Schatten classes for Hilbert modules over commutative C*-algebras

We define Schatten classes of adjointable operators on Hilbert modules over abelian $C^*$-algebras. Many key features carry over from the Hilbert space case. In particular, the Schatten classes form two-sided ideals of compact operators and are equipped with a Banach norm and a $C^*$-valued trace with the expected properties. For trivial Hilbert bundles, we show that our Schatten-class operators can be identified bijectively with Schatten-norm-continuous maps from the base space into the Schatten classes on the Hilbert space fiber, with the fiberwise trace. As applications, we introduce the $C^*$-valued Fredholm determinant and operator zeta functions, and propose a notion of $p$-summable unbounded Kasparov cycles in the commutative setting.

which clearly commutes with the entrywise lift of linear maps A → B and is cyclic if and only if the algebra A is commutative.
If E is a finitely generated projective Hilbert A-module, any compact adjointable endomorphism of E can be represented as an element of M (A) by a choice of isomorphism E ⊕ F A n . Whenever A is commutative, cyclicity implies that the trace of the resulting matrix is invariant under the choice of isomorphism and so constitutes the A-valued (Hattori-Stallings) trace on End A (E).
Chapter 3, below, introduces a robust framework that generalizes the construction to countably generated Hilbert modules.
Continuous families of Schatten-class operators. Under Gelfand duality and the Serre-Swan theorem, the finite-rank trace is just the fiberwise trace tr : Γ(End V ) → C 0 (X) of continuous endomorphisms of finite-rank Hermitian vector bundles over a locally compact Hausdorff space X.
If V → X is instead a continuous field of separable Hilbert spaces, then there is still a trace map on the fiberwise trace classes. The challenge is then to unify these fiberwise trace classes in such a way as to yield a C 0 (X)-valued fiberwise trace that retains the fundamental properties of the trace class on a fixed Hilbert space H.
As a fundamental example, consider the trivial bundle H × X → X with fiber a fixed separable Hilbert space H. The C * -algebra of its adjointable endomorphisms consists of all * -strongly continuous, bounded families of bounded operators on H. One wonders whether this algebra, denoted C str b (X, B(H)), contains two-sided ideals of "continuous Schatten-class operators" such that some or most of the usual theory of Schatten classes on Hilbert spaces is preserved.
In order to ensure continuity of the trace, the least one should demand of such an ideal is that the pointwise Schatten norms lie in C 0 (X). On the other hand, the strongest reasonable condition at hand is that the families are themselves Schattennorm continuous, that is, that they lie in the Banach space C 0 (X, L p (H)). Through careful control over the relation between the Schatten classes on the standard Hilbert space l 2 (C) and the complex matrix algebras M n (C), we are able in Theorem 3.18 to show that these conditions do in fact coincide on C str b (X, B(H)) and yield a two-sided ideal that is contained in the compact operators and is closed under its Banach norm.
Kasparov's stabilization theorem and unitary invariance of the Schatten norms allow us to easily generalize the trivial bundle example to all continuous fields of separable Hilbert spaces in Theorem 3.25. A further upshot of this approach is that much of the pointwise Schatten-class theory, including the Hölder-von Neumann inequality, carries over easily to the general case, cf. Theorem 3.28.
Frames and the fiberwise trace. The theory of Schatten classes on Hilbert spaces is often mediated through the language of orthonormal bases and diagonalization. The approach of Section 3.1 shows that one may very well work with (standard normalized) frames [18,33] instead. This allows for straightforward generalization of the familiar formulas to the theory of Hilbert modules, and indeed, the result is what one would hope for: the fiberwise trace turns out to be the norm-convergent sum over the diagonal in a given frame, cf. Theorem 3.5. In the context of frames, it was earlier remarked in [12], [18,Proposition 4.8] that the obvious notion of Hilbert-Schmidt inner product is invariant both under the choice of frame and under the adjoint. That observation is supplied with the necessary context as a special case of our Schatten-class operators in Section 3.3.
Applications. By the same principles as for the finite-rank trace, [2] introduced, in the context of K-theory, the Fredholm determinant of endomorphisms of finitely generated modules over unital commutative algebras. As the Fredholm determinant is interesting in its own right, Section 4.1 uses the result on the Schatten classes to extend its definition and basic properties to the setting of countably generated modules over unital commutative C * -algebras. A straightforward generalization of [3] remains however elusive, due to the conceptual problems in generalizing the relevant category.
Spectral geometry is the study of Riemannian manifolds M via the spectra of differential operators, such as spin c -Dirac operators D, on M . An important example of spectral invariant is the localized heat trace t → tr f e −tD 2 , f ∈ C(M ). It determines the volume and total scalar curvature of M , is strongly related to the Atiyah-Singer Index Theorem [19], and is able to describe classical field theories on M through the spectral action principle [11,Chapter 11]. The first step in generalizing the above to unbounded Kasparov cycles, that is, certain (C 0 (M ), C 0 (N ))-Hilbert bimodules carrying a selfadjoint, regular, C 0 (N )-linear unbounded operator S, is to make sense of the expressions tr f e −tS 2 and tr f |S| −z as elements of C 0 (N ). Sections 4.2,4.3 embark on the necessary theory. Open questions for further research remain, particularly in the direction of zeta residues and compatibility with the interior product in unbounded KK-theory. Moreover, the case where the target algebra is noncommutative is still wide open.

Preliminaries
We start by recalling the notion of frames on Hilbert C * -modules over C * -algebras. For basic definitions on Hilbert C * -modules, adjointable maps, tensor products, et cetera we refer to e.g. [40]. We also recall the definition of unbounded Kasparov cycles [4].
Keep in mind that we will, in later sections, specialize to the case of abelian C *algebras, that is, those of the form C 0 (X) for X a locally compact Hausdorff space. Hilbert C * -modules over such C * -algebras are given by the sections of continuous fields of Hilbert spaces; cf. [38,14].

2.1.
Frames on Hilbert C * -modules. We start this section by recalling two well-known results on Hilbert C * -modules. For completeness we include their (short) proofs.
A convenient basic fact about separable Hilbert spaces H is that they possess countable orthonormal bases {e i }. For one thing, this allows one to explicitly relate the compact operators B 0 (H) to the direct limit M (C) of matrix algebras over the base field C and in particular to treat the trace on L 1 (H) using the series expression tr T = i e i , T e i .
The situation for Hilbert C * -modules is slightly less straightforward: we will introduce the analogous but strictly weaker concept of a frame. In spite of the increased generality, we will see that frames provide sufficient flexibility to mimic standard treatments of trace-class operators on Hilbert spaces in the setting of Hilbert C 0 (X)-modules.
Such objects e were called 'standard normalized frames' in [18]. Note that the subsequent treatment in [33], which is very similar to the definition used here, is different for non-unital C * -algebras: we require the e i to be in E A , not in the 'multiplier module' L (A A , E A ). This choice will later imply, for instance, that we do not consider the identity on the C 0 (X)-module C 0 (X), for noncompact spaces X, to be in the trace class.
for all v, w ∈ K. That is, e = {P e i } is a frame of K. Note that e is not an orthonormal basis, because the e i might be neither orthogonal nor of norm 1. Now, in the context of trace-class operators on a separable Hilbert space H, frames 'are as good as orthonormal bases', in the sense of Corollary 2.6 below. Lemma 2.5. Let e, f be frames of a separable Hilbert space H and let T ∈ L (E A ). Then, the series T e i , T e i converges, and the limits agree.
