{'fraction_non_alphanumeric': 0.05395722505699707, 'fraction_numerical': 0.009271523178807948, 'mean_word_length': 3.8980112729979792, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Universal cycles and homological invariants of locally convex algebras

Using an appropriate notion of locally convex Kasparov modules, we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras; examples are obtained from spectral triples. Our considerations are based on the action of algebraic K-theory on these functors, and involve compatibility properties of the induction process with this action, and with Kasparov-type products. This is based on an appropriate interpretation of the Connes-Skandalis connection formalism. As an application, we prove Bott periodicity and a Thom isomorphism for algebras of Schwartz functions.


Introduction
The subject of this article are homology-type invariants of locally convex algebras. It analyses to what extend locally convex bimodules are universal for such homology theories.
Let us first recall the situation in the setting of (separable) C * -algebras. On this category, we have at our disposal Kasparov's bifunctor KK. When KK is restricted to the first or second variable, one obtains K-homology or Ktheory, respectively. The elements of KK may be described in essentially three ways: The extension picture, the module picture and the quasihomomorphism picture. The extension picture yields, when specialised to the first variable, the Ext-theory of Brown, Douglas and Fillmore ( [BDF77]), which is based on earlier work of Busby ([Bus68]); here classes in Ext(A, C) are described as extensions of A by the compact operators. In the module picture, classes in KK(A, B) are represented by bounded operators acting on a Hilbert Bmodule, which carries a left A-action ( [Kas80b]); in the so-called Baaj-Julg picture, the operator is replaced by a regular operator on the Hilbert module. In the quasihomomorphism picture, all information is contained in a pair of homomorphisms, both from A into an algebra containing K ⊗ B as an ideal, with difference in K ⊗ B ([Cun87]/ [Cun83]).
In each of these pictures, there is a corresponding description of the Kasparov product KK(A, B) × KK(B, C) → KK(A, C). In the extension picture, it corresponds to a splice, or Yoneda product, of extensions ( [Zek89]). In the module picture it can be defined rather explicitly using certain operators M and N , whose existence is guaranteed by Kasparov's technical theorem. The proof of this theorem is based on properties of the category of C * -algebras; so is the proof of the fundamental theorem in ( [Cun87]), which is used in the definition of the product in the quasihomomorphism picture. The most calculable description of the product is the one in the module picture, where it is a generalisation of the Atiyah-Singer sharp product.
It is well known that Kasparov's bivariant K-functor is universal among certain functors. If H is a split exact, homotopy invariant and stable functor, then every class in KK(A, B) induces a homomorphism H(A) → H(B). Furthermore, it can be shown that this induction process is compatible with Kasparov products: if we are given classes x ∈ KK(A, B) and y ∈ KK(B, C), then the homomorphism induced by the Kasparov product of x and y is the composition of the homomorphisms induced separately by x and y.
In the locally convex setting, the situation is different. We still have at our disposal a bivariant K-theory, namely the theory kk constructed by Cuntz ([Cun97]/ [Cun05]). The definition of kk is based on linearly split extensions of locally convex algebras of arbitrary length, and the composition corresponds again to the Yoneda product of extensions. However, elements of kk still appear naturally in form of what we will call locally convex Kasparov modules. For example, a spectral triple on a compact spin c -manifold M satisfies additional summability conditions, and therefore defines a locally convex (A, B)-Kasparov module, where A := C ∞ (M ) and I is some ideal in the algebra bounded operators on a Hilbert space. To this object, one may associate a quasihomomorphism ϕ of locally convex algebras, which in turn induces a homomorphism H(ϕ) : H(A) → H(B) for every split exact functor. Already at this stage, there are some unresolved issues. If we take H := kk( · , B), then it is not clear in how far elements of the form kk(ϕ, B), where ϕ is the quasihomomorphism associated to a locally convex (A, B)-Kasparov module x, exhaust the group kk(A, B). In [CMR07], it suggested that this is not the case. However, as we noted above, elements of kk can be represented as such locally convex bimodules, and we show that they suffice to obtain the isomorphisms under split exact, diffotopy invariant L p -stable functors we are interested in.
We therefore develop a complete framework of locally convex Kasparov modules, together with equivalence relations between them, and study how they induce morphisms under split exact functors on the category of locally convex algebras. We show that smooth versions of Kasparov's Dirac and dual-Dirac elements appear naturally as locally convex Kasparov modules.
The crucial, and most difficult question now is to what extent the above compatibility of the induction process with Kasparov products still holds in this more general context. Essentially, the fact that in the C * -setting any two composable classes in KK have a product which is again represented by a Kasparov product, corresponds to the fact that the splice of two extensions of length, say two, may again be represented by a length two extension; the analogous property for a functor on locally convex algebras is very likely false.
It is thus necessary to find sufficient conditions for a locally convex Kasparov module to represent the composition H(ψ) • H(ϕ) of two homomorphisms induced under a functor H by quasihomomorphisms coming from locally convex Kasparov modules. There are two closely related approaches to this problem. The first consists in finding operators M and N , as in Kasparov's original description of the product, that satisfy the correct algebraic relations with respect to locally convex algebras involved in the construction of the product. This approach is feasible at least to some extent, but it turns out to be very technical.
The second approach, which we will follow here, is based on a new interpretation of the Connes-Skandalis connection formalism ( [CS84]). Already in the C * -setting, it yields a genuine description of the Kasparov product, based only on Kasparov's version of the theorem of Voiculescu ( [Kas80a]). The essential idea is that the existence of a product for two (locally convex) Kasparov modules may be interpreted as the existence of an extension of a Kasparov module. Just like the connection formalism by Connes and Skandalis, this extension condition only yields a sufficient criterion for a cycle to represent a product; the existence of such an extension remains a consequence of Kasparov's technical theorem, whose validity is restricted to the C * -setting. However, the existence of the aforementioned extension is easier to show than the existence of the operators M and N . In the case where the first cycle is represented by a class in K-theory, we show that it is always possible to construct such an extension, and thus show the existence of a product in complete generality. This fact yields Bott periodicity. Using the formalism of locally convex Kasparov modules, we are also able to prove a smooth version of the Thom isomorphism theorem for the algebra of fibrewise Schwartz functions on a smooth vector bundle over a compact smooth manifold for every split exact, L p -stable, diffotopy invariant functor.
The formalism of locally convex Kasparov modules is further useful in order to define explicitly classes in kk, for example those related to pseudodifferential operators. This is quite difficult in general due to the very abstract definition of kk. Our considerations also show that kk is very likely not an analogue of Kasparov's KK-theory, but rather of the E-theory of Connes and Higson [CH90].
I would like to thank G. Skandalis and J. Cuntz for helpful discussions and remarks.

