Addendum to: Indefinite Kasparov modules and pseudo-Riemannian manifolds
aa r X i v : . [ m a t h . K T ] O c t Addendum to
Indefinite Kasparov modules and pseudo-Riemannian manifolds
Koen van den Dungen ∗ Mathematisches Institut , Universit¨at BonnEndenicher Allee 60, D-53115 Bonn
Abstract
We improve our previous results on indefinite Kasparov modules, which provide a generalisation ofunbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Inparticular, we can weaken the assumptions that are imposed on indefinite Kasparov modules. Usinga new theorem by Lesch and Mesland on the self-adjointness and regularity of the sum of two weaklyanticommuting operators, we show that we still have an equivalence between indefinite Kasparov modulesand pairs of Kasparov modules. Importantly, the weakened version of indefinite Kasparov modules nowincludes the main motivating example of the Dirac operator on a pseudo-Riemannian manifold. Theappendix contains a construction of an approximate identity for weakly commuting operators, which isdue to Lesch and Mesland.
Keywords : KK -theory; Lorentzian manifolds; noncommutative geometry. Mathematics Subject Classification 2010 : 19K35, 53C50, 58B34.
In a previous paper [DR16], we presented a definition of indefinite Kasparov modules, providing a generalisa-tion of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators.Our main theorem showed that to each indefinite Kasparov module we can associate a pair of (genuine)Kasparov modules, and that this process is reversible. The main assumption we imposed in the definition ofan indefinite Kasparov module ( A , E B , D ) is that Re D and Im D almost anticommute. This means, roughlyspeaking, that the anticommutator { Re D , Im D} is relatively bounded by Re D . The main tool we used is atheorem by Kaad and Lesch [KL12, Theorem 7.10], which states that the sum of two almost anticommutingoperators is regular and self-adjoint. The purpose of this short note is to improve on the results presentedin [DR16].The main issue with the results of [DR16] is that, unfortunately, our main motivating example, namelythe Dirac operator /D on a pseudo-Riemannian manifold, does not (in general) satisfy the definition of anindefinite Kasparov module. For such a Dirac operator, the real part Re /D contains the spacelike derivatives,while the imaginary part Im /D contains the timelike derivatives. The assumption that Re /D and Im /D almost anticommute then means that the anticommutator { Re /D, Im /D } contains only spacelike derivatives.In general, however, this anticommutator is a first-order differential operator containing both spacelike andtimelike derivatives.Thus, in order to improve our results, we need a generalisation of the theorem by Kaad and Lesch, inwhich the anticommutator { Re D , Im D} is allowed to be relatively bounded by the ‘combined graph norm’of Re D and Im D . This generalisation is now available thanks to recent work by Lesch and Mesland [LM19]. ∗ Email: [email protected] { Re /D, Im /D } is always a first-order differential operator.In Section 2, we will review the results of Lesch and Mesland [LM19]. Moreover, we will show that thesum of two weakly commuting (instead of anticommuting) operators is also essentially self-adjoint (thoughin general not closed). The proof of this fact relies on an alternative method of proof for the main resultof [LM19], which is also due to Lesch and Mesland, and which is included in the appendix. In Section 3,we will describe a natural example of weakly anticommuting operators, by decomposing the Dirac operatoron a Riemannian spin manifold with a given orthogonal direct sum decomposition of the tangent bundle.Finally, in Section 4, we will show that we can weaken the assumptions in the definitions given in [DR16],while all the results of [DR16] remain valid. Acknowledgements
The author thanks Matthias Lesch and Bram Mesland for interesting discussions, helpful suggestions, andfor kindly allowing the author to reproduce their proof of Proposition 2.3 in the Appendix. The author alsothanks the referees, whose comments have led to significant improvements in the article.
