A K-theoretic note on the spectral localiser

A K-theoretic note on the sp ectral lo caliser K o en v an den Dungen Mathematisc hes Institut , Universität Bonn Endenic her Allee 60, D-53115 Bonn kdungen@uni-b onn.de Abstract W e review the construction of the sp ectral lo caliser (due to Loring and Sch ulz-Baldes) from a K -theoretic p erspective. W e first giv e a K -theoretic argument pro viding a spec- tral flo w expression for the even or o dd index pairing in terms of the “infinite volume” sp ectral lo caliser. Our approac h tow ards this first step is more direct, treats the even and odd cases on an equal fo oting, and has the adv an tage that the construction of the sp ectral localiser b ecomes immediately apparent from the computation of the index pairing via a Kasparov pro duct. In a second step of “sp ectral truncation”, we then describ e how this sp ectral flo w expression can be computed in terms of the signature of the “finite volume” sp ectral lo caliser. Throughout, we do not require inv ertibility of the op erator represen ting the K -homology class, and the even index pairing then obtains an additional contribution coming from the F redholm index. Keywor ds : Sp ectral lo caliser, K -theory , index theory , sp ectral flo w. Mathematics Subje ct Classific ation 2020 : 46L80; 19K56, 58J30. 1 In tro duction Sp ectral lo calisers were recen tly introduced b y Loring and Sch ulz-Baldes [ LS17 , LS20 ] as a method for computing an ev en ( j = 0 ) or o dd ( j = 1 ) index pairing K j ( A ) × K K j ( A, C ) − → K 0 ( C ) ≃ Z , (1.1) where A is a unital C ∗ -algebra. Here the ev en or o dd K -homology class in K K j ( A, C ) is represen ted by a ‘generalised Dirac op erator’ D , and the K -theory class in K j ( A ) is represen ted in the even case ( j = 0 ) b y the p ositiv e sp ectral pro jection P > 0 ( H ) of a self- adjoin t inv ertible H ∈ M n ( A ) or in the odd case ( j = 1 ) b y an inv ertible G ∈ M n ( A ) . The image of such an index pairing [ P > 0 ( H )] ⊗ A [ D ] or [ G ] ⊗ A [ D ] can b e describ ed as a F redholm index on an infinite-dimensional Hilb ert space. Suc h index pairings are not only of purely mathematical interest, but they also app ear for instance in condensed matter ph ysics as topological in v arian ts in the theory of (dis- ordered) top ological insulators [ BES94 , PS16 ]. In this case, the v alue of the index pairing corresp onds to an exp erimen tally measurable quantit y . Loring and Sch ulz-Baldes intro- duced the sp ectral lo caliser as a p o w erful metho d for computing suc h quantities. Indeed, 1 2 K oen v an den Dungen the main purpose of the sp ectral lo caliser is to reduce the index pairing to the computation of a finite matrix. As such, this breakthrough makes the computation of the index in v arian t in K -theory accessible to n umerical calculations and has already seen v arious applications (see e.g. [ SS22 , CL22 , FG24 , CL24 , Sto25 ]). The main results on the (even or o dd) sp ectral lo caliser by Loring and Sch ulz-Baldes ha v e been reprov en in [ LS19 , LSS19 ] using a sp ectral flo w approac h (see also the b o ok [ DSW23 , Ch.10]). V ery recen tly , Li–Mesland [ LM25 ] and Kaad [ Kaa25 ] hav e studied sp ec- tral lo calisers from the p erspective of un b ounded K K -theory , considering a generalised index pairing K j ( A ) × K K j ( A, B ) → K 0 ( B ) , where B is another C ∗ -algebra. Li–Mesland [ LM25 ] use E-theory to describ e the o dd index pairing ( 1.1 ) (with j = 1 ) in terms of the asymptotic morphism K 1 ( A ) ≃ K 0 ( C 0 ( R , A )) → K 0 ( C ) asso ciated to the K -homology class in K K 1 ( A, C ) . Kaad [ Kaa25 ] fo cuses on generalising the even index pairing ( 1.1 ) (with j = 0 ) to the biv ariant setting K 0 ( A ) × K K 0 ( A, B ) → K 0 ( B ) . In this article, our main goal is to provide a new K -theoretic persp ectiv e on the (even or o dd) sp ectral lo caliser. As in [ LM25 ], we start in Section 2 by emplo ying the Bott p eri- o dicit y isomorphism K j ( A ) ≃ K j +1 ( C 0 ( R , A )) , but w e represen t the resulting K -theory class in K j +1 ( C 0 ( R , A )) by a F redholm op erator on a Hilb ert C 0 ( R , A ) -mo dule (instead of as a formal difference of pro jections, as in [ LM25 , Lemma 3.1]). Our F redholm rep- resen tativ e then allows us to obtain the (ev en or o dd) index pairing ( 1.1 ) via a relatively straigh tforw ard computation of the pairing K j +1 ( C 0 ( R , A )) × K K j ( A, C ) − → K 1 ( C 0 ( R )) ≃ K 0 ( C ) ≃ Z (1.2) as a Kasparo v product in the un b ounded picture of K K -theory . The result is describ ed as the sp ectral flow of a certain path of self-adjoint F redholm op erators, where the (self- adjoin t in vertible) spectral localiser app ears as one of the endpoints of this path (the other endpoint is also a sp ectral lo caliser, but comes from a trivial K -theory class). More precisely , for any κ > 0 , the ev en or o dd (infinite v olume) sp ectral lo caliser corresp onding to the pair ( H, D ) in the ev en case or ( G, D ) in the o dd case is defined b y L ev κ ( H , D ) :=  H κ D − κ D + − H  , L od κ ( G, D ) :=  κ D G G ∗ − κ D  , where in the even case we hav e decomp osed the Z 2 -graded operator D =  0 D − D + 0  . W e then pro v e in Theorem 2.5 that the even index pairing can b e computed (for sufficien tly small κ ) as [ P > 0 ( H )] ⊗ A [ D ] = sf  L ev κ ( − 1 , D ) → L ev κ ( H , D )  ∈ K 0 ( C ) ≃ Z , (1.3) where sf ( · → · ) denotes the spectral flow along the straigh t line path. Similarly , w e prov e in Theorem 2.10 that the o dd index pairing can b e computed (for sufficiently small κ ) as [ G ] ⊗ A [ D ] = sf  L od κ (1 , D ) → L od κ ( G, D )  ∈ K 0 ( C ) ≃ Z . (1.4) Compared to the existing literature on spectral lo calisers, our approac h is more direct (it do es not require an y in termediate homotopies) and has the particular adv an tage that the a k-theoretic note on the spectral localiser 3 construction of the spectral lo caliser becomes immediately apparent from the computation of the Kasparov product ( 1.2 ). Moreov er, our approach applies to b oth the even and the o dd index pairing, treating b oth cases on an equal footing. Next, in Section 3 , w e describ e the sp e ctr al trunc ation of the sp ectral lo caliser. The trunc ate d sp ectral lo caliser L ev κ,ρ ( H , D ) or L od κ,ρ ( G, D ) is a finite matrix, which is obtained b y compressing the sp ectral localiser to a finite-dimensional spectral subspace (sp ecified by the parameter ρ ) of the op erator D representing the K -homology class [ D ] ∈ K K j ( A, C ) . Loring and Sch ulz-Baldes [ LS17 , LS20 ] hav e sho wn that (for suitable κ, ρ ‘) this truncated sp ectral lo caliser is invertible . Com bining this fact with our spectral flo w expression ( 1.3 ) or ( 1.4 ), w e can express the (ev en or o dd) index pairing in terms of the signatur e of the truncated sp ectral lo caliser. In contrast to the work of Loring and Sch ulz-Baldes, we do not require that D is in v ertible. W e will pro ve in Theorem 3.8 that the expression for the ev en index pairing then receives an additional term given b y the F redholm index of D : [ P > 0 ( H )] ⊗ A [ D ] = 1 2 Sig  L ev κ,ρ ( H , D )  + 1 2 Index( D + ) . If D is in v ertible, then Index( D + ) = 0 and w e reco v er the main result from [ LS20 ]. F or the o dd index pairing, there is no such index correction, and we will pro v e in Theorem 3.15 that [ G ] ⊗ A [ D ] = 1 2 Sig  L od κ,ρ ( G, D )  . This recov ers the main result of [ LS17 ], without assuming D to be in vertible. F or the con venience of the reader, we briefly recall in App endix A some F redholm theory for op erators on Hilb ert C ∗ -mo dules ov er a C ∗ -algebra A , and in particular describe the (ev en or odd) relativ e index and the (ev en or o dd) sp ectral flo w, as defined in a general setting for op erators on Hilb ert A -mo dules. Finally , App endix B con tains a detailed computation of the Kasparo v pro duct pro viding the pairing ( 1.2 ). 2 Sp ectral lo calisers and index pairings 2.1 The even index pairing Setting 2.1. Consider a unital C ∗ -algebra A . Let A ⊂ A b e a dense unital ∗ -subalgebra, and consider an even spectral triple ( A , H , D ) ov er A representing a K -homology class [ D ] ∈ K K 0 ( A, C ) . T o b e precise, this means that we hav e a ∗ -representation π : A → B ( H ) on a Hilb ert space H , and a (densely defined) self-adjoint op erator D : Dom( D ) ⊂ H → H with compact resolven ts, suc h that for all a ∈ A w e ha ve π ( a ) · Dom( D ) ⊂ Dom( D ) and [ D , π ( a )] extends to a b ounded op erator on H . W e assume that the representation π is nonde gener ate , i.e., π ( A ) · H is dense in H . F urthermore, consider a pro jection p ∈ M n ( A ) for some 1 ≤ n ∈ N , representing a K -theory class [ p ] ∈ K 0 ( A ) . W e fix an inv ertible self-adjoin t element H ∈ M n ( A ) whose p ositiv e sp ectral pro jection P > 0 ( H ) equals p : P > 0 ( H ) := 1 2  1 + H | H | − 1  = p 4 K oen v an den Dungen (for instance, we could c ho ose H = 2 p − 1 , but we allow for non-unitary c hoices as well). W e denote the sp e ctr al gap of H by g := ∥ H − 1 ∥ − 1 . Since the sp ectral triple ( A , H , D ) is even, there is a Z 2 -grading on H , i.e. a direct sum decomp osition H = H + ⊕ H − along with a grading op erator Γ = 1 ⊕ ( − 1) . The op erator D is o dd and the representation π is ev en, so w e can write D =  0 D − D + 0  , π ( a ) =  π + ( a ) 0 0 π − ( a )  , ∀ a ∈ A. By extending the represen tation π = π + ⊕ π − to M n ( A ) → B ( H ⊕ n ) , w e obtain a pro jection π ( p ) ∈ B ( H ⊕ n ) as well as pro jections π + ( p ) ∈ B ( H ⊕ n + ) and π − ( p ) ∈ B ( H ⊕ n − ) . It is well-kno wn that the ev en index pairing of the K -theory class [ p ] with the K -homology class [ D ] can b e computed as a F redholm index as follo ws: [ p ] ⊗ A [ D ] = Index  π − ( p ) D + π + ( p )  . Alternativ ely , the ev en index pairing may b e expressed in terms of the sp ectral flo w, as follo ws. Consider the sp ecial case H + = H − and π + = π − . Assuming D is in v ertible, we ma y consider the unitary F + := D + |D + | − 1 . Since [ π ( H ) , F + ] is compact, we obtain from Corollary A.4 the equalities Index  π ( p ) F + π ( p )  = Index  P > 0 ( π ( H )) F + P > 0 ( π ( H ))  = sf  π ( H ) → F + π ( H ) F ∗ +  . 2.1.1 The even sp ectral lo caliser Our aim here is to provide a differen t sp ectral flow expression for the even index pairing in terms of the sp ectral localiser. W e will not assume that D is in v ertible. Definition 2.2. Let H and D b e as in Setting 2.1 . The even sp e ctr al lo c aliser of the pair ( H , D ) with tuning parameter κ ∈ (0 , ∞ ) is defined as the op erator L ev κ ≡ L ev κ ( H , D ) := κ D + Γ π ( H ) =  π + ( H ) κ D − κ D + − π − ( H )  on (Dom D ) ⊕ n ⊂ H ⊕ n . When no confusion arises, we will often suppress π from our notation. Lemma 2.3. Assume κ < g 2   [ D , H ]   − 1 . Then the even sp e ctr al lo c aliser L ev κ is invertible. Pr o of. The square can b e estimated b y  L ev κ  2 =  κ D + Γ H  2 = κ 2 D 2 + H 2 + κ [ D , H ]Γ ≥ g 2 − κ   [ D , H ]   > 0 , where in the third step we used κ 2 D 2 ≥ 0 and H 2 ≥ g 2 , and in the last step w e used the assumption κ < g 2   [ D , H ]   − 1 . 