The spectral localizer for semifinite spectral triples
aa r X i v : . [ m a t h - ph ] A p r The spectral localizer for semifinite spectral triples
Hermann Schulz-Baldes and Tom Stoiber
Department Mathematik, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, Germany
Abstract
The notion of spectral localizer is extended to pairings with semifinite spectral triples.By a spectral flow argument, any semifinite index pairing is shown to be equal to thesignature of the spectral localizer. As an application, a formula for the weak invariantsof topological insulators is derived. This provides a new approach to their numericalevaluation. MSC2010: 19K56, 46L80
In two recent works [11, 12], it was shown that the integer-valued pairing of a K -theory classwith a Fredholm module is equal to the signature of a certain gapped self-adjoint operatorcalled the spectral localizer. It is the object of this work to generalize this equality to semifiniteindex theory. Let us describe the result on integer-valued pairings more explicitly in the caseof an odd index pairing. Hence let a be an invertible operator on a Hilbert space and D = D ∗ an unbounded selfadjoint operator with compact resolvent and bounded commutator [ D, a ]. If P = χ ( D >
0) is the positive spectral projection of D expressed in terms of the characteristicfunction χ , then P aP + (1 − P ) is a Fredholm operator and the index pairing is h [ a ] , [ D ] i =Ind( P aP + (1 − P )). The main result of [11, 13] then states that this pairing can be computedwith the spectral localizer L κ = (cid:18) κ D aa ∗ − κ D (cid:19) , namely if L κ,ρ is the finite-dimensional restriction of L κ to the range of χ ( | D ⊕ D | < ρ ), thenprovided ρ is sufficiently large and κ sufficiently small, h [ a ] , [ D ] i = Sig( L κ,ρ ) . (1)The proof of (1) contains an analytic argument showing that 0 is not in the spectrum of L κ,ρ so that the signature is well-defined, and then a topological homotopy argument leading tothe equality (1). In the works [11, 12], this latter step was based on an evaluation of theindex map and actually (1) then appeared merely as corollary to a K -theoretic statement. Analternative argument for (1) using spectral flow was provided in [13, 14]. The main interest1f (1) is that the r.h.s. is amenable to numerical evaluation and is of use in connection withtopological insulators, see [9, 14]. In fact, in applications the finite volume restriction of L κ is readily obtained and no further functional calculus is involved so that merely the signatureof a finite-dimensional matrix has to computed, either by diagonalizing L κ,ρ or by the blockCholesky decomposition.For the generalization of (1), Section 2 first reviews the definitions and main results aboutFredholm operators and spectral flow in semifinite von Neumann algebras, based mainly on[16, 15, 1, 6, 8]. It then introduces the semifinite signature and analyses its link to spectralflow. With this arsenal established, the main results can be stated in Section 3. Section 4adapts the arguments of [12] to prove that the signature is well-defined and stable. Section 5then generalizes the spectral flow proof of (1) given in [13] to the present setting, while Section 6provides the proof that also the even index pairings can be calculated with the half-signature.The spectral flow argument in Section 6 is different from the one given in [14], and we feel thatit is more direct. Moreover, it does not require the normality of the off-diagonal entries of the(even) Dirac operator as in [12, 14], but instead supposes that the spectral triple is QC in thenotation of [5]. Finally Section 7 discusses applications to the numerical computation of weakinvariants of topological insulators. Let N be a von Neumann algebra with a semifinite faithful normal trace τ . A projection P ∈ N is called finite if τ ( P ) < ∞ . Let K denote the norm-closure of the smallest algebraicideal in N containing the finite projections. This C ∗ -algebra is called the ideal of τ -compactoperators in N and any projection in K is finite. Associated to K is a short exact sequence0 → K → N → N / K →