We consider regular self-adjoint operators S and T on a Hilbert B -module E (where B is a C ∗ -algebra),such that Dom S ∩ Dom T is dense. For x, y ∈ Dom S ∩ Dom T , we define the ‘combined graph inner product’ h x | y i S,T := h x | y i + h Sx | Sy i + h T x | T y i , and denote the corresponding norm by k · k S,T .We denote by [
S, T ] ± the (anti)commutator ST ± T S , which is defined on the natural initial domainDom([
S, T ] ± ) = (cid:8) x ∈ Dom S ∩ Dom T : Sx ∈ Dom T & T x ∈ Dom S (cid:9) . Rather than the notion of almost (anti)commuting operators given in [DR16, Definition 2.8] (whichwas based on [KL12, Assumption 7.1]), we will now consider the following (weaker) notion of weakly(anti)commuting operators.
Definition 2.1 ([LM19, Definition 2.1]) . Two regular self-adjoint operators S and T on a Hilbert B -module E are called weakly (anti)commuting if(1) there is a constant C > x ∈ Dom([
S, T ] ± ) we have (cid:10) [ S, T ] ± x | [ S, T ] ± x (cid:11) ≤ C h x | x i S,T ;(2) there is a core
E ⊂
Dom T such that ( S + λ ) − ( E ) ⊂ Dom[
S, T ] ± for λ ∈ i R , | λ | ≥ λ > S and T , it follows aposteriori that the assumption is also satisfied with S and T interchanged [LM19, § E = Dom T [LM19, Proposition 3.5]. The main result of [LM19] is the following: Theorem 2.2 ([LM19, Theorem 2.6]) . Let S and T be weakly anticommuting operators on a Hilbert B -module E . Then the operator S + T is regular and self-adjoint on the domain Dom( S + T ) = Dom S ∩ Dom T . This theorem can be proved in (at least) two different ways. One method is based on the followingproposition, which is also due to Matthias Lesch and Bram Mesland (but is not included in [LM19]). Theproof of this proposition is given in the Appendix. 2 roposition 2.3.
Let S and T be weakly commuting operators on a Hilbert B -module E . Then λ (cid:2) S, ( S + λ ) − ( T + λ ) − (cid:3) − and λ (cid:2) T, ( S + λ ) − ( T + λ ) − (cid:3) − are uniformly bounded (for λ ∈ i R , | λ | ≥ λ ) and convergestrongly to zero as λ → ± i ∞ . Given two weakly commuting operators S and T , one may check that S ± iT are closed operators. Thanksto Proposition 2.3, we have an approximate identity A n := − n ( S − in ) − ( T − in ) − such that [ S ± iT, A n ]converges strongly to zero as n → ∞ . By a standard argument (analogous to the proof of Proposition 2.4below) it then follows that ( S ± iT ) ∗ = ( S ∓ iT ). Using a doubling trick ( cf. [LM19, § S + T in case of two weakly anti commuting operators S and T . Finally, one can apply the local-global principle [Pie06, KL12] to prove regularity of S + T .The proof of Theorem 2.2 given in [LM19] is different, and in fact proves a stronger statement. Indeed,the proof in [LM19] not only shows that S + T is regular self-adjoint, but also that the resolvent ( S + T + µ ) − (with µ ∈ i R ) can be approximated by ( S + T + λ − ST + µ ) − as | λ | → ∞ ( λ ∈ i R ).The advantage of the method via Proposition 2.3 is that it also allows us to prove that the sum of twoweakly commuting operators (instead of anticommuting operators) is essentially self-adjoint. (Note that thesum of weakly commuting operators is in general not closed; the obvious example is T = − S .) Proposition 2.4.
