y ) , φ αβ ( x, y )( A )  , Then condition iii) of Definition 2.1 is e quiv a len t to φ αβ ( x, y )( A ) φ αβ ( y , z )( B ) = φ αβ ( x, z )( AB ) . In particular, (2.1) ( x, x ) 7→ φ αβ ( x, x ) : O αβ ∩ ∆( X ) − → Aut(Mat( k )) . Moreov er , Definition 2.2 on the level of the transition functions means that (2.2) φ αβ ( x, y )( A ∗ ) = φ αβ ( y , x )( A ) ∗ . Proposition 2.4 . L et F b e a ∗ -c onvolution bund le with tr ansition functions φ αβ as describ e d ab ove. Then (2.3) φ αβ ( x, y )( A ) = λ αβ ( x, y ) ϕ αβ ( x ) A ϕ αβ ( y ) − 1 with mappings ϕ αβ : O αβ − → S U ( k ) , λ αβ : O αβ − → C , satifying λ αβ ( x, x ) = 1 , λ αβ ( x, y ) λ αβ ( y , z ) = λ αβ ( x, z ) , λ αβ ( x, y ) = λ αβ ( y , x ) , and such t hat al l ϕ αβ ϕ β γ ϕ γ α ar e c onstant functions on their domain of definition, e qual to a k -th r o ot of unity t imes the identity matrix. Proof. Combining (2.1) with (2.2) we find ϕ αβ with φ αβ ( x, x )( A ) = ϕ αβ ( x ) A ϕ αβ ( x ) − 1 , since all automorphisms of Mat( k ) are inner . Now let us define φ ′ αβ ( x, y )( A ) = ϕ αβ ( x ) − 1 φ αβ ( x, y )( A ) ϕ αβ ( y ) . W e then hav e φ ′ αβ ( x, x )( A ) = A, φ ′ αβ ( A )( x, y ) φ ′ αβ ( y , z )( B ) = φ ′ αβ ( x, z )( AB ) . 6 J ¨ OR G SEILE R AND ALEXANDE R STROHMAIER It follows that φ ′ αβ ( x, y )( AB ) = φ ′ αβ ( x, y )( A ) φ ′ αβ ( y , y )( B ) = φ ′ αβ ( x, y )( A ) B a nd, analogo usly , φ ′ αβ ( x, y )( AB ) = Aφ ′ αβ ( x, y )( B ). The r efore, for all matric e s A , φ ′ αβ ( x, y )( 1 ) A = φ ′ αβ ( x, y )( A ) = Aφ ′ αβ ( x, y )( 1 ) , where 1 is the identit y matrix. This shows φ ′ αβ ( x, y )( 1 ) is a multiple of the identit y matrix. Denoting the corr espo nding facto r by λ αβ ( x, y ), the claim follows.  3. Pro jective Pseudo di fferen tial Op erators Pro jectiv e pseudo differential op erato rs have b een defined in [ MMS05 ]. W e adapt this definition to fit in our setting of conv olutio n bundles. 3.1. Pseudo differential op erators. T o clarify notation let us br iefly reca ll the definition of classic a l (or p olyhomogeneo us) pseudo differential op erators on a n op en subs e t Ω of R n . Let V ∼ = C k be a k -dimensiona l vector space. A symbol of order m ∈ R is a smo oth function a : Ω × Ω × R n → E nd( V ) = V ⊗ V ∗ satisfying estimates    ∂ α ξ ∂ β ( x,y ) a ( x, y , ξ )    ≤ C αβ K (1 + | ξ | ) m −| α | for any multi-indices α, β and any compact subset K of Ω × Ω, and having an asymptotic expansion a ∼ ∞ P j =0 χa m − j with a zero-excisio n function χ = χ ( ξ ) and homogeneous comp onent s a m − j , i.e., a m − j ( x, y , tξ ) = t m − j a m − j ( x, y , ξ ) for all ( x, ξ ) with ξ 6 = 0 a nd all t > 0. The pseudo differential o per a tor op( a ) : C ∞ 0 (Ω , V ) → C ∞ (Ω , V ) asso ciated with a is [op( a ) ϕ ]( x ) = Z Z e i ( x − y ) ξ a ( x, y , ξ ) ϕ ( y ) dy d ¯ ξ , ϕ ∈ C ∞ 0 (Ω , V ) . An op erator R : C ∞ 0 (Ω , V ) → C ∞ (Ω , V ) is ca lled smo othing if it ha s a smo oth int eg ral kernel k ∈ C ∞ (Ω × Ω , End( V )), i.e., ( Rϕ )( x ) = Z Ω k ( x, y ) ϕ ( y ) dy , ϕ ∈ C ∞ 0 (Ω , V ) . A pseudo differential o pera tor of order m ∈ R on Ω is a n op erator of the form A = o p( a ) + R , where a is a symbol of order m and R is smo othing. An y pseudo differential o pera tor A = op( a ) + R of or der m can b e represented in the for m op( a L ) + R ′ , whe r e a L ( x, ξ ) is a y -indep endent ‘left-symbol’ of order m ; up to order −∞ the le ft-s ym b ol is uniquely determined by the asymptotic