The bulk-edge correspondence for the quantum Hall effect in Kasparov theory
aa r X i v : . [ m a t h - ph ] J u l The bulk-edge correspondence for the quantum Hall effect inKasparov theory
Chris Bourne ∗†‡ , Alan L. Carey †‡ and Adam Rennie ‡† Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200,Australia ‡ School of Mathematics and Applied Statistics, University of Wollongong, Wollongong,NSW 2522, AustraliaJune 26, 2018
Abstract
We prove the bulk-edge correspondence in K -theory for the quantum Hall effect by constructingan unbounded Kasparov module from a short exact sequence that links the bulk and boundaryalgebras. This approach allows us to represent bulk topological invariants explicitly as a Kasparovproduct of boundary invariants with the extension class linking the algebras. This paper focuses onthe example of the discrete integer quantum Hall effect, though our general method potentially hasmuch wider applications. Keywords: quantum Hall effect, spectral triples, KK -theory, bulk-edge correspondence.Mathematics subject classification: Primary 81V70, Secondary 19K35. In this letter, we revisit the notion of the bulk-edge correspondence in the discrete (or tight binding)version of the integer quantum Hall effect as previously studied in [7, 8, 11, 12, 13, 14]. In these papers,the motivation is to incorporate the presence of a boundary or edge into Bellissard’s initial explanationof the quantum Hall effect [2]. This is done by introducing an ‘edge conductance’, σ e , which is thenshown to be the same as Bellissard’s initial expression for the (quantised) Hall conductance, σ H . Ourmotivation comes from the more K -theoretic arguments used in [12, 14].We propose a new method based on explicit representations of extension classes as Kasparov modules.Given a short exact sequence of C ∗ -algebras, → J → A → A/J → for some closed -sided ideal J , we know by results of Kasparov [10] that this gives rise to a class in Ext(
A/J, J ) , which is the same as KK ( A/J, J ) for the algebras we study. By representing our shortexact sequence as an unbounded Kasparov module, we can use the methods developed in [4, 9, 18] to takethe Kasparov product of our module with spectral triples representing elements in K j ( J ) ∼ = KK j ( J, C ) to give elements in K ( j +1) ( A/J, C ) .In this letter we focus on a simple case so as not to obscure the main idea with technical details.Thus we consider the short exact sequence representing the Toeplitz extension of the rotation algebra, A φ . An unbounded Kasparov module can be built from this extension by considering the circle actionon the rotation algebra A φ , as in [5].We outline an alternative method for constructing a Kasparov module representing an extensionclass (generalised in [22]) via a singular functional. We introduce this method with a view towards morecomplicated examples, where the circle-action picture breaks down. Such examples include the following. ∗ [email protected], [email protected], [email protected]
1. For the case of a finite group G with K ⊳ G , the short exact sequence → J ⋊ K → A ⋊ G → A/J ⋊ G/K → can no longer be represented by circle actions. Such crossed products may emerge by consideringthe symmetry group of topological insulator systems, for example.2. For models with internal degrees of freedom (such as a honeycomb lattice), we would no longer beworking with the Cuntz-Pimsner algebra of a self Morita-equivalence bimodule (as defined in [22,Section 2]) and so the singular functional method is necessary.See [22] for more examples of extensions requiring this viewpoint. The flexibility of our approach torepresenting extensions as Kasparov modules (with which products can be taken) will allow many moresystems-with-edge to be investigated, as we outline below. We begin with a Toeplitz-like extension of the rotation algebra A φ , and show how to construct anunbounded Kasparov module β = (cid:16) A φ , Z C ∗ ( b U ) , N (cid:17) that represents this extension in KK -theory. Here Z C ∗ ( b U ) is a Hilbert C ∗ -module coming from the extension, b U is the shift operator on ℓ ( Z ) along theboundary Z and the unbounded operator N is a number operator (defined later).We also introduce a ‘boundary spectral triple’ ∆ = (cid:16) C ∗ ( b U ) , ℓ ( Z ) , M (cid:17) , which we think of as thestandard spectral triple over the circle but in a Fourier transformed picture (so that M is the Fouriertransform of differentiation and b U is the bilateral shift). Our main result, Theorem 3.3, is as follows. Theorem.
The internal Kasparov product β ˆ ⊗ C ∗ ( b U ) ∆ is unitarily equivalent to the negative of the spectraltriple modelling the boundary-free quantum Hall effect. We note that the Kasparov product and unitary equivalence of the Kasparov modules considered inthe theorem is at the unbounded level, a stronger equivalence than in the bounded setting.Recall from the work of Bellissard [2] that the quantised Hall conductance in the case without bound-ary comes from the pairing of the Fermi projection with an element in K ( A φ ) . Our main result saysthat this K -homology class can be ‘factorised’ into a product of a K -homology class representing theboundary and a KK -class representing the short exact sequence linking the boundary and boundary-free systems. We can then use the associativity of the Kasparov product to immediately obtain an edgeconductance, and the equality of the bulk and edge conductances.It is in this point that our work differs from, but complements, the boundary picture developedin [12, 14], where the authors had to define a separate edge conductance and then show equality withthe usual Hall conductance. Instead, our method derives the bulk-edge correspondence as a directconsequence of the factorisation of the boundary-free K -homology class. Our work demonstrates howwe can obtain the bulk-edge correspondence of [12] without passing to cyclic homology and cohomology.This allows our method to be applied to systems with torsion invariants, which cannot be detected incyclic theory. This is essential for topological insulator theory, where torsion invariants arise naturally.We also note that by working in the unbounded KK picture, all computations are explicit. AsKasparov theory can also be extended to accommodate group actions and real/Real algebras this meansour method has potential applications to a much wider array of physical models. Topological insulatorsare an example of where the bulk-edge correspondence needs further work.The paper is organised into two major Sections. Section 2 contains the construction of the Kasparovmodule that is needed in Section 3 where the main theorem is proved. Some details are relegated to anAppendix. Acknowledgements
All authors acknowledge the support of the Australian Research Council. The authors are grateful to theHausdorff Institute for Mathematics for support to participate in the trimester program on ‘Noncom-mutative geometry and its applications’ where some of this work was completed. AC thanks the ErwinSchrödinger Institute for support. CB thanks Friedrich-Alexander Universität for hospitality during the2riting of this letter. All authors thank Koen van den Dungen and Hermann Schulz-Baldes for usefuldiscussions.
