The unbounded Kasparov product by a differentiable module

In this paper we investigate the unbounded Kasparov product between a differentiable module and an unbounded cycle of a very general kind that includes all unbounded Kasparov modules and hence also all spectral triples. Our assumptions on the differentiable module are as minimal as possible and we do in particular not require that it satisfies any kind of (smooth) projectivity conditions. The algebras that we work with are furthermore not required to possess a (smooth) approximate identity. The lack of an adequate projectivity condition on our differentiable module entails that the usual class of unbounded Kasparov modules is not flexible enough to accommodate the unbounded Kasparov product and it becomes necessary to twist the commutator condition by an automorphism. We show that the unbounded Kasparov product makes sense in this twisted setting and that it recovers the usual (bounded) Kasparov product after taking bounded transforms. Since our unbounded cycles are twisted (or modular) we are not able to apply the work of Kucerovsky (nor any of the earlier work on the unbounded Kasparov product) for recognizing unbounded representatives for the bounded Kasparov product. In fact, since we do not impose any twisted Lipschitz regularity conditions on our unbounded cycles, even the passage from an unbounded cycle to a bounded Kasparov module requires a substantial amount of extra care.

10. Relation to the bounded Kasparov product 37 11. Appendix: Norm estimates of error terms 41 11.1. Preliminary operator norm estimates 41 11.2. Norm-estimates of limit error terms 44 11.3. Operator norm estimates of truncated error terms 44 References 46

Introduction
In a series of papers from the early eighties, Kasparov proved the fundamental results on the KK-theory of C * -algebras, [Kas80a,Kas80b,Kas75]. One of the main inventions appearing in these papers is the interior Kasparov product which provides a bilinear and associative pairing between the KK-groups of three (separable) C * -algebras A, B and C. The interior Kasparov product of two KK-classes is computable in many cases, but the main construction remains inexplicit as it relies on Kasparov's absorption theorem and Kasparov's technical theorem.
One of the advantages of the KK-groups of C * -algebras is the wealth of explicit examples of elements arising from geometric data. Indeed, in the unbounded picture of KK-theory the cycles are unbounded Kasparov modules, which are bivariant versions of Connes concept of a spectral triple, and the unbounded Kasparov modules exhaust the KK-groups as was proved by Baaj and Julg, [BaJu83].
The problem that we are concerned with in this paper is to construct an unbounded version of the interior Kasparov product. More precisely, starting with two unbounded Kasparov modules, the aim is to find an explicit unbounded Kasparov module that represents the interior Kasparov product. In particular, this construction should bypass the need for invoking both the absorption theorem and the technical theorem. The problem of constructing the unbounded Kasparov product is currently receiving an increasing amount of attention, see [Con96,KaLe13,Mes14,MeRe15], as is also witnessed by the quantity of recent applications, see [BMS13,MeGo15,FoRe15,BCR15].
At a deeper level, the unbounded Kasparov product is important because of the loss of geometric information that is inherent in the passage from an unbounded Kasparov module to a class in KK-theory. It is thus in our interest to be able to perform a version of the interior Kasparov product while retaining a larger amount of geometric data (as for example the asymptotic behaviour of eigenvalues of differential operators).
In this paper we are focusing on the case where the class in the KK-group, KK(A, B), is represented by a C * -correspondence X from A to B and where the action of A from the left factorizes through the C * -algebra of compact operators on X. On the other hand, our class in the KK-group KK(B, C) will be represented by an unbounded selfadjoint operator D : D(D) → Y acting on a C * -correspondence from B to C. The unbounded operator D is required to satisfy a couple of extra conditions that will be detailed out in the main text. The first challenge is then to construct a new unbounded selfadjoint operator 1 ⊗ ∇ D : D(1 ⊗ ∇ D) → X ⊗ B Y that acts on the interior tensor product of the C * -correspondences X and Y . In the main part of the earlier works on the unbounded Kasparov product this step is accomplished by assuming the existence of a (tight normalized) frame {ζ k } for X (see [FrLa02]) such that the associated orthogonal projection P := ∞ n,m=1 ζ n , ζ m δ nm : ℓ 2 (Y ) → ℓ 2 (Y ) (which acts on the standard module over Y ) has a bounded commutator with the unbounded selfadjoint operator D : D(D) → Y (slightly weaker conditions are applied in [BMS13] and [MeRe15]). The unbounded selfadjoint operator 1 ⊗ ∇ D : D(1 ⊗ ∇ D) → X ⊗ B Y can then be expressed as the infinite sum T ζn DT * ζn where T ζn : Y → X ⊗ B Y , y → ζ n ⊗ B y, is the creation operator associated with the element ζ n ∈ X. It should be noted that the unbounded selfadjoint operator 1 ⊗ ∇ D can be described in an alternative way by using the notion of a densely defined covariant derivative ∇ on the C * -correspondence X. Indeed, the frame {ζ k } gives rise to a Grassmann covariant derivative ∇ Gr and the unbounded selfadjoint operator 1 ⊗ ∇ D is then given by the (closure of the) sum c(∇ Gr ) + 1 ⊗ D where the "c" refers to an appropriate notion of Clifford multiplication.
One of the main contributions of this paper is that we have been able to entirely remove the above smooth projectivity condition on the C * -correspondence X. This radical step is motivated by the detailed investigations of differentiable structures in Hilbert C * -modules carried out in [Kaa14,Kaa13]. In particular, we find that the removal of smooth projectivity is necessary for accommodating examples arising from non-complete manifolds.
Instead of smooth projectivity we will simply assume that there exists a sequence of generators {ξ k } for X such that the associated operator G := ∞ n,m=1 ξ n , ξ m δ nm : ℓ 2 (Y ) → ℓ 2 (Y ) has a bounded commutator with (the diagonal operator induced by) D : D(D) → Y . We then obtain a new unbounded selfadjoint operator T ξn DT * ξn on the interior tensor product X ⊗ B Y . We refer to this unbounded selfadjoint operator as the modular lift of D : D(D) → Y . The fact that our sequence {ξ k } is no longer a frame means that we obtain an extra (non-trivial) bounded adjointable operator on the interior tensor product. An investigation of the commutators between the algebra elements in A and the modular lift now shows that the usual straight commutator has to be replaced by a twisted commutator where the twist is given by the (modular) automorphism σ obtained from conjugation with the modular operator ∆. This modular automorphism corresponds to the analytic extension at −i ∈ C of the modular group of automorphisms σ t : T → ∆ it T ∆ −it , t ∈ R. We remark however that, in spite of the definitions in [CoMo08], we do not require that the modular automorphism σ is densely defined on the algebra A.
Our first main result can now be stated as follows (where we refer to the main text for the precise definitions): Theorem 1.1. Suppose that X is a differentiable C * -correspondence with left action factorizing through the compacts and that (Y, D, Γ) is an unbounded modular cycle (with modular operator Γ : Y → Y ). Then the triple is an unbounded modular cycle where the new modular operator is defined by ∆ := ∞ n=1 T ξn ΓT * ξn . The second central theme of this paper develops around the relationship between the assignment X, (Y, D, Γ) → X ⊗ B Y, D ∆ , ∆ and the interior Kasparov product KK 0 (A, B) × KK m (B, C) → KK m (A, C). In this respect it is first necessary to understand how to produce a class in KK-theory from an unbounded modular cycle. We announce the following theorem: Of course, this theorem is a direct analogue of the theorem of Baaj and Julg that shows how to construct a class in KK-theory from an unbounded Kasparov module. The proof of this result in the context of unbounded modular cycle is however far more involved. The reason for this extra difficulty can be found in the seemingly innocent change from straight commutators to twisted commutators. Indeed, an examination of the proof appearing in [BaJu83] shows that the crucial step fails for algebraic reasons when applied to unbounded modular cycles. An alternative approach would be to follow Connes and Moscovici's method and replace (1+D 2 ) −1/2 by (1 + |D|) −1 , see [CoMo08]. This alternative approach does however rely on an extra assumption of twisted Lipschitz regularity and we do not impose this kind of extra regularity conditions on our unbounded modular cycle. Indeed, it is unclear how twisted Lipschitz regularity behaves with respect to the unbounded Kasparov product given in Theorem 1.1. We have therefore found it necessary to develop a novel method of proof that can be applied to non-Lipschitz unbounded modular cycles.
The main new tool appearing in the proof of Theorem 1.2 is the modular transform G D,Γ : Γ(D(D)) → Y which is given by the (absolutely convergent) integral for all ξ ∈ D(D). The modular transform is obtained from the bounded transform by making a non-commutative change of variables corresponding to µ := λΓ 2 . This change of variables is motivated by the observation that the modular transform (contrary to the bounded transform) has the right commutator properties with elements in the algebra A. A substantial part of the proof of Theorem 1.2 is then devoted to a comparison between the bounded transform and the modular transform. Notice that the modular transform does not in general have a bounded extension to Y but that a sufficient condition for this to happen is that the modular operator Γ : Y → Y has a bounded inverse.
With the knowledge of the relationship between unbounded modular cycles and classes in KK-theory in place, we can state our second main result: in the KK-group KK m (A, B).
The proof of this theorem does again not follow the usual scheme in unbounded KK-theory. Indeed, the standard method that is available for recognizing an unbounded representative for the interior Kasparov product is to invoke the machinery invented by Kucerovsky, [Kuc97]. However, the results of Kucerovsky does not apply in the context of unbounded modular cycles because of our systematic use of twisted commutators instead of straight commutators. Instead of applying Kucerovsky's ideas we have found it necessary to rely directly on the notion of an F 2 -connection as introduced by Connes and Skandalis, [CoSk84].
Let us end this introduction by giving a more tangible corollary to our main theorems. Consider a countable union U := ∪ ∞ k=1 I k of bounded open intervals I k ⊆ R. For each k ∈ N we then choose a smooth function f k : R → R with support equal to the closure I k ⊆ R. After a rescaling we may assume that f k + df k dx ≤ 1/k for all k ∈ N (where · denotes the supremum norm). Define the first order differential operator and let D ∆ := (D ∆ ) 0 denote the closure. We then have the following result: is an odd spectral triple and the associated class in the odd K-homology group K 1 (C 0 (U)) agrees with the interior Kasparov product of (the KK-classes associated with) the C * -correspondence C 0 (U) and the (flat) Dirac operator on the real line.
Of course there is a similar kind of corollary where the setting is given by an arbitrary spectral triple (A , H, D) together with a sequence of elements {x k } in the algebra such that x k + [D, x k ] ≤ 1/k for all k ∈ N. When the algebra A is noncommutative it is however not true that one obtains a new spectral triple out of this construction. In the general case it becomes necessary to twist all the commutators appearing by the modular operator ∆ := ∞ k=1 x k x * k and the framework that we are developing here is therefore fine-tuned for treating this kind of examples.
1.1. Acknowledgement. The union U := ∪ ∞ k=1 I k appearing in the introduction is referred to as a fractral string when it is bounded and when the open intervals are disjoint. I am grateful to Michel Lapidus for making me aware of this example, [LavF13].