expansion a L ( x, ξ ) ∼ ∞ X | α | =0 1 α ! ∂ α ξ D α y a ( x, y , ξ )    x = y . NONCOMMUT A TIVE RESIDUE FOR PROJECTIVE ψ DO 7 The homogeneo us co mponents of A are by definition those of a L , σ m − j ( A )( x, ξ ) := ( a L ) m − j ( x, ξ ) . By the Sch warz kernel theor em, we can identify A with its dis tributional kernel K A ∈ D ′ (Ω × Ω , V ⊗ V ∗ ) , the top ological dual o f C ∞ 0 (Ω × Ω; V ∗ ⊗ V ). It is uniquely defined by the re lation h K A , ψ ⊗ ϕ i = h ψ , Aϕ i , ψ ∈ C ∞ 0 (Ω , V ∗ ) , ϕ ∈ C ∞ 0 (Ω , V ) . Denoting by tr : V ∗ ⊗ V → C the canonical contraction map, we have explicitly h K A , u i = Z Ω tr[ Au ( x, · )]( x ) dx, u ∈ C ∞ 0 (Ω × Ω; V ∗ ⊗ V ) . By pse udo -lo c alit y , K A ∈ C ∞ (Ω × Ω \ ∆(Ω) , V ⊗ V ∗ ). If U ⊂ X is a co or dina te neighborho o d, we ca n pull-ba ck the lo cal op erators under the c o or dinate map. The resulting space o f op erato r s we shall denote by ΨDO m cl ( U ; End( V )), the subspace o f smo othing op era to rs by ΨDO −∞ ( U ; End( V )). 3.2. Pro jective pseudo di fferen tial op erators. In the following choose a n atlas as explained in Remark 2.3. Definition 3.1 . L et F b e a ∗ -c onvolution bund le over U . A distribution A ∈ D ′ ( U , F ) is c al le d a pr oje ct ive pseudo differ ential op er ator of or der m ∈ R if (i) A is smo oth out s ide the diago n al, (ii) for any α the distribution  φ − 1 α  ∗ A   U α × U α is the distributional kernel of a pseudo differ ential op er ator A α ∈ Ψ DO m cl ( U α ; End( C k )) . We denote the ve ctor sp ac e of m -th or der pr oje ctive pseudo differ ential op er ators by ΨDO m cl ( U ; F ) , t he subsp ac e of smo othing elements by Ψ DO −∞ ( U ; F ) . The subspace Diff m ( U ; F ) of pro jective differential op erators c onsists o f a ll pro jec- tive pseudo differential o pera tors which are supp orted o n the diagona l. Remark 3.2 . If U = X × X and F = E ⊠ E ∗ for a bun d le E over X then ΨDO m cl ( U ; F ) c oincides with Ψ DO m cl ( X ; E , E ) , the pseudo differ ent ial op er ators of or der m acting on se ctions into E . Though pro jectiv e pseudo differential op erator s, in g eneral, are not op erators in the usual sense (i.e., acting b et ween s ections of vector bundles) all elements o f the standard calculus ca n b e generalized to this setting. In particular , the ∗ -str ucture gives r is e to a c o njugation on ΨDO m cl ( U ; F ), defined by A ∗ ( x, y ) := ( A ( y , x )) ∗ in the distributional sense. Let A b e a pro jective pseudo differential op erator with lo cal representativ es A α and A β , cf. Definition 3.1, where O αβ is not e mpty . By passing to lo cal co ordinates on 8 J ¨ OR G SEILE R AND ALEXANDE R STROHMAIER U α ∩ U β , we can as s ocia te with A α and A β lo cal symbols a α ( x, ξ ) and a β ( x, ξ ), resp ectively . These s ym b ols are then related by a β ( x, ξ ) = ∞ X | γ | =0 1 γ ! ∂ γ ξ D γ y    y = x φ αβ ( x, y )  a α ( x, ξ )  = ∞ X | γ | =0 1 γ ! ∂ γ ξ D γ y    y = x  λ αβ ( x, y ) ϕ αβ ( x ) a α ( x, ξ ) ϕ αβ ( y ) − 1  , (3.1) where the transitio n function φ αβ is as descr ibed in (2.2 ) a nd P r opo sition 2.4. Note that this b ehaviour, in g eneral, differs fro m the sta ndard case, due the fa c to r λ αβ ( x, y ). How ever, (3.1) tog ether with λ αβ ( x, x ) = 1 shows that with A we can asso ciate a well-defined ho mogeneous principa l symbol σ m ( A )( x, ξ ) ∈ C ∞ ( S ∗ X , π ∗ A ) , where π : S ∗ X → X is the canonica l co - sphere bundle ov er X . Vice versa, any given