arXiv:2108.06368v1 [math-ph] 13 Aug 2021 where h ala ne hoe [ theorem index Callias The Outline 1 h none vncs.Section case. even unbounded the Concre of triple. index spectral the the from express derived still pairing can index one an setting of elemen abstract terms be this will in potentials In m perturbing underlying the algebra. the and namely triple direction, spectral different a semifinite in generalizes paper This vnseta rpewt potential a with triple spectral even H operator Callias-type the then [ in theorem allow Robbin-Salomon can the one in example, as for bundles generalizations, bund possible vector ar many finite-dimensional which are a manifolds of There non-compact sections on on act operators that Dirac potentials of index the for utpirof trace multiplier with triple spectral semifinite ucin and function, nSection in M S21:1K6 68,46L80 46L87, 19K56, exam an non-commutative even MSC2010: and Both AMS commutative flow. both spectral ter and of considered, then in analogue is computed non-commutative result The a are triple. for indices spectral the their from derived and pairing introduced are triples ala-yeoeaosascae oseta triples spectral to associated operators Callias-type U exp( = ala-ye(rDrcSh¨dne)oeaosassociat Dirac-Schr¨odinger) (or operators Callias-type eatetMteai,Friedrich-Alexander-Universit¨ Mathematik, Department A 6 ı utemr,Section Furthermore, . = ıπG = √ C ( − H ∗ mi:[email protected] [email protected] [email protected], Email: .Tepeiesaeeti ie nSection in given is statement precise The 1. ( emn cuzBle n o Stoiber Tom and Schulz-Baldes Hermann A )i ntr enn a defining unitary a is )) hc sivril modulo invertible is which ) 14 aesr 1 -15 ragn Germany Erlangen, D-91058 11, Cauerstr. , 2 T κD , -Ind( 25 8 + n t vndmninlaaou [ analogue dimensional even its and ] T rsnscascladnwexamples. new and classical presents H ıH κD 5 n ala potential Callias a and aigafrhrsmer.Section symmetry. further a having ttsadpoe nee nlgefrpiig fan of pairings for analogue even an proves and states is + Abstract T ıH 43 Fehl o ml enough small for -Fredholm rHletmdl ude ffiietp [ type finite of bundles Hilbert-module or ] 1 = ) K ter ls in class -theory h [ D A ] , [ i h es fDefinition of sense the (in U dt btatsmfiiespectral semifinite abstract to ed nepee sa ne theorem index an as interpreted ] 1 i H , tErlangen-N¨urnbergat so nascae index associated an of ms eadaeivril tinfinity. at invertible are and le lsaegiven. are ples K d pcrltilsare triples spectral odd d sasl-don differentiable self-adjoint a is 3 1 o h aeo unbounded of case the for , ( etre yself-adjoint by perturbed e A so eti multiplier certain a of ts nt-iesoa vector finite-dimensional for ) ala-yeoperator Callias-type a 27 nfl srpae ya by replaced is anifold ey f( if tely, , > κ 11 G 6 , and 0 24 utbeswitch suitable a hnas covers also then N ieformulas give ] D, , 3 A below), 15 sa is ) , 7 ]. In the commutative setting one would take for A the algebra of smooth compactly supported fiberwise compact multiplication operators on sections of a vector bundle over a complete Riemannian manifold X, for D a weakly elliptic first-orderH differential operator and = T TrL2(X) Tr . Results comparable to our main result in that commutative setting with infinite- H dimensional⊗ vector bundles have been obtained by Kaad and Lesch [30] and more recently also with less regularity assumptions by van den Dungen [47]. These proofs rely on previous work on unbounded Kasparov products. In essence the strategy in [30, 47] is to prove that a Callias- type operator is an unbounded representative of the product of a K-homology class defined by a first-order differential operator and a class in the K1-group over the continuous functions on the manifold defined by a self-adjoint multiplication operator. As discussed in Section 7 this approach can also be made to work in the present more general noncommutative setting with some technical limitations. Instead, here we provide a new and rather elementary proof using semifinite spectral flow and explicit operator homotopies. Even for the classical case (e.g. [25, 27]), the argument constitutes a considerable simplification of the proof. In the special case where X is the real line the potential H represents a path of self-adjoint Fredholm operators and the index coincides with the spectral flow of the family. This analogy has been carried further by [30, 47] who call the index pairing [D], [U]1 a spectral flow also in higher dimensions, motivated by notions from KK-theory. Weh follow thati interpretation and therefore consider the index pairing as a non-commutative analogy of spectral flow.

2 Callias-type operators with bounded potentials

Let be a von Neumann algebra with semifinite normal faithful trace acting on a Hilbert spaceN . For the convenience of the reader, several facts about the trTace , the set of H T KT -compact operators and the notion of Breuer-Fredholm or -Fredholm operators and their Tindex -Ind are recalled in Appendix A. An (odd) semifinite spectralT triple ( ,D, A )[17, 16] consistsT of an unbounded self-adjoint operator D affiliated with and a N-algebra A N ∗ ⊂ N such that (i) Each A A preserves the domain of D and the hence densely defined operator [D, A] ∈ extends to a bounded element of . N 2 1 (ii) For each A A the product A(1 + D )− 2 is -compact. ∈ T Let = C∗(A ) be the C∗-algebra generated by A . By default a spectral triple produces an indexA pairing with the K-theory group K ( ) through [17] 1 A [D], [U] = -Ind(PUP + 1 P ) R , (1) h 1i T − ∈ where P = χ(D > 0) and the -index and the operator on the r.h.s. is -Fredholm for T T any representative U 1 + A defining a class in K1( ). The potentials for our Callias-type operators will be recruited∈ from a larger algebra: A

Definition 1 Let ( ,D, A ) be a semifinite spectral triple. N 2 (i) The multiplier algebra M( , ) is the idealizer of in , i.e. the largest C∗-subalgebra of such that M( , ) A N and M( , ) A .N Elements of M( , ) are also calledN -multipliers.A N A⊂A A A N ⊂ A A N A (ii) An -multiplier H M( , ) is differentiable w.r.t. D if H preserves Dom(D) and [D,HA] extends to a bounded∈ A operator.N

(iii) For a self-adjoint differentiable -multiplier H the associated Callias-type operator on the domain Dom(D) is defined by A

Dκ,H = κD + ıH, κ> 0 .

The parameter κ can be interpreted as the scale of the noncommutative space quanta. It plays a prominent role in the following. It will next be useful to pass to a self-adjoint supersymmetric operator Lκ,H which, due to the prior works [38, 39, 45], will also be referred to as the spectral localizer.

Proposition 2 For a self-adjoint -multiplier H, the adjoint of the Callias-type operator D A κ,H is (D )∗ = κD ıH and the spectral localizer κ,H − 0 D L = κ,H∗ κ,H D 0  κ,H  2 is self-adjoint on the domain Dom(D)× . Moreover, Dκ,H and Lκ,H are affiliated to and M ( ) respectively. N 2 N Proof. Note that 0 D 0 ıH L = κ + . κ,H D 0 ıH −0     The first summand is self-adjoint, and the second is a bounded self-adjoint perturbation leaving 2 the domain Dom(D)× invariant. Hence the self-adjointness of Lκ,H immediately follows from the Kato-Rellich theorem and this also implies that (Dκ,H )∗ = Dκ, H . As to the last claim, − recall that an equivalent condition for the affiliation of an operator T is that each unitary U ′ preserves the domain of T and commutes UT U ∗ = T (see e.g. [32]). ✷ ∈N The following often deals with operators in or affiliated to the matrix algebras M2( ) and M ( ). The former is supplied with the natural trace Tr, but for notational convenienceN 2 A T ◦ we will denote it by the same letter and speak of -compact and -Fredholm operators with no regard for the size of the matrices. T T The following provides a criterion for Callias-type operators to be -Fredholm: T Definition 3 A self-adjoint -multiplier H is called asymptotically invertible w.r.t. if there A A is a positive element V such that H2 + V > 0 is invertible. An asymptotically invertible and differentiable self-adjoint∈ A -multiplier H is called a Callias potential. A

3 In the classical case of a Riemannian manifold X where = C0(X, ( )) and H is given by an operator-valued bounded function x X H ( A), the asymptoticK H invertibility is ∈ 7→ x ∈ B H indeed equivalent to the uniform invertibility of Hx outside a compact subset K X, namely 2 ⊂ there is a positive constant c such that Hx c 1 for x X K. Proposition 5 shows that V in Definition 3 can always be chosen as a spectral≥ function∈ of\ H itself. This also implies that asymptotic invertibility of H is equivalent to the invertibility of π(H) in the quotient algebra M( , )/ . A N A

Proposition 4 If H is a Callias potential, then there exists a κ0 > 0 such that Lκ,H and therefore Dκ,H, Dκ,H∗ are -Fredholm for all κ (0, κ0]. Moreover, the -index of the Callias- type operator given by T ∈ T

-Ind(D ) = (Ker(D )) (Ker(D∗ )) R T κ,H T κ,H − T κ,H ∈ is independent of κ (0, κ ]. ∈ 0

One may also view -Ind(Dκ,H) as the supersymmetric index of Lκ,H . The proof of Proposi- tion 4 will use smoothT functional calculus of a self-adjoint operator via the well-known Helffer- Sj¨ostrand or Dynkin formula [22]. For later use in Section 6, let us recall the details for a possibly unbounded self-adjoint operator H. For ρ R, let ρ(R) denote the set of smooth ∈ S functions f : R R satisfying → − k 2 ρ k ∂ f(x) C (1 + x ) 2 , k N . | | ≤ k ∈

Then there exists for any N > 0 an almost analytic representation f˜N of f supported in a complex set G of the form G = x + ıy : y < 2√1+ x2 which coincides with f on R and { | | } − − 2 ρ 1 N N ∂ f˜(z) c C˜ (1 + x ) 2 m(z) (2) | z | ≤ N N+1 |ℑ | ˜ N+1 for a universal constant cN and CN+1 = k=1 Ck, (see e.g. [22, Lemma 2.2.1]). Provided ρ< 0, the Helffer-Sj¨ostrand representation P 1 f(H) = (∂ f˜ (z))(H z)− dz dz (3) z N − ∧ ZG is a norm-convergent integral for any N 1. For a complex-valued function f : R C this ≥ → can be done for real and imaginary part separately. Proof of Proposition 4: Let χ : [0, ) [0, 1] be a smooth function with χ(0) = 1 and g2 ∞ → 2 vanishing outside [0, 2 ] We have to show that χ(Lκ,H ) is -compact for which formula (3) is T 2 used with a quasianalytic extensionχ ˜N of χ. To control the resolvent of Lκ,H , let us note that κ2 D2 + H2 0 [D,H] 0 L2 = + ıκ . κ,H 0 κ2 D2 + H2 0 [D,H]    −  Now let V and g > 0 be such that H2 + V g21. Then with V˜ = V V ≥ ⊕ L2 + V˜ g2 κ [D,H] 1 , (4) κ,H ≥ − k k 2
4 2 ˜ which shows that Lκ,H + V is invertible for κ sufficiently small. Now replace the resolvent identity into the Helffer-Sj¨ostrand formula

2 2 ˜ 1 2 ˜ 1 ˜ 2 1 χ(Lκ,H ) = (∂zχ˜N (z)) (Lκ,H + V z)− + (Lκ,H + V z)− V (Lκ,H z)− dz dz . G − − − ∧ Z h i 2 g2 1 ˜ − The first summand is χ(Lκ,H + V ) and hence vanishes if κ κ0 where κ0 = 2 [D,H] . For the remaining term, applying the resolvent identity again shows≤ k k 2 1 2 2 1 2 2 1 V˜ (L z)− = V˜ (κ D 1 z)− 1 +(H 1 + ıκ[D,H] σ (L z)− . κ,H − ⊗ 2 − ⊗ 2 ⊗ 3 κ,H − 2 2 1 As V˜ M2( ), the factor V˜ (κ D 12 z)− is -compact due to the definition of the spectral triple.∈ As A is a norm closed ideal⊗ and− the integralT in the Helffer-Sj¨ostrand formula is norm T convergent,K this implies that χ(L2 ) is -compact. κ,H T The final claim follows from the fact that κ D is continuous in the graph topology so 7→ κ,H that the -index is constant along this path. ✷ T The next result is the last technical preparation. Proposition 5 Let H be a self-adjoint -multiplier satisfying H2 + V >g21 for some V = A g g V ∗ . Then for every function f : R C supported by [ , ] one has f(H) . ∈A → − 2 2 ∈A g2 Proof. Let χ : [0, ) [0, 1] be a smooth function satisfying χ(λ) = 1 for λ 4 and χ(λ)=0 for λ g2.∞ Then→χ(H2 + V ) = 0 by hypothesis so that the Helffer-Sj¨ostrand≤ formula and resolvent identity≥ imply

2 2 1 2 1 χ(H ) = (∂ χ˜ (z))(H + V z)− V (H z)− dz dz . z N − − ∧ ZG As V , this implies χ(H2) because is a norm-closed ideal in M( , ). Now by construction,∈ A f(H) = χ(H2)f(H∈) A and thereforeA invoking the ideal property onceA N again leads to f(H) . ✷ ∈A 3 Main result for bounded Callias potentials

