NONCOMMUTATIVE TOPOLOGY — HOMOTOPY FUNCTORS AND E-THEORY

KASPER K. S. ANDERSEN & JESPER GRODAL

Contents

Preface 1 Notation and Prerequisites 2 1. Asymptotic morphisms 3 1.1. Definitions and basic properties 3 1.2. Injectivity of asymptotic morphisms 5 1.3. Homotopies 6 1.4. Compositions and products 8 2. The relationship between asymptotic and *-homomorphisms 11 3. Cofibrations 14 4. The Puppe sequence 20 5. The category E 25 References 28

Date: March 13, 1997.

Preface

The main goal of this ‘fagprojekt’ is to explore the bifunctor E of Connes and Higson [17, 16, 24]. The Kasparov bifunctor KK has been known for a long time and is a powerful tool in the hands of the operator algebraist. However it has some drawbacks. First of all its definition is quite abstruse and unintuitive, and secondly it is not half exact in both variables for all (separable) C^{∗}-algebras.

The abstract existence of a universal (homotopy invariant, stable) bifunctor E which does satisfy half exactness for all separable C^{∗}-algebras was first proven by Higson [24], by a general categorical construction.

Higson had earlier characterized KK-theory as being the universal (homotopy invariant, stable) split exact bifunctor on the category of separable C^{∗}-algebras [23]. Thus we get a natural transformation KK → E. Higson also proved that it induces an isomorphism KK(A, B) → E(A, B), when B is (K-) nuclear.

This functor E was later given a concrete realization as homotopy classes of certain ‘almost *-homomorphisms’ (asymptotic morphisms) by Connes and Higson [17, 16]. In fact the functor E is simply given by

E(A, B) = [[ΣA ⊗ 𝕂, ΣB ⊗ 𝕂]]

where Σ is the suspension, 𝕂 the compact operators, and [[−, −]] the set of homotopy classes of asymptotic morphisms.

The homotopy category of the category with objects (say separable and compact) topological spaces and morphisms the asymptotic morphisms corresponds to some shape theory [12, 11, 21, 22, 4, 7]. Thus the asymptotic morphisms have a very concrete geometric interpretation

We will devote Section 1 to examining the properties of these asymptotic morphisms, using as our main references [2, 3, 15, 17, 16, 4, 6].

In Section 2 we establish conditions under which homotopy classes of asymptotic homomorphisms coincide with homotopy classes of *-homomorphisms, mainly following [3, 13].

In order to understand the role of the E bifunctor we are lead to examine the general properties of homotopy functors on the category of C^{∗}-algebras. We introduce the notion of a ‘cofibration half exact’ functor as being the natural generalization of the topological concept of an exact functor. We then show that a functor is half exact precisely if it is cofibration half exact and excisive. An excisive functor on the category of C^{∗}-algebras is one which does not distinguish between the kernel and the ‘homotopy kernel’ (the mapping cone) of a surjective *-homomorphism.

In Section 4 we then proceed to introduce the Puppe sequence associated to a morphism of C^{∗}-algebras mimicking the analogous topological constructions. Here we follow [36, 3, 39]. This allows us to establish the existence of long exact sequences corresponding to various functors.

Finally, in Section 5, we use the results obtained in the previous sections to introduce the bifunctor E and establish its basic properties.

Notation and Prerequisites

In our notation we will follow the conventions of [9], which also represents roughly the prerequisites on C^{∗}-algebras and K-theory which are assumed of the reader.

We employ the convention that all algebraic functors on categories of C^{∗}-algebras are assumed to be homotopy invariant. Furthermore all functors between pointed categories (i.e. categories with a zero object) will be assumed to preserve the zero object.

We will use facts from topology freely—a standard reference is Spanier [41]. It should be noted that we use [−, −] for (unpointed) homotopy classes of unpointed maps.

1. Asymptotic morphisms

In this section we will introduce the concept of an asymptotic morphism between two C^{∗}-algebras, and explore the basic properties.

We give a quick discussion of injectivity and weak injectivity of asymptotic morphisms following Loring [30]. Then we introduce a notion of homotopy equivalence and examine when the set of homotopy classes can be given a group structure. Finally we introduce a composition product on these classes.