such section can b e realized as the principal symbol of a pro jectiv e pseudo different ia l op erator. If the pro jectiv e pseudo differen tial op erators A 1 and A 2 are suppo rted in a suffi- ciently small neighborho o d of the diago nal in U their usual comp osition ( A 1 ◦ A 2 )( x, z ) = Z X m ( A 1 ( x, y ) ⊗ A 2 ( y , z )) dy is a distribution. By passing to lo cal co ordinates and using the co mpositio n theo- rems for pseudo differential o pera tors one ca n see that A 1 ◦ A 2 is a pro jectiv e pseu- do differen tial op erator. The homogeneo us principal symbol b ehav es m ultiplicative under comp osition. Of course, a n y pro jective pse udo differential op erato r c an b e written as a sum of tw o op erators , wher e one is s mo othing and the o ther is sup- po rted ne a r the diagona l. Summarizing, the coset space (3.2) L ∗ cl ( U , F ) := Ψ DO ∗ cl ( U , F ) / ΨDO −∞ ( U , F ) is a filter e d ∗ -alg ebra. As in the standar d cas e, a symptotic summations of sequences of pro jective op erator s of o ne - step decr easing or ders ar e p ossible and pa rametrices (i.e., inv erses mo dulo smo othing remainders) to elliptic elements ca n b e constructed. Theorem 3.3 . L et F b e a ∗ - c onvolution bun d le and let A b e a pr oje ct ive pseudo- differ ential op er ator. F or x ∈ X define WRes x ( A ) := Z S ∗ x X tr a − n ( x, ξ ) dσ ( ξ ) dx, wher e a − n ( x, ξ ) , n = dim X , is the homo gene ous c omp onent of or der − n of a symb ol of a lo c al r epr esentative A α with x ∈ O α , cf. Definition 3.1 . Then WRes x ( A ) is wel l-define d and defines a glob al density on X . Mor e over, WRes( A ) := Z X WRes x ( A ) NONCOMMUT A TIVE RESIDUE FOR PROJECTIVE ψ DO 9 defines a tr ac e fun ctional on the algebr a L ∗ cl ( U , F ) , the so-c al le d noncommutativ e residue or Wo dzicki r esidue. Proof. Let A β be another lo cal repr e s en tative and x ∈ O β . Fixing lo cal c o - ordinates o n O α ∩ O β , the lo cal symbols a α and a β are rela ted by the a symptotic expansion (3.1 ). F ollowing the pr o of in [ FGL S96 ] terms containing a deriv ative ∂ γ ξ , | γ | ≥ 1, v anish under integration. W e thus o btain the same v alue for WRes x ( A ) us- ing either a α ( x, ξ ) or a β ( x, ξ ). That WRes x ( A ) tra ns forms as density under changes of co ordinates is seen a s in the standard case, cf. [ F GLS96 ]. T o see that the integral of the r esidue density defines a trace functional we need to show that it v anishes on commutators [ A, B ]. T o this end fix a cov er { U ′ σ } of X b y co ordina te maps tog ether with a sub ordinate partition of unity , such that U ′ σ ∪ U ′ ρ is contained in some U α whenever U ′ σ ∩ U ′ ρ is not empt y . W e then ca n w r ite A = P σ A σ and B = P σ B σ mo dulo smo othing op erators, where the A σ and B σ are supp orted in O ′ σ := U ′ σ × U ′ σ . Then the commutator [ A, B ] can be written a s a sum of terms [ A σ , B ρ ]. Such a commutator is smo othing if O ′ σ ∩ O ′ ρ is empty . Otherwise it is contained in some set O α . Therefore the ca lc ula tion reduces to a lo cal one, which is not different from the one fo r usual ps eudodifferential o p erato rs that can b e found in [ FGL S9 6 ].  