Lefschetz numbers for C*-algebras

LEFSCHETZ NUMBERS F OR C ∗ -ALGEBRAS HEA TH EMERSON Abstra t. Using P oinaré dualit y , w e form ulate a form ula of Lefs hetz t yp e whi h omputes the Lefs hetz n um b er of an endomorphism of a separable, n u- lear C ∗ -algebra satisfying P oinaré dualit y and the Kunneth theorem. (The Lefs hetz n um b er of an endomorphism is the graded trae of the indued map on K -theory tensored with C , as in the lassial ase.) W e then examine endomorphisms of Cun tz-Krieger algebras O A . An endomorphism has an in- v arian t, whi h is a p erm utation of an innite set, and the on trating and expanding b eha vior of this p erm utation desrib es the Lefs hetz n um b er of the endomorphism. Using this desription w e deriv e a losed p olynomial form ula for the Lefs hetz n um b er dep ending on the matrix A and the presen tation of the endomorphism. 1. Intr odution Supp ose A and B are t w o separable, n ulear C ∗ -algebras. T o sa y that A and B are Poin ar é dual means that there is giv en a K -homology lass for A ⊗ B su h that up-ap pro dut with this lass indues an isomorphism b et w een the K -theory of A and the K - homolo gy of B . The homology lass pla ys the role of the orien tation lass of a ompat manifold. The idea in this form is due to Alain Connes (see [4 ℄). Sine the denition w as in v en ted, quite a n um b er of examples of P oinaré dual pairs ha v e app eared in the op erator algebra literature, onneted with dynamial systems, foliations, h yp erb oli groups, t wisted K -theory , C ∗ -algebras of disrete groups with nite B Γ , et . The ob jet of this note is to prop ose a simple appliation of the existene of dualit y b et w een a pair of algebras, whi h runs roughly along the lines of a lassial argumen t with de Rham ohomology and dieren tial forms. Supp ose φ : X → X is a smo oth self-map of a ompat, orien ted manifold. Assume that φ is in general p osition with regard to xed-p oin ts. Then φ indues a map on homology with rational o eien ts, and its Lefs hetz n um b er is tr s ( φ ∗ ) := trace( φ ∗ : H ev ( X ) → H ev ( X )  − trace( φ ∗ : H od d ( X ) → H od d ( X )  . The Lefs hetz xed-p oin t theorem states that this n um b er is equal to the n um b er of xed p oin ts of φ oun ted with appropriate m ultipliities. The pro of, whi h an b e found in an y textb o ok, in v olv es ideas onneted with P oinaré dualit y in de Rham theory: normal bundles, in tegration of forms, Thom lasses, and so on. The Kunneth form ula is a separate, additional ingredien t. It is sometimes therefore said that the Lefs hetz xed-p oin t form ula fol lows from P oinaré dualit y and the Kunneth form ula. In this artile, w e are going to formalize the exat onnetion b et w een P oinaré dualit y and the Lefs hetz xed-p oin t theorem in su h a w a y as to apply to the ategory of C ∗ -algebras, with K -theory and K -homology pla ying the role of ordinary homology and ohomology . P art of the pro of of the lassial Lefs hetz theorem is absorb ed in to our statemen t, so that the lassial theorem an b e dedued from 2000 Mathematis Subje t Classi ation. 19K35, 46L80. The author w as supp orted b y an NSER C Diso v ery gran t. 1 2 HEA TH EMERSON ours b y a simple, essen tially linear, index alulation. W e sho w via essen tially straigh tforw ard formal alulations with KK -theory , that if one has a P oinaré dualit y with `fundamen tal lass' ∆ ∈ KK n ( A ⊗ B , C ) , and if one has a morphism f ∈ KK( A, A ) , then the trae of the map on K -theory indued from f , is equal to the result of a ertain index pairing (see Setion 2) in v olving f and ∆ . This index pairing an in priniple b e omputed in geometri terms, pro vided that the yles underlying f and ∆ themselv es admit in teresting, `geometri' desriptions. Th us, to summarize, the Lefs hetz n um b er an b e realized as a Kasparo v pro dut. Of ourse there is more than one su h realization; in [8℄ w e pursue a similar idea to pro due other kinds of iden tities in equiv arian t KK -theory . Of ourse the main merit of the observ ation is that one no w has the p ossibilit y of pro ving analogues of the Lefs hetz theorem in man y dieren t settings, pro vided one has a v ailable an in teresting instane of nonomm utativ e P oinaré dualit y . The signiane of the lassial Lefs hetz form ula tends to b e explained in terms of the equalit y of a lo  al and a glob al in v arian t. In onnetion with C ∗ -algebras, this do es not en tirely mak e sense. What kind of Lefs hetz form ulas an w e exp et in onnetion with C ∗ -algebras? One example, based on the abstrat Lefs hetz form ula presen ted here, is w ork ed out in [6 ℄. This in v olv es prop er ations of disrete groups on manifolds. The primitiv e ideal spae of a ross pro dut C 0 ( X ) ⋊ G in this situation is the extended quotien t G \ ˆ X , where ˆ X := { ( x, h ) ∈ X × G | h ∈ Sta b G ( x ) } , where G ats on ˆ X b y g ( x, h ) = ( g x, g hg − 1 ) , and whi h, as a set, iden ties anon- ially with the primitiv e ideal spae of C 0 ( X ) ⋊ G and inherits a orresp onding h ull-k ernel top ology . It is a bundle o v er the ordinary spae G \ X with bre at Gx the irreduible dual of Stab G ( x ) , but it is not Hausdor. The Lefs hetz form ula for an automorphism of this situation has the orresp onding shap e: the geometri side of the form ula in v olv es xed p oin ts in the ordinary spae G \ X , and seondly , in v olv es represen tation theoreti data for the isotrop y of these xed-p oin ts. The seond purp ose of this note is to onsider the ase of a pair of simple algebras in dualit y , namely to