1.1. Definitions and basic properties.

Definition 1.1. Let A and B be C^{∗}-algebras. An asymptotic morphism from A to B is a family of (not necessarily continuous) maps φ_{t} : A → B, t ∈ T = [1; ∞[ that satisfy

• For all a ∈ A t ↦ φ_{t}(a) is continuous.

• For all a, b ∈ A and λ ∈ ℂ we have

φ_{t}(a + λb) − φ_{t}(a) − λφ_{t}(b) → 0 for t → ∞
φ_{t}(ab) − φ_{t}(a)φ_{t}(b) → 0 for t → ∞
φ_{t}(a^{∗}) − φ_{t}(a)^{∗} → 0 for t → ∞

Remark 1.2. Asymptotic morphisms generalize *-homomorphisms, since we can consider a *-homomorphism as an asymptotic morphism via φ_{t}(a) =^{def} φ(a), for all t ∈ T.

Lemma 1.3. For an asymptotic morphism (φ_{t}) we have lim sup_{t} ∥φ_{t}(a)∥ ≤ ∥a∥.

Proof. Define the asymptotic morphism φ_{t}^{+} : A^{+} → B^{+} as the unitization of φ_{t}, φ_{t}^{+}(a + λ1) = φ_{t}(a) + λ1. For a unitary u it follows from the definitions that ∥φ_{t}^{+}(u)∥ → 1. Since for any element a we can write \frac{1}{α}a, α > ∥a∥ as a convex combination of unitaries by the Russo-Dye-Gardner theorem, we get that lim sup_{t} ∥φ_{t}(a)∥ ≤ α, so we are done.

Remark 1.4. It follows directly from Lemma 1.3 that we can identify asymptotic morphisms with maps A → C_{b}(T, B) that satisfy

φ_{t}(a + λb) − φ_{t}(a) − λφ_{t}(b), φ_{t}(ab) − φ_{t}(a)φ_{t}(b), φ_{t}(a^{∗}) − φ_{t}(a)^{∗} ∈ C_{0}(T, B)

for all a, b ∈ A, λ ∈ ℂ.

Definition 1.5. Two asymptotic morphisms (φ_{t}) and (ψ_{t}) are said to be equivalent if we for all a have that φ_{t}(a) − ψ_{t}(a) → 0 for t → ∞. We write φ ≅ ψ.

Remark 1.6. Let B_{∞} = C_{b}(T, B)/C_{0}(T, B). Observe that B_{∞} is never separable. To an arbitrary asymptotic morphism φ we get an induced *-homomorphism φ̃ : A → B_{∞}. This gives a bijective correspondence between asymptotic morphisms modulo equivalence and the set of *-homomorphisms A → B_{∞}. Note that we have ∥φ̃(a)∥ = lim sup_{t} ∥φ_{t}(a)∥. In the following we will often identify equivalent asymptotic morphisms.

Remark 1.7. Note that we can embed (C_{b}(T)/C_{0}(T)) ⊗ B in B_{∞}, but in general they will not coincide. This can be seen from the computation

(C_{b}(T)/C_{0}(T)) ⊗ B ≅ C_{b}(T) ⊗ B/(C_{0}(T) ⊗ B)
= C_{b}(T) ⊗ B/C_{0}(T, B)
↪ C_{b}(T, B)/C_{0}(T, B)

In the case B = C_{b}(T) they do not coincide, as can be seen by considering a function in C_{b}(T, C_{b}(T)) = C_{b}(T × T) which is 1 along the diagonal and close to 0 away from it, e.g. f(t_{1}, t_{2}) = e^{−(t_{1}−t_{2})^{2}}.

We conclude this remark by mentioning the following amusing result (which we will not use): If B is a stable C^{∗}-algebra then the canonical morphism B → B_{∞} induces an isomorphism on K-theory. For a proof see Nagy [34].

We have the following theorem of Bartle and Graves about the existence of continuous cross-sections [10] [33, Prop. 7.2].

Theorem 1.8. Let X and Y be Banach spaces, and let π : X → Y be a continuous surjective linear operator. Then there exists a continuous homogeneous map f : Y → X such that πf = id_{Y}.