Proposition 6 Let H be a self-adjoint -multiplier which is asymptotically invertible w.r.t. and satisfies H2 + V >g21 for some Ag > 0 and self-adjoint V . Let G : R R be a A g ∈ A g → smooth nondecreasing odd function taking values 1 below 2 and 1 above 2 . Then the unitary operator − − U = eıπ(G+1)(H) defines a class in K ( ) which does not depend on the function G. It represents the image 1 A of the spectral projection [χ(H + < 0)]0 K0(M( , )/ ) under the exponential map in K-theory ∂ : K (M( , )/ ) AK ( ) associated∈ toA theN shortA exact sequence 0 0 A N A → 1 A 0 M( , ) M( , )/ 0 , (5) →A→ A N → A N A → namely [U] = ∂ [χ(H + < 0)] 1 0 A 0 5 Proof. By construction eıπ(G+1)(λ) 1 is supported in [ g , g ] and one therefore has U 1 − − 2 2 − ∈A by Proposition 5. Naturally it defines the class [U]1 in K1( ). The second claim results from the fact that 1 (G + 1)(H) is a lift of the projection 1 χ(HA+ < 0) into M( , ). ✷ 2 − A A N Clearly one can also replace M( , ) by any smaller C∗-algebra that contains H and has as an ideal, since by naturalityA allN computations pull back to the connecting map of A 0 C∗(H, ) C∗(H, )/ 0 where C∗(H, )= C∗(H)+ . →A→ A → A A→ A A Definition 7 For an asymptotically invertible -multiplier H, the D-spectral flow is defined as an index pairing in the sense of (1) by A Sf (H) = [D], [eıπ(G+1)(H)] R D h 1i ∈ for any admissible function G as specified in Proposition 6 above. Let us briefly justify why this definition applied to a particular set-up indeed reduces to the standard notion of semifinite spectral flow. Let (n, τ) be a semifinite von Neumann algebra and n the traceclass elements. Then a differentiable path x R Hx n of self-adjoint T Fredholm operators with invertible limits can be paired with a∈ winding7→ num∈ ber 1-cocycle to give the spectral flow in the formulation of Wahl [49], see Definition 41 in the appendix where this is spelled out for a finite interval. This latter spectral flow coincides with Definition 7 if one chooses A R n R n ( , , ,D) = Cc∞( , ), L∞( , ), = dx τ, ı∂x 1 . N T T T ⊗ ⊗  Z  The classical case is obtained when n = ( ) and τ = Tr. More generally, for differentiable d B H families x R Hx n with odd d, the above defintion reduces to a volume integral version of a generalized∈ 7→ multiparameter∈ spectral flow, see the discussion in Section 8.1. Definition 7 further conceptualizes these special cases, and is in the spirit of [30, Definition 8.9] and [47, Definition 2.18] where a K-theory valued spectral flow is introduced. The following main result shows that the index of a Callias-type operator is equal to the spectral flow in the sense of Definition 7: Theorem 8 Let H be a Callias potential for the semifinite spectral triple ( ,D, A ). Set N g2 κ = , g2 = min σ(H2 + ) . (6) 0 2 [D,H] A k k Then for all κ (0, κ0) ∈ -Ind(D ) = Sf (H) . T κ,H D Let us comment that the equality of index and spectral flow in general does not hold for large values of κ. Indeed, a counterexample (with X = R and = C (X, ( )) with an A 0 K H infinite dimensional fiber Hilbert space ) can be found in the work of Abbondandolo and Majer [1, Section 7]. Theorem 8 only concernsH the semiclassical regime of small κ, or otherwise stated the limiting index for small κ. For unbounded H the situation may be different. Indeed, the Robbin-Salamon theorem states that for one-dimensional potentials growing at infinity, all values of κ are allowed. This case is covered by Theorem 31 below.

6 4 Proof of the main result

The first step is make the Dirac operator invertible which can be achieved by a standard doubling trick. More precisely, set

D µ H 0 D˜ = , H˜ = (7) µ D 0 1  −    for some µ> 0 and also define

0 D˜ ˜ ˜ ˜ ˜ κ,∗ H˜ Dκ,H˜ = κD + ıH, Lκ,H˜ = ˜ . Dκ,H˜ 0 ! ˜ Self-adjointness of Lκ,H˜ follows again from the Kato-Rellich theorem, but the Fredholm property can depend nontrivially on κ and µ. In Lemma 11 below, we show that one may choose µ = (1) and then κ (µ). For µ = 0 the additivity of the Fredholm index gives O ≤ O

-Ind(D ) = -Ind(D˜ ˜ ) (8) T κ,H T κ,H and then the index stays unchanged for non-vanishing µ as long as the Fredholm-property is not violated. It is sufficient to prove the index formula for some sufficiently small κ because it then immediately holds for all κ as stated in Theorem 8. The next step is to express the index pairing as a spectral flow and to separate D˜ and H˜ in the 2 2 matrix. × ˜ Lemma 9 For any m > 0 and κ sufficiently small, the -index of Dκ,H˜ can be computed as spectral flow along a straight-line path: T

κD˜ H˜ ım κD˜ H˜ + ım -Ind(D˜ ˜ ) = Sf( − , ). T κ,H − H˜ + ım κD˜ H˜ ım κD˜  −   − −  1 ı Proof. By conjugation with the unitary matrix C = 1 , one has √2 1 ı  −  κD˜ H˜ ım κD˜ H˜ + ım Sf( − , ) H˜ + ım κD˜ H˜ ım κD˜  −   − −  m κD˜ + ıH˜ m κD˜ + ıH˜ = Sf( − , ) κD˜ ıH˜ m κD˜ ıH˜ m  −   − −  and so the claim follows from Proposition 44. ✷ The next step will be to deform the off-diagonal entries (more precisely the path in the lower left corner from H˜ + ım to H˜ ım) in the matrices on the r.h.s. of Lemma 9 into a unitary, − without changing the spectral flow. This will be done by functional calculus in H˜ (so for every spectral value λ R) by homotopically deforming the function ∈ t [0, 1] f (λ) = (1 t)(λ + ım)+ t(λ ım) (9) ∈ 7→ t,0 − − 7 into ı π G(λ) ı π G(λ) t [0, 1] f (λ) = (1 t)ıe− 2 + t( ı)e 2 . (10) ∈ 7→ t,1 − − The second path is constructed to contain a square root of the unitary U = eıπ(G(H)+1) appearing in the image of the exponential map in Proposition 6. The main analytical difficulty that has to be addressed next is that along such a deformation the Fredholm property has to be maintained. For this purpose, let use the odd spectral localizer

κD˜ f(H˜ ) Lo = ∗ κ,f f(H˜ ) κD˜  −  R C o for an arbitrary differentiable function f : . The Fredholm property of Lκ,f can again be checked by formally squaring →

2 2 κD˜ + f(H˜ ) κ[D,˜ f(H˜ )∗] (Lo )2 = | | , κ,f κ[f(H˜ ), D˜] κ2D˜ 2 + f(H˜ ) 2  | |  with modified commutator by the doubling given by [f(H),D] µ(f(1) f(H)) [f(H˜ ), D˜] = µ(f(1) f(H))− 0 − −  For the control of [f(H),D], let us recall:

Lemma 10 For every smooth function f, the commutator [D, f(H)] extends from Dom(D) to a bounded operator with norm bound

[D, f(H)] C˜ H [D,H] k k ≤ 3k kk k ˜ 3 (i) where C3 = C i f is a constant. =0 k k∞ Proof. Using theP Helffer-Sj¨ostrand formula (3) for N = 2, one has the norm convergent integral

1 1 [D, f(H)] = (∂ f (z))(H z)− [D,H](H z)− dz dz. − z 2 − − ∧ ZG and the claimed bound on the commutator follows immediately. An alternative proof can also be given using [D, f(H)] ( f ′) 1 R [D,H] where is the Fourier transform [26, k k≤k F kL ( )k k F Lemma 10.15]. ✷ Using Lemma 10, one obtains

(Lo )2 κ2µ2 + f(H˜ ) 2 κ [f(H˜ ), D˜] κ,f ≥ | | − k k κ2µ2 + f(H˜ ) 2 κ C˜ H [D,H] + µ f(H) f(1) . (11) ≥ | | − 3k kk k k − k From this, we will now need to derive a quantitative lower bound on the essental
spectrum o 2 o 2 of (Lκ,f ) , namely a lower bound on (Lκ,f ) + M2( ). For that purpose and the remainder KT of the section let us now assume that H > 1 mod which can be achieved without loss of generality by rescaling H and all other| parameters| (κA, µ, m, etc.).

8 Lemma 11 Associated to a smooth function f C(σ(H˜ ), C), let C˜3 be as in Lemma 10 and set ∈ c1 = min f(λ) , c2 = 2 f ,, λ 1 | | k k∞ | |≥ and κ such that 1 c2 + µ2κ2 κ(C˜ H [D,H] + µc ) > 0 , (12) 2 1 − 3k kk k 2 o 2 then (Lκ,f ) is a self-adjoint -Fredholm operator with spectrum mod M2( ) bounded from T KT below by 1 c2 + µ2κ2 κ(Cc H [D,H] + µc ). 2 1 − 3k kk k 2 Proof. Due to Lemma 10, there is a constant C > 0 such that [D, f(H)] C˜ H [D,H] . k k ≤ 3k kk k Moreover, f(H) c1 mod holds by functional calculus due to the normalization assump- tion σ(H +| ) | ≥( 1, 1) = A. Adding a spectral function V =χ ˜(H2) forχ ˜ a smooth 2 2 A ∩ − ∅ 2 c1 c1 ∈ A positive function supported in the interval [0,c1) and equal to 2 in [0, 2 ], it follows from (11)

(Lo )2 c2 1 c2 + µ2κ2 κ [D, H˜ ] µκc mod M ( ) κ,f ≥ 1 − 2 1 − | | − 2 2 A 1 c2 + µ2κ2 κ(C˜ H [D,H] + µc ) mod M ( ) . ≥ 2 1 − 3k kk k 2 2 A o 2 2 ˜ 2 Since (Lκ,f ) is a bounded perturbation of κ D it follows that is relatively -compact w.r.t. o 2 A T (Lκ,f ) . Hence, arguing as in the proof of Proposition 4, the same lower bound also holds modulo M2( ). ✷ KT

Lemma 12 Let G be any switch function as in Proposition 6 with g =1, namely G′ supported in ( 1, 1). The straight-line paths in (9) and (10) are homotopic via − s [0, 1] f (λ) = (1 s)f (λ)+ sf (λ) ∈ 7→ t,s − t,0 t,1 in such a way that (12) computed for fs,t is uniformly bounded from below by a strictly positive number for any small enough κ> 0. Moreover, the functions s [0, 1] f0,s and s [0, 1] f are invertible. ∈ 7→ | | ∈ 7→ | 1,s| Proof. By construction the parameter t merely flips the imaginary part ı f = e(f )+ (1 2t) m(f ). t,s ℜ 0,s 2 − ℑ 0,s We consider the homotopy for λ 1 where the functions simplify to | | ≥ f (λ) = sgn(λ)(1 + (1 s) λ )+ ım(1 s)(1 2t) t,s − | | − − and hence c1 1 and c2 2( H + m) uniformly in s, t [0, 1]. For µ and κ small enough the quantity (12)≥ is therefore≤ obviouslyk k bounded from below.∈ Checking pointwise invertibility for t = 0 and t = 1 is also simple: the imaginary part never changes sign and only ever vanishes when λ 1 where one always has an non-vanishing real part. ✷ | | ≥ Let us now fix µ, without restriction, to the value µ = 1. Smallness of κ is such that (12) in Lemma 12 holds.

