Localisations of half-closed modules and the unbounded Kasparov product
aa r X i v : . [ m a t h . K T ] J un Localisations of half-closed modulesand the unbounded Kasparov product
Koen van den Dungen ∗ Mathematisches Institut , Universität BonnEndenicher Allee 60, D-53115 Bonn
Abstract
In the context of the Kasparov product in unbounded KK -theory, a well-known theoremby Kucerovsky provides sufficient conditions for an unbounded Kasparov module torepresent the (internal) Kasparov product of two other unbounded Kasparov modules.In this article, we discuss several improved and generalised variants of Kucerovsky’stheorem. First, we provide a generalisation which relaxes the positivity condition, byreplacing the lower bound by a relative lower bound. Second, we also discuss Kucer-ovsky’s theorem in the context of half-closed modules, which generalise unboundedKasparov modules to symmetric (rather than self-adjoint) operators. In order to dealwith the positivity condition for such non-self-adjoint operators, we introduce a fairlygeneral localisation procedure, which (using a suitable approximate unit) provides a‘localised representative’ for the KK -class of a half-closed module. Using this localisa-tion procedure, we then prove several variants of Kucerovsky’s theorem for half-closedmodules. A distinct advantage of the localised approach, also in the special case of self-adjoint operators (i.e., for unbounded Kasparov modules), is that the (global) positivitycondition in Kucerovsky’s original theorem is replaced by a (less stringent) ‘local’ pos-itivity condition, which is closer in spirit to the well-known Connes-Skandalis theoremin the bounded picture of KK -theory. Keywords : Unbounded KK -theory; the Kasparov product; symmetric operators. Mathematics Subject Classification 2010 : 19K35. ∗ Email: [email protected] Koen van den Dungen
Let A , B , and C be C ∗ -algebras. Kasparov’s KK -theory [Kas80] provides the abelian group KK ( A, B ) of homotopy equivalence classes of (bounded) Kasparov A - B -modules. One ofthe main features of KK -theory is the existence of an associative bilinear pairing called theKasparov product: KK ( A, B ) × KK ( B, C ) → KK ( A, C ) . However, this Kasparov product can often be difficult to compute. An important improve-ment was made by Connes and Skandalis [CS84], who provided sufficient conditions (theso-called connection condition and positivity condition) which ensure that a certain Kas-parov module represents the Kasparov product of two other given Kasparov modules.It was shown by Baaj and Julg [BJ83] that elements in KK -theory can also be rep-resented by unbounded Kasparov modules. Many examples of elements in KK -theory areconstructed from geometric situations and are most naturally represented by unboundedKasparov modules. For example, the fundamental class in the K -homology of a spin man-ifold is naturally represented by the Dirac operator (viewed as an unbounded operator onthe Hilbert space of spinors). A distinct advantage of the unbounded picture is that theKasparov product is often easier to compute; in fact, under suitable assumptions, the un-bounded Kasparov product can be explicitly constructed [Mes14, KL13, BMS16].However, the unbounded Kasparov modules introduced by Baaj and Julg require theunbounded operators to be self-adjoint . This self-adjointness is rather natural in the caseof unital C ∗ -algebras (e.g. for compact manifolds), but imposes a completeness conditionin the non-unital case (e.g. for non-compact manifolds) [MR16]. Thus, unfortunately, theBaaj-Julg framework does not include typical examples such as Dirac-type operators onnon-compact, incomplete , Riemannian spin manifolds (which in general are only symmet-ric but not self-adjoint). Nevertheless, it was shown by Baum-Douglas-Taylor [BDT89]that any elliptic symmetric first-order differential operator D on a Riemannian manifold M yields a class [ D ] := [ F D ] in the K -homology of M , given by the bounded transform F D := D (1 + D ∗ D ) − . Hilsum [Hil10] later provided an abstract definition of half-closedmodules , which generalise unbounded Kasparov modules by replacing the self-adjointnesscondition by a more flexible symmetry condition. Moreover, Hilsum proved that the boundedtransform of a half-closed module still yields a well-defined class in KK -theory. Hilsum’sframework encompasses the symmetric elliptic operators of Baum-Douglas-Taylor as well asthe vertically elliptic operators on submersions of (possibly incomplete) smooth manifoldsstudied in [KS18, KS, Dun20].An important step in the development of the unbounded Kasparov product was providedby Kucerovsky’s theorem [Kuc97], which translated the Connes-Skandalis result to the un-bounded framework of Baaj and Julg. Thus, given two unbounded Kasparov modules ( A , ( E ) B , D ) and ( B , ( E ) C , D ) , Kucerovsky provided sufficient conditions for anotherunbounded Kasparov module ( A , E C , D ) to represent the (internal) Kasparov product. How-ever, while the positivity condition of Connes-Skandalis is a local condition, the positivitycondition of Kucerovsky is a global condition (and is therefore more restrictive). Moreover,Kucerovsky’s theorem only applies to the self-adjoint (complete) case.In this article, we discuss several improved and generalised variants of Kucerovsky’s the- ocalisations and the kasparov product relative lower bound with respect to the operatorrepresenting the Kasparov product;(II) we ensure that the positivity condition only needs to be checked locally (rather thanglobally), which is closer in spirit to the positivity condition of Connes-Skandalis.Our ‘local’ approach to the positivity condition in particular also allows us to consider thenon-self-adjoint (incomplete) case of half-closed modules . In fact, it was one of our maingoals in this article to enhance our understanding of the unbounded Kasparov product of twohalf-closed modules. The main motivating examples come from vertically elliptic operatorson submersions of open manifolds, for which the Kasparov product was already described in[Dun20]. We will show that similar results can be obtained in the context of noncommutative C ∗ -algebras as well. Specifically, we prove several localised versions of Kucerovsky’s theoremfor half-closed modules. We emphasise however that this localised approach can also be ofadvantage in the self-adjoint (complete) case of unbounded Kasparov modules, since the‘local’ positivity condition is more flexible and therefore more broadly applicable.We point out that Kaad and Van Suijlekom have also proved a variant of Kucerovsky’stheorem for half-closed modules [KS19, Theorem 6.10], which partially addresses item (II).However, in their work, if the left module is essential, then the required assumptions in[KS19, Theorem 6.10] imply that D is self-adjoint, so that the left module is in fact an un-bounded Kasparov module (see [KS19, Remark 6.5]). In the context of differential operatorson Riemannian submersions, to obtain the self-adjointness of D one needs to assume forinstance (cf. [KS]) that the submersion is proper (which means that the fibres are compact).It was one of the main goals of this article to obtain a version of Kucerovsky’s theoremwhich includes the case of submersions with non-compact and incomplete fibres as well, sothat in particular the results in [Dun20] are recovered as a special case.Let us summarise the contents and main results of this article. We start Section 2by recalling some preliminaries on KK -theory and the Kasparov product, and then reviewHilsum’s framework of half-closed modules in Section 2.2. In Section 2.3 we collect a fewuseful lemmas regarding regular symmetric operators on Hilbert modules. In Section 2.4we discuss the connection condition, which can be dealt with in the non-self-adjoint casein the same way as in the self-adjoint case, without additional difficulty. We will showthat the connection condition for half-closed modules implies the connection condition ofConnes-Skandalis.In Section 3, we first restrict our attention to the self-adjoint case of unbounded Kas-parov modules, and we show that the lower bound in Kucerovsky’s positivity conditioncan be replaced by a relative lower bound (as mentioned in (I) above). To motivate thisgeneralisation, let us first compare to the aforementioned Connes-Skandalis Theorem (seeTheorem 2.2). Consider three (bounded) Kasparov modules ( A , ( E ) B , F ) , ( B , ( E ) C , F ) ,and ( A , E B , F ) (where E = E ˆ ⊗ B E ). Roughly speaking, the positivity condition ofConnes-Skandalis requires the (graded) commutator [ F, F ˆ ⊗ to be positive modulo com-pact operators. If we think of this in terms of pseudodifferential operators, this means thereexists a positive zeroth-order pseudodifferential operator with the same principal symbol as [ F, F ˆ ⊗ . On the other hand, Kucerovsky’s positivity condition (roughly speaking) re- Koen van den Dungen quires the (graded) commutator [ D , D ˆ ⊗ to be positive modulo bounded operators (i.e.,modulo ‘zeroth-order’ operators). This seems somewhat unnatural, since if we think of thisin terms of differential operators, it means that the positivity condition depends not onlyon the (second-order) principal symbol of [ D , D ˆ ⊗ , but also on the first-order part of thesymbol (even though the KK -classes of D and D are determined by their principal symbolsalone). A more natural condition would therefore be to require [ D , D ˆ ⊗ to be positivemodulo ‘first-order’ operators, meaning that there exists a positive second-order differentialoperator with the same principal symbol as [ D , D ˆ ⊗ . We will prove in Theorem 3.3 thatKucerovsky’s theorem can indeed be generalised to this more natural positivity condition.The unbounded Kasparov product has been studied in [LM19] in the context of weaklyanti-commuting operators. As a corollary to Theorem 3.3, we obtain here an alternativeproof of [LM19, Theorem 7.4].In the remainder of this article, we consider the symmetric (non-self-adjoint) case ofhalf-closed modules. In Section 4 we describe our localisation procedure for half-closed mod-ules, and provide the construction of a localised representative e F D for a half-closed module ( A , E B , D ) . This construction was first given by Higson [Hig89] (see also [HR00, §10.8]) forthe case of elliptic symmetric first-order differential operators, and was generalised by theauthor [Dun20] to the case of vertically elliptic symmetric first-order differential operatorson submersions of open manifolds. We will show in Section 4.2 that Higson’s construction ofa localised representative can be generalised further to the abstract noncommutative settingof a half-closed module ( A , E B , D ) . The construction is based on the assumption that thereexists an almost idempotent approximate unit { u n } n ∈ N in the dense ∗ -subalgebra A ⊂ A .Such approximate units always exist in any σ -unital C ∗ -algebra A ; the only additional as-sumption here is that it must lie in the subalgebra A . Under this assumption, we show howto construct a localised representative for the KK -class of a half-closed module.A detailed treatment of the local version of the positivity condition is given in Section 5.The main technical result consists of showing that our ‘local’ positivity condition for half-closed modules implies that the positivity condition of Connes-Skandalis is satisfied ‘locally’.The proof relies on the following two features of the localised representative. First, it allowsus to work ‘locally’ with self-adjoint (rather than only symmetric) operators. Second, eachlocalised term can be rescaled independently (which is crucial in order to obtain a uniformconstant in the positivity condition).In Section 6, we then finally provide several localised versions of Kucerovsky’s the-orem for half-closed modules, which provide sufficient conditions for a half-closed mod-ule ( A , E C , D ) to represent the (internal) Kasparov product of two half-closed modules ( A , ( E ) B , D ) and ( B , ( E ) C , D ) . All these versions require the existence of a suitableapproximate unit { u n } ⊂ A as in Section 4.2. Our first main result (Theorem 6.3) requires(in addition to a domain condition) the following local version of the positivity condition:for each n ∈ N there exists c n ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D u n ) ∩ Ran( u n ) we have (cid:10) u n ( D ˆ ⊗ ψ (cid:12)(cid:12) D u n ψ (cid:11) + (cid:10) D u n ψ (cid:12)(cid:12) u n ( D ˆ ⊗ ψ (cid:11) ≥ − c n (cid:10) ψ (cid:12)(cid:12) (1 + ( u n D u n ) ) ψ (cid:11) . (1.1)Using the results from Section 5, Eq. (1.1) ensures that the positivity condition of Connes-Skandalis is satisfied ‘locally’. The construction of the localised representative e F D using a‘partition of unity’ then allows us to prove that the positivity condition is in fact satisfied ocalisations and the kasparov product (cid:2) D ˆ ⊗ , u n D u n (cid:3) is positive modulo ‘first-order operators’ (where ‘first-order’ is defined relative to u n D u n ).However, to obtain a better analogy to the Connes-Skandalis positivity condition, it wouldbe more natural to require u n [ D ˆ ⊗ , D ] u n to be positive modulo ‘first-order operators’(where ‘first-order’ should be defined relative to D ). Unless D commutes with u n (asin [KS19]), this additional step is non-trivial (indeed, although we know that [ D , u n ] isbounded, this alone does not guarantee that we may consider [ D ˆ ⊗ , u n ] D u n to be ‘first-order’). In Section 6.1, we will consider two possible sufficient conditions which allow us tomake this additional step.The first sufficient condition assumes, instead of (1.1), a strong local positivity condition,which requires that (for each n ∈ N ) there exist ν n ∈ (0 , ∞ ) and c n ∈ [0 , ∞ ) such that forall ψ ∈ Dom( D ) we have (cid:10) ( D ˆ ⊗ u n ψ (cid:12)(cid:12) D u n ψ (cid:11) + (cid:10) D u n ψ (cid:12)(cid:12) ( D ˆ ⊗ u n ψ (cid:11) ≥ ν n (cid:10) ( D ˆ ⊗ u n ψ (cid:12)(cid:12) ( D ˆ ⊗ u n ψ (cid:11) − c n (cid:10) u n ψ (cid:12)(cid:12) (1 + D ∗ D ) u n ψ (cid:11) . The second sufficient condition assumes, instead of (1.1), a local positivity condition ,which requires simply that (for each n ∈ N ) there exists c n ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D ) we have (cid:10) ( D ˆ ⊗ u n ψ (cid:12)(cid:12) D u n ψ (cid:11) + (cid:10) D u n ψ (cid:12)(cid:12) ( D ˆ ⊗ u n ψ (cid:11) ≥ − c n (cid:10) u n ψ (cid:12)(cid:12) (1 + D ∗ D ) u n ψ (cid:11) , along with a ‘differentiability’ condition , which requires that the operator u n [ D , u n ] u n +2 maps Dom( D u n +2 ) to Dom( D ˆ ⊗ .We note that the latter ‘differentiability’ condition is quite naturally satisfied in thecontext of first-order differential operators on smooth manifolds, when u n are compactlysupported smooth functions and D is elliptic (as in [Dun20]). In Section 6.2 we will showthat the strong local positivity condition is in fact fairly natural in the constructive approachto the unbounded Kasparov product. This article is a continuation of [Dun20], and the work on both these articles was initiatedduring a short visit to the Radboud University Nijmegen in late 2017, which was funded bythe COST Action MP1405 QSPACE, supported by COST (European Cooperation in Scienceand Technology). The author thanks Walter van Suijlekom for his hospitality during thisvisit, and for interesting discussions. Thanks also to Bram Mesland and Matthias Lesch forinteresting discussions.