Note that the lifting f can be chosen homogeneous (i.e. f(λy) = λf(y), y ∈ Y, λ ∈ ℂ) but in general not additive.

Consider the following diagram

[figure: C_{b}(T, B); π; A; φ̃; B_{∞}]

Choose by the Bartle-Graves theorem a cross-section f of the quotient map π : C_{b}(T, B) → B_{∞}. If we put ψ = fφ̃ we thus get a continuous homogeneous map ψ : A → C_{b}(T, B), which is equivalent to φ. An arbitrary asymptotic morphism is thus equivalent with a continuous, homogeneous asymptotic morphism. If A is separable and A or B is nuclear, then the asymptotic morphism can be chosen (completely) positive, linear and contractive. If furthermore A and B are unital, then the asymptotic morphism can be chosen (completely) positive linear and unital (and therefore contractive). This follows directly from the following theorem of Choi and Effros [14].

Theorem 1.9. Let A be separable and let J be an ideal in B. If φ : A → B/J is a completely positive linear contraction and either A, B, or B/J are nuclear, then there exist a completely positive linear contractive lift ψ : A → B.

[figure: B; π; A; ψ; φ; B/J]

If furthermore A, B and φ are unital, then we may also choose ψ to be unital.

1.2. Injectivity of asymptotic morphisms. We will now discuss two concepts of injectivity for asymptotic morphisms following Loring [30]. This subsection relates asymptotic morphisms to deformations and continuous fields of C^{∗}-algebras. Further information may be found in [5, 28]. However the results of this subsection will needed further on.

Definition 1.10. An asymptotic morphism φ is called weakly injective if lim sup_{t} ∥φ_{t}(a)∥ > 0 for all a ∈ A, a ≠ 0. We call φ injective if lim inf_{t} ∥φ_{t}(a)∥ > 0 for all a ∈ A, a ≠ 0.

Remark 1.11. Note that if φ ≅ ψ then φ injective iff ψ is injective. The same holds with injective replaced by weakly injective. It is clear from the definition that an injective asymptotic morphism is weakly injective. The opposite is not true. Let for example A be a nontrivial contractible C^{∗}-algebra, and let (φ_{t}) be an asymptotic morphism consisting of *-homomorphisms φ_{t} : A → A that for t even is the identity and for t odd is the null map. This construction is possible since A is contractible, and it is clearly an asymptotic morphism. This example also shows that an asymptotic morphism in no way has to ’converge’ to anything as t → ∞. It is on the other hand clear that if the (pointwise) limit function exists, i.e. φ_{t}(a) is convergent for each a, then the limit function is a *-homomorphism.

Theorem 1.12. For an asymptotic morphism (φ_{t}) the following conditions are equivalent:

1. φ is weakly injective

2. φ̃ is injective

3. lim sup_{t} ∥φ_{t}(a)∥ = ∥a∥ for all a ∈ A

Proof. This is obvious since ∥φ̃(a)∥ = lim sup_{t} ∥φ_{t}(a)∥, cf. Remark 1.6.

Definition 1.13. By a deformation from A to B we mean a continuous field (G(t)_{t∈I}, Γ) C^{∗}-algebras (see Dixmier [19, §10]) such that G(0) = A, G(t) = B for t ∈]0; 1] and C_{0}(]0; 1], B) ⊆ Γ|_{]0;1]} ⊆ C(]0; 1], B)

Remark 1.14. Let Γ be a deformation from A to B. We can for every a ∈ A find a cross-section α_{a} ∈ Γ with α_{a}(0) = a, cf. [19]. Let φ_{t}(a) = α_{a}(1/t). By this definition φ becomes an injective asymptotic morphism, since ∥φ_{t}(a + λb) − φ(a) − λφ_{t}(b)∥ → ∥a + λb − (a + λb)∥ = 0 because α_{a+λb}(1/t) − α_{a}(1/t) − λα_{b}(1/t) ∈ Γ. The other requirements are shown analogously and the injectivity is obvious. It is clear that φ is independent of the choice of cross-section α_{a} up to equivalence. We can thus talk about the asymptotic morphism associated to a deformation Γ.