F or purp oses below let us establish the following res ult: Proposition 3.4 . L et F b e a ∗ -c onvolution bund le and A b e the Azumaja bund le obtaine d by r estricting F to the diagonal. Mor e over, let p ∈ C ∞ ( S ∗ X , π ∗ A ) with p 2 = p . Then t her e exists a pr oje ctive pseudo differ ent ial op er ator P ∈ L 0 cl ( U , F ) which is a pr oje ction, i.e., P 2 = P , and which has p as its princip al symb ol. If, additiona l ly, p ∗ = p then P c an b e chosen such that P ∗ = P . Proof. If p 2 = p then e = 2 p − 1 is an idemp otent. W e now construct a pro jectiv e pseudo differen tial op erator E ∈ L 0 cl ( U , F ) which is an idemp o ten t and has e as its principal sy m b ol. Then P = (1 + E ) / 2 is the desired pro jection. Let e E ∈ L 0 cl ( U , F ) be any element having e as pr incipal s ym b ol. Then e E 2 = 1 − R with a r emainder R ∈ L − 1 cl ( U , F ). If P ∞ k =0 c k r k denotes the T a ylor series of f ( r ) = 1 / √ 1 − r let S ∈ L 0 cl ( U , F ) hav e the asymptotic expansio n ∞ P k =0 c k R k . Then (1 − R ) S 2 = 1 a nd S commutes with e E , s inc e R do es. Then define E = e E S . In cas e also p ∗ = p , first choo se E 0 having e a s principal symbol. Then set e E = E 0 E ∗ 0 and pro ceed as b efore; note that R ∗ = R and hence S ∗ = S .  10 J ¨ OR G SEILE R AND ALEXANDE R STROHMAIER 4. The noncommutativ e resi due in t wisted K -theo ry 4.1. Twisted K- the ory. Supp o se that A is an Azuma ja bundle over a com- pact manifold X . The twisted K-theory is defined to b e the K-theory of the C ∗ - algebra of co n tinuous sections C ( X ; A ) of A . If Y ⊂ X is a closed s ubs et then the set of sections C ( X, Y ; A ) v anishing on Y is a clo s ed tw o-sided ideal in C ( X, Y ; A ) and the quo tien t by this ideal can b e ident ified with the space of co n tinuous sections C ( Y ; A ) of the Azuma ja bundle A| Y . W e therefor e hav e the six term exact sequence as a conseq uence of the six term exact sequence in the theory of C ∗ -algebra s. K 0 ( X, Y ; A ) / / K 0 ( X ; A ) / / K 0 ( Y ; A )   K 1 ( Y ; A ) O O / / K 1 ( X ; A ) / / K 1 ( X, Y ; A ) where the relative K -gro ups K ∗ ( X, Y ; A ) a r e defined as K ∗ ( C ( X, Y ; A )). There is a natural map K ∗ ( C ( X, Y ; A )) ⊗ Z K ∗ ( C ( X )) 7→ K ∗ ( C ( X, Y ; A ) ˆ ⊗ C ( X )) . Here ˆ ⊗ is the tensor pro duct of C ∗ -algebra s which is w ell defined in this case as C ( X ) is nuclear. The usua l multiplication C ( X, Y ; A ) ˆ ⊗ C ( X ) → C ( X, Y ; A ) induces a map K ∗ ( C ( X, A ) ˆ ⊗ C ( X )) → K ∗ ( C ( X, A )). The comp osition of these t wo maps makes K ∗ ( X, Y ; A ) a mo dule over the Z 2 -graded ring K ∗ ( X ). Cho osing Y = ∅ de fines a K ∗ ( X ) mo dule structure on K ∗ ( X ; A ). Note that the morphis ms in the six ter m exa ct sequence are mo dule homo morphisms. These o bserv ations can b e used to prov e the following Leray-Hirsch theorem: Theorem 4.1 . L et R b e a c ommutative torsion-fr e e ring. Su pp ose t hat π : M F − → X is a c omp act smo oth fibr e bund le with fibr e F over X and let A b e an Azu maja bund le over X . As sume that K ∗ ( F ) ⊗ Z R is a fr e e R -mo dule and su pp ose ther e exist elements c 1 , . . . , c N ∈ K ∗ ( M ) ⊗ Z R such that the c j | M x form a b asis for K ∗ ( M x ) ⊗ Z R for every x ∈ X . Then the fol lowing map is an isomorphism : K ∗ ( X ; A ) ⊗ Z R N − → K ∗ ( M , π ∗ ( A )) ⊗ Z R, ( p, α ) 7→ N X j =1 α j π ∗ ( p ) · c j . Indeed, the usual pro of o f the Ler a y- Hirsch theor e m in top ologica l K -theor y (see e.g. [ H09 ], Theorem 2.25) can be ada pted to our setting in the following way . If NONCOMMUT A TIVE RESIDUE FOR PROJECTIVE ψ DO 11 Y ⊂ X is a clos ed subset of X , we have the following diag ram: / / K ∗ ( X, Y ; A ) ⊗ Z R N Φ   / / K ∗ ( X ; A ) ⊗ Z R N Φ   / / K ∗ ( Y ; A ) ⊗ Z R N Φ   / / / / K ∗ ( π − 1 X , π − 1 Y ; A ) ⊗ Z R / / K ∗ ( π − 1 X ; A ) ⊗ Z R / / K ∗ ( π − 1 Y ; A ) ⊗ Z R / / Here Φ is defined as in the theorem, Φ( p, α ) = N P j =1 α j π ∗ ( p ) · c j . The rows of this diagram are exact since tensoring with R N and R is an exact functor. All maps in the six ter m exa ct sequence a re natural a nd therefore the pull back π ∗ commutes with them. Moreover, the ma ps in the six term exact se q uence for the pair ( π − 1 X , π − 1 Y ) are K ∗ ( M ) mo dule homomo rphisms. Thus, the dia gram co mm utes. Since X is a finite cell complex o ne can pro ceed in the usua l wa y using the 5-lemma and induction in the num ber o f cells and the dimension to prove the theorem. 4.2. Residue-traces on filte red rings. Let L b e a r ing with filtr ation, i.e., L = L 0 ⊃ L − 1 ⊃ L − 2 ⊃ . . . w ith sub-r ings L − j and the mu ltiplica tio n induces maps L − i × L − j → L − i − j for a n y choice o f i, j . A trac e functional on L is a map τ : L → V for s ome vector spa ce V having the following prop erties: (1) τ is linear , τ ( A + B ) = τ ( A ) + τ ( B ) fo r all A, B ∈ L , (2) τ v anishes on commutators, τ ([ A, B ]) = τ ( AB − B A ) = 0 for all A, B ∈ L . W e call τ a r esidue-trace if, additio na lly , (3) there exis ts an N s uc h that τ ( A ) = 0 for a ll A ∈ L − N . W e shall now show that a residue-trac e res tricted to the set of pro jections in L is insensible for low er o rder terms. The pro of is elementary and pur ely a lgebraic. Theorem 4 .2 . L et τ b e a r esidue-tr ac e on L and P, e P ∈ L b e t wo pr oje ctions, i.e., P 2 = P and e P 2 = e P . If P − e P ∈ L − 1 then τ ( P ) = τ ( e P ) . Proof. Set R = e P − P and then define A = P RP , B = P R (1 − P ) , C = (1 − P ) RP , D = (1 − P ) R (1 − P ) . Obviously then e P = P + R and R = A + B + C + D . Using that P (1 − P ) = (1 − P ) P = 0 we obtain ( P + R )( P + R ) = P + 2 A + B + C + A 2 + AB + B C + + B D + C A + C B + D C + D 2 . On the o ther hand, using that e P is a pro jection, ( P + R )( P + R ) = ( P + R ) = P + A + B + C + D . 12 J ¨ OR G SEILE R AND ALEXANDE R STROHMAIER Equating these t wo expr essions and rear ranging of ter ms yields A 2 + A + B C + D 2 − D + C B + AB + B D + C A + D C = 0 . Multiplying this identit y from the left and the rig h t with P and 1 − P , resp ectively , yields A 2 + A + B C = 0 , D 2 − D + C B = 0 . The first identit y shows A ∈ L − 2 and A = − B C mo dulo L − 4 . Let us now r ewrite these equations a s A (1 + A ) = − B C, ( − D )  1 + ( − D )  = − C B . Multiplying the first equation by (1 − A ) yields A ≡ − B C − ( B C ) 2 mo dulo L − 6 . Multiplying it with (1 − A + A 2 ) then yields A ≡ − B C − ( B C ) 2 − 2( B C ) 3 mo dulo L − 8 . Pr ocee ding by induction we obtain A ≡ ℓ X k =1 c kℓ ( B C ) k mo d L − 2( ℓ +1) for any ℓ ∈ N with s uitable cons tan ts c kℓ . In the same w ay , with the same constants c kℓ , − D ≡ ℓ X k =1 c kℓ ( C B ) k mo d L − 2( ℓ +1) . Therefore we hav e A + D = ℓ X k =1 c kℓ  B , ( C B ) k − 1 C  mo d L − 2( ℓ +1) . Cho osing ℓ lar ge enough, we deduce that τ ( A ) + τ ( D ) = 0 . F urthermor e, τ ( B ) = τ ( P R (1 − P )) = τ ((1 − P ) P R ) = 0 and, analog ously , r es( C ) = 0. Altogether we obtain τ ( e P ) = τ ( P ) + τ ( A ) + τ ( D ) + τ ( B ) + τ ( C ) = τ ( P ) which is the cla im we wan ted to prov e.  4.3. The noncommuta tive residue. W e shall show that the noncommuta- tive residue induces a ma