Index formulas and charge deficiencies on the Landau levels

Index form ulas and c harge deficienc ies on the Landau lev els Magn us Goffeng Departmen t of M athematical Sciences, Division of Mathematics Chalmers unive rsit y of T ec hnology and Univ ersit y o f Gothenburg Abstract The notion of charge deficiency is studied from the view of K -theory of operator algebras and is applied to the Landau leve ls in R 2 n . W e calculate the charg e deficiencies at the higher Landau levels in R 2 n by means of an Atiy ah-Singer type index theorem. 1 In tro duction The pap er is a study of the charge deficiencies at the Landa u levels in R 2 n . The Landau levels ar e the eigenspaces of the Landa u Hamiltonian which is the energy o per ator for a q ua n tum par ticle moving in R 2 n under the influence of a constant magnetic field of full rank . In [1], the notion of charge deficiency was intro duce d as a measur e off how m uch a flux tub e changes a fermio nic s y stem in R 2 . The setting of [1] is a qua n- tum sy stem where the F ermi energy is in a gap and the question is what happens when the system is taken troug h a cy c le. Letting P denote the pro jection o n to the state space and U the unitary transformatio n repr esenting the cycle, the pro jection Q on to the new state space after it had b een tak en thro ugh a cycle can be expr essed a s Q = U P U ∗ . The relative index ind ( Q, P ) is defined as an infinite dimensional analogue of dim Q − dim P and is w ell defined whenever Q − P is a compact op era to r. T he co ndition that Q − P is compact is equiv alen t to that [ P, U ] is c ompact. In the setting of [1] the rela tiv e index r epresents the change in the num ber of fermions that U pro duces. In [1] the following formula was prov en: ind ( Q , P ) = ind ( P U P ) . F or sufficiently nice sy stems in R 2 one can choose the par ticular unitary given by mult iplication b y the b ounded function U := z / | z | . The condition on the system that is needed is that P commutes with U up to a compact o p era tor. The charge deficiency of a pro jection P in the s ense of [1] is then defined using U as c ( P ) := ind ( P U P ) . The viewp oint we will hav e in this pa p er is tha t the charge deficiency is a K -homolog y class . This viewp oint lies in line with the view on D -bra ne c harges in string theory , see mo re in [4 ], [10]. In the case studied in [1] the charge 1 deficiency is realized as an o dd K -homolo gy c lass on the circle T . The unitary U define a represe n tation of C ( T ) and using the fact that P commutes with U up to a compa ct op era tor we get a K -homolo gy class. Let us denote this K -homolog y class b y [ P ] and b y u we will denote the genera to r of C ( T ). In this notation, the charge deficiency is g iven b y c ( P ) = [ P ] ◦ [ u ] ∈ K K ( C , C ) ∼ = Z , the Ka sparov product betw een [ P ] ∈ K 1 ( C ( T )) and [ u ] ∈ K 1 ( C ( T )). Thus the charge deficiency is the image of [ P ] under the isomor phism K 1 ( C ( T )) = K K 1 ( C ( T ) , C ) ∼ = H om ( K 1 ( C ( T )) , K 0 ( C )) ∼ = Z , where the fir st isomorphism is the natur al mapping coming from the Universal Co efficient Theorem for K K - theory and the second isomorphism comes from choosing [ u ] as a gener ator for K 1 ( C ( T )). So a b etter pictur e is that the K - homology class [ P ] ∈ K 1 ( C ( T )) is the charge deficiency of P . The s ystem we will consider in this pap er consists of a particle moving in R 2 n under the influence of a cons tan t magnetic field B of full rank. If w e c ho ose a linear vector p o ten tial A satisfying d A = B the Ha miltonian of this system is given by H A := ( − i ∇ − A ) 2 , This L andau Hamiltonian should b e viewed as a densely defined op erator in the Hilber t space L 2 ( R 2 n ). T a king D ( H A ) = C ∞ c ( R 2 n ), the oper ator H A bec omes essentially self-adjoint, s ee more in [9]. Due to the identification R 2 n = C n we will use the co mplex structure and we will as s ume that B = i 2 P d z j ∧ d ¯ z j . The L a ndau Hamiltonian has a dis crete sp e c trum with eigenv alues Λ ℓ = 2 ℓ + n fo r ℓ ∈ N and the eig e nspaces L ℓ are infinite dimensional. Let P ℓ : L 2 ( R 2 n ) → L ℓ denote the o rthogonal pro jection to the ℓ :th eigenspace. Our po in t o f view o n the charge deficiencies for the Landau levels is that they ar e K -homology classes of the sphere S 2 n − 1 . F or a b ounded contin uous function a : R 2 n → M N ( C ) we define the contin uous function a r ∈ C ( S 2 n − 1 ) as a r ( v ) := a ( rv ) . W e le t A N be the subalgebra of C b ( R 2 n ) ⊗ M N ( C ) such that a r conv erges uni- formly in v to a contin uous function a ∂ on S 2 n − 1 . The mapping a 7→ a ∂ defines a ∗ - homomorphism A N → C ( S 2 n − 1 ) ⊗ M N ( C ). The pro jection P ℓ commutes up to a compact op erator with a ∈ A N (see b elow in Theor e m 3.2) a nd P ℓ a | L ℓ ⊗ C N : L ℓ ⊗ C N → L ℓ ⊗ C N is F r edholm if and o nly if a ∂ is inv ertible (see P rop osition 3.6). Now we may present the main theor em of this pap er: Theorem 1. If a ∂ is smo oth and invertible, t he index of P ℓ a | L ℓ ⊗ C N c an b e expr esse d as ind ( P ℓ a | L ℓ ⊗ C N ) = − ( ℓ + n − 1)! ℓ !