Recall [16] that in the discrete or ‘tight binding’ model of the quantum Hall effect without boundary, wehave magnetic translations b U and b V as unitary operators on ℓ ( Z ) . These operators commute with theunitaries U and V that generate the Hamiltonian H = U + U ∗ + V + V ∗ . We choose the Landau gaugesuch that ( b U λ )( m, n ) = λ ( m − , n ) , ( b V λ )( m, n ) = e − πiφm λ ( m, n − , ( U λ )( m, n ) = e − πiφn λ ( m − , n ) , ( V λ )( m, n ) = λ ( m, n − , where φ has the interpretation as the magnetic flux through a unit cell and λ ∈ ℓ ( Z ) . We are keepingthe model simple in order to make our constructions as clear as possible, though what we do extendsto more sophisticated models. We note that b U b V = e πiφ b V b U and U V = e − πiφ V U , so C ∗ ( b U , b V ) ∼ = A φ ,(the irrational rotation algebra when φ is irrational), and C ∗ ( U, V ) ∼ = A − φ . We can also interpret A − φ ∼ = A op φ , where A op is the opposite algebra with multiplication ( ab ) op = b op a op . Our choice of gaugealso means that C ∗ ( b U , b V ) ∼ = C ∗ ( b U ) ⋊ η Z , where b V is implementing the crossed-product structure via theautomorphism η ( b U m ) = b V ∗ b U m b V .We outline an idea loosely based on that of Kellendonk et al. [12, 14], who employed constructionsfrom Pimsner and Voiculescu [21]. The essence of the idea is to relate the bulk and edge algebras via aToeplitz-like extension. This viewpoint is also employed in [17]. Proposition 2.1 (§2 of [21]) . Let S be the usual shift operator on ℓ ( N ) with S ∗ S = 1 , SS ∗ = 1 − P n =0 .There is a short exact sequence, → C ∗ ( b U ) ⊗ K [ ℓ ( N )] ψ −→ C ∗ ( b U ⊗ , b V ⊗ S ) → C ∗ ( b U ) ⋊ η Z → . The map ψ given in Proposition 2.1 is such that ψ ( b U m ⊗ e jk ) = ( b V ∗ ) j b U m b V k ⊗ S j P n =0 ( S ∗ ) k for matrix units e jk in K [ ℓ ( N )] . It is then extended to the full algebra by linearity. One checks that ψ isan injective map into the ideal of C ∗ ( b U ⊗ , b V ⊗ S ) generated by ⊗ P n =0 . We also have the isomorphism C ( S ) ⋊ η Z ∼ = C ∗ ( b U ⊗ , b V ⊗ V ) ∼ = C ∗ ( b U , b V ) , where V is the image of S under the map to the Calkinalgebra. These alternate but equivalent presentations of A φ will be of use to us later. For convenience,we denote T = C ∗ ( b U ⊗ , b V ⊗ S ) . Remark . We see that in our exact sequence, we can think of the quotient A φ as representing our ‘bulkalgebra’ as it can be derived from a magnetic Hamiltonian on ℓ ( Z ) as in [16]. Our ideal C ∗ ( b U ) ⊗ K can be interpreted as representing the ‘boundary algebra’. To see this we put a boundary on our systemso that for the full system the Hilbert space is H = ℓ ( Z × N ) , while C ∗ ( b U ) acts on the boundary ℓ ( Z ) ,(this action being describable in terms of the bilateral shift operator). Tensoring by the compacts in thedirection perpendicular to the boundary has a physical interpretation as looking at observables actingon ℓ ( Z × N ) that act on the boundary and decay sufficiently fast away from it. We would intuitivelythink of the Hall current of such a system to be concentrated at the boundary with a fast decay into theinterior, so our boundary model lines up with this intuitive picture.We now recall some basic definitions from Kasparov theory; the reader may consult [3, 10] for amore complete overview. A right C ∗ - A -module is a space E with a right action by a C ∗ -algebra A and map ( · | · ) A : E × E → A , which we think of as an A -valued inner-product that is compatiblewith the right-action of A . We denote the set of adjointable operators on E with respect to this innerproduct by End A ( E ) . Within this space are the rank- endomorphisms, Θ e,f , where Θ e,f ( g ) = e · ( f | g ) A for e, f, g ∈ E , which generate the finite-rank endomorphsims End A ( E ) . The compact endomorphisms End A ( E ) are the closure of the finite-rank operators in the operator norm of End A ( E ) .3 efinition 2.3. Given Z -graded C ∗ -algebras A and B , an even unbounded Kasparov A - B -module ( A, E A , D ) is given by1. A Z -graded, countably generated, right C ∗ - B -module E B ;2. A Z -graded ∗ -homomorphism φ : A → End B ( E ) ;3. A self-adjoint, regular, odd operator D : Dom D ⊂ E → E such that the graded commutator [ D, φ ( a )] ± is an adjointable endomorphism, and φ ( a )(1 + D ) − / is a compact endomorphism forall a in a dense subalgebra A of A .If the module and algebras are trivially graded, then the Kasparov module is called odd.We can always pass from unbounded modules to bounded Kasparov modules via the mapping D D (1 + D ) − / [1]. In the last section, we introduced the short exact sequence → C ∗ ( b U ) ⊗ K ψ −→ T → A φ → . (2.1)We know that this sequence gives rise to a class in KK -theory using Ext groups, but in order tocompute the Kasparov product, it is desirable to have an explicit Kasparov module that represents aclass in KK ( A φ , C ∗ ( b U ) ⊗ K ) ∼ = KK ( A φ , C ∗ ( b U )) .To do this, we introduce our main technical innovation, a singular functional Ψ on the subalgebra C ∗ ( S ) of T , which is given by Ψ( T ) = res s =1 ∞ X k =0 h e k , T e k i (1 + k ) − s/ , where { e k } is any basis of ℓ ( N ) . Proposition 2.4.