0. The quotient N / K is called the Calkin algebra and the quotientmap is denoted by π . Associated to this exact sequence one has the standard notion of Breuer-Fredholm operator, but for the definition of spectral flow an extension to skew-corners is neededand will be described next. Associated to projections P, Q ∈ N , the skew-corner P N Q consistsof operators in N mapping Ran( Q ) to Ran( P ) and then T ∈ P N Q will be viewed as an operatorbetween these subspaces. The most basic example is P Q ∈ P N Q . Note that P N Q is an algebraonly if P = Q . For any T ∈ N we will denote by Ker( T ) both the kernel as a subspace as wellas the projection onto it, which is an element in N . Furthermore, for any pair of projections P, Q ∈ N , let P ∩ Q denote the projection onto the intersection of Ran( P ) and Ran( Q ). Definition 1 ([6, 1])
Let
P, Q ∈ N be projections and T ∈ N . Then T is called ( P · Q ) -Fredholm if Ker( T ) ∩ Q and Ker( T ∗ ) ∩ P are τ -finite projections and there exists a projection E ∈ N such that P − E is finite and Ran( E ) ⊂ Ran( T ) . Its (semifinite) index is then definedas Ind ( P · Q ) ( T ) = τ (cid:0) Ker( T ) ∩ Q (cid:1) − τ (cid:0) Ker( T ∗ ) ∩ P (cid:1) . If P = Q = , this is reduces to the standard semifinite index, which is sometimes denotedby τ -Ind. There exists the following generalization of Atkinson’s theorem.2 heorem 2 ([6]) Let
P, Q, R ∈ N be projections and T ∈ P N Q . (i) T is ( P · Q ) -Fredholm if and only if there exists S ∈ Q N P with T S − P ∈ P K P and ST − Q ∈ Q K Q . (ii) The set of ( P · Q ) -Fredholm operators is open in P N Q . (iii) If T is ( P · Q ) -Fredholm and S ∈ Q N R is ( Q · R ) -Fredholm, then T S is ( P · R ) -Fredholm Ind ( P · R ) ( T S ) = Ind ( P · Q ) ( T ) + Ind ( Q · R ) ( S ) . The following criterion is crucial for the definition of the spectral flow [15, 1].
Proposition 3 If P, Q ∈ N are projections with k π ( P − Q ) k < , then P Q is ( P · Q ) -Fredholm. Definition 4
For projections
P, Q ∈ N with k π ( P − Q ) k < , the essential codimension is ec( P, Q ) := Ind ( P,Q ) ( P Q ) = τ (cid:0) (1 − P ) ∩ Q (cid:1) − τ (cid:0) (1 − Q ) ∩ P (cid:1) . Denote by F sa ⊂ N the space of self-adjoint Fredholm operators. Definition 5
Let { T t } t ∈ [0 , be a norm-continuous path in F sa and t < t < ... < t n +1 = 1 be a partition such that k π ( p k +1 − p k ) k ≤ , for all k = 0 , . . . , n with p k := χ ( T t k ≥ . Then the spectral flow is defined as the real number Sf( { T t } t ∈ [0 , ) := n X k =0 ec( p k , p k +1 ) . If it is clear from the context, the index t ∈ [0 , is dropped. Let us quote the following basic properties of the spectral flow [1].
Proposition 6
Let { T t } and { T ′ t } be norm-continuous paths in F sa . (i) The spectral flow is well defined and does not depend on the choice of partition. (ii) (Homotopy invariance) If { T t } and { T ′ t } have the same endpoints and are connected by anorm-continuous homotopy within F sa , then Sf( { T t } ) = Sf( { T ′ t } ) . (iii) (Concatenation) If T = T ′ , then Sf( { T t } ∗ { T ′ t } ) = Sf( { T t } ) + Sf( { T ′ t } ) , with ∗ denoting concatenation of paths. { T t ⊕ T ′ t } ) = Sf( { T t } ) + Sf( { T ′ t } ) . If one has a path { D t } of self-adjoint unbounded operators affiliated to N (notably, eachspectral projection of D t lies in N ) such that its bounded transform f ( D t ) := D t (1 + D t ) − / (2)is a norm-continuous path in F sa , then its spectral flow can be defined bySf( { D t } ) := Sf( { f ( D t ) } ) . It then satisfies all the properties of Proposition 6.If T and T are in F sa and T − T is compact, one can always consider the spectral flowalong the straight-line path which will be denoted bySf( T , T ) := Sf( { tT + (1 − t ) T } ) . If T , T and T are in F sa and both T − T and T − T are compact, then one hasSf( T , T ) + Sf( T , T ) = Sf( T , T ) . This is also true for certain pairs of unbounded operators, e.g. if D = D ∗ is a selfadjointoperator affiliated to N and with compact resolvent and u unitary operator such that [ D, u ]extends to a bounded operator, then the straight-line path from D to uDu ∗ becomes under thebounded transform a norm-continuous path in F sa and one hasSf( D, uDu ∗ ) = Sf( { D + tu [ D, u ∗ ] } ) Definition 7
Let T ∈ N be self-adjoint with a τ -finite (i.e. τ -trace-class) support projection.Then the signature of T is defined by Sig( T ) := τ (sgn( T )) , sgn( T ) := χ ( T > − χ ( T < . Let us note that the two summands χ ( T >
0) and χ ( T <
0) are separately trace-class. Thefollowing generalizes Sylvester’s law of inertia.