Let S and T be weakly commuting operators on a Hilbert B -module E . Then S + T isessentially self-adjoint on Dom S ∩ Dom T .Proof. Since S + T is symmetric, it suffices to prove that Dom( S + T ) ∗ ⊂ Dom( S + T ). Let ξ ∈ Dom( S + T ) ∗ .Using the approximate identity A n := − n ( S − in ) − ( T − in ) − , we define the sequence ξ n := A n ξ ∈ Dom( S + T ) , which converges in norm to ξ (see Lemma A.1). For η ∈ Dom( S + T ), we can calculate h ξ n | ( S + T ) η i = (cid:10) A n ξ (cid:12)(cid:12) ( S + T ) η (cid:11) = (cid:10) ξ (cid:12)(cid:12) A ∗ n ( S + T ) η (cid:11) = (cid:10) ξ (cid:12)(cid:12) ( S + T ) A ∗ n η (cid:11) − (cid:10) ξ (cid:12)(cid:12) (cid:2) S + T, A ∗ n (cid:3) η (cid:11) = (cid:10) A n ( S + T ) ∗ ξ (cid:12)(cid:12) η (cid:11) − (cid:10)(cid:2) S + T, A ∗ n (cid:3) ∗ ξ (cid:12)(cid:12) η (cid:11) . Hence we find ( S + T ) ξ n = ( S + T ) ∗ ξ n = A n ( S + T ) ∗ ξ − (cid:2) S + T, A ∗ n (cid:3) ∗ ξ. (1)By Lemma A.1, the first term converges strongly to ( S + T ) ∗ ξ . Furthermore, on Dom( S + T ) we have theequality (cid:2) S + T, A ∗ n (cid:3) ∗ = − (cid:2) S + T, A n (cid:3) . Since both sides of this equality are bounded and adjointable, the left-hand-side equals the closure of theright-hand-side on all of E . Hence we know from Proposition 2.3 that the second term in Eq. (1) convergesto zero. Thus ( S + T ) ξ n converges, which proves that ξ ∈ Dom( S + T ).Again, one can try to apply the local-global principle to prove regularity of (the closure of) S + T .However, since we will not need regularity in the remainder of this article, we will not pursue this anyfurther. In this section we will describe a decomposition of the Dirac operator as a sum of two weakly anti-commutingoperators. Let (
M, g ) be an n -dimensional Riemannian spin manifold with the spinor bundle S . Supposethat we have an orthogonal decomposition T M = E ⊕ E , where E and E are oriented subbundles ofranks n and n , respectively. Locally, we can consider oriented orthonormal frames { e , . . . , e n } of T M such that e j ∈ Γ ∞ ( E ) for 1 ≤ j ≤ n and e j ∈ Γ ∞ ( E ) for n + 1 ≤ j ≤ n + n = n .3e consider the Clifford representation γ : Γ ∞ ( T M ) → Γ ∞ (End( S )) (our conventions are such that γ ( v ) = − g ( v, v ) and γ ( v ) ∗ = − γ ( v )). The Dirac operator /D on Γ ∞ c ( S ) is given locally by /D = n X j =1 γ ( e j ) ∇ Se j . We define a self-adjoint unitary operator Γ ∈ Γ ∞ (End( S )) on L ( S ) which is locally given byΓ := i n ( n +1) / γ ( e ) · · · γ ( e n ) , where { e , . . . , e n } is a local oriented orthonormal frame of E . We note that, in the case of a pseudo-Riemannian manifold, if the metric is negative-definite on E and positive-definite on E , then the operator( − i ) n Γ is the usual fundamental symmetry which turns the Hilbert space L ( S ) into a Krein space (see[DR16, § /D := 12 (cid:0) /D − ( − n Γ /D Γ (cid:1) , /D := 12 (cid:0) /D + ( − n Γ /D Γ (cid:1) . Then /D and /D are both symmetric operators on Γ ∞ c ( S ), and we have /D + /D = /D . In terms of a localorthonormal frame (corresponding to the decomposition T M = E ⊕ E ), we have the explicit expressions /D = n X j =1 (cid:16) γ ( e j ) ∇ Se j − γ ( e j ) (cid:2) ∇ Se j , Γ (cid:3) Γ (cid:17) + n X k = n +1 γ ( e k ) (cid:2) ∇ Se k , Γ (cid:3) Γ ,/D = n X j =1 γ ( e j ) (cid:2) ∇ Se j , Γ (cid:3) Γ + n X k = n +1 (cid:16) γ ( e k ) ∇ Se k − γ ( e k ) (cid:2) ∇ Se k , Γ (cid:3) Γ (cid:17) . Proposition 3.1.