(2 n − 1)!(2 π i ) n Z S 2 n − 1 tr(( a − 1 ∂ d a ∂ ) 2 n − 1 ) . The char ge deficiency [ P ℓ ] ∈ K 1 ( C ( S 2 n − 1 )) may b e ex pr esse d in terms of the Ber gman pr oje ct ion P B on the unit b al l in C n as [ P ℓ ] = ( ℓ + n − 1)! ℓ !( n − 1)! [ P B ] . 2 2 The particular Landau lev els The sp ectral theory of the Landau Hamiltonia n is w ell known and we will review it briefly . See more in [13]. W e will let ϕ := | z | 2 4 and assume that the magne tic field B is of the form B = i∂ ¯ ∂ ϕ . Here ∂ is the co mplex linear part of the e x terior differential d. Define the annihilation op era tors as q j := 2 ∂ ∂ ¯ z j + z j for j = 1 , . . . , n. The adjoints a re g iv en by the c r eation op erators q ∗ j := − 2 ∂ ∂ z j + ¯ z j . The annihi- lation and cre a tion op erator s satisfy the following for m ulas: [ q j , q i ] = [ q ∗ j , q ∗ i ] = 0 , [ q i , q ∗ j ] = 2 δ ij and H A = n X j =1 q ∗ j q j + n = n X j =1 q j q ∗ j − n. Here w e v iew H A as a dens ely defined op erator in L 2 ( C n ). Th us the low est eigenv a lue is n with corr espo nding eigens pa ce L 0 = e − ϕ F ( C n ) where F ( C n ) := L 2 ( C n , e − 2 ϕ ) ∩ O ( C n ) deno tes the F o ck space. Here O ( C n ) deno tes the space of holomo r phic functions in C n . In one complex dimension there is o nly one creation op erator q ∗ and the eigenspa ces a re given by L k = ( q ∗ ) k L 0 . Using m ulti-index notation, fo r k = ( k 1 , . . . , k n ) ∈ N n we define q k := q k 1 1 · · · q k n n and L k := q ∗ k L 0 = L k 1 ⊗ L k 2 ⊗ · · · ⊗ L k n . W e will call this space for the p articular Landau level of height k . Using that q j and q ∗ j define a repr esentation of the Heisenberg algebra in n dimension w e obtain the eigenv alues of H A as Λ ℓ = 2 ℓ + n with the cor resp onding eigenspaces L ℓ := M | k | = ℓ L k = M | k | = ℓ L k 1 ⊗ L k 2 ⊗ · · · ⊗ L k n . The ℓ :th eigenspace L ℓ is called the Landau level of height ℓ . Since the Hamil- tonian commutes with the repr esentation of S U ( n ) on C n , its eigenspa ces ar e S U ( n )-in v ariant. Also the o rthogonal pro jections P ℓ : L 2 ( C n ) → L ℓ are inv ari- ant under the S U ( n )-action. Recall that the v acuum s ubspace L 0 ⊆ L 2 ( C n ) has a repro ducing kernel induced b y the repro ducing kernel on the F o ck spac e. The r e pro ducing kernel of F ( C n ) is g iv en by K ( z , w ) = e w · ¯ z 4 . So the repro ducing kernel of L 0 is given by K 0 ( z , w ) := e 1 4 ( w · ¯ z −| z | 2 −| w | 2 ) . This e x pression for the repro ducing kernel implies that the orthogo nal pro jection P 0 : L 2 ( C n ) → L 0 is g iv en by P 0 f ( z ) = Z C n f ( w ) K 0 ( z , w )d V . By [1 2] the orthogo nal pr o jection P k : L 2 ( C n ) → L k onto the particula r La ndau level o f height k is a lso an int egra l op erator w ith kernel K k ( z , w ) = e 1 4 ( w · ¯ z −| z | 2 −| w | 2 ) n Y j =1 L k j  1 2 | z j − w j | 2  . (1) Here L k is the La guerre p olyno mial of order k . Notice tha t the pro jections P k are not S U ( n )-inv ar iant in g eneral. 3 3 T o eplitz op erators on the Landau l ev els W e want to study to polo gical prop erties of the particular Landa u levels using T o eplitz oper ators. The symbo ls will be ta k en from a suitable subalgebr a of C b ( C n ), the b ounded functions o n C n . The s ta ndard notatio n B ( H ) will b e used for the b ounded op erators on a sepa rable Hilb ert space H and the co mpact op erators will b e denoted by K ( H ). W e will le t π : C b ( C n ) → B ( L 2 ( C n )) denote the representation g iven by p oin twise m ultiplication. This is clearly an S U ( n )-equiv a riant mapping. Define the linear map T k : C b ( C n ) → B ( L k ) by T k ( a ) := P k π ( a ) | L k . Lemma 3.1. If a ∈ C 0 ( C n ) t hen T k ( a ) ∈ K ( L k ) for al l k ∈ N n . The pro of of this lemma is analogo us to the pro o f for the sa me statement for T o eplitz o p era tors on a pseudo conv ex domain from [1 4]. Pr o of. It is sufficient to prov e the claim for a ∈ C c ( C n ), since T k is contin uo us and C c ( C n ) ⊆ C 0 ( C n ) is dense. Define the compact set K := supp ( a ). Let R : L k → L 2 ( C n ) denote the op era tor given by multiplication by χ K , the characteristic function of K . W e hav e T k ( a ) = P k π ( a ) R so the Lemma holds if R is c o mpact. That R is co mpact follows from Cauch y estimates of holomorphic functions on a compa c t set. Define the S U ( n )-inv a r iant C ∗ -subalgebra A ⊆ C b ( C n ) a s consisting o f func- tions a such that a ( rv ) conv erges uniformly in v as r → ∞ to a contin uous function a ∂ : S 2 n − 1 → C when r → ∞ . Thus we obta in a surjective S U ( n )- equiv ar iant ∗ -homomo rphism π ∂ : A → C ( S 2 n − 1 ) given by π ∂ ( a )( v ) := lim r →∞ a ( rv ) . The ma pping π ∂ satisfies k er π ∂ = C 0 ( C n ). W e will henceforth co nsider T k as a ma pping from A to B ( L k ). If we let B n denote the op en unit ball in C n , a nother view on A is a s the image o f the S U ( n )-equiv a riant ∗ -monomorphism C ( B n ) → C b ( B n ) ∼ = C b ( C n ) where the last isomo r phism comes fro m a n S U ( n )-equiv ariant homeomo r phism B n ∼ = C n . Theorem 3. 