Preliminaries
In the sequel, we will work on the category LC of locally convex algebras. By a locally convex algebra we will mean a complete locally convex vector space that is at the same time a topological algebra, i.e., the multiplication is continuous (and associative). This means that for every continuous seminorm p on A there is a continuous seminorm q on A such that for all a, b ∈ A Recall that for two locally convex topological vector spaces V and W , their projective tensor product is defined as the algebraic tensor product V ⊙W of V and W with the locally convex structure induced from the family of seminorms defined for all x ∈ V ⊙ W by where p and q run through systems of seminorms defining the topologies of V and W (see [Gro55]). The completed projective tensor product of complete locally convex spaces as above will be denoted V ⊗ π W , the extension of p ⊙ q to V ⊗ π W will be denoted by p ⊗ q. The topology on V ⊗ π W is equal to the completion of the final topology for the canonical map The injective tensor product of locally convex spaces is defined in [Trè06], or again in the original [Gro55] . For complete locally convex spaces V and W , the completion of V ⊙ W in this topology will be denoted V ⊗ ε W ; if V and W are Banach spaces, V ⊗ ε W is a Banach space with a certain norm || · || ε . Recall that for complete subspaces V ′ ≤ V and W ′ ≤ W , ⊗ ε , in contrary to ⊗ π , does have the property that V ′ ⊗ ε W ′ ֒→ V ⊗ ε W (see [Trè06] Proposition 43.7).
For C * -algebras A and B, the minimal tensor product A ⊗ * B is obtained by taking the tensor product of the Hilbert spaces from a faithful representation of A and B and pulling back (and completing) the induced norms on A ⊙ B (see [Mur90] for details, e.g., the independence of the presentation, nuclearity, exactness and so on); we denote the obtained C * -norm by || · || * . The maximal tensor product A ⊗ max B is obtained as the completion in the norm || · || max given by the supremum over all norms coming from * -representations of A and B. If B is nuclear we will often not distinguish between these two C * -tensor products.
We denote by Z the locally convex algebra of differentiable functions from [0, 1] all of whose derivatives vanish at the endpoints, by C those that are zero in 1, and by S those that are zero in both endpoints; ; we define S n as the n-fold tensor product of S n with itself, similarly for C n and Z n . We further get functors mapping a locally convex algebra A to ZA, CA and SA as in the C * -setting. We write ev A t : ZA → A for the evaluations in t ∈ I.
Definition 1. Let ϕ 0 , ϕ 1 : A → B be homomorphisms of locally convex algebras. A diffotopy between ϕ 0 and ϕ 1 is by definition a homomorphism Φ : We note from [FS07] that this is not the same as a family of morphisms that is pointwise differentiable.
Remark 2. The well known decomposition for Hilbert spaces into matrices with respect to a projection carries over to locally convex algebras as follows: Let a ∈ A and p an idempotent in some larger algebra containing A. Set which entails a 1 +a 2 +a 3 +a 4 = a. As all of the subalgebras pAp, (1−p)A(1−p), (1 − p)Ap, and pA(1 − p) have empty intersection, A is isomorphic to the inner direct sum of them (as a locally convex space). If we write the a as a 2 × 2matrix a = a 1 a 2 a 3 a 4 , then the multiplication on A corresponds to the multiplication of matrices. This is usually called a Morita context (see e.g. [CST04]); we have reserved this expression for a more analytic version of a context below.

Functors and exact sequences
We collect some general statements concerning functors and exact sequences of locally convex algebras.
Definition 3. An exact sequence of locally convex algebras the sequence

Remark 5.
If H is split exact, then the sequence obtained by applying H automatically splits.

Definition 6.
Let H be a functor from locally convex algebras to abelian groups, and J a locally convex algebra containing an idempotent p. Then H is called is an isomorphism, denoted θ J A , for every locally convex algebra A, where ι : A → A ⊗ π J, a → a ⊗ p.