Preliminaries on operator spaces
We begin this paper by fixing our conventions for the analytic properties of the * -algebras appearing throughout this text. We have found that the conventional setup of Banach spaces is not adequate for capturing the relevant structure on our * -algebras. Indeed, it will soon become apparent that one needs to fix the analytic behaviour not only of the * -algebra itself but of all the finite matrices with entries in the * -algebra. The notion of operator spaces is therefore providing the correct analytic setting and we will now briefly survey the main definitions. For more details we refer the reader to the books by Blecher-Merdy and by Pisier, [BlLM04,Pis03].
Let H and G be Hilbert spaces, and let X ⊆ L (H, G) be a subspace (of the bounded operators from H to G) which is closed in the operator norm. Then the vector space M(X) := lim n→∞ M n (X) of finite matrices over X has a canonical norm · X coming from the identifications M n (X) ⊆ M n (L (H, G)) ∼ = L (H n , G n ). The properties of the pair M(X), · X are crystallized in the next definition.
Notice that the above construction yields a canonical norm · C : M(C) → [0, ∞) on the finite matrices over C since C ∼ = L (C, C). For each n ∈ N the norm · C : M n (C) ⊆ M(C) → [0, ∞) coincides with the unique C * -algebra norm.
Definition 2.1. An operator space is a vector space X over C with a norm · X on the finite matrices M(X) := lim n→∞ M n (X) such that (1) The normed space X ⊆ M(X) is a Banach space.
(2) The inequality v · ξ · w X ≤ v C · ξ X · w C holds for all v, w ∈ M(C) and all ξ ∈ M(X).
A morphism of operator spaces is a completely bounded linear map α : X → Y . The term completely bounded means that α n : M n (X) → M n (Y ) is a bounded operator for each n ∈ N and that sup n α n ∞ < ∞ (where · ∞ is the operator norm). The supremum is denoted by α cb := sup n α n ∞ and is referred to as the completely bounded norm.
By a fundamental theorem of Ruan each operator space X is completely isometric to a closed subspace of L (H) for some Hilbert space H. See [Rua88, Theorem 3.1].
We remark that any C * -algebra A carries a canonical operator space structure such that M n (A) becomes a C * -algebra for all n ∈ N.
We will in this text mainly be concerned with dense subspaces of operator spaces. On such a dense subspace X ⊆ X we will then refer to the norm on the surrounding operator space X as an operator space norm on X .
The next assumption will remain in effect throughout this paper: Assumption 2.2. Any * -algebra A encountered in this text will come equipped with an operator space norm · 1 : A → [0, ∞) and a C * -norm · : A → [0, ∞). We will denote the operator space completion of A by A 1 and the C * -algebra completion by A. It will then be assumed that the inclusion A → A extends to a completely bounded map A 1 → A.
In this text we will never assume the existence of a bounded approximate identity in A with respect to the norm · 1 : A → [0, ∞).
2.0.1. Stabilization of operator spaces. Let us consider an operator space X. The following stabilization construction will play a central role in this paper. It does of course not make any sense when X is merely a Banach space.
Definition 2.3. By the stabilization of X we will understand the operator space K(X) obtained as the completion of the vector space of finite matrices M(X) with respect to the canonical norm The matrix norms for K(X) comes from the matrix norms for X via the canonical identification (forgetting the subdivisions):