T e, T e .
Being bounded and monotone, the series ∞ i=1 T * f i , T * f i must converge. If we now switch T and T * , e and f and repeat the calculation, we see that the limits must in fact agree.
Corollary 2.6. Let e be a frame of a separable Hilbert space H. Then, for T ∈ L (H), Proof. For the first part, let f be an orthonormal basis and note that by Lemma 2.5, and the latter fact is equivalent to T * ∈ L 2 (H), which is equivalent to T ∈ L 2 (H).
For the second part, note that L 1 (H) = L 2 (H) L 2 (H). It is thus sufficient to consider an element T = |S| 2 ∈ L 1 (H) with S ∈ L 2 (H). Then, tr We will see later that the Example 2.4 is a very good prototype for the general situation for Hilbert C * -modules as well.
Example 2.7. Let A be a unital C * -algebra and let l 2 (A) = l 2 ⊗ C A A be its standard module. Let {e i } be the standard orthonormal basis of l 2 and define (v, w ∈ l 2 (A)).
Each frame of E A gives rise to a unitary θ e : E A → (θ e θ * e )l 2 (A), as follows. Proposition 2.8. Let e be a frame of E A . The frame transform θ e : E A → l 2 (A), given by θ e (v) := ( e i , v ) i is adjointable, and its adjoint satisfies θ * e (e k ⊗ a) = e k · a. Moreover, e is a frame if and only if θ * e θ e = id E . For the proof we refer to [33,Theorem 3.5].
Remark 2.9. Note that, unless A is unital, the converse does not hold: not every isometry sending E A to a complemented submodule of l 2 (A) is induced by a frame. The frame elements e i would be given by θ * e (δ ij 1 A ) j , but the latter is not an element of l 2 (A) unless A is unital. This is where our treatment differs from that of [33], which works with frames (in the present sense) of the multiplier module L (E A , A) instead.
Frames are compatible with * -homomorphisms. In particular, this means that characters of a commutative C * -algebra map frames of Hilbert C * -modules to frames of Hilbert spaces.  [27] shows that example 2.7 describes the general unital (and, as we will see, the non-unital) case very well.
Theorem 2.11. Let A be a unital C * -algebra and let E A be a countably generated Hilbert A-module. Then there exists a projection P 2 = P = P * in L l 2 (A) such that E A P l 2 (A) . In particular, E A possesses a frame.
For the proof we refer to [27] (see also [29,Theorem 6.2]). The non-unital case requires more effort, but the end result is the same. We refer to [22,Section 2] for a proof.
2.2. The standard module over abelian C * -algebras. For the rest of this subsection, let X be a locally compact Hausdorff space. The C * -algebra C 0 (X) is abelian -and, by Gelfand duality, all abelian C * -algebras are of this type.
We will investigate the Hilbert C 0 (X)-module C 0 (X, H), for H a separable Hilbert space, which will later provide a useful tool in investigating the Schatten classes of operators on more general Hilbert C 0 (X)-modules. We start with some basic definitions and results, whose proof we leave to the reader. Definition 2.13. Let f be a map from a locally compact topological space X to a normed space. We say that f vanishes at infinity whenever for all > 0, {x ∈ X : f (x) ≥ } is compact.
Definition 2.14. Let Y be a Banach space, equipped with its norm topology. The space C 0 (X, Y ) consists of the continuous functions from X to Y that vanish at infinity.
Proposition 2.15. Let Y be a Banach space and X be a locally compact topological space. Then, C 0 (X, Y ) is a Banach space when equipped with the norm f : Proposition 2. 16. Let H be a separable Hilbert space. Then, the Banach space C 0 (X, H) has the structure of a Hilbert C * -module when equipped with the C 0 (X)valued inner product v, w (x) := v(x), w(x) H .
Proposition 2.17. The Hilbert C * -module C 0 (X, H) is unitarily equivalent to the tensor product H ⊗ C C 0 (X) of Hilbert C * -modules.
Proof. Let {e i } i be an orthonormal basis of H, and for v ∈ C 0 (X, H) write v i : x → v(x), e i ; this defines a sequence of functions in C 0 (X).
i v i converges pointwise, is positive and lies in C 0 (X), it converges in the norm of C 0 (X) by Dini's theorem.
Consider then the C 0 (X)-linear map θ : Moreover, taking limits on both sides we see that θ is isometric. Now, the map m : . As m inverts θ by orthonormality of the {e i } i , we conclude that θ is a surjective C 0 (X)linear isometry, so that it is adjointable (with adjoint m) and moreover unitary. Therefore, C 0 (X, H) is unitarily equivalent to H ⊗ C C 0 (X).
Proposition 2.18. The C * -algebra L (C 0 (X, H)) of adjointable endomorphisms of the Hilbert C 0 (X)-module C 0 (X, H) is isomorphic to the C * -algebra C str b (X, B(H)) of bounded, * -strongly continuous maps from X to B(H).
Proof. Consider a point x in X as a character χ : C 0 (X) → C given by evaluation in x. By Proposition 2.2 there is an adjointable map χ * : C 0 (X, H) → C 0 (X, H) ⊗ C0(X,H) C whose image χ * (C 0 (X, H)) can be canonically identified with H. Accordingly, in terms of the map χ * : L (C 0 (X, H)) → L (C 0 (X, H)) given by T → T ⊗ 1, we have canonically χ * L (C 0 (X, H)) B(H). Now take T ∈ L (C 0 (X, H)) and a convergent sequence . As x and h were arbitrary, we conclude that T ∈ C str (X, B(H)). Moreover, if T (x) > C for some x ∈ X and C > 0, there is v 0 as above with v 0 ≤ 1 and T v 0 > C − for all > 0, so that we conclude that T ≥ C. Thus, if T preserves C 0 (X, H) it must lie in C str b (X, B(H)).
we conclude that the pointwise adjoint provides an adjoint of T as an operator of C 0 (X, H). That is, all such T are adjointable operators on C 0 (X, H).

2.2.1.
General Hilbert C * -modules over a commutative base. We now apply the above results to the case of general Hilbert C 0 (X)-modules. For a deeper topological understanding of such modules, see [38,14].
Proposition 2. 19. Let E be a Hilbert C 0 (X)-module. Then there exists a * -strongly continuous projection P ∈ C str b (X, B(H)) such that E is isomorphic to the subset Γ 0 (X, P ) ⊂ C 0 (X, H) of those elements h of C 0 (X, H) satisfying h(x) ∈ ran P x . Moreover, under this identification L (E) is isomorphic to the set of elements T ∈ C str b (X, B(H)) such that P T = T P = T . Proof. If X is compact, this is a direct consequence of Kasparov's stabilization Theorem 2.11 and Propositions 2.17 and 2.18. If not, consider E as an C 0 (X) +module and note that the endomorphism P from Theorem 2.11 is already in C str b (X + , B(H)) in terms of the one-point compactification X + of X. But since (e, e) ∈ C 0 (X) for e ∈ E, the map P must project into a subspace of C 0 (X, H).