2. The pr oje ction P k satisfies [ P k , π ( a )] ∈ K ( L 2 ( C n )) for al l a ∈ A . Ther efor e the ∗ -line ar mapping T k : A → B ( L k ) satisfies T k ( ab ) − T k ( a ) T k ( b ) ∈ K ( L k ) . The pr o o f is based on a similar re sult from [2] wher e the F o ck space was used to define a T o eplitz quantization of a ce r tain suba lg ebra of L ∞ ( C n ). The case of the F o ck space is mor e or less the same a s the case k = 0 for Landa u quantization. T o prove the Theor em we need a lemma s imilar to part ( iv ) o f Theorem 5 of [2]. Using the isomorphism A ∼ = C ( B n ) we define the dens e subalgebra A 1 ⊆ A a s the in verse imag e of the Lipschitz c o n tinuous functions in C ( B n ). Lemma 3.3 . F or a ∈ A 1 then for any ε > 0 we may write a = g ε + h ε wher e h ε ∈ C 0 ( C n ) and g ε ∈ A satisfies | g ε ( z ) − g ε ( w ) | ≤ ε | z − w | ∀ z , w ∈ C n . (2) 4 Pr o of. Let C denote the Lipschitz co nstant of π ∂ ( a ). T ake an ε > 0 a nd let χ ε be a Lipshitz contin uo us S U ( n )-in v aria n t cutoff such that χ ε ( z ) = 0 for | z | ≤ R and χ ε ( z ) = 0 for | z | ≥ 2 R where R = R ( ε, C ) is to b e defined later. T o sho rten notation, define a ∂ := π ∂ ( a ). Let g ε ( z ) := χ ε ( z ) · a ∂ ( z / | z | ) and h ε := a − g ε . Clearly h ε ∈ C 0 ( C n ) and g ε ∈ A so what rema ins to be proven is that R ca n b e chosen in such a way that g ε satisfies equa tion (2). W e have elementary e stimates     z | z | − w | w |     ≤ | z − w | | z | +     w | z | − w | w |     ≤ 2 | z − w | | w | . Thu s for z , w 6 = 0 the function a ∂ satisfies     a ∂  z | z |  − a ∂  w | w |      ≤ 2 C | w | | z − w | . The function χ ε has Lipschitz co efficient 1 / R so if we tak e R > 2 C /ε then g ε satisfies equa tion (2). Let C ( L 2 ( C n )) := B ( L 2 ( C n )) / K ( L 2 ( C n )) denote the Calkin alge bra and q the quotient mapping . Pr o of of The or em 3.2. Since Lipschitz con tinuous functions a re dense in A we may assume tha t a ∈ A 1 , so by Lemma 3.3 we can for any ε > 0 write a = g ε + h ε . In this cas e we hav e for f ∈ L 2 ( C n ) [ P k , π ( g ε )] f ( z ) = Z ( g ε ( z ) − g ε ( w )) K k ( z , w ) f ( w )d w . Define the o per ator B f ( z ) := Z | z − w | K k ( z , w ) f ( w )d w . By equatio n (1) we hav e that for some C the integral k ernel of B is bo unded by | z − w || K k ( z , w ) | ≤ C | z − w | | k | +1 e − 1 8 | z − w | 2 . Therefore the kernel of B is do minated by the kernel of a b ounded c o n volution op erator and k B k < ∞ . The estimate (2) for g ε implies that k [ P k , π ( g ε )] k ≤ ε k B k . Using that [ P k , π ( g ε )] = [ P k , π ( a )] mo dulo compact op erators , by Lemma 3 .1, we have the inequality k q ([ P k , a ]) k C ( L 2 ( C n )) ≤ ε k B k ∀ ε > 0 . Therefore q ([ P k , a ]) = 0 and [ P k , a ] is co mpact. 5 Theorem 3.2 implies that the mapping ˜ β k := q ◦ T k : A → C ( L k ) is a well defined ∗ -homomor phism. Define the C ∗ -algebra ˜ T k := { a ⊕ x ∈ A ⊕ B ( L k ) : ˜ β k ( a ) = q ( x ) } . This C ∗ -algebra contains K as a n ideal via the embedding k 7→ 0 ⊕ k and we obtain a sho r t exac t sequence 0 → K → ˜ T k → A → 0 . (3) Lemma 3.4. L et ( k p ) N p =1 ⊆ N n b e a finite c ol le ction of distinct n -tu ples of inte gers. Then the mapping A ∋ a 7→ q N X p =1 P k p ! π ( a ) N X p =1 P k p !! ∈ C ( ⊕ N p =1 L k p ) c oincides with the mapping A ∋ a 7→ ⊕ N p =1 ˜ β k p ( a ) ∈ C ( ⊕ N p =1 L k p ) . Pr o of. The Lemma follows if we show that P k π ( a ) P k ′ ∈ K ( L 2 ( C n )) for k 6 = k ′ . But Theo rem 3 .2 implies tha t P k π ( a )(1 − P k ) ∈ K ( L 2 ( C n )). So the Lemma follows fro m P k π ( a ) P k ′ = P k π ( a )(1 − P k ) P k ′ . In pa rticular we ca n lo ok at the c ollection of all k :s such that | k | = ℓ . W e will define the S U ( n )-equiv ar ia n t mapping ˜ β ℓ : A → C ( L ℓ ) as a 7→ ⊕ | k | = ℓ ˜ β k ( a ) . Just a s for the particular Landau levels we define ˜ T ℓ := { a ⊕ x ∈ A ⊕ B ( L ℓ ) : ˜ β ℓ ( a ) = q ( x ) } . The pro jection map ˜ T ℓ → A given by a ⊕ x 7→ a defines a n S U ( n )-equiv a riant extension 0 → K → ˜ T ℓ → A → 0 . Lemma 3.5. The kernel of ˜ β ℓ is C 0 ( C n ) . Pr o of. Lemma 3.1 implies that C 0 ( C n ) ⊆ k er ˜ β ℓ . T o prov e the reverse inclusion we o bserve that the mapping ˜ β ℓ is a unital S U ( n )-equiv a riant ∗ -ho momorphism. Since ˜ β ℓ is equiv ariant, the ideal ker ˜ β ℓ ⊆ A is