9 ı π G(H˜ ) Corollary 13 Let us introduce the unitary W˜ = ıe 2 . Then for κ small enough − ˜ ˜ ˜ ˜ ˜ κD W κD W ∗ -Ind(Dκ,H˜ ) = Sf( , ). T − W˜ ∗ κD˜ W˜ κD˜  −   −  Proof. Start out with Lemma 9 and note that this straight line path there is given in (9). As the Fredholm property holds troughout the square (t, s) [0, 1]2 by Lemma 11, the homotopy ∈ invariance of the spectral flow as stated in Proposition 43(ii) allows to deform the path (9) into (10) by respecting the invertibility of the end points, see Lemma 12. ✷ Proof of Theorem 8: As already stated above, it is sufficient to prove the equality for some κ> 0 because then the constancy of the -index along paths of Fredholm operators allows to T conclude, due to the bound (4). Now let us start out with (8) and then invoke Corollary 13:

κD˜ W˜ κD˜ W˜ ∗ -Ind(Dκ,H ) = Sf( , ). T − W˜ ∗ κD˜ W˜ κD˜  −   −  Set U˜ = eıπG(H˜ ) = eıπ(G(H˜ )+1). Then by construction U˜ = W˜ 2 and therefore the adjoint action of the unitary diag(1, W˜ ) transforms the spectral flow to

κD˜ 1 κD˜ U˜ ∗ -Ind(Dκ,H) = Sf( , ) . T − 1 κW˜ D˜W˜ ∗ U˜ κW˜ D˜W˜ ∗  −   − 

The next aim is to replace W˜ D˜W˜ ∗ by D˜ by a homotopy s [0, 1] (1 s)W˜ D˜W˜ ∗ + sD˜ ∈ 7→ − leading to a homotopy of straight line paths. The difference W˜ D˜W˜ ∗ D˜ = W˜ [D,˜ W˜ ∗] is a bounded operator by Lemma 10, and therefore for κ small enough the− invertibility of the end points remains valid along the homotopy as does the lower bound on the essential spectrum so that the Fredholm property is conserved throughout. Therefore

κD˜ 1 κD˜ U˜ -Ind(D ) = Sf( , ∗ ) T κ,H − 1 κD˜ U˜ κD˜  −   −  Now we are in the situation to apply Proposition 47 which gives

-Ind(D ) = -Ind(χ(D>˜ 0)Uχ˜ (D>˜ 0)+1 χ(D˜ > 0)) T κ,H T − where we took into account that compared to the formulation of Proposition 47 the spectral projection is flipped, which cancels the factor of -1. The last expression is the index pairing between D˜ and U˜ which is equal to the pairing between the undoubled Dirac operator D and U˜ 1, see [21]. Since U˜ 1 = U = exp(ıπ(G(H) + 1)) the expression is equal to the spectral ⊖ ⊖ flow SfD(H). ✷

5 Even version

A spectral triple is called even if there is a proper self-adjoint unitary γ that anti-commutes with D, but commutes with all elements of . As matrices with respect∈N to the projections A 10 π = 1 (γ 1) induced by the grading γ one then has the decompositions ± 2 ± 0 D 0 F A 0 D = 0∗ , sgn(D) = ∗ , A = + , D0 0 F 0 0 A      − for each A and a partial isometry F . In the following we denote T = π T π for any ∈ A ± ± ± operator where such a decomposition makes sense, and also set Σ = sgn(D). For even spectral triples the index pairing with any unitary vanishes, but instead there is a pairing with K ( ) given by the skew-corner index [21, 31] 0 A

[D0], [P ]0 = -IndP+ P− (P+F ∗P ) (13) h i T · − where P ∼ is a projection representing the class [P ] [s(P )] K ( ) (and the formulas ∈A 0 − 0 ∈ 0 A adapt to matrices in the obvious manner). Any Callias-type operator in the sense of Definition 1 has a vanishing index since one has Dκ,H = γDκ,H∗ γ. To obtain an even analogue for the index theorem, let us therefore shift to non-self-adjoint− potentials T , which form the off-diagonal part of a doubled potential H M(M ( ), M ( )), or alternatively and more in the spirit of physical systems having an ∈ 2 A 2 N extra (so-called chiral) symmetry, the self-adjoint potential H is required to be a 2 2 matrix that is off-diagonal w.r.t. a natural extra grading by the third Pauli matrix J = diag(× 1, 1). − 0 T Definition 14 An -multiplier T M( , ) is an even Callias potential if H = ∗ A ∈ A N T 0 ∈   M(M2( ), M2( )) is a Callias potential for the spectral triple (M2( ),D 12, A 12). The associatedA evenN Callias-type operator is N ⊗ ⊗

e T+ κD0∗ Dκ,T = , κD0 T ∗  − − acting on the domain Dom(D ) Dom(D∗). 0 ⊕ 0 Let us note the differentiability of H w.r.t. D 1 is equivalent to the differentiability of ⊗ T w.r.t. D. The asymptotic invertibility of H (contained in the notion of Callias potential) 2 2 requires that there is a self-adjoint operator V M2( ) and a g > 0 such that H + V g 12. Moreover, the off-diagonal nature of J is equivalent∈ toA the (chiral) symmetry ≥

JHJ = H, J = diag(1, 1) . − −

Note that also J = J+ J and then J H J = H . ⊕ − ± ± ± − ± e Next let us show how Dκ,T naturally arises from the Callias operator Dκ,H as given in Definition 1. In fact, one readily checks

0 ıT+∗ κD0∗ 0 e ıT+ 0 0 κD0∗ 0 (Dκ,T ∗ ) Dκ,H =   = Π∗3π e − Π 3π , (14) κD0 0 0 ıT ∗ 2 Dκ,T 0 2 −    0 κD0 ıT 0   −    11 where 1 0 0 0 0 0 0 eıϕ Π = . (15) ϕ 0 eıϕ 0 0  0 0 1 0      Hence Dκ,H is block off-diagonal in an appropriate basis and one of the off-diagonal entries is e indeed Dκ,T , hence motivating Definition 14. The above identity also allows to deduce several e analytic properties of Dκ,T from corresponding statements for the odd case. Proposition 2 e e e implies that (Dκ,T )∗ = Dκ,T ∗ , while Proposition 4 shows that Dκ,T is a -Fredholm operator. The following K-theoretic result now corresponds to Proposition 6. T

Proposition 15 Let T be an even Callias potential such that the associated H M2(M( , )) satisfies H2 + V >g21 for some g > 0 and self-adjoint V M ( ). For∈ an oddA switchN 2 ∈ 2 A function G : R R as in Proposition 6, define the following self-adjoint unitary S M ( ∼) → ∈ 2 A and projection P M ( ∼) ∈ 2 A

ı π G(H) ı π G(H) 1 S = e− 2 Je 2 , P = (1 S) . 2 2 − Then the index map ∂ : K (M( , )/ ) K ( ) associated to the exact sequence (5) gives 1 1 A N A → 0 A [P ] [diag(1, 0)] = ∂ ([T ] ) . 0 − 0 0 1 Proof. This is exactly the definition of the index map. ✷ Note that JHJ = H implies − ıπG(H) ıπG(H) S = Je = e− J.

Definition 16 For an even Callias potential H = JHJ with off-diagonal entry T , the D- spectral flow is defined as a skew-corner index pairing− (13) by

Sf (T ) = [D ], [P ] R , D h 0 0i ∈ where P is as in Proposition 15.

Now the main result of this section can be stated.

Theorem 17 Let H be an even Callias potential with off-diagonal entry T and let κ0 be as in (6). Then for all κ (0, κ0) ∈ -Ind(De ) = Sf (T ). T κ,T D The l.h.s. can also be understood as the supersymmetric index of the odd self-adjoint operator κ(D 12)+ γH (in the sense of [11]), though one may prefer the formulation in terms of T due to the⊗ homomorphism property:

12 Corollary 18 If T1,T2 are even Callias potentials then T1T2 is an even Callias potential with -Ind(De ) = -Ind(De )+ -Ind(De ) T κ,T1T2 T κ,T1 T κ,T2 for small enough κ and therefore

SfD(T1T2)= SfD(T1)+SfD(T2).

T T 0 T 0 Proof. There is a standard homotopy between 1 2 and 1 and it is not difficult 0 1 0 T2 to check that differentiability and asymptotic invertibility  are satisfie d along such a path. ✷ If the Dirac operator is not invertible, it is again necessary to regularize it by adding a mass term µ. This can in principle be done by the usual doubling procedure (7), but it is more convenient to work with a unitarily equivalent representation in which the regularized Dirac operator is again off-diagonal, namely by setting ˜ 0 D0∗ µ D0∗ π+ π 0 D˜ = , D˜ 0 = − , γ˜ = ⊕ − , D˜ 0 0 D0 µ 0 π π+      − − ⊕  and diag(T , 1) 0 0 T˜ T˜ = + , H˜ = ∗ . 0 diag(T , 1) T˜ 0  −    It is then again possible to decompose H˜ M2( ∼) as H˜ = H˜+ H˜ by applyingπ ˜ = π 12 − ± ± to each matrix entry. As before the index∈ of theA Callias operators⊕ does not depend on µ unless⊗ the mass term is too large and breaks the Fredholm property. In order to avoid clumsy notations, let us from now on simply suppose without loss of generality that D is invertible with a lower bound D µ. This also leads to some minor simplification in Lemma 19 below compared to | | ≥ Lemma 11. From now on, we thus suppress all tildes on D, H, etc. Moreover, we will assume that a scaling as in Section 4 has been carried out, assuring that H 1 mod . ≥ A The proof of Theorem 17 starts out again by applying Proposition 44 which allows to e compute the index of Dκ,A as a spectral flow

e e m (D )∗ m (D )∗ -Ind(De ) = Sf( − κ,T , κ,T ) . T κ,T De m De m  κ,T   κ,T − 

A permutation Π0 defined via (15) mixing the spectral eigenspaces of γ and J leads to

m T+∗ κD0∗ 0 e m (Dκ,T )∗ T+ m 0 κD0∗ Π0∗ e Π0 =  −  . Dκ,T m κD0 0 m T ∗  −  − − −  0 κD0 T m   − −    Using J = diag(1, 1) as a matrix also in the mixed eigenspaces, this can be written as ± − e m (Dκ,T )∗ H+ + mJ+ κD0∗ 12 0 T ∗ Π0∗ e Π0 = ⊗ , H = ± . Dκ,T m κD0 12 H mJ ± T 0  −   ⊗ − − − −  ±  13 Hence by the unitary invariance of the spectral flow

e H+ mJ+ κD0∗ 12 H+ + mJ+ κD0∗ 12 -Ind(Dκ,H ) = Sf( − ⊗ , ⊗ ) . (16) T κD0 12 H + mJ κD0 12 H mJ  ⊗ − − −  ⊗ − − − − This turns out to be a better starting point for the homotopy arguments. More precisely, the operators H + J m will be deformed within the set of operators of the form f(H )+ J g(H ) π ± ± ı G(H±) ± ± ± to the operator J e 2 , along a path that conserves the Fredholm property, for details see ± Lemma 20 below.− For that purpose, one needs a Fredholm criterion for the homotopy of paths which is the next result, a modification of Lemma 11. For a smooth odd function f : R R → and a smooth function g : R C satisfying g( λ) = g(λ), both compactly supported, let us introduce the associated even→ spectral localizer −

e f(H+)+ J+ g(H+) κD0∗ 12 Lκ,f,g = ⊗ . κD0 12 f(H ) J g(H )  ⊗ − − − − − 

Due to J H J = H and the symmetry of g, one has (J g(H ))∗ = J g(H ) and therefore ± ± ± − ± e ± ± ± ± 2 by the Kato-Rellich theorem also Lκ,f,g is a self-adjoint operator with domain Dom(D0)× 2 ⊕ Dom(D0∗)× .

2 Lemma 19 Let T be an even Callias potential such that H + V 1 for some V = V ∗ M ( ). For f and g as above, associated constants ≥ ∈ 2 A 2 2 2 c1 = min f(λ) + g(λ) , λ 1 | | | | | |≥
as well as C˜3 = C˜3(f)+ C˜3(g) in terms of the constants in Lemma 10, suppose that κ is such that 1 c2 + µ2κ2 κ C˜ H [D,H] > 0 . (17) 2 1 − 3k kk k e 2 Then (Lκ,f,g) is a self-adjoint -Fredholm operator with spectrum mod bounded from below T KT by 1 c2 + µ2κ2 κC˜ H [D,H] . 2 1 − 3k kk k Proof. One computes

2 2 2 e 2 f(H+) + g(H+) + κ D0∗D0 κB∗ Lκ,f,g = | | | | 2 2 2 κB f(H ) + g(H ) + κ D0D0∗  | − | | − | 
with B = D0(f(H+)+ J+g(H+)) (f(H )+ J g(H ))D0. Noting that − − − − 0 B ∗ = [D, f(H)]+ J[D,g(H)] γ , B 0  
one deduces Le 2 c2 + κ2µ2 κ C˜ H [D,H] , κ,f,g ≥ 1 − 3k kk k and can conclude the proof by the
same arguments as in the proof of Lemma 11. ✷

14 Lemma 20 The straight-line path t [0, 1] f (λ)+ Jg (λ) = (1 t)(λ Jm)+ t(λ + Jm) ∈ 7→ t,0 t,0 − − is homotopic to the straight-line path

ı π G(λ) ı π G(λ) t [0, 1] f (λ)+ Jg (λ) = (1 t)( J)e− 2 + tJe 2 ∈ 7→ t,1 t,1 − − via s [0, 1] f (λ)+ Jg (λ) = (1 s) f (λ)+ Jg (λ) + s f (λ)+ Jg (λ) ∈ 7→ t,s t,s − t,0 t,0 t,1 t,1 in such a way that (17) computed for fs,t and gs,t is uniformly bounded
from below by a strictly
positive number for any small enough κ> 0. Moreover, for t 0, 1 the two paths s [0, 1] f (λ) 2 + g (λ) 2 are uniformly bounded away from 0. ∈{ } ∈ 7→ | t,s | | t,s | Proof. In the statement J is merely used as a symbol to join the two functions fs,t and gs,t. One expands f (λ) 2 + g (λ) 2 = (1 s)λ 2 + (1 2t)(m(1 s)+ s cos( π G(λ))) 2 + s sin( π G(λ)) 2 , | s,t | | s,t | − − − 2 2 which upon substituting the value
of G for λ 1 reduces to

| | ≥ 2 2 2 2 2 1 min fs,t(λ) + gs,t(λ) = min ((1 s)λ) + ((1 2t)m(1 s)) + s . λ 1 | | | | λ 1 − − − ≥ 2 | |≥ | |≥