For the last statement, let T ∈ C str b (X, B(H)) such that P T = T P = T . Then P T P = T so that T preserves Γ 0 (X, P ). Conversely, let S ∈ L (Γ 0 (X, P )). Then, the map T = SP is a composition of adjointable operators and is therefore an element of L (C 0 (X, H)) C str b (X, B(H)) using Proposition 2.18. We clearly have P T = T P = T so that the claim follows.
Note that the fibers P x H of E may vary quite wildly with x ∈ X, as the following example shows.
Example 2.20. Let U ⊂ X be open and let P be the orthogonal projection onto a closed subspace V ⊂ H. Then, E = C 0 (U, V ) is (in particular) a Hilbert C 0 (X)module because the action of C 0 (X) on C 0 (X, V ) by pointwise multiplication preserves the subspace C 0 (U, V ). The fibers of E are V for x ∈ U and {0}, for x ∈ U .
The following example illustrates how the projections associated to such bundles behave.
Example 2.21. Let U ⊂ X be open and let V = span v 0 ⊂ H, with v 0 H = 1. We will investigate the projection associated to the Hilbert C 0 (X)-module E = C 0 (U, V ) by Proposition 2.19.
Let {η i } i be a compactly supported partition of unity on U , so that i η 2 In fact, any frame f is of this form: we have η i = v 0 , f i . Now, let {w i } i be an orthonormal basis of H. Then, θ e , the frame transform of e, maps w ∈ C 0 (U, V ) to i w i e i , w .
Although the image Γ 0 (X, θ e θ * e ) of θ e (consisting of those elements w of C 0 (X, H) for which w i , w has support contained in that of e i ) is isomorphic to C 0 (U, V ) through the map θ * e , it looks decidedly different from the isomorphic subspace C 0 (U, V ) of C 0 (X, H) we started with. The associated projection P = θ e θ * e maps w to i w i η i j η j w, w j . In the previous Example, note that P (x) = 1 for x ∈ U and P (x) = 0 for h} is locally trivial. This illustrates a general criterion for local triviality: Conversely, if the bundle is locally trivial, at least when the fibers are constant, we can choose P to be norm continuous. More precisely, if we let p : F → X be a locally trivial bundle of Hilbert spaces with separable, infinite-dimensional fiber H. By [34,Corollary 4 If instead H is finite-dimensional, then by the Serre-Swan theorem [20, Theorem 2.10] there exists a projection p ∈ M n (A) with pA n Γ 0 (F ). Thus, in Proposition 2.19 we may choose H = C n and P = p.

Schatten classes for Hilbert C 0 (X)-modules
When A is abelian, i.e. A C 0 (X) for some locally compact Hausdorff space X, each adjointable operator T on a Hilbert A-module E A can be localized by the pure states x of A to yield a family χ * T of operators on the Hilbert spaces χ * E A . We will unify the pointwise Schatten classes L p (χ * E A ) into a two-sided ideal L p (E A ) ⊂ L (E A ) and define an A-valued trace on L 1 (E A ).
Assumption 3.1. We will require all of our Hilbert A-modules to be countably generated in order to ensure access to frames using Proposition 2.12.
We will denote the character space of a commutative C * -algebra A equipped with the weak * topology by A, so that A C 0 ( A) with A a locally compact Hausdorff space.
The least we should demand of 'Schatten-class operators' T on E A is that their pointwise Schatten norm, i.e. the trace of |x * T | p , varies continuously with x ∈ A. In fact, this is the way to ensure that the 'trace-class operators' have traces with values in A and that the other Schatten classes respect this property in their pairing. The most we could reasonably demand, in contrast, is that the operators χ * T are continuous in Schatten norm with respect to some trivialization of E A (see Definition 3.8, below). It will turn out that these requirements, properly understood, are equivalent and yield a well-behaved Schatten class.
The following proposition, familiar from the Hilbert space case, is immediate from the definition: A. This theorem plays a crucial role throughout, because it allows us to relate the fiberwise Schatten norms on bundles of Hilbert spaces to various expressions for the element tr |T | p ∈ A, for T ∈ L p (E A ).
We will use the existence of frames (cf. Proposition 2.12), to relate L p (E A ) to the Schatten classes L p l 2 (A) on the standard module l 2 (A) and to relate the trace tr |T | p to a series expression in terms of (arbitrary) frames.
converges in norm; the limit equals tr |T | p = tr θ e |T | p θ * e . Proof. We start with the second part. As χ * (e) is a frame of the Hilbert space For the first part, let {e i } i be the standard orthonormal basis of l 2 , so that θ * e (e i ⊗ 1) = e i . Because θ * e θ e = id E A , we have |θ e T θ * e | p = θ e |T | p θ * e . Then, for any The series must then converge in norm by Dini's theorem.
Proof. Let e be a frame of E A . Then i∈F e i , |S| p e i ≤ i∈F e i , |T | p e i for all finite F ⊂ N; in particular, the left-hand side is Cauchy whenever the right-hand side is. By Theorem 3.5, this will suffice.
Remark 3.7. The above Corollary is weaker than the Hilbert space version (cf. Lemma 3.14(3) below. Instead, Corollary 3.23 below gives a stronger result but an additional assumption on S is required. Note that this is the only point in the treatment of this chapter where such a difference between the Hilbert module and Hilbert space Schatten classes appears.
The most straightforward road to analyzing the structure and properties of L p (E A ) now lies open: we will investigate L p l 2 (A) and use the pullback by the frame transforms to transfer its properties to L p (E A ). It will turn out that L p l 2 (A) is indeed very well-behaved, so that this allows us to recover many of the familiar properties of the Schatten classes of operators on Hilbert spaces.
3.1. The Schatten class on the standard module. Let H be a separable Hilbert space and let A = C 0 (X) be an abelian C * -algebra. Recall from Section 2.2 that the Hilbert A-module H ⊗ C A is isomorphic to C 0 (X, H) through the canonical isomorphism χ * (H ⊗ C A) H (with a character χ ∈ A always corresponding via the Gelfand transform to a point x ∈ X). Recall, moreover, that its endomorphism space is given by C str b (X, H) with the fiberwise action given simply by Because all fibers are identified canonically with H, we may canonically compare the localizations of an operator T ∈ L (H ⊗ C A) between different fibers. This technically useful difference between H ⊗ C A and other Hilbert A-modules will allow us to use topologies on B(H) to define particular subsets of L (H ⊗ C A). Most importantly, . Note that where the requirement that tr |T | p ∈ A is the least restrictive among reasonable criteria for a 'Schatten-class operator' T , as discussed above Definition 3.2, the condition of Definition 3.8 is arguably the most restrictive.
However, we will prove that the demands are, in fact, equivalent, so that the space L p (H ⊗ C A) of Schatten-class operators can be identified with the Banach space C 0 (X, L p (H)) of continuous Schatten-class operators. This will later -specifically in Theorem 3.25-allow us to combine the properties that follow straightforwardly from either of the two definitions.