• invariant under inner automorphisms if H(Ad
• pointwise diffotopy invariant, if for every family ϕ t : A → B of morphisms of locally convex algebras such that ϕ t (a) is smooth for all a ∈ A, H(ϕ 0 ) = H(ϕ 1 ).
• diffotopy invariant if for every homomorphism ϕ : Remark 7. If P is a class of idempotents in J that are all conjugate to another, in the sense that J is an ideal in a larger algebraĴ such that for any two minimal projections p and q there is an invertible element u ∈Ĵ with u −1 pu = q, then the definition does not depend on the choice of idempotent in P if we suppose that H is invariant under conjugation by such u (of course it suffices that conjugation by u is continuous B → B, so one can weaken the ideal condition). For example, the minimal projections in the compacts or the p-summable operators have this property.
We then have the following properties:  (iii) is also proved in [CT06]: If U ∈ A is invertible, then U 1 defines an automorphism of A ⊗ M 2 . As the two natural inclusions of A in A ⊗ M 2 are the same under any M 2 -stable functor, U acts trivially.
If H is a separable Hilbert-space, then factoring the natural inclusion ι : M n → L p (H) as the stabilisation M n → M n ⊗ π L p (H) followed by the canonical identification L p (H n ) ≈ L p (H) -which both induce isomorphisms -we get (iv) in the scalar case; tensoring by A gives the general statement.
(v) follows now from commutativity of The first part of (vi) follows from (iii), and entails the second.
We restate the following fundamental result from [CT06], Theorem 4.2.1, in the form appropriate for later use (using Proposition 8 -(v)): Theorem 9. Every functor from the category of locally convex algebras to the category of abelian groups which is split exact and L p -stable for some p ≥ 2 is diffotopy invariant.

Morita contexts and split exact functors
The following definition of Morita context is basically from [Cun05], but we carry it over to arbitrary functors. We also add some isomorphisms to the definitions in [Cun05] in order to make the existence of a Morita-context a weaker condition than being isomorphic.
Definition 10. Let A and B be locally convex algebras. Then a Morita-context from A to B is given by data (ϕ, D, ψ, ξ i , η i ), where D is a locally convex algebra, ϕ : A → D, ψ : B → D are isomorphisms onto subalgebras of D, and sequences η i , ξ i in D such that With this definition: • If ϕ : A → B is an isomorphism, then we get a Morita context (ϕ + , B + , · + ) from A to B, where B + is the unitization of B, b → b + the canonical embedding, and ϕ + = ϕ • · + • In particular, there is now a canonical Morita-context from A to A, for any locally convex algebra A • If B ⊆ K ⊗ π C is a subalgebra, and we call a corner in B a subalgebra Definition 11. Let H be a functor on the category of locally convex algebras.
Note that for a, a ′ ∈ A we have

Definition 12. A Morita-bicontext between locally convex algebras A and B is given by two Morita-contexts from
Theorem 13. Given two Morita contexts as in the above definition, and a diffotopy invariant, K-stable functor H, we have is invertible. We suppose left compatibility; then denoting θ A and θ B the maps A → K ⊗ π B and B → K ⊗ π A determined by the two contexts, and multiplying by ε K⊗πA on the left, we see that it suffices to show that the and it is diffotopic to the stabilisation as follows. The L×L matrix with entries yields a diffotopy of the map in equation 1.

Quasihomomorphisms and induced morphisms
The notion of quasihomomorphisms was introduced in [Cun83], and used for example in [CT06] to induce elements in kk. We modify it according to our needs.

Definition 14. Let A and B be locally convex algebras,B an algebra containing
B as a subalgebra. Then an LC-quasihomomorphism from A to B (relative tô B) is given by a pair (α,ᾱ) of morphisms A →B such that the maps are all three actually B-valued and continuous (with respect to the topology on A and B). We denote such a quasihomomorphism by (α,ᾱ) : A ⇉B B.
Split exact functors have additional functoriality properties with respect to quasihomomorphisms. We first define the objects involved in the construction.
is a locally convex algebra D α . B identifies to an ideal in D α that carries the subspace topology via the inclusion ι B : Further D α fits into a split short exact sequence Proof. By hypothesis, the multiplication is continuous. Further, we may identify D α with the subalgebra of A ×B, and hence the multiplication is associative.
Remark 16. If A,B, B and α come from an LC-quasihomomorphism (α,ᾱ) : : A → D α defines another split of the above extension. Further we see that it is actually possible to suppose thatB is a locally convex algebra and α,ᾱ are morphisms of locally convex algebras. However, the elements we will use do not naturally have this form.
We will usually identify B with its image in D and write α andᾱ instead of α ′ ,ᾱ ′ .
commutes in the usual sense and

Proof. As H(ι ′ ) is injective and
the result follows.
Remark 19. It suffices to suppose that we are given a commutative diagram such as the one in 2 that commutes after applying H.
Remark 20. In the setting of C * -algebras, one usually applies a slightly different construction (e.g. [CT06]). If we fix a C * -quasihomomorphism (α,ᾱ) : A ⇉ M(B) B and denote byD α the subalgebra of A ⊕ C * M(B) generated by B and (a, α(a)), then there is a unique morphism θ yielding a commutative diagram That is, because the lefthand side commutes, θ has to fix B, and as the right hand side commutes, (a, 0) → (a, α(a)). It is easily calculated that the morphism thus determined actually is a * -homomorphism (if we equip D α with the obvious involution). Therefore the construction coincides with the one in the C * -setting.
Proposition 21. Let (α,ᾱ) : A ⇉B B be an LC-quasihomomorphism, H a split exact functor on the category of locally convex algebras with values in abelian groups. Then the following properties hold: is a morphism of the double split exact sequences associated to D α•ϕ and D α , and apply Lemma 18. Similar reasoning applies to the morphism (ψ, ψ ⊕ id A , id A ) and proves (ii).
we have a morphism (id B * , ϕ 2 * , − id A * ) after applying H and use Remark 19.