The functional Ψ is a well-defined trace on C ∗ ( S ) such that Ψ (cid:0) S l ( S ∗ ) l S n ( S ∗ ) n (cid:1) = δ l − l ,n − n , where δ a,b is the Kronecker delta. Moreover, Ψ( T ) = 0 for any compact T .Proof. That Ψ is a trace is straightforward from its definition and the properties of the usual trace andcomplex residues. Thus, for S α ( S ∗ ) β ∈ C ∗ ( S ) , we see that h e k , S α ( S ∗ ) β e k i = δ α,β h ( S ∗ ) α e k , ( S ∗ ) α e k i = δ α,β χ [ k, ∞ ) ( α ) , where χ [ k, ∞ ) is the indicator function. Hence Ψ (cid:2) S α ( S ∗ ) β (cid:3) = res s =1 ∞ X k =0 δ α,β χ [ k, ∞ ) ( α )(1 + k ) − s/ = res s =1 ∞ X k = α δ α,β (1 + k ) − s/ = δ α,β . Similarly Ψ (cid:0) ( S ∗ ) α S β (cid:1) = δ α,β . From this we have that, for l ≥ n , Ψ (cid:0) S l ( S ∗ ) l S n ( S ∗ ) n (cid:1) = Ψ (cid:0) S l ( S ∗ ) l − n + n (cid:1) = δ l ,l − n + n = δ l − l ,n − n ; or, for l ≤ n , Ψ (cid:0) S l ( S ∗ ) l S n ( S ∗ ) n (cid:1) = Ψ (cid:0) S l − l + n ( S ∗ ) n (cid:1) = δ l − l + n ,n = δ l − l ,n − n . Since ( S ∗ ) α S α = 1 C ∗ ( S ) , one now readily checks that Ψ( T ) ≤ k T k Ψ (cid:0) C ∗ ( S ) (cid:1) = k T k (2.2)for all T ∈ C ∗ ( S ) and so Ψ extends by continuity to C ∗ ( S ) . For any finite-rank operator, F ∈ C ∗ ( S ) , h e k , F e k i 6 = 0 for finitely many k . This tells us that P k h e k , F e k i (1 + k ) − s/ is holomorphic at s = 1 ,whence Ψ( F ) = 0 . By (2.2), Ψ vanishes on all the compacts operators on ℓ ( N ) .4n order to simplify computations, we realise T as the norm closure of the linear span of the operators ( b V ⊗ S ) n [( b V ⊗ S ) ∗ ] n ( b U ⊗ m = b V n − n b U m ⊗ S n ( S ∗ ) n for m ∈ Z and n , n ∈ N . We put the b U on the right as we are going to construct a right C ∗ ( b U ) -moduleusing this presentation.The first step is the inner product: ( · | · ) : T × T → C ∗ ( b U ) given by (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) := (cid:16) b V l − l b U m (cid:17) ∗ b V n − n b U m Ψ h(cid:0) S l ( S ∗ ) l (cid:1) ∗ S n ( S ∗ ) n i . To show this actually takes values in C ∗ ( b U ) , we use Proposition 2.4 to compute that (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = b U − m b V l − l b V n − n b U m δ l − l ,n − n = b U m − m δ l − l ,n − n , which is in C ∗ ( b U ) . With this in mind we construct, in the next result, a right C ∗ ( b U ) module. Proposition 2.5.
The map ( · | · ) : T × T → C ∗ ( b U ) together with an action by right multiplicationmakes T a right C ∗ ( b U ) -inner-product module. Quotienting by vectors of zero length and completingyields a right C ∗ ( b U ) -module.Proof. Using the equation (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = b U m − m δ l − l ,n − n most of the requirements for ( · | · ) to be a C ∗ ( b U ) -valued inner-product follow in a straightforward way.We will check compatibility with multiplication on the right by elements of C ∗ ( b U ) . We compute that (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) · ( b U α ⊗ (cid:17) = (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m + α ⊗ S n ( S ∗ ) n (cid:17) = b U m − m + α δ l − l ,n − n = (cid:16) b U m − m δ l − l ,n − n (cid:17) b U α = (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) · b U α for α ∈ Z . Obtaining the result for arbitrary elements in C ∗ ( b U ) is a simple extension of this.We denote our C ∗ -module by Z C ∗ ( b U ) and inner-product by ( · | · ) C ∗ ( b U ) . The point of the singulartrace Ψ becomes apparent in the next proposition where we construct a left action of A φ on Z C ∗ ( b U ) . Proposition 2.6.
There is an adjointable representation if A φ on Z C ∗ ( b U ) .Proof. Clearly we can multiply elements of Z C ∗ ( b U ) by T on the left, but by Proposition 2.4, we knowthat ( b U j b V k ⊗ k ) · Z C ∗ ( b U ) = 0 if k ∈ K . Therefore the representation of T descends to a representationof T /ψ [ C ( S ) ⊗ K ] ∼ = A φ . This gives us the explicit left-action by ( b U α b V β ) · (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = ( b U α b V β b V n − n b U m ) ⊗ S n + β ( S ∗ ) n = e πiφα ( n − n + β ) b V β + n − n b U m + α ⊗ S β + n ( S ∗ ) n for α, β ∈ Z with β ≥ and ( b U α b V β ) · (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = e πiφα ( n − n + β ) b V β + n − n b U m + α ⊗ S β ( S ∗ ) n + | β | β < . It follows that, as operators on Z C ∗ ( b U ) , b U b V = e πiφ b V b U . Next we just need to verify that theaction is adjointable as a module over C ∗ ( b U ) . For this it suffices to check that multiplication by b U and b V are adjointable. We compute that (cid:16) b U · (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:17)(cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) C ∗ ( b U ) = (cid:16) e πiφ ( l − l ) b V l − l b U m +1 ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) C ∗ ( b U ) = e − πiφ ( l − l ) b U m − − m δ l − l ,n − n = (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) e − πiφ ( n − n ) b V n − n b U m − ⊗ S n ( S ∗ ) n (cid:17) C ∗ ( b U ) = (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b U − · (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17)(cid:17) C ∗ ( b U ) and then (cid:16) b V · (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:17)(cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) C ∗ ( b U ) = (cid:16) b V l − l +1 b U m ⊗ S l +1 ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) C ∗ ( b U ) = b U m − m δ l − l +1 ,n − n = b U m − m δ l − l ,n − n − = (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n − b U m ⊗ S n ( S ∗ ) n +1 (cid:17) C ∗ ( b U ) = (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V − · (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17)(cid:17) C ∗ ( b U ) . and so our generating elements are adjointable and unitary on the dense span of monomials in Z C ∗ ( b U ) .Thus if b U , b V are bounded, they will generate an adjointable representation of A φ . To consider theboundedness of b U and b V , we first note that the inner-product in Z C ∗ ( b U ) is defined from multiplicationin T and the functional Ψ , which has the property Ψ( T ) ≤ k T k , by Equation (2.2). These observationsimply that k a k End( Z ) = sup z ∈ Z k z k =1 ( a · z | a · z ) C ∗ ( b U ) ≤ sup z ∈ Z k z k =1 k aa ∗ k ( z | z ) C ∗ ( b U ) = k aa ∗ k . Therefore the action of A φ is bounded, and so extends to an adjointable action on Z C ∗ ( b U ) .In Section 2.3, we show that by considering a left module C ∗ ( b U ) Z , we may also obtain an adjointablerepresentation of A op φ . Before we finish building our Kasparov module, we need some further resultsarising from properties of the singular trace Ψ . Proposition 2.7.