Assume that M is complete and has bounded geometry. Then (the closures of ) theoperators /D and /D are self-adjoint and weakly anticommuting.Proof. The completeness of M implies that the symmetric operators /D and /D are self-adjoint. Theassumption of bounded geometry ensures that the coefficients of /D and /D are globally bounded. Sinceboth /D + /D and /D − /D are elliptic, there exists C > ψ ∈ Dom /D ∩ Dom /D we have k ψ k + k ( /D ± /D ) ψ k ≤ k ψ k + k /D ψ k + k /D ψ k ≤ C k ψ k /D ± /D . where k · k /D ± /D denotes the graph norm of /D ± /D . Thus the graph norm of /D ± /D is equivalentto the combined graph norm k · k /D , /D , and we have the equality Dom /D ∩ Dom /D = Dom( /D ± /D ).Since the principal symbols of /D and /D anticommute, we know that the anticommutator [ /D , /D ] + is afirst-order differential operator, which (by bounded geometry) has globally bounded coefficients. Using againellipticity of /D ± /D and the equivalence k · k /D ± /D ∼ k · k /D , /D , there exists a C ′ > ψ ∈ Dom[ /D , /D ] ± we have the inequality (cid:13)(cid:13) [ /D , /D ] + ψ (cid:13)(cid:13) ≤ C ′ k ψ k /D , /D . It follows that condition (1) of Definition 2.1 is satisfied.To prove the domain condition (2), we will make use of a clever Clifford matrix trick, inspired by theproof of [LM19, Theorem 5.1]. Consider the operators S := (cid:18) /D /D (cid:19) , T := (cid:18) /D + /D /D − /D (cid:19) . We note that the combined graph norm k · k
S,T is equivalent to the graph norm of the elliptic operator T ,and the anticommutator [ S, T ] + = (cid:18) /D , /D ] + [ /D , /D ] + (cid:19)
4s relatively bounded by T . Hence S and T satisfy condition (1) of Definition 2.1. Furthermore, usingagain the assumption of bounded geometry, we note that Dom T is a core for S , and that the second-orderdifferential operators ST and T S are well-defined on the domain ( T ± i ) − · Dom T = Dom T of the second-order elliptic operator T . So S and T also satisfy condition (2) of Definition 2.1. Thus S and T weaklyanticommute, and it follows from [LM19, Theorem 2.6.(2)] that we also have the domain condition( S ± i ) − · Dom T = Dom[ S, T ] ± . Rephrasing this in terms of /D and /D , we obtain( /D ± i ) − · (Dom /D ∩ Dom /D )= (cid:8) ψ ∈ Dom /D ∩ Dom /D : /D ψ ∈ Dom /D ∩ Dom /D , /D ψ ∈ Dom /D (cid:9) . Since Dom /D ∩ Dom /D is a core for /D , this proves that /D and /D also satisfy condition (2) of Definition2.1. First, let us briefly recall our notion of (reverse) ‘Wick rotations’ [DR16, Definitions 2.2 & 2.5]. Given aclosed operator D on a Hilbert B -module E such that Dom D ∩
Dom D ∗ is dense, we define the real andimaginary parts of D as the closures ofRe D := 12 ( D + D ∗ ) , Im D := − i D − D ∗ ) , on the initial domain Dom D ∩
Dom D ∗ . Furthermore, we define the ‘Wick rotations’ of D as the closures of D + := Re D + Im D , D − := Re D − Im D , on the initial domain Dom Re D ∩
Dom Im D .Conversely, given two closed symmetric operators D and D on E such that Dom D ∩ Dom D is dense,we define the reverse Wick rotation of the pair ( D , D ) as the closure of D := 12 ( D + D ) + i D − D )on the initial domain Dom D ∩ Dom D .We now replace our former definitions of indefinite Kasparov modules [DR16, Definition 3.1] and pairsof Kasparov modules [DR16, Definition 3.6], using the weaker notion of weakly anticommuting operatorsdescribed above. Definition 4.1.