2.1.2 The sp ectral flow expression Our starting p oin t is an alternativ e represen tation of the even K -theory class [ p ] ∈ K 0 ( A ) based on the Bott p eriodicity isomorphism K 0 ( A ) ≃ K 1  C 0 ( R ) ⊗ A  . F or this purp ose, we consider contin uous functions χ ± : R → [0 , 1] with the following prop erties: supp χ + ⊂ [0 , ∞ ) and supp χ − ⊂ ( −∞ , 0] , and χ + ( t ) = 1 for t ≥ 1 and χ − ( t ) = 1 for t ≤ − 1 . (2.1) a k-theoretic note on the spectral localiser 5 The follo wing construction of the op erator S H is analogous to the construction in the pro of of [ Dun25b , Lemma B.3] (the idea go es bac k to [ W ah09 , App endix A.1]). Recall that the Bott p eriodicity isomorphism K 1  C 0 ( R , A )  ≃ − → K 0 ( A ) is implemented by taking the Kasparo v product with [ − i∂ t ] ∈ K K 1  C 0 ( R ) , C  (see App endix A ). Lemma 2.4. L et χ ± : R → [0 , 1] b e c ontinuous functions satisfying the pr op erties in Eq. ( 2.1 ) . Consider the (b ounde d) self-adjoint F r e dholm op er ator S H := − χ − + χ + H on C 0 ( R , A ) ⊕ n , r epr esenting a class [ S H ] ∈ K 1  C 0 ( R , A )  . Then we have the e quality [ p ] = [ S H ] ⊗ C 0 ( R ) [ − i∂ t ] ∈ K 0 ( A ) . Thus the map [ p ] 7→ [ S H ] implements the natur al isomorphism K 0 ( A ) ≃ − → K 1  C 0 ( R , A )  . Pr o of. F rom Prop ositions A.3 and A.5 we obtain [ S H ] ⊗ C 0 ( R ) [ − i∂ t ] = sf  { S H ( t ) } t ∈ [ − 1 , 1]  = rel-ind  P > 0 ( S H (1)) , P > 0 ( S H ( − 1))  = rel-ind  P > 0 ( H ) , P > 0 ( − 1)  = rel-ind  p, 0  = [ p ] . Recall the notation sf ( T 0 → T 1 ) for the sp ectral flo w of the straight line path from T 0 to T 1 (see Eq. ( A.3 )). W e are now ready to provide a sp ectral flow expression for the ev en index pairing in terms of the sp ectral lo caliser. The pro of relies on the computation of the Kasparo v product [ S H ] ⊗ A [ D ] , which is described in Appendix B . Theorem 2.5. Assume κ < g 2   [ D , H ]   − 1 . The p airing of [ p ] ∈ K 0 ( A ) with [ D ] ∈ K K 0 ( A, C ) is given by [ p ] ⊗ A [ D ] = sf  L ev κ ( − 1 , D ) → L ev κ ( H , D )  = sf  κ D − Γ → κ D + Γ H  ∈ K 0 ( C ) ≃ Z . Mor e over, if D is invertible, we obtain [ p ] ⊗ A [ D ] = sf  κ D → L ev κ ( H , D )  . Pr o of. Using Lemma 2.4 along with the prop erties of the Kasparov product, we can rewrite [ p ] ⊗ A [ D ] =  [ S H ] ⊗ C 0 ( R ) [ − i∂ t ]  ⊗ A [ D ] = [ S H ] ⊗ C 0 ( R ,A )  [ − i∂ t ] ⊗ [ D ]  = [ S H ] ⊗ C 0 ( R ,A )  [ D ] ⊗ [ − i∂ t ]  =  [ S H ] ⊗ A [ D ]  ⊗ C 0 ( R ) [ − i∂ t ] . By Corollary B.2 , the Kasparo v pro duct [ S H ] ⊗ A [ D ] is given b y [ L ev κ ] ∈ K 1  C 0 ( R )  , where the op erator L ev κ on the Hilb ert C 0 ( R ) -mo dule C 0 ( R , H ⊕ n ) is given b y L ev κ ( t ) := κ D + Γ S H ( t ) . Noting that the isomorphism ⊗ C 0 ( R ) [ − i∂ t ] : K 1  C 0 ( R )  ≃ − → K 0 ( C ) ≃ Z is given b y the sp ectral flow (see Proposition A.5 ), w e hav e  [ S H ] ⊗ A [ D ]  ⊗ C 0 ( R ) [ − i∂ t ] = [ L ev κ ] ⊗ C 0 ( R ) [ − i∂ t ] = sf  {L ev κ } t ∈ [ − 1 , 1]  . 6 K oen v an den Dungen Since D has compact resolv ents and S H is bounded, L ev κ ( t ) is a relativ ely D -compact p erturbation of κ D (for eac h t ∈ [ − 1 , 1] ). By Prop osition A.3 , the sp ectral flow dep ends only on the endpoints, and w e obtain the first statement: [ p ] ⊗ A [ D ] = sf  L ev κ ( − 1) → L ev κ (1)  = sf  κ D − Γ → κ D + Γ H  . Assuming D is in vertible, the middle point L ev κ (0) = κ D is also in v ertible. F or t ∈ [ − 1 , 0] , w e compute L ev κ ( t ) 2 =  κ D − χ − ( t )Γ  2 = κ 2 D 2 + χ − ( t ) 2 ≥ κ 2 D 2 , and we see that L ev κ ( t ) is in v ertible for all t ∈ [ − 1 , 0] . Hence sf  {L ev κ ( t ) } t ∈ [ − 1 , 0]  = 0 , and w e obtain the second statemen t: [ p ] ⊗ A [ D ] = sf  κ D − Γ → κ D  + sf  κ D → κ D + Γ H  = sf  κ D → κ D + Γ H  . 2.2 The o dd index pairing Setting 2.6. Consider a unital C ∗ -algebra A . Let A ⊂ A b e a dense unital ∗ -subalgebra, and consider an o dd sp ectral triple ( A , H , D ) ov er A represen ting a K -homology class [ D ] ∈ K K 1 ( A, C ) . T o b e precise, this means that w e hav e a ∗ -representation π : A → B ( H ) and a (densely defined) self-adjoint op erator D : Dom( D ) ⊂ H → H with compact resolv en ts, such that for all a ∈ A w e hav e π ( a ) · Dom( D ) ⊂ Dom( D ) and [ D , π ( a )] extends to a bounded op erator on H . W e assume that the represen tation π is nonde gener ate , i.e., π ( A ) · H is dense in H . F urthermore, consider a unitary u ∈ M n ( A ) for some 1 ≤ n ∈ N , representing a class [ u ] ∈ K 1 ( A ) . W e fix an inv ertible elemen t G ∈ M n ( A ) whose phase equals u : G | G | − 1 = u (for instance, we could simply c ho ose G = u ). W e denote the sp e ctr al gap of G by g := ∥ G − 1 ∥ − 1 . W e obtain a unitary π ( u ) and an in v ertible π ( G ) in B ( H ⊕ n ) b y extending the rep- resen tation π to M n ( A ) → B ( H ⊕ n ) . It is well-kno wn that the odd index pairing of the K -theory class [ u ] with the K -homology class [ D ] can be computed as a F redholm index as follo ws: [ u ] ⊗ A [ D ] = Index  P > 0 ( D ) π ( u ) P > 0 ( D )  . Using Corollary A.4 , w e may also express the index pairing as a sp ectral flo w: [ u ] ⊗ A [ D ] = − sf  D → π ( u ∗ ) D π ( u )  = sf  D → π ( u ) D π ( u ∗ )  . 2.2.1 The o dd sp ectral lo caliser Our aim here is to provide a different spectral flo w expression for the o dd index pairing in terms of the spectral lo caliser. W e will not assume that D is inv ertible. Definition 2.7. Let G and D b e as in Setting 2.6 . The o dd sp e ctr al lo c aliser of the pair ( G, D ) with tuning parameter κ ∈ (0 , ∞ ) is defined as the op erator L od κ ≡ L od κ ( G, D ) :=  κ D π ( G ) π ( G ∗ ) − κ D  on (Dom D ) ⊕ n ⊕ (Dom D ) ⊕ n ⊂ H ⊕ n ⊕ H ⊕ n . When no confusion arises, we will often suppress π from our notation. a k-theoretic note on the spectral localiser 7 Lemma 2.8. Assume κ < g 2   [ D , G ]   − 1 . Then the o dd sp e ctr al lo c aliser L od κ is invertible. Pr o of. As in the pro of of Lemma 2.3 , the square can b e estimated b y  L od κ  2 =  κ 2 D 2 + GG ∗ κ [ D , G ] − κ [ D , G ∗ ] κ 2 D 2 + G ∗ G  ≥ g 2 − κ   [ D , G ]   > 0 . 