pairs A = O A , B = O A T of Cun tz-Krieger algebras (see [11 ℄). Here, in on trast to the example of the previous paragraph, here there are no p oin ts at all. Giv en an endomorphism of O A arising from ertain geometri data, w e will solv e the index problem on the geometri side of the formal Lefs hetz form ula. The endomorphisms with whi h w e w ork orresp ond to n -tuples of on tin uous, partially dened homeomorphisms ϕ : Z ⊂ Σ + A → Σ + A , where Σ + A is the sym b ol spae of sequenes ( x i ) su h that A x i ,x i +1 = 1 for all i . The information in v olv ed in su h an n -tuple an b e summarized in a single map on the oun table set of paths in the graph orresp onding to A . The geometri omputation of the Lefs hetz n um b er turns out to dep end, roughly , on the dierene b et w een the n um b er of strings whose length is shrunk b y the map, and the n um b er of strings whose length is expanded b y the map. This ev en tually leads to a desription of the Lefs hetz n um b er roughly in the follo wing terms: if w e write t i = P s µ s ∗ ν for w ords µ, ν , for one of the images of the generators of O A under the endomorphism, then an app earane of ( µ, ν ) with | µ | ≤ | ν | on tributes +1 to the Lefs hetz n um b er and | µ | > | ν | + 1 on tributes − 1 (and there are no on tributions when | µ | = | ν | + 1 ). As a result of this deription w e an, if w e w an t, write do wn an expliit, losed form ula for the Lefs hetz n um b er, whi h is a p olynomial expression in the en tries of the matrix A . This example th us sho ws that the `Lefs hetz tri k' results in in teresting form ulas ev en in what one migh t lo osely refer to as a `v ery' nonomm utativ e situation. LEFSCHETZ NUMBERS F OR C ∗ -ALGEBRAS 3 The idea of formalizing the Lefs hetz xed-p oin t theorem's pro of using P oinaré dualit y (and the Kunneth theorem) in order to w ork in a more general on text, is due to André W eil, though not of ourse in onnetion with C ∗ -algebras and K -theory . It w as used b y him in onnetion with the so-alled W eil onjetures (see the App endix to [9℄). So in this sense, w e ha v e rediso v ered an old tri k. Ho w ev er, ev en so it seems w orth making it expliit in the op erator algebrai on text in view of the v ariet y of Lefs hetz-t yp e form ulas one an reasonably hop e to a hiev e b y using C ∗ -algebras and KK -theory , whi h em brae su h a wide seletion of geometri situations. 2. The abstra t Lefshetz theorem Kasparo v's KK -theory is a realization of an additiv e, Z / 2 -graded ategory with ob jets C ∗ -algebras and morphisms A → B the elemen ts of KK • ( A, B ) , dened as a quotien t of a ertain set of yles, b y a ertain equiv alene relation (see [10 ℄.) In addition to its struture of an additiv e ategory , KK is a symmetri monoidal ategory with unit ob jet the C ∗ -algebra C and bifuntor giv en b y the tensor pro d- ut of C ∗ -algebras on ob jets and the "external pro dut" (1) KK • ( D 1 , D ′ 1 ) × KK • ( D 2 , D ′ 2 ) → KK • ( D 1 ⊗ D 2 , D ′ 1 ⊗ D ′ 2 ) , ( f 1 , f 2 ) 7→ f 1 ˆ ⊗ C f 2 . on morphisms, and the ip Σ : A ⊗ B → B ⊗ A induing the braiding. The in teration b et w een the ip, the monoidal struture, and the grading in KK is summarized b y the follo wing diagram, whi h gr ade d  ommutes , for all D i , D ′ i : KK • ( D 1 , D ′ 1 ) × KK • ( D 2 , D ′ 2 ) ip   ˆ ⊗ C / / KK • ( D 1 ⊗ D 2 , D ′ 1 ⊗ D ′ 2 ) ip   KK • ( D 2 , D ′ 2 ) × K • ( D 1 , D ′ 1 ) ˆ ⊗ C / / KK • ( D 2 ⊗ D 1 , D ′ 2 ⊗ D ′ 1 ) In other w ords, (2) f 1 ˆ ⊗ C f 2 = ( − 1) ∂ f 1 ∂ f 2 [Σ] ˆ ⊗ D 2 ⊗ D 1 ( f 2 ˆ ⊗ C f 1 ) ˆ ⊗ D ′ 2 ⊗ D ′ 1 [Σ] for all f 1 ∈ K K • ( D 1 , D ′ 1 ) and f 2 ∈ K K • ( D 2 , D ′ 2 ) . Of ourse a ategory with similar prop erties is the ategory of omplex Z / 2 - graded v etor spaes and v etor spae maps, where for the monoidal struture w e use graded tensor pro dut of v etor spaes, and for the braiding w e use the gr ade d ip Σ s ( a ˆ ⊗ C b ) := ( − 1) ∂ a∂ b b ˆ ⊗ C a instead of the ordinary ip. The ation of a linear transformation T 1 ˆ ⊗ C T 2 on V 1 ˆ ⊗ C V 2 , where T i : V i → V ′ i , V i , V ′ i graded v etor spaes, is dened b y ( T 1 ˆ ⊗ C T 2 )( a ˆ ⊗ C b ) := ( − 1) ∂ T 1 ∂ b T 1 ( a ) ˆ ⊗ C T 2 ( b ) . Then a short alulation sho ws that dep ending on the fat that ∂ x 1 ∂ x 2 + ∂ T 1 ∂ x 2 + ( ∂ T 1 + ∂ x 1 )( ∂ T 2 + ∂ x 2 ) = ∂ T 1 ∂ T 2 + ∂ x 1 ∂ x 2 mo d (2) sho ws that the monoidal struture on V s is also graded omm utativ e, in the sense desrib ed ab o v e for KK • . These denitions ensure that the K -theory funtor KK → V s , A 7→ K C • ( A ) , f ∈ KK • C ( A, B ) 7→ f ∗ : K C • ( A ) → K C • ( B ) , whi h asso iates to a C ∗ -algebra A the omplex, Z / 2 -graded v etor spae K C • ( A ) := K • ( A ) ⊗ Z C , is ompatible with the symmetri monoidal strutures on ea h at- egory , at least on a b o otstrap ategory N (the Kunneth theorem) where it is an isomorphism (the Univ ersal Co eien t theorem.) In order to illustrate these fats in a onrete w a y w e pro v e the follo wing simple lemma. 