The ostensibly more restrictive definition of the continuous Schatten class has some advantages to that of L p (C 0 (X, H)). For instance, it is immediate from the definition that C 0 (X, L p (H)) is closed under addition, as we have not yet shown for L p (C 0 (X, H)). Moreover, we can easily obtain a continuous version of the Hölder-von Neumann inequality: Proposition 3.10. Let p, q, r ≥ 1 such that 1 p + 1 q = 1 r , and let S ∈ C 0 (X, L p (H)) and T ∈ C 0 (X, L q (H)). Then ST ∈ C 0 (X, L r (H)) and ST r ≤ S p T q .
Proof. For any x, y ∈ X we have for all x, the statement on the norms follows as well as the claim that ST ∈ C 0 (X, L r (H)).
We now identify a subset of C 0 (X, L p (H)) ⊆ L p (H ⊗ C A) (cf. Remark 3.9 for the latter inclusion) whose completion in the Banach norm of C 0 (X, L p (H)) is all of L p (H ⊗ C A). This will show that L p (H ⊗ C A) coincides with the Banach space C 0 (X, L p (H)). Moreover, the fact that this common subset consists of finite-rank operators, in the Hilbert module sense, allows us to show that L p (E A ) ⊂ K (E A ) in Theorem 3.25.
Proposition 3.11. The finite-rank operators (in the Hilbert C * -module sense) on C 0 (X, H) lie in C 0 (X, L p (H)).
Pointwise in H, however, we have |ξ η| p = ξ η , so that norm continuity of v and w finish the proof.
. Then, all elements of C 0 (X, B(V )) are finite rank operators on the Hilbert modules C 0 (X, H). In particular, we have C 0 (X, The purpose is to show that one can use the series that defines the trace of |T | p to control the rate at which certain finite-rank approximations of T will converge to T in Schatten norm. Proof. Let {e i } i be an eigenbasis of (the compact, normal operator) T * T , ordered by decreasing of the corresponding eigenvalues {λ i }.
For p ≥ 2 the function x → x p/2 is convex on R + and we find, mutatis mutandis in the argument as above that now For the final claim, if |S| 2 ≤ R * R, one has Se i p ≤ Re i p so that S  Proof. As in Lemma 3.14, let T * T have eigenbasis {g i } with eigenvalues {λ i }. Then, for any v ∈ H, we have T ev, T ev = |T | 2 ev, ev = j λ j | ev, g j | 2 . In particular, But from the Lemma it then follows that With respect to a choice of orthonormal basis on H, we can view L p (H) as a completion of the direct limit Mat C of finite matrix algebras Mat n (C) in the Schatten norm. This can be done 'uniformly', where the convergence of the limit is controlled by the trace, as we will show in Proposition 3.17 below.
Given T ∈ L (H) and a sequence of increasing finite-dimensional subspaces P H ⊂ H, the operator P T P converges to T * -strongly as P → id H . The following Lemma allows us to control the p-norm of the difference when increasing the rank of P by one, such that P T P → T in p-norm precisely when T ∈ L p (H). Proof. One has QT Q − P T P = P T e + eT e + eT P = P T Q + eT P , so that QT Q − P T P p ≤ eT Q p + P T e p by cyclicity of the Schatten norms. Now compose T as T = S|T | 1 2 with |S| 2 = |T |. Then, by the Hölder-von Neumann inequality, we have eT Q p ≤ eS 2p |T | 1/2 Q 2p and similarly P T e p ≤ P S 2p |T | 1/2 e 2p . Moreover, P S 2p ≤ T 1/2 p and |T | 1/2 Q 2p ≤ T 1/2 p , and as eS 2p = S * e 2p , we may now apply Corollary 3.15 directly (to S * e 2p and |T | 1/2 e 2p ) in order to finish the proof. L p (H)). It is a well-known fact that the finite-rank operators are dense in L p (H) for any p, as can easily be seen from the spectral theorem for compact self-adjoint operators. This argument, however, does not extend uniformly to self-adjoint T ∈ C 0 (X, L p (H)) unless T is continuously diagonalizable. The explicit, albeit apparently somewhat clumsy, result of Lemma 3.16 presents a solution to this problem. Namely, orthogonal projections P ∈ B(H) can be lifted to constant projections C str b (X, B(H)). This allows the Lemma to be applied uniformly to all of C str b (X, B(H)), as we do in Proposition 3.17 below. This will then provide the main ingredient of main result of this section, to wit, the identification of L p (H ⊗ C A) with C 0 (X, L p (H)) (Theorem 3.18).
Proposition 3.17. Let T ∈ C str b (X, B(H)), let p ∈ [1, ∞) and assume that the function tr |T | p : x ∈ X → tr |T (x)| p is defined everywhere and lies in A = C 0 (X).
Let {e i } i be an orthonormal basis of H and let, for any n ≥ 0, P n := n i=1 |e i e i | be the corresponding spectral projections. Then the operators T n := P n T P n (as in Lemma 3.16) are elements in C 0 (X, L p (H)) and, for m ≥ n, Proof. Note that T n is a finite-rank operator and in particular T n ∈ C 0 (X, L p (H)), by Lemma 3.12. Moreover, by Lemma 3.16 applied to the projections P = P n , Q = P m and e = P m − P n , and by convexity of B(H)). Then T ∈ C 0 (X, L p (H)) if and only if T ∈ L p (C 0 (X, H)).
Proof. The implication ⇒ was already established in Remark 3.9. For the converse, assume that T ∈ C str b (X, B(H)) and that x → tr |T (x)| p lies in C 0 (X). Then, pick an orthonormal basis {e i } i of H and write again T n = P n T P n as in Proposition 3.17.
| p e i must converge uniformly on compact subsets of X by Dini's theorem. That, in turn, implies that goes to zero for large n. Since C 0 (X, L p (H)) is a Banach space, the sequence T n thus converges (to T ) in Schatten p-norm, so that T ∈ C 0 (X, L p (H)).   (X, B(H)). Proof. Let T ∈ C 0 (X, L p (H)) and let T ∈ C str b (X, B(H)). Then, for any basis {e i } i of H the operators T n := P n T P n (p ≥ 2) or T n := P n SP n |T | 1 2 P n (p < 2), as in Proposition 3.17, converge to T in the continuous Schatten p-norm. Now, note that T (T m − T n ) p ≤ T T m − T n p and (T m − T n )T p ≤ T T m − T n p ; thus, T T n and T n T converge to T T and T T , respectively, in the norm of C 0 (X, L p (H)).  Proof. The operators T n of Proposition 3.17 are of finite rank in the Hilbert module sense, because they are contained in C 0 (X, B(V )) for some finite-dimensional V ⊂ H so that we can apply Lemma 3.12. As T n → T for T ∈ C 0 (X, L p (H)) in the Schatten p-norm, T n → T in operator norm as well. We conclude that T is compact in the Hilbert module sense.