Given (separable) Z -graded C ∗ -algebras A and B , an indefinite unbounded Kasparov A - B -module ( A , E B , D ) is given by • a Z -graded, countably generated, right Hilbert B -module E ; • a Z -graded ∗ -homomorphism π : A → End B ( E ); • a separable dense ∗ -subalgebra A ⊂ A ; • a closed odd operator D : Dom D ⊂ E → E such that(1) there exists a linear subspace E ⊂
Dom
D ∩
Dom D ∗ which is dense with respect to k · k D , D ∗ , andwhich is a core for both D and D ∗ ;(2) the operators Re D and Im D are regular and essentially self-adjoint on E ;(3) the operators Re D and Im D are weakly anticommuting ;(4) we have the inclusion π ( A ) · E ⊂ Dom
D ∩
Dom D ∗ , and the graded commutators [ D , π ( a )] ± and[ D ∗ , π ( a )] ± are bounded on E for each a ∈ A ;55) the map π ( a ) ◦ ι : Dom D ∩
Dom D ∗ ֒ → E → E is compact for each a ∈ A , where ι : Dom D ∩
Dom D ∗ ֒ → E denotes the natural inclusion map, and Dom D ∩
Dom D ∗ is considered as a Hilbert B -module with the inner product h·|·i D , D ∗ .If B = C and A is trivially graded, we will write E = H and refer to ( A , H , D ) as an even indefinite spectraltriple over A . Remark 4.2.
In contrast with [DR16, Definition 3.1], we no longer assume that D is regular (i.e. that1 + D ∗ D has dense range), since this assumption is not used anywhere. Definition 4.3.
We say ( A , E B , D , D ) is a pair of unbounded Kasparov A - B -modules if ( A , E B , D ) and( A , E B , D ) are unbounded Kasparov A - B -modules such that:(1) there exists a linear subspace E ⊂
Dom D ∩ Dom D which is a common core for D and D ;(2) the operators D + D and D − D are regular and essentially self-adjoint on E ;(3) the operators D + D and D − D are weakly anticommuting.If B = C and A is trivially graded, we will write E = H and refer to ( A , H , D , D ) as an even pair of spectraltriples over A .Using Theorem 2.2 instead of [DR16, Corollary 2.12], the proof of [DR16, Proposition 3.8] carries through.Furthermore, using Proposition 2.4 instead of [DR16, Proposition 2.13], the proof of [DR16, Proposition 3.9]also carries through. Thus we still have an equivalence between indefinite Kasparov modules and pairs ofKasparov modules. Theorem 4.4 (cf. [DR16, Theorem 3.11]) . The procedure of (reverse) Wick rotation implements a bijectionbetween indefinite unbounded Kasparov A - B -modules ( A , E B , D ) and pairs of unbounded Kasparov A - B -modules ( A , E B , D , D ) . This bijection also descends to the corresponding unitary equivalence classes. The main advantage of the new version of indefinite Kasparov modules is that the definition now incor-porates any pseudo-Riemannian manifold (with only mild assumptions).
Proposition 4.5.