4 HEA TH EMERSON Lemma 1. Supp ose c = P a i ˆ ⊗ C b i ∈ K • ( A ⊗ B ) is written as a sum with a i , b i homo gene ous. 1 L et f ∈ KK • ( A, A ′ ) and g ∈ KK • ( B , B ′ ) b e homo gene ous. Then c ˆ ⊗ A ⊗ B ( f ˆ ⊗ C g ) = X ( − 1) ∂ b i ∂ f ( a i ˆ ⊗ A f ) ˆ ⊗ C ( b i ˆ ⊗ B g ) ∈ K • ( A ′ ⊗ B ′ ) . Pr o of. Suppressing subsripts, supp ose a ∈ KK • C ( C , A ) , b ∈ KK • C ( C , B ) , f and g as ab o v e. Then (3) ( a ˆ ⊗ C b ) ˆ ⊗ A ⊗ B ( f ˆ ⊗ C g ) = a ˆ ⊗ A (1 A ˆ ⊗ C b ) ˆ ⊗ A ⊗ B ( f ˆ ⊗ C 1 B ) ˆ ⊗ A ′ ⊗ B (1 A ′ ˆ ⊗ C g ) = a ˆ ⊗ A Σ ∗ ( b ˆ ⊗ C 1 A ) ˆ ⊗ A ⊗ B ( f ˆ ⊗ C 1 B ) ˆ ⊗ A ′ ⊗ B (1 A ′ ˆ ⊗ C g ) . Sine Σ ∗ ( f ˆ ⊗ C 1 B ) = Σ ∗ (1 B ˆ ⊗ C f ) , w e an write the ab o v e (4) = a ˆ ⊗ A ( b ˆ ⊗ C 1 A ) ˆ ⊗ B ⊗ A (1 B ˆ ⊗ C f ) ˆ ⊗ B ⊗ A ′ Σ ∗ (1 A ′ ˆ ⊗ C g ) . Using graded omm utativit y w e ha v e (5) ( b ˆ ⊗ C 1 A ) ˆ ⊗ B ⊗ A (1 B ˆ ⊗ C f ) = b ˆ ⊗ C f = ( − 1) ∂ b∂ f Σ ∗ ( f ˆ ⊗ C b ) = ( − 1 ) ∂ b∂ f Σ ∗  f ˆ ⊗ A ′ (1 A ′ ˆ ⊗ C b )  . Putting this in to ( 4 ) and mo ving the ip aross the tensor pro dut giv es (6) = ( − 1 ) ∂ b∂ f a ˆ ⊗ A f ˆ ⊗ A ′ (1 A ′ ˆ ⊗ C b ) ˆ ⊗ A ′ ⊗ B (1 A ′ ˆ ⊗ C g ) = ( − 1 ) ∂ b∂ f ( a ˆ ⊗ A f ) ˆ ⊗ C ( b ˆ ⊗ B g ) , as required.  W e next state the essen tial denition of this note (see [4℄, [11 ℄, [5 ℄, [1 ℄.) Denition 2. Let A and B b e C ∗ -algebras. Then A and B ar e dual in KK (with a dimension shift of n ) if there exists ∆ ∈ KK n ( A ⊗ B , C ) su h that the omp osition (7) KK • ( D 1 , D 2 ⊗ A ) − ˆ ⊗ C 1 B − → KK • ( D 1 ⊗ B , D 2 ⊗ A ⊗ B ) ˆ ⊗ A ⊗ B ∆ − → KK • + n ( D 1 ⊗ B , D 2 ) is an isomorphism for ev ery D 1 , D 2 . W e all ∆ the fundamental lass of the dualit y . Supp ose A and B are dual with lass ∆ . In the ab o v e notation, set D 1 = C and D 2 = B . Then there is a unique lass b ∆ ′ ∈ KK − n ( C , B ⊗ A ) su h that the isomorphism (8) KK − n ( C , B ⊗ A ) ∼ = − → KK 0 ( B , B ) arries b ∆ ′ to 1 B . W e all b ∆ ′ the dual fundamental lass . By denition, w e ha v e the equation (9) ( b ∆ ′ ˆ ⊗ C 1 B ) ˆ ⊗ B ⊗ A ⊗ B (1 B ˆ ⊗ C ∆) = 1 B . A simple omputation sho ws that the map (10) KK • ( D 1 ⊗ B , D 2 ) ˆ ⊗ C 1 A − → KK • ( D 1 ⊗ B ⊗ A, D 2 ⊗ A ) b ∆ ′ ˆ ⊗ B ⊗ A − → KK •− n ( D 1 , D 2 ⊗ A ) . is an in v erse to ( 7). W e obtain a seond equation (11) (1 A ˆ ⊗ C b ∆ ′ ) ˆ ⊗ A ⊗ B ⊗ A (∆ ˆ ⊗ C 1 A ) = 1 A . 1 An elemen t of a graded set is homo gene ous if it has a denite degree. LEFSCHETZ NUMBERS F OR C ∗ -ALGEBRAS 5 If one prefers to arrange things in a dieren t logial pattern, one an start with a pair of lasses ∆ and b ∆ ′ and insist that they satisfy the equations (9 ) and (11 ) . Then the map as in (7) an b e sho wn to b e an isomorphism with in v erse (10 ). R emark 3 . In the ab o v e notation, (12) (1 A ˆ ⊗ C b ∆ ′ ) ˆ ⊗ A ⊗ B ⊗ A (∆ ˆ ⊗ C 1 A ) = ( − 1 ) n  Σ ∗ ( b ∆ ′ ) ˆ ⊗ C 1 A  ˆ ⊗ A ⊗ B ⊗ A  1 A ˆ ⊗ C Σ ∗ (∆)  . In [5 ℄ the denition of P oinaré dualit y in v olv ed lasses b ∆ ∈ KK − n ( C , A ⊗ B ) and ∆ n ( A ⊗ B , C ) satisfying appropriate equations. T o onnet our urren t disussion with that denition, set b ∆ = Σ ∗ ( b ∆ ′ ) . Then b y (12), the analogues of equations (9 ) and (11) are (13) (Σ ∗ ( b ∆) ˆ ⊗ C 1 B ) ˆ ⊗ B ⊗ A ⊗ B (1 B ˆ ⊗ C b ∆) = 1 B , ( b ∆ ˆ ⊗ C 1 A ) ˆ ⊗ A ⊗ B ⊗ A  1 A ˆ ⊗ C Σ ∗ ( b ∆)  = ( − 1) n 1 A whi h is as in [5 ℄. Notie also that the roles of A and B are symmetri when n is ev en and an ti- symmetri when n is o dd. Giv en A and B dual as ab o v e, dene a Z -bilinear map (14) K • ( A ) × K • ( B ) → Z , ( x | y ) := y ˆ ⊗ B ˆ x, where ˆ x denotes the P oinaré dual of x . Lemma 4. With the Poin ar é duality p airing dene d in ( 14 ) , ( x | y ) = ( − 1) ∂ x∂ y ( x ˆ ⊗ C y ) ˆ ⊗ A ⊗ B ∆ for homo gene ous elements x ∈ K • ( A ) , y ∈ K • ( B ) . Pr o of. Expanding the denitions, w e ha v e (15) y ˆ ⊗ B ˆ x = y ˆ ⊗ B ( x ˆ ⊗ C 1 B ) ˆ ⊗ A ⊗ B ∆ = y ˆ ⊗ B Σ ∗ (1 B ˆ ⊗ C x ) ˆ ⊗ A ⊗ B ∆ = y ˆ ⊗ B (1 B ˆ ⊗ C x ) ˆ ⊗ A ⊗ B Σ ∗ (∆) = ( y ˆ ⊗ C x ) ˆ ⊗ B ⊗ A Σ ∗ (∆) = ( − 1 ) ∂ x∂ y Σ ∗ ( x ˆ ⊗ C y ) ˆ ⊗ B ⊗ A Σ ∗ (∆) = ( − 1) ∂ x∂ y ( x ˆ ⊗ C y ) ˆ ⊗ A ⊗ B ∆ .  T ensoring with the omplex n um b ers w e obtain a dualit y pairing ( | ) : K C • ( A ) × K C • ( B ) → C . This pairing is non-degenerate if B satises the Univ ersal Co eien t theorem. It is supp orted on { ( x, y ) | ∂ ( x ) + ∂ ( y ) = n } . No w note that if A and B are P oinaré dual, then K • ( A ) and K • ( B ) are nitely generated ab elian groups (and for the same reason, if A and B are dual in KK C then K C • ( A ) and K C • ( B ) ha v e nite rank.) By elemen tary metho ds one an th us nd a basis ( x ǫ,i ) for K C • ( A ) and a dual basis ( x ∗ n − ǫ,j ) for K C • ( B ) with resp et to ( | ) , i.e. so that w e ha v e (16) ( x ǫ,i | x ∗ η, j ) = δ η, n − ǫ δ ij . Lemma 5. In terms of the b ases ( x ǫ,i ) and ( x ∗ η, j ) , the lass b ∆ ′ is given by b ∆ ′ = X i,ǫ ( − 1) n − ǫ x ∗ n − ǫ,i ˆ ⊗ C x ǫ,i . 6 HEA TH EMERSON Pr o of. It sues to sho w that the map (17) KK − n C ( C , B ⊗ A ) ˆ ⊗ C 1 B − → K K − n C ( B , B ⊗ A ⊗ B ) ˆ ⊗ A ⊗ B ∆ − → KK 0 C ( B , B ) sends P i,ǫ ( − 1) n − ǫ x ∗ n − ǫ,i ˆ ⊗ C x ǫ,i to the iden tit y in KK • C ( B , B ) . Sine w e are o v er C , the UCT giv es that KK • C ( B , B ) ∼ = Hom C  K C • ( B ) , K C • ( B )  . If x ∈ K C • ( B ) , and denoting our prop osed form ula for b ∆ b y b δ , then w e ha v e (18) x ˆ ⊗ B ( b δ ˆ ⊗ 1 B ) = ( − 1) ∂ x∂ b δ b δ ˆ ⊗ C x b y Lemma 1. Hene the image of b δ under (17 ) sends x ∈ K C • ( B ) to (19) ( − 1) n∂ x X i,ǫ ( − 1) n − ǫ  x ∗ n − ǫ,i ˆ ⊗ C x ǫ,i ˆ ⊗ C x  ˆ ⊗ B ⊗ A ⊗ B (1 B ⊗ ∆) = ( − 1 ) n∂ x X ( − 1) n − ǫ x ∗ n − ǫ,i ·  ( x ǫ,i ˆ ⊗ C x ) ˆ ⊗ A ⊗ B ∆  = X ( − 1) n∂ x +( n − ǫ )+ ǫ∂ x x ∗ n − ǫ,i · ( x ǫ,i | x ) . No w setting x = x ∗ γ ,j , ea h term v anishes sa v e when ǫ = n − γ , in whi h ase the sign is ( − 1) nγ + γ +( n − γ ) γ = +1 .  