The following slight strengthening of Corollary 3.6 to C 0 (X, L p (H)) now translates to L p (C 0 (X, H)). Corollary 3.23. If 0 ≤ S ≤ T ∈ C str b (X, B(H)) and T ∈ L p (C 0 (X, H)) and additionally we have S ∈ C norm b (X, B(H)), then S ∈ L p (C 0 (X, H)) and tr |S| p ≤ tr |T | p .
Proof. The operators S, T are pointwise compact, norm continuous (this is where we use the additional assumption on S) and positive, so that their individual eigenvalues λ k (ordered decreasingly) are continuous ([28, p. IV.3.5], see also Lemma 4.16 below).
Then, by the min-max theorem, the k'th singular value of χ * S is bounded by the k'th singular value of χ * T , so that the same holds for their p'th powers. As the Schatten norm of χ * T is the sum of those p'th powers, which converges to a continuous function, the convergence must be uniform by Dini's theorem. Thus, the series k λ k (T ) p of elements of A is Cauchy, so that the series k λ k (S) p must be Cauchy as well. We conclude that x → χ * S p lies in A.
Remark 3.24. The additional assumption in Corollary 3.23 is necessary because the positive compact operators on a Hilbert C * -module, in contrast to those on a Hilbert space, may not necessarily form an order ideal: there is an additional continuity requirement on the (pointwise compact) localizations. In contrast, as in Corollary 3.6, the positive trace-class operators on a Hilbert C * -module do form an order ideal.

3.2.
Properties of the Schatten classes on Hilbert modules. We now return to the general setup of countably generated Hilbert C * -modules over commutative C *algebras. For the case of the standard module l 2 (A) with A = C 0 (X) Theorem 3.18 shows that L p l 2 (A) is a Banach space and a two-sided ideal of L l 2 (A) that is moreover contained in K l 2 (A) . These are very desirable properties for general, countably generated Hilbert A-modules. Fortuitously, the existence of frames (Proposition 2.12) and the pull-back criterion of Theorem 3.5 allows us to easily establish equivalent properties of L p (E A ) for all countably generated Hilbert Amodules E A .
Proof. Choose a frame e of E A and let φ e be the * -homomorphism induced by the frame transform: φ e : L (E A ) → L l 2 (A) , T → θ e T θ * e . By Theorem 3.5, we have T ∈ L p (E A ) if and only if φ e (T ) ∈ L p l 2 (A) . By Theorem 3.18, then, L p (E A ) is closed under finite linear combinations. Moreover, for S ∈ L (E A ) , T ∈ L p (E A ), we have φ e (ST ) ∈ L p l 2 (A) and φ e (T S) ∈ L p l 2 (A) by Corollary 3.20. We conclude that L p (E A ) is a two-sided ideal.
Moreover, as in the previous paragraph, T ∈ L p (E A ) iff θ e T θ * e ∈ L p l 2 (A) ⊂ K l 2 (A) by Corollary 3.22. But if θ e T θ * e is compact, then the operator T = θ * e θ e T θ * e θ e is compact as well (essentially because θ * e (|e i e j |)θ e = |e i e j | for any i, j ∈ N). Remark 3.26. Dixmier's definition of continuous-trace C * -algebras [13, Chapter 4.5] applies to K (E A ). By Theorem 3.25, the corresponding trace and Hilbert-Schmidt classes agree with our L 1 (E A ) and L 2 (E A ) respectively. The projections satisfying Fell's criterion are then the compact finite-rank ones, that is, those that correspond to finitely generated projective modules.
Remark 3.27. In the light of Theorem 3.25, we have obtained an a fortiori method of determine whether a positive operator T on a Hilbert C * -module over an abelian C * -algebra is compact: it suffices that tr T p lies in A for some p ≥ 1. Compare e.g. the proof of [24, Proposition 7] and [ibid., Remark 6] to see that showing such compactness directly can be a nontrivial undertaking.
Using Theorem 3.5, we can pull back the Banach norm on C 0 (X, L p (H)) L p (H ⊗ C A) to L p (E A ), and this turns out rather well: Theorem 3.28. The function · p : T → tr |T | p 1/p A is a norm that turns L p (E A ) into a normed vector space. Moreover, for all T ∈ L p (E A ), Proof. Let e be a frame of E A and consider the projection P e := θ e θ * e in L l 2 (A) . Recall that the invertible * -homomorphism φ e : T → θ e T θ * e maps L p (E A ) into P e L p l 2 (A) P e by Theorem 3.5. Moreover, elementary calculation shows that φ e | L p (E A ) is an isomorphism of normed spaces.
For the properties of the norm, recall that χ(tr |T | p ) = tr |χ * T | p = χ * T p p . Thus, tr |T | p 1/p A = sup χ∈ A χ * T 1/p p . Therefore, by the analogous properties of L p (H), we see that For property 5, recall that θ e ST θ * e = θ e Sθ * e θ e T θ * e ∈ L r l 2 (A) by Proposition 3.10, and so ST ∈ L r (E A ) by Theorem 3.5. The norm inequality then follows from the Hölder-von Neumann inequality for operators on Hilbert spaces.
Finally, to establish completeness of L p (E A ) it is enough to prove that its pullback P e L p l 2 (A) P e is a closed subspace of L p l 2 (A) . But if P e T n P e → T in L p l 2 (A) then P e T n P e − P e T P e p = P e (P e T n P e − T )P e p ≤ P e 2 (P e T n P e − T ) p → 0 as n → ∞, in virtue of the just-proved inequality 3. Hence, P e T n P e converges to P e T P e ∈ P e L p l 2 (A) P e as desired.
3.3. The Hilbert module of Hilbert-Schmidt operators. The Hilbert-Schmidt class L 2 (E A ) is a somewhat special case among the Schatten classes, because the map T → tr T * T is a positive definite quadratic form. That is, it induces an inner product as we will now explore.
Definition 3.29. The pairing ·, · 2 : When viewed fiberwise, this is just the ordinary Hilbert-Schmidt inner product: Proposition 3.30. For S, T ∈ L 2 (E A ) and a character χ of A the pairing S, T 2 satisfies χ ( S, T 2 ) = tr((χ * S) * χ * T ). Moreover, if e is a frame of E A , then the series ∞ i=1 Se i , T e i converges in norm to S, T 2 . Proof. Since χ * is a homomorphism, the first part follows from the fact that tr χ * (|T + i k S| 2 ) = χ(tr |T + i k S| 2 ) by the polarization identity for the fiberwise Hilbert-Schmidt inner product. Since i k e i , |T + i k S| 2 e i , the second part follows from Theorem 3.5.
Next, because A is commutative E A is automatically an A-bimodule. In fact, there is a * -homomorphism ρ : Proposition 3.32. The A-bimodule structure of L (E A ) restricts to L p (E A ) for all 1 ≤ p < ∞ and satisfies T · a p ≤ a T p as well as a · T p ≤ a T p .
Proof. Since any * -homomorphism between C * -algebras is norm decreasing, this follows from Theorem 3.25 since T ρ(a) p ≤ ρ(a) T p ≤ a T p .
All this leads to the following result for the case that p = 2: Proposition 3.33. With the above right A-action and the inner product of Definition 3.29, L 2 (E A ) becomes a Hilbert A-module.