Proposition 8
Let T ∈ N be self-adjoint with a τ -finite support projection. Further let A ∈ N be invertible. Then Sig( A ∗ T A ) = Sig( T ) . Proof.
Decomposing into positive and negative part T = T + − T − , it is enough to prove thestatement for T ≥
0. In that case, one hasSig( T ) = τ (Supp( T )) , Sig( A ∗ T A ) = τ (Supp( A ∗ T A )) , T A = v | T A | the unique polar decomposition withpartial isometry v , one has A ∗ T = | T A | v ∗ = v ∗ v | T A | v ∗ and thusRan( v ) = Ran( T A ) = Ran( T ) , Ran( v ∗ ) = Ran( A ∗ T ) = Ran( A ∗ T A ) . Since T and A ∗ T A are self-adjoint, their support projections are given by Supp( T ) = v ∗ v andSupp( A ∗ T A ) = vv ∗ . Hence the claim follows from τ ( v ∗ v ) = τ ( vv ∗ ). ✷ Certain spectral flows can be computed in terms of the signature:
Proposition 9
Let { T t } be a norm-continuous path of self-adjoints in N such that the supportprojections satisfy Supp( T t ) ≤ P and the range projections satisfy Ran( T t ) ≤ P for all t and asingle τ -finite projection P ∈ N . Then { F t } := { P T t P + 1 − P } is a norm-continuous path in F sa and Sf( { F t } ) = (cid:16) Sig( T ) − Sig( T ) (cid:17) + (cid:16) τ (cid:0) P Ker( T ) (cid:1) − τ (cid:0) P Ker( T ) (cid:1)(cid:17) . (3) In particular, if the endpoints T and T are invertible elements of the algebra P N P , then thesecond term vanishes. Proof.
The Fredholm-property and continuity are obvious. Since π ( F t ) = 1, the two pointpartition is sufficiently fine and hence the spectral flow is given bySf( { F t } ) = ec( p , p ) , with p t as in Defintion 5. Since P and T t commute, one has p t = χ ( F t >
0) = χ (cid:0) P T t P ⊕ (1 − P ) > (cid:1) = P χ ( T t > P ⊕ (1 − P ) , and hence τ ((1 − p ) ∩ p ) = τ (cid:0) ( P (1 − χ ( T > P ) ∩ ( P χ ( T > P (cid:1) = τ (cid:0) ( P − P χ ( T > ∩ χ ( T > (cid:1) = τ (cid:0) P χ ( T > (cid:1) − τ (cid:0) P ( χ ( T > ∩ χ ( T ) > (cid:1) . Switching 0 and 1 and taking the difference leads toSf( { F t } ) = τ (cid:0) P χ ( T > − P χ ( T > (cid:1) . Finally, noting that
P χ ( T t >
0) = χ ( T t >
0) and thus τ (cid:0) P Ker( T t ) (cid:1) + τ (cid:0) χ ( T t > (cid:1) = τ (cid:0) P χ ( T t > (cid:1) = τ ( P ) − τ (cid:0) χ ( T t < (cid:1) , one obtains the formula (3). ✷ The signature also has an additional invariance property that is somewhat inconvenient toexpress in terms of the spectral flow: 5 roposition 10 If { T t } t ∈ [0 , is a continuous path of self-adjoints all of which have compactsupport projections and such that for every t ∈ [0 , there is an open interval ∆ t around suchthat ∆ ∩ σ ( T t ) ⊂ { } , then for all t, t ′ ∈ [0 , T t ) = Sig( T t ′ ) . Proof.