Let A and B denote σ -unital Z -graded C ∗ -algebras. By an approximate unit for A wewill always mean an even, positive, increasing, and contractive approximate unit for the C ∗ -algebra A . Let E be a Z -graded Hilbert module over B (for an introduction to Hilbert Koen van den Dungen modules and further details, see for instance [Lan95, Bla98]). For ξ ∈ E , we consider theshort-hand notation ⟪ ξ ⟫ := h ξ | ξ i , where h·|·i denotes the B -valued inner product on E . We denote the set of adjointableoperators on E as End B ( E ) , and the subset of compact endomorphisms as End B ( E ) . Forany operator T on E , we write deg T = 0 if T is even, and deg T = 1 if T is odd. The gradedcommutator [ · , · ] is defined (on homogeneous operators) by [ S, T ] := ST − ( − deg S · deg T T S .For any
S, T ∈ End B ( E ) we will write S ∼ T if S − T ∈ End B ( E ) . Similarly, forself-adjoint S, T we will write S & T if S − T ∼ P for some positive P ∈ End B ( E ) ; in thiscase we will say that S − T is positive modulo compact operators .Given a ∗ -homomorphism A → End B ( E ) , an operator T ∈ End B ( E ) is called locallycompact if aT is compact for every a ∈ A .Given any regular operator D , we define the bounded transform F D := D (1 + D ∗ D ) − .The graph inner product of D is given for ψ ∈ Dom D by h ψ | ψ i D := h ψ | ψ i + hD ψ |D ψ i , and the corresponding graph norm is given by k ψ k D := kh ψ | ψ i D k . KK -theory Kasparov [Kas80] defined the abelian group KK ( A, B ) as a set of homotopy equivalenceclasses of Kasparov A - B -modules. We start by briefly recalling the main definitions; formore details we refer to e.g. [Bla98, §17]. Definition 2.1.
A (bounded)
Kasparov A - B -module ( A, π E B , F ) is given by a Z -gradedcountably generated right Hilbert B -module E , a ( Z -graded) ∗ -homomorphism π : A → End B ( E ) , and an odd adjointable endomorphism F ∈ End B ( E ) such that for all a ∈ A : π ( a )( F − F ∗ ) , [ F, π ( a )] , π ( a )( F − ∈ End B ( E ) . Two Kasparov A - B -modules ( A, π E B , F ) and ( A, π E B , F ) are called unitarily equi-valent (denoted with ≃ ) if there exists an even unitary in Hom B ( E , E ) intertwining the π j and F j (for j = 0 , ).A homotopy between ( A, π E B , F ) and ( A, π E B , F ) is given by a Kasparov A - C ([0 , , B ) -module ( A, e π e E C ([0 , ,B ) , e F ) such that (for j = 0 , ) ev j ( A, e π e E C ([0 , ,B ) , e F ) ≃ ( A, π j E jB , F j ) . Here ≃ denotes unitary equivalence, and ev t ( A, e π e E C ([0 , ,B ) , e F ) := ( A, e π ˆ ⊗ e E ˆ ⊗ ρ t B B , e F ˆ ⊗ ,where the ∗ -homomorphism ρ t : C ([0 , , B ) → B is given by ρ t ( b ) := b ( t ) .A homotopy ( A, e π e E C ([0 , ,B ) , e F ) is called an operator-homotopy if there exists a Hil-bert B -module E with a representation π : A → End B ( E ) such that e E equals the Hilbert C ([0 , , B ) -module C ([0 , , E ) with the natural representation e π of A on C ([0 , , E ) in-duced from π , and if e F is given by a norm -continuous family { F t } t ∈ [0 , . A module ( π, E, F ) is called degenerate if π ( a )( F − F ∗ ) = [ F, π ( a )] = π ( a )( F −
1) = 0 for all a ∈ A . ocalisations and the kasparov product KK -theory KK ( A, B ) of A and B is defined as the set of homotopy equivalenceclasses of (bounded) Kasparov A - B -modules. Since homotopy equivalence respects directsums, the direct sum of Kasparov A - B -modules induces a (commutative and associative)binary operation (‘addition’) on the elements of KK ( A, B ) such that KK ( A, B ) is in factan abelian group [Kas80, §4, Theorem 1].If no confusion arises, we often simply write ( A, E B , F ) instead of ( A, π E B , F ) , and itsclass in KK -theory is simply denoted by [ F ] ∈ KK ( A, B ) . Let A be a ( Z -graded) separable C ∗ -algebra, and let B and C be ( Z -graded) σ -unital C ∗ -algebras. It was shown by Kasparov [Kas80, §4, Theorem 4] that there exists an associativebilinear pairing, called the (internal) Kasparov product : KK ( A, B ) × KK ( B, C ) → KK ( A, C ) . Given two KK -classes [ F ] ∈ KK ( A, B ) and [ F ] ∈ KK ( B, C ) , the Kasparov product isdenoted by [ F ] ⊗ B [ F ] .Though the Kasparov product always exists, it is often not easy to compute. A moretractable approach was provided by Connes and Skandalis [CS84], who gave sufficient con-ditions which allow to check whether a given Kasparov module represents the Kasparovproduct. For convenience, let us first introduce some notation. Given a Hilbert B -module E and a Hilbert C -module E with a ∗ -homomorphism B → End C ( E ) , we consider the(graded) internal tensor product E := E ˆ ⊗ B E . For any ψ ∈ E , we define the operator T ψ : E → E as T ψ η = ψ ˆ ⊗ η for any η ∈ E . The operator T ψ is adjointable, and its adjoint T ∗ ψ : E → E is given by T ∗ ψ ( ξ ⊗ η ) = h ψ | ξ i · η . Furthermore, we also introduce the operator e T ψ on the Hilbert C -module E ⊕ E given by e T ψ := (cid:18) T ψ T ∗ ψ (cid:19) . We cite here a slightly more general version of the theorem by Connes and Skandalis.First, as explained by Kucerovsky [Kuc97, Proposition 5], it suffices to check the connectioncondition for ψ ∈ φ ( A ) E (rather than all ψ ∈ E ). Second, as described in the commentsfollowing [Bla98, Definition 18.4.1], the positivity condition in fact only requires a lowerbound greater than − . Theorem 2.2 ([CS84, Theorem A.3]) . Consider two Kasparov modules ( A, π ( E ) B , F ) and ( B, π ( E ) C , F ) , and consider the Hilbert C -module E := E ˆ ⊗ B E and the ∗ -homo-morphism π := π ˆ ⊗ A → End C ( E ) . Suppose that ( A, π E C , F ) is a Kasparov module suchthat the following two conditions hold: Connection condition: for any ψ ∈ π ( A ) · E , the graded commutator [ F ⊕ F , e T ψ ] iscompact on E ⊕ E ; Positivity condition: there exists a ≤ κ < such that for all a ∈ A we have that π ( a )[ F ˆ ⊗ , F ] π ( a ∗ ) + κπ ( aa ∗ ) is positive modulo compact operators on E . Koen van den Dungen
Then ( A, π E C , F ) represents the Kasparov product of ( A, π E B , F ) and ( B, π E C , F ) : [ F ] = [ F ] ˆ ⊗ B [ F ] ∈ KK ( A, C ) . Moreover, an operator F with the above properties always exists and is unique up to operator-homotopy. Let A and B be Z -graded σ -unital C ∗ -algebras, and let E be a Z -graded countably gener-ated Hilbert B -module. For any densely defined operator D on E , we consider the followingsubspaces of End B ( E ) : Lip( D ) := (cid:8) T ∈ End B ( E ) : T · Dom
D ⊂
Dom D , and [ D , T ] is bounded on Dom D (cid:9) , Lip ∗ ( D ) := (cid:8) T ∈ Lip( D ) : T · Dom D ∗ ⊂ Dom D (cid:9) . If D ∗ is also densely defined, we note that T ∈ Lip( D ) implies T ∗ ∈ Lip( D ∗ ) , and then − [ D , T ] ∗ equals the closure of [ D ∗ , T ∗ ] (see [Hil10, Lemma 2.1]). Moreover, if D and T aresymmetric, then we have [ D ∗ , T ] = [ D , T ] . Definition 2.3 ([Hil10, §2]) . A half-closed A - B -module ( A , E B , D ) is given by a Z -gradedcountably generated Hilbert B -bimodule E , an odd regular symmetric operator D on E , a ∗ -homomorphism A → End B ( E ) , and a dense ∗ -subalgebra A ⊂ A such that(1) A ⊂
Lip ∗ ( D ) ;(2) (1 + D ∗ D ) − is locally compact.If furthermore D is self-adjoint, then ( A , E B , D ) is called an unbounded Kasparov A - B -module .Unbounded Kasparov modules were first introduced by Baaj and Julg [BJ83], who provedthat their bounded transforms yield Kasparov modules. This statement was generalised tohalf-closed modules by Hilsum. Theorem 2.4 ([Hil10, Theorem 3.2]) . Let ( A , E B , D ) be a half-closed A - B -module, andconsider a closed extension D ⊂ ˆ D ⊂ D ∗ . Then the bounded transform F ˆ D = ˆ D (1 + ˆ D ∗ ˆ D ) − yields a Kasparov A - B -module ( A, E B , F ˆ D ) , and its class is independent of the choice of theextension ˆ D . The following theorem by Kucerovsky provides an analogous version of the Connes-Skandalis result (Theorem 2.2) for the Kasparov product of unbounded Kasparov modules .(Note that we have simplified the domain condition using [Kuc97, Lemma 10(i)].)
Theorem 2.5 ([Kuc97, Theorem 13]) . Let ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) be un-bounded Kasparov modules. Suppose that ( A , π ˆ ⊗ ( E ˆ ⊗ B E ) C , D ) is an unbounded Kasparovmodule such that: (1) for all ψ in a dense subspace of A ·
Dom D , we have e T ψ ∈ Lip(
D ⊕ D ) . (2) we have the domain inclusion Dom
D ⊂
Dom D ˆ ⊗ ; (3) there exists c ∈ R such that for all ψ ∈ Dom( D ) we have h ( D ˆ ⊗ ψ | D ψ i + hD ψ | ( D ˆ ⊗ ψ i ≥ c h ψ | ψ i . ocalisations and the kasparov product Then ( A , π ˆ ⊗ ( E ˆ ⊗ B E ) C , D ) represents the Kasparov product of ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) . In Section 3 we will show that the lower bound in the positivity condition (3) can beweakened to a relative bound. Moreover, using the localisation procedure from Section 4,we will provide several variants of the above theorem for half-closed modules in Section 6.
Let B be a Z -graded σ -unital C ∗ -algebra, and let E be a Z -graded countably generatedHilbert B -module. Throughout this subsection, we consider a regular symmetric operator D on E and a positive number r ∈ (0 , ∞ ) . For λ ∈ [0 , ∞ ) , we introduce the notation R r D ( λ ) := ( r + λ + D ∗ D ) − . We recall that we have the integral formula ( r + D ∗ D ) − = 1 π ∞ Z λ − / R r D ( λ ) dλ, (2.1)where the integral converges in norm. Let us consider the ‘bounded transform’ F r D := D ( r + D ∗ D ) − . The following lemma is exactly as in [Dun20, Lemma 2.7], except that we have replaced D ∗ D by r + D ∗ D . Lemma 2.6.
For all ψ ∈ E we have π ∞ Z λ − / D R r D ( λ ) ψdλ = F r D ψ. Moreover, for any continuous function f : R → R such that f ( x )(1 + x ) − is bounded, wealso have π ∞ Z λ − / f ( D ∗ D ) R r D ( λ ) ψdλ = f ( D ∗ D )( r + D ∗ D ) − / ψ. The following lemma was proven on Hilbert spaces in [Les05, Proposition A.1], and itwas shown in [LM19, Lemma 7.7] that the argument can be generalised to Hilbert modules.
Lemma 2.7 (cf. [LM19, Lemma 7.7]) . Let P be an invertible regular positive self-adjointoperator on E , and let T be a symmetric operator on E with Dom P ⊂ Dom T . If T P − isbounded, then the densely defined operator P − T P − extends to an adjointable endomorph-ism on E , and k P − T P − k ≤ k T P − k . Lemma 2.8.
Let a = a ∗ ∈ Lip( D ) , and consider b ∈ End B ( E ) such that (1 + D ∗ D ) − b iscompact on E . Then the following statements hold: (1) The operators [ D R r D ( λ ) , a ] b and (1 + λ ) [ R r D ( λ ) , a ] b are compact and of order O ( λ − ) . Koen van den Dungen (2)
The operator [ F r D , a ] b is compact.Proof. The proof of (1) is an abstract generalisation of [Dun20, Lemma 2.9]. We have [ R r D ( λ ) , a ] b = (cid:2) ( r + λ + D ∗ D ) − , a (cid:3) b = − ( r + λ + D ∗ D ) − [ D ∗ D , a ]( r + λ + D ∗ D ) − b. We note that, a priori, [ D ∗ D , a ] may not be well-defined, since it is not clear if a preserves Dom D ∗ D . However, rewriting [ D ∗ D , a ] = D ∗ [ D , a ] + ( − deg a [ D ∗ , a ] D , we obtain the well-defined expression [ R r D ( λ ) , a ] b = − R r D ( λ ) (cid:16) R r D ( λ ) [ D ∗ , a ] (cid:0) D R r D ( λ ) (cid:1) + ( − deg a (cid:0) R r D ( λ ) D ∗ (cid:1) [ D , a ] R r D ( λ ) (cid:17) R r D ( λ ) b. Since R r D ( λ ) b is compact, one sees that the right-hand-side of this expression is compactand of order O ( λ − ) . Moreover, D [ R r D ( λ ) , a ] b is also well-defined, compact, and of order O ( λ − ) . Finally, we see that [ D R r D ( λ ) , a ] b = [ D , a ] R r D ( λ ) b + D [ R r D ( λ ) , a ] b is compact and of order O ( λ − ) . Thus we have proven (1).Using Lemma 2.6, we have for any ψ ∈ E that (cid:2) F r D , a (cid:3) bψ = − π ∞ Z λ − / [ D R r D ( λ ) , a ] bψdλ. By (1), [ D R r D ( λ ) , a ] b is compact and of order O ( λ − ) . Hence the above integral convergesin norm to a compact operator, which proves (2). Lemma 2.9.
Let a ∈ End B ( E ) be such that a (1 + D ∗ D ) − is compact, and let b ∈ Lip ∗ ( D ) .Then ab (1 + DD ∗ ) − is compact.Proof. Consider the domain inclusions ι : Dom D ֒ → E and ι : Dom D ∗ ֒ → E . By assump-tion, a ◦ ι is compact. Furthermore, b maps Dom D ∗ into Dom D , and b : Dom D ∗ → Dom D is bounded with respect to the graph norms (indeed, its norm is bounded by k b k + k [ D , b ] k ).Moreover, by [Hil10, Lemma 2.2 & Remark 2.4], b : Dom D ∗ → Dom D is adjointable. Hence ab ◦ ι = ( a ◦ ι ) ◦ b : Dom D ∗ → E is the composition of an adjointable and a compact operator,and therefore it is compact. Since (1 + DD ∗ ) − is a bounded map from E to Dom D ∗ , thestatement follows. Definition 2.10.