Proposition 22. Let
in other words, the composition of the morphisms induced by the outer edges in the diagram are the same.
and both sides are injective, it suffices to show This follows from the hypotheses by an easy diagram chase.
If A and B are locally convex algebras, we define Note that as the projective tensor product preserves split exactness, tensoring by C yields again a quasihomomorphism, which is furthermore again in standard form.
If we start with an arbitrary quasihomomorphism (α,ᾱ) : A ⇉B B, then we may replaceB by D α to obtain a quasihomomorphism in standard form.
As the projective tensor product preserves exactness of linearly split extensions, we obtain double split exact sequences containing E 1 ⊗ π B 2 , E 1 ⊗ π D 2 , E 1 ⊗ π A 2 , and similarly B 1 ⊗ π E 2 , D 1 ⊗ π E 2 and A 1 ⊗ π E 2 that fit into a diagram as in the above Lemma, and obviously satisfy the commutativity conditions. Identifying D 1 ⊗Ĉ with the algebra associated to the quasihomomorphism (α 1 ,ᾱ 1 ) ⊗ C : A 1 ⊗ π C ⇉ B 1 ⊗ π C B 1 ⊗ π C for any locally convex algebra C, and similarly for the others, we have proved most of the following Proposition 24.
or, in other words, the obvious outer product of quasihomomorphisms is compatible with the induction process. In particular If (α,ᾱ) : A ⇉B B is a quasihomomorphism in standard form and C a supplementary locally convex algebra, then Proof. The last statement follows by identifying D α⊗C ≈ D α ⊗ π C, applying Lemma 18 to In particular, for suitably nice ideals and algebras, it is easy to see that for Hence we may apply the above to obtain We will make use of the following result, which shows that the induction process is automatically compatible with stabilisation for stable functors, later on: Proposition 25. Let H be a J-stable, split exact functor. Then for every quasihomomorphism (α,ᾱ) from A to B the following diagram commutes: Proof. Identify D α⊗J ≈ D α ⊗ π J and apply Lemma 18 to where the maps ι are inducing the stabilisation isomorphism under H and make the diagram commute.
LetB be an algebra containing ZB as an ideal. Set x : Proof. Denote π i :B →B/C i B the quotient maps, and apply Lemma 18 to

The action of K-theory
In this section, K(A) denotes the algebraic K-theory of an algebra A, ∪ the outer product K(A) ⊗ K(B) → K (A ⊗ π B). The notion of invariance under inner automorphisms was given in Definition 6.
Proposition 28. Let H be a split exact, M 2 -stable functor from locally convex algebras to abelian groups. For every locally convex algebras A and B there is a product: If A = C and x = 1 ∈ K(A), then Furthermore, if f : If C is another locally convex algebra, x ∈ K(A), y ∈ K(B) and c ∈ H(C) Proof. By Proposition 8, H is invariant under inner automorphisms and M nstable for all n. Assume first that A is unital. Let p be an idempotent in M n (A). Then define a morphism This yields (by the universal property of the enveloping group) a homomorphism of groups which defines the product x · b := F B A (x)(b). The compatibility with unital morphisms is straightforward. If C is another unital algebra, then for all x ∈ K(A), y ∈ K(B) we have directly from the definitions.
To treat the nonunital case, it suffices to apply the following Lemma.

Lemma 29. If
is a split exact sequence of locally convex algebras with A 1 and A 2 unital, then It remains to compare the two ways a given quasihomomorphism acts on H:  H((α,ᾱ) coincide.
Proof. We may suppose, replacing by D α , thatÂ is locally convex and A a closed ideal in it. Then for B = C this follows directly from the definitions. The general case follows from the scalar case by applying it to the functor This shows that quasihomomorphisms from C to B are determined by their action on K-theory: B be two quasihomomorphisms. Then H(α 1 ,ᾱ 1 ) = H(α 2 ,ᾱ 2 ) for all split exact M 2 stable functors if and only if the equality holds for the specific functor K.