Unbounded modular cycles
Throughout this section we let A be a * -algebra which satisfies the conditions in Assumption 2.2. We let A 1 denote the operator space completion of A and A denote the C * -completion of A . We let B be an arbitrary C * -algebra.
Let us recall some basic constructions for a Hilbert C * -module X over B, for more details the reader may consult the book by Lance, [Lan95].
The standard module over X is the Hilbert C * -module ℓ 2 (X) over B consisting of all sequences ∞ n=1 x n δ n in X such that the sequence of partial sums N n=1 x n , x n converges in the norm on B. The right module structure is given by ∞ n=1 x n δ n · b := ∞ n=1 (x n · b)δ n and the inner product is given by x n , y n (where the convergence of the last sum follows from the Cauchy-Schwartz inequality).
The bounded adjointable operators on X is the C * -algebra L (X) consisting of all the bounded operators on X that admit an adjoint with respect to the inner product on X.
For a bounded adjointable operator T : X → X we let C * (T ) ⊆ L (X) denote the C * -subalgebra generated by T .
We are now ready to introduce the first of the main new concepts of the present paper: Definition 3.1. An odd unbounded modular cycle from A to B is a triple (X, D, ∆) where (1) X is a countably generated Hilbert C * -module over B which comes equipped with a * -homomorphism π : A → L (X); (2) D : D(D) → X is an unbounded selfadjoint and regular operator on X; (3) ∆ : X → X is a bounded positive and selfadjoint operator with dense image, such that the following holds: (1) π(a) · (i + D) −1 : X → X is a compact operator for all a ∈ A; (2) T ∆(D(D)) ⊆ D(D) and is contained in the image of ∆ 1/2 and the unbounded operator ∆ −1/2 d ∆ (T )∆ −1/2 : Im(∆ 1/2 ) → X has a bounded adjointable extension to X for all T ∈ π(A ) + C · Id X ; (4) The linear map ρ ∆ : A → L (X) defined by is completely bounded; (5) There exists a countable approximate identity {V n } ∞ n=1 for the C * -algebra C * (∆) such that the sequence V n π(a) ∞ n=1 converges in operator norm to π(a) for all a ∈ A. We will refer to ∆ : X → X as the modular operator of our unbounded modular cycle.
An even unbounded modular cycle from A to B is an odd unbounded modular cycle equipped with a Z/2Z-grading operator γ : X → X such that γπ(a) = π(a)γ γ∆ = ∆γ and γD = −Dγ for all a ∈ A.
Remark 3.2. The definition of an unbounded Kasparov module (see [BaJu83]) is a special case of the above definition. Indeed, it corresponds to the case where the modular operator ∆ = Id X . The concept of a twisted spectral triple (see [CoMo08]) is closely related to the above definition. Indeed, one of the main examples of a twisted spectral triple is obtained by starting from a unital spectral triple (A , H, D) together with a fixed positive and invertible element g ∈ A . One then forms the twisted spectral triple (A , H, gDg) where the modular automorphism σ : A → A is given by σ(a) := g 2 ag −2 (in this case we have that ∆ = g 2 ). This procedure corresponds to making a conformal change of the underlying metric, see for example [Hij86,Proposition 4.3.1].
Our definition of an unbounded modular cycle is inspired by this construction but there are three important differences: (1) We are considering a bivariant theory, thus the scalars can consist of an arbitrary C * -algebra and not just the complex numbers; (2) The modular operator ∆ : X → X can have zero in the spectrum (thus allowing for a treatment of non-compact manifolds); (3) The modular automorphism σ given by conjugation with ∆ need not map the algebra A into itself, in fact it need not even be defined on A .
For more information about twisted spectral triples we refer to [FaKh11,Mos10,PoWa15].
Let us spend a little extra time commenting on the conditions in Definition 3.1. We let π : A 1 → L (X) denote the completely bounded map induced by the inclusion A → A and the * -homomorphism π : A → L (X). It then follows by a density argument that the conditions (2) and (3) also hold for all T ∈ π(A 1 ) + C · Id X . Furthermore, we obtain a completely bounded map ρ ∆ : A 1 → L (X) which is induced by ρ ∆ : A → L (X). For condition (5) we notice that the sequence {V n } converges strictly to the identity on X (this holds since Im(∆) is dense in X). Furthermore, condition (5) automatically holds for any countable approximate identity for C * (∆) (once it holds for one of them). In particular we could choose V n = ∆(∆ + 1/n) −1 for all n ∈ N. In general we have that condition (3) and (5) are automatic when ∆ : X → X is invertible as a bounded operator.
For later use we introduce the following terminology: Definition 3.3. When (X, D, ∆) is an unbounded modular cycle (from A to B) we will say that a bounded adjointable operator T : X → X is differentiable (with respect to (X, D, ∆)) when the following holds: (1) T ∆(D(D)) ⊆ D(D) and DT ∆ − ∆T D : D(D) → X extends to a bounded adjointable operator d ∆ (T ) : X → X.
(2) The image of d ∆ (T ) : X → X is contained in the image of ∆ 1/2 : X → X and the unbounded operator has a bounded adjointable extension ρ ∆ (T ) : X → X.
We remark that the adjoint of a differentiable operator T : X → X is automatically differentiable as well and that the identities d ∆ (T ) * = −d ∆ (T * ) and ρ ∆ (T ) * = −ρ ∆ (T * ) are valid.
3.1. Stabilization of unbounded modular cycles. Let us fix an unbounded modular cycle (X, D, ∆) from the * -algebra A to the C * -algebra B. We let γ : X → X denote the grading operator in the even case.
The aim of this subsection is to construct a stabilization of (X, D, ∆) which is an unbounded modular cycle from the finite matrices over A to B. The parity of the stabilization is the same as the parity of (X, D, ∆).
To this end, we first notice that the finite matrices over A comes equipped with a canonical operator space norm and a canonical C * -norm (see Definition 2.3): The respective completions are the operator space K(A 1 ) and the C * -algebra K(A). We remark that K(A) is isomorphic to the compact operators on the standard module ℓ 2 (A) where A is considered as a Hilbert C * -module over itself.
We now consider the standard module ℓ 2 (X) over B and we equip it with the * -homomorphism K(π) : K(A) → L (ℓ 2 (X)) given by where a · δ nm ∈ K(A) denotes the finite matrix with a ∈ A in position (n, m) and zeroes elsewhere.
Furthermore, on the standard module over X, we have the diagonal operators induced by the unbounded selfadjoint and regular operator D : D(D) → X and the modular operator ∆ : X → X. The diagonal operator induced by D : D(D) → X is given by x n δ n ∈ ℓ 2 (X) | x n ∈ D(D) and ∞ n=1 D(x n )δ n ∈ ℓ 2 (X) The diagonal operator induced by ∆ : X → X is given by ∆(x n )δ n Likewise (in the even case) we have the diagonal operator diag(γ) : ℓ 2 (X) → ℓ 2 (X) induced by the grading operator γ : X → X.
It is well-known (and a good exercise) to check that diag(D) : D diag(D) → ℓ 2 (X) is again a selfadjoint and regular operator. We also note that diag(D) has a core given by the algebraic direct sum ⊕ ∞ n=1 D(D) ⊆ ℓ 2 (X). To ease the notation, we write Definition 3.4. By the stabilization of (X, D, ∆) we will understand the triple (ℓ 2 (X), 1 ⊗ D, 1 ⊗ ∆) with Z/2Z-grading operator 1 ⊗ γ in the even case.

Differentiable Hilbert C * -modules
Throughout this section A and B will be * -algebras which satisfy the conditions in Assumption 2.2. We let A 1 and B 1 denote the operator space completions and we let A and B denote the C * -completions of A and B, respectively. The next definition is the second main new concept which we introduce in this paper: Definition 4.1. A Hilbert C * -module X over B which comes equipped with a *homomorphism π : A → L (X) is said to be differentiable (from A to B) when there exists a sequence {ξ n } ∞ n=1 in X such that the following holds: (2) ξ n , T ξ m ∈ B for all T ∈ π(A ) + C · Id X and all n, m ∈ N.
(3) The sequence of finite matrices N n,m=1 n,m=1 ξ n , π(a)ξ m δ nm is completely bounded (with respect to the operator space norm on A ).
We will refer to a sequence {ξ n } ∞ n=1 in X satisfying the above conditions as a differentiable generating sequence.
Given a sequence {ξ n } that satisfies (1), (2), (3a), and (4a) we obtain a sequence satisfying (1), (2), (3), and (4) by rescaling each ξ n ∈ X by 1 n , for example. 4.0.1. Example: Finitely generated Hilbert C * -modules. Let us consider a * -algebra B which satisfies the conditions of Assumption 2.2. Let us also consider a dense * -subalgebra A of a C * -algebra A. Let now X be a finitely generated Hilbert C * -module over B with generators ξ 1 , . . . , ξ N ∈ X and let π : A → L (X) be a * -homomorphism. By "finitely generated" we mean that the subspace is dense in the norm-topology on X. Thus, in our context, finitely generated does not imply that X is finitely generated projective as a right module over B.
Suppose that We then have a linear map Using this linear map we obtain an operator space norm on A by defining 3). By construction we then get that X is a differentiable Hilbert C *module from A to B.