Consider three half-closed modules ( A , π ( E ) B , D ) , ( B , π ( E ) C , D ) ,and ( A , π E C , D ) , where E := E ˆ ⊗ B E and π = π ˆ ⊗ , and suppose that π is essential.The connection condition requires that for all ψ in a dense subspace E of Dom D , we have e T ψ := (cid:18) T ψ T ∗ ψ (cid:19) ∈ Lip(
D ⊕ D ) . ocalisations and the kasparov product π is essential (since this is our case of interest in later sections).However, we note that the case of non-essential representations π can be dealt with in thesame way as in [Kuc97, Proposition 14]. Proposition 2.11.
The connection condition of Definition 2.10 implies the connection con-dition of Theorem 2.2 for F = F D and F = F D .Proof. It suffices to consider elements aψb ∈ E , for even elements a ∈ A and b ∈ B , andhomogeneous ψ ∈ E ⊂ Dom D , where E is the dense subset from the connection condition.Since T aψb = aT ψ b and T ∗ aψb = b ∗ T ∗ ψ a ∗ , we have e T aψb = (cid:18) aT ψ bb ∗ T ∗ ψ a ∗ (cid:19) = (cid:18) a b ∗ (cid:19) (cid:18) T ψ T ∗ ψ (cid:19) (cid:18) a ∗ b (cid:19) =: e a e T ψ e a ∗ . Hence the graded commutator with F D⊕D is given by [ F D⊕D , e T aψb ] = [ F D⊕D , e a e T ψ e a ∗ ]= [ F D⊕D , e a ] e T ψ e a ∗ + e a [ F D⊕D , e T ψ ] e a ∗ + ( − deg ψ e a e T ψ [ F D⊕D , e a ∗ ] . By Definition 2.10, we know that e T ψ ∈ Lip(
D ⊕ D ) , so it follows from Lemma 2.8.(2)that the second term is compact. Since the first and third terms are also compact (byTheorem 2.4), this proves the statement. In this section, we consider only the special case of unbounded Kasparov modules (i.e., thecase of self-adjoint operators). Thus we consider the following setting.
Assumption 3.1.
Let A be a ( Z -graded) separable C ∗ -algebra, let B and C be ( Z -graded) σ -unital C ∗ -algebras, and let A ⊂ A and B ⊂ B be dense ∗ -subalgebras. Consider threeunbounded Kasparov modules ( A , π ( E ) B , D ) , ( B , π ( E ) C , D ) , and ( A , π E C , D ) , where E := E ˆ ⊗ B E and π = π ˆ ⊗ .Our aim in this section is to improve Kucerovsky’s Theorem 2.5 by weakening the pos-itivity condition, as follows. Definition 3.2.
In the setting of Assumption 3.1, the positivity condition requires that thefollowing assumptions hold:(1) we have the domain inclusion
Dom( D ) ⊂ Dom( D ˆ ⊗ ;(2) there exists c ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D ) we have (cid:10) ( D ˆ ⊗ ψ (cid:12)(cid:12) D ψ (cid:11) + (cid:10) D ψ (cid:12)(cid:12) ( D ˆ ⊗ ψ (cid:11) ≥ − c (cid:10) ψ (cid:12)(cid:12) (1 + D ) ψ (cid:11) . We then obtain the following improvement of Theorem 2.5.
Theorem 3.3.
In the setting of Assumption 3.1, assume furthermore that the connectioncondition (Definition 2.10) and the positivity condition (Definition 3.2) are satisfied. Then ( A , π E C , D ) represents the Kasparov product of ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) . Koen van den Dungen
Before proving the theorem, we first describe a few consequences. First of all, in the bounded picture, we know that two Kasparov modules ( A, π E B , F ) and ( A, π E B , F ′ ) areequivalent if for each a ∈ A , the operator a ∗ [ F, F ′ ] a is positive modulo compact operators(see [Ska84, Lemma 11] or [Bla98, Proposition 17.2.7]). The following statement givesan unbounded analogue , which says, roughly speaking, that two unbounded Kasparovmodules ( A , π E B , D ) and ( A , π E B , D ′ ) are equivalent if [ D , D ′ ] is positive modulo ‘first-order operators’ (this generalises [Kuc97, Corollary 17]). Corollary 3.4.
Let ( A , π E B , D ) and ( A , π E B , D ′ ) be unbounded Kasparov modules suchthat Dom
D ⊂
Dom D ′ , and suppose there exists c ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D ) we have hD ′ ψ | D ψ i + hD ψ | D ′ ψ i ≥ − c (cid:10) ψ (cid:12)(cid:12) (1 + D ) ψ (cid:11) . Then the two unbounded Kasparov modules ( A , π E B , D ) and ( A , π E B , D ′ ) are homotopy-equivalent, i.e. [ D ′ ] = [ D ] ∈ KK ( A, B ) .Proof. We claim that [ D ] equals the internal Kasparov product (over B ) of [ D ′ ] with B ∈ KK ( B, B ) . The class B is represented by the unbounded Kasparov module ( B, B B , .We can identify E ˆ ⊗ B B ≃ E , and then for each ψ ∈ E , the map T ψ : B → E is given by b ψb . Its adjoint T ∗ ψ : E → B is given by φ
7→ h ψ | φ i . For each ψ ∈ Dom D , we then seethat the operators D T ψ = T D ψ and T ∗ ψ D = T ∗D ψ are both bounded. Hence the connectioncondition is satisfied. Since the positivity condition holds by hypothesis, the claim followsfrom Theorem 3.3.The following lemma provides a sufficient condition for the positivity condition, givenin terms of operators instead of form estimates. This sufficient condition is in particularuseful for the construction of the unbounded Kasparov product from weakly anti-commutingoperators (see Corollary 3.6). Lemma 3.5.
Let D and S be odd regular self-adjoint operators on a Z -graded Hilbert C -module E , such that Dom
D ⊂
Dom S . Suppose there exists a core F ⊂
Dom D such that S · F ⊂
Dom D and D · F ⊂
Dom S , so that [ D , S ] is well-defined on F . Assume that on F we have the equality [ D , S ] = P + R , where P is a densely defined positive symmetricoperator on F (i.e. F ⊂
Dom P and h ψ | P ψ i ≥ for all ψ ∈ F ), and R is a densely definedsymmetric operator which is relatively bounded by D (i.e. Dom
D ⊂
Dom R ). Then for all ψ ∈ Dom D we have hS ψ | D ψ i + hD ψ | S ψ i ≥ −k R (1 + D ) − k (cid:10) ψ (cid:12)(cid:12) (1 + D ) ψ (cid:11) . Proof.
From Lemma 2.7 we know that (cid:13)(cid:13) (1 + D ) − R (1 + D ) − (cid:13)(cid:13) ≤ c := k R (1 + D ) − k ,and hence we have for ψ ∈ Dom D the inequality ±h ψ | Rψ i = ± (cid:10) (1 + D ) ψ (cid:12)(cid:12) (1 + D ) − R (1 + D ) − (1 + D ) ψ (cid:11) ≤ c (cid:10) ψ (cid:12)(cid:12) (1 + D ) ψ (cid:11) . Using also the positivity of P , we then find for all ψ ∈ F that hS ψ | D ψ i + hD ψ | S ψ i = h ψ | P ψ i + h ψ | Rψ i ≥ − c (cid:10) ψ (cid:12)(cid:12) (1 + D ) ψ (cid:11) . Albeit a ‘global’ analogue, since we no longer ‘localise’ by elements a ∈ A . ocalisations and the kasparov product ψ ∈ Dom D , we choose a sequence ψ n ∈ F such that k ψ n − ψ k D → as n → ∞ .Since Dom
D ⊂
Dom S , we note that we then also have the convergence kS ψ n − S ψ k → .Applying the above inequality to ψ n we obtain hS ψ | D ψ i + hD ψ | S ψ i = lim n →∞ hS ψ n | D ψ n i + hD ψ n | S ψ n i≥ − c lim n →∞ (cid:10) ψ n (cid:12)(cid:12) (1 + D ) ψ n (cid:11) = − c (cid:10) ψ (cid:12)(cid:12) (1 + D ) ψ (cid:11) . Let S and T be odd regular self-adjoint operators on a Z -graded Hilbert C -module E .We consider the linear subspace F ( S , T ) := Dom( ST ) ∩ Dom(
T S ) = { ψ ∈ Dom
S ∩
Dom T : S ψ ∈ Dom T , T ψ ∈ Dom S} . Then S and T are called weakly anti-commuting [LM19, Definition 2.1] if • there is a constant C > such that for all ψ ∈ F ( S , T ) we have (cid:10) [ S , T ] ψ (cid:12)(cid:12) [ S , T ] ψ (cid:11) ≤ C (cid:0) h ψ | ψ i + hS ψ |S ψ i + hT ψ |T ψ i (cid:1) ; • there is a core E ⊂
Dom T such that ( S + λ ) − ( E ) ⊂ F ( S , T ) for λ ∈ i R , | λ | ≥ λ > .We now obtain a different proof of the following result due to Lesch and Mesland. Corollary 3.6 ([LM19, Theorem 7.4]) . Suppose we are given two unbounded Kasparovmodules ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) . We write E := E ˆ ⊗ B E , π = π ˆ ⊗ , and S := D ˆ ⊗ . Let T be an odd regular self-adjoint operator on E , and consider the operator D := S + T on the domain Dom D := Dom S ∩
Dom T . We assume that the followingconditions hold: (1) for all ψ in a dense subset of Dom D , we have e T ψ := (cid:18) T ψ T ∗ ψ (cid:19) ∈ Lip(
T ⊕ D ); (2) we have A ⊂
Lip( T ) ; (3) S and T are weakly anti-commuting.Then ( A , π ( E ) C , D ) is an unbounded Kasparov module that represents the Kasparov productof ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) .Proof. By (3) and [LM19, Theorem 2.6], D is regular and self-adjoint. Using (2) we clearlyhave A ⊂
Lip( T ) ∩ Lip( S ) ⊂ Lip( D ) . As in the proof of [LM19, Theorem 7.4], we know that a ( D ± i ) − is compact for any a ∈ A . Thus ( A , π ( E ) C , D ) is indeed an unbounded Kasparovmodule. For any ψ ∈ Dom D we have bounded operators S T ψ = T D ψ and T ∗ ψ S = T ∗D ψ ,which means that e T ψ ∈ Lip(
S ⊕ . Combined with (1) this ensures that the connectioncondition (Definition 2.10) is satisfied.We note that F := F ( S , T ) is a core for D [LM19, Theorem 2.6.(4)], that S · F ⊂
Dom S [LM19, Theorem 5.1], and that [ S , T ] is relatively bounded by D [LM19, Theorem2.6.(1)]. Thus Lemma 3.5 applies with F = F ( S , T ) , P = S and R = [ S , T ] , and wesee that the positivity condition (Definition 3.2) is also satisfied. Hence it follows fromTheorem 3.3 that ( A , π ( E ) C , D ) represents the Kasparov product of ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) .4 Koen van den Dungen
Remark 3.7.
Under suitable assumptions [Mes14, KL13, BMS16], one can construct anoperator T of the form T = 1 ˆ ⊗ ∇ D , where ∇ is a suitable ‘connection’ on E , and prove that T satisfies the conditions ofCorollary 3.6. In many geometric examples, the connection ∇ is naturally determined bythe given geometry [BMS16, KS18, KS, Dun20]. In fact, such an operator T always exists,if one is willing to allow ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) to be replaced by homotopy-equivalent modules [MR16]. Before we proceed with the proof, let us first introduce some notation.
Notation 3.8.
Let D and S be regular self-adjoint operators on a Hilbert C -module E ,such that Dom
D ⊂
Dom S . We consider the quadratic form Q defined for ψ ∈ Dom D by Q ( ψ ) := 2 ℜhD ψ |S ψ i = hD ψ |S ψ i + hS ψ |D ψ i . For λ, µ ∈ [0 , ∞ ) , we use the notation R D ( λ ) := (1 + λ + D ) − , R S ( µ ) := (1 + µ + S ) − . We introduce the following bounded operators: k D ( λ ) := √ λR D ( λ ) , k S ( µ ) := p µR S ( µ ) ,h D ( λ ) := D R D ( λ ) , h S ( µ ) := S R S ( µ ) . Furthermore, we define M ( λ, µ ) := h D ( λ ) h S ( µ ) , M ( λ, µ ) := k D ( λ ) h S ( µ ) ,M ( λ, µ ) := h D ( λ ) k S ( µ ) , M ( λ, µ ) := k D ( λ ) k S ( µ ) . Lemma 3.9.
For ψ ∈ E , we have the inequality X m =1 π ∞ Z ∞ Z ( µλ ) − (cid:10) M m ( λ, µ ) ψ (cid:12)(cid:12) (1 + D ) M m ( λ, µ ) ψ (cid:11) dλdµ ≤ h ψ | ψ i . Proof.
Let us consider the four integrals (for m = 1 , , , ) given by I m := 1 π ∞ Z ∞ Z ( µλ ) − (cid:10) M m ( λ, µ ) ψ (cid:12)(cid:12) (1 + D ) M m ( λ, µ ) ψ (cid:11) dλdµ. We note that h D ( λ ) + k D ( λ ) = R D ( λ ) . Moreover, by Lemma 2.6 we have the stronglyconvergent integral π ∞ Z λ − (1 + D ) (cid:0) h D ( λ ) + k D ( λ ) (cid:1) dλ = 1 π ∞ Z λ − (1 + D ) R D ( λ ) dλ = 1 . ocalisations and the kasparov product λ , we thus obtain the equalities I + I = 1 π ∞ Z µ − (cid:10) h S ( µ ) ψ (cid:12)(cid:12) h S ( µ ) ψ (cid:11) dµ,I + I = 1 π ∞ Z µ − (cid:10) k S ( µ ) ψ (cid:12)(cid:12) k S ( µ ) ψ (cid:11) dµ. Summing up the latter two equalities and computing the remaining norm-convergent integralover µ , we obtain X m =1 I m = 1 π ∞ Z µ − (cid:10) ψ (cid:12)(cid:12) R S ( µ ) ψ (cid:11) dµ = (cid:10) ψ (cid:12)(cid:12) (1 + S ) − ψ (cid:11) ≤ (cid:10) ψ (cid:12)(cid:12) ψ (cid:11) . The following result relates our positivity condition to the positivity condition of Connes-Skandalis (Theorem 2.2).
Proposition 3.10.
Let D and S be odd regular self-adjoint operators on a Z -graded Hilbert C -module E , such that Dom
D ⊂
Dom S . Suppose there exists a constant c ∈ [0 , ∞ ) suchthat for all ψ ∈ Dom D we have hS ψ | D ψ i + hD ψ | S ψ i ≥ − c (cid:10) ψ (cid:12)(cid:12) (1 + D ) ψ (cid:11) . (3.1) Then for any < κ < there exists an α ∈ (0 , ∞ ) such that [ F D , F α S ] + κ is positive: [ F D , F α S ] ≥ − κ. Proof.