LC-Kasparov modules and abstract Kasparov modules
We will now introduce abstract Kasparov modules, an intermediate step between Kasparov-modules and quasihomomorphisms. There are several versions of this notion of abstract Kasparov module, compare [CT06] and [CMR07]. In the latter, constructions for abstract Kasparov modules (in the C * -setting) are done by simply lifting an abstract Kasparov module back to a classical one, and then transferring the construction back to the abstract setting (compare for example [CMR07], Lemma 8.33). For the product, this leads to the notion of double Kasparov module, and as noted again in [CMR07], the conditions in the non-C * -setting get extremely technical, and it is not clear how useful this construction is for bornological algebras. We will describe the product in a different manner below, similar to the approach we sketched for KK-theory. Again, we have to modify the definitions to make them applicable to our settings. We proceed to define an appropriate notion of locally convex bimodule: Definition 35. Let A and B be locally convex algebras. An LC-Kasparov module from A to B is given by an algebraB containing B, elements ε, F ∈B and an algebra morphism ϕ : A →B such that ε 2 = 1, F ε = −εF , [ϕ(a), ε] = 0 and such that the maps are actually B-valued and continuous. We then call ε a grading and denote the LC-Kasparov (A, B)-module by (B, ϕ, F ).
Note that it suffices to suppose the map a → ϕ(a)(1 − F 2 ) be continuous B-valued and (1 − F 2 )ϕ(a) ∈ B, continuity of the second in a then follows; similarly for the maps a → ϕ(a)b and a → bϕ(a)in (ii). There is also an obvious notion of representation of LC-Kasparov modules which we will use in the sequel. We also identify LC-Kasparov modules that are isomorphic in the obvious sense.
Proof. W F is it's own inverse: and modulo a continuous B-valued map we have As follows from the hypothesis, multiplication by W F is continuous on B, thus the result follows by multiplying equation 5 by W F on the left.
The point of this definition is the following: Using the usual transformation F → U F (see below) is adequate in the setting of p-summable Fredholm modules, but it is not, for example, on smooth functions.
Then (E ⊕ E op , ϕ ⊕ 0, U ′ F ) and (E ⊕ E op , ϕ ⊕ 0, W ′ F ) are perturbations of the same module, hence equivalent. A straightforward calculation shows that the quasihomomorphism induced by (E ⊕ E op , ϕ ⊕ 0, U ′ F ) is equivalent to (P ε α, U F P ⊥ ε αU F ); similarly for W F and W ′ F . One may also see this by remarking that U F and W F can be obtained from passing to the Fredholm picture, i.e., regrouping E ⊕ E op into even and odd parts by applying a permutation matrix. The conjugation of U ′ F and W ′ F by this permutation matrix yields operators with off diagonal entries U F , U * F and W F , W * F , respectively.
There is an obvious sum operation on abstract Kasparov modules, given by Proposition 40. B-perturbation is a weaker equivalence relation than diffotopy, and the transformation Qh from LC-Kasparov modules to LC-quasihomomorphisms preserves diffotopy.
Proof . Let (B, ϕ, F ), (B, ϕ, G) be B-perturbations of one another. Let g ∈ C with g(1) = 1, set f := 1 − g, defineF := f F + gG ∈ ZB. Then multiplication byF on ZB is continuous because multiplication by F and G and elements of Z is. Further The rest is obvious.
The next example shows that the induction process is correctly normalized.
Example 41. Let (B, ϕ, F ) be an LC-Kasparov (C, B)-module with grading ε, where B is some locally convex algebra, such that ϕ is unital andSS = 1, with respect to P ε . Then we get Hence the map (P ⊥ ε ϕ) W F −P ε ϕ is a homomorphism orthogonal to P ε ϕ, namely the map Thus for every split exact functor H by Proposition 21 -(iv) and (iii). If, on the other hand, SS = 1, we obtain the homomorphism and by Proposition 21 (iv)

H(B, ϕ, F ) = H(1 −SS).
One could also deduce the second from the first by a rotation in matrices and using diffotopy invariance.
The following is an abstract variant of [Kas80b], Theorem 5: Proof. It suffices to prove the case with A = C, the general case follows by applying the result to the functor H( · ⊗ π A). Setting P := ϕ(1) and replacing by the module (P B(H)P, P ϕ, P F P ), we may assume that ϕ is unital (because H(P ⊥ B(H)P ⊥ , P ⊥ ϕ, P ⊥ F P ⊥ ) = 0).
With respect to the grading, F = S T . By hypothesis, T is Fredholm and hence has closed cokernel.
Assume that T has negative index. Without loss of generality, it is injectif, and we can define the bounded operator T ′ as T −1 on Im(T ) and zero on Im(T ) ⊥ . We have and as T is invertible modulo L p , T ′ − S ∈ L p follows. Therefore the module In case that T has negative index, we reduce to the former case by passing to the module (B, ϕ, −F ) with grading −ε.

Example: The Bott element
We now construct the smooth analogue of the Bott elements. We assume in this section that the dimension n is even. The point is that otherwise we would have to work with graded functors, as certain gradings would not be inner.
We recall that the C * -Kasparov (C, S n ⊗ C n )-module y n was defined by Kasparov ([Kas80b], see also [Greb]) as (S n ⊗ C n , 1, q(D 2 )), where D 2 denotes the operator of multiplication by the inclusion R n ֒→ C n and 1 denotes the action of C given by scalar multiplication. In [Kas80b], the grading is the one coming from the natural grading of the Clifford algebras. We may assume that the algebra S n ⊗ C n is trivially graded and the grading on the Hilbert module S n ⊗ C n is given by left multiplication with the volume element. This does not change the class in KK-theory, because it corresponds to applying a graded Morita context. In fact, more generally: If A is a C * -algebra with an inner grading κ U such that U 2 = 1, then there is a graded Morita context between A and A with the trivial grading.
The context is given by letting (A, κ U ) (with grading) act on the Hilbert (A, id A )-module (trivially graded) A with grading induced by left multiplication with U . Taking Kasparov products with the corresponding KK-equivalence transforms the grading exactly to the ones used above. Note that this does not signify forgetting the grading (in fact, x n is trivial in KK if we just view it as ungraded).
As we need to define an LC-Kasparov (C, S n ⊗ C n )-module, the operator q(D 2 ) used in the C * -setting is inadequate, as 1 − q(D 2 ) 2 = (1 + D 2 2 ) −1 is not a Schwartz function. The idea, in dimension 1, is that it suffices to replace the inclusion f : R → C 1 by the function t → (0, te t 2 ), or any other odd real-valued function h growing sufficiently fast at infinity; in arbitrary dimension we define LC-Kasparov (C, S n ⊗ C n )-modules We then have the following weak multiplicativity property (in contrast to the classical setting) Lemma 43. Let H be a split exact, diffotopy invariant M 2 -stable functor, m, n ∈ 2N. Denote D n,2 the operator D 2 in dimension n. Then Proof. The first equality follows by factoring over K-theory or from a diffotopy using the operator q(e t||y|| 2 h(c + (x)) + e t||x|| 2 h(c + (y))), To see the second equality, note that in K-theory the class of y ∞ m+n is the outer product of y ∞ m and y ∞ n . Further, H(y ∞ m+n ) coincides with the action of the K-theory class [y ∞ m+n ] by multiplication, as defined in Proposition 28. As the pairing is compatible with outer products (Proposition 28), we get the above statement.
Note also that the operators q(h(D 2 )) are equivariant with respect to the O(n)-action; this is not true for the operators built up from the one-dimensional ones, which do however have a representative given by an equivariant module according to the above Lemma.