Let l − l = n − n . Then b V n − n b U m ⊗ S n ( S ∗ ) n = b V l − l b U m ⊗ S l ( S ∗ ) l aselements in Z C ∗ ( b U ) .Proof. We can assume without loss of generality that l = n + k and l = n + k for some k ∈ Z . Asa preliminary, we compute Ψ (cid:2) ( S n ( S ∗ ) n − S n + k ( S ∗ ) n + k ) ∗ ( S n ( S ∗ ) n − S n + k ( S ∗ ) n + k ) (cid:3) . Firstly weexpand (cid:0) S n ( S ∗ ) n − S n + k ( S ∗ ) n + k (cid:1) ∗ (cid:0) S n ( S ∗ ) n − S n + k ( S ∗ ) n + k (cid:1) = S n ( S ∗ ) n S n ( S ∗ ) n − S n ( S ∗ ) n S n + k ( S ∗ ) n + k − S n + k ( S ∗ ) n + k S n ( S ∗ ) n + S n + k ( S ∗ ) n + k S n + k ( S ∗ ) n + k = S n ( S ∗ ) n − S n + k ( S ∗ ) n + k − S n + k ( S ∗ ) n + k + S n + k ( S ∗ ) n + k = S n ( S ∗ ) n − S n + k ( S ∗ ) n + k .
6e now recall that Ψ( S α ( S ∗ ) β ) = δ α,β , so that Ψ (cid:2) ( S n ( S ∗ ) n − S n + k ( S ∗ ) n + k ) ∗ ( S n ( S ∗ ) n − S n + k ( S ∗ ) n + k ) (cid:3) = Ψ( S n ( S ∗ ) n ) − Ψ( S n + k ( S ∗ ) n + k ) = 0 . From this point, it is a simple task to show that b V n − n b U m ⊗ S n ( S ∗ ) n = b V n − n b U m ⊗ S n + k ( S ∗ ) n + k in the norm induced by ( · | · ) C ∗ ( b U ) . Lemma 2.8.
Let e n ,n ,m denote the element b V n − n b U m ⊗ S n ( S ∗ ) n ∈ Z C ∗ ( b U ) . Then for all k ∈ Z Θ e l ,l ,k ,e l ,l ,k ( e n ,n ,m ) = δ l − l ,n − n e n ,n ,m , where Θ e,f ( g ) = e ( f | g ) C ∗ ( b U ) are the rank- endomorphisms that generate End C ∗ ( b U ) ( Z ) .Proof. We check that Θ e l ,l ,k ,e l ,l ,k ( e n ,n ,m ) = b V l − l b U k ⊗ S l ( S ∗ ) l × (cid:16) b V l − l b U k ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) C ∗ ( b U ) = (cid:16) b V l − l b U k ⊗ S l ( S ∗ ) l (cid:17) · b U m − k δ l − l ,n − n = b V n − n b U m ⊗ S n ( S ∗ ) n δ l − l ,n − n , where we have used Proposition 2.7.With these preliminary results out the way, we now state the main result of this subsection. Proposition 2.9.
Define the operator N : Dom( N ) ⊂ Z → Z such that N (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) =( n − n ) b V n − n b U m ⊗ S n ( S ∗ ) n . Then (cid:16) A φ , Z C ∗ ( b U ) , N (cid:17) is an unbounded, odd Kasparov module.Proof. Lemma 2.8 shows that for any n , n with n − n = k , the operator Φ k = Θ e n ,n , ,e n ,n , is anadjointable projection. These projections form an orthogonal family Φ l Φ k = δ l,k Φ k by Lemma 2.8, and it is straightforward to show that P k ∈ Z Φ k is the identity of Z (convergence in thestrict topology). The arguments used in [20] show that given z ∈ Z and defining Φ k z = z k , we have that z = X k ∈ Z z k . This allows us to define a number operator
N z = X k ∈ Z kz k for those z ∈ Dom( N ) = nP k z k : P k k ( z k | z k ) C ∗ ( b U ) < ∞ o . As N is given in in its spectral represen-tation, standard proofs show that N is self-adjoint (again, see [20] for an explicit proof).To show that N is regular, we observe that N (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = ( n − n ) b V n − n b U m ⊗ S n ( S ∗ ) n and so N has the spanning set of T as eigenvectors. Therefore (1 + N ) has dense range and so N isregular. 7o check that we have an unbounded Kasparov module, we need to show that [ N, a ] is a boundedendomorphism for a in a dense subset of A φ and that (1 + N ) − / ∈ End C ∗ ( b U ) ( Z ) . We have that, for β ≥ N ( b U α b V β ) (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = N (cid:16) e πiφα ( n − n + β ) b V n − n + β b U m + α ⊗ S n + β ( S ∗ ) n (cid:17) = ( n − n + β ) e πiφα ( n − n + β ) b V n − n + β b U m + α ⊗ S n + β ( S ∗ ) n and ( b U α b V β ) N (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = ( n − n ) e πiφα ( n − n + β ) b V n − n + β b U m + α ⊗ S n + β ( S ∗ ) n , which implies that [ N, b U α b V β ] = β b U α b V β since the span of b V n − n b U m ⊗ S n ( S ∗ ) n is dense in thedomain of N in the graph norm. Hence for an element a = P α,β a α,β b U α b V β in a dense subset of A φ with ( a α,β ) ∈ S ( Z ) , the Schwartz class sequences, we have that [ N, a ] = X α,β βa α,β b U α b V β which is in A φ as βa α,β ∈ S ( Z ) and therefore is bounded. An entirely analogous argument also worksfor β < .Finally, we recall that N has a set of eigenvectors given by the spanning functions n b V n − n b U m ⊗ S n ( S ∗ ) n : n , n ∈ N , m ∈ Z o . This means that we can write N = M k ∈ Z k Φ k where Φ k is the projection on onto span { b V n − n b U m ⊗ S n ( S ∗ ) n ∈ Z C ∗ ( b U ) : n − n = k, m ∈ Z } . Asthe projections Φ k can be written as a rank one operator Θ e n ,n , ,e n ,n , ∈ End C ∗ ( b U ) ( Z ) , we have that (1 + N ) − / = M k ∈ Z (cid:0) k (cid:1) − / Φ k is a norm-convergent sum of elements in End C ∗ ( b U ) ( Z ) and is therefore in End C ∗ ( b U ) ( Z ) . A op φ -action The module Z C ∗ ( b U ) has more structure. It is in fact a left