2 The constant C˜3 is obviously bounded by compactness. It remains to show that fs,t(λ) + 2 | | gs,t(λ) is invertible for t 0, 1 and all λ. Invertibility can only fail at λ = 0 since that is | | ∈ { } 2 2 the only point where the first and third summand of fs,t(λ) + gs,t(λ) have a common zero. But then | | | | f (0) 2 + g (0) 2 = (1 2t)2(m(1 s)+ s)2 = (m(1 s)+ s)2 min m2, 1 , | s,t | | s,t | − − − ≥ { } which by continuity shows the last claim. Let us stress that it is the required invertibility that effectively fixes the signs of the coefficents of gt,1. ✷

π ı G(H±) Corollary 21 Set W = e 2 . Then, for κ small enough, ±

e J+W+∗ κD0∗ 12 J+W+ κD0∗ 12 -Ind(Dκ,H ) = Sf( − ⊗ , ⊗ ) . T κD0 12 J W ∗ κD0 12 J W  ⊗ − −   ⊗ − − −  Proof. Start out with (16), which with the notations of Lemma 20 can be written out with diagonal entries f0,0(H )+ J g0,0(H ) and f1,0(H )+ J g1,0(H ). Now the straight line path ± ± ± ± ± ± can be deformed due to Lemmata 19 and 20 combined with the homotopy invariance of the spectral flow under homotopies with invertible end points. Therefore, -Ind(De ) is equal to T κ,H f (H )+ J g (H ) κD∗ 1 f (H )+ J g (H ) κD∗ 1 Sf( 0,s + + 0,s + 0 ⊗ 2 , 1,s + + 1,s + 0 ⊗ 2 ) κD0 12 f0,s(H ) J g0,s(H ) κD0 12 f1,s(H ) J g1,s(H )  ⊗ − − − − −   ⊗ − − − − −  ı π G(λ) ı π G(λ) for all s [0, 1]. Use this for s = 1. As ft,1(λ) = 0 and gt,1(λ) = (1 t)e− 2 + te 2 , replacing∈ the definition of W shows the claim. − − ✷ ± 15 Lemma 22 For κ small enough,

2 e 0 κD0∗ 12 J+W+ κD0∗ 12 -Ind(Dκ,H ) = Sf( ⊗ , ⊗ 2 ) . T κD0 12 0 κD0 12 J W  ⊗   ⊗ − − −  Proof. Let us introduce the unitary

π π ı G(H) U+ 0 ı G(H±) U = e 4 = , U = e 4 . 0 U ±  − 2 Then U ∗ J = J U again due to J H J = H , and U = W . Applying the adjoint action ± ± ± ± ± ± ± ± of U to± the formula in Corollary 21 leads to − ±

2 e J+ κD0∗ 12 J+W+ κU+∗ (D0∗ 12)U -Ind(Dκ,H) = Sf( − ⊗ , ⊗ 2 − ) . T κD0 12 J κU ∗ (D0 12)U+ J W  ⊗ −   − ⊗ − − −  Now

0 U+∗ (D0∗ 12)U ⊗ − = U ∗(D 12)U = D 12 + U ∗[D 12, U] . U ∗ (D0 12)U+ 0 ⊗ ⊗ ⊗  − ⊗ 

As [D 12, U] is bounded and then multiplied by κ, a homotopy as in the proof of Theorem 8 allows⊗ to remove the commutator so that

2 e J+ κD0∗ 12 J+W+ κD0∗ 12 -Ind(Dκ,H ) = Sf( − ⊗ , ⊗ 2 ) . T κD0 12 J κD0 12 J W  ⊗ −   ⊗ − − −  Finally, one can also homotopically remove the diagonal entries J of the left entry since ± these entries are required for neither the invertibility nor the Fred∓holm property. In fact, as ıπG(H) J( 1+ e ) M2( ) is relatively compact to D, the Fredholm property of all involved operators− can readily∈ beA checked. ✷ Let us now complete the proof under the additional assumption that the spectral triple ( ,D, A ) is Lipshitz regular, which by definition means that ( , D , A ) is also a spectral N N | | triple. Likewise a self-adjoint -multiplier is said to be Lipshitz-differentiable if it is differen- tiable w.r.t. both D and D .A Once the proof is achieved for Lipshitz regular spetral triples, it will be shown in Lemma| 25| below that any spectral triple can be deformed into a Lipshitz regular one. For an even spectral triple with grading γ consider π = 1 (γ 1) and Σ = sgn(D). If ± 2 ± the triple is Lipshitz regular, then one can consider the representation ρ : given by A→N ρ(a)= π+aπ+ + π ΣaΣπ . Then ( ,D,ρ( )) is again an even spectral triple because − − N A

[D, ρ(A)] = π+Σ[ D , A]π+ + π [ D , A]Σπ | | − | | − is bounded if [ D , A] is bounded. The following lemma repackages a similar spectral flow argument from| Section| 6 of [45] that eventually connects to the index pairing:

16 Lemma 23 Let ( ,D, A ) be a Lipshitz regular even spectral triple with invertible Dirac op- N erator. Assume that S = 1 2P for a projection P = P0 + A where A with [D, A] and [ D , A] bounded and P −a projection∈ N with [D,P ] = 0 = [γ,P ], i.e. P is the∈ A scalar part of | | 0 0 0 0 P . Setting ρ(S)= 1 2(P + ρ(A)), one then has − 0 Sf(Sγ,ρ(S)γ) = Sf(κD + Sγ, κD) for κ small enough. Proof. Let us consider the two-parameter family (s, t) [0, 1] [0, 1] T = sκD + (1 t)S + tρ(S) γ . ∈ × 7→ s,t − Note that ρ(S) S holds due to [Σ,S] and hence t T0,t is
a path of Fredholm − ∈ KT ∈ KT 7→ operators. For s> 0 note that Ts,t = sκD + t(1 2P0)γ is invertible for all values of t due to ˜2 2 2 2 2 − Ts,t = κ s D + t and the invertibility of D. This implies that Ts,t is Fredholm for s> 0 since it is a relatively -compact perturbatione of T . Applying the usual resolvent identities a few T s,t times, one shows that Ts,t gives a graph-continuous homotopy in the Fredholm operators (there is a technical error in [45, Lemma 16] where ite was tacitly assumed that this homotopy is Riesz- continuous; that can actually never be the case for unbounded D). Its endpoints t 0, 1 are invertible for small enough κ since ∈{ } (κsD + Sγ)2 = s2κ2D2 + S2 κs[S, D]γ 1 sκ [D,S] − ≥ − k k and likewise for the other path. Hence Sf(Sγ,ρ(S)γ) = Sf(κD + Sγ, κD + ρ(S)γ) , and then by concatenation Sf(κD + Sγ, κD + ρ(S)γ) = Sf(κD + Sγ, κD) + Sf(κD, κD + ρ(S)γ) .

Finally, define the unitary U = π Σπ+ + π+Σπ for which one checks that UDU ∗ = D and − − − − Uρ(S)γU ∗ = ρ(S)γ and hence using invariance under unitary conjugation − Sf(κD, κD + ρ(S)γ) = Sf( κD, (κD + ρ(S)γ)) = Sf(κD, κD + ρ(S)γ) =0 , − − − concluding the proof. ✷ Proof of Theorem 17. Due to Lemma 25 below, one may assume without loss of generality that T is a Lipshitz differentiable -multiplier and the Dirac operator D is Lipshitz regular. Then Lemma 23 can be applied toA the expression in Lemma 22, by choosing S = Jeıπ G(H) = 2 2 1 0 diag(J+W+,J W ) and P0 = 0 0 . Thus − −

e
0 κD0∗ 12 S+ κD0∗ 12 -Ind(Dκ,T ) = Sf( ⊗ , ⊗ ) T κD0 12 0 κD0 12 S  ⊗   ⊗ − −  S 0 S 0 = Sf( + , + ) − 0 S 0 FS+F ∗  − −  −  = Sf(S ,FS+F ∗) −

17 where F is the phase of D0. Recalling Proposition 15, Lemma 24 below now implies e -Ind(Dκ,T ) = -IndP+ P− (P+F ∗P ) , T T · − which according to (13) and Definition 16 concludes the proof. ✷ Lemma 24 The skew-corner index can also be computed as a spectral flow via

-IndP+ P− (P+F ∗P ) = Sf(1 2P , F (1 2P+)F ∗) . T · − − − − Proof. By definition (22) of the skew-corner index,

-IndP+ P− (P+F ∗P ) = (Ker(P+F ∗P ) P ) (Ker(P FP+) P+) , T · − T − ∩ − − T − ∩ while the spectral flow on the r.h.s. can be computed from the Definition 42 using only the endpoints

Sf(1 2P , F (1 2P+)F ∗) = (P (F (1 P+)F ∗)) ((1 P ) (FP+F ∗)) , − − − T − ∩ − − T − − ∩ because P FP+F ∗ follows from [P, Σ] which holds in any spectral triple. Since − T T F is unitary− here, one∈K can check that ∈K

Ker(P+F ∗P ) P = (F (1 P+)F ∗) P − ∩ − − ∩ − and Ker(P FP+) P+ = (F ∗(1 P )F ) P+ = F ∗ ((1 P ) (FP+F ∗) F, − ∩ − − ∩ − − ∩ such that taking traces gives the desired equality. ✷ To complete the proof of Theorem 17, it remains to show that the Lipshitz regularity can be assumed without loss of generality. The gist of the argument, going back to a trick by Kaad 2 r [28, Proposition 5.1], is that by replacing the Dirac operator D with D(1 + D )− 2 for some 0

1 [f (D),H]ψ = (∂ f˜ (z)) [(D z)− ,H]ψ dz dz R z R,N − ∧ ZG 1 1 = (∂ f˜ (z))(D z)− [D,H](D z)− ψ dz dz − z R,N − − ∧ ZG and since f (D) and H are bounded the latter expression also holds for all ψ . Since R ∈ H − − 1 1 2 ρ 1 N 2+N (∂ f˜ (z))(D z)− [D,H](D z)− c C˜ (1 + x ) 2 m(z) − [D,H] z R,N − − ≤ n R,N+1 |ℑ | k k

˜ with CR,N+1 bounded uniformly in R, the integral is also bounded uniformly in R when sub- stituting N 2 and hence supR 1 [fR(D),H] < . For ψ one also has Hψ ≥ ≥ k k ∞ ∈ E ∈ E and the spectral representation shows Hf(D) = H limR fR(D)ψ = limR HfR(D)ψ and →∞ →∞ f(D)H = limR fR(D)Hψ. Hence [f(D),H]ψ = limR [fR(D),H]ψ for all ψ which →∞ →∞ 1 ∈ E implies that the commutator extends to a bounded operator. Finally (f(D)+ ı)− A for 1 ∈KT H again follows from the functional calculus since (f(D)+ ı)− can be expressed as a ∈ A 1 norm-convergent integral of terms (D + z)− A . ✷ ∈KT Assuming that the Dirac operator D is invertible and let 0 <ρ< 1 then this Lemma (ρ) 2 ρ (ρ) (ρ) implies that for D = D(1 + D )− 2 one has spectral triples ( ,D , A ) and ( , D , A ) 2 ρ N N (for the latter note that λ λ (1+λ )− 2 may be replaced by a smooth function as 0 / σ(D)). 7→ | | ∈ Moreover, any bounded D-differentiable multiplier is D(ρ)- and D(ρ) -differentiable. Hence one can replace the spectral triple with a Lipshitz regular one for which H is Lipshitz differentiable.

(ρ) e Lemma 26 If T is a bounded Callias potential and (D )κ,T the even Callias-type operator (ρ) (ρ) e obtained from pairing with the Dirac operator D , then -Ind((D )κ,T ) does not depend on ρ [ 1 , 1] for small enough κ. T ∈ 2 (ρ) ρ Proof. Due to the inequality [D , T ] D− [D, T ] one can choose κ so small that ≤ k k k k (D(ρ))e is -Fredholm for all ρ [ 1 , 1] and then the result follows from homotopy invariance κ,T T ∈ 2 since the path r [ 1 , 1] D(r) is graph-continuous and T a bounded perturbation. ✷ ∈ 2 7→ 6 Callias-type operators with unbounded potentials

This section introduces a class of unbounded Callias potentials for which it is possible to reduce the computation of the index to the bounded case. This then allows to state and prove unbounded versions of Theorems 8 and 17.