The modular lift
In this section we will consider two Hilbert C * -modules X and Y with the same base C * -algebra A. We will then fix an unbounded selfadjoint and regular operator D : D(D) → Y on the Hilbert C * -module Y together with a bounded selfadjoint and positive operator Γ : Y → Y with dense image. Furthermore, we will consider a bounded adjointable operator Φ : X → Y such that the adjoint Φ * : Y → X has dense image.
The main concern of this section is to "transport" the unbounded selfadjoint and regular operator D : D(D) → Y to an unbounded selfadjoint and regular operator D ∆ : D(D ∆ ) → X. This transportation will happen via the bounded adjointable operator Φ : X → Y . We will apply the notation: We remark that ∆ : X → X is bounded selfadjoint and positive and that Im(∆) ⊆ X is norm-dense. The following standing assumptions will be in effect: the unbounded operator The main aim of this section is to show that the composition is essentially selfadjoint and regular, where the domain is given by We immediately remark that D(Φ * DΦ) ⊆ X is norm-dense. Indeed, this follows since Φ * Γ(D(D)) ⊆ D(Φ * DΦ). Furthermore, it is evident that the unbounded operator Φ * DΦ : D(Φ * DΦ) → X is symmetric.
We notice that ∆ D(Φ * DΦ) ⊆ D(Φ * DΦ) and that for all η ∈ D(Φ * DΦ). In particular, this shows that the straight commutator has a bounded adjointable extension to X.
5.1. Selfadjointness. In order to show that the modular lift is selfadjoint we need a few preliminary lemmas.
Proof. Let η ∈ D(D) and compute as follows: Using the selfadjointness assumption on D : D(D) → Y , this implies that Φ∆(ξ) ∈ D(D) and furthermore that This clearly implies the result of the lemma.
Proof. We define the two holomorphic functions g and h : Let now η ∈ D Φ * DΦ be fixed. By the uniqueness of holomorphic extensions it is then enough to prove that with |z| > ∆ be given. We then have that where the sum converges absolutely in operator norm. Furthermore, by Lemma 5.3 we obtain that − N n=0 ∆ n z −n−1 (ξ) ∈ D (Φ * DΦ) * and that for all N ∈ N. Since the right hand side converges to This ends the proof of the present lemma.
We are now ready to show that the modular lift D ∆ : D(D ∆ ) → X is selfadjoint: and furthermore by Lemma 5.3 and Lemma 5.4 that To show that ξ ∈ D(D ∆ ) it therefore suffices to prove that the sequence For each n ∈ N we use Lemma 5.3 and Lemma 5.4 to compute in the following way: Since the sequence ∆(∆ + 1/n) −1 converges strictly to the identity operator on X, the result of the proposition is proved, provided that the sequence 1 converges to zero in the norm on X. But this is a consequence of the next lemma.
Lemma 5.6. The sequence 1 is bounded in operator norm and converges strictly to the zero operator on X.
Proof. We first show that our sequence is bounded in operator norm. To this end, we simply notice that 1 To prove the lemma, we may now limit ourselves to showing that 1 for all ξ in a dense subspace of X. Since Im(∆) ⊆ X is dense in X we let η ∈ X and remark that 1 for all n ∈ N. This computation ends the proof of the present lemma.

5.2.
Regularity. In order to show that the modular lift D ∆ : D(D ∆ ) → X is regular we will use the local-global principle for unbounded regular operators, see [Pie06,KaLe12]. We will thus pause for a second and remind the reader how this principle works. Let ρ : A → C be a state on the C * -algebra A. We may then define the pairing, The completion of X/N ρ is then a Hilbert space with inner product induced by ·, · ρ . We denote this Hilbert space by X ρ and let [·] : X → X ρ denote the canonical map (quotient followed by inclusion). The unbounded selfadjoint operator D ∆ : D(D ∆ ) → X yields an induced unbounded symmetric operator where the domain is given by We denote the closure of this unbounded symmetric operator by The local-global principle states that the unbounded selfadjoint operator D ∆ is regular if and only if D ∆ ⊗ 1 is selfadjoint for each state ρ : A → C, see [KaLe12, Theorem 4.2]. We remark that an even stronger result is proved in [Pie06]: It does in fact suffice to prove selfadjointness for every pure state on A.
Let us from now on fix a state ρ : A → C. We are interested in showing that D ∆ ⊗ 1 : D(D ∆ ⊗ 1) → X ρ is selfadjoint. We remark that it already follows by the local-global principle that the unbounded operator D ⊗ 1 : The next lemma is left as an exercise to the reader: Lemma 5.7. The triple (D ⊗ 1, Γ ⊗ 1, Φ ⊗ 1) satisfies the conditions (1), (2), and (3) stated in Assumption 5.1 (where Γ ⊗ 1 : Y ρ → Y ρ and Φ ⊗ 1 : X ρ → Y ρ are defined using the same recipe as in the unbounded case). Furthermore, we have the identities It is a consequence of the above lemma and Proposition 5.5 that the composition is essentially selfadjoint. We will denote the closure by (D ⊗ 1) ∆⊗1 . We may thus focus our attention on proving the identity We start by proving the easiest of the two inclusions: This proves the lemma.
The proof of the reverse inclusion is more subtle. It will rely on the following lemma: We now compute as follows: This shows that We thus have that (Φ * Γ ⊗ 1)(η) ∈ D(D ∆ ⊗ 1) and furthermore that . It then follows from the above considerations (∆ ⊗ 1)(ξ) ∈ D(D ∆ ⊗ 1) and furthermore that The result of the lemma now follows by using that (Φ * ⊗ 1)(D ⊗ 1)(Φ ⊗ 1) = (D ⊗ 1) ∆⊗1 by definition.
We are now ready to prove the reverse inclusion which (together with Lemma 5.8) will imply the following: Proposition 5.10. We have the identity of unbounded operators on the Hilbert space X ρ . In particular we obtain that D ∆ ⊗ 1 is selfadjoint.
Proof. By Lemma 5.8 we only need to show that Let thus ξ ∈ D (D ⊗ 1) ∆⊗1 be given. For each n ∈ N, it is then a consequence of Lemma 5.4 and Lemma 5.9 that Furthermore, these two lemmas allow us to compute as follows: Together with Lemma 5.6 (and Lemma 5.7) this computation shows that This proves the present proposition.
The main theorem of this section is now a consequence of the above considerations and Proposition 5.10: Theorem 5.1. Suppose that the conditions in Assumption 5.1 hold. Then the modular lift D ∆ : D(D ∆ ) → X is selfadjoint and regular.