C ∗ -module over C ∗ ( b U ) where we define aninner-product by C ∗ ( b U ) (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = b V l − l b U m (cid:16) b V n − n b U m (cid:17) ∗ × Ψ (cid:2) S l ( S ∗ ) l ( S n ( S ∗ ) n ) ∗ (cid:3) = b V l − l b U m − m b V n − n δ l − l ,n − n = η − n − n ( b U m − m ) δ l − l ,n − n , recalling that η n ( b U m ) = b V − n b U m b V n is the automorphism defining the crossed-product structure. Wecheck compatibility of C ∗ ( b U ) ( · | · ) with left-multiplication by C ∗ ( b U ) , where C ∗ ( b U ) (cid:16) b U b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = b U b V l − l b U m − m b V n − n δ l − l ,n − n = b U · C ∗ ( b U ) (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) . The other axioms for a left C ∗ ( b U ) -valued inner-product are straightforward. We complete in the inducednorm and denote our left-module by C ∗ ( b U ) Z . 8 roposition 2.10. There is an adjointable representation of A − φ ∼ = A op φ on C ∗ ( b U ) Z .Proof. We construct an action by C ∗ ( U, V ) ∼ = A op φ by defining U · (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) · b U = b V n − n b U m +1 ⊗ S n ( S ∗ ) n ,V · (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) · b V = e πiφm b V n − n +1 b U m ⊗ S n +1 ( S ∗ ) n and extending to the whole algebra. One finds that, as operators on C ∗ ( b U ) Z , U V = e − πiφ V U . Aspreviously, we check adjointability on generating elements, where C ∗ ( b U ) (cid:16) U · (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:17)(cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = η − n − n ( b U m +1 − m ) δ n − n ,l − l = C ∗ ( b U ) (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m − ⊗ S n ( S ∗ ) n (cid:17) = C ∗ ( b U ) (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) U − · (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17)(cid:17) as expected. For V , we find that C ∗ ( b U ) (cid:16) V · (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:17)(cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = C ∗ ( b U ) (cid:16) e πiφm b V l − l +1 b U m ⊗ S l +1 ( S ∗ ) l (cid:12)(cid:12)(cid:12) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = e πiφm b V l − l +1 b U m − m b V n − n δ l − l +1 ,n − n = e πiφm e − πiφ ( m − m ) b V l − l b U m − m b V n − n +1 δ l − l ,n − n − = C ∗ ( b U ) (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) e − πiφm b V n − n − b U m ⊗ S n ( S ∗ ) n +1 (cid:17) = C ∗ ( b U ) (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) ( b V n − n b U m ⊗ S n ( S ∗ ) n ) b V − (cid:17) = C ∗ ( b U ) (cid:16) b V l − l b U m ⊗ S l ( S ∗ ) l (cid:12)(cid:12)(cid:12) V − · (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17)(cid:17) and so our generating elements are adjointable and unitary on the dense span of monomials in C ∗ ( b U ) Z .Thus if U, V are bounded, they will generate an adjointable representation of A op φ . To consider theboundedness of U and V , we first note that the inner-product in C ∗ ( b U ) Z is defined from multiplicationin T and the functional Ψ , which has the property Ψ( T ) ≤ k T k , by Equation (2.2). These observationsimply that k a op k End( Z ) = sup z ∈ Z k z k =1 C ∗ ( b U ) ( a op · z | a op · z ) ≤ sup z ∈ Z k z k =1 k a op ( a op ) ∗ k C ∗ ( b U ) ( z | z ) = k a op ( a op ) ∗ k . Therefore the action of A op φ is bounded, and so extends to an adjointable action on C ∗ ( b U ) Z . Remark . Our construction of C ∗ ( b U ) Z shows that Z can be equipped with a bimodule structure over C ∗ ( b U ) . Proposition 2.6 and 2.10 show that the right (resp. left) module comes with an adjointablerepresentation of A φ (resp. A op φ ). While it may be tempting to think so, we emphasise that theserepresentations are not adjointable on the left (resp. right) module.Another thing to note is that the actions of A φ and A op φ on Z commute. The proof of this is acomputation; the only part that requires some work is to show that [ b U , V ] = 0 . Since b U V (cid:16) b V n − n b U m ⊗ V n ( V ∗ ) n (cid:17) = e πiφ ( n − n +1) e πiφm b V n − n +1 b U m +1 ⊗ V n ( V ∗ ) n and V b U (cid:16) b V n − n b U m ⊗ V n ( V ∗ ) n (cid:17) = e πiφ ( m +1) e πiφ ( n − n ) b V n − n +1 b U m +1 ⊗ V n ( V ∗ ) n , we find that, as required, [ b U , V ] = 0 . Once again, we reiterate that these actions cannot be consideredas simultaneous representations on the level of right or left C ∗ ( b U ) -modules.9ll the technical results in Section 2.2 about the singular trace Ψ still hold in the left-module setting.In particular, a completely analogous argument to the proof of Proposition 2.9 gives us the following. Proposition 2.12.
The tuple (cid:16) A op φ , C ∗ ( b U ) Z, N (cid:17) is an odd, unbounded A op φ - C ∗ ( b U ) op Kasparov module.