S U ( n )-in v aria n t. The inclusion C 0 ( C n ) ⊆ ker ˜ β k implies tha t there is an equiv a r iant surjection C ( S 2 n − 1 ) → A/ ker ˜ β ℓ which must be an isomor phism since C ( S 2 n − 1 ) is S U ( n )-simple and ˜ β ℓ is unital. It follows that ker ˜ β ℓ = C 0 ( C n ). It is interesting that although the statement of Lemma 3 .5 sounds algebraic , it is really the ana lytic statemen t that T ℓ ( a ) is compact if a nd only if a v anishes at infinity . And this is prov en with algebraic methods ! Prop osition 3.6. If u ∈ A ⊗ M N , the op era tor T ℓ ( u ) is F r e dholm if and only if π ∂ ( u ) is invertible. Pr o of. By Atkinson’s Theor em T ℓ ( u ) is F r edholm if a nd o nly if ˜ β ℓ ( u ) is inv e r t- ible. Lemma 3 .5 implies that ker π ∂ = ker ˜ β ℓ so ˜ β ℓ ( u ) is inv ertible if and o nly if π ∂ ( u ) is inv ertible. 6 4 Pulling sym b ols bac k from S 2 n − 1 T o put the T o eplitz o per ators on a La ndau level in a suitable homo lo gical pic- ture, we must pa s s fro m A to C ( S 2 n − 1 ). This is a consequence o f the circum- stance that A is homo top y equiv alen t to C , so A do es not contain any r elev ant top ological info r mation. With Lemma 3 .5 in mind we de fine the T o eplitz alg e - bra T k for C ( S 2 n − 1 ) as if β k were injective. So let λ : C ( S 2 n − 1 ) → B ( L 2 ( C n )) denote the ∗ - representation defined by λ ( a ) f ( z ) = a  z | z |  f ( z ) . (4) T ake χ 0 ∈ C ∞ ( R ) to b e a smo oth function such that χ 0 ( x ) = 0 for | x | ≤ 1 and 1 − χ 0 ∈ C ∞ c ( R ). W e define the cut-off χ ( z ) := χ 0 ( | z | ) and the o p era tor ˜ P k := P k χ. (5) F or the op erato r ˜ P k , q ( ˜ P k ) is a pro jection by Lemma 3.1. W e let T k be the C ∗ -algebra generated b y ˜ P k λ ( C ( S 2 n − 1 )) ˜ P ∗ k . Theorem 4.1. F or any k , k ′ ∈ N n ther e exist a unitary Q k , k ′ : L k ′ → L k such t hat Ad ( Q k , k ′ ) : T k → T k ′ is an isomorph ism satisfying q ( ˜ P k ′ λ ( a ) ˜ P ∗ k ′ ) = q ◦ Ad ( Q k , k ′ )( ˜ P k λ ( a ) ˜ P ∗ k ) . (6) F u rt hermor e, for any k ∈ N n , t he r epr esentation of T k on L k given by the inclusion T k ⊆ B ( L k ) is irr e ducible and has the cyclic ve ctor ξ k define d by ξ k ( z ) := q ∗ k (e −| z | 2 / 4 ) . Up t o normalization the cyclic ve ctors satisfy Q k , k ′ ξ k ′ = ξ k . Pr o of. Let us start with obs erving that for any a , b ∈ C ( S 2 n − 1 ) w e have ˜ P k λ ( ab ) ˜ P ∗ k − ˜ P k λ ( a ) ˜ P ∗ k P k λ ( b ) ˜ P ∗ k ∈ K . So if T k acts irreducibly on L k , then K ⊆ T k . First w e will construct a cyclic v ector for the T k -action on L k and use the cyclic vector in L 0 to sho w that T 0 acts ir r educibly on L 0 . Then we will show that for k such that T k acts irreducibly on L k and 1 ≤ j ≤ n there is a n isomorphism T k ∼ = T k + e j induced by a unitary intert wining the T k -action o n L k with the T k + e j -action on L k + e j . Consider the elements ξ m , k ∈ L k for m ∈ N n defined by ξ m , k ( z ) := q ∗ k ( z m e −| z | 2 / 4 ) . The elements ξ m , k form an o rthogonal basis for L k . As in the statement of the theorem, w e de fine ξ k := ξ 0 , k . F or a ∈ C ( S 2 n − 1 ) w e hav e h ξ m , k , ˜ P k a ˜ P ∗ k ξ k i = h ξ m , k , χ 2 aξ k i = Z C n ¯ q ∗ k ( ¯ z m e −| z | 2 / 4 ) q ∗ k (e −| z | 2 / 4 ) χ 2 ( z ) a  z | z |  d V = Z S 2 n − 1 p m ( ¯ z ) a ( z )d S, 7 for some p olynomia ls p m of deg ree at most 2 | k | + | m | . It follows that T k ξ k span L k and ther e fore T k ξ k = L k . Thus ξ k is a cyclic vector for the T k -action. By standard theor y T 0 acts irreducibly o n L 0 if and o nly if there are no non-zero ξ ′ 0 , ξ ′′ 0 ∈ L 0 such that ξ 0 = ξ ′ 0 + ξ ′′ 0 and T 0 ξ ′ 0 ⊥ T 0 ξ ′′ 0 . Assume that for some ξ ′ 0 ∈ L 0 we have T 0 ξ ′ 0 ⊥ T 0 ( ξ 0 − ξ ′ 0 ). The ortho g onality conditio n implies that h ˜ P 0 a ˜ P ∗ 0 ( ξ 0 − ξ ′ 0 ) , ξ ′ 0 i = 0 for all a ∈ C ( S 2 n − 1 ) and P 0 is self-adjoint s o this relation is equiv alent to h χ 2 aξ 0 , ξ ′ 0 i = h χ 2 aξ ′ 0 , ξ ′ 0 i for all a ∈ C ( S 2 n − 1 ). There exist a holomor phic function f 0 such tha t ξ ′ 0 ( z ) = f 0 ( z )e −| z | 2 / 4 and the equation h χ 2 aξ 0 , ξ ′ 0 i = h χ 2 aξ ′ 0 , ξ ′ 0 i implies Z C n f 0 ( z )e −| z | 2 / 2 χ 2 ( z ) a  z | z |  d V = Z C n | f 0 ( z ) | 2 e −| z | 2 / 2 χ 2 ( z ) a  z | z |  d V . Hence f 0 m ust b e real, and since it is holomorphic it must be co nstant . Th us ξ ′ 0 is in the linea r spa n of ξ 0 and ξ 0 defines a pure state. Since the T 0 -action o n L 0 has a pure state, it is irreducible. Assume that T k acts irreducibly on L k . Consider the p olar deco mp ositio n of the unbounded op erator q j on L 2 ( C n ), that is q ∗ j = E j Q j where Q j is a coisometry a nd E j is a s trictly p ositive unbo unded op erator. Clearly E j is diagonal on the energ y levels and E j = M k ′ ∈ N n q k ′ j P k ′ . W e define the ∗ -homomorphism ρ j : T k + e j → B ( L k ) by ρ j ( T ) := Q ∗ j T