Recall that for ψ ∈ E we have by Lemma 2.6 that F D ψ = π R ∞ λ − D R D ( λ ) ψdλ ,and similarly for F S . Applying Lemma 2.6 twice, we can then rewrite (cid:10) ψ (cid:12)(cid:12) [ F D , F S ] ψ (cid:11) = 2 ℜ (cid:10) ψ (cid:12)(cid:12) F D F S ψ (cid:11) = 1 π ∞ Z ∞ Z ( µλ ) − ℜ (cid:10) ψ (cid:12)(cid:12) D R D ( λ ) S R S ( µ ) ψ (cid:11) dλdµ. By the same computation as in [Kuc97, Lemma 11] (or as in Lemma 5.5 below, taking thespecial case v = φ = ρ = 1 ), the integrand on the right-hand-side can be rewritten as ℜ (cid:10) ψ (cid:12)(cid:12) D R D ( λ ) S R S ( µ ) ψ (cid:11) = X m =1 Q (cid:0) M m ( λ, µ ) ψ (cid:1) . By Eq. (3.1) we have Q ( ψ ) ≥ − c h ψ | (1 + D ) ψ i , and therefore we obtain (cid:10) ψ (cid:12)(cid:12) [ F D , F S ] ψ (cid:11) ≥ − c X m =1 π ∞ Z ∞ Z ( µλ ) − (cid:10) M m ( λ, µ ) ψ (cid:12)(cid:12) (1 + D ) M m ( λ, µ ) ψ (cid:11) dλdµ. Koen van den Dungen
Applying the inequality from Lemma 3.9, we conclude that (cid:10) ψ (cid:12)(cid:12) [ F D , F S ] ψ (cid:11) ≥ − c h ψ | ψ i . Finally, if we replace S by α S for some α > , then we see from Eq. (3.1) that c should bereplaced by αc . Thus, by choosing α small enough, we can ensure that αc < κ < . Proof of Theorem 3.3 . We can represent [ D ] , [ D ] = [ α D ] (for some α ∈ (0 , ∞ ) ), and [ D ] by their bounded transforms F D , F α D , and F D , respectively. The statement of thetheorem follows from Theorem 2.2, where the connection condition is satisfied by [Kuc97,Proposition 14] (see also Proposition 2.11 if π is essential), and (choosing α small enough)the positivity condition is satisfied by Proposition 3.10. Let D be a regular symmetric operator on a Hilbert B -module E . For any b = b ∗ ∈ Lip ∗ ( D ) ,we will consider the localisation of D given by the operator b D b . We recall the followinglemma. Lemma 4.1 ([KS19, Lemma 3.2]) . Let D be a regular symmetric operator on a Hilbert B -module E , and let b = b ∗ ∈ Lip ∗ ( D ) . Then (the closure of ) b D b is regular and self-adjoint,and Dom D is a core for b D b . Lemma 4.2.
Let ( A , E B , D ) be a half-closed module, and consider homogeneous self-adjointelements a, b ∈ A such that ab = a . Then a ( b D b ± i ) − is a compact endomorphism.Proof. Consider the regular self-adjoint operator e D := (cid:18) D ∗ D (cid:19) , and write a = (cid:18) a a (cid:19) and b = (cid:18) b b (cid:19) . Repeatedly using ab = a , we see that ( e D ± i ) a = ( e D ± i ) ab = [ e D , a ] b + ( − deg a a ( e D ± i ) b = [ e D , a ] b + ( − deg a a ( b e D b ± i ) . We multiply from the right by ( b D b ± i ) − and from the left by ( e D ± i ) − = (cid:18) ± i D ∗ D ± i (cid:19) − = (cid:18) ∓ i (1 + D ∗ D ) − (1 + D ∗ D ) − D ∗ (1 + DD ∗ ) − D ∓ i (1 + DD ∗ ) − (cid:19) . Using that [ D ∗ , a ] = [ D , a ] (more precisely, their closures are equal) and that b D ∗ b = b D b on Dom b D b , this yields a ( b D b ± i ) − = b a ( b D b ± i ) − = b (cid:18) ∓ i (1 + D ∗ D ) − (1 + D ∗ D ) − D ∗ (1 + DD ∗ ) − D ∓ i (1 + DD ∗ ) − (cid:19) ×× (cid:18) [ D , a ] b (cid:18) (cid:19) + ( − deg a a (cid:18) ± i b D bb D b ± i (cid:19)(cid:19) ( b D b ± i ) − . ocalisations and the kasparov product b (1 + D ∗ D ) − is compact, and by Lemma 2.9 also b (1 + DD ∗ ) − is compact.Since (cid:18) ± i b D bb D b ± i (cid:19) ( b D b ± i ) − is bounded, it follows that a ( b D b ± i ) − is compact. Lemma 4.3.
Let ( A , E B , D ) be a half-closed module. Consider homogeneous self-adjointelements a ∈ A and c, b ∈ A such that b, c are even, ac = a , and cb = c . Let D b := b D b .Then a ( F D − F D b ) is compact on E .Proof. The proof closely follows the argument of [Dun20, Lemma 2.10] (which in turn wasinspired by [Hil10, Lemma 3.1]). The main difference here is that we have to take care ofthe fact that the operators c ( D b − D ) and c ( D b − D ∗ ) do not vanish (see below).Since a ( F D − F ∗D ) is compact by Theorem 2.4, it suffices to show that a ( F ∗D − F D b ) iscompact. We can rewrite a ( F ∗D − F D b ) = a (cid:16) (1 + D ∗ D ) − D ∗ − D b (1 + D b ) − (cid:17) = a (cid:16) D ∗ (1 + DD ∗ ) − − D b (1 + D b ) − (cid:17) . Using Lemma 2.6, we have for any ψ ∈ E that a ( F ∗D − F D b ) ψ = 1 π ∞ Z λ − T ( λ ) ψdλ, where T ( λ ) := a (cid:0) D ∗ (1 + λ + DD ∗ ) − − (1 + λ + D b ) − D b (cid:1) . We claim that T ( λ ) is a compact operator on E , and that k T ( λ ) k is of order O ( λ − ) as λ → ∞ . It then follows that π R ∞ λ − T ( λ ) dλ is in fact a norm -convergent integral ofcompact operators, which proves the statement. To prove the claim, we rewrite T ( λ ) = a (1 + λ + D b ) − (1 + λ + D b ) D ∗ (1 + λ + DD ∗ ) − − a (1 + λ + D b ) − D b (1 + λ + DD ∗ )(1 + λ + DD ∗ ) − = a (1 + λ + D b ) − D b ( D b − D ) D ∗ (1 + λ + DD ∗ ) − + (1 + λ ) a (1 + λ + D b ) − ( D ∗ − D b )(1 + λ + DD ∗ ) − . We note that the operators on the last line are still well-defined. For instance, we have
Ran (cid:0) D ∗ (1 + λ + DD ∗ ) − (cid:1) ⊂ Dom
D ⊂
Dom D b , so that ( D b − D ) D ∗ (1 + λ + DD ∗ ) − isa well-defined bounded operator. We also note that b · Dom D ∗ ⊂ Dom D , so that D b iswell-defined on Dom D ∗ .Next, since cb = c , we note that c ( D b − D ) = c ( b − D + cb [ D , b ] = c [ D , b ] and similarly c ( D ∗ − D b ) = − c [ D ∗ , b ] . Noting that [ D ∗ , b ] = [ D , b ] and using that ac = a , we find that T ( λ ) = a (cid:2) c, (1 + λ + D b ) − D b (cid:3) ( D b − D ) D ∗ (1 + λ + DD ∗ ) − + a (1 + λ ) (cid:2) c, (1 + λ + D b ) − (cid:3) ( D ∗ − D b )(1 + λ + DD ∗ ) − + a (1 + λ + D b ) − D b c [ D , b ] D ∗ (1 + λ + DD ∗ ) − Koen van den Dungen − a (1 + λ )(1 + λ + D b ) − c [ D , b ](1 + λ + DD ∗ ) − . From Lemma 4.2 we know that c ( D b ± i ) − and a ( D b ± i ) − are compact. In particular,also a (1 + λ + D b ) − is compact. Furthermore, we may apply Lemma 2.8.(1) to see that a (cid:2) c, (1 + λ + D b ) − D b (cid:3) is compact and of order O ( λ − ) , and that a (cid:2) c, (1 + λ + D b ) − (cid:3) iscompact and of order O ( λ − ) . Using these facts, we see that T ( λ ) is indeed compact andof order O ( λ − ) . We show here that the construction of a localised representative for vertical operators onsubmersions of open manifolds, as described in [Dun20, §2.4], generalises to the abstract(noncommutative) setting of half-closed modules. Recall that a (positive, increasing, con-tractive) approximate unit { u n } n ∈ N is called almost idempotent if u n +1 u n = u n for all n ∈ N [Bla06, Definition II.4.1.1]. Assumption 4.4.
We consider a half-closed A - B -module ( A , π E B , D ) for which the repres-entation π : A → End B ( E ) is essential. We assume that the ∗ -subalgebra A ⊂ A containsan (even) almost idempotent approximate unit { u n } n ∈ N for A . Remark 4.5. (1) Since π is essential, it follows that π ( u n ) converges strongly to theidentity on E (as n → ∞ ).(2) We know from [Bla06, Corollary II.4.2.5] that a σ -unital C ∗ -algebra A always containsan almost idempotent approximate unit { u n } . Our main assumption is that we canfind such { u n } inside the dense ∗ -subalgebra A .(3) In the special case where A is unital, we can of course consider u n = 1 A for all n ∈ N . Definition 4.6.
The ‘partition of unity’ { χ k } k ∈ N corresponding to the approximate identity { u n } n ∈ N is defined by χ := u , χ k := ( u k − u k − ) ( k > . (4.1)While χ k = u k − u k − always lies in A for each k ∈ N , we note that we do not know if also χ k lies in A (we only know χ k ∈ A ). Lemma 4.7.
The following statements hold: (1) χ j χ k = 0 for all j > k + 1 ; (2) u n χ k = χ k for all k < n ; (3) u n χ k = 0 for all k > n + 1 .Proof. (1) Since { u n } is almost idempotent, we know that u j u k = u k for all j > k . Thenfor j > k + 1 we have χ j χ k = ( u j − u j − )( u k − u k − ) = ( u j − u j − ) u k − ( u j − u j − ) u k − = u k − u k − u k − + u k − = 0 . In particular, χ j commutes with χ k , and therefore their square roots χ j and χ k alsocommute and we see that χ j χ k = ( χ j χ k ) = 0 . ocalisations and the kasparov product n > k we have u n χ k = u n ( u k − u k − ) = u k − u k − = χ k . In particular, u n commutes with χ k , which implies that u n commutes with χ k and u n χ k = χ k .(3) For k > n + 1 we have u n χ k = u n ( u k − u k − ) = u n − u n = 0 . In particular, u n commutes with χ k , which implies that u n commutes with χ k and u n χ k = u n (cid:0) u n χ k (cid:1) = 0 .We pick a sequence of elements v k ∈ { u n } n ∈ N such that v k u k +1 = u k +1 for each k ∈ N (the simplest choice is of course to take v k = u k +2 , but in later sections it will be convenientto choose v k = u k +3 or v k = u k +4 , so we allow for this additional flexibility). Pick a sequence { α k } k ∈ N ⊂ (0 , ∞ ) of strictly positive numbers, and consider the operators D k := v k D v k , F α k D k := α k D k (1 + α k D k ) − . Since { u n } n ∈ N ⊂ A ⊂ Lip ∗ ( D ) , we know from Lemma 4.1 that (the closure of) D k isregular and self-adjoint, and that Dom D is a core for D k . In particular, the operator F α k D k is well-defined via continuous functional calculus. Definition 4.8.
For any sequence { α k } k ∈ N ⊂ (0 , ∞ ) of strictly positive numbers, we definethe localised representative of D as e F D ( α ) := ∞ X k =0 χ k F α k D k χ k . Lemma 4.9.
The operator e F D ( α ) is well-defined as a strongly convergent series.Proof. First, since for each n the sum P ∞ k =0 χ k u n is finite (see Lemma 4.7), we see that e F D ( α ) u n ψ is a finite (hence convergent) sum for each ψ ∈ E . Hence e F D ( α ) convergesstrongly on the dense subset { u n ψ | n ∈ N , ψ ∈ E } . Second, using the operator inequalities ± F α k D k ≤ k F α k D k k ≤ , we see for K ∈ N that ± K X k =0 χ k F α k D k χ k ≤ K X k =0 χ k = u K ≤ . Hence the partial sums are uniformly bounded, and therefore the series converges stronglyon all of E . Lemma 4.10 ([Dun20, Lemma 2.8]) . Let D be a regular self-adjoint operator on a Hilbert B -module E . Let a ∈ End B ( E ) such that a ( D ± i ) − is compact. Then for any α > , theoperator a ( F D − F α D ) is compact. Koen van den Dungen
Theorem 4.11.
Consider the setting of Assumption 4.4. Then for any a ∈ A , the operator a ( e F D ( α ) − F D ) is compact. Hence ( A, E B , e F D ( α )) is a (bounded) Kasparov A - B -module,and [ e F D ( α )] = [ F D ] ∈ KK ( A, B ) . In particular, the class [ e F D ( α )] is independent of thechoices made in the construction.Proof. Since we have norm-convergence au n → a , it suffices to prove the compactness of u n ( e F D ( α ) − F D ) . Recall that e F D ( α ) = P ∞ k =0 χ k F α k D k χ k , where D k = v k D v k . By applyingLemma 4.2 (using χ k = χ k u k +1 from Lemma 4.7) we know that χ k ( D k ± i ) − is compact;hence we can apply Lemma 4.10 to see that χ k ( F α k D k − F D k ) is compact. We know fromTheorem 2.4 that the commutator [ F D , χ k ] is compact. Furthermore, applying Lemma 4.3(with a = χ k , c = u k +1 , and b = v k ) we know that χ k ( F D k − F D ) is compact. FromLemma 4.7 we know that u n e F D ( α ) is given by a finite sum, and therefore u n ( e F D ( α ) − F D ) = X k u n χ k F α k D k χ k − u n F D . ∼ X k u n (cid:0) χ k F D k χ k − χ k F D (cid:1) . ∼ X k u n χ k ( F D k − F D ) χ k . ∼ . The aim in this section is to show that the local positivity condition (see Definition 6.6below) implies a localised version of the positivity condition in Theorem 2.2. This wasalready proven by the author for the case of first-order differential operators on smoothmanifolds [Dun20, Proposition 3.1], and in fact, many of the arguments of [Dun20, §3] canbe adapted to the following more abstract context.
Assumption 5.1.