Compactness of resolvents
We will in this section remain in the general setting presented in Section 5 and the conditions in Assumption 5.1 will therefore be in effect. In particular, it follows by Theorem 5.1 that the modular lift D ∆ := Φ * DΦ : D(D ∆ ) → X is a selfadjoint and regular unbounded operator. We recall that ∆ := Φ * ΓΦ.
Our principal interest is now to study the compactness properties of the resolvent (i + D ∆ ) −1 : X → X of the modular lift.
Lemma 6.1. We have the identity Let ξ ∈ D(DΦ). Since the unbounded operator (i + Φ * DΦ) : D(DΦ) → X has dense image (by Theorem 5.1) it is enough to verify that But this follows from the computation Proof. It is an immediate consequence of Lemma 6.1 that The result of the lemma therefore follows by noting that the sequence ∆ 2 (∆ + 1/n) −1 converges to ∆ : X → X in operator norm.
For later use, we shall also be interested in the relationship between the resolvents of the squares D 2 ∆ : D(D 2 ∆ ) → X and D 2 : D(D 2 ) → Y . In order to study these two resolvents we will need the following extra assumption: We start with a preliminary lemma: Lemma 6.4. We have the identity for all ξ ∈ D(D 2 ∆ ). Proof. Consider first an element η ∈ D(DΦ). We then have that Hence, since D(DΦ) ⊆ X is a core for the modular lift D ∆ : D(D ∆ ) → X we obtain that for all ξ ∈ D(D 2 ∆ ). Thus to prove the lemma we only need to show that We will prove the stronger statement that this identity holds for all ξ ∈ D(D ∆ ).
To obtain this, we may focus on the case where ξ ∈ D(DΦ). A straightforward computation then implies that Since GΓGΓΦ = Φ∆ 2 this proves the identity in (6.1) and hence the lemma.
Let us apply the notation The next result will play an important role in our later investigations of the relationship between the unbounded Kasparov product and the interior Kasparov product: Proposition 6.5. We have the identity Proof. Let ξ ∈ D(D 2 ∆ ). To prove the lemma, it clearly suffices to check that However, by Lemma 6.4 we have that The result of the present lemma then follows since (1 + D 2 )T λ * Φ∆ 2 = Φ∆ 2 − T λ Γ 2 Φλ∆ 2 /r

The unbounded Kasparov product
Throughout this section we let A and B be * -algebras which satisfy the conditions in Assumption 2.2. As usual we denote the C * -completions by A and B and the operator space completions by A 1 and B 1 . Furthermore, we will fix a third C *algebra C.
On top of this data, we shall consider: (1) An unbounded modular cycle (Y, D, Γ) from B to C (with grading operator γ : Y → Y in the even case).
(2) A differentiable Hilbert C * -module X from A to B with a fixed differentiable generating sequence {ξ n } ∞ n=1 . We let π A : A → L (X) and π B : B → L (Y ) denote the * -homomorphisms associated with the above data. It will then be assumed that To explain the aims of this section we form the interior tensor product X ⊗ B Y of Hilbert C * -modules. We recall that this is the Hilbert C * -module over C defined as the completion of the algebraic tensor product of modules X ⊗ B Y with respect to the norm coming from the C-valued (pre) inner product The interior tensor product comes equipped with a * -homomorphism It is the principal goal of this section to apply the above data to construct a new (and explicit) unbounded modular cycle from A to C: We shall refer to this new unbounded modular cycle as the unbounded Kasparov product of the differentiable Hilbert C * -module X and the unbounded modular cycle (Y, D, Γ).
Let us return to the interior tensor product X ⊗ B Y . For each ξ ∈ X we have a bounded adjointable operator where the adjoint is given explicitly by For each N ∈ N we may then define the bounded adjointable operator T * ξn (z)δ n Lemma 7.1. The sequence of bounded adjointable operators converges in operator norm to a bounded adjointable operator Φ : Proof. Let us prove that the sequence {Φ N } is Cauchy in operator norm. To this end, we let M > N be given and notice that is a Cauchy sequence in K(B 1 ) (by our assumption on the differentiable generating sequence {ξ n }) and since the canonical map B 1 → B is completely bounded, this shows that {Φ N } is a Cauchy sequence as well.
To see that the image of Φ * : Lemma 7.2. Let T ∈ π A (A ) + C · Id X be given. Then the following holds: (1) The bounded adjointable operator extends to a bounded adjointable operator of (1 ⊗ Γ) 1/2 : ℓ 2 (Y ) → ℓ 2 (Y ) and the unbounded operator extends to a bounded adjointable operator.
Furthermore, the linear map A → L (ℓ 2 (Y )) defined by is completely bounded (with respect to the operator space norm on A ).
Proof. Let τ : A → K(B 1 ) denote the completely bounded map defined by (where the complete boundedness is understood with respect to the operator space norm on A ). Let also g ∈ K(B 1 ) be given by g := ∞ n,m=1 ξ n , ξ m δ nm . Finally, we let K(π B ) : K(B) → L (ℓ 2 (Y )) denote the * -homomorphism defined by (where we are suppressing the canonical map K(B 1 ) → K(B)). The result of the lemma is now a consequence of Proposition 3.5 (and the remarks following Definition 3.1).
It follows by Lemma 7.2 (with T = Id X ) that the triple Φ, (1⊗Γ), (1⊗D) satisfies the conditions applied in Section 5. In particular, we may form the modular lift We define the bounded adjointable operator Theorem 7.1. Suppose that the conditions outlined in the beginning of this section are satisfied. Then the triple (X ⊗ B Y, (1 ⊗ D) ∆ , ∆) is an unbounded modular cycle from A to C. The parity of (X ⊗ B Y, (1 ⊗ D) ∆ , ∆) is the same as the parity of (Y, D, Γ) and the grading operator is given by 1 ⊗ γ : X ⊗ B Y → X ⊗ B Y in the even case.
Proof. We will verify each of the points in Definition 3.1 separately.
The fact that X ⊗ B Y is a countably generated Hilbert C * -module follows since both X and Y are countably generated by assumption.
We will now focus on the conditions (1)-(5) in Definition 3.1.