Now we put the pieces together. By [10, Section 7], the extension class associated to (cid:16) A φ , Z C ∗ ( b U ) , N (cid:17) comes from the short exact sequence → End C ∗ ( b U ) ( P Z ) → C ∗ ( P A φ P ) → A φ → , (2.3)where P = χ [0 , ∞ ) ( N ) is the non-negative spectral projection.We have that the map W : Z → ℓ ( Z ) ⊗ C ∗ ( b U ) given by W (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n (cid:17) = e n − n ⊗ b U m is an adjointable unitary isomorphism. Conjugation by the unitary W gives an explicit isomorphism End C ∗ ( b U ) ( P Z ) ∼ = K [ ℓ ( N )] ⊗ C ∗ ( b U ) . This isomorphism is compatible with the sequence in equation (2.3)in that the commutators [ P, S k ] and [ P, ( S ∗ ) k ] generate K [ ℓ ( N )] . With a suitable identification, the map End C ∗ ( b U ) ( P Z ) ι ֒ −→ C ∗ ( P A φ P ) is just inclusion.Now define the isomorphism ζ : C ∗ ( P A φ P ) → T by ζ ( P b V n P ) = ( b V ⊗ S ) n , ζ ( P b V − n P ) = [( b V ⊗ S ) ∗ ] n for n ≥ and ζ ( b U m ) = ∞ X j =0 ( b V ∗ ) j b U m b V j ⊗ S j (1 − SS ∗ )( S ∗ ) j and then extend accordingly. Then we have that the diagram / / K ⊗ C ∗ ( b U ) / / T / / A φ / / / / End C ∗ ( b U ) ( P T ) / / ∼ = Ad W O O C ∗ ( P A φ P ) / / ∼ = ζ O O A φ / / commutes, and so these extensions are unitarily equivalent. We summarise this Section by the following. Proposition 2.13.
The extension class representing the short exact sequence of Equation (2.1) is thesame as the class represented by the Kasparov module (cid:16) A φ , Z C ∗ ( b U ) , N (cid:17) in KK ( A φ , C ∗ ( b U )) . Once again recall the short exact sequence → C ∗ ( b U ) ⊗ K [ ℓ ( N )] ψ −→ T → A φ → . The ideal is regarded as our boundary data, as we can consider it acting on ℓ ( Z × N ) but with compactoperators acting in the direction perpendicular to the boundary. The quotient A φ describes a quantumHall system in the absence of the boundary. 10here is an obvious spectral triple in the work of Bellissard et al. [2] for the boundary-free quantumHall system. We use the notation (cid:0) A − φ , ℓ ( Z ) ⊕ ℓ ( Z ) , X (cid:1) for this triple which represents a classin KK ( A − φ , C ) . Here we have X = (cid:18) X − iX X + iX (cid:19) , where X and X are position (or,equivalently, number) operators on ℓ ( Z ) . We think of this as a ‘Dirac-type’ operator.We also have the natural spectral triple on C ∗ ( b U ) that gives us a class h ( C ∗ ( b U ) , ℓ ( Z ) C , M ) i ∈ KK ( C ( S ) , C ) ∼ = KK ( C ∗ ( b U ) ⊗ K , C ) for M the position/number operator on ℓ ( Z ) . Our idea is touse the Kasparov module that represents the Toeplitz extension to relate the bulk and boundary spectraltriples via the internal Kasparov product. Namely, we claim that, under the map KK ( A φ , C ( S )) × KK ( C ( S ) , C ) → KK ( A φ , C ) , we have that h ( A φ , Z C ∗ ( b U ) , N ) i ˆ ⊗ C ∗ ( b U ) h ( C ∗ ( b U ) , ℓ ( Z ) C , M ) i = − (cid:2) ( A φ , ℓ ( Z ) C , X, Γ) (cid:3) . Of course, our original boundary-free spectral triple is in K ( A − φ ) , not K ( A φ ) . By using the extrastructure coming from the left-module (cid:16) A op φ , C ∗ ( b U ) T , N (cid:17) , we are able to resolve this discrepancy andobtain the Bellissard spectral triple from the product module up to an explicit unitary equivalence. We have our module β = (cid:16) A φ , Z C ∗ ( b U ) , N (cid:17) giving rise to a class in KK ( A φ , C ∗ ( b U )) . We now obtain our‘boundary module’ by considering the space ℓ ( Z ) with action of C ∗ ( b U ) by translations; i.e, ( b U λ )( m ) = λ ( m − . We have a natural spectral triple in this setting denoted by ∆ = (cid:16) C ∗ ( b U ) , ℓ ( Z ) , M (cid:17) , where M :Dom( M ) → ℓ ( Z ) is given by M λ ( m ) = mλ ( m ) . It is a simple exercise to show that (cid:16) C ∗ ( b U ) , ℓ ( Z ) , M (cid:17) is indeed a spectral triple and therefore an odd, unbounded C ∗ ( b U ) - C Kasparov module. This is alsowhat we would expect for a boundary system as the operator M becomes the Dirac operator on thecircle if we switch to position space. Our goal is to take the internal Kasparov product over C ∗ ( b U ) andobtain a class in KK ( A φ , C ) , which we then link to Bellisard’s spectral triple modelling a boundarylessquantum Hall system.Whilst computing the product β ˆ ⊗ C ∗ ( b U ) ∆ is relatively straight-forward, we relegate the details to theappendix and state the result. Lemma 3.1.
The Kasparov product of the unbounded modules β = (cid:16) A φ , Z C ∗ ( b U ) , N (cid:17) and ∆ = (cid:16) C ∗ ( b U ) , ℓ ( Z ) , M (cid:17) is given by β ˆ ⊗ C ∗ ( b U ) ∆ = − " A φ , Z ⊗ C ∗ ( b U ) ℓ ( Z ) Z ⊗ C ∗ ( b U ) ℓ ( Z ) ! C , (cid:18) ⊗ ∇ M − iN ˆ ⊗
11 ˆ ⊗ ∇ M + iN ˆ ⊗ (cid:19)! , where A φ acts diagonally and ∇ : Z → Z ⊗ poly( b U ) Ω (poly( b U )) is a connection on a smooth submodule Z of Z (see the Appendix). The overall minus sign means the negative of this class in KK ( A φ , C ) . Our task now is to relate the product spectral triple of Lemma 3.1 to the boundary-free quantumHall system. The authors of [2] actually deal with Fredholm modules, but there is a very natural extension to the setting of spectraltriples. .2.2 Equivalence of the product triple and boundary-free triple Recall once again [2, 16] our ‘bulk’ spectral triple (cid:18) A − φ , (cid:18) ℓ ( Z ) ℓ ( Z ) (cid:19) C , (cid:18) X − iX X + iX (cid:19)(cid:19) , where ( X ± iX ) λ ( m, n ) = ( m ± in ) λ ( m, n ) for λ ∈ Dom( M ± iN ) ⊂ ℓ ( Z ) and A − φ ∼ = C ∗ ( U, V ) hasthe representation generated by ( U λ )( m, n ) = e − πiφn λ ( m − , n ) , ( V λ )( m, n ) = λ ( m, n − , with H = U + U ∗ + V + V ∗ and λ ∈ ℓ ( Z ) . Our quantum Hall system without boundary also comeswith a representation of A φ ∼ = C ∗ ( b U , b V ) by magnetic translations such that the two representationscommute. To put this another way (cf [16]), let σ ( k, k ′ ) = e πiφk ′ k be a group 2-cocycle for Z . Then C ∗ ( U, V ) gives a right σ -representation of Z and there is a corresponding left σ -representation of Z by C ∗ ( b U , b V ) which commutes with the right representation. Because C ∗ ( U, V ) ∼ = A − φ ∼ = A op φ , we obtainthe following. Proposition 3.2.