Theorem 1.15. For an asymptotic morphism (φ_{t}) : A → B the following conditions are equivalent:

1. φ is injective

2. φ is the asymptotic morphism associated to a deformation Γ from A to B.

3. lim_{t} ∥φ_{t}(a)∥ = ∥a∥ for all a ∈ A

Proof. It is clear that (2) ⇒ (3) ⇒ (1). We now show (3) ⇒ (2). Let Γ consist of the set of cross-sections α ∈ A × ∏_{t∈]0;1]} B, that satisfy that α(0) = a and α(t) − φ_{1/t}(a) ∈ C_{0}(]0; 1], B) for an a ∈ A. By Lemma 1.3 it is clear that C_{0}(]0; 1], B) ⊆ Γ|_{]0;1]} ⊆ C(]0; 1], B), so we just have to show that Γ is a continuous field. Since φ is an asymptotic morphism it is easily seen that Γ is closed under pointwise operations.

Let β ∈ A × ∏_{t∈]0;1]} B be a cross-section, that locally can be approximated by a cross-section in Γ. We have to show that β ∈ Γ. It is clear that β is continuous on ]0; 1]. Now, let ϵ > 0 be given. By the assumption there exist a ∈ A, f ∈ C_{0}(]0; 1], B) such that ∥β(t) − φ_{1/t}(a) − f(t)∥ < ϵ/2 for t > 0 and ∥β(0) − a∥ < ϵ/2. We thus get ∥β(t) − φ_{1/t}(β(0)) − f_{1}(t)∥ < ϵ, for a function f_{1} ∈ C_{0}(]0; 1], B). This gives that β(t) − φ_{1/t}(β(0)) ∈ C_{0}(]0; 1], B), and therefore β ∈ Γ. This shows that Γ is a continuous field, and φ is obviously the asymptotic morphism associated to it.

We conclude the proof by showing (1) ⇒ (3). If lim inf_{t} ∥φ_{t}(a)∥ < ∥a∥, we can choose a sequence t_{n} → ∞ such that sup_{n} ∥φ_{t_{n}}(a)∥ < ∥a∥. Let B̂ = ∏_{n=1}^{∞} B/⊕_{n=1}^{∞} B, and define the morphism φ̂ : A → B̂, by φ̂(a) = (φ_{t_{n}}(a)). We have ∥φ̂(a)∥ ≤ sup_{n} ∥φ_{t_{n}}(a)∥ < ∥a∥, so there exists an x ≠ 0 such that φ̂(x) = 0. This implies that lim inf_{t} ∥φ_{t}(x)∥ ≤ lim inf_{n} ∥φ_{t_{n}}(x)∥ = ∥φ̂(x)∥ = 0, which contradicts the injectivity of φ. Therefore we must have lim inf_{t} ∥φ_{t}(a)∥ ≥ ∥a∥, i.e. lim_{t} ∥φ_{t}(a)∥ = ∥a∥.

1.3. Homotopies. We remind the reader that a pair of *-homomorphisms φ, ψ : A → B is called homotopic, if there exists a *-homomorphism F : A → IB such that F (a)(0) = φ(a) and F (a)(1) = ψ(a). The set of equivalence classes is denoted [A, B]. In case of commutative unital C^{∗}-algebras we have [C(X), C(Y)] = [Y, X], when X and Y are compact Hausdorff, which is a direct consequence of the Gelfand transform and the exponential lemma.

We now make the following analogous definition in the case of asymptotic morphisms.

Definition 1.16. Two asymptotic morphisms (φ_{t}), (ψ_{t}) : A → B are said to be homotopic, if there exists an asymptotic morphism F : A → IB such that F_{t}(a)(0) = φ_{t}(a) and F_{t}(a)(1) = ψ_{t}(a) for all a ∈ A, t ∈ T. This is obviously an equivalence relation. We write φ ≃ ψ. The set of equivalence classes of asymptotic morphisms is denoted [[A, B]]. The equivalence classes are denoted [[φ]].

We denote by Sep-C^{∗} the full subcategory of C^{∗} with objects the separable C^{∗}-algebras. We then have the quotient category Ho(Sep-C^{∗}) of C^{∗} where the morphisms are homotopy classes of *-homomorphisms. We will later show that it is possible to define a composition of homotopy classes of asymptotic morphisms between separable C^{∗}-algebras. We can therefore define a category Asymp whose objects are all separable C^{∗}-algebras and whose morphisms are homotopy classes of asymptotic morphisms.