Let ( M, g ) be an n -dimensional time- and space-oriented pseudo-Riemannian spin man-ifold of signature ( t, s ) , with a given spinor bundle S → M . Let r be a spacelike reflection, such that theassociated Riemannian metric g r is complete. Assume furthermore that ( M, g, r, S ) has bounded geometry(as in [DR16, Definition 4.1]). Then the canonical Dirac operator /D on S → M yields an indefinite spectraltriple ( C ∞ c ( M ) , L ( S ) , /D ) .Proof. The main thing to check is that Re /D and Im /D weakly anticommute ( cf. [DR16, § { e j } corresponding to the decomposition T M = E t ⊕ E s as the direct sum of a timelike and a spacelike subbundle(of rank t and s = n − t , respectively), and defining the fundamental symmetry J M := i t ( t − / γ ( e ) · · · γ ( e t ),we have the explicit expressions [DR16, § /D = n X j = t +1 γ ( e j ) ∇ Se j + 12 n X j =1 γ ( e j ) J M (cid:2) ∇ Se j , J M (cid:3) , Im /D = i t X j =1 γ ( e j ) ∇ Se j + i n X j =1 γ ( e j ) J M (cid:2) ∇ Se j , J M (cid:3) . We observe that Re /D and Im /D are symmetric operators whose principal symbols anti-commute, and thatRe /D + Im /D and Re /D − Im /D are both elliptic. Hence the same argument as in the proof of Proposition3.1 applies, and it follows that Re /D and Im /D are self-adjoint and weakly anti-commuting.6 An approximate identity for weakly commuting operators
Based on unpublished notes by Matthias Lesch and Bram Mesland
This Appendix contains the proof of Proposition 2.3. The argument is due to Matthias Lesch and BramMesland, and the author kindly thanks them for their permission to reproduce their proof here.To remind ourselves, and for the convenience of the reader, we recall the following facts: • It is a consequence of the Banach-Steinhaus Theorem that a strongly convergent sequence of boundedoperators on a Banach space is uniformly norm bounded. • Given a uniformly bounded sequence ( A n ) ⊂ L ( X ) of operators on a Banach space X , then for ( A n )being strongly continuous it suffices to show pointwise convergence on a dense subspace. • As a consequence of uniform boundedness, if ( A n ) converges strongly to A and if ( B n ) convergesstrongly to B , then ( A n · B n ) converges strongly to A · B .These facts will be used repeatedly without mentioning.Now let S and T be weakly commuting, regular self-adjoint operators on E , and denote by [ S, T ] =[
S, T ] − = ST − T S the ordinary commutator. Let λ, µ ∈ i R with | λ | , | µ | ≥ λ . Lemma A.1.
The operators λ ( S + λ ) − , λ ( T + λ ) − , and λ ( S + λ ) − ( T + λ ) − converge strongly to theidentity as | λ | → ∞ .Proof. The family (cid:0) S ( S + λ ) − (cid:1) | λ |≥ λ is uniformly bounded, and for ψ ∈ Dom S we have k S ( S + λ ) − ψ k = k ( S + λ ) − Sψ k ≤ | λ | k Sψ k . Hence S ( S + λ ) − converges to zero strongly. Consequently, λ ( S + λ ) − =1 − S ( S + λ ) − converges strongly to the identity. Thus the product λ ( S + λ ) − λ ( T + λ ) − also convergesstrongly to the identity. Lemma A.2.