p on twisted K -theor y . Proposition 4.3 . L et A b e the Azumaja bun d le obtaine d by r estricting a ∗ -c on- volution bu n d le F to the diagonal. The nonc ommutative r esidue fr om The or em 3.3 desc ends to a gr oup homomorp hism (4.1) WRes : K 0 ( S ∗ X , π ∗ A ) → C , wher e π : S ∗ X → X denotes the c o-spher e bund le over X . NONCOMMUT A TIVE RESIDUE FOR PROJECTIVE ψ DO 13 Proof. A typical element in K 0 ( S ∗ X , π ∗ A ) c a n be r e pr esent ed by a se c tion p ∈ C ∞ ( S ∗ X , π ∗ Mat N ( A )) w hich is (po in twise) a pro jection. This is p ossible, since the na tural inclusion of the K -theor y o f the lo cal C ∗ -algebra C ∞ ( S ∗ X , π ∗ A ) into that of C ( S ∗ X , π ∗ A ) is an isomor phism, cf. [ Bl 98 ]. By P rop osition 3.4 ea c h such section is the principal symbol o f a pr o jective pseudo differential op erator P ∈ L 0 ( X ; Mat N ( F )) which is a pro jection. The noncommutativ e r esidue of the K - group element is then defined a s WRes( P ) in the sense of Theorem 3.3. W e hav e to show that this map is well-defined. So let e p ∈ C ∞ ( S ∗ X , π ∗ Mat M ( A )) represent the same element a s p do es. Let e P ∈ L 0 ( X ; Mat M ( F )) b e asso ciated with e p . Since p and e p ar e eq uiv a le n t ther e exists a unitary u ∈ C ∞ ( S ∗ X , π ∗ Mat M + N ( A )) such that u ( p ⊕ 0 M ) u − 1 coincides with 0 N ⊕ e p . Let U ∈ L 0 ( X ; Mat M + N ( F )) hav e u a s its principal symbol. Then WRes( U ( P ⊕ 0 M ) U − 1 ) = WRes( P ⊕ 0 M ) = WRes( P ) . On the other hand U ( P ⊕ 0 M ) U − 1 is a pro jection having the same principa l symbol as 0 N ⊕ e P . Thus, by Theo rem 4.2, WRes( U ( P ⊕ 0 M ) U − 1 ) = WRes(0 N ⊕ e P ) = WRes( e P ) . This shows that the no ncomm utative r esidue is indep endent of the choice of the representative.  5. Twisted Dirac op erators and connections Let A b e the Azuma ja bundle o bta ined by restricting a ∗ -co nvolution bundle F to the diagonal. Definition 5.1 . A pr oje ctive c onne ction ∇ = ∇ F on F is a line ar map Y 7→ ∇ Y : C ∞ ( X ; T X ) − → Diff 1 ( U ; F ) satisfying, for any ve ctor field Y ∈ C ∞ ( X, T X ) and any funct ion f ∈ C ∞ ( X ) , (1) ∇ f Y = f ∇ Y , (2) [ ∇ Y , f ] = Y f for any f ∈ C ∞ ( X ) . It is c al le d a hermitian c onne ction if additionally (3) ∇ ∗ Y + ∇ Y + div Y = 0 ( her e, f and div Y ar e c onsider e d as elements of Diff 0 ( U ; F )) . Note that in ca se U = X × X and F = E ⊠ E ∗ for a vector bundle E over X we just recov er a usual her mitian co nnection on E . One can always co nstruct a pr o jective hermitian connectio n from lo cal hermitian connectio ns b y gluing with a partition of unity . 14 J ¨ OR G SEILE R AND ALEXANDE R STROHMAIER If ∇ = ∇ F is a pro jectiv e connection and φ α is a lo cal trivia lization o f F over U α × U α as de s cribed in Remark 2.3, the cor respo nding lo cal differential op erator ∇ α Y ∈ Diff m ( U α , End( C k )) is of the fo r m ∇ α Y = Y + Γ α Y ( x ) , Γ α Y ∈ C ∞ ( U α , Mat( k )) . If we use a nother trivialisa tion φ β of F on U β × U β , we hav e the relatio n Γ β Y ( x ) = φ αβ ( x, x )(Γ α Y ( x )) + Y y φ αβ ( x, y )( 1 )   y = x , x ∈ U α ∩ U β , where 1 is the identit y matrix. T hus, ana logous to the theory of sta nda rd c onnec- tions, we may de s cribe pro jective connections by ‘co nnection matr ic es’ Γ α Y asso- ciated to a c overing X = ∪ α U α satisfying the above compatibility rela tions. F or a hermitian connection the connection matrices a ls o ha ve to b e