With these preliminaries out of the w a y , w e an no w state and pro v e the for- mal Lefs hetz theorem for P oinaré dual pairs of C ∗ -algebras alluded to in the in tro dution. Supp ose w e ha v e a dualit y with fundamen tal lasses b ∆ ′ ∈ K K − n ( B ⊗ A, C ) and ∆ ∈ KK n ( A ⊗ B , C ) . Let f ∈ KK( B , B ) . Dene Ind (∆ , f ) := b ∆ ′ ˆ ⊗ B ⊗ A ( f ˆ ⊗ C 1 A ) ˆ ⊗ B ⊗ A Σ ∗ ∆ ∈ KK ( C , C ) ∼ = Z . As the notation suggests, this `index' only dep ends on f and ∆ sub jet to the ondition that ∆ implemen t a P oinaré dualit y . Ho w ev er, the w a y Ind is dened in v olv es b oth ∆ and the dual lass b ∆ ′ , so that if one  hanges the dualit y , it is easy to  he k that t w o anelling  hanges are in tro dued in to Ind , so that Ind (∆ , f ) do es not dep end on the  hoie of ∆ . Motiv ated b y the lassial ase, w e dene the L efshetz numb er Lef ( f ) of f ∈ KK 0 ( B , B ) in the standard w a y as the graded trae (20) Lef ( f ) := tr s ( f ∗ : K C • ( B ) → K C • ( B )) := tra ce C  f ∗ : K C 0 ( B ) → K C 0 ( B )  − trace C  f ∗ : K C 1 ( B ) → K C 1 ( B ) .  . of f ating on the omplexied K -theory of B . Theorem 6. L et A and B b e C ∗ -algebr as satisfying the Universal  o eient the o- r ems and the Kunneth the or em. Supp ose that A and B ar e dual with fundamental lass ∆ ∈ KK n ( A ⊗ B , C ) and dual lass b ∆ ′ ∈ KK − n ( C , B ⊗ A ) . Then for any f ∈ KK 0 ( B , B ) , the L efshetz numb er of f is e qual to the index (21) Lef ( f ) = Ind (∆ , f ) . In p artiular, the zeta funtion ζ f ( t ) := ∞ X n =0 Ind (∆ , f n ) t n . is r ational. LEFSCHETZ NUMBERS F OR C ∗ -ALGEBRAS 7 Pr o of. Let f ∈ KK 0 ( B , B ) . W e an write f ∗ ( x ∗ ǫ,i ) = P r f ǫ ir x ∗ ǫ,r . Hene (22) (1 A ˆ ⊗ C f ) ∗ ( b ∆) = X ( − 1) n − ǫ f n − ǫ ir x ∗ n − ǫ,r ˆ ⊗ C x ǫ,i . Applying the ip giv es (23) Σ ∗ (1 A ˆ ⊗ C f ) ∗ ( b ∆) = X ( − 1) n − ǫ + ǫ ( n − ǫ ) f n − ǫ ir x ǫ,i ˆ ⊗ C x ∗ n − ǫ,r . Finally , pairing this expression using ∆ giv es (24) < Σ ∗ (1 A ˆ ⊗ C f ) ∗ ( b ∆) , ∆ > = X ( − 1) n − ǫ + ǫ ( n − ǫ ) f n − ǫ ir ( x ǫ,i ˆ ⊗ C x ∗ n − ǫ,r ) ˆ ⊗ A ⊗ B ∆ = X ( − 1) n − ǫ f n − ǫ ir ( x ∗ n − ǫ,r | x ǫ,i ) = X ( − 1) n − ǫ f n − ǫ ii = tr( f 0 ∗ ) − tr( f 1 ∗ ) = tr s ( f ∗ ) as required. The statemen t regarding the zeta funtion is an elemen tary onsequene, see [ 9℄.  Finally , w e note that it is rather natural to all the Lefs hetz n um b er of the iden tit y morphism 1 B ∈ KK • ( B , B ) the Euler har ateristi of B ; it is the dierene in ranks of K 0 ( B ) and K 1 ( B ) , and b y our formal Lefs hetz theorem it is the index (25) Eul B = < b ∆ , ∆ >, whi h is a sort of formal Gauss-Bonnet theorem. Example 7 . Let A b e the C ∗ -algebra of setions C τ ( X ) of the Cliord algebra of a ompat manifold X , and B = C ( X ) . The b est-kno wn example of K -theoreti P oinaré dualit y is in this situation. The lass ∆ is represen ted b y the un b ounded self-adjoin t op erator D := d + d ∗ ating on the bundle Λ ∗ C ( X ) of dieren tial forms on X , where d is the de Rham dieren tial, and the additional datum of the Cliord m ultipliation C τ ( X ) ⊗ C ( X ) → B ( L 2 (Λ ∗ C ( X ))  . The lass b ∆ in v olv es a Cliord m ultipliation b y an appropriate v etor eld on X × X , ating on a submo dule of C ( X ) ⊗ C τ ( X ) , but the imp ortan t p oin t is that this v etor eld v anishes on the diagonal. It is immediate that when w e tak e the Kasparo v pro dut of ∆ and b ∆ , w e get simply the op erator D ating on forms L 2 (Λ ∗ C ( X )) . Hene (25 ) sa ys that Eul X = Index( D dR ) , with D dR the de Rham op erator on X . See [7℄ for a losely related omputation. It is also a simple manner to dedue the lassial Lefs hetz xed-p oin t form ula in the same w a y . The fat that the lass b ∆ is supp orted in a neigh b ourho o d of the diagonal in X × X means that if w e t wist b ∆ b y a smo oth map whose graph X → X × X is transv erse to the diagonal, and pair with ∆ , the result is supp orted in a small neigh b ourho o d of the xed-p oin t set of the map. The latter is a disrete set. Th us, the formal Lefs hetz theorem gets us this far, and to nish the omputation w e need to arry out a lo al index omputation. (See [6 ℄ for the details, in a more general on text.) The reader w an ting other simple examples ma y wish to onsider an automor- phism of a nite group. A pleasan t nonomm utativ e Lefs hetz form ula for this situation an b e dedued using the formal Lefs hetz form ula. This form ula giv es the w ell-kno wn relationship b et w een the n um b er of xed p oin ts of the indued map 8 HEA TH EMERSON ˆ ζ : b G → b G on the (nite) spae of irreduible represen tations, and the n um b er of ` ζ -t wisted onjugay lasses' in G . The referene [6 ℄ also on tains this result. In the remainder of this note, w e are going to w ork out a highly nonomm utativ e example (the algebras A and B are simple ). The merit of onsidering an example lik e this is that w e do get a gen uinely new equalit y of in v arian ts  a gen uinely new Lefs hetz theorem. The diult y with the example is that it is not so easy to see what its meaning is. So that it is helpful to ha v e the lassial examples at hand, for omparison. 