As [0 ,
1] is compact, the spectra σ ( T t ) \ { } have a common gap ∆ and hence one canchoose continuous functions f, g such that χ ( T t >
0) = f ( T t ) , χ ( T t <
0) = g ( T t ) , ∀ t ∈ [0 , . Therefore the paths t χ ( T t >
0) and t χ ( T t <
0) are actually norm-continuous paths ofprojections. Since projections that are close in norm must be unitarily equivalent, this impliesthat the signature is constant along the path. ✷ A semifinite spectral triple [7] (also called an unbounded semifinite Fredholm module) is a tuple( D, A , N ) consisting of a semifinite von Neumann algebra N with a trace τ , a ∗ -subalgebra A ⊂ N and a self-adjoint operator D affiliated to N (namely, each spectral projection of D lies in N ), which satisfy the following conditions:(i) For all a ∈ A the commutator [ D, a ] is densely defined and extends to a bounded operator(which is then an element of N ).(ii) For any a ∈ A and z / ∈ R , the products R ( z ) a = ( D − z ) − a are in K , i.e. are τ -compact.(iii) The triple is called even if there is a self-adjoint unitary γ ∈ N that commutes with all a ∈ A and anticommutes with D , otherwise it is called odd.It will also be assumed that 0 is not an eigenvalue of D . This can be done without loss ofgenerality by adding a term proportional to χ ( D = 0) to D , which is merely a τ -compactperturbation. If A has no unit, let A + ⊂ N be its minimal unitization.For an odd spectral triple and a unitary u ∈ A + , one has the index pairing h [ u ] , [ D ] i := Ind ( P · P ) ( P uP ) , where P = χ ( D > e.g. [8]): h [ u ] , [ D ] i = Sf( u ∗ Du, D ) , (4)For an even spectral triple, the Dirac operator D anti-commutes with a self-adjoint unitary γ which is represented by the matrix diag(1 , −
1) in the grading provided by γ . As D and all6dd functions of D are off-diagonal in this representation, there is an unbounded operator D and a unitary F such that D = (cid:18) D ∗ D (cid:19) , sgn( D ) = (cid:18) F ∗ F (cid:19) . (5)In the same sense, any element a ∈ A + decomposes with respect to the grading of γ as a = (cid:18) a + a − (cid:19) . For a projection p ∈ A + , the index pairing is defined by [6] h [ p ] , [ D ] i := Ind ( p + · p − ) ( p + F ∗ p − ) , (6)where the properties of the spectral triple show that the skew-corner index is well-defined. Itcan be written as a spectral flow by h [ p ] , [ D ] i = ec( p + , F ∗ p − F ) = Sf( F ∗ (1 − p − ) F, − p + ) . If p is given by p = χ ( h <
0) for a self-adjoint invertible operator h ∈ A + , then continuouslydeforming (1 − p ) to h shows h [ p ] , [ D ] i = Sf(( F ∗ (1 − p − ) F, − p + ) = Sf( F ∗ h − F, h + ) . (7)The straight-line paths (and the homotopy) are well-defined since [ a, sgn( D )] is τ -compact forany a ∈ A and hence F ∗ a − − a + F ∗ is also τ -compact. We now define the spectral localizer: Definition 11
Let ( N , A , D ) be a spectral triple. (i) If the triple is odd, assume that a ∈ A + is invertible and define the spectral localizer by L κ := (cid:18) κD aa ∗ − κD (cid:19) , as an operator affiliated to M ( C ) ⊗ N . With the abbreviations h = (cid:18) aa ∗ (cid:19) , D ′ = (cid:18) D − D (cid:19) , the spectral localizer can also be written as L κ = h + κD ′ . (ii) For an even spectral triple let us suppose that the commutators [ | D | , a ] are densely definedand extend to bounded operators for all a ∈ A , notably the triple ( N , A , D ) is QC in thenotation of [5] . If h ∈ A + is invertible and self-adjoint, the associated spectral localizer isdefined by L κ := κD + hγ = (cid:18) h + κD ∗ κD − h − (cid:19) , with the last expression again as a matrix w.r.t. the grading γ . urther set P ρ := χ (( D ′ ) < ρ ) in the odd case and P ρ := χ ( D < ρ ) in the even case. Thenin both cases the reduced spectral localizer is defined by L κ,ρ := P ρ L κ P ρ . The two cases of odd and even pairings are similar: there is a self-adjoint unitary thatanti-commutes with h and commutes with D ′ in the odd case, respectively that anti-commuteswith D and commutes with h in the even case. Both of the associated pairings can now beread off the spectral localizer, as shows the main result of the paper: Theorem 12
Let ( N , A , D ) be a spectral triple and h = (cid:18) aa ∗ (cid:19) with a ∈ A + invertible inthe odd case and, respectively, h ∈ A + invertible in the even case. Further let g = k h − k − bethe size of the spectral gap of h . If κ > is chosen so small that κ ≤ g k [ D, h ] k k h k , (8) and ρ so large that ρ > gκ , (9) then the spectral localizer L κ,ρ as defined above is invertible and satisfies h [ a | a | − ] , [ D ] i = Sig( L κ,ρ ) in the odd case and h [ χ ( h < , [ D ] i = Sig( L κ,ρ ) in the even case. For a given ρ >
0, let us introduce a smooth cut-off function G ∈ C ∞ ( R ) such that G ρ ( x ) = ( , for | x | ≤ ρ , , for | x | > ρ , and whose Fourier transform satisfies (cid:13)(cid:13) F ( G ′ ρ ) (cid:13)(cid:13) ≤ ρ − . A standard estimate ( e.g. [11]) shows k [ G ρ ( D ) , a ] k ≤ ρ k [ D, a ] k . Lemma 13
A pair ( κ, ρ ) is called admissible if it satisfies the inequalities (8) and (9) . (i) If ( κ, ρ ) is admissible, then ( L κ,ρ ) > g P ρ . If ( κ, ρ ) and ( κ ′ , ρ ′ ) are admissible, then Sig( L κ,ρ ) = Sig( L κ ′ ,ρ ′ ) . Proof.