Spectral triples
We come now to the notion of spectral triple, which is central in noncommutative geometry. Recall that [T, D] is called bounded, where D is a regular self adjoint operator on a Hilbert space, and T a bounded operator, if T preserves the domain of D and [T, D] extends to an element of B(H). If H is graded and ϕ is even with respect to the induced grading on B(H), and D is odd, then the spectral triple is called even. If H is ungraded, the spectral triple is called an odd spectral triple.
, we see that it is equivalent to demand a(1 + D 2 ) − 1 /2 ∈ L p or aR i (D) ∈ L p . In [Con08], the topology on the algebra A is reconstructed from the axioms. We go the other way, defining a spectral triple as an "unbounded LC-Kasparov (A, L p )-module" represented on some Hilbert space: continuous p-summable spectral triple is a spectral triple  over a locally convex algebra A such that [D, ϕ(a)] is continuous if viewed as a  function of a with values in the bounded operators, and ϕ(a)(1 + D 2 ) −1/2 , (1 + D 2 ) −1/2 ϕ(a) are continuous as functions of a with values in L p .
We will need a consequence of the Baaj-Julg integral formula for the proof of Proposition 49 below. We set q(x) := x √ 1+x 2 for all x ∈ R. q is called the Woroniwicz transform -just like the Cayley-transform, it associates a bounded operator to an unbounded one, hence it can be used to define a functional calculus for unbounded operators ( [Wor91]). We have the Baaj-Julg formula from [BJ83] Lemma 47. Let D be a regular self adjoint unbounded operator on a Hilbert B-module E. Then as an integral in B(E) Hence for every ξ ∈ E Lemma 48. Let E be a Hilbert module, D a regular self-adjoint operator, a ∈ B(E). If [D, a](1 + D 2 ) ε−1 2 is bounded for some ε > 0, then as a uniformly converging integral of bounded operators.
Here the hypothesis that [D, a](1+D 2 ) ε−1 2 is bounded means that a preserves the domain of D and that [D, a](1 + D 2 ) ε−1 2 is an adjointable operator on Proof. Set T t := (1 + t + D 2 ) −1 , let a ∈ A be homogeneous. Then we have Estimating the norms, we see that for every t for a positive constant C, and similarly for the second term. Thus as an integral with values in the bounded operators.
Note that a linear function ϕ : A → L p is continuous iff a → ϕ(a) * ϕ(a) is continuous into L p/2 at 0.

Proposition 49. If (H, ϕ, D) is a continuous p-summable spectral triple over a locally convex algebra A such that the multiplication A × A → A is surjective and open, then (B(H), ϕ, q(D)) is an
Proof. We set T t := (1 + t + D 2 ) −1 . As A carries the identification topology for the product, it suffices to show that (a, b) → [q(D), ab] is continuous on A × A.
These are norm-convergent integrals in L 2p ; for example: for a constant C. Furthermore, the integrands are continuous functions of a and b, hence the result follows from dominated convergence.

Example: The Dirac element
We assume again that the dimension n is even in order that the grading on the Clifford algebras be inner.
Recall again from [Kas80b] or [Greb] that x n denotes the C * -Kasparov (S n ⊗ C n , C)-module defined by letting S n ⊗ C n act on L 2 -forms on R n and with operator the Dirac operator. The grading on S n ⊗ C n is again inner; and we may therefore pass again to to a C * -Kasparov (S n ⊗ C n , C)-module x ′ n , where S n ⊗ C n is viewed as trivially graded. Denote ι : S n ⊗ C n → S n ⊗ C n the inclusion.
Proof. This follows from Proposition 49 once we have checked the hypotheses. That the commutators are bounded holds even for all smooth functions with bounded derivatives, and continuity of f → [f, D] is obvious. It suffices to show f (1 + D 2 ) −1/2 ∈ L (n,+) for scalar f ; for every function of compact support, the operator f (1 + D 2 ) −1 f * is an element of L (n+1)/2 , being a pseudodifferential operator of order −1 on a compact manifold (see for example [Con88], Theorem 1), hence f (1+ D 2 ) −1/2 ∈ L n+1 . In order to extend this to S, let h be a smooth function of compact support K that is constant 1 on the unit-cube; for α ∈ Z n , we denote by t α the operator of translation by α -an isometry when restricted to L 2 -and set K α := t α (K). Define H := α t α h, and g := h/H; then α t α (g) = 1. As the Dirac operator commutes with translations, for every smooth scalar f and α ∈ Z n : where P α = χ(K α ) is the projection determined by the support of t α (g). Now if f is rapidly decreasing, then it is easily seen that for an appropriate k and a constant C, and therefore which shows continuity of f → f (1 + D 2 ) −1/2 . Alternatively, one may use the integral kernels of the operators and the following fact of independent interest from [Sim05]: If K(x, y) = f (x)g(x − y), then ||T K || p ≤ ||f || p ||g|| p , where T K denotes the operator with kernel K.