3. Example  endomorphisms of Cuntz-Krieger algebras Let O A (see [2℄, [3℄) b e the Cun tz-Krieger algebra with (irreduible) matrix A , the univ ersal C ∗ -algebra generated b y n nonzero partial isometries s 1 , . . . , s n su h that X A ij s j s ∗ j = s ∗ i s i . W e are going to illustrate the formal Lefshetz theorem giv en in the previous setion b y pro ving an analogue of the Lefs hetz xed-p oin t theorem for (ertain) endomor- phisms of O A . W e rst remind the reader of the follo wing theorem of Cun tz and Krieger. Lemma 8. (se e [ 2 ℄ ). The gr oup K 0 ( O A ) is Z / (1 − A t ) Z , and the gr oup K 1 ( O A ) is the quotient of Z / (1 − A t ) Z by its torsion sub gr oup. T o ompute the Lefs hetz n um b er of an endomorphism of O A , w e m ust therefore split o the free part of the K 0 group and ompute the images of a set of generators, and similarily , nd free generators for K 1 and ompute their generators. Example 9 . A standing n umerial example will b e the ase A =   1 1 0 1 1 1 0 1 1   . By Lemma 8, K -theory of O A is innite yli in ea h of dimensions 0 and 1 , with free generator [ s 1 s ∗ 1 ] the lass of the pro jetion s 1 s ∗ 1 in ev en degree, and free generator the lass [ s 1 + s ∗ 3 ] of the unitary s 1 + s ∗ 3 in o dd degree. Note that [ s 2 s ∗ 2 ] = 0 and [ s 3 s ∗ 3 ] = − [ s 1 s ∗ 1 ] in K 0 . An y (unital) endomorphism α : O A → O A maps ea h generator s i to a partial isometry t i ∈ O A su h that t 1 , . . . , t n satisfy the same relations. Con v ersely , b y the univ ersal prop ert y of O A , an y  hoie of t 1 , . . . , t n satisfying the Cun tz-Krieger relations giv es rise to a `symmetry' of O A , i.e. an endomorphism. A w ell-kno wn family of is the p erio di 1 -parameter family giv en b y the irle ation s i 7→ z s i , where z ∈ S 1 , i = 1 , 2 , . . . , n. These endomorphisms are, ho w ev er, ob viously homotopi to the iden tit y , whene w e annot exp et v ery in teresting Lefs hetz n um b ers (they will all b e zero, sine the Euler  harateristi of O A is zero.) Instead w e are in terested in more om bina- torially dened endomorphisms. The follo wing denition is v aguely analogous to the assumption that one has an innitely dieren tiable map, in the setting of the lassial Lefs hetz theorem. Let Σ + A denote the Can tor set of sequenes ( x i ) in the graph Λ determined b y the matrix A . Denition 10. Let Z ⊂ Σ + A b e an op en subset and ϕ : Z → Σ + A b e a on tin uous map with domain Z . Then ϕ is smo oth if • Z is a ylinder set. • There exists k ∈ N and a map ψ ′ : P ≤ k → P su h that ψ ( x 1 , x 2 , x 3 , . . . ) = ( ψ ′ ( x 1 , . . . , x k ) , x k +1 , x k +2 , . . . ) , for all x = ( x 1 , x 2 , x 3 , . . . ) ∈ Z , LEFSCHETZ NUMBERS F OR C ∗ -ALGEBRAS 9 where P := { ( x 1 , . . . , x m ) | A x i ,x i +1 = 1 , m ≥ 0 } is the set of nite allo w able strings in the alphab et determined b y A , and P ≤ k is the set of strings of length at most k . W e allo w the empt y string ∅ . With this on v en tion, the left shift σ A : Z := Σ + A → Σ + A is smo oth, sine σ A ( x 1 , x 2 , . . . ) = ( σ ′ A ( x 1 ) , x 2 , x 3 , . . . ) where σ ′ A ( x ) = ∅ for ev ery string of length 1 . Our denition is atually loser to the idea of a quasi-onformal map. Note that P is the v ertex set of the tree ˜ Λ whi h is the univ ersal o v er of Λ . As su h, it admits a anonial path metri: it is `Gromo v h yp erb oli' as a metri spae, and so has a Gromo v b oundary . Lemma 11. A smo oth map ϕ : Z → Σ + A is the b oundary value of a quasi-isometry ϕ ′ : Z ′ → ˜ Λ , wher e Z ′ is a subset of P . Pr o of. Supp ose w e are giv en a smo oth map ψ : Z → Σ + A in the ab o v e sense, with k and ψ ′ as in the denition. W e an tak e the ylinder set Z to b e of the form Z = { x ∈ Σ + A | π N ( x ) ∈ F } , where π N : Σ + → P N is the pro jetion, and F is a nite subset of P N , and where N is larger than k . No w whether an innite string is or is not in the domain of ϕ only dep ends on the rst N letters. Geometrially , the set of innite strings x with rst N letters b elonging to a giv en xed, nite set of nite strings, is a lop en set of Σ + A , and is the losure in the ompatiation of the tree, of the set of nite strings of length at least N , and with rst N letters in the giv en set. Hene the set Z is the b oundary v alues of a subset Z ′ ⊂ P , i.e. Z = Z ′ ∩ ∂ ˜ Λ , where Z ′ is the set of nite strings of length at least N with rst N letters in F . Assuming no w that the nite string ( x 1 , . . . , x m ) is in Z ′ , whene that an y b ound- ary p oin t x = ( x 1 , . . . , x m , x m +1 , . . . ) is in Z , w e ha v e ψ ( x ) = ( ψ ′ ( x 1 , . . . , x k ) , x k +1 , . . . , x m , . . . ) whi h sa ys that the last letter of ϕ ′ ( x 1 , . . . , x k ) is allo w ed to b e follo w ed b y x k +1 . Therefore w e an dene ψ ′ ( x 1 , . . . , x m ) := ( ψ ′ ( x 1 , . . . , x k ) , x k +1 , . . . , x m ) . It is lear that ψ ′ is a quasi-isometry of the tree. It therefore extends to the b oundary ∂ ˜ Λ = Λ and it is lear that its b oundary v alues giv es preisely ψ : Z → Σ + A .  A t ypial `geometri' endomorphism of O A will b e sp eied b y the follo wing denition. Denition 12. A ge ometri endomorphism of O A , where A is n -b y- n , shall refer to the data of a partition Σ + A = Z 1 ∪ · · · ∪ Z n of the sym b ol spae, and an n - tuple Ψ = ( ψ 1 , . . . , ψ n ) of smo oth homeomorphisms ψ i : W i ∼ = → Z i ⊂ Σ + A , su h that W i = S A ij =1 Z i . It is lear that su h partially dened maps determine elemen ts of O A ; w e dene (26) t i := X µ ∈ W ′ i , | µ | = k s ψ ′ i ( µ ) s ∗ µ , where the summation is o v er the w ords of length k in W ′ i , with W ′ i ⊂ P with W ′ ∩ ∂ ˜ Λ = W i as explained ab o v e, and where ψ ′ i are the extensions of the ψ i to P , and, of ourse, where k is suien tly large. Then the range pro jetion of t i iden ties, in the ob vious sense, with the image Z i of ψ i , and the ok ernel pro jetion iden ties with the domain of denition W i of ψ i . Hene, due to ondition 3), w e get an endomorphism α Ψ ( s i ) := t i of O A . 