Let us focus on the proof in the odd case, following [13]. The even case can be dealtwith by similar means as in [12] (it is enough to replace H ⊗ σ and H ⊗ Γ appearing in [12]with h and hγ respectively). For ρ ≤ ρ ′ , let us introduce the path λ ∈ [0 , L κ,ρ,ρ ′ ( λ ) := κP ρ ′ D ′ P ρ ′ + P ρ ′ G λ,ρ hG λ,ρ P ρ ′ , (10)with G λ,ρ = (1 − λ ) + λG ρ ( D ′ ). By literally the same computation as in [13], one obtains( L κ,ρ,ρ ′ ( λ )) + (1 − P ρ ′ ) > , and ( L κ,ρ,ρ ′ (0)) > g P ρ ′ , provided that ( κ, ρ ) is admissible. In particular, ( L κ,ρ ) = ( L κ,ρ,ρ (0)) > g P ρ ′ . Next let usshow that Sig( L κ,ρ ) = Sig( L κ ′ ,ρ ′ ) for any admissible pairs ( κ, ρ ) and ( κ ′ , ρ ′ ) with ρ ≤ ρ ′ . As L λκ ′ +(1 − λ ) κ,ρ ′ is gapped around 0 for any λ , the signature remains unchanged by Proposition 10,hence one may assume κ = κ ′ . Since the path (10) is continuous and also satisfies the conditionsof Proposition 10, it is sufficient to prove Sig( L κ,ρ,ρ (1)) = Sig( L κ,ρ,ρ ′ (1)). As P ρ ′ G ρ = P ρ G ρ , onehas L κ,ρ,ρ ′ (1) = κP ρ D ′ P ρ + P ρ ′ G ρ hG ρ P ρ ′ = L κ,ρ,ρ (1) + κ ( P ρ ′ − P ρ ) D ′ ( P ρ ′ − P ρ ) , and, moreover, the last sum is direct. HenceSig( L κ,ρ,ρ ′ (1)) = Sig( L κ,ρ,ρ (1)) + Sig(( P ρ ′ − P ρ ) D ′ ( P ρ ′ − P ρ )) = Sig( L κ,ρ,ρ (1)) , where the second equality holds obviously due to the definition of D ′ . ✷ For a affiliated to N or matrices over N , let us write a ρ = P ρ aP ρ and a ρ c = (1 − P ρ ) a (1 − P ρ ).Then D = D ρ + D ρ c and, in the odd and even case respectively, L κ = L κ,ρ ⊕ L κ,ρ c + P ρ hP ρ c + P ρ c hP ρ ,L κ = L κ,ρ ⊕ L κ,ρ c + P ρ ( hγ ) P ρ c + P ρ c ( hγ ) P ρ . Lemma 14 If ( κ, ρ ) is admissible and ρ is large enough, then L κ,ρ c is invertible in P ρ c N P ρ c and Sf( L κ , L κ,ρ ⊕ L κ,ρ c ) = 0 . Proof.