For λ large enough and for λ, µ ∈ i R with | λ | , | µ | > λ , the operator families [ S, T ]( S + λ ) − ( T + µ ) − and [ S, T ]( T + λ ) − ( S + µ ) − converge strongly to zero as | λ | → ∞ (for fixed µ ).Proof. Since S ( S + λ ) − converges strongly to zero and ( T + µ ) − is bounded, we know that also S ( S + λ ) − ( T + µ ) − converges strongly to zero as | λ | → ∞ . We will show that also T ( S + λ ) − ( T + µ ) − convergesstrongly to zero as | λ | → ∞ . We write T ( S + λ ) − ( T + µ ) − = ( S + λ ) − [ S, T ]( S + λ ) − ( T + µ ) − + ( S + λ ) − T ( T + µ ) − . (2)Since T ( T + µ ) − is bounded, the second summand is of order | λ | − and therefore converges in norm to zero.For the first summand, we note that by [LM19, Lemma 3.2] we have for λ large enough and ψ ∈ Dom[
S, T ]that there exists c > (cid:13)(cid:13) [ S, T ] ψ (cid:13)(cid:13) ≤ c (cid:18) | λ | + 1 | µ | (cid:19) (cid:13)(cid:13) ( T + µ )( S + λ ) ψ (cid:13)(cid:13) . In particular, [
S, T ]( S + λ ) − ( T + µ ) − is uniformly bounded, so the first summand in Eq. (2) also convergesstrongly to zero. Finally, by condition (1) of Definition 2.1 we have (cid:13)(cid:13) [ S, T ]( S + λ ) − ( T + µ ) − ψ (cid:13)(cid:13) ≤ C (cid:16)(cid:13)(cid:13) ( S + λ ) − ( T + µ ) − ψ (cid:13)(cid:13) + (cid:13)(cid:13) S ( S + λ ) − ( T + µ ) − ψ (cid:13)(cid:13) + (cid:13)(cid:13) T ( S + λ ) − ( T + µ ) − ψ (cid:13)(cid:13) (cid:17) , which shows that [ S, T ]( S + λ ) − ( T + µ ) − ψ converges strongly to zero as well. Interchanging S and T , thesame result also applies to [ S, T ]( T + λ ) − ( S + µ ) − . Lemma A.3.
The operators λ (cid:2) S, ( S + λ ) − ( T + λ ) − (cid:3) and λ (cid:2) T, ( S + λ ) − ( T + λ ) − (cid:3) are uniformly boundedfor | λ | ≥ λ . roof. We write λ (cid:2) T, ( S + λ ) − ( T + λ ) − (cid:3) = λ ( S + λ ) − [ S, T ]( S + λ ) − ( T + λ ) − . The first factor λ ( S + λ ) − is of order | λ | in norm. The second factor [ S, T ]( S + λ ) − ( T + λ ) − is of order | λ | − by [LM19, Lemma 3.2]. Hence λ (cid:2) T, ( S + λ ) − ( T + λ ) − (cid:3) is uniformly bounded (in norm) for | λ | ≥ λ .By interchanging S and T , we see that also λ (cid:2) S, ( T + λ ) − ( S + λ ) − (cid:3) is uniformly bounded (in norm). Itthen follows that λ (cid:2) S, ( S + λ ) − ( T + λ ) − (cid:3) = − (cid:16) λ (cid:2) S, ( T + λ ) − ( S + λ ) − (cid:3)(cid:17) ∗ is uniformly bounded as well. Proof of Proposition 2.3 . By Lemma A.3 it suffices to establish strong convergence on a dense submoduleof E . With λ as in Lemma A.2, and for some µ ∈ i R with | µ | > λ , we rewrite λ (cid:2) T, ( S + λ ) − ( T + λ ) − (cid:3) = λ ( S + λ ) − · [ S, T ]( S + λ ) − ( T + µ ) − · λ ( T + λ ) − · ( T + µ ) . As | λ | → ∞ , the first and third factors converge strongly to the identity by Lemma A.1, while the secondfactor converges strongly to zero by Lemma A.2. Thus this proves that λ (cid:2) T, ( S + λ ) − ( T + λ ) − (cid:3) convergesstrongly to zero on the dense submodule Dom T , and hence on E . Next, we rewrite λ (cid:2) S, ( S + λ ) − ( T + λ ) − (cid:3) = λ ( S + λ ) − ( T + λ ) − · [ T, S ]( T + λ ) − ( S + µ ) − · ( S + µ ) . Again, by Lemmas A.1 and A.2, as | λ | → ∞ the first factor converges strongly to the identity, while thesecond factor converges strongly to zero. Thus λ (cid:2) S, ( S + λ ) − ( T + λ ) − (cid:3) converges strongly to zero onDom S , and hence on E . References [DR16] K. van den Dungen and A. Rennie,
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