10 HEA TH EMERSON R emark 13 . The iden tit y endomorphism orresp onds to the eviden t partition with Z i = { x | x b egins with i } and ψ i ( x ) = ( i, x ) for x ∈ W i := ∪ A ij =1 Z j . F rom no w on w e will abuse notation and denote b y the same letter the partially dened maps ψ i : P → P , and the maps ψ i : Σ + A → Σ + A . Of ourse there is am biguit y in the  hoie of the lifts ψ : P → P , but w e x  hoies one and for all. Similarly w e will write W i instead of W ′ i and Z i instead of Z ′ i . Example 14 . A go o d example of a geometri endomorphism for A =   1 1 0 1 1 1 0 1 1   is (27) t 1 = s 2 1 s ∗ 1 s ∗ 2 + s 1 s 2 ( s ∗ 2 ) 2 + s 2 s 2 3 s ∗ 3 s ∗ 2 + s 2 s 3 s 2 s ∗ 2 s ∗ 3 + s 2 s ∗ 1 , t 2 = s 3 s 2 , t 3 = s 2 3 s ∗ 3 . The orresp onding partition and triple of smo oth maps is as follo ws. 1) Z 1 is all sequenes ( x n ) b eginning with 1 or 2 . 2) Z 2 is all sequenes ( x n ) b eginning with 32 . 3) Z 3 is all sequenes ( x n ) b eginning with 33 . ψ 1 W e require ψ 1 : W 1 := Z 1 ∪ Z 3 ∼ = − → Z 1 . If a sequene b egins with 1 , then w e replae the initial 1 b y a 2 . If a sequene b egins with 2 then w e replae the initial 2 b y a 1 , unless the seond o ordinate is 3 . In that ase, w e replae the initial 23 b y 233 . Finally , on sequenes b eginning with 32 , w e replae the initial 32 b y 232 . Observ e that the image of ψ 1 is all strings b eginning with 2 or 1 . ψ 2 W e require ψ 2 : W 2 := Z 1 ∪ Z 2 ∪ Z 3 = Σ + A ∼ = − → Z 2 . W e add 32 to the b eginning of an y sequene. ψ 3 W e require ψ 3 : W 3 := Z 2 ∪ Z 3 ∼ = − → Z 3 . T o an y sequene b eginning with 3 w e add an additional 3 . R emark 15 . The endomorphism α Ψ ab o v e sends the range pro jetion of s 1 to the range pro jetion of t 1 , whi h is s 1 s ∗ 1 + s 2 s ∗ 2 . Hene ( α Ψ ) ∗ ([ s 1 s ∗ 1 ]) = [ s 1 s ∗ 1 ] + [ s 2 s ∗ 2 ] = [ s 1 s ∗ 1 ] , so the indued map ( α Ψ ) ∗ : K 0 ( O A ) → K 0 ( O A ) is the iden tit y . T o see the ation on K 1 ( O A ) , one an  he k that the map ( t 1 + t ∗ 3 | · ) : K 0 ( O A ) → Z indued from the P oinaré dualit y pairing (see the end of this setion) v anishes iden tially . Therefore [ t 1 + t ∗ 3 ] = 0 ∈ K 1 ( O A ) and ( α Ψ ) ∗ : K 1 ( O A ) → K 1 ( O A ) is the zero map. So the Lefs hetz n um b er of α Ψ : O A → O A is equal to 1 . (In partiular, α Ψ is not an automorphism .) W e no w desrib e an in v arian t of an y geometri endomorphism, whi h will b e a single partially dened map P → P . Denition 16. Let Ψ = ( ψ 1 , . . . , ψ n ) b e a geometri endomorphism. Extend the ψ i to partially dened self-maps of P . W e let ˙ Ψ : P → P b e the partially dened map dened b y ˙ Ψ( x 1 , . . . , x n ) := ψ x n ( x 1 , . . . , x n − 1 ) if ( x 1 , . . . , x n − 1 ) ∈ Dom( ψ x n ) . Example 17 . The map ˙ Ψ of Example 14 is dene on paths of length 2 b y (28) ˙ Ψ(11) = 2 , ˙ Ψ(21) = 1 , ˙ Ψ(12) = (321) , ˙ Ψ(22) = (322) , ˙ Ψ(32) = (323) , ˙ Ψ(33) = (33) , LEFSCHETZ NUMBERS F OR C ∗ -ALGEBRAS 11 On paths of length 3 . (29) ˙ Ψ(111) = (21 ) , ˙ Ψ(121) = (22 ) , ˙ Ψ(211) = (11 ) , ˙ Ψ(221) = (12 ) . (30) ˙ Ψ( ⋆ ⋆ 2) = (3 2 ⋆ ⋆ ) for an y ( ⋆⋆ ) allo w able, (31) ˙ Ψ(323) = (3 3) , ˙ Ψ(333) = (3 33) . Finally , on w ords of length 4 , Ψ is dened b y (32) ˙ Ψ(1111 ) = (211) , ˙ Ψ(1121 ) = (212) , ˙ Ψ(1221 ) = (222) , ˙ Ψ(2111 ) = (111) , ˙ Ψ(1211 ) = (221) , ˙ Ψ(2211 ) = (121) , ˙ Ψ(3211 ) = (2321 ) , ˙ Ψ(1121) = (212) , ˙ Ψ(2121) = (112) , (33) ˙ Ψ( ⋆ ⋆ ⋆ 2) = (32 ⋆ ⋆⋆ ) , for an y ⋆ ⋆ ⋆ allo w able, and nally , (34) ˙ Ψ(3223 ) = (3322 3) , ˙ Ψ(3233 ) = (3323 3) , ˙ Ψ(3323 ) = (3332 3) , ˙ Ψ(3333) = (33 333) , In these form ulas, an y string not men tioned is not in the domain. Sa y that t w o partially dened maps ˙ Ψ and ˙ Ψ are equiv alen t if ˙ Ψ = ˙ Ψ ′ on suien tly long strings. Then it is only the lass of ˙ Ψ mo dulo ∼ whi h will matter to us for what is oming. Denote b y [ ˙ Ψ] the lass of ˙ Ψ . W e are going to asso iate to a geometri endomorphism an in teger in v arian t. This in v arian t will only dep end on the equiv alene lass [ ˙ Ψ] and not on ˙ Ψ itself. By formal series P ∞ k =1 a k , where a k are real n um b ers, w e will refer to the se- quene of of its terms, mo dulo the equiv alene relation P ∞ k =1 a k ∼ P ∞ k =1 b k if P m k =1 a k = P m k =1 b k for m suien tly large. F or example 1 + 2 + 3 + 4 + · · · ∼ 2 + 0 + 3 + 4 + · · · . The ondition ∼ implies, ob viously , that a k = b k for large enough k , but it is stronger. Denition 18. Let Ξ : P → P b e a partially dened bijetion with nite propa- gation: that is, there exists N := Pro p(Ξ ) su h that Ξ( P k ) ⊂ [ | l − k |≤ N P l , for all k . Let Dom(Ξ) ⊂ P b e its domain and Im(Ξ) its range. W e set (35) Index k (Ξ) := ca rd  P k ∩ Im(Ξ)  −  P k ∩ Dom(Ξ)  . W e let Index(Ξ) b e the formal series (36) Index(Ξ) = ∞ X i =1 Index i (Ξ) . W e sho w b elo w that the index only dep ends on [Ξ] and on v erges if Ξ = ˙ Ψ for some geometri endomorphism Ψ . Lemma 19. The index has the fol lowing pr op erties. 1) If Ξ and Ξ ′ ar e two p artial ly dene d maps whih agr e e on P k for al l su- iently lar ge k , then Index(Ξ) = Index(Ξ ′ ) as formal series. Hen e, Index is  omp atible with ∼ . 