Remark 1.17. Since two *-homomorphisms φ, ψ : A → B which are homotopic as *-homomorphisms are also homotopic as asymptotic morphisms, we can define a canonical map c : [A, B] → [[A, B]]. This map is in general neither surjective nor injective. For example consider the Warsaw circle W. We show later that C(W) and C(S^{1}) are equivalent in Asymp (cf. also Dădărlat [3, cor. 19]). This shows that [[C(W), C(S^{1})]] = [[C(S^{1}), C(S^{1})]]. We will later show that in the case when X is metrizable ANR and X, Y are locally compact Hausdorff we have [[C_{0}(X), C_{0}(Y)]] = [C_{0}(X), C_{0}(Y)], cf. Corollary 2.6. Thus we conclude that [[C(W), C(S^{1})]] = [S^{1}, S^{1}] = [S^{1}, S^{1}]_{pt} = π_{1}(S^{1}) = ℤ. On the other hand [C(W), C(S^{1})] = π_{1}(W) = 0. This shows that c is not surjective in general. A variation over this theme can likewise be used to give an example where c is not injective. Take the topologist’s sine curve V (i.e. the Warsaw circle before it’s joined together). As before we have that C(V) is equivalent with C(pt) = ℂ in Asymp so that [[C(V), C(pt)]] = 0 whereas [C(V), C(pt)] = π_{0}(V) is a set with two points.

These examples show that the concept of asymptotic morphisms is really a new one, even when restricting to the subcategory of (locally compact Hausdorff) topological spaces. In fact the homotopy category of the full subcategory of Asymp having as objects locally compact Hausdorff spaces is equivalent to some strong shape category (cf. [4, 7, 11, 12, 21, 22]).

Remark 1.18. If φ, ψ are equivalent then they are also homotopic, since we can use the homotopy F_{t}(a)(s) = (1 − s)φ_{t}(a) + sψ_{t}(a). The converse is of course not true. If B is contractible then [A, B] = [[A, B]] = 0. For a contractible C^{∗}-algebra B we therefore have that id_{B} and 0 are homotopic (both as asymptotic and as ordinary morphisms), but they are obviously not equivalent unless B is trivial.

Furthermore notice that if φ is an asymptotic morphism, such that φ_{t} is an *-homomorphism for all t ∈ T, then φ is homotopic to φ_{t_{0}} for every t_{0} ∈ T via the homotopy F_{t}(a)(s) = φ_{st_{0}+(1−s)t}(a). Said in another way, the asymptotic morphisms which are *-homomorphisms for all t are in the image of the canonical map c.

Proposition 1.19. An asymptotic morphism φ : A → B induces homomorphisms φ_{∗} : K_{∗}(A) → K_{∗}(B) in K-theory. Homotopic asymptotic morphisms induce the same homomorphisms.

Proof. It enough to prove the case of K_{0}, since K_{1} = K_{0}Σ. Let p ∈ 𝕄_{∞}(A^{+}) be a projection, then for large t, φ_{t}^{+}(p) is close to the selfadjoint (φ_{t}^{+}(p) + φ_{t}^{+}(p)^{∗})/2, which is again close to the projection q_{t} = f((φ_{t}^{+}(p) + φ_{t}^{+}(p)^{∗})/2), where f(x) = (1 + sign(x − \frac{1}{2}))/2 which is continuous on the spectra for large t. We define φ_{∗}^{+}([p]) = [q_{t}]. This is really well defined since by construction all the projections q_{t} are homotopic. Now define φ_{∗} as the restriction to K_{0}(A) of φ_{∗}^{+}.

The last part of the proposition follows by using the fact that K_{∗}(IB) ≃ K_{∗}(B).

Theorem 1.20. If for two asymptotic morphisms φ, ψ, the *-homomorphisms φ̃ and ψ̃ : A → B_{∞} are homotopic then φ, ψ are homotopic as asymptotic morphisms.