2.2.2 The sp ectral flow expression W e closely follow the discussion of the ev en case given in Section 2.1 , and we will therefore b e brief regarding some of the details. Our starting p oin t is no w an alternativ e represen t- ation of the o dd K -theory class [ u ] ∈ K 1 ( A ) based on the Bott p erio dicit y isomorphism K 1 ( A ) ≃ K 0  C 0 ( R ) ⊗ A  . Recall the contin uous functions χ ± : R → [0 , 1] with the prop- erties given in Eq. ( 2.1 ). The follo wing construction of the op erator S G can b e found in the pro of of [ Dun25b , Lemma B.3] in the sp ecial case G = u (the idea of the construction go es back to [ W ah09 , Appendix A.1]). Lemma 2.9. L et χ ± : R → [0 , 1] b e c ontinuous functions satisfying the pr op erties in Eq. ( 2.1 ) . Consider the (b ounde d) self-adjoint F r e dholm op er ator S G :=  0 χ − + χ + G χ − + χ + G ∗ 0  on C 0 ( R , A ) ⊕ n ⊕ C 0 ( R , A ) ⊕ n , r epr esenting a class [ S G ] ∈ K 0  C 0 ( R , A )  . Then we have the e quality [ u ] = − [ S G ] ⊗ C 0 ( R ) [ − i∂ t ] ∈ K 1 ( A ) . Thus the map [ u ] 7→ − [ S G ] implements the natur al isomorphism K 1 ( A ) ≃ − → K 0  C 0 ( R , A )  . Pr o of. W e first compute P > 0  0 G G ∗ 0  = 1 2  1 0 0 1  +  0 G G ∗ 0       0 G G ∗ 0      − 1 ! = 1 2  1 G | G | − 1 G ∗ | G ∗ | − 1 1  = 1 2  1 u u ∗ 1  . F rom Prop ositions A.3 and A.5 w e then obtain [ S G ] ⊗ C 0 ( R ) [ − i∂ t ] = sf 1  { S G ( t ) } t ∈ [ − 1 , 1]  = rel-ind 1  P > 0 ( S G (1)) , P > 0 ( S G ( − 1))  = rel-ind 1  1 2  1 u u ∗ 1  , 1 2  1 1 1 1  = [ u ∗ ] = − [ u ] . Theorem 2.10. Assume κ < g 2   [ D , G ]   − 1 . The p airing of [ u ] ∈ K 1 ( A ) with [ D ] ∈ K K 1 ( A, C ) is given by [ u ] ⊗ A [ D ] = sf  L od κ (1 , D ) → L od κ ( G, D )  = sf  κ D 1 1 − κ D  →  κ D G G ∗ − κ D  ∈ Z . Mor e over, if D is invertible, we obtain [ u ] ⊗ A [ D ] = sf  κ D 0 0 − κ D  →  κ D G G ∗ − κ D  . 8 K oen v an den Dungen Pr o of. As in the pro of of Theorem 2.5 , w e can now use Lemma 2.9 along with the properties of the Kasparov pro duct to rewrite [ u ] ⊗ A [ D ] = −  [ S G ] ⊗ C 0 ( R ) [ − i∂ t ]  ⊗ A [ D ] = − [ S G ] ⊗ C 0 ( R ,A )  [ − i∂ t ] ⊗ [ D ]  = [ S G ] ⊗ C 0 ( R ,A )  [ D ] ⊗ [ − i∂ t ]  =  [ S G ] ⊗ A [ D ]  ⊗ C 0 ( R ) [ − i∂ t ] . where in the third step w e used that the exterior Kasparo v product of o dd Kasparo v mo dules is antic ommutative . By Corollary B.3 , the Kasparo v pro duct [ S G ] ⊗ A [ D ] is given b y [ L od κ ] ∈ K 1  C 0 ( R )  , where the op erator L od κ on the Hilb ert C 0 ( R ) -mo dule C 0 ( R , H ⊕ 2 n ) is giv en b y L od κ ( t ) := L od ( S G ( t ) , D ) . Combined with Proposition A.5 , w e obtain  [ S G ] ⊗ A [ D ]  ⊗ C 0 ( R ) [ − i∂ t ] = [ L od κ ] ⊗ C 0 ( R ) [ − i∂ t ] = sf  {L od κ } t ∈ [ − 1 , 1]  . Since L od κ ( t ) is a relativ ely D -compact perturbation of ( κ D ) ⊕ ( − κ D ) (for eac h t ∈ [ − 1 , 1] ), the spectral flow dep ends only on the endpoints (see Prop osition A.3 ), and we obtain the first statement: [ u ] ⊗ A [ D ] = sf  L od κ (1 , D ) → L od κ ( G, D )  . Assuming D is in v ertible, the middle p oin t L od κ (0) = ( κ D ) ⊕ ( − κ D ) is also in vertible. F or t ∈ [ − 1 , 0] , we compute L od κ ( t ) 2 = κ 2 D 2 + χ − ( t ) 2 ≥ κ 2 D 2 , and we see that L od κ ( t ) is in v ertible for all t ∈ [ − 1 , 0] . Hence sf  {L od κ ( t ) } t ∈ [ − 1 , 0]  = 0 , and we obtain the second statemen t: [ u ] ⊗ A [ D ] = sf  ( κ D ) ⊕ ( − κ D ) → L od κ ( G, D )  . Remark 2.11. The sp ectral flow expression of Theorem 2.10 for in vertible D already app eared in [ LS19 , §3], but it was noted by Loring and Sch ulz-Baldes in their pro of that “one should, strictly sp eaking, alw ays replace D b y F ρ ( D ) ” (where F ρ ( D ) is a b ounde d op erator obtained from D via con tin uous functional calculus). The pro of giv en here w orks directly for unbounded operators and does not in volv e any intermediate homotopies. 3 Sp ectral truncation of the index pairing W e consider an (even or o dd) sp ectral triple ( A , H , D ) . F or any sp e ctr al r adius ρ ∈ (0 , ∞ ) w e consider the sp ectral pro jection P [ − ρ,ρ ] ( D ) . W e in tro duce the short notation H ρ := Ran P [ − ρ,ρ ] ( D ) and H ρ c := ( H ρ ) ⊥ . W e denote b y π ρ : H → H ρ the surjectiv e partial isometry with Ker π ρ = H ρ c , and note that π ∗ ρ : H ρ  → H is the natural inclusion. F or an op erator T on H , w e denote by T ρ := π ρ T π ∗ ρ its compression on H ρ (and similarly for T ρ c ). 3.1 The even index pairing W e consider again an even sp ectral triple ( A , H , D ) o ver A and a self-adjoin t in vertible H ∈ M n ( A ) with sp ectral gap g := ∥ H − 1 ∥ − 1 whose p ositiv e sp ectral pro jection equals p , as in Setting 2.1 . Recall the definition of the even sp e ctr al lo c aliser L ev κ of the pair ( H , D ) with tuning parameter κ ∈ (0 , ∞ ) from Definition 2.2 . a k-theoretic note on the spectral localiser 9 Definition 3.1. The even trunc ate d sp ectral lo caliser of the pair ( H, D ) with tuning parameter κ and spectral radius ρ ∈ (0 , ∞ ) is defined as the op erator L ev κ,ρ ≡ L ev κ,ρ ( H , D ) := π ρ L ev κ ( H , D ) π ∗ ρ = κ D ρ + Γ ρ H ρ . Assumption 3.2. W e assume that the tuning parameter κ ∈ (0 , ∞ ) and the sp ectral radius ρ ∈ (0 , ∞ ) satisfy the following conditions: κ ≤ g 3 12 ∥ H ∥∥ [ D , H ] ∥ , 2 g κ < ρ. Theorem 3.3 (cf. [ DSW23 , Theorem 10.3.1]) . The even trunc ate d sp e ctr al lo c aliser L ev κ,ρ is invertible with sp e ctr al gap 1 2 g . F urthermor e, the signatur e of L ev κ,ρ is indep endent of the choic es of κ and ρ (satisfying Assumption 3.2 ). Remark 3.4. The inv ertibilit y of the truncated sp ectral lo caliser is w ell-established in the literature; in the ev en case, it w as first pro ven in [ LS20 ], and w e cite here the result from the textb o ok [ DSW23 ]. Although [ DSW23 , Theorem 10.3.1] assumes D to b e inv ertible, w e note that the first part of its proof, which yields the ab ov e theorem, does not actually require this assumption. The inv ertibility of D is only used in the pro of of the second state- men t in [ DSW23 , Theorem 10.3.1], which we will repro ve in Theorem 3.8 b elo w without assuming inv ertibility of D . (Alternativ ely , see the pro of of [ Kaa25 , Prop osition 10.1], whic h also do es not require in vertibilit y of D .) Lemma 3.5. The op er ator L ev κ,ρ c := π ρ c L ev κ π ∗ ρ c is invertible with sp e ctr al gap q 47 48 κρ . Pr o of. F rom Assumption 3.2 we obtain the inequalities κ 2 ρ 2 > 4 g 2 ≥ 4 12 κ ∥ H ∥∥ [ D , H ] ∥ g ≥ 48 κ ∥ [ D , H ] ∥ , where in the last step w e used ∥ H ∥ ≥ g . The square of L ev κ,ρ c can then b e estimated b y  L ev κ,ρ c  2 =  κ D ρ c + Γ ρ c H ρ c  2 = κ 2 D 2 ρ c + H 2 ρ c + κ [ D ρ c , H ρ c ]Γ ρ c ≥ κ 2 ρ 2 − κ   [ D , H ]   > 47 48 κ 2 ρ 2 , where we used D 2 ρ c ≥ ρ 2 , H 2 ρ c ≥ 0 , and ∥ [ D ρ c , H ρ c ] ∥ ≤ ∥ [ D , H ] ∥ . Next, we need to prov e that the sp ectral flow from L ev κ to L ev κ,ρ ⊕ L ev κ,ρ c v anishes. The follo wing lemma can b e extracted from the pro ofs of [ DSW23 , Theorems 10.3.1 & 10.4.1]. Lemma 3.6. F or j = 1 , 2 , let L j b e densely define d self-adjoint invertible op er ators on a Hilb ert sp ac e H , with sp e ctr al gaps g j := ∥ L − 1 j ∥ − 1 . If B ∈ B ( H ) satisfies ∥ B ∥ < √ g 1 g 2 , then the op er ator  L 1 B ∗ B L 2  on Dom L 1 ⊕ Dom L 2 ⊂ H ⊕ H is also invertible. 