We will show that for ρ large enough the term of L κ that is off-diagonal with respectto the decomposition = P ρ ⊕ P ρ c can be shrunk to zero with a linear path t ∈ [0 , L κ ( t )in the invertible operators given by L κ ( t ) = L κ,ρ ⊕ L κ,ρ c + t ( P ρ L κ P ρ c + P ρ c L κ P ρ ) . | L κ,ρ | > g P ρ by Lemma 13 and( L κ,ρ c ) = (cid:18) κ D ,ρ c + a ρ c a ∗ ρ c κ [ D ,ρ c , a ρ c ] κ [ D ,ρ c , a ρ c ] ∗ κ D ,ρ c + a ∗ ρ c a ρ c (cid:19) ≥ κ ρ P ρ c + κ (cid:18) D ,ρ c , a ρ c ][ D ,ρ c , a ρ c ] ∗ (cid:19) ≥ ( κ ρ − κ k [ D , a ] k ) P ρ c ≥ κ ρ P ρ c , (11)due to (8) and for ρ sufficiently large. The inverse is again diagonal in the decomposition = P ρ ⊕ P ρ c and given by | L κ,ρ ⊕ L κ,ρ c | − = P ρ | L κ,ρ | − ⊕ P ρ c | L κ,ρ c | − , such that L κ ( t ) is equal to | L κ,ρ ⊕ L κ,ρ c | (cid:16) S + t (cid:16) | L κ,ρ | − P ρ hP ρ c | L κ,ρ c | − + | L κ,ρ c | − P ρ c hP ρ | L κ,ρ | − (cid:17) (cid:17) | L κ,ρ ⊕ L κ,ρ c | , with S being the unitary from the polar decomposition of L κ,ρ ⊕ L κ,ρ c . By a Neumann seriesargument, this is invertible for t ≤ (cid:13)(cid:13)(cid:13) | L κ,ρ | − P ρ hP ρ c | L κ,ρ c | − (cid:13)(cid:13)(cid:13) ≤ k h k g ( ρ κ ) , this is the case for ρ large enough. ✷ This section provides the proof of the odd case of Theorem 12. Thus let us consider an oddspectral triple and suppose that ( κ, ρ ) is admissible. By Lemma 13 (ii) it is enough to prove thesignature formula for some admissible ( κ, ρ ) and we assume ρ to be large enough for Lemma 14to hold. As a is invertible, it is connected to its polar decomposition u = a | a | − by a continuouspath and hence the spectral flow formula (4) gives h [ u ] , [ D ] i = Sf (cid:18)(cid:18) u ∗
00 1 (cid:19) (cid:18) κD − κD (cid:19) (cid:18) u
00 1 (cid:19) , (cid:18) κD − κD (cid:19)(cid:19) = Sf (cid:18)(cid:18) κD − κD (cid:19) , (cid:18) u
00 1 (cid:19) (cid:18) κD − κD (cid:19) (cid:18) u ∗
00 1 (cid:19)(cid:19) . Noting that the path is in the invertibles except at the left endpoint, which has by assumptiona trivial kernel, one must haveSf (cid:18)(cid:18) κD − κD (cid:19) , (cid:18) κD − κD (cid:19)(cid:19) = 0 , (cid:18)(cid:18) u
00 1 (cid:19) (cid:18) κD − κD (cid:19) (cid:18) u ∗
00 1 (cid:19) , (cid:18) u
00 1 (cid:19) (cid:18) κD − κD (cid:19) (cid:18) u ∗
00 1 (cid:19)(cid:19) = 0 . Using the concatenation property of the spectral flow and deforming the resulting path againto a straight-line path with a homotopy in the space of Fredholm operators implies h [ u ] , [ D ] i = Sf (cid:18)(cid:18) κD − κD (cid:19) , (cid:18) u
00 1 (cid:19) (cid:18) κD − κD (cid:19) (cid:18) u ∗
00 1 (cid:19)(cid:19) = Sf (cid:18)(cid:18) κD − κD (cid:19) , (cid:18) κ uDu ∗ uu ∗ − κD (cid:19)(cid:19) . Now κuDu ∗ = κD + κu [ D, u ∗ ] and κu [ D, u ∗ ] is a bounded summand that for κ sufficiently smalldoes not alter the invertibility of the spectral localizer of u so thatSf (cid:18)(cid:18) κuDu ∗ uu ∗ − κD (cid:19) , (cid:18) κD uu ∗ − κD (cid:19)(cid:19) = 0 . Therefore again by concatenation and deforming the paths one obtains h [ u ] , [ D ] i = Sf (cid:18)(cid:18) κD − κD (cid:19) , (cid:18) κD uu ∗ − κD (cid:19)(cid:19) . Finally one can connect u to a by the path t ∈ [0 , a ( t + (1 − t ) | a | − ) leading to a path ofinvertible spectral localizers. Hence again by concatenation and deformation of the straight-linepaths h [ u ] , [ D ] i = Sf (cid:18)(cid:18) κD − κD (cid:19) , (cid:18) κD aa ∗ − κD (cid:19)(cid:19) = Sf (cid:18)(cid:18) κD − κD (cid:19) , L