Proof. Note that χ * (T • ρ(a)) = χ(a)χ * T for all a ∈ A and characters χ of A. Thus, with Proposition 3.30, the inner product is A-sesquilinear. All that is left to show, therefore, is that L 2 (E A ) is complete. That, however, was proven already in Theorem 3.25.
Proposition 3.34. Let H be a separable Hilbert space. Then, Theorem 3.18. Under this identification, the Hilbert C * -module structure of L 2 (H ⊗ C A) coincides with the canonical Hilbert C * -module structure on C 0 (X, L 2 (H)) induced by the inner product on the Hilbert space L 2 (H). Thus, we have L 2 (H ⊗ C A) C 0 (X, L 2 (H)) as Hilbert C * -modules. Now, invoke once again the isomorphism L 2 (H) ⊗ C A C 0 (X, L 2 (H)), this time for the Hilbert space L 2 (H), to complete the proof.
3.4. The trace class and the trace. We will refer to the ideal L 1 (E A ) ⊂ L (E A ) consisting of those operators T for which tr |T | is given by an element of A, as the trace class. In the case of Schatten classes of Hilbert spaces, i.e. A = C and E A = H, it is customary to identify the trace class as the ideal generated by squares of elements of the Hilbert-Schmidt class, in order to relate the Hilbert-Schmidt inner product to a linear function, the trace, on L 1 (H). The situation here is completely analogous: This furnishes us with a way to turn the bilinear map ·, · 2 on L 2 (E A ) into a linear map tr on L 1 (E A ) called the trace:  Proof. Assume T = RS with R, S ∈ L 2 (E A ) and let e be a frame. Then R * , S 2 = i R * e i , Se i , which converges in norm by Proposition 3.30. As R * e i , Se i = e i , T e i , we have two expressions for tr T : one independent of the decomposition T = RS and one independent of the choice e of frame. The proposition follows.
Corollary 3.38. Let χ be a character of A and let T ∈ L 1 (E A ). Then, tr χ * T = χ(tr T ).
Proof. Note that χ( e i , T e i ) = χ * e i , χ * T χ * e i . Since χ * e i is a frame of χ * E A the result then follows from Corollary 2.6.
Proof. For all χ ∈ A we have χ(| tr T |) = | tr χ * T | ≤ tr |χ * T | = χ(tr |T |) by the inequality | tr S| ≤ tr |S| on L 1 (H) for Hilbert spaces H. As the characters separate A, the first statement follows immediately. The last statement follows from the inequality a ≤ a 1 A for positive elements of any unital C * -algebra. Now, we can finally show that the trace is cyclic. The standard approach is as follows: Proof. Consider the value of a character χ ∈ A on the difference tr ST − tr T S ∈ A and use Corollary 3.38 above: = tr(χ * (ST )) − tr(χ * (T S)) = tr(χ * (S)χ * (T )) − tr(χ * (T )χ * (S)) We may now use the tracial property of the trace on L 1 (χ * E A ) (cf. [36,Corollary 3.8]) and the fact that χ separates A to conclude the proof.

Applications of Schatten classes
4.1. The Fredholm determinant. As a first application of the above theory of Schatten classes for Hilbert modules over unital abelian C * -algebras, we consider the Fredholm determinant. Let A be a commutative C * -algebra and let E A be a countably generated A-module.
Note that G(E A ) is a group under the multiplication of L (E A ) because we have (id +T ) −1 = id −T (id +T ) −1 and L 1 (E A ) is an ideal. We will define the Fredholm determinant, first on the standard module, and then by pullback by a frame transform on general countably generated Hilbert A-modules.
The following definition of the Fredholm determinant on a Hilbert space H is well-known. See e.g. [36, Chapter 3] for a brief discussion in the context of Lidskii's theorem. Recall that the series converges by the estimate k T 1 ≤ T k 1 /k!.
The Fredholm determinant of id +T is invariant under conjugation of T by partial isometries that commute with T : Lemma 4.4. If u : H → K is a partial isometry of Hilbert spaces, and T ∈ L 1 (H) is such that u * uT = T u * u = T , then id +uT u * ∈ G(K C ) and in fact det(id K +uT u * ) = det(id H +T ).
Proof. Note that we have so that id +uT u * ∈ G(K C ).
Next, note that tr k uT u * = tr Proposition 4.5. If K is a separable Hilbert space equipped with a frame e and T ∈ L 1 (K), then det(1 + θ e T θ * e ) = det(1 + T ) in terms of the corresponding frame transform θ e : K → l 2 .
Proof. Since θ * e θ e = id K commutes with T we can apply Lemma 4.4. Definition 4.6. Let E A be a countably generated Hilbert C * -module over a unital and abelian C * -algebra A. For T ∈ L 1 (E A ), the Fredholm determinant det(id +T ) of id +T is the function on A given by χ → det(χ * (id +T )).
Proposition 4.7. Let A be unital and abelian as above. For T ∈ L 1 (E A ), the Fredholm determinant lies in A ≡ C( A) and as such we have χ(det(id +T )) = det(χ * (id +T )).
Proof. Let e be a frame of E A . Note that, for all χ ∈ A, det(χ * (id +T )) = det(χ * (id +θ e T θ * e )) by Proposition 4.5. That is, det(id +T ) = det(id +θ e T θ * e ). Now, θ e T θ * e ∈ L 1 l 2 (A) by Theorem 3.5. With Remark 4.3 we see that χ → det(id +χ * S) is continuous whenever S ∈ L 1 l 2 (A) . Thus, since A is unital (and thus A compact) we find that det(id +T ) ∈ C( A) = A. Proposition 4.8. Let T ∈ C(X, L 1 (H)) with X = A. Then we have k (T ) ∈ C X, L 1 k H . In particular, one has det(id +zT ) = k≥0 z k tr k (T ), and z → det(id +zT ) is entire (as an A-valued function on C).
Proof. Let A, B ∈ L 1 (H) and note that As a consequence, we see that k (T ) ∈ C X, L 1 k H whenever T lies in C(X, L 1 (H)). Moreover, we have the pointwise series expression det(χ * (id +zT )) = det(id +z as in [36,Lemma 3.3]. Since tr k (χ * T ) ≤ χ * T k 1 /k! the series k≥0 z k tr k (T ) in A converges absolutely for all z ∈ C. This implies in particular that z → det(id +zT ) is entire.
is a (nonzero) eigenvalue of χ * T for all χ ∈ A (corresponding to the point x ∈ X).
As the characters separate A, it will suffice to prove the formula for the case E A = H C , where it is well-known. For |z| T 1 < 1, by absolute convergence of the trace and Lidskii's theorem, Proposition 4.12. Let T ∈ L 1 (E A ). Then id +T is invertible (that is, id +T ∈ G(E A )) if and only if det(id +T ) ∈ A is invertible.
Proof. By Proposition 4.10, det(id +T ) is invertible whenever id +T is. For the converse statement, in view of Proposition 4.5 we may assume without loss of generality that E A = l 2 (A) C(X, H). So, if det(id +T ) is invertible then it follows that id +χ * T is invertible for all χ by [36, Theorem 3.5b)]. Moreover, as is contained in the closed set consisting of the non-invertible elements of A, which contradicts the assumpion.