Definition 27 An unbounded -multiplier T is a closed operator affiliated to in such a way A N that the bounded transform 1 F (T ) = T (1 + T ∗T )− 2

1 is a multiplier in M( , ) and (1 + T ∗T )− 2 is a dense subset of . A N A A 19 If and act non-degenerately on , then M( , ) is the usual multiplier algebra and T A N H A N is affiliated to in the C∗-algebraic sense of Woronowicz [50], however, we do not ask for that since one mayA want to pass to proper non-dense subalgebras of later. Left multiplication by A an unbounded multiplier A T A gives a closed operator from a dense subset of to . Since 7→ A A functional calculus factors through the bounded transform each C0-function of a self-adjoint multiplier lies in M( , ). A N Definition 28 A self-adjoint operator H affiliated to is said to be D-differentiable (with N respect to the self-adjoint operator D) if there is a core for D such that the following holds for each µ R 0 : E ∈ \{ } 1 1 (i) (H ıµ)− Dom(D) Dom(H) and D(H ıµ)− Dom(H). − E ⊂ ∩ − E ⊂ 1 (ii) The operator [D,H](H ıµ)− extends from to a bounded operator in . − E N If the two conditions hold for some core , then they also hold for = Dom(D)[29, E 1 E Proposition 7.3]. The dense subspace =(H + ı)− Dom(D) is dense in Dom(D) Dom(H) DH ∩ w.r.t. the graph norm ψ D,H = ψ + Dψ + Hψ . In particular, the commutator [D,H] is also densely definedk andk symmetrick k onk k. Ak boundedk operator H is differentiable if and DH only if it preserves Dom(D) and [D,H] extends to a bounded operator. The above notion of differentiability is chosen precisely such that the self-adjointness criteria from [29] imply well-definedness of the following: Proposition 29 For a differentiable -multiplier H introduce the Callias-type operators A D = κD + ıH , D∗ = κD ıH , κ,H κ,H − on the domain Dom(D) Dom(H) as well as ∩ 0 D L = κ,H∗ , κ,H D 0  κ,H  2 on the domain (Dom(D) Dom(H))× . Then Lκ,H is self-adjoint and therefore Dκ,H and Dκ,H∗ are adjoints to each other.∩ Since D and H are affiliated to one can check from the domains that the Callias-type N operators and L are affiliated to and M ( ) respectively (again, a closed operator T is κ,H N 2 N affiliated if it commutes with each unitary U ′). ∈N Definition 30 An unbounded self-adjoint -multiplier H is asymptotically invertible if there is a positive self-adjoint element V suchA that H2 +V has a bounded inverse (which then lies ∈A in M( , )). A self-adjoint D-differentiable -multiplier H that is asymptotically invertible will beA calledN an (unbounded) Callias potential.A The main result is that the index theorem as stated in Theorem 8 extends to unbounded Callias potentials. For the formulation, let us note that Proposition 6 remains valid if [χ(H + < 0)]0 K0(M( , )/ ) is replaced by [χ(F (H)+ < 0)]0. In particular, the index Apairing [D∈], [eıπ(G+1)(AH)N] AR is well-defined. A h 1i ∈ 20 Theorem 31 Let H be a (possibly unbounded) Callias potential for the semifinite spectral triple ( ,D, A ). Then there is a κ > 0 such that for all κ (0, κ ], N 0 ∈ 0 -Ind(D ) = [D], [eıπ(G+1)(H)] T κ,H h 1i and G any switch function chosen as in Proposition 6. More precisely, if there exists some g > 0 and positive self-adjoint V such that H2 + V >g21, one can choose ∈A 2 1 1 g κ0 = [D,H](H + ı)− − . k k 1+ g2

1 Such a V exists for all g > 0 if and only if the resolvent ofpH is -compact, i.e. (H +µı)− , A ∈A in which case all κ (0, ) are allowed and the resolvent of L is -compact. ∈ ∞ κ T Let us briefly discuss the last mentioned situation for the classical case of a Riemannian manifold X. Then = C0(X, ( )) and H is given by an operator-valued bounded function x X H (A). If nowKx H H grows at infinity, one can indeed choose g arbitrarily ∈ 7→ x ∈ B H 7→ x large and still find V such that H2 + V >g21. This is the situation considered in the work of Robbin and Salamon [43]. Checking the Fredholm property is more difficult in the unbounded case, since there are domain issues and also the commutator [D,H] is not bounded, but only relatively H-bounded:

Proposition 32 For a Callias potential H there exists some κ0 > 0 such that Dκ,H is - Fredholm for all 0 < κ κ . In particular, κ can be chosen as in Theorem 31. T ≤ 0 0 1 Proof. As recalled above, H = (H + ı)− Dom(D) is a core for both Dκ,H and Dκ,H∗ and contained in the domain of [D,HD ]. For ψ let us consider the quadratic form ∈DH (κD + ıH)ψ, (κD + ıH)ψ Hψ,Hψ + κ ψ, ı[H,D]ψ , h i≥h i h i which is estimated as in the proof of [29, Lemma 7.5]

κ2b2 ψ, ıκ[H,D]ψ (s + ) ψ, ψ + s Hψ,Hψ ±h i ≤ 4s h i h i for any 0 g21 for some positive self-adjoint V . g2 1 ∈ A Fixing s = 2 and setting b = [D,H](H + ı)− one checks that 2(1+g ) k k s κ2 H2 + V >g21 + b2 1 ≥ 1 s 4s(1 s)  − −  g2 for all 0 κ κ0 = (which was obtained by maximizing the r.h.s. over all 0

D ψ, D ψ + (1 s) ψ,Vψ (1 s)( Hψ,Hψ + ψ,Vψ g2 ψ, ψ ) > 0 , h κ,H κ,H i − h i ≥ − h i h i − h i 21 2 and hence the strict positivity of H +V g1 implies that D∗ D +(1 s)V 1 is invertible. − κ,H κ,H − ⊗ 2 A similar argument also yields D D∗ + (1 s)V > 0. κ,H κ,H − Also V is -compact relative to L and thus L2 , since T κ,H κ,H 1 1 1 1 V˜ (L + ı)− = V˜ (D σ + ı)− + V˜ (D σ + ı)− (H σ )(L + ı)− , κ,H ⊗ 1 ⊗ 1 ⊗ 1 κ,H 1 1 1 where V˜ (D σ1 + ı)− M2( ) and (H σ1)(Lκ,H + ı)− since (Lκ,H + ı)− is bounded as ⊗ ∈ KT ⊗ an operator from to Dom(H). Since is an ideal this completes the proof. ✷ H KT The last statement in Theorem 31 follows immediately from the above, since one can take for V a spectral function of H2 as in Proposition 5, respectively for each function f C (R) ∈ c one can find g so large that the proof implies f(H) and f(Lκ) M2( ). Hence the same ∈A ∈ KT holds for all C0-functions.

The dependence of Dκ,H on κ is still graph-continuous and so the index again does not depend on κ as long as it is small enough. Let us now proceed to prove that the bounded transform maps unbounded Callias potentials to bounded ones with the same index. This fact ıπ(G+1)(H) immediately concludes the proof of Theorem 31 since one can obviously write [e ]1 = ıπ(G˜+1)(F (H)) [e ]1 for another switch function G˜ and then apply Theorem 8. The technically most difficult part of the proof is to verify the differentiability of F (H). For decaying functions the differentiability of spectral functions of H again follows from the Helffer-Sj¨ostrand calculus:

Lemma 33 If H is a self-adjoint differentiable multiplier and f ρ(R) for some ρ< 0, then f(H) preserves the domain of D and [D, f(H)] extends from Dom(∈ SD) to a bounded operator.

1 Proof. Since H is D-differentiable one has (H z)− ψ Dom(D) for any ψ . Let us − ∈ ∈ E estimate

1 1 1 1 D(H z)− ψ (H z)− Dψ + (H z)− [H,D](H z)− ψ k − k≤ k − k k − − k 1 1 1 1 mz − Dψ + m(z) − (1 + ı + z m(z) − ) [H,D](H ı)− ψ , ≤ |ℑ | k k |ℑ | | ||ℑ | k − kk k due to 1 1 1 1 [H,D](H z)− = [H,D](H + ı)− + [H,D](H + ı)− (ı + z)(H z)− − − Using an extension f˜ of f satisfying (2) with N 2 and ρ < 0, the integral representation ≤ (3) therefore also converges in the graph norm of D and defines a bounded operator f(H): Dom(D) Dom(D). Using →

1 1 [D, f(H)] = (∂ f˜(z))(H z)− [D,H](H z)− dz dz , − z − − ∧ ZG the boundedness of the commutator follows similarly. ✷ The following result is morally similar to bounds obtained in [19], but the proof presented here avoids the use of double operator integrals.

Lemma 34 If H an unbounded D-differentiable -multiplier, the bounded transform F (H) is A 1 also D-differentiable with [D, F (H)] [D,H](H + ı)− . k k≤k k 22 Proof. One needs to check that F (H) maps a core of D into Dom(D) and extends from there to a bounded operator. As recalled below Definition 28, H is also differentiable with the core 1 = Dom(D) and H =(H + ı)− Dom(D) is a core of D. E D 1 1 Applying Lemma 33 (with ρ = 1) to the smooth function (F (H)+ ıµ)− (H + ı)− implies 1 − (F (H)+ ıµ)− H Dom(D). It remains to show that the commutator [D, F (H)] extends from to a boundedD ⊂ operator. For the commutator one has from the integral representation DH of fractional powers (compare [18, Proposition 2.10]) an integral formula:

1 ∞ 1 2 1 2 1 [D, F (H)]ψ = (1 + H + λ)− (1 + λ)[D,H](1 + H + λ)− π 0 √λ Z  2 1 2 1 H(1 + H + λ)− [D,H]H(1 + H + λ)− ψ dλ . −  It converges absolutely in the norm of for each ψ Dom(D). For the bounded self-adjoint 1 H1 ∈ operator T = ı(H ı)− [D,H](H + ı)− , one has − 2 1 2 2 1 T [D,H](H + ı)− (1 + H )− , ≤ and hence 1 2 1 T T [D,H](H + ı)− (1 + H )− 2 ≤ | | ≤ by operator monotonicity of the square root. Therefore

1 2 1 ψ, ı[D,H]ψ = (H + ı)ψ, T (H + ı)ψ [D,H](H + ı)− ψ, (1 + H ) 2 ψ h i h i ≤ h i holds for all ψ . On the same domain, one then has ∈DH 1 [D,H](H + ı)− ∞ ψ, ı[D, F (H)]ψ k k ψ, f (H)ψ dλ h i ≤ π h λ i Z0 with the positive continuous function

1 2 1 2 1 2 1 fλ(H) = (1 + H + λ)− (1 + λ)(1 + H ) 2 (1 + H + λ)− √λ  2 1 2 1 2 1 + H(1 + H + λ)− (1 + H ) 2 H(1 + H + λ)− √1+ H2  = . √λ(1 + H2 + λ)

Using the spectral measure µψ w.r.t. H, the integral becomes

2 ∞ ∞ √1+ x ψ, fλ(H)ψ dλ = dλ dµψ(x) = π dµψ(x) = π , 2 0 h i R 0 √λ(1 + x + λ) R Z Z Z Z which shows that the commutator defines a bounded quadratic form and hence has a bounded 1 extension with [D, F (H)] [D,H](H + ı)− . ✷ k k≤k k

23 Proposition 35 If H is an unbounded Callias potential, then the bounded transform F (H) is also a Callias potential. Furthermore there exists a constant κ > 0 such that for all 0 < κ κ 0 ≤ 0 -Ind(D ) = -Ind(D ) . T κ,F (H) T κ,H Proof. By assumption there is some self-adjoint V and g > 0 such that H2 +V >g21 > 0 ∈A g2 holds. Let χ be a smooth non-increasing function which is equal to 1 on [0, 4 ) and vanishes outside [0,g2). We note that the proof of Proposition 5 can be adapted for unbounded - 2 1 2 1 A multipliers since the resolvents (H + V z)− and (H z)− are readily checked to lie in the − − norm-closed algebra M( , ). Hence one concludes V˜ = χ(H2) lies in . A N A The spectral mapping property implies F (H)2+V˜ > g˜2 for some positive constantg ˜ > 0 and therefore F (H) is asymptotically invertible. Lemma 34 also shows that F (H) is differentiable. To show the equality of indices it is enough to prove that the joint potential H ( F (H)) ⊕ − has index 0, due to the additivity of the index. Consider the modified potential

H m V˜ Hˆ = , m m V˜ F (H)  −  which we check to be a differentiable self-adjoint multiplier w.r.t. D 1 . That statement is ⊗ 2 clear for m = 0 and that Hˆm is again a multiplier follows easily from perturbation formulas for the bounded transform (such as [18, Lemma 2.7]). Let = Dom(D 12), then due to the domain of self-adjointness one has E ⊗

1 (Hˆ ı)− (Dom(D) Dom(H)) Dom(D) = Dom(D 1 ) Dom(Hˆ ) , m − E ⊂ ∩ ⊕ ⊗ 2 ∩ m and using the resolvent identity to compare with Hˆ0, one also has ˜ 1 1 1 0 mV 1 D(Hˆ ıµ)− D(Hˆ ıµ)− + D(Hˆ ıµ)− (Hˆ ıµ)− m − E ⊂ 0 − E 0 − mV˜ 0 m − E ⊂E   1 1 since V˜ preserves Dom(D) and (Hˆ ıµ)− preserves . Finally, since [D 1 , Hˆ ](Hˆ ıµ)− m − E ⊗ 2 0 0 − and [D, V˜ ] extend to bounded operators, another application of the resolvent identity implies 1 that [D 1 , Hˆ ](Hˆ ıµ)− extends to a bounded operator as well. ⊗ 2 m m − Since Hˆ is asymptotically invertible and Hˆ Hˆ M ( ), we have shown that Hˆ is a 0 m − 0 ∈ 2 A m Callias potential for any m 0. From the above one also sees ≥ 1 max [D, Hˆm](Hˆm + ı)− < m [0,1] ∞ ∈ such that the proof of Proposition 32 implies that there is some κ0 such that Dκ,Hˆm is - Fredholm for all 0 < κ κ and all 0 m 1. T ≤ 0 ≤ ≤ For any m 0 one can check that Hˆ is invertible from the square ≥ m 2 2 ˜ 2 ˜ 2 H + m V m(H F (H))V (Hˆm) = − , m(H F (H))V˜ F (H)2 + m2V˜ 2  −  24 which can be diagonalized in the spectral representation. The off-diagonal part mV˜ does not affect the index since it is relatively -compact w.r.t. D. Fixing any m > 0 one has T -Ind(Dκ,Hˆ ) = 0 for small enough κ since D∗ ˆ Dκ,Hˆ and similarly Dκ,Hˆ D∗ ˆ become T m κ,Hm m m κ,Hm ˆ 2 2 invertible. More precisely, this follows from the proof of Proposition 32, since Hm >cm1 allows one to choose V = 0 there. Since the index does not depend on m and κ,

0 = -Ind(Dκ,Hˆ ) = -Ind(Dκ,H ( F (H)) = -Ind(Dκ,H ) -Ind(Dκ,F (H)) , T m T ⊕ − T − T concluding the proof. ✷ Finally let us turn to the even unbounded case.