κ (cid:19) . Now by Lemma 14 one can decouple L κ to L κ,ρ ⊕ L κ,ρ c so that, once again by concatenationand deforming the linear paths, h [ u ] , [ D ] i = Sf (cid:18)(cid:18) κD − κD (cid:19) , L κ,ρ ⊕ L κ,ρ c (cid:19) . Finally also the first entry is diagonal in the decomposition P ρ ⊕ P ρ c so that h [ u ] , [ D ] i = Sf (cid:18)(cid:18) κD ρ − κD ρ (cid:19) ⊕ (cid:18) κD ρ c − κD ρ c (cid:19) , L κ,ρ ⊕ L κ,ρ c (cid:19) = Sf (cid:18)(cid:18) κD ρ − κD ρ (cid:19) , L κ,ρ (cid:19) , because one can use the homomorphism property of Proposition 6(iv) and the fact that thespectral flow on P ρ c vanishes since the path is in the invertibles due to (11). Finally, thesignature formula Proposition 9 completes the proof of the odd case of Theorem 12 by notingthat the signature of the left endpoint vanishes.11 Proof of the even index formula
In the even case not only [
D, a ] is assumed to be bounded for every a ∈ A , but [ | D | , a ] as well.This implies that the (in general inequivalent) representation of A in N given by a ∈ A 7→ π + ( a ) = (cid:18) a + F a + F ∗ (cid:19) also defines an even spectral triple ( N , π + ( A ) , D ). However, writing out (6) one finds that itsindex pairing vanishes. The spectral localizer with respect to this triple is still useful and isdenoted by L + κ := (cid:18) h + κD ∗ κD − F h + F ∗ (cid:19) , and analogously L + κ,ρ = P ρ L + κ P ρ . Note that in a spectral triple [ a, sgn( D )] is τ -compact for all a ∈ A and hence also a − π + ( a ) ∈ K , so e.g. the spectral flow from L κ to L + κ is well-defined.The starting point for the proof of the even signature formula is the spectral flow formula h [ p ] , [ D ] i = Sf( h − , F h + F ∗ ) given in (7). The additivity of the spectral flow leads to h [ p ] , [ D ] i = Sf (cid:18)(cid:18) h + − F h + F ∗ (cid:19) , (cid:18) h + − h − (cid:19) (cid:19) = Sf (cid:0) π + ( h ) γ, hγ (cid:1) . Again by Lemma 13(ii) it is enough to prove the signature formula for some admissible ( κ, ρ ),so one can assume that κ is as small and ρ as large as necessary. Lemma 15
For κ small enough, one has Sf( hγ, L κ ) = 0 and Sf( π + ( h ) γ, L + κ ) = 0 . Proof.
Considering that (cid:0) (1 − t )( hγ ) + tL κ (cid:1) = h + t κ D + tκ [ h, D ] ≥ ( g − κ k [ D, h ] k ) , the straight-line path connecting hγ and L κ is invertible for κ k [ D, h ] k < g . The same holdsfor the other path since ( N , π + ( A ) , D ) is also a spectral triple. ✷ Since the intermediate paths have compact differences, we can concatenate and then deformback to a straight-line path, hence h [ p ] , [ D ] i = Sf (cid:0) L + κ , L κ (cid:1) . According to Lemma 14 the endpoints decouple, namelySf( L κ , L κ,ρ ⊕ L κ,ρ c ) = 0 = Sf( L + κ , L + κ,ρ ⊕ L + κ,ρ c )for sufficiently large ρ . The paths that shrink the off-diagonal parts of the localizers to zeroagain have compact differences, so the spectral flow can be decomposed into two summands12sing the additivity. The contribution on P ρ can be expressed in terms of the signature due toProposition 9: Sf( L + κ , L κ ) = Sf( L + κ,ρ , L κ,ρ ) + Sf( L + κ,ρ c , L κ,ρ c )= (cid:0) Sig( L κ,ρ ) − Sig( L + κ,ρ )) + Sf( L + κ,ρ c , L κ,ρ c ) . Considering that L + κ,ρ is unitarily equivalent to its negative due to − L + κ,ρ = (cid:18) F ∗ − F (cid:19) L + κ,ρ (cid:18) − F ∗ F (cid:19) , the signature Sig( L + κ,ρ ) vanishes. It only remains to show that the last summand also vanishes: Lemma 16
For κρ large enough, one has Sf( L + κ,ρ c , L κ,ρ c ) = 0 . Proof.