skew-symmetric, Γ α Y ( x ) ∗ = − Γ α Y ( x ). Suppo se now S is a Cliffor d mo dule ov er X a nd let γ denote the Clifford m ultiplica- tion. Mor eov er, let ∇ S be a connection on S which is compatible with the Clifford structure. W riting e F := S ⊠ S ∗ it is ea s y to see that F ⊗ e F is a ∗ -convolution bundle ov er U and we can define the pro jective her mitian connection ∇ := ∇ F ⊗ 1 + 1 ⊗ ∇ S by cho o sing the co r resp onding co nnec tion matrices as Γ F, α Y ( x ) ⊗ 1 + 1 ⊗ Γ S,α Y ( x ) , x ∈ U α , where the U α are chosen in such a wa y that b o th F and e F a re lo cally trivial ov er U α × U α . Then we can define the twisted Dirac op erator D := (1 ⊗ γ ) ◦ ∇ ∈ Diff 1 ( U ; F ⊗ e F ); in fact, in ea ch loc al trivialisatio n ∇ is a usual hermitean connection a nd w e can comp ose it lo cally with 1 ⊗ γ . 6. V anishing of the W o dzic ki residue Theorem 6.1 . If X is an o dd dimensional oriente d manifold t he map WRes of (4.1) vanishes identic al ly. As a dir ect consequence , WRes( P ) = 0 for any pro jection P ∈ L 0 ( U , F ). Proof o f Theorem 6.1. Suppo se the dimension of X is n = 2 ℓ − 1. Let S = ⊕ k even Λ k ( T ∗ X ) deno te the bundle o f even-degree forms ov er X and le t ∗ : Λ k ( T ∗ X ) → Λ n − k ( T ∗ X ) the Ho dge star op erator and denote by d and δ the exterior differential and the co-differential res pectively . Define the op erator D S acting on sections of S as D S = i ℓ ∗ ( δ + ( − 1) k +1 d ) on k -forms . NONCOMMUT A TIVE RESIDUE FOR PROJECTIVE ψ DO 15 Then by Prop osition 1.22 a nd 2 .8 in [ BGV04 ] this is a gener alized Dirac op erator , where the Cliffo r d action on S is g iv en by γ ( ξ ) = i ℓ ∗  int ( ξ ) + ( − 1) k +1 ext( ξ )  for ξ ∈ T ∗ X a nd the compatible connection is the Levi-Civita co nnection. The pr incipal symbol of D S restricted to the co-spher e bundle is a s e lf-adjoin t inv olution a nd the pro jection σ + ( D S ) = 1 2 ( σ ( D S ) + 1) ont o its +1 eigens pace defines an ele ment in K 0 ( S ∗ X ). It is well known (see for instance [ APS76 ]) that restriction of this element to each co-sphere S ∗ x X equals 2 ℓ times the Bott element on S n − 1 which together with the class of the trivial line bundle genera tes K 0 ( S n − 1 ). F or notational convenience denote by K ∗ R ( X ) the gro ups K ∗ ( X ) ⊗ Z R . By Theor em 4.1 applied to the co-spher e bundle of X , any element of K 0 R ( S ∗ X , π ∗ A ) can b e represented in the fo rm α 0 π ∗ ( p ) · [ 1 ] + α 1 π ∗ ( p ) · [ σ + ( D S )] for some α 0 , α 1 ∈ R and so me p ∈ K 0 ( X, A ). Here bo th the clas s [ 1 ] of the trivial line bundle and the class [ σ + ( D S )] ar e unders too d as elemen ts in K 0 R ( S ∗ X ). The elements in α 0 π ∗ ( p ) · [ 1 ] can b e represented by pro jections in C ∞ ( X ; Mat N ( A )). Therefore, the noncommutativ e res idue o f these elements v anishes. It r emains to show that this is a lso true for the second summand. T o this end let p b e a pro jection in Mat N ( C ∞ ( X ; A )). Let us define the new con- volution bundle F p having fibre p ( x )Mat N ( F ) ( x,y ) p ( y ) ⊂ Mat N ( F ) ( x,y ) in ( x, y ). W e now apply the ab ov e construction and build a twisted Dirac op erator D p with resp ect to F p ⊗ e F , e F = S ⊠ S ∗ . Then σ + ( D p ) repr esen ts the class π ∗ ([ p ]) · [ σ + ( D S )] in K 0 ( S ∗ X , π ∗ A ). The pro jective differential op erator D p can now b e use d to constr uct a certain pro- jection Q ∈ L 0 cl ( X, F p ) whic h has pr incipal symbol