The modular transform
Throughout this section we will consider the following data: (1) An unbounded selfadjoint and regular operator D : D(D) → Y acting on a fixed Hilbert C * -module Y .
(2) A positive selfadjoint bounded operator ∆ : Y → Y such that Im(∆) ⊆ Y is dense. We will then make the following standing assumption: Let us choose r ∈ ( ∆ 2 , ∞) For each λ ≥ 0 we then apply the notation: S λ := (λ∆ 2 /r + 1 + D 2 ) −1 and R λ := (λ + 1 + D 2 ) −1 We are then interested in studying the modular transform of the pair (D, ∆). This is the unbounded operator defined by The modular transform will play a key role in our later proof of one of the main theorems in this paper, namely that the bounded transform of an unbounded modular cycle yields a bounded Kasparov module (Theorem 9.1) and hence a class in KK-theory.
We notice that the modular transform has been obtained from the bounded transform by making a non-commutative change of variables in the integral over the halfline. Indeed, the idea is just to replace the scalar-valued variable λ ≥ 0 by the operator-valued variable λ · ∆ 2 /r. In the case where D and ∆ actually commute it can therefore be proved that the modular transform is just a restriction of the bounded transform to ∆ D(D) ⊆ Y . However, in the case of real interest, thus when d(∆) = 0, there is a substantial error-term appearing and a great deal of this section is devoted to controlling the size of this error-term. There are easier proofs of the main results of this section when the modular operator ∆ : Y → Y is assumed to be invertible (as a bounded operator). One of the important points of the whole theory that we are developing here does however lie in the fact that ∆ : Y → Y is allowed to have zero in the spectrum. This condition should therefore not be relaxed.
8.1. Preliminary algebraic identities. Let us apply the notation We start our work on understanding the modular transform by rewriting the (modular) resolvent S λ = (λ∆ 2 /r + 1 + D 2 ) −1 in a way that is more amenable to a computation of the integral appearing in the expression for the modular transform. More precisely, we will first expand the resolvent S λ : Y → Y as a power-series involving the (standard) resolvent R λ : Y → Y and the bounded adjointable operator K : Y → Y . We will then reorganize this power-series by moving all the K-terms to the left and all the R λ -terms to the right (and hence picking up an error-term). This will be accomplished in the present subsection.
Lemma 8.2. For each λ ≥ 0 we have the identities where the sum converges absolutely.
Proof. Let λ ≥ 0 be given. By the resolvent identity we have that Since ∆ 2 < r we have that X λ ≤ λ(1 + λ) −1 < 1. We may thus conclude that where the sum converges absolutely. From the above we deduce that This proves the lemma.
We will from now on apply the notation Lemma 8.3. Let λ ≥ 0, n ∈ N and k ∈ N 0 . Then we have that The proof runs by induction on n ∈ N using the identity Notice that it is convenient to do the cases k = 0 and k ∈ N separately, starting with k = 0.
Lemma 8.4. Let λ ≥ 0, n ∈ N and k ∈ N 0 be given. Then we have that Proof. The proof runs by induction using the identity in Lemma 8.3.
For each m ∈ N and each λ ≥ 0 we define the bounded adjointable operator Proof. By an application of Lemma 8.4 (and a reordering of terms) we obtain that The result of the lemma now follows by noting that For each λ ≥ 0 we define the bounded adjointable operator Lemma 8.6. Let λ ≥ 0 be given. Then the sequence {L λ (m)} ∞ m=1 converges to L λ : Y → Y in operator norm.
Proof. Using the Leibniz rule we see that it suffices to verify that the sequence I(X m λ )S λ ∞ m=1 converges to zero in operator norm. However, using the Leibniz rule one more time, we obtain that The result of the lemma now follows easily by noting that X λ ≤ λ(1 + λ) −1 < 1. Indeed, we may then find a constant C > 0 such that We are now ready to prove the main result of this subsection. It provides an expansion of S λ ∆ 3 : Y → Y where the first power-series appearing can be directly related (after integration over the half-line) to the bounded adjointable operator (1 + D 2 ) −1/2 : Y → Y . The exponent 3 that appears here (and earlier in this section) is not special, we will only need that it is large enough for certain estimates to carry through later on.
Proposition 8.7. Let λ ≥ 0 be given. Then we have the identity where each of the sums converges absolutely in operator norm.
To continue, we notice that Now, by an application of Lemma 8.5, we see that we may restrict our attention to proving that the sequence converges in operator norm to ∞ j=0 K j L λ R j+1 λ λ j . To this end, we define Both of these constants are of course finite. Let now ε > 0 be given. By Lemma 8.6 we may then choose N 0 , M 0 ∈ N such that

It is then straightforward to verify that
This proves the present proposition.
8.2. Integral formulae for the square root. The aim of this subsection is to compute the integral over the half line of the continuous map which appears (up to a factor of (λr) −1/2 ) in the expression for ∆ 3 S λ ∆ 3 : Y → Y obtained in Proposition 8.7. The main result of this subsection is then the explicit which is proved in Proposition 8.13. We start by recalling a general result on integral formulae for powers of resolvents: Lemma 8.8. Let Λ : D(Λ) → Y be an unbounded selfadjoint regular operator and let p, q > 0. Then we have the identity where the integral converges absolutely and where is the beta function.
Proof. Notice that a change of variables (λ = µ · t) implies that for all t > 0. The result now follows by an application of the continuous functional calculus for unbounded selfadjoint regular operators, see [Wor91,WoNa92].
Let us fix two elements ξ, η ∈ Y together with a state ρ : B → C on the base C * -algebra. We will often apply the notation y 0 , y 1 ρ := ρ y 0 , y 1 y 0 , y 1 ∈ Y for the localized inner product. The next lemma reduces the computation of the integral 1 π ∞ 0 f (λ) dλ to a (delicate) matter of interchanging an infinite sum and an integral.
In order to compute the integral of f : (0, ∞) → L (X) (and to show that this function is integrable) we now want to apply the Lebesgue dominated convergence theorem. Or in other words we need to find a positive integrable function g : (0, ∞) → [0, ∞) such that for all λ > 0 , N ∈ N This turns out to be a subtle problem and the solution will rely on the algebraic identities of Subsection 8.1 and the detailed estimates that we carry out in the appendix to this paper. On top of these estimates we will need the following two lemmas: Lemma 8.10. Let p ∈ [0, 2] be given. Then we have that where the sum converges absolutely in operator norm for all λ ≥ 0.
Proof. It is clear the the sum converges absolutely for all λ ≥ 0. To prove the relevant identity we let λ ≥ 0 be given and compute as follows: Lemma 8.11. The sequence of partial sums is bounded in operator norm.
Proof. By an application of Lemma 8.5 we obtain that N n=0 for all λ ≥ 0 and all N ∈ N. We estimate the operator norm of each of these terms separately.
For the first term in (8.1) we apply Lemma 11.3 to obtain that for all λ ≥ 0 and all N ∈ N.
For the second term in (8.1) we apply Lemma 11.3 and Lemma 11.1 to find a constant C 1 > 0 such that for all λ ≥ 0 and all N ∈ N (recall that d(∆ 3 ) = ∆ 1/2 ρ(∆ 3 )∆ 1/2 ). For the third term in (8.1) we apply the Cauchy-Schwartz inequality to obtain that To continue, we note that Proposition 11.9 and Lemma 8.11 implies that there exists a constant C 2 > 0 such that Furthermore, by the identity in Lemma 8.10 we have that (1 + D 2 )R 2n+2 λ λ 2n 1/2 ≤ (2λ + 1) −1/2 for all N ∈ N and all λ ≥ 0. This provides an adequate norm-estimate of the final term in (8.1) and the lemma is therefore proved.
The main result of this subsection now follows by Lemma 8.9, Lemma 8.12, and the Lebesgue dominated convergence theorem: Proposition 8.13. The continuous function is absolutely integrable (with respect to Lebesgue measure on [0, ∞) and the operator norm). Furthermore, the integral is given explicitly by 1 π ∞ 0 f (λ) dλ = ∆ 5 (1 + D 2 ) −1/2 8.3. Comparison with the bounded transform. We are now ready to prove the main theorem of this section. The interpretation of this result is that the bounded transform F D has the same summability properties as the modular transform after multiplication from the left with a sufficiently large power of the modular operator. It is also appropriate to remark that the exponent p ∈ [0, 1/2) appearing in the theorem below is the "best" exponent possible (a part from possibly the limit case p = 1/2). Indeed, if we were interested in carrying out a more detailed analysis of summability properties in relation to the unbounded Kasparov product we would be able to show that, in the situation we consider, there is only an infinitesimal loss of summability. For the present study it does however largely suffice to limit ourselves to the question of compactness of resolvents and we will therefore (for the moment) not go into a deeper study of the decay properties of eigenvalues.
Theorem 8.1. Let p ∈ [0, 1/2) be given. Then the difference of unbounded operators has a bounded extension to Y .
Proof. By an application of Proposition 8.7 and Proposition 8.13 we may focus our attention on proving that the unbounded operator has a bounded extension to Y . To this end, we apply the Cauchy-Schwartz inequality to obtain that for all λ ≥ 0. Next, by an application of Proposition 11.8 and Lemma 8.11 we may find a constant C 1 > 0 such that Furthermore, by Lemma 8.10 we have that These estimates imply that the integral converges absolutely in operator norm and the theorem is therefore proved.