Q j | L k . Since Q j is a coisometr y this is clearly a ∗ -mo nomorphism. It follows from the fact that q ∗ j | : L k → L k + e j is an isomorphism, that Q j | : L k → L k + e j is unitary , so ρ j is unital. If a ∈ C ∞ ( S 2 n − 1 ) then for some non-zero constant c we have ρ j ( ˜ P k + e j λ ( a ) ˜ P ∗ k + e j ) = cq j ˜ P k + e j λ ( a ) ˜ P ∗ k + e j q ∗ j | L k = = cP k  ∂ ∂ ¯ z j , χ 2 λ ( a )  P k + e j q ∗ j | L k + ˜ P k λ ( a ) ˜ P ∗ k ∈ T k , bec ause Theorem 3 .2 implies P k bP k + e j ∈ K ( L 2 ( C n )) for b ∈ A and by the induction assumption K ⊆ T k . So we obtain a ∗ -monomor phism ρ j : T k + e j → T k . How ever, we have cy c lic v ectors ξ k and ξ k + e j for T k resp ectively T k + e j . F or these vectors, Q j ξ k is a multiple of ξ k + e j so L k + e j = T k + e j ξ k + e j Q ∗ j − − → T k ξ k . Therefore ρ j is surjective and an isomorphism. W e conclude that T k is indepen- dent of k and the repr esent ations o n L k are irreducible since ξ 0 is pure and the T k -actions a re a ll equiv alen t. In [5] a weaker, but more ex plicit, statement was pr ov e n in complex dimen- sion 1. Lemma 9 . 2 of [5] gives an explicit ex pression of Q ∗ k, 0 T k ( a ) Q k, 0 if a ∈ A is smo o th as Q ∗ k, 0 T k ( a ) Q k, 0 = T 0 ( D k ( a )) , where D k := id + P k j =1 d j,k ∆ j , for some ex plicit constants d j,k and ∆ is the Laplacian o n C . 8 F or i = 1 , . . . , n we let z i : S 2 n − 1 → C denote the co ordinate functions of the em b edding S 2 n − 1 ⊆ C n . Clear ly z i ∈ C ( S 2 n − 1 ). Corollary 4.2. The op er ators P k λ ( z i ) P ∗ k to gether with K gener ate T k as a C ∗ -algebr a. Pr o of. Let U denote t he C ∗ -algebra gener ated by P k λ ( z i ) P k and K . The C ∗ -algebra T k is constructed as the C ∗ -algebra generated by the linear spac e P k λ ( C ( S 2 n − 1 )) P k bec ause P k λ ( a ) P k − ˜ P k λ ( a ) ˜ P ∗ k ∈ K . So it is sufficient to prove P k λ ( C ( S 2 n − 1 )) P k ⊆ U . Given a function a ∈ C ( S 2 n − 1 ) the Stone-W eier strass theorem implies that there is a seq uence of polyno mials R j = R j ( z , ¯ z ) such that R j → a in C ( S 2 n − 1 ). The functions R j are p olynomia ls so it follows that P k λ ( R j ) P k − R j ( P k λ ( z ) P k , P k λ ( z ∗ ) P k ) ∈ K and P k λ ( R j ) P k ∈ U . Fina lly k P k λ ( R j ) P k − P k λ ( a ) P k k B ( L k ) ≤ k R j − a k C ( S 2 n − 1 ) which implies P k λ ( a ) P k ∈ U . Corollary 4.3 . The mapping β k : C ( S 2 n − 1 ) → C ( L k ) induc e d fr om ˜ β k is inje ctive, so if u ∈ A ⊗ M N the op er ator T k ( u ) is F r e dholm if and only if π ∂ ( u ) is invertible. Pr o of. Due to equation (6) in Theor e m 4.1, the Cor ollary follows fro m Lemma 3.5. The pro of of the second statement of the Corollar y is proven in the same fashion as P rop osition 3.6. F rom the fact that the mapping β k is injective it follows that the symbol mapping ˜ P k λ ( a ) ˜ P ∗ k 7→ a gives a well defined surjection σ k : T k → C ( S 2 n − 1 ). Clearly the k ernel of σ k is non-zero and k er σ k ⊆ K , so b y Theorem 4.1 k er σ k = K . Ther efore we ca n constr uct the exa ct sequence 0 → K → T k σ k − − → C ( S 2 n − 1 ) → 0 . (7) A completely positive s plitting of the symbo l mapping σ k : T k → C ( S 2 n − 1 ) is given by a 7→ ˜ P k λ ( a ) ˜ P ∗ k . The exact s equence (7) defines an extension cla s s [ T k ] ∈ E xt ( C ( S 2 n − 1 )). T o read more ab out E xt , K - theory and K -homolog y we refer the reader to the references. Since C ( S 2 n − 1 ) is a nuclear C ∗ -algebra there is an isomorphism E xt ( C ( S 2 n − 1 )) ∼ = K 1 ( C ( S 2 n − 1 )) and we can descr ibe the K -homolog y class of [ T k ] explicitely by a F redho lm module as follows; we let λ : C ( S 2 n − 1 ) → B ( L 2 ( C n )) b e as in equation (4) and define the op erato r F k = (1 + ˜ P k ) 2 where ˜ P k is as in equatio n (5). Clearly , ( L 2 ( C n ) , λ, F k ) defines a F redholm mo dule w hich re pr esents the image o f [ T k ] in K 1 ( C ( S 2 n − 1 )). Corollary 4. 4. Th e class [ T k ] ∈ E xt ( C ( S 2 n − 1 )) is indep endent of k . Pr o of. The extension T k is equiv alent to T k ′ since it follows from equation (6) that the following dia g ram with exact r ows commute 0 − − − − → K − − − − → T k ′ − − − − → C ( S 2 n − 1 ) − − − − → 0   y Ad ( Q k , k ′ )   y Ad ( Q k , k ′ )    0 − − − − → K − − − − → T k − − − − → C ( S 2 n − 1 ) − − − − → 0 . 