Let D be an odd regular symmetric operator on a Z -graded Hilbert C -module E , let S be an odd regular self-adjoint operator on E , and let χ, ρ, φ, v ∈ End C ( E ) be even and self-adjoint. We define the operator D v := v D v . We assume that the followingconditions hold:(L1) Dom( D v ) ∩ Ran( v ) ⊂ Dom( S v ) ;(L2) ρχ = χ , φρ = ρ , vρ = ρ , vφ = φv , k ρ k ≤ , and k φ k ≤ ;(L3) ρ, φ ∈ Lip( D ) ∩ Lip( S ) and v ∈ Lip ∗ ( D ) ∩ Lip( S ) ;(L4) χ (1 + D v ) − is compact;(L5) there exists a constant c ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D v ) ∩ Ran( v ) we have Q v ( φψ ) := (cid:10) D vφψ (cid:12)(cid:12) v S φψ (cid:11) + (cid:10) v S φψ (cid:12)(cid:12) D vφψ (cid:11) ≥ − c (cid:10) φψ (cid:12)(cid:12) (1 + D v ) φψ (cid:11) . Since v ∈ Lip ∗ ( D ) , we note that D v is regular and self-adjoint (see Lemma 4.1). Since ρ, φ ∈ Lip D and ρ, φ commute with v , we know that we also have ρ, φ ∈ Lip D v . Further-more, since φ preserves Dom( D v ) ∩ Ran( v ) and we have Dom( D v ) ∩ Ran( v ) ⊂ Dom( S v ) =Dom( v S ) , we see that Q v is well-defined. Notation 5.2.
For λ, µ ∈ [0 , ∞ ) and for r ∈ (0 , ∞ ) , we use the notation R r D v ( λ ) := ( r + λ + D v ) − , R S ( µ ) := (1 + µ + S ) − . ocalisations and the kasparov product k r D v ( λ ) := p r + λR r D v ( λ ) , k S ( µ ) := p µR S ( µ ) ,h r D v ( λ ) := D v R r D v ( λ ) , h S ( µ ) := S R S ( µ ) . We also define the operator B ( λ ) := [ S , ρ ] φ D v R r D v ( λ ) ρ. We now redefine the operators M m ( λ, µ ) , using D v instead of D , and inserting ρ : M ( λ, µ ) := h r D v ( λ ) ρh S ( µ ) , M ( λ, µ ) := k r D v ( λ ) ρh S ( µ ) ,M ( λ, µ ) := h r D v ( λ ) ρk S ( µ ) , M ( λ, µ ) := k r D v ( λ ) ρk S ( µ ) . Furthermore, we define ˆ B ( λ, µ ) := 2 ℜ (cid:10) B ( λ ) k S ( µ ) χψ (cid:12)(cid:12) k S ( µ ) χψ (cid:11) + 2 ℜ (cid:10) B ( λ ) h S ( µ ) χψ (cid:12)(cid:12) h S ( µ ) χψ (cid:11) , ˆ M ( λ, µ ) := X m =1 ℜ (cid:10) [ φ, D ] vM m ( λ, µ ) χψ (cid:12)(cid:12) v S φM m ( λ, µ ) χψ (cid:11) , ˆ Q ( λ, µ ) := X m =1 Q v (cid:0) φM m ( λ, µ ) χψ (cid:1) . Lemma 5.3 ([KS19, Lemma 3.4]) . For any r > (cid:13)(cid:13) [ D , v ] v (cid:13)(cid:13) , the operator ir + D v : Dom( D v ) → E is bijective, its inverse ( ir + D v ) − is adjointable on E , and we have the equality ( ir + D v ) − v = v ( ir + D v ) − . Lemma 5.4 (cf. [KS19, Lemma 6.15]) . For any r > (cid:13)(cid:13) [ D , v ] v (cid:13)(cid:13) , we have the followinginclusions: Ran (cid:0) k r D v ( λ ) v (cid:1) ⊂ Dom( D v ) ∩ Ran( v ) , Ran (cid:0) h r D v ( λ ) v (cid:1) ⊂ Dom( D v ) ∩ Ran( v ) . Proof.
From Lemma 5.3 we know for any r > (cid:13)(cid:13) [ D , v ] v (cid:13)(cid:13) that Ran (cid:0) ( ir + D v ) − v (cid:1) = v · Dom( D v ) = Dom( D v ) ∩ Ran( v ) . The inclusions then follow, because we can rewrite R r D v ( λ ) v = (cid:0) i p r + λ + D v (cid:1) − v (cid:0) − i p r + λ + D v (cid:1) − , D v R r D v ( λ ) v = (cid:0) − i p r + λ + D v (cid:1) − v − i p r + λR r D v ( λ ) v. Koen van den Dungen
We will study the positivity (modulo compact operators) of the operator χ [ F D v , F S ] χ .Using ρχ = χ , we can rewrite χ [ F D v , F S ] χ = χ ( ρF D v F S + F S F D v ρ ) χ = χ ( F D v ρF S + F S ρF D v ) χ + χ ([ ρ, F D v ] F S + F S [ F D v , ρ ]) χ. We note that the commutator [ F D v , ρ ] χ is compact by Lemma 2.8.(2) (using ρ ∈ Lip( D v ) and condition (L4)), and therefore it suffices to consider instead the operator χ ( F D v ρF S + F S ρF D v ) χ . Furthermore, since the function x x (1 + x ) − − x ( r + x ) − lies in C ( R ) ,we know that also ( F D v − F r D v ) χ is compact. Hence we may replace F D v by F r D v . ApplyingLemma 2.6 twice, we then rewrite (cid:10) ψ (cid:12)(cid:12) χ ( F r D v ρF S + F S ρF r D v ) χψ (cid:11) = 2 ℜ (cid:10) χψ (cid:12)(cid:12) F r D v ρF S χψ (cid:11) (5.1) = 1 π ∞ Z ∞ Z ( λµ ) − ℜ (cid:10) χψ (cid:12)(cid:12) D v R r D v ( λ ) ρ S R S ( µ ) χψ (cid:11) dλdµ. Our first task is to study the integrand on the right-hand-side. Via a straightforward butsomewhat tedious calculation, we will rewrite this integrand in terms of the operators definedabove.
Lemma 5.5.
For any ψ ∈ E we have ℜ (cid:10) χψ (cid:12)(cid:12) D v R r D v ( λ ) ρ S R S ( µ ) χψ (cid:11) = ˆ B ( λ, µ ) + ˆ M ( λ, µ ) + ˆ Q ( λ, µ ) . Proof.
We compute: (cid:10) χψ (cid:12)(cid:12) D v R r D v ( λ ) ρ S R S ( µ ) χψ (cid:11) = (cid:10) ρ (1 + µ + S ) R S ( µ ) χψ (cid:12)(cid:12) D v R r D v ( λ ) ρ S R S ( µ ) χψ (cid:11) = (cid:10) ρk S ( µ ) χψ (cid:12)(cid:12) D v R r D v ( λ ) φ ρ S k S ( µ ) χψ (cid:11) + (cid:10) φ ρ S h S ( µ ) χψ (cid:12)(cid:12) D v R r D v ( λ ) ρh S ( µ ) χψ (cid:11) = (cid:10) ρk S ( µ ) χψ (cid:12)(cid:12) D v R r D v ( λ ) φ [ φρ, S ] k S ( µ ) χψ (cid:11) + (cid:10) ρk S ( µ ) χψ (cid:12)(cid:12) D v R r D v ( λ ) φ S φρk S ( µ ) χψ (cid:11) + (cid:10) φ [ φρ, S ] h S ( µ ) χψ (cid:12)(cid:12) D v R r D v ( λ ) ρh S ( µ ) χψ (cid:11) + (cid:10) S φρh S ( µ ) χψ (cid:12)(cid:12) φ D v R r D v ( λ ) ρh S ( µ ) χψ (cid:11) = (cid:10) B ( λ ) k S ( µ ) χψ (cid:12)(cid:12) k S ( µ ) χψ (cid:11) + (cid:10) h S ( µ ) χψ (cid:12)(cid:12) B ( λ ) h S ( µ ) χψ (cid:11) + (cid:10) φ D v R r D v ( λ ) ρk S ( µ ) χψ (cid:12)(cid:12) S φρk S ( µ ) χψ (cid:11) + (cid:10) S φρh S ( µ ) χψ (cid:12)(cid:12) φ D v R r D v ( λ ) ρh S ( µ ) χψ (cid:11) , (5.2)where we have inserted the definition of B ( λ ) . To rewrite the terms on the last line, weconsider either ξ = ρk S ( µ ) χψ or ξ = ρh S ( µ ) χψ . Since vρ = ρ , we note that in both caseswe have ξ = vξ ∈ Ran( v ) , so from Lemma 5.4 and condition (L1) we know that k r D v ( λ ) ξ and h r D v ( λ ) ξ lie in Dom( v S ) . Since φ preserves Dom( v S ) , we can compute (cid:10) φ D v R r D v ( λ ) ξ (cid:12)(cid:12) S φξ (cid:11) = (cid:10) φ D vR r D v ( λ ) ξ (cid:12)(cid:12) v S φ ( r + λ + D v ) R r D v ( λ ) ξ (cid:11) = (cid:10) φ D vk r D v ( λ ) ξ (cid:12)(cid:12) v S φk r D v ( λ ) ξ (cid:11) + (cid:10) v S φh r D v ( λ ) ξ (cid:12)(cid:12) φ D vh r D v ( λ ) ξ (cid:11) = (cid:10) [ φ, D ] vk r D v ( λ ) ξ (cid:12)(cid:12) v S φk r D v ( λ ) ξ (cid:11) + (cid:10) D φvk r D v ( λ ) ξ (cid:12)(cid:12) v S φk r D v ( λ ) ξ (cid:11) ocalisations and the kasparov product + (cid:10) v S φh r D v ( λ ) ξ (cid:12)(cid:12) [ φ, D ] vh r D v ( λ ) ξ (cid:11) + (cid:10) v S φh r D v ( λ ) ξ (cid:12)(cid:12) D φvh r D v ( λ ) ξ (cid:11) . Taking twice the real part, and inserting the definition of Q v , this yields ℜ (cid:10) φ D v R r D v ( λ ) ξ (cid:12)(cid:12) S φξ (cid:11) = 2 ℜ (cid:10) [ φ, D ] vk r D v ( λ ) ξ (cid:12)(cid:12) v S φk r D v ( λ ) ξ (cid:11) + 2 ℜ (cid:10) [ φ, D ] vh r D v ( λ ) ξ (cid:12)(cid:12) v S φh r D v ( λ ) ξ (cid:11) + Q v (cid:0) φk r D v ( λ ) ξ (cid:1) + Q v (cid:0) φh r D v ( λ ) ξ (cid:1) . Inserting the latter into Eq. (5.2), we thus obtain ℜ (cid:10) χψ (cid:12)(cid:12) D v R r D v ( λ ) ρ S R S ( µ ) χψ (cid:11) = 2 ℜ (cid:10) B ( λ ) k S ( µ ) χψ (cid:12)(cid:12) k S ( µ ) χψ (cid:11) + 2 ℜ (cid:10) h S ( µ ) χψ (cid:12)(cid:12) B ( λ ) h S ( µ ) χψ (cid:11) + 2 ℜ (cid:10) [ φ, D ] vk r D v ( λ ) ρk S ( µ ) χψ (cid:12)(cid:12) v S φk r D v ( λ ) ρk S ( µ ) χψ (cid:11) + 2 ℜ (cid:10) [ φ, D ] vh r D v ( λ ) ρk S ( µ ) χψ (cid:12)(cid:12) v S φh r D v ( λ ) ρk S ( µ ) χψ (cid:11) + Q v (cid:0) φk r D v ( λ ) ρk S ( µ ) χψ (cid:1) + Q v (cid:0) φh r D v ( λ ) ρk S ( µ ) χψ (cid:1) + 2 ℜ (cid:10) [ φ, D ] vk r D v ( λ ) ρh S ( µ ) χψ (cid:12)(cid:12) v S φk r D v ( λ ) ρh S ( µ ) χψ (cid:11) + 2 ℜ (cid:10) [ φ, D ] vh r D v ( λ ) ρh S ( µ ) χψ (cid:12)(cid:12) v S φh r D v ( λ ) ρh S ( µ ) χψ (cid:11) + Q v (cid:0) φk r D v ( λ ) ρh S ( µ ) χψ (cid:1) + Q v (cid:0) φh r D v ( λ ) ρh S ( µ ) χψ (cid:1) = 2 ℜ (cid:10) B ( λ ) k S ( µ ) χψ (cid:12)(cid:12) k S ( µ ) χψ (cid:11) + 2 ℜ (cid:10) B ( λ ) h S ( µ ) χψ (cid:12)(cid:12) h S ( µ ) χψ (cid:11) + X m =1 ℜ (cid:10) [ φ, D ] vM m ( λ, µ ) χψ (cid:12)(cid:12) v S φM m ( λ, µ ) χψ (cid:11) + X m =1 Q v (cid:0) φM m ( λ, µ ) χψ (cid:1) . Lemma 5.6.
We have the inequality ± π ∞ Z ∞ Z ( λµ ) − ˆ B ( λ, µ ) dλdµ ≤ C B (cid:10) χψ (cid:12)(cid:12) χψ (cid:11) , where C B := 2 (cid:13)(cid:13) [ S , ρ ] (cid:13)(cid:13) .Proof. We note that the integral over λ of π − λ − D v R r D v ( λ ) converges strongly to F r D v = D v ( r + D v ) − ≤ , and thus we have the operator inequality ± π ∞ Z λ − (cid:0) B ( λ ) + B ( λ ) ∗ (cid:1) dλ = ± [ S , ρ ] φF r D v ρ ± ρF r D v φ [ ρ, S ] ≤ (cid:13)(cid:13) [ S , ρ ] (cid:13)(cid:13) = C B . Inserting this into the definition of ˆ B ( λ, µ ) , the remaining integral over µ is norm-conver-gent, and using k S ( µ ) + h S ( µ ) = R S ( µ ) we find ± π ∞ Z ∞ Z ( λµ ) − ˆ B ( λ, µ ) dλdµ ≤ π C B ∞ Z µ − (cid:10) k S ( µ ) χψ (cid:12)(cid:12) k S ( µ ) χψ (cid:11) dµ + 1 π C B ∞ Z µ − (cid:10) h S ( µ ) χψ (cid:12)(cid:12) h S ( µ ) χψ (cid:11) dµ Koen van den Dungen = 1 π C B ∞ Z µ − (cid:10) χψ (cid:12)(cid:12) R S ( µ ) χψ (cid:11) dµ = C B (cid:10) χψ (cid:12)(cid:12) (1 + S ) − χψ (cid:11) ≤ C B (cid:10) χψ (cid:12)(cid:12) χψ (cid:11) . The following lemma generalises Lemma 3.9.
Lemma 5.7.
Let T be any densely defined symmetric operator on E such that T ( r + D v ) − is bounded. For ψ ∈ E , we have the inequality X m =1 π ∞ Z ∞ Z ( µλ ) − (cid:10) M m ( λ, µ ) ψ (cid:12)(cid:12) T M m ( λ, µ ) ψ (cid:11) dλdµ ≤ (cid:13)(cid:13) T ( r + D v ) − (cid:13)(cid:13) k ρ k h ψ | ψ i . Proof.