We conclude that there exists µ > 0 with inf n |1 + λ n (χ * T )| > µ uniformly for χ ∈ A. In particular, (id +χ * T ) −1 = sup n |1 + λ n (χ * T )| −1 < 1/µ for all x. We conclude that χ → (id +χ * T ) −1 is bounded and therefore continuous.  A)). Remark 4.14. In [2,3] it was shown that the Fredholm determinant det(id +zT ) is the unique additive invariant of an endomorphism T : E → E on finitely generated projective A-modules. It is an interesting open question to see under which additional conditions this result extends to the countably generated Hilbert module context. Clearly, the Fredholm determinant discussed above gives rise to a additive map from a countably generated Hilbert A-module E equipped with a trace-class operator T to analytic A-valued functions.
4.2. The zeta function. As a second application we consider zeta functions associated to positive Schatten class operators on a Hilbert module. Again, A is a commutative C * -algebra and we identify A = X so that A ∼ = C 0 (X).
For z ∈ C with z > p, define T z using the continuous functional calculus in L (E A ). Then, the associated zeta function is the function on the complex half-plane z > p given by ζ(z, T ) := tr T z .
We will show that the zeta function is in fact holomorphic (in the sense of Banach space-valued holomorphic functions, see e.g. [35,Definition 3.30]) on the defining half-plane.
First, let us recall that the individual (fiberwise) eigenvalues of positive compact operators are themselves continuous.
. For k ≥ 0 and χ ∈ A be the character corresponding to the point x ∈ X, let {λ k (x)} k be the eigenvalues of χ * T , in decreasing order with multiplicity. Then, the map x → λ k (x) is an element of A.
Remark 4.17. Note that it is important to know that both T is pointwise compact and that T is norm continuous. Dropping the latter assumption is fatal (see e.g. Example 2.21).
By self-adjointness and norm continuity, the full spectrum of T is continuous in norm topology, cf. [28,Remark IV.3.3]. However, the spectral projections (or even the eigenvectors) can in general not be continuously extended over any open neighbourhood, even in the case where the module is finitely generated: see e.g. [26]. A fortiori, continuous diagonalizability ('diagonability') is entirely out of the question in general.
The classical treatment of zeta functions of operators on Hilbert spaces is in terms of the Dirichlet series tr T z = k λ z k . The Jensen-Cahen theorem (see e.g. [21]) shows that these series converge uniformly on angular regions contained in the defining half-plane, so that the limit is in fact holomorphic. The theorem translates very well to the Hilbert module setting. Now, because the series ∞ k=0 λ k (x) p converges pointwise to tr T p , Dini's theorem shows that ∞ k=0 λ p k converges in norm. In particular, for all > 0 there exists m 0 such that A(m, q) < cos α for all q ≥ m ≥ m 0 .
By [21,Lemma 2], we have ∆ z,k ≤ |z|/p∆ p,k . Thus, as |z|/p ≤ sec α throughout C α , we have As in the classical case, the Jensen-Cahen theorem paves the way for a holomorphic zeta function. In particular, we have φ µ (λ z k ) = λ k (x) z dµ(x). Now, if C is a contour around z 0 in C z>p , the contour integral C λ k (x) z vanishes for all x ∈ X. Note that |λ z k | = λ z k and so, by Fubini's theorem, C λ k (x) z dµ(x) = C λ k (x) z dµ(x) = 0. By Morera's theorem, we conclude that φ µ (ζ n ) is holomorphic on C z>p . Moreover, as ζ n (·, T ) → ζ(·, T ) uniformly on compact subsets of C z>p by Lemma 4.18, the function φ µ (ζ(·, T )) is holomorphic on C z>p as well. By [35,Theorem 3.31], we may conclude that ζ(·, T ) is holomorphic, as an A-valued function, on the half-plane z > p.
For the second statement, note that by Lemma 4.18, for all > 0 there is, for all compact subsets K ⊂ C z>p , some m 0 with ζ m (z, T ) − ζ(z, T ) A < for all z ∈ K and all m ≥ m 0 .
Remark 4.20. It would be desirable to extend the previous Lemma and Theorem to the functions tr aT z , for a ∈ L (E A ). However, since the spectral projections of T are in general not even weakly continuous, this would be a nontrivial extension. We expect that a possibility for such an extension would be to investigate the functions x → tr p k (x)a(x)p k (x)/ rank(p k (x)), where p k (x) is the spectral projection on the eigenspace belonging to the eigenvalue λ k (x) of T (x), and then use these expressions as coefficients in the continuous Dirichlet series.
The next step in the classical case would be to show that certain operators have zeta functions that can be continued meromorphically to all of C, with a discrete set of poles. The residues at these poles then yields interesting information about the operator T . For instance, if T is the bounded transform (1 + D 2 ) −1/2 of a pseudodifferential operator these residues give geometric information about the pertinent background manifold (cf. [5] for more details). For this reason, it would be very desirable to have a reasonable criterion under which our zeta function of operators on Hilbert C * -modules can be continued meromorphically to all of C, but further research in that direction is beyond the scope of the present work. 4.3. Summability of unbounded Kasparov cycles. We now apply the justdeveloped theory of Schatten classes to arrive at a notion of summability for unbounded Kasparov cycles over a commutative C * -algebra. This notion is supposed to generalize summability for spectral triples (as unbounded Kasparov cycles over C) [9, Section IV.2] (cf. [20,Definition 10.8]) . We refer to [4,31,23,32] for all relevant notions of unbounded Kasparov cycles, external and internal Kasparov product, and to [25,37] for the specific application to Riemannian submersions and immersions to be discussed below.
In order to set the notation, for A, B two C * -algebras we let ( A E B , S) be an • S is a regular, self-adjoint (and odd) operator, densely defined on dom(S) ⊂ E A . • a(1 + S 2 ) −1 is in K (E B ) for all a ∈ A.
• There exists a dense subalgebra A ⊂ A that preserves dom(S) and is such that for all a ∈ A, the commutator [S, a] extends to an adjointable operator on E B .
Definition 4.21. Let B be a unital, commutative C * -algebra and let ( A E B , S) be an unbounded Kasparov A − B cycle. We say that For simplicity, we restrict to the case where A is unital. See [7, Section 2] for an understanding of the nonunital case. By the very definition of the Schatten classes, the localization of an unbounded p-summable Kasparov (A, B)-cycle along a character of B yields a p-summable spectral triple. In the other direction, it is not true that if all such localized spectral triples are p-summable, then the original unbounded KK-cycle is p-summable: one needs the fiberwise L p -norms of the resolvents to be continuous over the base. 4.3.1. Summability and the external Kasparov product. One of the key results in [4] was an explicit and linear formula for the external Kasparov product. More precisely, they showed that two unbounded Kasparov cycles (restricting to the even-odd case for simplicity) (E B , γ, S) and (F C , T ) can be combined into an external product unbounded KK-cycle over the minimal tensor product B ⊗ C: For spectral triples this can be understood as the direct product of the corresponding (noncommutative) spaces. In any case, it is desirable to have an additive property of summability for this external product in the case of a commutative base.