as σ + ( D p ) on S ∗ X . In the case o f a Dirac type op erator D acting on a vector bundle the pro jection w o uld just be the op erator 1 2 ( | D | − 1 D + 1). The symbol of this pr o jection can b e constructed from the a parametrix of D and this construction is lo cal mo dulo smo othing op erato r s. That is the full symbol of 1 2 ( | D | − 1 D + 1) mo dulo smo othing ter ms in lo cal co ordinates depe nds o nly on the full s y m b ol of D in these lo c al c o or dinates. Thu s, the construc- tion can be r e peated for the op erato r D p to yield an element in L 0 cl ( X, F p ) which w e denote b y Q or for mally 1 2 ( | D p | − 1 D p + 1 ). By construction [ σ ( Q )] ∈ K 0 R ( S ∗ X ; A ) is equal to π ∗ ([ p ]) · [ σ + ( D S )]. In [ BG92 ] (Theorem 3.4) Br anson and Gilkey have used inv ariant theor y to show that the residue density of the pos itiv e spe c tr al pro jection for any g e neralized Dira c op erator v anishes identically . Lo cally , D p is a generaliz e d Dirac op erator and since the co ns truction of the r esidue density is lo cal the r esidue density of Q v anishes as well. So we ca n conclude that the no ncomm utative res idue of Q v anishes which completes o ur pro of.  16 J ¨ OR G SEILE R AND ALEXANDE R STROHMAIER Ac kno wled g emen ts. The authors would like to thank Thomas Schick for com- men ts a nd for p oin ting o ut a gap in an e a rlier version o f this pap er. References [APS76] M .F. At i y ah, V. K. P ato di, and I.M. Singer, Sp e ctr al asymmetry and Riemannian ge om- etry II I, Math. Pro c. Camb r idge Phil os. So c. 79 (1976) , no.1, 71–99. [BGV04] N. Berline, E. Getzler and M. V ergne, He at kernels and Dir ac op er ators , Grundlehren T ext Edi tions, Spri nger- V erlag, Berlin, 2004. [Bl98] B. Black adar, K -the ory for op er ator algebr as , M athemat ical Sciences Researc h Institute Publications, V ol. 5, Second. Ed., Cambridge Universit y Press, 1998. [BG92] T.P . Br anson and P .B. Gilke y , R esidues of t he eta function for an op er ator of Dir ac typ e , J. F unct . Anal. 108 (1992), no. 1, 47–87. [BL99] J. Br ¨ uning and M. Lesch, O n the η -invariant of certain nonlo c al b oundary v alue pr ob- lems , Duke Math. J. 96 (1999), no. 2, 425–468. [FGLS 96] B . F edosov, F. Golse, E. Leich tnam and E. Schrohe, The nonc ommutative r esidue for manifolds with bo undary . J. F unct. Anal. 142 (1996) , no. 1, 1–31. [Gi79] P . Gilkey , The r esidue of the lo c al eta function at the origin , Math. Ann. 240 (1979), no. 2, 183–189. [Gi81] P . Gilkey , The r esidue of t he glob al η function at the origin , Adv. in Math. 4 0 (1981), no. 3, 290–307. [H09] A. Hatc her, V e ctor Bund les & K -The ory , unpublished man uscri pt, a v ailable at http://w ww.math.cornell.edu/ ∼ hatcher/VBKT/VBpage.html , version 2.1, May 2009. [MMS05] V. Mathai, R.B. M elrose and I.M. Singer, The index of pr oje ctive families of el liptic op er ators , Geom. T opol. 9 (2005), 341–373. [MMS06] V. M athai, R.B. Melrose and I.M . Singer, F r actional analytic i ndex , J. Di ffer ential Geom. 74 (2006), no. 2, 265–292. [W o84] M. W odzicki, L o c al invariants of sp e ctr al asymmetry , In ven t. Math. 75 (1984), no. 1,143– 177. Loughborough University, Dep ar tment of Ma thema tical S ciences, Leicestershire LE11 3TU (UK) E-mail addr ess : j.seiler@lboro. ac.uk Loughborough University, Dep ar tment of Ma thema tical S ciences, Leicestershire LE11 3TU (UK) E-mail addr ess : a.strohmaier@lb oro.ac.uk