2) F or any ge ometri endomorphism Ψ , Index k ( ˙ Ψ) = 0 for k suiently lar ge. Hen e the formal series in (36 )  onver ges in this  ase. 12 HEA TH EMERSON Pr o of. W e pro v e 2) rst. F or simpliit y , w e assume that a giv en partially dened map Ξ has propagation at most 1 . W e start b y assuming that Ξ has no strings in its domain of length ≤ m − 1 , for some m ≥ 2 . No w w e remo v e an y strings of length m from the domain of Ξ . Let the new partially dened map b e alled Ξ ′ . W e laim that the index (or more preisely the formal sum ( 36 ) ) has not  hanged. The index in dimension m − 1 has learly b een shrunk b y the n um b er of elemen ts in dimension m whi h previously mapp ed to dimension m − 1 . Call this a ( m, m − 1) . Th us, Index m − 1 (Ξ ′ ) = Index m − 1 (Ξ) − a ( m, m − 1) . On the other hand, the domain in dimension m has b een redued b y card(Dom(Ξ) ∩ P m ) , while the image in dimension m has b een redued b y a ( m, m ) . So Index m (Ξ ′ ) = Index m (Ξ) − a ( m, m ) + Dom(Ξ) ∩ card( P m ) . Finally , the image in dimension m + 1 is redued b y a ( m, m + 1) . Mean while, the index in dimension < m − 1 has not  hanged, nor has the index in dimensions > m + 1 , sine Ξ  hanges lengths of strings b y at most 1 . So (37) Index(Ξ ′ ) = Index(Ξ) − a ( m, m − 1) − a ( m, m ) + − a ( m, m ) + card(Dom( ˙ Ψ) ∩ P m ) = Index(Ξ) . This pro v es the result. No w this means that for an y Ξ an ha v e its domain suessiv ely shrunk b y eliminating strings of length 1 , then 2 , and so on, without altering its index. The rst assertion is no w immediate, sine without  hanging the index, w e an alter b oth maps un til they agree as partially dened maps. The seond assertion is left to the reader (it follo ws from the analyti onsider- ations disussed in the last setion of the pap er.)  Example 20 . Consider Example 14 , 17 . The domain in dimension 1 has 0 elemen ts in it. The image has 2 elemen ts in it. So Index 1 ( ˙ Ψ) = 2 . The domain in dimension 2 has 6 elemen ts in it and the image has 6 elemen ts in it. Hene Index 2 ( ˙ Ψ) = 0 . In dimension 3 there are 13 elemen ts in the domain and 12 in the image. So Index 3 ( ˙ Ψ) = − 1 . One  he ks that Index k ( ˙ Ψ) = 0 for k > 3 . Hene Index( ˙ Ψ) = 2 + 0 − 1 = 1 . R emark 21 . Altering the domain of a ˙ Ψ on a nite piee is analogous to altering a map f : M → M up to homotop y , whilst retaining transv ersalit y . The net eet on the xed p oin ts (with signs) is zero. R emark 22 . 1) The iden tit y morphism of O A orresp onds to the partially dened map ˙ Ψ id with domain of denition the set of paths ( x 1 , . . . , x n ) su h that A x n ,x 1 = 1 , i.e. the set of lo ops in the graph. The ation of ˙ Ψ id is b y shifting the parameteriza- tion of lo ops. In partiular, Index(Ψ id ) = 0 . 2) If the graph orresp onding to A is  omplete , then Index( ˙ Ψ) = 0 for every ˙ Ψ . This follo ws from the Lefs hetz theorem. The p oin t ab out the index is that there is a lot of anellation in the expres- sion (36 ). T aking in to aoun t this anellation, w e get a m u h more omputable desription of the index Lemma 23. Let Ξ : P → P b e a partially dened homeomorphism with nite propagation. Let m > 0 . Dene γ m (Ξ) := ♯ { x ∈ P | | x | > m, | Ξ( x ) | ≤ m } − ♯ { x ∈ P | | x | ≤ m, | Ξ( x ) | > m } . LEFSCHETZ NUMBERS F OR C ∗ -ALGEBRAS 13 Then Index 1 (Ξ) + Index 2 (Ξ) + · · · + Index m (Ξ) = γ m (Ξ) . In partiular, if Ξ = ˙ Ψ for some geometri endomorphism Ψ , then γ m = γ m +1 = · · · = Index ( ˙ Ψ) for m suien tly large. Pr o of. Let a ( i, j ) denote the n um b er of strings of length i whi h are mapp ed b y Ξ to strings of length j . Let δ ( i, j ) := a ( i, j ) − a ( j, i ) . Assume that Ξ alters lengths of strings b y at most N . Cho ose k > 0 . By denition, Index m (Ξ) = ♯ (Im(Ξ) ∩ P m ) − ♯ (Dom(Ξ) ∩ P m ) . On the other hand, ♯ Im(Ξ) ∩ P m ) = P N k = − N a ( m + k , m ) while ♯ (Dom(Ξ) ∩ P ) = P N k = − N a ( m, m + k ) , whene Index m (Ξ) = N X k = − N δ ( m + k , m ) . Of ourse δ ( i, j ) = − δ ( j, i ) . Hene when w e tak e the (formal) sum (38) Index(Ξ) = ∞ X m =1 N X k = − N δ ( m + k, m ) , a term δ ( i, j ) app ears exatly t wie with opp osite signs, if i and j are small enough relativ e to m . It follo ws that m X k =1 Index k (Ξ) = γ m (Ξ) b eause of telesoping. The last assertion follo ws from Lemma 19 .  Example 24 . F or instane, in Example 14 , 17 , m = 3 is large enough, note Prop( ˙ Ψ) ≤ 1 . There are 8 strings of length 4 mapp ed to strings of length 3 and 7 strings of length 3 mapp ed to strings of length 4 , so Index( ˙ Ψ) = 8 − 7 = 1 . Based on Lemma 23 , w e an giv e a p o ynomial form ula for the index as follo ws. Fix m large. Fix j . W e oun t the n um b er of strings of length m + j (for j = 1 , 2 , . . . , N , whi h are mapp ed to strings of length ≤ m . W e refer to the presen tation (26 ). Fix i and µ with | µ | = k . Supp ose that | ψ i ( µ ) | ≤ | µ | − j + 1 . Consider a string w = ( µ, u ) of length m + j , where | u | = m + j − | µ | is a string (path) from the termin us t ( µ ) of µ to i . Then this is mapp ed under ˙ Ψ to a string of length ≤ m + j − 1 − j + 1 = m . Hene for ea h su h i, µ and u w e get a p ositiv e on tribution to the index. F or xed µ and i the n um b er of p ossible u 's is equal to the n um b er of paths of length m + j − k from t ( µ ) to i , whi h equals A m + j − k t ( µ ) i . Hene the total p ositiv e on tribution to the index is n X i =1 N X j =1 X µ ∈ W i , | ψ i ( µ ) |≤| µ |− j +1 A m + j − k t ( µ ) ,i . F or the negativ e on tributions, for j = 0 , 1 , . . . , N − 1 x i and µ su h that | ψ i ( µ ) | ≥ | µ | + j + 2 . Then for ea h w = ( µ, u ) of length m − j , so that u is a string from t ( µ ) to i of length m − j − k , the length of ˙ Ψ( w ) is ≥ m − j − 1 + j + 2 = m + 1 . 