Of course, the real challenge is to establish the compatibility of summability with the internal unbounded Kasparov product. Clearly, Lemma 4.23 is then not sufficient but we leave its (challenging) extension for future research. Instead, we limit ourselves to establishing such summability results for certain classes of geometric examples. When combined with [6,25,15,37] where these examples were described in terms of unbounded Kasparov cycles, one may conclude the sought-for compatibility of summability with the internal product at least for these geometric examples.

4.3.2.
Example: Riemannian submersions. In [25] the factorization of the Dirac operator D Y on Y in terms of a vertical operator S and the Dirac operator D X on X was studied for a Riemannian submersion π : Y → X of compact spin c manifolds (more general proper Riemannian submersions were considered in [24,15,16]).
We let L 2 (S X ) and L 2 (S Y ) denote the Hilbert space completions of the spinor modules Γ ∞ (S X ) and Γ ∞ (S Y ), respectively. Based on a certain C ∞ (Y )-module of smooth sections of the vertical spinor bundle S V one then defines a Hilbert C * -module E C(X) between C(Y ) and C(X), together with a self-adjoint and regular unbounded operator D V on E, such that and in such a way that the operator D Y corresponds to the tensor sum D V ⊗ γ E + 1 ⊗ ∇ D X for some metric connection ∇ on E C(X) and the grading operator γ E on E (up to an explicit error term related to the curvature).
Let us analyze here the summability aspects of the operator D V : dom(D V ) → E. The main property that we will use below is that D V is the closure of a socalled vertically elliptic operator D : Γ ∞ (S V ) → Γ ∞ (S V ). This means that for all f ∈ C ∞ (Y ), [D, f ] is an endomorphism of Γ ∞ (S V ) that is invertible at all points where df | ker dπ is nonzero (see [25,24] for more details). In fact, this allows one to prove [24,Theorem 3] that the pair (E, D V ) is an unbounded Kasparov C(Y )−C(X) cycle.
As far as summability is concerned, note that the restrictions χ * D = D x of D to the fibers of x ∈ X (for the character χ : C(X) → C) are elliptic Dirac type operators. Since the dimension of the fibers is constant and equal to dim F for the typical fiber F , one has (1 + D 2 x ) −1/2 p < ∞ for all p > dim F [10] (cf. [20,Theorem 11.1]). The question, then, is whether this pointwise trace is continuous.
Proposition 4.25. The L p -norm of (1 + D 2 x ) −1/2 defines (as x varies over X) a continuous function on X for any p > dim F . Consequently, the unbounded Kasparov C(Y ) − C(X) cycle (E, D V ) is p-summable for all p > dim F .
Proof (based on [24,Section 2.3]). For simplicity of exposition, we will assume that the bundle S Y → Y is locally trivial over X; that is, we assume that around each point x 0 ∈ X there exists a neighbourhood U ⊂ X such that 1) there exists a diffeomorphism ψ : π −1 (U ) → U × F , and 2) the bundle S Y → Y can be smoothly and unitarily trivialized over U . If we pull sections of S Y over U back through this trivialization, we obtain a family of Dirac-type differential operators {D x } on the trivial bundle C k over the compact Riemannian manifold F .
In particular, these operators can be written as with A j , B symmetric matrix-valued smooth functions on U × F and A j invertible. Thus, there are smooth families Z x,x (z) = A j (x , z)A −1 j (x, z), W x,x (z) = B(x , z) − Z x,x B(x, z) of matrices such that D x = Z x,x D x + W x,x , with in particular lim x →x (id −Z x,x ) = lim x →x W x,x = 0. Now denote the closure of D x by D x . Note that D x is selfadjoint by compactness of F . Moreover, because D x is an elliptic differential operator of order 1, the resolvent (D x + i) −1 lies in L p L 2 (F, C k ) for all p > dim F .
Then, D x (D x + i) −1 being bounded, we see that lim x →x (D x − D x )(D x + i) −1 ≤ lim x →x id −Z x,x + W x,x = 0. Thus, by the resolvent identity, we conclude that lim x →x (D x + i) −1 − (D x + i) −1 p = 0 for all p > dim F , and so (1 + D 2 ) −1/2 ∈ L p (E) if and only if p > dim F .
Note that when we combine this result with the factorization result [25] of D Y in terms of D V and D X we obtain for Riemannian submersions of spin c manifolds the desired additivity of summability for the unbounded interior Kasparov product.

4.3.3.
Example: Embedding spheres in Euclidean space. We consider a special class of immersions, given by the embedding of spheres S n in Euclidean space R n+1 . This is based on [8,37,39]. As in [8] the embedding S n → R n+1 gives rise to an immersion class in KK-theory. For spheres, the unbounded representative is given by the module C 0 (S n × (− , )) based on a normal neighborhood of S n ⊂ R n+1 , equipped with the regular self-adjoint operator S given by the multiplication operator with a suitable function f : (− , ) → R. For convenience, we will take S to be multiplication by the function f (s) = π 2 tan πs 2 ; (s ∈ (− , )).
Hence the unbounded Kasparov C(S n ) − C 0 (R n+1 ) is p-summable for all p > 0.
Again, this is a confirmation of additivity for summability under the unbounded interior Kasparov product. Indeed, in [37] it was shown that D S n can be related to the immersion class defined by S as above and D R n+1 in the following way. Namely, the unbounded interior product of S and D R n+1 is equal to the unbounded interior product of D S n with a so-called index cycle T . The latter represents the identity at the bounded level but is in fact a p-summable Kasparov cycle for all p > 1.
is p-summable for all p > 1.
As such, the summability of D R n+1 plus that of the immersion cycle (i.e. 0 + ) indeed coincides with the summability of D S n plus that of the index cycle. Then tr χ * |D| −p = tr |D S 1 | −p < ∞ for all characters χ and p > 1 so that we have |D| −1 ∈ L p (E) for p > 1.
As an example of an unbounded Kasparov cycle, consider a homeomorphism on X and consider the action n · f := f • φ n of Z on C(X). Let C(X) φ Z be the corresponding full crossed product C * -algebra. Consider the unitary U ∈ L (E) given by U = S ⊗ 1, where S is multiplication by θ → e iπθ on L 2 (S 1 ). Then, the map ρ defined on finite sums in C(X) φ Z by ρ : where the left action of C(X) is just given by pointwise multiplication, extends by universality to a representation of C(X) φ Z. Moreover, [D, k f k U k ] = k kf k U k because [D S 1 , S k ] = kS, so that there is a dense subalgebra of C(X) φ Z with [D, a] bounded. We conclude that (E C(X) , D) is an unbounded p-summable Kasparov C(X) φ Z − C(X) cycle for all p > 1.
In particular, one has ζ D (f u k , z)(x) = tr χ * (f U k D −z ) = f (x) tr L 2 (S 1 ) S k |D S 1 | −z , so that ζ D ( k f k u k , z) extends meromorphically to C \ {1} and in fact Res z=1 ζ D ( k f k u k , z) ∝ f 0 .