Let us consider the four integrals (for m = 1 , , , ) given by I m := 1 π ∞ Z ∞ Z ( µλ ) − (cid:10) M m ( λ, µ ) ψ (cid:12)(cid:12) T M m ( λ, µ ) ψ (cid:11) dλdµ. Since T ( r + D v ) − is bounded and T is symmetric, we know from Lemma 2.7 that also ( r + D v ) − T ( r + D v ) − is bounded, and (cid:13)(cid:13) ( r + D v ) − T ( r + D v ) − (cid:13)(cid:13) ≤ (cid:13)(cid:13) T ( r + D v ) − (cid:13)(cid:13) . We obtain the operator inequality h r D v ( λ ) T h r D v ( λ ) = ( r + D v ) h r D v ( λ ) ( r + D v ) − T ( r + D v ) − h r D v ( λ )( r + D v ) ≤ ( r + D v ) h r D v ( λ ) (cid:13)(cid:13) ( r + D v ) − T ( r + D v ) − (cid:13)(cid:13) h r D v ( λ )( r + D v ) ≤ (cid:13)(cid:13) T ( r + D v ) − (cid:13)(cid:13) ( r + D v ) h r D v ( λ ) . Similarly, we also have the operator inequality k r D v ( λ ) T k r D v ( λ ) ≤ (cid:13)(cid:13) T ( r + D v ) − (cid:13)(cid:13) ( r + D v ) k r D v ( λ ) . We note that h r D v ( λ ) + k r D v ( λ ) = R r D v ( λ ) . Moreover, by Lemma 2.6, the integral of λ − ( r + D v ) R r D v ( λ ) converges strongly to π , and we find π ∞ Z λ − (cid:0) h r D v ( λ ) T h r D v ( λ ) + k r D v ( λ ) T k r D v ( λ ) (cid:1) dλ ≤ (cid:13)(cid:13) T ( r + D v ) − (cid:13)(cid:13) . Thus we obtain the inequalities I + I ≤ π (cid:13)(cid:13) T ( r + D v ) − (cid:13)(cid:13) k ρ k ∞ Z µ − (cid:10) h S ( µ ) ψ (cid:12)(cid:12) h S ( µ ) ψ (cid:11) dµ,I + I ≤ π (cid:13)(cid:13) T ( r + D v ) − (cid:13)(cid:13) k ρ k ∞ Z µ − (cid:10) k S ( µ ) ψ (cid:12)(cid:12) k S ( µ ) ψ (cid:11) dµ. ocalisations and the kasparov product µ , we obtain X m =1 I m ≤ π (cid:13)(cid:13) T ( r + D v ) − (cid:13)(cid:13) k ρ k ∞ Z µ − (cid:10) ψ (cid:12)(cid:12) R S ( µ ) ψ (cid:11) dµ = (cid:13)(cid:13) T ( r + D v ) − (cid:13)(cid:13) k ρ k (cid:10) ψ (cid:12)(cid:12) (1 + S ) − ψ (cid:11) ≤ (cid:13)(cid:13) T ( r + D v ) − (cid:13)(cid:13) k ρ k (cid:10) ψ (cid:12)(cid:12) ψ (cid:11) . Lemma 5.8.
We have the inequality π ∞ Z ∞ Z ( λµ ) − ˆ Q ( λ, µ ) dλdµ ≥ − C Q (cid:10) χψ (cid:12)(cid:12) χψ (cid:11) , where C Q := c (cid:13)(cid:13) φ (1 + D v ) φ ( r + D v ) − (cid:13)(cid:13) .Proof. By condition (L5) in Assumption 5.1, we have the inequality Q v (cid:0) φM m ( λ, µ ) χψ (cid:1) ≥ − c (cid:10) M m ( λ, µ ) χψ (cid:12)(cid:12) φ (1 + D v ) φM m ( λ, µ ) χψ (cid:11) . The statement then follows by applying Lemma 5.7 with the operator T = φ (1 + D v ) φ .It remains to consider the term ˆ M ( λ, µ ) in Lemma 5.5. Of course, in the special casewhere φ = Id , we have ˆ M ( λ, µ ) = 0 , and then we obtain the following result. Proposition 5.9.
Consider the setting of Assumption 5.1. Suppose furthermore that φ =Id . Then for any < κ < there exists an α > such that the operator χ [ F D v , F α S ] χ + κχ is positive modulo compact operators: χ [ F D v , F α S ] χ & − κχ . Proof.
As already mentioned earlier, we know that χ [ F D v , F S ] χ is equal to χ (cid:0) F r D v ρF S + F S ρF r D v (cid:1) χ modulo compact operators. For any ψ ∈ E , we thus need to estimate theintegral in Eq. (5.1). From Lemma 5.5 we obtain the equality (cid:10) ψ (cid:12)(cid:12) χ (cid:0) F r D v ρF S + F S ρF r D v (cid:1) χψ (cid:11) = 1 π ∞ Z ∞ Z ( λµ ) − (cid:0) ˆ B ( λ, µ ) + ˆ M ( λ, µ ) + ˆ Q ( λ, µ ) (cid:1) dλdµ. Since φ = Id , we have ˆ M ( λ, µ ) = 0 . Using Lemmas 5.6 and 5.8, we then obtain the estimate (cid:10) ψ (cid:12)(cid:12) χ (cid:0) F r D v ρF S + F S ρF r D v (cid:1) χψ (cid:11) ≥ − ( C B + C Q ) ⟪ χψ ⟫ . Since this holds for any ψ , we have the operator inequality χ (cid:0) F r D v ρF S + F S ρF r D v (cid:1) χ ≥ − ( C B + C Q ) χ . Koen van den Dungen
We have therefore shown that χ [ F D v , F S ] χ & − ( C B + C Q ) χ . Next, we fix < κ < . If we replace S by α S for some α > , then the constants C B and C Q are replaced by αC B and αC Q , respectively. Indeed, for C B = 2 (cid:13)(cid:13) [ S , ρ ] (cid:13)(cid:13) this is obvious, andwe note that C Q is proportional to the constant c from condition (L5) in Assumption 5.1.Thus, by choosing α small enough, we can ensure that α ( C B + C Q ) < κ < .In the following, we provide two different sufficient conditions which allow us to dealwith the term ˆ M ( λ, µ ) also for non-trivial φ . Proposition 5.10.
In addition to the setting of Assumption 5.1, suppose furthermore thatthe following strengthening of condition (L5) is satisfied:(L5’) there exist constants ν ∈ (0 , ∞ ) and c ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D v ) ∩ Ran( v ) we have Q v ( φψ ) ≥ ν ⟪ v S φψ ⟫ − c (cid:10) φψ (cid:12)(cid:12) (1 + D v ) φψ (cid:11) . (5.3) Then for any < κ < there exists an α > such that the operator χ [ F D v , F α S ] χ + κχ ispositive modulo compact operators: χ [ F D v , F α S ] χ & − κχ . Proof.
First of all, for any β > we can estimate ± ˆ M ( λ, µ ) = ± X m =1 ℜ (cid:10) β − [ φ, D ] vM m ( λ, µ ) χψ (cid:12)(cid:12) βv S φM m ( λ, µ ) χψ (cid:11) ≤ X m =1 β ⟪ v S φM m ( λ, µ ) χψ ⟫ + β − ⟪ [ φ, D ] vM m ( λ, µ ) χψ ⟫ . Combined with condition (L5’) this yields ˆ M ( λ, µ ) + ˆ Q ( λ, µ ) ≥ ( ν − β ) ⟪ v S φM m ( λ, µ ) χψ ⟫ − β − ⟪ [ φ, D ] vM m ( λ, µ ) χψ ⟫ − c (cid:10) φM m ( λ, µ ) χψ (cid:12)(cid:12) (1 + D v ) φM m ( λ, µ ) χψ (cid:11) . Taking the double integral of this inequality, and estimating the second and third terms byapplying Lemma 5.7 with T = β − v [ D , φ ][ φ, D ] v + cφ (1 + D v ) φ , we obtain π ∞ Z ∞ Z ( λµ ) − (cid:0) ˆ M ( λ, µ ) + ˆ Q ( λ, µ ) (cid:1) dλdµ ≥ ( ν − β ) P − C M ( β, c ) ⟪ χψ ⟫ , (5.4)where we define P := 1 π ∞ Z ∞ Z ( λµ ) − ⟪ v S φM m ( λ, µ ) χψ ⟫ dλdµ ≥ , ocalisations and the kasparov product C M ( β, c ) := β − (cid:13)(cid:13) v [ D , φ ][ φ, D ] v ( r + D v ) − (cid:13)(cid:13) + c (cid:13)(cid:13) φ (1 + D v ) φ ( r + D v ) − (cid:13)(cid:13) . As in the proof of Proposition 5.9, we need to consider the integral (cid:10) ψ (cid:12)(cid:12) χ (cid:0) F r D v ρF S + F S ρF r D v (cid:1) χψ (cid:11) = 1 π ∞ Z ∞ Z ( λµ ) − (cid:0) ˆ B ( λ, µ ) + ˆ M ( λ, µ ) + ˆ Q ( λ, µ ) (cid:1) dλdµ. Combining Eq. (5.4) with Lemma 5.6, we obtain the estimate (cid:10) ψ (cid:12)(cid:12) χ (cid:0) F r D v ρF S + F S ρF r D v (cid:1) χψ (cid:11) ≥ ( ν − β ) P − (cid:0) C B + C M ( β, c ) (cid:1) ⟪ χψ ⟫ . (5.5)We now replace S by α S for some α ∈ (0 , ∞ ) . We recall that C B is then replaced by αC B .Moreover, we note that condition (L5’) implies (cid:10) D v φψ (cid:12)(cid:12) α S φψ (cid:11) + (cid:10) α S φψ (cid:12)(cid:12) D v φψ (cid:11) ≥ α − ν (cid:10) αv S φψ (cid:12)(cid:12) αv S φψ (cid:11) − αc (cid:10) φψ (cid:12)(cid:12) (1 + D v ) φψ (cid:11) , and hence ν and c are replaced by α − ν and αc . We choose β large enough such that β − (cid:13)(cid:13) v [ D , φ ][ φ, D ] v ( r + D v ) − (cid:13)(cid:13) < κ. We then choose α small enough such that α − ν − β > , αc (cid:13)(cid:13) φ (1 + D v ) φ ( r + D v ) − (cid:13)(cid:13) < κ, αC B < κ. These choices ensure that αC B + C M ( β, αc ) < κ . Thus from Eq. (5.5) we obtain (cid:10) ψ (cid:12)(cid:12) χ (cid:0) F r D v ρF α S + F α S ρF r D v (cid:1) χψ (cid:11) ≥ − κ ⟪ χψ ⟫ . Since this holds for any ψ , we have the operator inequality χ (cid:0) F r D v ρF α S + F α S ρF r D v (cid:1) χ ≥ − κχ . Since χ [ F D v , F α S ] χ is equal to χ (cid:0) F r D v ρF α S + F α S ρF r D v (cid:1) χ modulo compact operators, thiscompletes the proof. Proposition 5.11.
In addition to the setting of Assumption 5.1, suppose furthermore thatthe following condition is satisfied:(L6) The operators φ and φ [ D v , φ ] map Dom( D v ) to Dom S .Then for any < κ < there exists an α > such that the operator χ [ F D v , F α S ] χ + κχ ispositive modulo compact operators: χ [ F D v , F α S ] χ & − κχ . Proof.
First, we will derive an estimate for the double integral of ˆ M ( λ, µ ) . Since by as-sumption φ maps Dom( D v ) to Dom( S ) , we know that S φ ( r + D v ) − is bounded, andtherefore also [ D v , φ ] S φ ( r + D v ) − is bounded. Furthermore, we have assumed that φ [ D v , φ ] also maps Dom( D v ) to Dom( S ) , which ensures that φ S [ φ, D v ]( r + D v ) − = Koen van den Dungen S φ [ φ, D v ]( r + D v ) − + [ φ, S ][ φ, D v ]( r + D v ) − is bounded as well. By applying Lemma 5.7with the symmetric operator T = [ D v , φ ] S φ + φ S [ φ, D v ] , we obtain the inequality ± π ∞ Z ∞ Z ( λµ ) − ˆ M ( λ, µ ) dλdµ ≤ C M (cid:10) χψ (cid:12)(cid:12) χψ (cid:11) , where C M := (cid:13)(cid:13)(cid:0) [ D v , φ ] S φ + φ S [ φ, D v ] (cid:1) ( r + D v ) − (cid:13)(cid:13) . As in the proof of Proposition 5.9, weneed to consider the integral (cid:10) ψ (cid:12)(cid:12) χ (cid:0) F r D v ρF S + F S ρF r D v (cid:1) χψ (cid:11) = 1 π ∞ Z ∞ Z ( λµ ) − (cid:0) ˆ B ( λ, µ ) + ˆ M ( λ, µ ) + ˆ Q ( λ, µ ) (cid:1) dλdµ. Combining Lemmas 5.6 and 5.8 with the above inequality for ˆ M ( λ, µ ) , we now obtain theestimate (cid:10) ψ (cid:12)(cid:12) χ (cid:0) F r D v ρF S + F S ρF r D v (cid:1) χψ (cid:11) ≥ − ( C B + C Q + C M ) ⟪ χψ ⟫ . The proof then proceeds exactly as in Proposition 5.9.
Remark 5.12.
The condition (L6) is quite naturally satisfied in the context of first-orderdifferential operators S and D on smooth manifolds, when φ, v are compactly supportedsmooth functions and D v is elliptic on supp( φ ) (this is the setting considered in [Dun20]). Assumption 6.1.
Let A be a ( Z -graded) separable C ∗ -algebra, let B and C be ( Z -graded) σ -unital C ∗ -algebras, and let A ⊂ A and B ⊂ B be dense ∗ -subalgebras. Consider three half-closed modules ( A , π ( E ) B , D ) , ( B , π ( E ) C , D ) , and ( A , π E C , D ) , where E := E ˆ ⊗ B E and π = π ˆ ⊗ , and suppose that π is essential. We assume that the ∗ -subalgebra A ⊂ A contains an (even) almost idempotent approximate unit { u n } n ∈ N for A .For ease of notation, we will usually identify u n ≡ u n ˆ ⊗ ≡ π ( u n ˆ ⊗ on E . We recall the‘partition of unity’ { χ k } k ∈ N from Definition 4.6. Proposition 6.2.