The data (cid:18) A φ ⊗ A op φ , ℓ ( Z ) ⊕ ℓ ( Z ) , (cid:18) X − iX X + iX (cid:19) , γ = (cid:18) − (cid:19)(cid:19) de-fines an even spectral triple.Proof. The only thing we need to check is that our Dirac-type operator has bounded commutators witha smooth subalgebra of C ∗ ( b U , b V ) , which is an easy computation.Our aim is to reproduce this spectral triple via an explicit unitary equivalence with the module wehave constructed via the Kasparov product. We state our central result. Theorem 3.3.
Let ̺ : Z C ∗ ( b U ) ⊗ C ∗ ( b U ) ℓ ( Z ) → ℓ ( Z ) be the map ̺ (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n ⊗ C ∗ ( b U ) e j (cid:17) = e − πiφ ( j + m )( n − n ) e j + m,n − n , where e j and e j,k are the standard basis elements of ℓ ( Z ) and ℓ ( Z ) respectively. Then there is arepresentation of A φ ⊗ A op φ on Z ⊗ C ∗ ( b U ) ℓ ( Z ) such that ̺ gives a unitary equivalence between the spectraltriple A φ ⊗ A op φ , Z ˆ ⊗ C ∗ ( b U ) ℓ ( Z ) Z ˆ ⊗ C ∗ ( b U ) ℓ ( Z ) ! , (cid:18) ⊗ ∇ M − iN ˆ ⊗
11 ˆ ⊗ ∇ M + iN ˆ ⊗ (cid:19)! arising from the product triple of Lemma 3.1 and the bulk quantum Hall triple in Proposition 3.2.Proof. We first check that, by moving elements of C ∗ ( b U ) across the internal tensor product, b V n − n b U m ⊗ S n ( S ∗ ) n ⊗ C ∗ ( b U ) e j = ( b V n − n ⊗ S n ( S ∗ ) n ) · b U m ⊗ C ∗ ( b U ) e j = b V n − n ⊗ S n ( S ∗ ) n ⊗ C ∗ ( b U ) b U m · e j = b V n − n ⊗ S n ( S ∗ ) n ⊗ C ∗ ( b U ) e j + m , we see that the map ̺ respects the inner-products on Z ˆ ⊗ C ∗ ( b U ) ℓ ( Z ) and on ℓ ( Z ) . Hence ̺ is anisometric isomorphism between Hilbert spaces.Next we need to define a commuting representation of A op φ on our product module. We can do thisby pulling back the representation of C ∗ ( U, V ) on ℓ ( Z ) via the isomorphism ̺ . Alternatively, the samerepresentation comes from the left action of A op φ on C ∗ ( b U ) Z , the module we constructed in Section 2.3.We first note that generating elements of Z C ∗ ( b U ) ⊗ C ∗ ( b U ) ℓ ( Z ) can be written as b V n − n ⊗ S n ( S ∗ ) n ⊗ e j for some j ∈ Z and n , n ∈ N . Then U α V β · (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17) = e πiφβj b V n − n + β ⊗ S n + β ( S ∗ ) n ⊗ e j + α β ≥ . A similar formula but replacing S n + β ( S ∗ ) n with S n ( S ∗ ) n + | β | gives the action for β < .This left-action of A op φ is compatible with the isomorphism, that is, ̺ h U α V β · (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n ⊗ e j (cid:17)i = U α V β · ̺ (cid:16) b V n − n b U m ⊗ S n ( S ∗ ) n ⊗ e j (cid:17) and this relation extends appropriately.What remains to check is that the map ̺ is compatible with the representation of A φ and theDirac-type operator. That is, we need to show that ̺ h b U α b V β · (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17)i = b U α b V β · ̺ (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17) ,̺ h (1 ˆ ⊗ ∇ M ± iN ˆ ⊗ (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17)i = ( X ± iX ) ̺ (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17) . For the first claim, more computations give that, for β ≥ , ̺ h b U α b V β · (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17)i = ̺ (cid:16) e πiφα ( β + n − n ) b V n − n + β ⊗ S n + β ( S ∗ ) n ⊗ e j + α (cid:17) = e πiφα ( β + n − n ) e − πiφ ( j + α )( β + n − n ) e j + α,n − n + β = e − πiφφj ( β + n − n ) e j + α,n − n + β and b U α b V β · ̺ (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17) = b U α b V β e − πiφj ( n − n ) e j,n − n = e − πiφjβ e − πiφj ( n − n ) e j + α,n − n + β . Again, the case for β < is basically identical. Because the result holds on generating elements, whichare represented as shift operators, the result extends to the whole algebra and space. For the secondclaim, we once more check the result on spanning elements. We recall from the appendix that (1 ˆ ⊗ ∇ M ) (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17) = b V n − n ⊗ S n ( S ∗ ) n ⊗ M b U e j = j (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17) . Therefore, ̺ h (1 ˆ ⊗ ∇ M ± iN ˆ ⊗ (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17)i = ( j ± i ( n − n )) ̺ (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e j (cid:17) = ( j ± i ( n − n )) e − πiφj ( n − n ) e j,n − n = ( X ± iX ) ̺ (cid:16) b V n − n ⊗ S n ( S ∗ ) n ⊗ e m (cid:17) and the main result follows by extending in the standard way. Remark . In the proof of Theorem 3.3 the bimodule structure of Z could be used to obtain the left-action of A op φ on the product module. An important observation is thatwe can either take the Kasparov product of (cid:16) A φ , Z C ∗ ( b U ) , N (cid:17) or (cid:16) A op φ , C ∗ ( b U ) Z, N (cid:17) with our boundarymodule and the resulting module is the same . Hence we pick up an extra representation on our productmodule, which is necessary in order to completely link up the product module to the bulk spectral triple.The deeper meaning behind this extra structure is related to Poincaré duality for A φ : see [6] for moreinformation.By setting up a unitary equivalence of spectral triples, we can conclude that the K -homological datapresented in Bellissard’s spectral triple is the same as that presented by the product module we haveconstructed. The unitary equivalence is of course much stronger than just stable homotopy equivalenceon the level of K -homology. 