9 So we know that [ T k ] is indep endent of k , this implies that the index o f T k ( u ) for u ∈ M n ⊗ A is indep endent of k . But how do we calc ulate it? The index theorem that allows the ca lculation involv es studying how the co ordinate functions on S 2 n − 1 act on the monomial base o f L 0 . W e will first review some theory of T o eplitz ope rators on the Ber gman spa ce a nd then study what hap- pens in complex dimension 1 and 2 . The Berg ma n space on the unit ball B n ⊆ C n is defined as A 2 ( B n ) := L 2 ( B n ) ∩ O ( B n ), that is; ho lo morphic functions on B n which are squar e inte- grable. The Ber gman space is a closed subspace of L 2 ( B n ) a nd w e will denote the orthogona l pro jection L 2 ( B n ) → A 2 ( B n ) by P B . The Berg man pro jection defines a K -homology class [ P B ] ∈ K 1 ( C ( S 2 n − 1 )) in the same fashion as for the La ndau pro jections. That is, for a ∈ C ( B n ) the op erator [ P B , a ] ∈ B ( L 2 ( B n )) is compact. The reaso n that w e can use P B to define a K -homolo gy clas s for S 2 n − 1 instead of B n is analog o usly to ab ov e that P B a | A 2 ( B n ) is compact if and only if a ∈ C 0 ( B n ), se e more in [14]. Thus P B a | A 2 ( B n ) is F redholm if and o nly if a | S 2 n − 1 is inv ertible. F urthermor e [ P B , a ] is compact. So [ P B ] is a well defined K -homology class in K 1 ( C ( S 2 n − 1 )). By [3] the following index formula holds for the T o eplitz op erator P B a | A 2 ( B n ) if the s y m bo l a ∂ := a | S 2 n − 1 is smo o th: ind ( P B a | A 2 ( B n ) ) = − ( n − 1)! (2 n − 1)!(2 π i ) n Z S 2 n − 1 tr(( a − 1 ∂ d a ∂ ) 2 n − 1 ) . (8) This formula was a lso prov en in [8] by an elegant use of Atiy ah-Singer s index theorem. W e will by T n denote the C ∗ -algebra gener ated by P B C ( B n ) P B in B ( A 2 ( B n )). The K -ho mo logy clas s [ P B ] ∈ K 1 ( C ( S 2 n − 1 )) ca n be represented by the exten- sion class [ T n ] ∈ E xt ( C ( S 2 n − 1 )) defined by means of the short exact s equence 0 → K → T n σ n − − → C ( S 2 n − 1 ) → 0 . (9) 5 The sp ecial cases C and C 2 In this chapter w e will study the sp ecial c ases of co mplex dimension 1 and 2. Dimension 1 has been studied previously in [1] a nd provides a simpler picture than in higher dimensio ns. In the 1- dimensional case w e hav e that K 1 ( C ( T )) ∼ = Z and we can take the co ordina te function z : T → C to b e a g enerator. So when we want to determine the cla s s [ T k ] we only need to calculate the index of P k λ ( z ) P k where λ is as in eq uation (4). W e r ecall the following Prop osition from [1 ]: Prop osition 5.1 (Pro pos ition 7 . 3 fro m [1]) . F or any k ∈ N we have that ind ( P k λ ( z ) P k ) = − 1 . The metho d used in [1] to prov e this Pr opo sition was to s how that in a suitable basis P k λ ( z ) P k was up to some co efficients a unilatera l shift. In higher dimension the pr o o f is based on similar ideas . 10 Theorem 5.2. F or n = 1 ther e is an isomorphism T k ∼ = T 1 making [ T k ] = [ T 1 ] ∈ K 1 ( C ( T )) . Pr o of. By Prop osition 7 . 3 of [1] [ T k ] . [ u ] = ind ( P k λ ( u ) P k ) = − wind ( u ) = [ T 1 ] . [ u ] (10) for an inv ertible function u ∈ C ( T ). Here wind ( u ) denotes the winding num b er of u which is defined for smo oth u as wind ( u ) := 1 2 π i Z T u − 1 d u and defines an isomor phism K 1 ( C ( T )) → Z . By the Universal Co efficient The- orem for K K -theory (see Theo rem 4 . 2 of [11]) the mapping K 1 ( C ( T )) → H om ( K 1 ( C ( T )) , Z ) is a n is omorphism so equation (10) implies that [ T k ] = [ T 1 ]. By Theorem 13 of [6 ], the sho rt exact sequence 0 → K → T k → C ( T ) → 0 is characterized by an isometry v such that v v ∗ − 1 is compact and T k is genera ted by v . Then z 7→ v defines a splitting and the symbol mapping T k → C ( T ) is just v 7→ z . By equation (10), 1 − v v ∗ is a rank one pro jection, so the theor em follows. Also in dimension 2 we can find a generato r fo r the o dd K - theory . As generator for K 1 ( C ( S 3 )) ∼ = Z we ca n ta k e the diffeomorphism u : S 3 → S U (2) defined as u ( z 1 , z 2 ) :=  z 1 z 2 − ¯ z 2 ¯ z 1  . Prop osition 5.3. The extension class [ T 2 ] gener ate K 1 ( C ( S 3 )) and [ u ] gener- ate K 1 ( C ( S 3 )) . Pr o of. Recalling that P B denotes the Bergman pro jection we will s ta rt by c a lcu- lating the index of the T o eplitz ope rator P B uP B : A 2 ( B 2 ) ⊗ C 2 → A 2 ( B 2 ) ⊗ C 2 . Using the index theorem by Boutet de Monv el ([3]) re v iew ed ab ov e in equation (8), the following index formula holds for s moo th u : ind ( P B uP B ) = − 1 3!(2 π i ) 2 Z S 3 tr(( u ∗ d u ) 3 ) . (11) A straightforw ard ca lculation gives that tr(( u ∗ d u ) 3 ) = 3( z 1 d ¯ z 1 − ¯ z 1 d z 1 ) ∧ d z 2 ∧ d ¯ z 2 + 3( z 2 d ¯ z 2 − ¯ z 2 d z 2 ) ∧ d z 1 ∧ d ¯ z 1 . Inv o king Stokes Theorem on eq uation (11) gives that − 1 3!(2 π i ) 2 Z S 3 tr(( u ∗ d u ) 3 ) = 1 48 · v ol ( B 2 ) Z B 2 dtr(( u ∗ d u ) 3 ) = = 1 4 · v ol ( B 2 ) Z B 2 d z 1 ∧ d ¯ z 1 ∧ d z 2 ∧ d ¯ z 2 = − 1 v ol ( B 2 ) Z B 2 d V = − 1 This eq ua tion shows that [ T 2 ] . [ u ] = ind ( P B uP B ) = − 1 . (12) 11 Consider the s plit-exact sequence 0 → C 0 ( R 3 ) → C ( S 3 ) → C → 0 where the mapping C ( S 3 ) → C is p oint ev aluation. Since the s equence splits, and K 1 ( C ) = K 1 ( C ) = 0 the em b edding C 0 ( R 3 ) → C ( S 3 ) induces isomor phisms K 1 ( C ( S 3 )) ∼ = K 1 ( C 0 ( R 3 )) = Z and K 1 ( C ( S 3 )) ∼ = K 1 ( C 0 ( R 3 )) = Z . So the Kaspar ov pro duct K 1 ( C ( S 3 )) × K 1 ( C ( S 3 )) → Z is just a pairing Z × Z → Z , and since [ T 2 ] . [ u ] = − 1 it follows that [ T 2 ] g enerates K 1 ( C ( S 3 )) and [ u ] gener ates K 1 ( C ( S 3 )). Theorem 5.4. F or any k ∈ N 2 we have ind ( P k λ ( u ) P k ) = − 1 . (13) Ther efor e [ T 2 ] = [ T k ] . Pr o of. If e quation (13) holds, [ T 2 ] = [ T k ] follows directly from equation (12) using the Universal Coe fficie n t Theorem for K K - theory (see Theorem 4 . 