In the setting of Assumption 6.1, assume that the connection condi-tion (Definition 2.10) is satisfied. For each k ∈ N , consider elements v k , w k ∈ { u n } with v k u k +1 = w k u k +1 = u k +1 , and write D k := v k D v k and D ,k := w k D w k . We assumefurthermore that for some < κ < the following condition is satisfied: • for each k ∈ N there exists α k ∈ (0 , ∞ ) such that χ k [ F D k , F α k D ,k ˆ ⊗ χ k + κχ k ispositive modulo compact operators.Then ( A , π E C , D ) represents the Kasparov product of ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) .Proof. We can represent D and D by their bounded transforms F D and F D , and (byTheorem 4.11) we can represent D by a localised representative e F D ( α ) . We have seen inProposition 2.11 that the connection condition of Theorem 2.2 is satisfied. Thus it remains ocalisations and the kasparov product F D and e F D ( α ) (for the given sequence { α k } k ∈ N ) satisfy the positivity conditionof Theorem 2.2.To prove the positivity condition, it suffices to consider u n [ F D , e F D ( α ) ˆ ⊗ u n , sincewe have norm-convergence au n → a . From Lemma 4.7 we know that P k χ k u n is a finitesum. We know from Lemma 4.3 (applied with a = χ k , c = u k +1 , and b = v k ) that χ k ( F D − F D k ) ∼ . Using furthermore that [ F D , χ k ] ∼ by Theorem 2.4, we have u n [ F D , e F D ( α ) ˆ ⊗ u n = X k u n [ F D , χ k ( F α k D ,k ˆ ⊗ χ k ] u n . ∼ X k u n χ k [ F D , F α k D ,k ˆ ⊗ χ k u n . ∼ X k u n χ k [ F D k , F α k D ,k ˆ ⊗ χ k u n . By hypothesis, we have a sequence { α k } k ∈ N ⊂ (0 , ∞ ) such that χ k [ F D k , F α k D ,k ˆ ⊗ χ k + κχ k is positive modulo compact operators. We then conclude that u n [ F D , e F D ( α ) ˆ ⊗ u n ∼ X k u n χ k [ F D k , F α k D ,k ˆ ⊗ χ k u n & − X k κu n χ k u n = − κu n . Thus the positivity condition of Theorem 2.2 also holds, and the statement follows.
Theorem 6.3.
In the setting of Assumption 6.1, assume that the connection condition(Definition 2.10) is satisfied. We assume furthermore that for each n ∈ N the followingconditions are satisfied: • we have the domain inclusion Dom( D u n ) ∩ Ran( u n ) ⊂ Dom( D u n ˆ ⊗ ; • there exists c n ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D u n ) ∩ Ran( u n ) we have (cid:10) u n ( D ˆ ⊗ ψ (cid:12)(cid:12) D u n ψ (cid:11) + (cid:10) D u n ψ (cid:12)(cid:12) u n ( D ˆ ⊗ ψ (cid:11) ≥ − c n (cid:10) ψ (cid:12)(cid:12) (1 + ( u n D u n ) ) ψ (cid:11) . (6.1) Then ( A , π E C , D ) represents the Kasparov product of ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) .Proof. Let us write v k := u k +2 and w k := u k +3 . For any k ∈ N , we will check thatAssumption 5.1 is satisfied by the operators D , S = D ,k ˆ ⊗ w k D w k ˆ ⊗ , χ = χ k , ρ = u k +1 , φ = Id , and v = v k . We shall write D k := D v = v k D v k .We have by assumption the domain inclusion Dom( D v k ) ∩ Ran( v k ) ⊂ Dom( D v k ˆ ⊗ .Noting that Dom( D v k ˆ ⊗
1) = Dom( D w k v k ˆ ⊗
1) = Dom( S k v k ) , we see that condition (L1)is satisfied. Since φ = Id and { u n } is almost idempotent, and using Lemma 4.7, we seethat condition (L2) is satisfied. By assumption, we have u n ∈ Lip ∗ ( D ) ∩ Lip ∗ ( D ˆ ⊗ ,and since { u n } is commutative, it follows that we also have u n ∈ Lip( S k ) , so condition(L3) is satisfied. Condition (L4) follows from Lemma 4.2 (using that χ k u k +1 = χ k and u k +1 v k = u k +1 ). Finally, for ψ ∈ Dom( D v k ) ∩ Ran( v k ) we have w k ψ = ψ , and then itfollows from Eq. (6.1) that Q v ( ψ ) = (cid:10) D v k ψ (cid:12)(cid:12) v k S k ψ (cid:11) + (cid:10) v k S k ψ (cid:12)(cid:12) D v k ψ (cid:11) Koen van den Dungen = (cid:10) D v k ψ (cid:12)(cid:12) v k ( D ˆ ⊗ ψ (cid:11) + (cid:10) v k ( D ˆ ⊗ ψ (cid:12)(cid:12) D v k ψ (cid:11) ≥ − c k +2 (cid:10) ψ (cid:12)(cid:12) (1 + D k ) ψ (cid:11) , which shows condition (L5). Thus Assumption 5.1 is indeed satisfied.Hence we can apply Proposition 5.9 for each k ∈ N , and for any < κ < we obtaina sequence { α k } k ∈ N ⊂ (0 , ∞ ) such that χ k [ F D k , F α k D ,k ˆ ⊗ χ k + κχ k is positive modulocompact operators. The statement then follows from Proposition 6.2. Remark 6.4.
We note that Eq. (6.1) may be replaced by (cid:10) u n ( D ˆ ⊗ u n ψ (cid:12)(cid:12) u n D u n ψ (cid:11) + (cid:10) u n D u n ψ (cid:12)(cid:12) u n ( D ˆ ⊗ u n ψ (cid:11) ≥ − c n (cid:10) ψ (cid:12)(cid:12) (1 + ( u n D u n ) ) ψ (cid:11) . In this case the proof of Theorem 6.3 may be repeated, now choosing w k = u k +2 insteadof w k = u k +3 (indeed, condition (L5) then follows from the above inequality, while forcondition (L1) we note that Dom( D v k ˆ ⊗ ⊂ Dom( v k D v k ˆ ⊗ ). Remark 6.5.
In the setting of Theorem 6.3, suppose furthermore that D commutes withthe approximate identity u n . Then it follows from a standard argument that D is in fact self-adjoint. Indeed, if [ D , u n ] = 0 then also [ D ∗ , u n ] = 0 . Using that u n : Dom D ∗ → Dom D ,we then see for any ξ ∈ Dom D ∗ that D u n ξ = u n D ∗ ξ converges to D ∗ ξ , which proves that ξ lies in the domain of D . (For another argument for the self-adjointness of D , see also[KS19, Remark 6.5].)Furthermore, we note that { u n } n ∈ N is a localising subset in the sense of [KS19, Definition6.2]. The statement of Theorem 6.3 is then very similar to [KS19, Theorem 6.10], whichrequires the following ‘local positivity condition’ [KS19, Definition 6.3]: (cid:10) ( D ˆ ⊗ u n ψ (cid:12)(cid:12) D u n ψ (cid:11) + (cid:10) D u n ψ (cid:12)(cid:12) ( D ˆ ⊗ u n ψ (cid:11) ≥ − c n h ψ | ψ i . However, we note that our local positivity condition is a weaker assumption, since we allowfor the additional factor (1 + ( u n D u n ) ) in the inner product on the right-hand-side ofthe above inequality. Perhaps even more importantly, the major advantage of our proof ofTheorem 6.3 is that we do not require u n to commute with D . This allows us to genuinelydeal with the case of non-self-adjoint D , whereas [KS19] deals ‘essentially’ only with thecase where D is self-adjoint (as explained in [KS19, Remark 6.5]). As already explained in the Introduction, we would like to replace the conditions in The-orem 6.3 by the following more natural condition (and in §6.1.1 and §6.1.2 we will considertwo possible sufficient conditions which allow us to make this additional step).
Definition 6.6.
In the setting of Assumption 6.1, the local positivity condition requiresthat for each n ∈ N the following assumptions hold:(1) we have the inclusion u n · Dom( D ) ⊂ Dom( D ˆ ⊗ ;(2) there exists c n ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D ) we have (cid:10) ( D ˆ ⊗ u n ψ (cid:12)(cid:12) D u n ψ (cid:11) + (cid:10) D u n ψ (cid:12)(cid:12) ( D ˆ ⊗ u n ψ (cid:11) ≥ − c n (cid:10) u n ψ (cid:12)(cid:12) (1 + D ∗ D ) / u n ψ (cid:11) . ocalisations and the kasparov product Remark 6.7.
For the domain condition (1) in Definition 6.6, it is of course sufficient (butnot necessary!) to have the domain inclusion
Dom( D ) ⊂ Dom( D ˆ ⊗ . Lemma 6.8.
Consider D k := u k +4 D u k +4 and S k := u k +4 D u k +4 ˆ ⊗ , and ψ ∈ Dom D .Then we have ℜ (cid:10) D k u k +2 ψ (cid:12)(cid:12) S k u k +2 ψ (cid:11) = 2 ℜ (cid:10) D u k +2 ψ (cid:12)(cid:12) ( D ˆ ⊗ u k +2 ψ (cid:11) + 2 ℜ (cid:10) [ u k +4 , D ] u k +2 ψ (cid:12)(cid:12) [ D ˆ ⊗ , u k +3 ] u k +2 ψ (cid:11) ≥ ℜ (cid:10) D u k +2 ψ (cid:12)(cid:12) ( D ˆ ⊗ u k +2 ψ (cid:11) − (cid:13)(cid:13) [ D, u k +4 ][ D ˆ ⊗ , u k +3 ] (cid:13)(cid:13) (cid:10) u k +2 ψ (cid:12)(cid:12) u k +2 ψ (cid:11) . Proof.
We compute (cid:10) D k u k +2 ψ (cid:12)(cid:12) S k u k +2 ψ (cid:11) = (cid:10) u k +4 D u k +2 ψ (cid:12)(cid:12) u k +4 ( D ˆ ⊗ u k +3 u k +2 ψ (cid:11) = (cid:10) u k +4 D u k +2 ψ (cid:12)(cid:12) u k +4 [ D ˆ ⊗ , u k +3 ] u k +2 ψ (cid:11) + (cid:10) D u k +2 ψ (cid:12)(cid:12) u k +3 ( D ˆ ⊗ u k +2 ψ (cid:11) = (cid:10) [ u k +4 , D ] u k +2 ψ (cid:12)(cid:12) [ D ˆ ⊗ , u k +3 ] u k +2 ψ (cid:11) + (cid:10) D u k +2 ψ (cid:12)(cid:12) [ D ˆ ⊗ , u k +3 ] u k +2 ψ (cid:11) + (cid:10) D u k +2 ψ (cid:12)(cid:12) u k +3 ( D ˆ ⊗ u k +2 ψ (cid:11) = (cid:10) D u k +2 ψ (cid:12)(cid:12) ( D ˆ ⊗ u k +2 ψ (cid:11) + (cid:10) [ u k +4 , D ] u k +2 ψ (cid:12)(cid:12) [ D ˆ ⊗ , u k +3 ] u k +2 ψ (cid:11) . The inequality is then clear.
In the setting of Assumption 6.1, the strong local positivity condition re-quires that for each n ∈ N the following assumptions hold:(1) we have the inclusion u n · Dom( D ) ⊂ Dom( D ˆ ⊗ ;(2) there exist ν n ∈ (0 , ∞ ) and c n ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D ) we have (cid:10) ( D ˆ ⊗ u n ψ (cid:12)(cid:12) D u n ψ (cid:11) + (cid:10) D u n ψ (cid:12)(cid:12) ( D ˆ ⊗ u n ψ (cid:11) ≥ ν n ⟪ ( D ˆ ⊗ u n ψ ⟫ − c n (cid:10) u n ψ (cid:12)(cid:12) (1 + D ∗ D ) u n ψ (cid:11) . Lemma 6.10.
In the setting of Assumption 6.1, assume that the strong local positivitycondition is satisfied. Then writing S k := u k +4 D u k +4 ˆ ⊗ , D k := u k +4 D u k +4 , and φ k := u k +2 , there exists for each k ∈ N a constant d k ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D ) wehave (cid:10) S k φ k ψ (cid:12)(cid:12) D k φ k ψ (cid:11) + (cid:10) D k φ k ψ (cid:12)(cid:12) S k φ k ψ (cid:11) ≥ ν k +2 ⟪ u k +4 S k φ k ψ ⟫ − d k (cid:10) φ k ψ (cid:12)(cid:12) (1 + D k ) / φ k ψ (cid:11) . Proof.
Combining Lemma 6.8 with the strong local positivity condition, we have ℜ (cid:10) D k φ k ψ (cid:12)(cid:12) S k φ k ψ (cid:11) ≥ ℜ (cid:10) D φ k ψ (cid:12)(cid:12) ( D ˆ ⊗ φ k ψ (cid:11) − (cid:13)(cid:13) [ D, u k +4 ][ D ˆ ⊗ , u k +3 ] (cid:13)(cid:13) ⟪ φ k ψ ⟫ ≥ ν k +2 ⟪ ( D ˆ ⊗ φ k ψ ⟫ − c k +2 (cid:10) φ k ψ (cid:12)(cid:12) (1 + D ∗ D ) φ k ψ (cid:11) − (cid:13)(cid:13) [ D, u k +4 ][ D ˆ ⊗ , u k +3 ] (cid:13)(cid:13) ⟪ φ k ψ ⟫ . (6.2)The first term in the last expression can be estimated by ⟪ D ˆ ⊗ φ k ψ ⟫ = (cid:10) ( D ˆ ⊗ u k +4 φ k ψ (cid:12)(cid:12) ( D ˆ ⊗ φ k ψ (cid:11) Koen van den Dungen = (cid:10) u k +4 ( D ˆ ⊗ φ k ψ (cid:12)(cid:12) u k +4 ( D ˆ ⊗ φ k ψ (cid:11) + (cid:10) [ D ˆ ⊗ , u k +4 ] φ k ψ (cid:12)(cid:12) ( D ˆ ⊗ u k +3 φ k ψ (cid:11) = ⟪ u k +4 S k φ k ψ ⟫ + (cid:10) [ D ˆ ⊗ , u k +4 ] φ k ψ (cid:12)(cid:12) [ D ˆ ⊗ , u k +3 ] φ k ψ ⟫ + (cid:10) [ D ˆ ⊗ , u k +4 ] φ k ψ (cid:12)(cid:12) u k +3 ( D ˆ ⊗ φ k ψ ⟫ ≥ ⟪ u k +4 S k φ k ψ ⟫ − (cid:13)(cid:13) [ D ˆ ⊗ , u k +4 ][ D ˆ ⊗ , u k +3 ] (cid:13)(cid:13) ⟪ φ k ψ ⟫ , where we used that u k +3 [ D ˆ ⊗ , u k +4 ] φ k = 0 . The second term in Eq. (6.2) can be estimatedas follows. Since u k +4 : Dom( u k +4 D u k +4 ) → Dom D , we know that the symmetric operator T := u k +4 (1 + D ∗ D ) u k +4 is relatively bounded by D k . Using Lemma 2.7, we find that (cid:10) φ k ψ (cid:12)(cid:12) (1 + D ∗ D ) / φ k ψ (cid:11) = (cid:10) φ k ψ (cid:12)(cid:12) u k +4 (1 + D ∗ D ) / u k +4 φ k ψ (cid:11) ≤ (cid:13)(cid:13) T (1 + D k ) − (cid:13)(cid:13) (cid:10) φ k ψ (cid:12)(cid:12) (1 + D k ) / φ k ψ (cid:11) . (6.3)We then obtain the desired inequality with d k = ν k +2 (cid:13)(cid:13) [ D ˆ ⊗ , u k +4 ][ D ˆ ⊗ , u k +3 ] (cid:13)(cid:13) + c k +2 (cid:13)(cid:13) T (1 + D k ) − (cid:13)(cid:13) + 2 (cid:13)(cid:13) [ D, u k +4 ][ D ˆ ⊗ , u k +3 ] (cid:13)(cid:13) . Theorem 6.11.