13 .3 Pairings with K -Theory and the edge conductance We know abstractly that the KK class defined by the Kasparov module ( A φ , Z C ∗ ( b U ) , N ) represents theboundary map in K -homology [10, Section 9]. We examine this more closely by considering the pairingsrelated to the quantisation of the Hall conductance.We recall that the bulk spectral triple ( A φ , ℓ ( Z ) ⊕ ℓ ( Z ) , X, γ ) pairs with elements in K ( A φ ) ∼ = Z [1] ⊕ Z [ p φ ] , where p φ is the Powers-Rieffel projection. For simplicity, we denote the corresponding K -homology class of our spectral triple by [ X ] , where we know that [ X ] = [ β ] ˆ ⊗ C ∗ ( b U ) [∆] . Now, [ X ] pairsnon-trivially with [ p F ] , the Fermi projection, to give the Hall conductance up to a factor of e /h . Hencewe have that σ H = e h (cid:0) [ p F ] ˆ ⊗ A φ [ X ] (cid:1) = − e h (cid:16) [ p F ] ˆ ⊗ A φ (cid:16) [ β ] ˆ ⊗ C ∗ ( b U ) [∆] (cid:17)(cid:17) , where the minus sign arises from Lemma 3.1. We can now use the associativity of the Kasparov productto rewrite this equation as [ p F ] ˆ ⊗ A φ (cid:16) [ β ] ˆ ⊗ C ∗ ( b U ) [∆] (cid:17) = (cid:0) [ p F ] ˆ ⊗ A φ [ β ] (cid:1) ˆ ⊗ C ∗ ( b U ) [∆] . We see that this new product [ p F ] ˆ ⊗ A φ [ β ] is in KK ( C , C ∗ ( b U )) ∼ = K ( C ∗ ( b U )) ∼ = Z , where the last grouphas generator b U . So [ p F ] ˆ ⊗ A φ [ β ] is represented by b U m ∈ C ∗ ( b U ) for some m ∈ Z and we are now takingan odd index pairing.Next we note that the map K ( C ∗ ( b U )) × K ( C ∗ ( b U )) → Z where (cid:0) [ p F ] ˆ ⊗ A φ [ β ] (cid:1) × [∆] (cid:0) [ p F ] ˆ ⊗ A φ [ β ] (cid:1) ˆ ⊗ C ∗ ( b U ) [∆] depends only on our boundary data, so this pairing is the mathematical formulation of the so-called edgeconductance which, as we have seen, is the same as our bulk Hall conductance up to sign.Now, our definition of the edge conductance is purely mathematical, but one can see that the unitariesand spectral triples being used come quite naturally from considering the algebra C ∗ ( b U ) acting on ℓ ( Z ) ,which is exactly what we would consider as a ‘boundary system’ in the discrete picture. Hence ourapproach to the edge conductance is physically reasonable. Furthermore, the computation of the edgeconductance boils down to computing Index (cid:16) Π b U m Π (cid:17) = − m for Π : ℓ ( Z ) → ℓ ( N ) , which is a mucheasier computation than [ p F ] ˆ ⊗ A φ [ X ] . Appendix: Computing the odd Kasparov product
It is proved in [9, Theorem 7.5] that the KK -class of the product h(cid:16) A φ , Z C ∗ ( b U ) , N (cid:17)i ⊗ C ∗ ( b U ) h(cid:16) C ∗ ( b U ) , ℓ ( Z ) , M (cid:17)i is represented by A φ , Z ⊗ C ∗ ( b U ) ℓ ( Z ) Z ⊗ C ∗ ( b U ) ℓ ( Z ) ! C , (cid:18) N ˆ ⊗ − i ⊗ ∇ MN ˆ ⊗ i ⊗ ∇ M (cid:19)! . There are several conditions to check in order to apply [9, Theorem 7.5], but the product we are takingturns out to be of the simplest kind, and we omit these simple checks. Here A φ acts diagonally oncolumn vectors, and the grading is (cid:18) − (cid:19) . To define ⊗ ∇ M , we let Z C ∗ ( b U ) be the submodule of Z given by finite sums of elements b V n − n b U m ⊗ S n ( S ∗ ) n and take the connection ∇ : Z → Z ⊗ poly( b U ) Ω (poly( b U )) given by ∇ X n ,n ,m z n ,n ,m ! = X n ,n z n ,n , ⊗ δ ( b U m ) , δ is the universal derivation, and we represent -forms on ℓ ( Z ) via ˜ π ( a δ ( a )) λ = a [ M, a ] λ for λ ∈ ℓ ( Z ) . We define (1 ⊗ ∇ M )( z ⊗ λ ) := ( z ⊗ M λ ) + (1 ⊗ ˜ π ) ◦ ( ∇ ⊗ x ⊗ λ ) . The need to use a connection to correct the naive formula ⊗ M is because ⊗ M is not well-defined onthe balanced tensor product. Computing yields that (1 ⊗ ∇ M ) X n ,n ,β z n ,n ,β ⊗ λ = X n ,n z n ,n , ⊗ b U β M λ + X n ,n z n ,n ⊗ [ M, b U β ] λ = X n ,n z n ⊗ M b U β λ. Now conjugating the representation, operator and grading by (cid:18) i (cid:19) yields the unitarily equivalentspectral triple A φ , Z ⊗ C ∗ ( b U ) ℓ ( Z ) Z ⊗ C ∗ ( b U ) ℓ ( Z ) ! C , (cid:18) − (1 ˆ ⊗ ∇ M − iN ˆ ⊗ − (1 ˆ ⊗ ∇ M + iN ˆ ⊗
1) 0 (cid:19)! with grading (cid:18) − (cid:19) . In turn, the KK -class of this spectral triple is given by − " A φ , Z ⊗ C ∗ ( b U ) ℓ ( Z ) Z ⊗ C ∗ ( b U ) ℓ ( Z ) ! C , (cid:18) ⊗ ∇ M − iN ˆ ⊗
11 ˆ ⊗ ∇ M + iN ˆ ⊗ (cid:19)! with grading (cid:18) − (cid:19) . References
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