10 K oen v an den Dungen Pr o of. W e conjugate b y | L 1 | − 1 2 ⊕ | L 2 | − 1 2 to obtain a new op erator L 1 | L 1 | − 1 | L 1 | − 1 2 B ∗ | L 2 | − 1 2 | L 2 | − 1 2 B | L 1 | − 1 2 L 2 | L 2 | − 1 ! . Since the op erators L j | L j | − 1 are unitary and   | L 2 | − 1 2 B | L 1 | − 1 2   ≤ √ g 2 − 1 ∥ B ∥ √ g 1 − 1 < 1 , this new op erator is in v ertible, and the statemen t follo ws. No w let us consider the straigh t line path L ( t ) := L ev κ,ρ ⊕ L ev κ,ρ c + tπ ρ Γ H π ∗ ρ c + tπ ρ c Γ H π ∗ ρ , t ∈ [0 , 1] , b et ween L (0) = L ev κ,ρ ⊕ L ev κ,ρ c and L (1) = L ev κ . Lemma 3.7. F or sufficiently lar ge ρ , the op er ator L ( t ) is invertible for e ach t ∈ [0 , 1] , and we have the e quality sf  κ D − Γ → L ev κ  = sf  κ D − Γ → L ev κ,ρ ⊕ L ev κ,ρ c  . Pr o of. Since π ρ and π ρ c are partial isometries, w e hav e   tπ ρ Γ H π ∗ ρ c + tπ ρ c Γ H π ∗ ρ   ≤ ∥ H ∥ , for all t ∈ [0 , 1] . F rom Theorem 3.3 and Lemma 3.5 we ha ve   L ev κ,ρ   ≥ g 1 := g 2 and   L ev κ,ρ c   ≥ g 2 := q 47 48 κρ . F or an y fixed κ , w e may choose ρ to b e as large as needed to ensure that ∥ H ∥ < √ g 1 g 2 = 4 r 47 192 √ g κρ. It then follows from Lemma 3.6 that L ( t ) is in v ertible for all t ∈ [0 , 1] . Since π ρ has finite- dimensional range, L ( t ) is a compact p erturbation of L ev κ , so that the straigh t line path from κ D − Γ to L ( t ) lies within the F redholm op erators for all t ∈ [0 , 1] . The homotopy in v ariance of the sp ectral flow then yields the second part of the statement. Theorem 3.8. Consider an even sp e ctr al triple ( A , H , D ) and a pr oje ction p ∈ M n ( A ) as in Setting 2.1 . L et H ∈ M n ( A ) b e invertible and self-adjoint with sp e ctr al gap g = ∥ H − 1 ∥ − 1 such that P > 0 ( H ) = p . Assume the p ar ameters κ and ρ satisfy Assumption 3.2 . Then the even index p airing of [ p ] ∈ K 0 ( A ) with [ D ] ∈ K K 0 ( A, C ) is given by [ p ] ⊗ A [ D ] = 1 2 Sig  L ev κ,ρ ( H , D )  + 1 2 Index( D + ) . Remark 3.9. If D is in vertible, then Index( D + ) v anishes, and we reco v er the main result from [ LS20 ]. If D is not inv ertible, the correction term given b y the F redholm index also app eared in [ Kaa25 , Theorem 10.2], albeit in sligh t disguise (in the form 1 2 Sig  Γ ρ  ). Recall that [1] ⊗ A · : K K 0 ( A, C ) → K 0 ( C ) is the index map. Choosing H = 1 w e reco ver from Theorem 3.8 the equality [1] ⊗ A [ D ] = 1 2 Sig  L ev κ,ρ (1 , D )  + 1 2 Index( D + ) = Index( D + ) . a k-theoretic note on the spectral localiser 11 Indeed, since D is odd with respect to the Z 2 -grading Γ (and Γ comm utes with P [ − ρ,ρ ] ( D ) = π ∗ ρ π ρ ), w e note that L ev κ,ρ (1 , D ) = κ D ρ + Γ ρ is inv ertible for all κ ∈ [0 , ∞ ) , and in particular its signature is independent of κ . W e then compute Sig  L ev κ,ρ (1 , D )  = Sig  Γ ρ  = Sig  Γ Ker D  = dim Ker( D + ) − dim Ker( D − ) = Index( D + ) . Similarly , for H = − 1 w e compute Sig  L ev κ,ρ ( − 1 , D )  = Sig  − Γ ρ  = − Index( D + ) , (3.1) so that, in this case, Theorem 3.8 reduces to the ob vious equalit y [0] ⊗ A [ D ] = 0 . Pr o of. By Theorem 3.3 , the signature of L ev κ,ρ is independent of the choices of κ and ρ , so w e may assume ρ to b e as large as necessary . F rom Theorem 2.5 and Lemma 3.7 w e obtain the direct sum decomposition [ p ] ⊗ A [ D ] = sf  κ D − Γ → L ev κ  = sf  κ D ρ − Γ ρ → L ev κ,ρ  + sf  κ D ρ c − Γ ρ c → L ev κ,ρ c  . The second summand is given by the spectral flo w along the straigh t line path κ D ρ c + π ρ c Γ  t − 1 + tH  π ∗ ρ c , t ∈ [0 , 1] . Cho osing ρ to b e sufficiently large suc h that max  1 , ∥ H ∥  < κρ , it follo ws from   κ D ρ c   ≥ κρ that this path lies within the inv ertibles. Hence w e obtain [ p ] ⊗ A [ D ] = sf  κ D ρ − Γ ρ → L ev κ,ρ  . No w the Hilb ert space H ρ is finite-dimensional . The spectral flo w can then easily b e computed in terms of the signatures of the in v ertible op erators at the endp oin ts: sf  κ D ρ − Γ ρ → L ev κ,ρ  = 1 2  Sig  L ev κ,ρ  − Sig  κ D ρ − Γ ρ   . F rom Eq. ( 3.1 ) we kno w that Sig  κ D ρ − Γ ρ  = − Index( D + ) , and the statemen t follows. 3.2 The o dd index pairing W e consider again an o dd sp ectral triple ( A , H , D ) o ver A and an inv ertible G ∈ M n ( A ) with spectral gap g := ∥ G − 1 ∥ − 1 whose phase equals u , as in Setting 2.6 . Recall the definition of the o dd sp e ctr al lo c aliser L od κ of the pair ( G, D ) with tuning parameter κ ∈ (0 , ∞ ) from Definition 2.7 . Definition 3.10. The o dd trunc ate d sp ectral localiser of the pair ( G, D ) with tuning parameter κ and spectral radius ρ ∈ (0 , ∞ ) is defined as the op erator L od κ,ρ ≡ L od κ,ρ ( G, D ) := π ρ L od κ ( G, D ) π ∗ ρ =  κ D ρ G ∗ ρ G ρ − κ D ρ  . W e assume that the tuning parameter κ ∈ (0 , ∞ ) and the sp ectral radius ρ ∈ (0 , ∞ ) satisfy the conditions from Assumption 3.2 . 