In the setting of Assumption 6.1, assume that the connection condition(Definition 2.10) and the strong local positivity condition (Definition 6.9) are satisfied. Then ( A , π E C , D ) represents the Kasparov product of ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) .Proof. Let us write v k := w k := u k +4 . For any k ∈ N , we will check that Assumption 5.1is satisfied by the operators D , S = D ,k ˆ ⊗ w k D w k ˆ ⊗ , χ = χ k , ρ = u k +1 , φ = u k +2 ,and v = v k . We shall write D k := D v = v k D v k .We have by assumption the domain inclusion u n · Dom( D ) ⊂ Dom( D ˆ ⊗ for any n ∈ N .For any ψ ∈ Dom( D v ) we note that vψ = u k +5 vψ ∈ u k +5 · Dom( D ) ⊂ Dom( D ˆ ⊗ , andtherefore ψ ∈ Dom( S v ) , which shows condition (L1). Conditions (L2)-(L4) follow as inthe proof of Theorem 6.3. Furthermore, for any ψ ∈ Dom( D v ) we have φψ = φvψ , where vψ ∈ Dom D . Condition (L5’) is then satisfied by Lemma 6.10. So by Proposition 5.10 weknow that there exists a sequence { α k } k ∈ N ⊂ (0 , ∞ ) such that χ k [ F D k , F α k D ,k ˆ ⊗ χ k + κχ k is positive modulo compact operators. The statement then follows from Proposition 6.2. In the setting of Assumption 6.1, assume that the local positivity conditionis satisfied. Then writing S k := u k +4 D u k +4 ˆ ⊗ , D k := u k +4 D u k +4 , and φ k := u k +2 , thereexists for each k ∈ N a constant d k ∈ [0 , ∞ ) such that for all ψ ∈ Dom( D ) we have (cid:10) S k φ k ψ (cid:12)(cid:12) D k φ k ψ (cid:11) + (cid:10) D k φ k ψ (cid:12)(cid:12) S k φ k ψ (cid:11) ≥ − d k (cid:10) φ k ψ (cid:12)(cid:12) (1 + D k ) / φ k ψ (cid:11) . Proof.
Combining Lemma 6.8 with the local positivity condition, we have ℜ (cid:10) D k φ k ψ (cid:12)(cid:12) S k φ k ψ (cid:11) ocalisations and the kasparov product ≥ ℜ (cid:10) D φ k ψ (cid:12)(cid:12) ( D ˆ ⊗ φ k ψ (cid:11) − (cid:13)(cid:13) [ D, u k +4 ][ D ˆ ⊗ , u k +3 ] (cid:13)(cid:13) (cid:10) φ k ψ (cid:12)(cid:12) φ k ψ (cid:11) ≥ − c k +2 (cid:10) φ k ψ (cid:12)(cid:12) (1 + D ∗ D ) φ k ψ (cid:11) − (cid:13)(cid:13) [ D, u k +4 ][ D ˆ ⊗ , u k +3 ] (cid:13)(cid:13) (cid:10) φ k ψ (cid:12)(cid:12) φ k ψ (cid:11) . As in Eq. (6.3), an application of Lemma 5.7 with T := u k +4 (1 + D ∗ D ) u k +4 yields (cid:10) φ k ψ (cid:12)(cid:12) (1 + D ∗ D ) / φ k ψ (cid:11) ≤ (cid:13)(cid:13) T (1 + D k ) − (cid:13)(cid:13) (cid:10) φ k ψ (cid:12)(cid:12) (1 + D k ) / φ k ψ (cid:11) . Thus we obtain the desired inequality with d k = c k +2 (cid:13)(cid:13) T (1 + D k ) − (cid:13)(cid:13) + 2 (cid:13)(cid:13) [ D, u k +4 ][ D ˆ ⊗ , u k +3 ] (cid:13)(cid:13) . Theorem 6.13.
In the setting of Assumption 6.1, assume that the connection condition(Definition 2.10) and the local positivity condition (Definition 6.6) are satisfied. Further,assume that the following condition holds: ( ⋆ ) for each n ∈ N , u n [ D , u n ] u n +2 maps Dom( D u n +2 ) to Dom( D ˆ ⊗ .Then ( A , π E C , D ) represents the Kasparov product of ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) .Proof. Most of the proof is similar to the proof of Theorem 6.11, so we shall be briefhere. Conditions (L1)-(L4) in Assumption 5.1 are again satisfied by the operators D , S = D ,k ˆ ⊗ w k D w k ˆ ⊗ , χ = χ k , ρ = u k +1 , φ = u k +2 , and v = v k , where v k = w k = u k +4 . Condition (L5) is shown in Lemma 6.12. Thus Assumption 5.1 isindeed satisfied. Furthermore, since φ k [ D k , φ k ] = u k +2 [ D , u k +2 ] u k +4 maps Dom( D k ) to Dom( D ˆ ⊗ ⊂ Dom( S k ) by condition ( ⋆ ), also condition (L6) is satisfied. Hence we canapply Proposition 5.11 for each k ∈ N , and we obtain a sequence { α k } k ∈ N ⊂ (0 , ∞ ) suchthat χ k [ F D k , F α k D ,k ˆ ⊗ χ k + κχ k is positive modulo compact operators. The statementthen follows from Proposition 6.2. Remark 6.14.
If for each n ∈ N , u n [ D , u n ] maps Dom( D ) to Dom( D ˆ ⊗ , and u n +2 commutes with [ D , u n ] , then one can check that the condition ( ⋆ ) is satisfied. This situationfor instance occurs if D and D are first-order differential operators on smooth manifolds(with D elliptic) and u n are compactly supported smooth functions. This is precisely thesetting studied in [Dun20], and Theorem 6.13 can be viewed as a generalisation of thestatement and proof of [Dun20, Theorem 4.2]. In this subsection we show that the strong local positivity condition is quite naturallysatisfied in (a localised version of) the constructive approach to the unbounded Kasparovproduct. In the following, we will in particular assume the existence of a suitable operator T . As explained in Remark 3.7, one should keep in mind here the case where (under suitableassumptions) T = 1 ˆ ⊗ ∇ D is constructed from a suitable ‘connection’ ∇ on E . Assumption 6.15.
Let A be a ( Z -graded) separable C ∗ -algebra, let B and C be ( Z -graded) σ -unital C ∗ -algebras, and let A ⊂ A and B ⊂ B be dense ∗ -subalgebras. Considertwo half-closed modules ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) , and suppose that π is es-sential. We write E := E ˆ ⊗ B E , π = π ˆ ⊗ , and S := D ˆ ⊗ , and we consider an odd4 Koen van den Dungen symmetric operator T on E . We denote by D := S + T the closure of the operator S + T on the initial domain Dom( S ) ∩ Dom( T ) . We assume that the following conditions aresatisfied:(A1) the intersection Dom( S ) ∩ Dom( T ) is dense in E , and the triple ( A , π E C , D ) is ahalf-closed module;(A2) for all ψ in a dense subset of A ·
Dom D , we have e T ψ := (cid:18) T ψ T ∗ ψ (cid:19) ∈ Lip(
T ⊕ D ); (A3) the ∗ -subalgebra A ⊂ A contains an almost idempotent approximate unit { u n } n ∈ N for A , such that we have the inclusion u n · Dom( D ) ⊂ Dom( S ) ∩ Dom( T ) . Theorem 6.16.
Consider the setting of Assumption 6.15. We assume furthermore thatthere exists a closed symmetric operator C on E with u n · Dom( D ) ⊂ Dom( C ) , such that forall ψ ∈ Dom( D ) we have the equality (cid:10) S u n ψ (cid:12)(cid:12) T u n ψ (cid:11) + (cid:10) T u n ψ (cid:12)(cid:12) S u n ψ (cid:11) = (cid:10) u n ψ (cid:12)(cid:12) C u n ψ (cid:11) . (6.4) Then ( A , π E C , D ) represents the Kasparov product of ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) .Proof. For any ψ ∈ Dom D we have bounded operators S T ψ = T D ψ and T ∗ ψ S = T ∗D ψ ,which means that e T ψ ∈ Lip(
S ⊕ . Combined with condition (A2) this ensures that e T ψ preserves (cid:0) Dom( S ) ∩ Dom( T ) (cid:1) ⊕ Dom( D ) , and that [ D ⊕ D , e T ψ ] is bounded on thisdomain. Since Dom( S ) ∩ Dom( T ) is a core for D , it follows that the connection condition(Definition 2.10) is satisfied.Our main task is to check the strong local positivity condition (Definition 6.9). Forthis purpose we can estimate the right-hand-side of Eq. (6.4) as follows. Since C u n +1 (1 + D ∗ D ) − is bounded, and writing c n := (cid:13)(cid:13) C u n +1 (1 + D ∗ D ) − (cid:13)(cid:13) , we know from Lemma 2.7that (cid:13)(cid:13) (1 + D ∗ D ) − u n +1 C u n +1 (1 + D ∗ D ) − (cid:13)(cid:13) ≤ c n . For any ψ ∈ Dom( D ) we then have theinequality ± (cid:10) u n ψ (cid:12)(cid:12) C u n ψ (cid:11) = ± (cid:10) (1 + D ∗ D ) u n ψ (cid:12)(cid:12) (1 + D ∗ D ) − u n +1 C u n +1 (1 + D ∗ D ) − (1 + D ∗ D ) u n ψ (cid:11) ≤ c n (cid:10) u n ψ (cid:12)(cid:12) (1 + D ∗ D ) u n ψ (cid:11) . Thus we obtain (cid:10) S u n ψ (cid:12)(cid:12) D u n ψ (cid:11) + (cid:10) D u n ψ (cid:12)(cid:12) S u n ψ (cid:11) = 2 ⟪ S u n ψ ⟫ + (cid:10) S u n ψ (cid:12)(cid:12) T u n ψ (cid:11) + (cid:10) T u n ψ (cid:12)(cid:12) S u n ψ (cid:11) = 2 ⟪ S u n ψ ⟫ + (cid:10) u n ψ (cid:12)(cid:12) C u n ψ (cid:11) ≥ ⟪ S u n ψ ⟫ − c n (cid:10) ψ (cid:12)(cid:12) (1 + D ∗ D ) ψ (cid:11) . We conclude that the strong local positivity condition is also satisfied (with ν n = 2 for all n ∈ N ). The statement then follows from Theorem 6.11. ocalisations and the kasparov product C in the above theorem should of course be equal to the graded commutator [ S , T ] whenever the latter is defined. However, in general it is not clear if the domain Dom( ST ) ∩ Dom(
T S ) is dense in E . The following result provides a sufficient conditionwhich ensures that the hypothesis of the above theorem is satisfied by the closure C = [ S , T ] . Corollary 6.17.
Consider the setting of Assumption 6.15. We assume furthermore thatthere exists a core
F ⊂
Dom D such that for each n ∈ N we have u n · F ⊂ Dom( ST ) ∩ Dom(
T S ) , and there exists C n ∈ [0 , ∞ ) such that for all η ∈ F we have k [ S , T ] u n η k ≤ C n k u n η k D . (6.5) Then ( A , π E C , D ) represents the Kasparov product of ( A , π ( E ) B , D ) and ( B , π ( E ) C , D ) .Proof. We will show that the hypothesis of Theorem 6.16 is satisfied by the closed symmetricoperator C := [ S , T ] . Fix n ∈ N , and let ψ ∈ Dom( D ) . Since F is a core for D , we canchoose a sequence { ψ k } k ∈ N ⊂ F such that k ψ k − ψ k D → as k → ∞ . The inequality (cid:13)(cid:13) u n ψ k − u n ψ (cid:13)(cid:13) D ≤ (cid:0) k u n k + k [ D , u n ] k (cid:1)(cid:13)(cid:13) ψ k − ψ (cid:13)(cid:13) D ensures that we also have the convergence k u n ψ k − u n ψ k D → as k → ∞ . In particular, u n ψ k is Cauchy with respect to k · k D , and from Eq. (6.5) we see that u n ψ k is also Cauchywith respect to the graph norm of [ S , T ] . Hence u n ψ = lim k →∞ u n ψ k lies in the domain ofthe closure [ S , T ] , and we have shown the inclusion u n · Dom( D ) ⊂ Dom([ S , T ]) .Furthermore, since u n ψ ∈ Dom( D ) , we can also choose a sequence { η k } k ∈ N ⊂ F suchthat k η k − u n ψ k D → as k → ∞ . The inequality (cid:13)(cid:13) u n +1 η k − u n ψ (cid:13)(cid:13) D = (cid:13)(cid:13) u n +1 ( η k − u n ψ ) (cid:13)(cid:13) D ≤ (cid:0) k u n +1 k + k [ D , u n +1 ] k (cid:1)(cid:13)(cid:13) η k − u n ψ (cid:13)(cid:13) D ensures that we then have the convergence k u n +1 η k − u n ψ k D → as k → ∞ . Since u n +1 · Dom( D ) ⊂ Dom( S ) by condition (A3), we can estimate (cid:13)(cid:13) S u n +1 η k − S u n ψ (cid:13)(cid:13) ≤ (cid:13)(cid:13) S u n +1 (1 + D ∗ D ) (cid:13)(cid:13) (cid:13)(cid:13) η k − u n ψ (cid:13)(cid:13) D . This ensures that we have norm-convergence S u n +1 η k → S u n ψ as k → ∞ . Similarly wealso have norm-convergence T u n +1 η k → T u n ψ and [ S , T ] u n +1 η k → [ S , T ] u n ψ . Hence wehave (cid:10) S u n ψ (cid:12)(cid:12) T u n ψ (cid:11) + (cid:10) T u n ψ (cid:12)(cid:12) S u n ψ (cid:11) = lim k →∞ (cid:10) S u n +1 η k (cid:12)(cid:12) T u n +1 η k (cid:11) + (cid:10) T u n +1 η k (cid:12)(cid:12) S u n +1 η k (cid:11) = lim k →∞ (cid:10) u n +1 η k (cid:12)(cid:12) [ S , T ] u n +1 η k (cid:11) = (cid:10) u n ψ (cid:12)(cid:12) [ S , T ] u n ψ (cid:11) . References [BDT89] P. Baum, R. G. Douglas, and M. E. Taylor,
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