Again let us consider the square((1 − t ) L + κ,ρ c + tL κ,ρ c ) = κ D ρ c + tκ [ h ρ c , D ρ c ] + (1 − t ) κ [ π + ( h ) ρ c , D ρ c ]+ (cid:18) ( h + ) ρ c t ( h − ) ρ c + (1 − t ) ( F h + F ∗ ) ρ c + t (1 − t )[( h − ) ρ c , ( F h + F ∗ ) ρ c ] (cid:19) ≥ ( κ ρ − κ k [ D, h ] k − κ (cid:13)(cid:13) [ D, π + ( h )] (cid:13)(cid:13) − k h k ) ρ c where h ρ c = P ρ c hP ρ c and ( h ± ) ρ c = P ± ρ c h ± P ± ρ c with P ρ c = diag( P + ρ c , P − ρ c ). This shows that thepath lies within the invertibles for large κρ . ✷ If the semifinite von Neumann algebra ( N , τ ) is of type I ∞ , namely given by a pair ( B ( H ) , Tr),then the finite volume spectral localizer L κ,ρ is a finite-dimensional matrix and it is immediatelypossible to use it for numerical computation of the index pairing based on Theorem 12, see[9, 10, 14]. In the setting of a type II ∞ -von Neumann algebra, the spectral localizer L κ,ρ isin general an operator of infinite rank, but its signature may still be well approximated byfinite-dimensional quantities that are accessible to numerical computations. Here we sketchhow this can be done for a large class of Schr¨odinger-type operators describing topologicalinsulators. A typical example for an observable algebra is the disordered non-commutativetorus A = C (Ω) ⋊ ξ Z d , constructed from an invariant ergodic probability space (Ω , Z d , P )describing homogeneous disorder and a twist ξ provided by the magnetic field [18]. In therepresentation on ℓ ( Z d ), one considers for n lattice directions e , . . . , e n the Dirac operator D = P nk =1 σ k ⊗ X k with σ k a representation of the complex Clifford algebra of n generatorsand X , . . . , X n the unbounded position operators. For n < d , the non-integer-valued indexpairing with D is a so-called weak Chern number and is localized by a spectral triple ( A , N , D )( cf. [2, 3]), where N is the von Neumann algebra generated by A and bounded functions of13 , . . . , X n equipped with a trace τ that can be interpreted as an average trace per volume. Aself-adjoint invertible h ∈ M N ( A d ), assumed to take the form h = (cid:18) a ∗ a (cid:19) for n odd, describesa random family ( h ω ) ω ∈ Ω of Hamiltonians on ℓ ( Z d , C N ) modeling a topological insulator andthe index pairings h [ χ ( h ≤ , [ D ] i , respectively h [ a | a | − ] , [ D ] i , are related to linear responsecoefficients and the appearance of topologically protected boundary states [18].The spectral localizer L κ,ρ = ( L κ,ρ,ω ) ω ∈ Ω can be considered a random family of Schr¨odinger-type Hamiltonians that describes the restriction of h to a cylinder B nρ × Z d − n with an additionalpotential κD ρ . For numerical computations, one further truncates to operators L ( V ℓ ) κ,ρ,ω acting onthe finite-dimensional space ℓ ( B nρ × V ℓ ) with V ℓ being a cube with sides ℓ and supplied with e.g. periodic or Dirichlet boundary conditions. For h satisfying the usual smoothness conditionsand f ∈ C ( R ), one can show that almost surely with respect to P τ ( f ( L κ,ρ )) a.s. = lim ℓ →∞ (cid:12)(cid:12) B nρ (cid:12)(cid:12) | V ℓ | Tr( f ( L ( V ℓ ) κ,ρ,ω )) . Since L κ,ρ has a spectral gap, one can replace Sig( L κ,ρ ) = τ (sgn( L κ,ρ )) = τ ( f ( L κ,ρ )) for asuitable continuous function f and hence the signature can be approximated using only finite-dimensional algebra. When choosing restrictions with periodic boundary conditions, one canadapt methods from [17] to show that the approximations L ( V ℓ ) κ,ρ,ω have a uniform spectral gapandSig( L κ,ρ ) a.s. = lim ℓ →∞ (cid:12)(cid:12) B nρ (cid:12)(cid:12) | V ℓ | h L ( V ℓ ) κ,ρ,ω ) − L ( V ℓ ) κ,ρ,ω ) i , where the deterministic component (cid:12)(cid:12) Sig( L κ,ρ ) − | B nρ | | V ℓ | − E Sig( L ( V ℓ ) κ,ρ,ω ) (cid:12)(cid:12) of the finite volumeerror is exponentially small in ℓ for typical Hamiltonians. We expect that the method describedhere generalizes to compute weak invariants of other aperiodic quantum systems which are e.g. described by point patterns [2]. References [1] M. T. Benameur, A. L. Carey, J. Phillips, A. Rennie, F. A. Sukochev, K. P. Wojciechowski,
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