14 HEA TH EMERSON Hene w e get a negativ e on tribution to the index. Therefore the total negativ e on tributions is n X i =1 N − 1 X j =0 X µ ∈ W i | ψ i ( µ ) |≥| µ | + j +2 A m − j − k t ( µ ) ,i . Therefore w e get the follo wing urious, ompletely expliit, p olynomial form ula for the index: it is giv en expliitly b y the form ula (39) n X i =1 N X j =1 X µ ∈ W i , | ψ i ( µ ) |≤| µ |− j +1 A m + j − k t ( µ ) ,i − n X i =1 N − 1 X j =0 X µ ∈ W i | ψ i ( µ ) |≥| µ | + j +2 A m − j − k t ( µ ) ,i , for an y N > Pro p( ˙ Ψ) = max i,µ  | µ | − | ψ i ( µ ) | + 1  for an y m large enough. Example 25 . F or instane, in our main example the ab o v e form ula with k = 2 , m = 3 , N = 1 giv es (40) Index( ˙ Ψ) = ( A 2 11 + A 2 21 + A 2 11 + A 2 21 ) − ( A 12 + A 22 + A 12 + A 22 + A 32 + A 22 + A 32 ) = 8 − 7 = 1 . W e an no w state our Lefs hetz form ula for Cun tz-Krieger algebra endomor- phisms, at least those oming from simple om binatoris of generators and rela- tions. Theorem 26. (L efshetz The or em for Cuntz-Krie ger endomorphisms.) L et Ψ ∈ G A and α Ψ : O A → O A b e the  orr esp onding endomorphism. F or suiently lar ge m and N , the L efshetz numb er of α e quals the index of Ψ . Thus, the L efshetz formula (41) T ra c e s  ( α Ψ ) ∗ : K ∗ ( O A ) Q → K ∗ ( O A ) Q  = n X i =1 N X j =1 X µ ∈ W i , | ψ i ( µ ) |≤| µ |− j +1 A m + j − k t ( µ ) ,i − n X i =1 N − 1 X j =0 X µ ∈ W i | ψ i ( µ ) |≥| µ | + j +2 A m − j − k t ( µ ) ,i , holds. R emark 27 . The reader will learly see the dierene b et w een omputing the trae of an endomorphism using the p olynomial form ula (39 ), and b y w a y of omputing the ation on K -theory . Compare for instane the expression (27 ) with the alu- lations in Remark 15 , whi h of ourse need the w ork of Cun tz and Krieger just to get o the ground. T o pro v e this, w e need to sho w that Index( ˙ Ψ) = Ind (∆ , [ α Ψ ]) for an appropriate ∆ induing a dualit y . Kamink er and Putnam pro v ed su h a dualit y in [11℄. W e refer the reader to their pap er for further details, and merely sk et h the omputation here. Let s 1 , . . . , s n denote the generators for O A and t 1 , . . . , t n the generators for O A t . Dene H A := ℓ 2 ( P ) , where P is the set of strings, as ab o v e. Let S i : H A → H A , S i ( e w ) := A i o( w ) e iw , R j ( e w ) := A t( w ) j e wj . Clearly [ S i , R j ] = 0 , while S ∗ i , R j ] = 0 mo dulo nite-rank op erators. It is also easy to  he k that P j A ij S j S ∗ j = S ∗ i S i mo dulo nite rank op erators, and similarly the R j satisfy the relations for O A t . W e obtain the Busb y in v arian t O A ⊗ O A T → B ( H A ) / K ( H A ) of an extension of O A ⊗ O A t b y the ompat op erators and hene (sine O A is n ulear) a lass in KK 1 ( O A ⊗ O A t , C ) . Kamink er and Putnam pro v e that ∆ indues a dualit y with dual lass the elemen t w = P s i ⊗ t ∗ i . Then ww ∗ = w ∗ w and ea h LEFSCHETZ NUMBERS F OR C ∗ -ALGEBRAS 15 are pro jetions. Therefore w + 1 − w w ∗ is a unitary in O A ⊗ O t A and so denes an b ∆ of KK 1 ( C , O A , ⊗ O A t ) . No w supp ose w e ha v e an endomorphism s i 7→ t i := X µ ∈ W i , | µ | = k s ψ i ( µ ) s ∗ µ . Then under the endomorphism, b ∆ is mapp ed to X i, µ ∈ W i , | µ | = k s ψ i ( µ ) s ∗ µ ⊗ t i . T o ompute the pairing Ind (∆ , [ α ]) = < ( α Ψ ⊗ 1 O A t ) ∗ ( b ∆) , ∆ > w e need to ompute the index of the ob vious lift F redholm index of W Ψ + (1 − W Ψ W ∗ Ψ ) . Ho w ev er, it is lear that W Ψ W ∗ Ψ is e qual to W ∗ Ψ W Ψ on ℓ 2 ( P m ) for m ≥ dmin(Ψ) + 1 . Hene the sum ∞ X j =1 dim ker(( W Ψ ) | ℓ 2 ( P j ) ) − dim ran( W Ψ ) ∩ ℓ 2 ( P j )) on v erges, and eviden tly on v erges to the analyti index. No w w e an regard W Ψ as the op erator indued b y the partial p erm utation ˙ Ψ of P , in whi h p oin t masses e w in the k ernel of W Ψ orresp ond to w ords w not in the domain of W Ψ . W e are no w in the setting of our earlier disussion of partially dened maps P → P , and it is lear that the index of W Ψ is exatly the same as the index dened in Denition 18 and w e are done b y Setion 2. R emark 28 . It w as men tioned ab o v e that the geometri index of an arbitr ary en- domorphism m ust v anish in the ase of a Cun tz algebra. This is of ourse ob vious from the Lefs hetz form ula sine the K -theory of Cun tz algebras v anishes rationally . On the other hand, it do es not seem v ery ob vious from a geometri p oin t of view. This sort of thing happ ens in lassial top ology of ourse: one pro v es existene of xed p oin ts b y homology omputations. Referenes [1℄ J. Bro dzki, V. Mathai, J. Rosen b erg, and R. Szab o, D-br anes, RR-elds, and duality on non ommutative manifolds. (2007), Preprin t. [2℄ J. Cun tz, A lass of C ∗ -algebr as and top olo gi al Markov hains. II. R e duible hains and the Ext-funtor for C ∗ -algebr as , In v en t. Math. 63 (1981), no. 1, 2540. MR 608527 (82f: 46073b) [3℄ Joa him Cun tz and W olfgang Krieger, A lass of C ∗ -algebr as and top olo gi al Markov hains , In v en t. Math. 56 (1980), no. 3, 251268. MR 561974 (82f: 46073a) [4℄ Alain Connes, Non ommutative ge ometry , A ademi Press In., San Diego, CA, 1994. MR 1303779 (95j: 46063) [5℄ Heath Emerson, Non ommutative Poin ar é duality for b oundary ations of hyp erb oli gr oups , J. Reine Angew. Math. 564 (2003), 133. [6℄ Heath Emerson, Siegfried E h terho, and Hyun-Jeong Kim, Fixe d p oint formulas for pr op er ations (2007), preprin t. [7℄ Heath Emerson and Ralf Mey er, Euler har ateristis and Gysin se quen es for gr oup ations on b oundaries , Math. Ann. 334 (2006), no. 4, 853904. MR 2209260 (2007b: 19006) [8℄ , L efshetz maps for simpliial  omplexes and smo oth manifolds (2007), preprin t. [9℄ Robin Hartshorne, A lgebr ai ge ometry , Springer-V erlag, New Y ork, 1977. Graduate T exts in Mathematis, No. 52. MR 0463157 (57 #3116) [10℄ G. G. Kasparo v, Equivariant K K -the ory and the Novikov  onje tur e , In v en t. Math. 91 (1988), no. 1, 147201. MR 88j: 58123 [11℄ Jerome Kamink er and Ian Putnam, K -the or eti duality of shifts of nite typ e , Comm. Math. Ph ys. 187 (1997), no. 3, 509522. MR 1468312 (98f: 46056) E-mail addr ess : hemersonuvi.a University of Vitoria, Vitoria, BC