2 of [11]). This is a co nsequence of the fact that the natura l mapping K 1 ( C ( S 3 )) → H om ( K 1 ( C ( S 3 )) , Z ) is a n isomo rphism. The injectivity o f this map implies that if [ T 2 ] . [ u ] = [ T k ] . [ u ] for a ge ne r ator [ u ] then [ T 2 ] = [ T k ]. T o pr ov e equation (1 3) we take k = 0, s inc e Cor ollary 4.4 implies tha t the integer ind ( P k λ ( u ) P k ) is independent of k . W e claim that P 0 λ ( u ) P 0 is an injectiv e ope rator and the cokernel of P 0 λ ( u ) P 0 is spanned by the C 2 -v alued function z 7→ e −| z | 2 / 4 ⊕ 0. This statement will prov e the theorem. T o prove that P 0 λ ( u ) P 0 is injective, assume f ∈ ker( P 0 λ ( u ) P 0 ). Define the functions ξ m ( z ) := z m e −| z | 2 / 4 for m ∈ N 2 . The functions ξ m form an or thogonal basis for L 0 by Theorem 1 . 63 o f [7]. Expand the function f in an L 2 -conv ergent series f = X m ∈ N 2 c m ξ m , where c m = c (1) m ⊕ c (2) m ∈ C 2 . Since f ∈ ker( P 0 λ ( u ) P 0 ) we hav e the following orthogo nalit y condition 0 = h ξ m ′ ⊕ 0 , λ ( u ) f i = X m Z C 2 c (1) m ¯ z m ′ z m + e 1 | z | + c (2) m ¯ z m ′ z m + e 2 | z | ! e | z | 2 / 2 d V = = X m t m , m ′ Z S 3  c (1) m ¯ z m ′ z m + e 1 + c (2) m ¯ z m ′ z m + e 2  d S, for some co efficient s t m , m ′ , for a detailed calcula tion of t m , m ′ see below in Propo - sition 6.1. Using that the functions ξ m are orthogonal we obtain that there exist a C m > 0 s uc h that c (1) m − e 1 = − C m c (2) m − e 2 . (14) On the other hand, we hav e 0 = h 0 ⊕ ξ m ′ , λ ( u ) f i = X m Z C 2 − c (1) m ¯ z m ′ + e 2 z m | z | + c (2) m ¯ z m ′ + e 1 z m | z | ! e | z | 2 / 2 d V = 12 = X m t m , m ′ Z S 3  − c (1) m ¯ z m ′ + e 2 z m + c (2) m ¯ z m ′ + e 1 z m  d S. Again using or thogonality of the functions ξ m we obtain that there is a C ′ m > 0 such that c (1) m + e 2 = C ′ m c (2) m + e 1 . (15) Equation (1 4) implies c (1) m = 0 fo r m 2 = 0. F or m 2 > 0 e q uation (14) implies c (1) m = − C m + e 1 c (2) m − e 2 + e 1 . Then equation (15) for m − e 2 gives c (1) m  1 + C m + e 1 C ′ m − e 2  = 0 . So c (1) m = 0 for all m . E quation (14) implies c (2) m = 0 for all m . Thus f = 0 and ker( P 0 λ ( u ) P 0 ) = 0. The second s tatemen t, that the cokernel of P 0 λ ( u ) P 0 is spanned by the C 2 - v alued function z 7→ e −| z | 2 / 4 ⊕ 0 , is pr ov en analogo usly . Ther e is a natural isomorphism coker P 0 λ ( u ) P 0 ∼ = (im P 0 λ ( u ) P 0 ) ⊥ = ker P 0 λ ( u ∗ ) P 0 . Analogously to the re asoning a b ove, for g ∈ ker P 0 λ ( u ∗ ) P 0 we expand the func- tion g in an L 2 -conv ergent series g = X m ∈ N 2 d m ξ m , where d m = d (1) m ⊕ d (2) m ∈ C 2 . After taking scalar pro duct by ξ m ′ , for so me D m , D ′ m > 0 we o bta in the following conditions on the co efficients: d (1) m + e 1 = D m d (2) m − e 2 and (16) d (1) m + e 2 = − D ′ m d (2) m − e 1 . (17) The seco nd of these equations implies d (1) m = 0 for m 1 = 0 and m 2 > 0. Also, the first o f these equations implies d (1) m = 0 for m 2 = 0 a nd m 1 > 0. F o r m 1 , m 2 > 0, putting in m − e 1 in the first equa tion, gives d (1) m = D m − e 1 d (2) m − e 1 − e 2 . Finally , combining this relation with the second equation for m − e 2 we o btain d (1) m  1 + D m − e 1 D ′ m − e 2  = 0 for m 1 , m 2 > 0 . Therefore d (1) m = 0 for all m 6 = 0 . The equations in (16) imply d (2) m = 0 for all m . How ever, the function z 7→ e −| z | 2 / 4 ⊕ 0, co rresp onding to d (1) 0 = 1, is in the space ker ( P 0 λ ( u ∗ ) P 0 ) whic h co mpletes the pro of. 13 6 The index form ula on the particular Landau lev els In this section we will pr ov e a n index formula for the particular Landau lev- els. On S 2 n − 1 we hav e the c o mplex co ordinates z 1 , . . . , z n and we denote by Z 1 , . . . , Z n the image of these co ordinate functions under the representation λ which was defined in equatio n (4). So Z i is the o per ator on L 2 ( C n ) given by m ultiplication by the almost everywhere defined function z 7→ z i | z | . Consider the po lar decomp ositions P 0 Z i P 0 = V i, 0 S i, 0 , where V i, 0 are pa rtial isometries a nd S i, 0 > 0 . An or thonormal basis for L 0 is given by η m ( z ) := z m e −| z | 2 / 4 √ π n 2 | m | + n m ! , see mo re in [7]. Prop osition 6.1. The op er ator V i, 0 is an isometry describ e d by the e quation V i, 0 η m = η m + e i and the op er ator S i, 0 is diagonal in the b asis η m