The Kasparov module of an unbounded modular cycle
Throughout this section we let A be a * -algebra which satisfies the conditions of Assumption 2.2. We will then consider a fixed unbounded modular cycle (X, D, ∆) from A to an arbitrary C * -algebra B. As usual we will assume that (X, D, ∆) is either of even or odd parity and in the even case we will denote the Z/2Z-grading operator by γ : X → X. We will apply the notation for the bounded transform of the unbounded selfadjoint and regular operator D : The aim of this section is to show that the pair (X, F D ) is a bounded Kasparov module from A to B and hence that our unbounded modular cycle gives rise to a class in the KK-group, KK p (A, B) (where p = 0, 1 according to the parity of (X, D, ∆)).
We will thus prove (see Theorem 9.1) that the following holds for all a ∈ A: (1) π(a)(F 2 (4) F D γ = −γF D and π(a)γ = γπ(a) in the even case. For more information about KK-theory we refer the reader to the book by Blackadar, [Bla98].
The main difficulty is to show that the commutator condition (3) and it is to this end that we have introduced and studied the modular transform in Section 8. To explain why this was necessary we first recall the notation S λ := (λ∆ 2 /r + 1 + D 2 ) −1 : X → X where r ∈ ( ∆ 2 , ∞) is a fixed constant. The next lemma then presents the main algebraic reason for working with the modular resolvent S λ instead of the ordinary resolvent R λ = (λ + 1 + D 2 ) −1 . Indeed, when the computation below is carried out with R λ in the place of S λ then the commutator [∆ 2 , T ] has to be replaced by the commutator [(1 + λ)∆ 2 , T ] and there is then no gain in the decay properties when the variable λ tends to infinity. This makes the usual proof ([BaJu83]) of condition (3) from the above list fail utterly.
Lemma 9.1. Let T : X → X be differentiable with respect to (X, D, ∆) (as in Definition 3.3). We then have the identity Proof. Let first ξ ∈ D(D 2 ) and notice that The result of the lemma then follows since In the next two lemmas we show that we may replace the bounded transform F D (up to a compact perturbation) by the modular transform G D,∆ (in a slight disguise).
Lemma 9.2. Let T ∈ L (X) and suppose that (1 + D 2 ) −1 T ∈ K (X) and that (T ∆)(D(D)) ⊆ D(D). Then the unbounded operator has a compact extension to X.
Proof. It follows by Theorem 8.1 that the difference has a compact extension to X. Furthermore, we notice that the difference is a compact operator for all λ ≥ 0 (in fact each of the two terms is compact).
To prove the lemma, it therefore suffices to find a constant C > 0 such that for all λ ≥ 0. This amounts to providing operator norm estimates of the three bounded adjointable operators This can be carried out by an application of the results in the Appendix (Subsection 11.1). The details are left to the careful reader.
Lemma 9.3. Let T ∈ L (X) and suppose that (1 + D 2 ) −1 T ∈ K (X) and that (T ∆)(D(D)) ⊆ D(D). Then the unbounded operator has a compact extension to X.
Proof. We start by noting that Indeed, this follows by using the integral formula λ −1/2 (λ + 1 + D 2 ) −1 dλ and the fact that [D, ∆] : D(D) → X has a bounded extension to X. Now, by Lemma 9.2 we obtain that the difference of unbounded operators has a compact extension to X. We then remark that the difference is a compact operator (again we do in fact have that each of the two terms is compact).
To prove the lemma, it therefore suffices to find a constant C > 0 such that But this follows again by the techniques developed in the Appendix (Subsection 11.1) and the details are therefore not provided here.
Proof. By Lemma 9.2 and Lemma 9.3 it suffices to show that the difference To this end, we notice that is compact for all λ ≥ 0. In order to prove the proposition, it therefore suffices to find a constant C > 0 such that for all λ ≥ 0. To show that this is indeed possible, we notice that The relevant estimate then follows by Lemma 9.1 and the results in Subsection 11.1.
Theorem 9.1. Let (X, D, ∆) be an unbounded modular cycle from A to the C *algebra B (with grading operator γ : X → X in the even case). Then the bounded transform X, D(1 + D 2 ) −1/2 is a bounded Kasparov module from the C * -algebra A to the C * -algebra B of the same parity as (X, D, ∆) and with grading operator γ : X → X in the even case.
Proof. The only non-trivial issue is the compactness of the commutator [F D , π(a)] : X → X for all a ∈ A. However, it already follows by Proposition 9.4 that [F D , ∆ 5 π(a)∆ 5 ]π(b) : X → X is compact for all a, b ∈ A . Using the density of A in A and the fact that ∆(1/n + ∆) −1 π(a) → π(a) in operator norm for all a ∈ A we obtain that [F D , π(a)]π(b) ∈ K (X) for all a, b ∈ A. It then follows that [F D , π(a)] ∈ K (X) for all a ∈ A by a standard trick in KK-theory.
Remark 9.5. There is a much easier proof of Theorem 9.1 in the case where the unbounded modular cycle is Lipschitz regular thus when the twisted commutator |D|π(a)∆ − ∆π(a)|D| : D(D) → X has a bounded extension for all a ∈ A . Indeed, it is then possible to follow [CoMo08, Proposition 3.2] more or less to the letter. It is however highly unclear whether the condition of Lipschitz regularity is compatible with the unbounded Kasparov product construction given in Section 7. In fact, to our knowledge, this problem is not even decided in the case of the passage from D to gDg (see Remark 3.2 and [CoMo08, Section 2.2]). We have therefore in this text chosen to avoid the extra Lipschitz regularity condition altogether.