12 K oen v an den Dungen Theorem 3.11 ([ DSW23 , Theorem 10.4.1]) . The o dd trunc ate d sp e ctr al lo c aliser L od κ,ρ is invertible with sp e ctr al gap 1 2 g . F urthermor e, the signatur e of L od κ,ρ is indep endent of the choic es of κ and ρ (satisfying Assumption 3.2 ). Remark 3.12. The inv ertibilit y of the truncated sp ectral lo caliser is well-established in the literature; it was first prov en in [ LS17 ], and we cite here the result from the textb o ok [ DSW23 ]. Although [ DSW23 , Theorem 10.4.1] assumes D to b e inv ertible, we note that the first part of its pro of, whic h yields the ab o v e theorem, do es not actually require this assumption. The inv ertibilit y of D is only used in the pro of of the second statement in [ DSW23 , Theorem 10.4.1], which w e will reprov e in Theorem 3.15 b elo w without assuming in v ertibilit y of D . Lemma 3.13. The op er ator L od κ,ρ c := π ρ c L od κ π ∗ ρ c is invertible with sp e ctr al gap q 47 48 κρ . Pr o of. The pro of is similar to the pro of of Lemma 3.5 . W e no w consider the straigh t line path L ( t ) := L od κ,ρ ⊕ L od κ,ρ c + tπ ρ  0 G G ∗ 0  π ∗ ρ c + tπ ρ c  0 G G ∗ 0  π ∗ ρ , t ∈ [0 , 1] , b et ween L (0) = L od κ,ρ ⊕ L od κ,ρ c and L (1) = L od κ . Lemma 3.14. F or sufficiently lar ge ρ , the op er ator L ( t ) is invertible for e ach t ∈ [0 , 1] , and we have the e quality sf  L od κ (1 , D ) → L od κ ( G, D )  = sf  L od κ (1 , D ) → L od κ,ρ ( G, D ) ⊕ L od κ,ρ c ( G, D )  . Pr o of. The pro of is similar to the pro of of Lemma 3.7 . Theorem 3.15. Consider an o dd sp e ctr al triple ( A , H , D ) and a unitary u ∈ M n ( A ) as in Setting 2.6 . L et G ∈ M n ( A ) b e invertible with sp e ctr al gap g = ∥ G − 1 ∥ − 1 such that G | G | − 1 = u . Assume the p ar ameters κ and ρ satisfy Assumption 3.2 . Then the o dd index p airing of [ u ] ∈ K 1 ( A ) with [ D ] ∈ K K 1 ( A, C ) is given by [ u ] ⊗ A [ D ] = 1 2 Sig  L od κ,ρ ( G, D )  . Remark 3.16. This reco vers the main result of [ LS17 ], without assuming D to b e inv ert- ible. In con trast with Theorem 3.8 , there is no correction term for non-inv ertible D , due to the simple fact that the matrix  0 1 1 0  has v anishing signature. Pr o of. The pro of is similar to the pro of of Theorem 3.8 , so w e shall b e somewhat brief. By Theorem 3.11 , the signature of L od κ,ρ ( G, D ) is indep enden t of the choices of κ and ρ , so w e ma y assume ρ to b e as large as necessary . F rom Theorem 2.10 and Lemma 3.14 w e obtain the direct sum decomposition [ u ] ⊗ A [ D ] = sf  L od κ,ρ (1 , D ) → L od κ,ρ ( G, D )  + sf  L od κ,ρ c (1 , D ) → L od κ,ρ c ( G, D )  . a k-theoretic note on the spectral localiser 13 Cho osing ρ to b e sufficien tly large suc h that max  1 , ∥ G ∥  < κρ , the second summand v anishes and we obtain [ u ] ⊗ A [ D ] = sf  L od κ,ρ (1 , D ) → L od κ,ρ ( G, D )  = 1 2  Sig  L od κ,ρ ( G, D )  − Sig  L od κ,ρ (1 , D )   . Finally , we note that L od κ,ρ (1 , D ) is inv ertible for all κ , and therefore its signature is inde- p enden t of κ . The statement then follo ws from Sig  L od κ,ρ (1 , D )  = Sig  0 1 1 0  = 0 . A F redholm op erators and K -theory Let E be a Hilb ert C ∗ -mo dule o ver a C ∗ -algebra A . W e denote by L A ( E ) the C ∗ -algebra of adjoin table op erators on E , and b y K A ( E ) the C ∗ -subalgebra of compact ( = ‘generalised compact’ or A -compact) op erators. In this App endix, we will consider F redholm op erators on E and in particular their index and spectral flo w taking v alues in the K -theory of A . Some computations make use of Kasparo v’s biv arian t K K -theory and the Kasparov pro duct [ Kas80 ], and we refer to the b ook [ Bla98 ] for further details on this theory . A regular op erator D on E is called F r e dholm if there exist a left p ar ametrix Q l and a right p ar ametrix Q r suc h that (the closure of ) Q l D − 1 and D Q r − 1 are compact endomorphisms on E . If D is regular and F redholm, w e denote by Index( D ) ∈ K 0 ( A ) its F redholm index (for its construction, we refer to [ Dun19 , §2.2] and references therein). Let no w D be a regular self-adjoint F redholm op erator on E . W e consider b oth the Z 2 -graded case ( j = 0 ) and the ungraded case ( j = 1 ). In the graded case ( j = 0 ), w e assume that D is o dd with resp ect to the direct sum decomp osition E = E + ⊕ E − giv en b y the Z 2 -grading, so that w e may write D =  0 D − D + 0  . By [ Dun19 , Prop osition 2.14], D yields a well-defined class [ D ] in K K 0 ( C , A ) ≃ K 0 ( A ) resp. K K 1 ( C , A ) ≃ K 1 ( A ) (see [ Dun19 , §2.2] for the construction of this class). F urthermore, if t wo suc h op erators D and D ′ are homotopic, then [ D ] = [ D ′ ] . In the graded case ( j = 0 ), the class [ D ] ∈ K K 0 ( C , A ) corresp onds to Index( D + ) ∈ K 0 ( A ) under the standard isomorphism K K 0 ( C , A ) ≃ K 0 ( A ) . W e recall from [ Dun25a , Proposition A.11] the stabilit y under relativ ely compact p erturbations: if R is a symmetric op erator on E whic h is relativ ely D -compact (i.e., R ( D ± i ) − 1 is compact), then D + R is also regular, self-adjoin t, and F redholm, and [ D + R ] = [ D ] ∈ K K j ( C , A ) (where j = 0 if R, D are odd, and j = 1 otherwise). A.1 Relativ e index and sp ectral flo w W e briefly recall the definitions and basic prop erties of the (even or odd) relative index and the (ev en or o dd) spectral flow on a Hilb ert A -mo dule E , taking v alues in the K - theory group K j ( A ) . F or a more detailed exp osition, we refer to [ Dun25b , App endix A] and [ W ah07 , §3 & §8]. 14 K oen v an den Dungen Consider t wo pro jections P , Q ∈ L A ( E ) , suc h that the difference P − Q is a c omp act endomorphism on E . In the ungraded case ( j = 0 ), we note that compactness of P − Q implies that Q : Ran( P ) → Ran( Q ) is a F redholm operator and th us has a K 0 ( A ) -v alued index. In the graded case ( j = 1 ), we assume E = E + ⊕ E − is Z 2 -graded, and w e require in addition that 2 P − 1 and 2 Q − 1 are o dd. In this case, w e can write P = 1 2  1 U ∗ P U P 1  , and Q = 1 2  1 U ∗ Q U Q 1  , (A.1) where U P , U Q : E + → E − are unitaries suc h that U P U ∗ Q lies in the minimal unitisation of the compact op erators on E − . Definition A.1. Consider pro jections P, Q ∈ L A ( E ) with P − Q ∈ K A ( E ) . W e define the (even or o dd) relative index rel-ind j ( P , Q ) ∈ K j ( A ) as follows. In the ungraded case j = 0 , we define the (even) r elative index of ( P , Q ) by rel-ind( P , Q ) ≡ rel-ind 0 ( P , Q ) := Index  Q : Ran( P ) → Ran( Q )  ∈ K 0 ( A ) . In the graded case j = 1 , we additionally require 2 P − 1 and 2 Q − 1 to b e o dd, and define the o dd r elative index of ( P , Q ) b y rel-ind 1 ( P , Q ) :=  1 0 0 U P U ∗ Q  ∈ K 1 ( A ) , where U P and U Q are obtained from P and Q as in Eq. ( A.1 ). No w consider a regular self-adjoint op erator D ( · ) on the Hilbert C ([0 , 1] , A ) -mo dule C ([0 , 1] , E ) corresp onding to a family of op erators {D ( t ) } t ∈ [0 , 1] on E . W e can define its (even or o dd) sp ectral flow, pro vided that there exist lo cally trivialising families (see [ Dun25a , Definition 2.6]). Note that, in our main case of in terest, suc h trivialising families alw a ys exist (see Prop osition A.3 below). Giv en a regular self-adjoint F redholm op erator D on E and t w o trivialising op erators B 0 and B 1 for D , we note that P > 0 ( D + B 1 ) − P > 0 ( D + B 0 ) is compact (and if j = 1 , the op erators 2 P > 0 ( D + B 0 ) − 1 and 2 P > 0 ( D + B 1 ) − 1 are odd), and we introduce the notation ind j ( D , B 0 , B 1 ) := rel-ind j  P > 0 ( D + B 1 ) , P > 0 ( D + B 0 )  . (A.2) The spectral flo w is then defined as follo ws. Definition A.2. Let j = 0 or j = 1 . Let D ( · ) = {D ( t ) } t ∈ [0 , 1] b e a regular self-adjoin t op- erator on the Hilbert C ([0 , 1] , A ) -mo dule C ([0 , 1] , E ) , for which locally trivialising families exist. (If j = 1 , we require E to b e Z 2 -graded and D (as w ell as all trivialising families) to b e o dd.) Let 0 = t 0 < t 1 < . . . < t n = 1 be suc h that there is a trivialising family {B i ( t ) } t ∈ [ t i ,t i +1 ] of {D ( t ) } t ∈ [ t i ,t i +1 ] for each i = 0 , . . . , n − 1 . Assume that the endp oin ts D (0) and D (1) are in v ertible. Then we define sf j  {D ( t ) } t ∈ [0 , 1]  := ind j  D (0) , 0 , B 0 (0)  + n − 1 X i =1 ind j  D ( t i ) , B i − 1 ( t i ) , B i ( t i )  + ind j  D (1) , B n − 1 (1) , 0  ∈ K j ( A ) , a k-theoretic note on the spectral localiser 15 where ind j is defined in Eq. ( A.2 ). W e call sf 0 the (even) sp e ctr al flow and sf 1 the o dd sp e ctr al flow of the family {D ( t ) } t ∈ [0 , 1] . W e will often simply write sf ≡ sf 0 for the (even) sp ectral flow. The definition of the sp ectral flo w is indep enden t of the c hoice of subdivision and the choice of trivialising families {B i ( t ) } t ∈ [ t i ,t i +1 ] . Prop osition A.3 ([ Dun25a , Proposition 2.8] & [ Dun25b , Prop osition A.11]) . L et D ( · ) = {D ( t ) } t ∈ [0 , 1] b e a r e gular self-adjoint op er ator on the Hilb ert C ([0 , 1] , A ) -mo dule C ([0 , 1] , E ) . (In the gr ade d c ase j = 1 , E is Z 2 -gr ade d and D is o dd.) Assume that the endp oints D (0) and D (1) ar e invertible, D ( t ) : Dom D (0) → E dep ends norm-c ontinuously on t , and D ( t ) − D (0) is r elatively D (0) -c omp act for e ach t ∈ [0 , 1] . Then ther e exists a trivialising family for {D ( t ) } t ∈ [0 , 1] and sf j  {D ( t ) } t ∈ [0 , 1]  = rel-ind j  P > 0 ( D (1)) , P > 0 ( D (0))  ∈ K j ( A ) . In the setting of Prop osition A.3 , it follows in particular that the sp ectral flow dep ends only on the endp oints D (0) and D (1) . W e then in tro duce the notation sf j  D (0) → D (1)  for the sp ectral flow of the straight line path from D (0) to D (1) : sf j  D (0) → D (1)  := sf j  (1 − t ) D (0) + t D (1)  t ∈ [0 , 1]  . (A.3) Corollary A.4 ([ Dun25b , Corollary A.12]) . L et D b e an invertible r e gular self-adjoint op er ator on the Hilb ert A -mo dule E . Consider a unitary op er ator u ∈ L A ( E ) such that u : Dom D → Dom D and [ D , u ] is r elatively D -c omp act. L et χ : [0 , 1] → R b e any c on- tinuous function satisfying χ (0) = 0 and χ (1) = 1 . F or t ∈ [0 , 1] we define D ( t ) := (1 − χ ( t )) D + χ ( t ) u ∗ D u = D + χ ( t ) u ∗ [ D , u ] . Then we have the e quality sf 0  D → u ∗ D u  = sf 0  {D ( t ) } t ∈ [0 , 1]  = − Index  P > 0 ( D ) uP > 0 ( D )  ∈ K 0 ( A ) . The following result shows that the (even or o dd) sp ectral flow implements the Bott p eriodicity isomorphism K j +1  C 0 ( R , A )  ≃ − → K j ( A ) , whic h is giv en by taking the Kasparo v pro duct with the K -homology class [ − i∂ t ] ∈ K K 1  C 0 ( R ) , C  . The latter is represented b y the sp ectral triple  C ∞ c ( R ) , L 2 ( R ) , − i∂ t  . Prop osition A.5 ([ Dun25b , Prop osition A.13], cf. [ W ah07 , §4 & §8]) . Consider a r e gu- lar self-adjoint F r e dholm op er ator D ( · ) = {D ( t ) } t ∈ [0 , 1] on the Hilb ert C ([0 , 1] , A ) -mo dule C ([0 , 1] , E ) , with D (0) and D (1) invertible. (In the gr ade d c ase j = 1 , E is Z 2 -gr ade d and e ach D ( t ) is o dd.) W e extend the family to R by setting D ( t ) := D (0) for al l t < 0 and D ( t ) := D (1) for al l t > 1 , and we view D ( · ) as a r e gular self-adjoint F r e dholm op er ator on the Hilb ert C 0 ( R , A ) -mo dule C 0 ( R , E ) , defining a class [ D ( · )] ∈ K j +1  C 0 ( R , A )  . If ther e exist lo c al ly trivialising families for {D ( t ) } t ∈ R , then sf j  {D ( t ) } t ∈ [0 , 1]  = [ D ( · )] ⊗ C 0 ( R ) [ − i∂ t ] ∈ K j ( A ) . B Computation of a Kasparo v pro duct In this section, we will compute a particular case of a Kasparo v pro duct in the unbounded picture [ BJ83 ] of Kasparov’s biv ariant K K -theory [ Kas80 ]. The computation makes use of 16 K oen v an den Dungen the description of the un b ounded Kasparov pro duct giv en b y Lesc h–Mesland [ LM19 , The- orem 7.4]. The (infinite volume) sp ectral lo caliser naturally app ears from the construction of this Kasparo v pro duct, and this observ ation is the crucial ingredient in the results of Section 2 . Consider unital C ∗ -algebras A and B . Let A ⊂ A b e a dense unital ∗ -subalgebra, and consider an (even) unbounded Kasparov A - B -mo dule ( A , E , D ) ov er A representing a K K -theory class [ D ] ∈ K K 0 ( A, B ) . W e assume that the corresponding represen tation π : A → L B ( E ) is nonde gener ate , i.e., π ( A ) · E is dense in E . On the Z 2 -graded Hilb ert B -mo dule E = E + ⊕ E − w e ma y write the op erator D and the Z 2 -grading Γ D as D =  0 D − D + 0  , Γ D =  1 0 0 − 1  . Consider also a norm-contin uous family { S ( t ) } t ∈ R ⊂ M 2 n ( A ) with the following prop erties: • S ( t ) = S ( − 1) for all t ≤ − 1 and S ( t ) = S (1) for all t ≥ 1 ; • S ( − 1) and S (1) are inv ertible; • S ( t ) is odd w.r.t. the Z 2 -grading Γ S = 1 ⊕ ( − 1) on A ⊕ 2 n = A ⊕ n ⊕ A ⊕ n ; • π ( S ( t )) preserv es Dom( D ) ⊕ 2 n , the comm utator [ D , π ( S ( t ))] is b ounded, and t 7→ [ D , π ( S ( t ))] is norm-con tin uous. W e then obtain an odd self-adjoin t F redholm operator S ∈ C 0  R , M 2 n ( A )  ⊂ L C 0 ( R ,A )  C 0 ( R , A ⊕ 2 n )  , whic h represen ts a K -theory class [ S ] ∈ K 0  C 0 ( R , A )  (see App endix A ). W e consider a real parameter κ satisfying 0 < κ < min n   S ( − 1) − 1   − 2   [ D , S ( − 1)]   − 1 ,   S (1) − 1   − 2   [ D , S (1)]   − 1 o . (B.1) Prop osition B.1. The p airing of [ S ] ∈ K 0  C 0 ( R ) ⊗ A  with [ D ] ∈ K K 0 ( A, B ) over A is given by [ S ] ⊗ A [ D ] = [ L κ ] ∈ K 0  C 0 ( R ) ⊗ B  , wher e the self-adjoint F r e dholm op er ator L κ on the Hilb ert C 0 ( R , B ) -mo dule C 0 ( R , E ⊕ 2 n ) is given by L κ ( t ) := κ Γ S D + S ( t ) , and wher e the p ar ameter κ satisfies Eq. ( B.1 ) . Pr o of. The op erator L κ is giv en by a family {L κ ( t ) } t ∈ R of self-adjoin t op erators with constan t domain Dom L κ ( t ) = (Dom D ) ⊕ 2 n , such that L κ ( t ) ∈ L B  (Dom D ) ⊕ 2 n , E ⊕ 2 n  dep ends norm-contin uously on t . Hence L κ defines a regular self-adjoint op erator on the Hilb ert C 0 ( R , B ) -mo dule C 0 ( R , E ⊕ 2 n ) , whic h is o dd with resp ect to the Z 2 -grading Γ S Γ D . Since D has compact resolv en ts and L κ ( t ) − κ Γ S D is bounded, w e see that L κ ( t ) also has compact resolv ents, and in particular eac h L κ ( t ) is F redholm. Moreov er, the operators L κ ( ± 1) = κ Γ S D + S ( ± 1) are inv ertible by the same argument as in Lemmas 2.3 and 2.8 (using the assumption ( B.1 )). W e then obtain a (left and righ t) parametrix Q for L κ giv en b y Q ( t ) = (1 − χ ( t )) L κ ( t ) − 1 + χ ( t )( L κ ( t ) + i ) − 1 , for an y χ ∈ C c ( R ) such that χ ( t ) = 1 for all t ∈ [ − 1 , 1] . Thus L κ is F redholm and therefore yields a well-defined class [ L κ ] ∈ K 0  C 0 ( R , B )  . a k-theoretic note on the spectral localiser 17 W e ma y c ho ose a smooth function ϕ : R → [1 , ∞ ) suc h that ϕ ( t ) = 1 for all t ∈ [ − 1 , 1] and lim | t |→∞ ϕ ( t ) = ∞ . Then the op erator S ′ giv en by S ′ ( t ) := ϕ ( t ) S ( t ) has compact resolv en ts and therefore defines an un b ounded Kasparov C - C 0 ( R ) ⊗ A -module  C , C 0 ( R , A ⊕ 2 n ) , S ′  , such that [ S ′ ] = [ S ] ∈ K 0  C 0 ( R ) ⊗ A  . Indeed, the norm-contin uous family [0 , 1] × R ∋ ( s, t ) 7→ ϕ ( t ) s S ( t ) ≡ S s ( t ) yields a regular self-adjoint operator S • on C 0  [0 , 1] × R , A ⊕ 2 n  , which is F redholm with a parametrix giv en b y ( s, t ) 7→ (1 − χ ( s, t )) S s ( t ) − 1 , for any χ ∈ C c  [0 , 1] × R  with χ ( s, t ) = 1 for all ( s, t ) ∈ [0 , 1] × [ − 1 , 1] . Th us S • pro vides a homotopy b et ween S and S ′ , so that [ S ] = [ S ′ ] . Similarly , w e may replace L κ b y the op erator L ′ κ := κ Γ S D + S ′ . Indeed, since [ D , S ] is b ounded, the graded commutator [Γ S D , S ′ ] is relativ ely b ounded by S ′ , so that L ′ κ is regular and self-adjoint on the domain Dom( D ) ∩ Dom( S ′ ) (cf. [ KL12 , Theorem 7.10]). Since S ′ has compact resolven ts, it then follows that also L ′ κ has compact resolven ts and therefore defines an unbounded Kasparov C - C 0 ( R , B ) -mo dule  C , C 0 ( R , E ⊕ 2 n ) , L ′ κ  . The regular self-adjoin t operator L κ, • giv en b y L κ,s ( t ) := κ Γ S D + S s ( t ) has a parametrix given b y ( s, t ) 7→ (1 − χ ( s, t )) L κ,s ( t ) − 1 + χ ( s, t )( L κ,s ( t ) + i ) − 1 , so it provides a homotopy betw een L κ and L ′ κ and we therefore ha v e that [ L ′ κ ] = [ L κ ] ∈ K 0  C 0 ( R , B )  . It remains to show that L ′ κ represen ts the (unbounded) Kasparov pro duct of S ′ with D . W e will compute the Kasparo v pro duct b y applying [ LM19 , Theorem 7.4] to the operators S ′ and T := κ Γ S D on C 0  R , E ⊕ 2 n  . Note that A is unital, and there is a dense ∗ -subalgebra A ⊂ A suc h that [ D , a ] is b ounded for all a ∈ A . Since D acts diagonally on E ⊕ 2 n , condition (i) of [ LM19 , Theorem 7.4] is satisfied b y the dense submo dule C c  R , A ⊕ 2 n  . Condition (ii) is trivially satisfied ( C commutes with T ). Finally , condition (iii) is satisfied b ecause the comm utator [ D , S ] is b ounded, so that the an ticommutator [ T , S ′ ] + is relativ ely b ounded b y S ′ . Hence the statement of [ LM19 , Theorem 7.4] shows that S ′ + T = ϕS + κ Γ S D = L ′ κ indeed represents the Kasparo v product of [ S ′ ] = [ S ] with [ κ D ] = [ D ] . Corollary B.2. In the setting of Pr op osition B.1 , c onsider now inste ad a trivially graded element S ∈ C 0  R , M n ( A )  . The p airing of [ S ] ∈ K 1  C 0 ( R ) ⊗ A  with [ D ] ∈ K K 0 ( A, B ) over A is then given by [ S ] ⊗ A [ D ] = [ L ev κ ] ∈ K 1  C 0 ( R ) ⊗ B  , wher e the self-adjoint F r e dholm op er ator L ev κ on the Hilb ert C 0 ( R , B ) -mo dule C 0 ( R , E ⊕ n ) is given by L ev κ ( t ) := κ D + Γ D S ( t ) , and wher e the p ar ameter κ satisfies Eq. ( B.1 ) . Pr o of. The statemen t follo ws b y com bining Prop osition B.1 with the description of the o dd-ev en Kasparo v product giv en in [ BMS16 , Example 2.38]. Corollary B.3. In the setting of Pr op osition B.1 , c onsider now inste ad an o dd un- b ounde d Kasp ar ov A - B -mo dule ( A , E , D ) . The p airing of [ S ] ∈ K 0  C 0 ( R ) ⊗ A  with [ D ] ∈ K K 1 ( A, B ) over A is then given by [ S ] ⊗ A [ D ] = [ L od κ ] ∈ K 1  C 0 ( R ) ⊗ B  , wher e the self-adjoint F r e dholm op er ator L od κ on the Hilb ert C 0 ( R , B ) -mo dule C 0 ( R , E ⊕ 2 n ) is given by L od κ ( t ) := κ Γ S D + S ( t ) , and wher e the p ar ameter κ satisfies Eq. ( B.1 ) . Pr o of. The statemen t follo ws b y com bining Prop osition B.1 with the description of the ev en-o dd Kasparov pro duct given in [ BMS16 , Example 2.37]. 18 K oen v an den Dungen References [BES94] J. Bellissard, A. v an Elst, and H. Sc hulz-Baldes, The nonc ommutative ge ometry of the quantum Hal l effe ct , J. Math. Phys. 35 (1994), no. 10, 5373–5451 . [BJ83] S. Baa j and P . 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