with eigenvalues given by λ η i, m = Γ  | m | + n + 1 2  √ m i + 1 ( | m | + n )! . (18) Pr o of. F or m , m ′ ∈ N we hav e h η m ′ , Z i η m i = Z C n 1 π n √ 2 | m + m ′ | +2 n m ! m ′ ! ¯ z m ′ z m + e i | z | e −| z | 2 / 2 d V = = 1 π n √ 2 | m + m ′ | +2 n m ! m ′ ! Z ∞ 0 r | m | + | m ′ | + n − 1 e − r 2 / 2 d r Z S 2 n − 1 ¯ z m ′ z m + e i d S = = δ m ′ , m + e i Γ  | m | + n + 1 2  2 π n m ! p ( m j + 1) Z S 2 n − 1 ¯ z m ′ z m + e i d S = = δ m ′ , m + e i Γ( | m | + n + 1 2 ) √ m i + 1 ( | m | + n )! . It follows that V i, 0 η m = η m + e i and S i, 0 η m = λ η i, m η m , where λ η i, m is as in equa- tion (18 ). On the other hand, we ca n, just a s on L 0 , let ˜ Z 1 , . . . , ˜ Z n ∈ B ( L 2 ( B n )) b e the o per ators on L 2 ( B n ) defined by the mult iplication b y the almost everywhere defined function z 7→ z i | z | . Consider the po lar de c o mpo s itions P B ˜ Z i P B = V i,B S i,B , where ag ain V i,B are par tial isometries and S i,B > 0. An or thonormal basis for A 2 ( B n ) is given by µ m ( z ) := π − n/ 2 r ( n + | m | )! m ! z m . Similar to the low es t Landau level, the par tia l isometr ie s V i,B are just shifts in this basis: 14 Prop osition 6.2. The op er ator V i,B is an isometry describ e d by the e quation V i,B µ m = µ m + e i and the op er ator S i,B is diagonal in the b asis µ m with eigenvalues given by λ µ i, m = √ m i + 1 p n + | m | + 1 . (19) Pr o of. The pro of is the ana logous to that of Pro po s ition 6 .1. F or m , m ′ ∈ N we hav e h µ m ′ , ˜ Z i µ m i = Z B n π − n r ( n + | m | )!( n + | m ′ | )! m ! m ′ ! ¯ z m ′ z m + e i | z | d V = = π − n r ( n + | m | )!( n + | m ′ | )! m ! m ′ ! Z 1 0 r | m | + | m ′ | +2 n − 1 d r Z S 2 n − 1 ¯ z m ′ z m + e i d S = = δ m ′ , m + e i ( n + | m | )! p n + | m | + 1 (2 | m | + 2 n ) m ! √ m i + 1 Z S 2 n − 1 ¯ z m ′ z m + e i d S = = δ m ′ , m + e i √ m i + 1 p n + | m | + 1 . It follows that V i,B µ m = µ m + e i and S i,B µ m = λ µ i, m µ m where the eige nv alues λ µ i, m are g iven in equa tion (19). Lemma 6.3. If a is a r e al numb er then Γ( x + a ) Γ( x ) = x a + O ( x − 1+ a ) as x → + ∞ . Pr o of. By Stirling’s formula ln Γ ( x ) =  x − 1 2  ln x − x + ln 2 π 2 + O ( x − 1 ) . After T aylor ex panding ln Γ( x + a ) ar ound a = 0 we obta in that ln Γ ( x + a ) − ln Γ( x ) = a ln x + O ( x − 1 ) . Lemma 6.4. With t he unitary U : A 2 ( B n ) → L 0 define d by µ m 7→ η m , the op er ators S i, 0 and S i,B satisfy U ∗ S i, 0 U − S i,B ∈ K . Pr o of. The op erators U ∗ S i, 0 U and S i,B are bo th diag onal in the basis µ m . So it is sufficient to prov e that | λ η m − λ µ m | → 0. The pro of of this statement is based on the estima te from Lemma 6.3. When | m | → ∞ , Lemma 6.3 implies | λ η m − λ µ m | =      Γ  | m | + n + 1 2  √ m i + 1 ( | m | + n )! − √ m i + 1 p | m | + n − 1      = = √ m i + 1      Γ  ( | m | + n + 1) − 1 2  Γ ( | m | + n + 1) − ( | m | + n − 1) − 1 / 2      = O ( | m | − 1 ) . Therefore we hav e that U ∗ S i, 0 U − S i,B ∈ L n + ( A 2 ( B n )), the n :th Dixmier idea l. In particular U ∗ S i, 0 U − S i,B is co mpact. 15 Theorem 6 .5. The unitary U induc es an isomorphism Ad ( U ) : T 0 ∼ − → T n such that σ n ◦ Ad ( U ) = σ 0 . wher e σ n and σ 0 ar e the symb ol mappings. Pr o of. Lemma 6.4 and the Pr opo sitions 6.1 and 6 .2 imply U ∗ ( P 0 Z i P 0 ) U = P B ˜ Z i P B + K i , (20) for some compact op erato rs K i . Since T n contains the co mpact o p era tors, U ∗ ( P 0 Z i P 0 ) U ∈ T n . Corolla ry 4.2 therefor e implies U ∗ T 0 U ⊆ T n . Theorem 4.1 states that T 0 acts irreducibly o n L 0 , so U ∗ T 0 U acts irr educibly on A 2 ( B n ). Therefore K ⊆ U ∗ T 0 U and P B ˜ Z i P B ∈ U ∗ T 0 U . The op era to rs P B ˜ Z i P B together with K ge ner ate T n so U ∗ T 0 U ⊇ T n . The relation σ n ◦ Ad ( U ) = σ 0 holds since by eq ua tion (20) it holds on the ge nerators of C ( S 2 n − 1 ). Corollary 6.6. L et [ T n ] ∈ E xt ( C ( S 2 n − 1 )) denote the T o eplitz quantization of t he Ber gman sp ac e define d in e quation (9) and [ T k ] ∈ E xt ( C ( S 2 n − 1 )) the T o eplitz quantization of the p articular La ndau level of height k define d in e qua- tion (7) . Then [ T n ] = [ T k ] . So for u ∈ A ⊗ M N such t hat u ∂ := π ∂ ( u ) is invertible and smo oth ind ( P k u | L k ⊗ C N ) = − ( n − 1)! (2 n − 1)!(2 π i ) n Z S 2 n − 1 tr(( u − 1 ∂ d u ∂ ) 2 n − 1 ) . (21) Pr o of. By Co rollary 4 .4 the class [ T k ] is indep endent of k , so take k = 0. In this case Theo rem 6.5 implies that the unitary U makes the following diagra m commutativ e: 0 − − − − → K − − − − → T 0 σ 0 − − − − → C ( S 2 n − 1 ) − − − − → 0   y Ad ( U )   y Ad ( U )    0 − − − − → K − − − − → T n σ n − − − − → C ( S 2 n − 1 ) − − − − → 0 . Therefore [ T n ] = [ T 0 ] = [ T k ] and the index formula (21) follows fr o m [8]. 16 References [1] J. E. Avron, R. Seiler, B. 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