Relation to the bounded Kasparov product
Throughout this section we let A and B be two * -algebras which satisfy the conditions in Assumption 2.2.
We will consider an unbounded modular cycle (Y, D, Γ) from B to an auxiliary C * -algebra C. The parity of (Y, D, Γ) is denoted by p ∈ {0, 1}. Furthermore, we let X be a differentiable Hilbert C * -module from A to B with differentiable generating sequence {ξ n } ∞ n=1 . We will finally suppose that the * -homomorphism π A : A → L (X) factorizes through the compact operators K (X) ⊆ L (X).
As a consequence of Theorem 7.1 we then obtain that the triple is an unbounded modular cycle from A to C of the same parity as (Y, D, Γ). Thus, by an application of Theorem 9.1 we obtain a bounded Kasparov module from A to C and hence a class [F ∆ ] in the KK-group KK p (A, C).
On the other hand, since π A (a) ∈ K (X) for all a ∈ A, our differentiable Hilbert C * -module X defines an even bounded Kasparov module X, 0 from A to B, and hence a class [X] in the even KK-group KK 0 (A, B). The grading operator is here just the identity operator on X. On top of this, we know from Theorem 9.1 that our original unbounded modular cycle (Y, D, Γ) yields a bounded Kasparov module denotes the interior Kasparov product in KK-theory.
To ease the notation, we define and For the rest of this paper we will assume that the C * -algebra A is separable and that the C * -algebra B has a countable approximate identity (thus that B is σ-unital).
Remark 10.1. We would like to emphasize that even though the interior Kasparov product in KK-theory is only constructed under the assumption that A is separable and B is σ-unital we do not rely on these assumptions for the construction of the unbounded Kasparov product. The bounded Kasparov module (X ⊗ B Y, F ∆ ) therefore exists regardless of these assumptions on the C * -algebras A and B.
Due to a result of Connes and Skandalis we may focus on proving that F ∆ is an F -connection, [CoSk84,Theorem A.3]. Or in other words, if we can show that for all ξ ∈ X we may conclude that the identity in (10.1) holds. We recall here that Remark 10.2. In the work of Kucerovsky, [Kuc97, Theorem 13], conditions are given for recognizing unbounded representatives for the interior Kasparov product. These conditions can not be applied in our setting since our unbounded cycles are not unbounded Kasparov modules in the sense of [BaJu83]. Indeed, the main difference is that we are considering a twisted commutator condition (see Definition 3.1) instead of the straight commutator condition applied in [BaJu83].
Lemma 10.4. Suppose that there exists a k ∈ N such that Proof. We first show that To this end, we notice that for all n ∈ N. Indeed, this is a consequence of the assumptions of the present lemma and the fact that ∆[F ∆ , ∆] : X ⊗ B Y → X ⊗ B Y is compact (this last assertion follows by Lemma 10.3 and Proposition 9.4). The inclusion in (10.3) then follows by noting that the sequence (1 ⊗Γ) 1/2 Φ∆ k (∆ k + 1/n) −1 ∞ n=1 converges to (1 ⊗Γ) 1/2 Φ : Our next step is to show that In this respect, we remark that for all n ∈ N. Indeed, this is a consequence of the inclusion in (10.3) and the fact that 1 ⊗ [F, Γ k ]Γ k (Γ k + 1/n) −1 Φ ∈ K (X ⊗ B Y, ℓ 2 (Y )) (as above this last assertion follows by Proposition 9.4). The inclusion in (10.4) now follows since the sequence 1 ⊗Γ k (Γ k + 1/n) −1 Φ ∞ n=1 converges to Φ : X ⊗ B Y → ℓ 2 (Y ) in the operator norm. By the definition of Φ : X ⊗ B Y → ℓ 2 (Y ) we see from (10.4) that for all n ∈ N. Let now b ∈ B and n ∈ N be given. We then have that Thus, since (Y, F ) is a bounded Kasparov module we deduce from (10.5) that Since the sequence {ξ n } ∞ n=1 generates X as a Hilbert C * -module over B we conclude from (10.6) that This proves the lemma. Let us apply the notation where r ∈ ( ∆ 2 + Γ 2 , ∞) is a fixed constant. The next lemma relates these two modular resolvents to one another. We will in the following often put D := 1 ⊗ D : D(1 ⊗ D) → ℓ 2 (Y ) and Γ := 1 ⊗ Γ : ℓ 2 (Y ) → ℓ 2 (Y ) Lemma 10.5. The difference extends to a compact operator K λ : X ⊗ B Y → ℓ 2 (Y ) and there exists a constant C > 0 such that Proof. It is not hard to see that the difference in (10.7) has a compact extension K λ : X ⊗ B Y → ℓ 2 (Y ) for all λ ≥ 0 (in fact we have that this holds for each of the two terms). We may thus focus our attention on providing the operator norm-estimate in (10.8) Our first step in this direction is to notice that it is enough to consider the difference of unbounded operators. This follows since we may dominate the operator norm (uniformly in λ ≥ 0) of each of the bounded adjointable operators by C 0 · (1 + λ) −3/4 for some constant C 0 > 0. To see that this is indeed the case it suffices to apply the elementary estimates in the Appendix (Subsection 11.1). Our next step is to define the unbounded operator (where we recall the notation G := ΦΦ * : ℓ 2 (Y ) → ℓ 2 (Y )). It then follows by the estimates in the Appendix (Subsection 11.1) that there exists a constant C 1 > 0 such that M λ (ξ) ≤ C 1 (1 + λ) −3/4 · ξ (10.9) for all λ ≥ 0 and all ξ ∈ D(D ∆ ) ⊆ X ⊗ B Y . Furthermore, by Proposition 6.5 we have that for all λ ≥ 0. In order to provide the relevant estimate on K λ : X ⊗ B Y → ℓ 2 (Y ) it therefore suffices to analyze the difference of unbounded operators. However, we have that for all ξ ∈ D(DΦ) and the result of the lemma therefore follows by one more application of the operator norm estimates in the Appendix (Subsection 11.1).
Lemma 10.6. The unbounded operator is the restriction of an operator in K (X ⊗ B Y, ℓ 2 (Y )).
Proof. This follows in a straightforward way by an application of Lemma 10.5.
We are now ready to prove our final main theorem: Theorem 10.1. The bounded adjointable operator In particular, we have the identity inside the KK-group KK p (A, C).
Proof. By Lemma 10.4, we only need to show that However, by Lemma 9.3 we see that it suffices to check that the difference is the restriction of an element in K (X ⊗ B Y, ℓ 2 (Y )). But this is a consequence of Lemma 10.6.

Appendix: Norm estimates of error terms
In this appendix we have collected various operator norm estimates needed in the treatment of the modular transform (Section 8) and for the comparison result between the unbounded Kasparov product and the bounded Kasparov product (Section 10).
11.1. Preliminary operator norm estimates. We start with a string of elementary operator norm estimates that will be needed throughout this appendix (and in many places in the main text as well).
Using that D(D 2 ) ⊆ Y is a core for D : D(D) → Y it follows that Ω * λ Ω λ D = D − DS λ (λ∆ 2 /r + 1) on the common domain D(D) ⊆ Y . The desired identity then follows by a direct computation.
Lemma 11.6. Let m ≥ 2 be given. There exists a constant C > 0 such that for all λ ≥ 0.
Proof. Let λ ≥ 0. We compute as follows: Since D∆ m−2 (i + D) −1 : Y → Y is a bounded adjointable operator by Assumption 8.1, we obtain the relevant estimate by an application of Lemma 11.1.
Lemma 11.7. Let m ≥ 3 be given. There exists a constant C > 0 such that Proof. To prove the first of the two estimates we apply Lemma 11.2 to obtain that After a consultation of Lemma 11.3 and Lemma 11.6 (together with Assumption 8.1) we then see that it suffices to find a constant C 1 > 0 such that S 1/2 λ Ω λ ∆ 1/2 ≤ C 1 · (1 + λ) −1/8 But this follows by noting that S 1/2 λ Ω λ ∆ 1/2 2 = S λ D∆DS λ (see the proof of Lemma 11.5).
Proof. For each λ ≥ 0 we rewrite L λ : Y → Y in the following way: It is then not hard to see that the desired estimate follows by the results in Subsection 11.1.
To take care of the second term in (11.3) we let l ≥ 3 be given. It then suffices to estimate the norm of the operator ∆ 2 I(X m λ )S λ ∆ l (i + D) −1 : Y → Y uniformly in m ∈ N and λ ≥ 0. In order to achieve this goal we notice that We now define Our next step is to estimate the operator norm of each of the terms A λ (m) : Y → Y and B λ (m) : Y → Y uniformly in λ ≥ 0 and m ∈ N. We start with A λ (m). Using the Cauchy-Schwartz inequality we obtain that It then follows by Lemma 11.3 and Lemma 11.7 that there exists a constant C 1 > 0 such that A λ (m) ≤ C 1 · (1 + λ) −1−1/8 for all m ∈ N and all λ ≥ 0.
We continue with B λ (m). Another application of the Cauchy-Schwartz inequality yields that As a consequence of Lemma 11.3 and Lemma 11.7 we may then find a constant C 2 > 0 such that B λ (m) ≤ C 2 (1+λ) −1−1/8 for all m ∈ N and all λ ≥ 0. Combining these estimates we find that ∆ 2 I(X m λ )S λ ∆ l (i + D) −1 ≤ (C 2 /r + C 3 /r) · (1 + λ) −1/8 for all m ∈ N and all λ ≥ 0. This ends the proof of the proposition.