0 T Definition 36 An unbounded -multiplier T is an even Callias potential if H = ∗ is A T 0 an unbounded Callias potential for the spectral triple (M ( ),D 1 , A 1 ). The associated 2 N ⊗ 2 ⊗ 2 even Callias-type operator is defined as

e T+ κD0∗ Dκ,T = κD0 T ∗  − − on the domain (Dom(D0) Dom(T+)) (Dom(D0∗) Dom(T ∗ )). ∩ ⊕ ∩ − e The unitary equivalence (14) and the self-adjointness of Lκ,H again implies that Dκ,T is a e e closed affiliated operator on the stated domain and (Dκ,T )∗ = Dκ,T ∗ . Now the generalization of Theorem 17 to unbounded potentials reads as follows:

0 T Theorem 37 Let T be an even Callias potential with H = ∗ . Then T 0   -Ind(De ) = [D ], [ 1 (1 JeıπG(H))] T κ,T h 0 2 − 0i for each κ and G as in Theorem 31.

The proof is immediate from the bounded version and the next result.

e Proposition 38 If T is an unbounded even Callias potential, there exists κ0 such that Dκ,T is e -Fredholm for all 0 < κ κ0 and -Ind(Dκ,T ) does not depend on 0 < κ κ0. Also, the T ≤ T 1 ≤ bounded transform F (T )= T (1+ T ∗T )− 2 is a Callias-admissible potential with the same index

-Ind(De ) = -Ind(De ) , T κ,T T κ,F (T ) for small enough κ.

Proof. Due to the block decomposition (14), De is -Fredholm if L is -Fredholm. Hence κ,T T κ,H T the existence of κ0 follows from the odd case (Proposition 32). That F (T ) is again a Callias potential is also clear from considerations of the odd case and the fact that we defined differentiability using a doubling construction. The doubled

25 Hamiltonian Hˆm from the proof of Proposition 35 is unitarily equivalent to an off-diagonal matrix 0 Tˆ Hˆ m∗ , m ∼ Tˆ 0  m  with the operator T mχ(T ∗T ) Tˆm = − , mχ(T T ∗) F (T ∗)   which is therefore an even Callias-potential and invertible for any m> 0. One argues as in the odd case that

e e e e ∗ ∗ 0 = -Ind(Dκ,Tˆ ) = -Ind(Dκ,T ( F (T )) = -Ind(Dκ,T ) + -Ind(Dκ, F (T )) T m T ⊕ − T T − for small enough κ. Replacing a Callias potential T by λT with λ = 1 again gives a Callias potential and since 1 | | S is connected, -Ind(Dκ,T )= -Ind(Dκ, T ) by homotopy. Conjugating the potential gives a T T − factor of 1 and thus − e e e -Ind(Dκ,T ) =( 1) -Ind(Dκ, T ∗ ) = -Ind(Dκ,F (T )) , T − T − T concluding the proof. ✷

7 Comparison with the unbounded Kasparov product

This section outlines how the Callias index arises as an unbounded representative of a KK- group. For simplicity, it will be assumed that the Dirac operator of the semifinite spectral triple (A , ,D) is invertible (see Section 4 on how to achieve this). N

Definition 39 Let , be separable C∗-algebras. An unbounded Kasparov cycle (A ,E,D) A B is a tuple consisting of a countably generated - -Hilbert-C∗-module E together with an odd regular self-adjoint unbounded operator D : Dom(A DB ) E E such that ⊂ → (i) A is a dense -subalgebra such that each a A preserves Dom(D) and the graded commutator⊂A [D, a] extends∗ to a bounded operator∈ on E.

1 (ii) The products a(D ı)− are -compact for all a . − B ∈A The remainder of the section considers a semifinite spectral triple (A , ,D) with separable N C∗-algebra = A . Such a spectral triple naturally defines an unbounded or bounded Kasparov A cycle if and only if is σ-unital. Since that condition does not generally hold in the semifinite T setting one passes [K31, 17] to the norm-closed sub-algebra generated by all -algebraic T combinations of elements C⊂K ∗

A[F (D), B] , [F (D), B] , F (D)A[F (D), B] , ϕ(D)A (18)

26 2 1 with A, B , F (D) = D(1 + D )− 2 and ϕ C (R). Then ( , , F (D)) KK( , ) ∈ A ∈ 0 A C ∈ A C defines a bounded ungraded Kasparov cycle and is the smallest C∗-algebra for which this is the case. An important technical point is that theC unbounded Kasparov cycle is constructed precisely from D and A since only then the Callias operators can be interpreted as unbounded Kasparov products. In most cases (A , , F (D)) is already the bounded transform of a well- defined Kasparov cycle: C

ıDt ıDt Proposition 40 Set α (A) = e Ae− and let be the smallest α-invariant C∗- t Aα ⊂ N algebra containing . If α acts non-degenerately on , then (A C1, C2, Dσ2) is an unbounded KasparovA cycle.A H ⊗ C ⊗

Proof. The only non-trivial point in Definition 39 is the regular self-adjointness of D on or, equivalently, that D is affiliated to . To prove the latter we derive a better characterizationC of the algebra . C C The affiliation of D to implies that α defines a weak- -continuous action on . Since A N t ∗ N any A is differentiable, the identity αt(A) A = 0 ıαs([D, A]) ds (with convergence in the weak- ∈-topology) implies that the orbit under α−is norm-continuous, hence is still separable R α and α∗extends to a strongly continuous R-action on . Now define toA be the separable Aα Cα C∗-subalgebra of spanned by the elements (18), but with replaced by α. KT A A One can next form the crossed product algebra α ⋊α R for which we now recall some A ⋊ R R universal properties [44]: There are embeddings ı α : α M( α α ) and ıR : Cc( ) ⋊ R ⋊ R A A → A R → M( α α ) such that α α is generated by the linear span of ı α ( α)ıR(C0( )). Further- A more,A given a non-degenerateA covariant representation (π, U) on a HilbertA space , there is a ⋊ R H non-degenerate representation (π U): α α ( ) for which (π U) ı α = π and × A → B H × ◦ A ((π U) ıR)(ϕ) = ϕ(D) for D the self-adjoint generator of U. Also ıR extends uniquely to × ◦ Cb(R) with the same property. The identical map π : ( ) and U : R exp(ıD ) trivially form a covariant Aα → B H → · representation (π, U). By definition one has (π U)( α ⋊α R) α since the latter contains the × A ⋊⊂R C generators π(a)ϕ(D). Moreover one has equality (π U)( α α )= α, since [ıR(F ), ı α (a)] A ⋊ R holds for any smooth switch function like×F (seeAe.g. [35]). C ∈ Aα α From the above one concludes that a dense subset of is given by all elements of the form Cα N N ıDtk ıDtk ϕk(D)αtk (Ak) = ϕk(D)e Ake− k k X=1 X=1 with ϕ1, ..., ϕN Cc(R) and A1, ..., AN . Therefore D is densely defined on α and clearly 1 ∈ ∈A C (D+ı)− α α is also a norm-dense subset. Hence D is affiliated to α and regular self-adjoint in the Hilbert-moduleC ⊂ C sense. C

To complete the proof let us show that α = , for which it is only necessary to verify ıDt C C ϕ(D)Ae− for all ϕ Cc(R), t R, A . Choose approximate units (Bn)n N for ∈ C ∈ ∈ ∈ A ∈ A and (Ψm)m N for C0(R). Then ∈ ıDt ıDt ıDt ϕ(D)Ae− = lim ϕ(D)ABne− = lim lim ϕ(D)ABne− Ψm(D) , n n m →∞ →∞ →∞ 27 converges in norm, since the Bn and Ψm(D) are also approximate units for (π U)( α ⋊α R). That shows that is norm-dense in . × A ✷ C Cα Let us now compare our main result for unbounded Callias operators to approaches using the unbounded Kasparov product (specifically [47, 30] which treat the classical case). Semifinite spectral triples often arise in noncommutative geometry from an unbounded Kasparov cycle 1 (A C1, E C2, Dσ2) KK− ( , ) where is a separable C∗-algebra that carries a densely B defined⊗ faithful⊗ lower semi-continuous∈ A B trace B, E a countably generated right -module and T B B D a regular self-adjoint unbounded operator on E . Both A and act naturally on the B B Hilbert space obtained by completing the submodule of E for which ( e, e EB ) < in B B the obvious norm.H In that situation one obtains a semifinite spectral tripleT h ( ,D,i A )∞ with N = ′′ to which extends as a normal semifinite faithful trace . All examples described B Nin SectionB 8 belowT can be written in that form for some naturalT algebra . If acts non- degenerately, it is completely general since, as shown above, for any semifiniteB spectralA triple the minimal choice = = E is available. B C B If the potential H is a self-adjoint unbounded -multiplier with resolvent in , then it de- A 1 A fines an odd unbounded Kasparov cycle (C, C1, Hσ2) KK (C, ). The Callias operator A ⊗ ∈ A C Hσ1 + Dσ2 on E should then represent the product class [Hσ2] C1 [Dσ1] KK( , ) B ⊗A⊗ ∈ B ≃ K0( ), given some compatibility conditions between H and D which are similar to the differen- tiabilityB that we impose here (a possible set of conditions may be derived from e.g. [37, Theorem 7.4]). Composing the product class with the homomorphism : K0( ) C computes the T∗ B → -index, while applying to the product of the bounded transforms [F (Hσ1)] C1 [F (Dσ2)] T T∗ ıπF (H) ⊗A⊗ recovers the index pairing [e ]1, [D] (see [17]). Since the bounded and unbounded picture of KK-theory are isomorphich [28, 48], onei concludes that Theorem 31 holds in that special case. For the case of a spectral triple over ( ), a more detailed proof can also be found in [8], though compared to our notations the regularityB H assumptions are formulated in terms of ıπF (H) the Cayley transform of H, which is another unitary representing the class [e ]1. The even case can be handled similarly with the product represented by the self-adjoint operator D +γH (compare [11]). The KK-theoretic approach has certain advantages, in particular, the class of the Cal- lias operator may carry finer topological invariants besides the numerical index and also the associativity of the Kasparov product can then be further applied to prove more specialized formulas for the index. A severe limitation is that apparently the potential H must always be 1 unbounded with -compact resolvent, i.e. (H +ı)− which is a stronger condition than our asymptotic invertibility.A That is necessary for the obvious∈A cycle to define a class KK1(C, ) in unbounded KK-theory, though there might be more complicated constructions to handleA a bounded potential that is invertible up to . In the classical commutative case of potentials on a manifold, one can already treat the largerA class of asymptotically invertible potentials with pointwise compact resolvents by amplifying them with a growing function on the underlying manifold (as it is done in [47, 30]).