Products of Kasparov modules and induced morphisms
For brevity, we will call two quasihomomorphisms (α,ᾱ) and (β,β) composable, if the target of the first is the domain of the second. We say they have a product for a given class of split exact functors if there is a quasihomomorphism (γ,γ) such that for every functor in the class. We use the analogous wordings for abstract Kasparov modules and LC-Kasparov modules. Recall that for a given quasihomomorphism (α,ᾱ) : A ⇉ B B, the algebra D α is defined as the topological vector space A ⊕ B with a twisted multiplication, and that it comes equipped with a canonical inclusion ι of B into D α . Qh(B, ϕ 1 , F 1 ), D α and ι the to α associated algebra and inclusion. Let H be a split exact functor. If there is a Kasparov In particular, (B, ϕ 1 , F 1 ) and (Ĉ, ϕ 2 , F 2 ) then have a product with respect to the class of split exact, M 2 -stable functors.
Proof. This follows from If H is M 2 -stable, we may use an obvious diagonal sum to present the latter.
Theorem 53. Let H be a split exact, L p -stable functor. Then H(x ∞ n ) and H(y ∞ n ) are inverse to one another. More generally, for any locally convex algebra A: Proof. By Theorem 9, H is diffotopy invariant. We first compute H(x ∞ n ) • H(y ∞ n ); denote by (α,ᾱ) the LC-quasihomomorphism associated to x ∞ n and by (β,β) the LC-quasihomomorphism associated to y ∞ n . Then (β,β) and (α,ᾱ) have a product by Corollary 50.
By Proposition 42, it suffices to calculate the index of the corresponding operator. Now the product-quasihomomorphism defines a class in KK-theory. It has index one by Proposition 38, Remark 20, the description of products in [Grea] and Bott periodicity in KK. Thus We now show that H(x ∞ n ) also has a left inverse. We proceed to calculate the product with the morphism induced by as defined in Definition 23.
To unclutter notation, set S := S n ⊗ C n , p := 2(n + 1). Note that σ S ,S (the shift µ ⊗ ν → ν ⊗ µ) is diffotopic to id S ⊗θ, where θ is the tensor product of f (x) → f (−x) with the automorphism of C n determined by − id R . We get, using Proposition 24 Switching the factors in the tensor product (see again Proposition 24), we have Now we apply the diffotopy to replace H(σ S ,S ): Another application of Proposition 24 yields As we have shown above that H(α,ᾱ) • H(β,β) = 1 for any functor, we may apply this to H S := H( · ⊗ π S ) above to deduce that ρ is invertible. Because H(x ∞ n ) has right inverse H(y ∞ n ) by 6, and is left invertible, because ρ is invertible, . If A is any locally convex algebra, we may apply the result to H( · ⊗ π A) to obtain the result in general.
Remark 54. If one does not want to refer to Bott periodicity in KK, one can proceed as follows: Denote D β ⊆ S⊗C n ⊕ C the algebra associated to (β,β) (Lemma 15); then D β acts in a natural way on H for it may be viewed as a subalgebra of C ∞ b (R n )⊗ C n .
As (α,α) extends to D β , we see that (β,β) and (α,α) have a product with respect to H (Proposition 51). By Proposition 42, it suffices to calculate the index of the corresponding operator. Now one may calculate, using the basis of Hermite polynomials, the product directly.
To calculate the product the other way around, one performs the rotation argument as in the proof above. Even though it is not clear that ρ in the above proof has a product, the morphism obtained at the end then has a product by the first half of the proof, and we are done.
If we are willing to use kk L p , then it is therefore possible to use a "shortcut" and avoid the rotation trick as follows. We know that the coefficients kk L p (C, C) are Z (see [CT06]) and that kk L p satisfies Bott periodicity and is M 2 stable, and hence kk L p (S(R n ), S(R n )) = Z. Consequently where 1 denotes the unit in kk(S(R n ), S(R n )), is an idempotent in Z. If it was zero, it would be zero on all of Z = kk(S(R n ), S(R n )), and we would get a contradiction by However, this argument does not carry over to the Thom isomorphism, and already fails for the theory kk alg (of [Cun97]) stabilized only by the smooth compact operators (one may adopt the above arguments to kk alg ).