28 8 Examples

8.1 The classical Callias index theorem The first example, which was already discussed briefly above is the classical geometric situ- ation with Callias-type operators on a Riemannian manifold X. We will consider a possi- bly infinite dimensional vector bundle E over X with typical fiber isomorphic to a Hilbert space 0. For D a weakly elliptic first-order differential operator we have a spectral triple 2 H K (L (X, 0)),D,Cc∞(X) ( 0) . If H = (Hx)x X is a self-adjoint fibered operator that B H ⊗ H ∈ is differentiable (with derivative ( Hx)x X bounded relative to H) and invertible modulo ∇
∈ C (X) K( ) then the technical conditions of our index theorem can all be verified and 0 ⊗ H0 it reproduces the result of Kaad and Lesch [30] which used the Kasparov product. If 0 is finite-dimensional and H invertible outside a compact set K, there are other ways to computeH the index, for example, the original index theorem by Callias [14] on Rn expresses the index as an integral over the boundary of a large enough sphere and more generally the index theorem by Anghel [2] gives a similar generalization to manifolds with warped ends. In both of these situations one only needs to know the potential H on a lower-dimensional submanifold that envelops all singular points of the potential. Explicitly, in the case X = Rn, n odd, with D the Euclidean Dirac operator one has

(n 1) Ind(κD + H) Tr((QdQ)∧ − ) (19) ∼ Z∂BR(0) 1 for Q = H H − invertible outside B (0). In contrast, our index formula with standard index | | R computations would give a volume integral:

n Ind(κD + H) Tr((U ∗dU)∧ ) , (20) ∼ Rn Z with U = eıπ(G+1)(H) or some other representative with sufficiently fast decaying derivatives. There is a simple K-theoretical relation between those formulations. Let us assume that H is not only invertible up to = C (X), but already up to ′ = C (K) for K X a compact A 0 A 0 ⊂ set. Since ′ is an ideal in , one then also has H M( ′, ) and so the exponential map in A A ∈ A N K-theory gives an element of K1(C0(K)). One can do even better since there is a commutative diagram q 0 C0(K) Cb(X) Cb(X)/C0(K) 0

ρ ρ r q 0 C0(K◦) C(K) C(∂K) 0 where r is the restriction and ρ acts as the identity on C0(K). Hence the naturality of the exponential map implies

∂ [χ(H + < 0)] = ∂ [χ(r(H) < 0)] . 0 A 0 0 0

29 This shows that, as expected, the spectral flow SfD(H) naturally depends only on the class that r(H) = H ∂K defines in K0(C(∂K)). For a more detailed examination of those K-theoretical aspects of| Callias-type operators we refer to the work of Bunke [11]. The equality between the expressions (20) and (19) can be derived from the boundary maps of K-homology [12], which are dual to the K-theoretic connecting maps, however, in a more general setting constructing an explicit representative for the odd K-homology class on C(∂K) that computes the same pairing as the spectral flow can involve subtle geometric and analytic issues. In that sense our main index theorem on Callias-type operators contains only part of the information of the original theorem by Callias. It is an interesting problem to find additional analytic and algebraic data from which one can canonically construct a spectral triple for the boundary class also in the noncommutative case. Partial solutions may be provided by the construction of relative spectral triples [23]. Another natural question is whether it is possible to reduce the index computation to a compact hypersurface also in the case of an infinite-dimensional vector bundle. In general the answer is negative, though, since due to Kuipers’ theorem any potential H that is invertible on ∂K may be homotopic to another potential H˜ that has the same index but is flat in the sense that H˜ = 1 2P with P ( ) some fixed projection. Hence the non-trivial index |∂K − 0 0 ∈ B H0 is invisible on ∂K. One way to evade such counterexamples is to consider a more restricted class of Callias-type operators, for example, those of the form H = H0 + V with H0 a fixed self-adjoint operator with compact resolvent and V : X ( ) a norm-continuous family → B H0 of self-adjoint operators. In that case again one can compute the index from a K1-class over C∗(H0)+ C0(K) K obtained through the exponential map from a K0-class over C∗(H0) ∂K + C(∂K) K. It must⊗ therefore be possible to compute the index using only the potential| on ⊗ the boundary, though we are not aware of any known formulas.

Finally, let us note an interpretation as spectral flow under the additional assumption Hx is invertible for all but finitely many isolated points x , ..., x X. In that case (19) decomposes 0 N ∈ into a sum of contributions of small spheres around each xi, each of which is individually integer-valued. This is analogous to the way that usual spectral flow counts the number of eigenvalue crossings. In physics one uses such expressions to assign topological charges to stable band-touching points and the spectral flow is therefore the total charge. A particular set-up of this type is considered in [34] where the potential H is a linear combination of Clifford algebra generators with essentially commuting coefficients. If the critical points of H are not isolated or 0 is infinite-dimensional the spectral flow is more difficult to interpret. For even dimensionsH our analogue of Section 5 can be used to define topological charges for potentials satisfing a chiral symmetry of the type JHJ = H. − 8.2 The Boutet de Monvel index theorem The framework of Section 8.1 can be transposed to an even 2n-dimensional complete Rie- mannian manifold X. Let then E be a possibly infinite dimensional vector bundle over X with typical fiber Hilbert space . Furthermore there is supposed to exist an involution H0 γ = (γx)x X : E E and a weakly elliptic first-order differential operator D satisfying ∈ →

30 2 γDγ = D and which results in an even spectral triple (L (X, )),D,C∞(X) K( ) . − B H0 c ⊗ H0 The even Callias potential is then given by a self-adjoint multiplication operator H =(Hx)x X ∈
on E E satisfying γHγ = H, JHJ = H with J = diag(1 , 1 ) and being invertible ⊕ − E − E modulo C0(X) K( 0). In the grading of γ one then has H = H+ H and both H are − ± off-diagonal in the⊗ gradingH of J with lower left entry T . ⊕ ± The set-up of the index theorem of Boutet de Monvel [10, 27] assumes that X is a strongly pseudoconvex domain in Cn equipped with the Bergmann metric, E is finite-dimensional and given by the differential forms on X of type (n, p) graded by the parity of p and then D is the Dolbeaut operator. Furthermore H+ = H is supposed to extend smoothly to the boundary − ∂X. A related setting is obtained by simply choosing X = R2n with the euclidean metric and D the associated Dirac operator on E = R2n CN where CN is the representation space of the Clifford algebra with N generators. Then the⊗ index theorem states [10, 27, 11]

e (2n 1) -Ind(D ) Tr(J(QdQ)∧ − ) , T κ,T ∼ Z∂X 1 2n where again Q = H H − and in the case X = R one replaces ∂X by ∂B (0) with R | | R sufficiently large. The r.h.s. is an odd Chern number in the representation given, e.g. [13]. e Theorems 17 and 37 connect -Ind(Dκ,T ) to an integral over X rather than its boundary, and hence do not directly provideT the right side. However, in the case of finite-dimensional fibers one can again argue as in Section 8.1. To determine conditions under which such a formula holds also with infinite dimensional fibers remains an open problem.

8.3 A generalized Robbin-Salamon theorem As a first non-commutative example we consider a generalized Robbin-Salamon theorem. Let be a separable C∗-algebra with a densely defined faithful lower semicontinuous trace τ and a C strongly continuous automorphic R-action α that leaves τ invariant (everything below directly n transposes to R -actions). Then the crossed product algebra = ⋊α R has an induced dual trace and a dual R-actionα ˆ. Let further be the vonA NeumannC algebra generated T M by in the semicyclic GNS representation for and the corresponding representation space.C The regular representation π of acts onT the HilbertH space L2(R, ) such that α is generated by right translation, i.e. π αA = Ad(U ) π with U = et∂ . ThenH extends to a ◦ t t ◦ t T semifinite normal faithful trace on the von Neumann algebra = π( )′′. Let A be the dense -subalgebra of elements A that can be written in the formN π(A)=A π(f(t))U dt with a ∗ ∈A R t function f : R Dom(τ) that is smooth and rapidly decaying in the norm A + τ( A ) R and D = ı∂ →then ( ,D,⊂AA ) is a semifinite spectral triple (compare e.g. [17]).k k If H |is| a differentiable− self-adjointN multiplier invertible modulo , then the index theorem implies that A ˆ-Ind(κD + H) = Sf (H) , T D and by choosing a representative U 1+A of the class [eıπG(H)] , one can compute the spectral ∈ 1 flow [40] Sf (H) = [D], [U] = ((1 U ∗)[∂, U]) . (21) D h 1i T −

31 The r.h.s. is the non-commutative winding number as is expected for an analytic formula for spectral flow. The appropriate setting for the theorem in [43] is a trivial action α in which case = C (R, ) consists of paths in the von Neumann algebra = π( )′′. Since τ-traceclass A 0 C M C elements are dense in one has τ and hence invertibility of H M( ) Cb(R, ) modulo means that HC is a continuousC ⊂K path of τ-Fredholm operators. In∈ thatA case,⊂ the r.h.s.M of (21) computesA the usual semifinite spectral flow, in fact, it is almost exactly the definition of spectral flow for graph-continuous paths (see Appendix B, note however that for an unbounded H to be a multiplier here, it must describe a Riesz-continuous path).

8.4 Index theorems for topological insulators Let us now discuss a more complicated non-commutative example coming from the theory of topological insulators [42]. To keep the discussion simple we consider a two-dimensional example with magnetic field, but no disordered potential. Thus the observable algebra is the two-dimensional non-commutative torus aθ with twisting angle θ generated by two unitaries ıθ with the commutation relation v1v2 = e v2v1. Let c (Z) be the algebra of sequences which ∗ admit limits for and c (Z) the subalgebra for which those limits vanish. Let then be ±∞ 0 A the C∗-algebra generated by c (Z) and the unitaries v1,v2 with the additional commutation ∗ relations fv1 =(f T1)v1 and fv2 = v2f with T1 : c (Z) c (Z) left translation. Each elementb ◦ ∗ → ∗ x y a has a representation as a formal sum a = x,y Z fx,yv1 v2 . Consider the ideal of ∈ A ∈ A ⊂ A those elements for which the coefficient functions are in c0(Z). Then one has an exact sequence P b b 0 a a 0 →A→ A → θ ⊕ θ → obtained by evaluation at . On aθ one can introduce a finite trace τ and on a densely defined lower semi-continuous±∞ traceτ ˆ such thatb A x y x y τ(v1 v2 ) = δx,0 δy,0 , τˆ(fv1 v2 ) = δx,0 δy,0 f(k) , k Z X∈ Z 1 Z A x y holds for all x, y and f ℓ ( ). Let be the subalgebra of all x,y Z fx,yv1 v2 for ∈ ∈ ∈ ∈ A which fx,y(k) decays faster than any inverse polynomial in x,y,k. One can represent on the | | 2 2 P A Hilbert space ℓ (Z ) in such a way that c (Z) acts by multiplication and v1,v2 as magnetic shifts. ∗ 2 2 Furthermore there are the two position operators X1,X2 acting on the standard basis ofb ℓ (Z ) by Xiex = xiex. The commutators [Xi, ] produce densely defined derivations on and aθ. As · 2 A 2 the Dirac operator, let us use D = X2 which results in a spectral triple ( (ℓ (Z, C )),D, A ). One can therefore consider Callias-type operator with potentials H in theB multiplier algebra x y M( ). Any such multiplier has a representation H = x,y Z hx,yv1 v2 with coefficient functions A ∈ hx,y ℓ∞(Z). Assume that hx,y decays faster than any inverse polynomial in x, y from which ∞ one can∈ check that a H isk boundedk and differentiableP in our sense. If H is invertible modulo , then by Theorem 8 its index is given by A Ind(κD + ıH) = Sf (H) = [X ], [U] , D h 2 1i with U = eıπfexp(H) 1 + . In fact one has U 1 + A and this index can be computed ∈ A ∈ explicitly [41] [X ], [U] =τ ˆ((1 U ∗)[X , U]) . h 2 1i − 2 32 In physics the algebra aθ describes an observable algebra for two-dimensional tight-binding models which are invariant under magnetic translations, while M( ) more generally allows A modulations with respect to the x -direction. In particular, a self-adjoint multiplier H 1 ∈ A ⊂ M( ) represents the Hamiltonian for a system with an interface at the line x = 0 between A 1 two asymptotic ”bulk” Hamiltonians H aθ which describe the local Hamiltonian far awayb ± from the interface for x . The number∈ Sf (H) again makes sense as a non-commutative 1 → ±∞ D spectral flow: if the flow along the ”path” H connecting the invertible Hamiltonians H+, H − is non-trivial, then H itself cannot be invertible (”the spectral gap closes”) and this fact only depends on H up to homotopy. The known results on the bulk-boundary correspondence of such operators form a close analogue of the Callias-index formula. Indeed, for H the class [U] K ( ) of the ∈ A 1 ∈ 1 A spectral flow is the image under the exponential map of the class in [P+ P ] K0(aθ aθ) ⊕ − ∈ ⊕ of the Fermi projections P = χ(H+ < 0). It is then knownb (e.g. [33]) that ± [X ], ∂ [P ] = [X σ + X σ ], [P ] h 2 0 0i h 1 ⊗ 1 2 ⊗ 2 0i = ı τ(P+[[X1,P+][X2,P+]]) ı τ(P [[X1,P ][X,P ]]) − − − − which shows that the index can be computed from the boundaries at . Compared to the Callias index formula the situation seems inverted since the boundary now±∞ actually represents a higher dimensional space. This is not too unusual since cyclic cohomology is 2-periodic and so dualities can affect the apparent dimensions of cocycles and algebras. This example can be generalized in different ways. The noncommutative torus does not really play a role in the arguments. One can construct analogous spectral triples and bulk- boundary sequences for any twisted crossed product ⋊Rd or ⋊ Zd where the base algebra C C admits a densely defined faithful lower semicontinuous trace. When one generalizes, this naturallyC leads to semifinite spectral triples, e.g. if one chooses = C(K) for some compact metric space K with measure µ, the spectral triple will be basedC on the type-I-von Neumann 2 Z2 algebra L∞(K,µ) (ℓ ( )) with trace K dµ . For higher dimension d> 2 one can also construct spectral⊗ triples B with Dirac operators that⊗ T involve less than d 1 spatial directions, so R − that the spectral triple is then naturally based on a type II -von Neumann algebra. Examples ∞ for multipliers H with non-trivial indices can then be given in terms of Hamiltonians of so-called weak topological insulators, see [45].