C ∞ (X)-linearity and a smooth Thom isomorphism
Definition 56. If A is a locally convex algebra and X a closed manifold, then we call A a C ∞ (X)-algebra if there is a homomorphism µ A : C ∞ (X) ⊗ π A → A that endows A with the structure of a left C ∞ (X)-module.
Definition 57. Let A and B be locally convex C ∞ (X)-algebras,B an algebra that carries a left C ∞ (X)-module structure. An LC-quasihomomorphism (α,ᾱ) : A ⇉B B is called C ∞ (X)-linear, if α andᾱ are C ∞ (X)-linear.
With this definition, we have the following Lemma 58. Let (α,ᾱ) : A ⇉B B be a C ∞ (X)-linear LC-quasihomomorphism. Then for every split exact functor We suppose throughout this section that E is an oriented bundle of even dimension.
For any bundle E ։ X we denote by E E the canonical Hilbert C 0 (X)-module of longitudinal differential forms on E that are locally continuous functions in the X-direction with values in the L 2 -forms along the fibres. We denote by h ∈ L 2 (R n ) the function defined as h(ξ) := exp(− ||ξ|| 2 2 ). For every bundle, we get an inclusion Thus we get an embedding We also denote by D E the longitudinal Dirac operator on E E . In order to avoid the choice of an algebra of Schwartz sections, we assume for the rest of this paragraph that the base space is a compact manifold. By S(E) we denote the smooth functions on E that are Schwartz functions along the fibres, and by Γ ∞ (E) the smooth sections of E. We denote S E the space of smooth sections of the longitudinal Clifford bundle of E that are Schwartz in the direction of the fibres. We also denote c + := ε + ε * the fibrewise action of S(E) on E E .
Remark 59. If E = P × O(p) R p is a decomposition of E as an associated bundle for a principal O(p)-bundle P over X obtained by choosing a Riemannian metric, then and S E is thus the analogue of the function space used in [Greb] and [Kas80b].
Using the identification from equation 7, we get the following Definition 60. We define x ∞ E as the LC-Kasparov (S E , S X ⊗ π L p )-module and set (α,ᾱ) := Qh(x ∞ E ). We denote y ∞ E the LC-Kasparov (S X , S E )-module obtained by taking y ∞ n fibrewise, and denote (β,β) the associated quasihomomorphism.
We will use the following factorisation result in the proof of the Thom isomorphism: Lemma 61. Let E, E ′ ։ X be two smooth bundles over a compact manifold X. If ϕ E,E ′ : S E×E ′ → S E⊕E ′ is the restriction of forms on E × E ′ to E ⊕ E ′ , then there is a quasihomomorphism (α ′ ,ᾱ ′ ) such that Proof. This follows from Proposition 21-(i), because (α,ᾱ) ⊗ S E ′ extends to a quasihomomorphism of C * -algebras whose composition with the extension of L p ⊗ ϕ X,E factors over the extension of ϕ E,E ′ .
Theorem 62. Let H be a split exact, L p stable functor, X a closed manifold and E ։ X an orientable real vector bundle of even rank n; let A be a supplementary locally convex algebra. Then there is an isomorphism If E is a spin c -bundle, then H(A ⊗ π S(X)) ≈ H(A ⊗ π S(E)).
Proof. Using the functor H A := H( · ⊗ π A), we may reduce the case for a general algebra A to the case A = C. We also recall that H is diffotopy invariant by Theorem 9.
Let B ∈ K(S E ) denote the class obtained by restricting y ∞ E to C. Then B acts as multiplication as in F X E (B) : H(S X ) → H(S X ⊗ π S E ) by Proposition 28. Combining this with the multiplication S X ⊗ π S E → S E , we obtain a map where we apply Proposition 28 and 31 to see the last inequality.
Using again the rotation trick, but this time fibrewise, we get that H(α,ᾱ) is invertible. More in detail: Thus again H(α,ᾱ) is left and right invertible, hence invertible with inverse H(β,β).
For the second part of the theorem, one applies the Morita context coming from the spin c structure.
Remark 63. As in the case of Bott periodicity, one may also calculate the product more directly: Let (α,ᾱ) := Qh(x ∞ n ) and (β,β) := Qh(y ∞ n ). Then there are obvious fibred versions (γ,γ) := ((α,ᾱ)⊗C n ) P ∞ and (δ,δ) := ((β,β)⊗ C n ) P ∞ of these quasihomomorphisms obtained as in the C * -case. As in the proof of smooth Bott periodicity, (δ,δ) may be replaced with (δ,δ) which is the fibred version of the quasihomomorphism associated to the LC-Kasparov module obtained as before by replacing the operator in the Dirac element byD. Then (δ,δ) extends to (D α⊗id Cn ) P ∞ . Hence (γ,γ) and (δ,δ) have a product. As the equivariant index of their product over a point is one, we get the result.
Using a fibred version of the rotation argument, we see that (δ,δ) and (γ,γ) are inverse to each other.

Applications
We now sketch the possible applications of the results proved in this paper. Details will appear elsewhere.
(i) Much effort has gone into developing an analogue of Kasparov's bivariant K-functor for more general algebras, see for example the papers by Cuntz concerning the functor kk, as well as those of Weidner ([Wei89]), or with a more moderate scope Phillips monovariant K-theory for Frechet algebras ( [Phi91]). The Bott periodicity theorem 53 can be used to develop bivariant K-theories on categories more general than C * -algebras. The recipe is similar to the techiques used by Higson in [Hig90] to construct E-theory. However, it seems necessary to approach the construction from a more "homotopic" viewpoint, and to avoid stabilisations. Essentially, one proceeds as follows: One forms the stable diffotopy category of the category of locally convex algebra, as put forward in [Ros82]. One then inverts a certain class of morphisms, in order to obtain a split exact functor; stabilizing this functor by an appropriate operator ideal yields a stable homotopy invariant split exact functor. In a forthcoming paper, we show that, as a corollary of Theorem 53, this is the universal split exact, homotopy invariant and stable functor.
It seems very unlikely at the moment that this functor coincides with Cuntz's kk, and thus that kk is not the universal split exact functor, as one might expect if it was the analogue of Kasparov's KK.
(ii) The Connes-Thom isomorphism for locally convex algebras: This important result due to Connes ([Con81]) relates the K-theory of a crossed product of a C * -algebras A ⋊ α R n by R n from the K-theory of A. The result was later proved elegantly by Fack and Skandalis in the bivariant setting ( [FS80]). Using the techniques developed so far and the Bott periodicity theorem 53, it is possible to prove an analogous result for a smooth version of such crossed products.
(iii) Further, there here are applications to index theory and pseudodifferential operators. The passsage from locally convex algebras to double split extensions used above may be used to associate elements in bivariant K-theory groups to pseudodifferential operators. Using the Thom isomorphism (Theorem 62), it is now possible to state an index theorem for any split exact functor with certain properties. In particular, this includes the program outlined recently by Cuntz which analyses index-theory in the setting of his functor kk.