A Semifinite index

Let be a semifinite von Neumann algebra with normal semifinite faithful trace . This appendixN briefly reviews the theory of semifinite index and its continuity properties.T The domain of is by definition given by = A : ( A ) < . It is a -algebra and T an ideal in T . It becomes a Banach- -algebraN { when∈ N suppliedT | with| the∞} submultiplicative∗ norm N ∗ A = A + ( A ) and its C∗-completion is the algebra of -compact operators. T kThekT quotientk k T/ | |is called the Calkin algebraK and⊂N the quotient mapT will be denoted by π. N KT The -essential spectrum of A is then defined by σess(A)= σ(π(A)) = σ(A + ). T ∈N KT

33 A possibly unbounded operator T affiliated to is called -Fredholm if there is a continuous N T function χ : [0, ) [0, 1] with χ(0) = 1 such that χ(T ∗T ) and χ(T T ∗) . This is ∞ → ∈ KT ∈ KT equivalent to the existence of a pair of operators K,K′ such that T ∗T + K and T T ∗ + K′ ∈KT are invertible. The set of -Fredholm operators will be denoted ( ) and intersection with the self-adjoint operators byT ( ). F N Fsa N If T ( ), then also T ∗ ( ) and furthermore the kernel projection Ker(T ) lies in . T It is important∈F N to note that any∈F -compactN projection is automatically -finite and thereforeK T T one has a well-defined index

-Ind(T ) = (Ker(T )) (Ker(T ∗)) R . T T − T ∈ The index is invariant under addition of perturbations and constant on norm-connected T components of ( ) . K F N ∩N An element T P Q for two projections P, Q is called P Q-Fredholm if T ∗T and ∈ N ∈ N · T T ∗ are -Fredholm in the corner algebras Q Q and P P respectively. One then defines more generallyT the skew-corner index [20] by N N

-IndP Q(T ) = (Ker(T ) Q) (Ker(T ∗) P ) . (22) T · T ∩ − T ∩ For the study of the spectral flow of unbounded Fredholm operators in Appendix B, one needs to introduce topologies on sa( ), of which there are several distinct ones [36, 49]. The most important ones are the Riesz-topologyF N and the graph topology, induced respectively by the metrics d : , d (T , T ) = F (T ) F (T ) , R Fsa ×Fsa R 1 2 k 1 − 2 k where F is the bounded transform and

1 1 d : , d (T , T ) = (T ) (T ) = (T + ı)− (T + ı)− , R Fsa ×Fsa G 1 2 kC 1 − C 2 k k 1 − 2 k 1 with (T )=(T ı)(T + ı)− the Cayley transform. A sequence of self-adjoint operators C − converges in the graph topology if and only if it converges in the norm-resolvent sense. Hence if (Tt)t [0,1] is a path in sa( ) is graph continuous, then (f(Tt))t [0,1] is therefore a norm- ∈ ∈ continuous path in forF anyN f C (R). The graph topology is weaker than the Riesz N ∈ 0 topology since it does not imply continuity under the bounded transform. All those topologies are equivalent to the norm-topology when restricting to bounded self- adjoint operators. The graph topology can be extended to non-self-adjoint Fredholm operators by setting 0 T 0 T d˜ (T , T ) = d ( 1∗ , 2∗ ) , G 1 2 G T 0 T 0  1   2  and the -index of unbounded -Fredholm operators stays constant under graph-continuous homotopiesT [49, Proposition 5.2].T

34 B Semifinite spectral flow

This appendix recalls the definition and properties of the spectral flow in a semifinte von Neumann algebra. For paths of bounded self-adjoint operators this is reviewed in [5]. For paths of unbounded self-adjoint operators, depending on the notion of continuity there are different possible ways to define spectral flow which we now describe in some detail since it is relevant for the main part of this article. Spectral flow for graph-continuous paths using the notion of a non-commutative winding number has apparently been introduced in the Hilbert space setting by [6] and extended to the semifinite setting by [49]. 1 Consider the Banach -algebra of differentiable paths C0 ([0, 1], ) with norm ∗ NT

f C = sup f(t) + f ′(t) , k k t [0,1] k kT k kT ∈ which is dense in the C∗-algebra C0([0, 1], ) with spectrally invariant inclusion (the latter KT follows from the inequality fg C f g C + f C g via a standard argument using geometric series, see also [46]).k Thek ≤ noncommutative k k k k k windingk k k number defined by

1 1 1 C wind : C0 ([0, 1], ) C0 ([0, 1], ) , wind(f1, f2) = ı (f1(t)f2′ (t)) , T NT × NT → T Z0 is a cyclic 1-cocycle and therefore pairs with odd K-theory groups. Due to spectral invari- 1 ance, one has K1(C0 ([0, 1], ))) K1(C0([0, 1], )) and more strongly any class [f]1 NT ≃ KT ∈ K1(C0([0, 1], )) defined by a unitary path f 1 + C0([0, 1], ) can be represented by a K˜T 1 ∈ KT unitary path f 1 + C0 ([0, 1], ) such that the real-valued pairing ∈ NT

wind , [f]1 = wind (f˜∗ 1, f˜ 1) h T i T − − is well-defined and does not depend on the choice of representative.

Definition 41 ([49]) Let t [0, 1] Tt be a graph-continuous path in sa( ) with invertible endpoints. One can always choose∈ a7→ so-called switch function G : R RFforN the path, which is → 2 a smooth function with supp(G′) ( 1, 1), G( 1) = 1 and 1 G(Tt) for all t. Then ⊂ − ± ±1 − ∈KT the norm-continuous unitary path t [0, 1] eıπ(G(Tt)+ ) lies in 1 + and the spectral flow ∈ 7→ KT ıπG(T ) Sf( Tt t [0,1]) = wind , [e ]1 { } ∈ h T i is well-defined and does not depend on the choice of G.

The important technical point here is that eıπG 1 is a continuous compactly supported − C0(R)-function which vanishes on the -essential spectrum. Therefore graph-continuity is sufficient, but in exchange the endpointsT have to be invertible. For Riesz-continuous paths the spectral flow can also be defined more directly as the flow of spectrum from the negative to the positive:

35 Definition 42 ([5]) For projections P, Q with π(P Q) < 1 define the essential codi- mension ∈N k − k ec(P, Q) = (1 P ) Q (1 Q) P . T − ∩ − T − ∩ For a Riesz-continuous path t [0, 1] T in ( ) one can always choose a partition ∈ 7→ t
Fsa N
0= t0 < t1 < ... < tK+1 =1 such that π (p p ) 1 k k+1 − k k ≤ 2 holds for all k =0,...,K with p = χ(T 0). In that case the spectral flow is given by k tk ≥ K

Sf( Tt t [0,1]) = ec(pk,pk+1) . { } ∈ k X=0 If the endpoints of the path are invertible both notions coincide. Therefore the spectral flow for graph-continuous paths is well-defined by choosing for each endpoint a -compact T perturbation Q0, Q1 such that Ti + Qi is invertible and to set

Sf( Tt t [0,1]) =Sf( Tt + (1 t)Q0 + tQ1 t [0,1]) + Sf( T0 + tQ0 t [0,1]) { } ∈ { − } ∈ { } ∈ + Sf( T1 + (1 t)Q1 t [0,1]) , { − } ∈ where the second two terms use the definition for Riesz-continuous paths and only the former the one for graph-continuous ones. One has the following properties:

Proposition 43 Let t [0, 1] T and t [0, 1] T ′ be graph-continuous paths in ( ). ∈ 7→ t ∈ 7→ t Fsa N (i) (Triviality) If Tt has a bounded inverse for each t [0, 1] then Sf( Tt t [0,1])=0. ∈ { } ∈ (ii) (Homotopy invariance) If the two paths are connected by a graph-continuous (respectively Riesz-continuous) homotopy (t, s) [0, 1] [0, 1] T within ( ) with T = T , ∈ × 7→ s,t Fsa N 0,t t T = T ′ and such that the endpoints T and T are invertible for each s [0, 1], then 1,t t s,0 s,1 ∈

Sf( Tt t [0,1]) = Sf( Tt′ t [0,1]) . { } ∈ { } ∈

(iii) (Concatenation) If T1 = T0′, then

Sf( Tt t [0,1] Tt′ t [0,1]) = Sf( Tt t [0,1]) + Sf( Tt′ t [0,1]) , { } ∈ ∗{ } ∈ { } ∈ { } ∈ with denoting concatenation of paths. ∗ (iv) (Homomorphism)

Sf( Tt Tt′ t [0,1]) = Sf( Tt t [0,1]) + Sf( Tt′ t [0,1]) . { ⊕ } ∈ { } ∈ { } ∈ For straight-line paths the spectral flow will be abbreviated by

Sf(T0, T1) = Sf( (1 t)T0 + tT1 t [0,1]) . { − } ∈ Let us now recall some relations between spectral flow and the -Fredholm index: T 36 Proposition 44 (Proposition 5.1 [49]) For T a possibly unbounded -Fredholm operator and any m> 0, T m T ∗ m T ∗ -Ind(T ) = Sf( − , ) . T T m T m  −   −  Then there are more specific spectral flow formulas for unitary conjugates [16, Theorem 4.2]

Theorem 45 Let D be a self-adjoint invertible -Fredholm operator affiliated to . If U is a unitary that preserves Dom(D), [D, U] extendsT to a bounded operator in andN such∈N that 1 1 N (D + ı)− (U 1) and (D + ı)− [D, U] , then − ∈KT ∈KT

-Ind(PUP + 1 P ) = Sf(U ∗DU, D) T − where P = χ(D> 0).

The bounded version of this formula is [5, Section 5]

Proposition 46 If T is a self-adjoint involution and U a unitary with [T, U] , then ∈N ∈KT

-Ind(PUP + 1 P ) = Sf(T, U ∗T U) T − where P = χ(T < 0).

Theorem 45 requires that [D, U] must be relatively D-compact for the path to be Fredholm. While such a condition often is satisfied in applications, it is sometimes inconvenient e.g. in the setting of spectral triples without smoothness assumptions. We therefore provide an alternative:

Proposition 47 Let D be a self-adjoint invertible -Fredholm operator affiliated to and T N U a unitary that preserves Dom(D), [D, U] extends to a bounded operator in [D, U] ∈N 1 N ∈N and (U 1)(D + ı)− . Then − ∈KT κD 1 κD U -Ind(PUP + 1 P ) = Sf( , ∗ ) T − 1 κD U κD  −   −  1 1 1 1 holds for κ< min , [D, U] − , [D, U] D− and where P = χ(T < 0). { 4 4 k k k k k k} Proof. By a standard argument [P, U] and thus Proposition 46 and additivity imply ∈KT

κ(1 2P ) 0 κU ∗(1 2P )U 0 -Ind(PUP + 1 P ) = Sf( − , − ). T − 0 κ(1 2P ) 0 κ(1 2P )  − −   − −  The endpoints of the path are invertible and introducing an off-diagonal constant term only increases the spectral gap, thus

κ(1 2P ) 1 κU ∗(1 2P )U U ∗ -Ind(PUP + 1 P ) = Sf( − , − ). T − 1 κ(1 2P ) U κ(1 2P )  − −   − −  37 For the Fredholm conditions along that straight-line homotopy one checks

2 κ(1 2P ) κtU[P, U ∗] s(t + (1 t)U ∗) − − − κ2 + s2(1 2t)2 mod s(t + (1 t)U) κ(1 2P ) ≥ − KT  − − −  The right endpoint has a spectral gap in 1 κ [D, U] (which includes the interval ± − k k ( 3 , 3 ) by assumption) that is not closed when one replaces U (1 2P )U =(1 2P ) 2U [P, U] 4 4 p ∗ ∗ by− 1 2P (which is a compact perturbation and so does not affect− the Fredho−lm properties):− −

κ(1 2P ) 1 κ(1 2P ) U ∗ -Ind(PUP + 1 P ) = Sf( − , − ). T − 1 κ(1 2P ) U κ(1 2P )  − −   − −  The proof is complete once one shows that the homotopy

1 γ κD D − t1 + (1 t)U ∗ (γ, t) [0, 1] [0, 1] | | − 1 γ ∈ × 7→ t1 + (1 t)U κD D −  − − | |  is graph-continuous, takes values in the Fredholm operators and the endpoints at t 0, 1 ∈ { } are invertible. The off-diagonal terms are norm-continuous paths of bounded operators and the graph-continuity of the diagonal terms is easily asserted using the spectral mapping property of continuous functional calculus (here the invertibility of D is essential). The Fredholm property was already asserted for γ =0 and for 0 <γ 1 the relative -compactness of 1 U w.r.t. D 1 γ ≤ T − and thus D D − reduces it to the trivial observation that | | 1 γ 2 κD D − 1 2 2 2γ | | 1 γ = (κ D − + 1) 12 1 κD D − | | ⊗  − | |  is invertible. For the right endpoint one has

1 γ 2 κD D − U ∗ 2 2 2γ 1 γ | | 1 γ (κ D − +1 κ [D D − , U] 12 U κD D − ≥ | | − | |   − | | 1 γ γ which is invertible by assumption since [D D − , U] D − [D, U] as a consequence of the integral formula for fractional powers| (see| e.g. [26,≤ Lemma| | 10.18]).k k ✷

Acknowledgements: T. S. received funding from the Studienstiftung des deutschen Volkes. This work was also supported by the DFG grant SCHU 1358/6-2.

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