Proof. Note that a homotopy of *-homomorphisms A → B_{∞} is a *-homomorphism A → I(B_{∞}), and that a homotopy of asymptotic morphisms A → B is a *- homomorphism A → (IB)_{∞}. Therefore we just have to prove that we have a natural inclusion I(B_{∞}) ↪ (IB)_{∞} such that the diagram

[figure: I(B_{∞}); e_{s}; (IB)_{∞}; e_{s}; B_{∞}]

commutes. We have

I(B_{∞}) ≅ C(I) ⊗ (C_{b}(T, B)/C_{0}(T, B))
= (C(I) ⊗ C_{b}(T, B))/(C(I) ⊗ C_{0}(T, B))
= C(I, C_{b}(T, B))/C(I, C_{0}(T, B))
↪ C_{b}(I × T, B)/C_{0}(I × T, B)
= C_{b}(T, IB)/C_{0}(T, IB)
= (IB)_{∞}

where the first equality follows because C([0; 1]) is nuclear, and therefore exact.

Remark 1.21. The C^{∗}-algebras I(B_{∞}) and (IB)_{∞} are in general not naturally equivalent, even for B = ℂ. This is because the inclusion C(I, C_{b}(T)) ↪ C_{b}(I × T) is not an isomorphism. To see this note that the function

f(s, t) = sin(st) , (s, t) ∈ I × T

gives an example of a continuous bounded function of two variables, which is not continuous as a function I → C_{b}(T, B).

1.4. Compositions and products.

Proposition 1.22. Let A, B be two C^{∗}-algebras. If B is a suspension then [[A, B]] is a group under the loop composition. If B is stable then [[A, B]] is an abelian monoid where the composition of φ_{t} and ψ_{t} is given by

A →^{(φ_{t},ψ_{t})} B ⊕ B → 𝕄_{2}(B) →^{≃} B.

If B is a double suspension or a stable suspension then [[A, B]] is an abelian group and the two possible compositions coincide.

Proof. The claim in the case where B is a suspension or a double suspension is totally analogous to the proof that the fundamental group actually is a group, and that the higher homotopy groups are abelian (for a proof of these facts see for example [8]).

We now look at the case where B is stable. It is clear that the composition is well defined.

Keeping your identifications straight, associativity is no problem. To show commutativity note that the multiplier algebra, M(B), of a stable algebra, B, is again stable. Commutativity is now clear, since conjugation by the path of unitaries

s → [ \begin{matrix}cos(πs/2) & − sin(πs/2) \\ sin(πs/2) & cos(πs/2) \end{matrix} ] ∈ 𝕄_{2}(M(B)), s ∈ I

gives a homotopy between the diagonal matrices diag(b_{1}, b_{2}) and diag(b_{2}, b_{1}) in 𝕄_{2}(B).

The zero map is a unit since if B is stable the multiplier algebra M(B) contains a copy of 𝔹 (since we have that 𝔹 = M(𝕂) ↪ M(𝕂 ⊗ M(B)) = M(B)). Hence we can choose an isometry u ∈ 𝕄_{2}(𝔹) ⊆ 𝕄_{2}(M(B)) such that

u^{∗} [ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ] u = [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] .

We can use conjugation by u to identify B with 𝕄_{2}(B). Since 𝕄_{2}(𝔹) is a von Neumann algebra, we can choose a continuous path of unitaries u_{s} ∈ 𝕄_{2}(𝔹) such that u_{0} = I and u_{1} = u. Then s ↦ u_{s}φ_{t}(a)u_{s}^{∗} ∈ 𝕄_{2}(B) is a homotopy between

φ_{t}(a) ∈ 𝕄_{2}(B) and [ \begin{matrix}φ_{t}(a) & 0 \\ 0 & 0 \end{matrix} ] ∈ 𝕄_{2}(B).

The last part of the proposition follows from a general categorical argument. In both cases we have two compositions which have a common neutral element, namely the zero map, and are mutually distributive. This assures that the compositions coincide and are commutative, cf. [41, Thm. 1.6.8]. Therefore in these cases [[A, B]] is an abelian group.

We now want to define a product [[A, B]] × [[B, C]] → [[A, C]] when A, B and C are separable C^{∗}-algebras. So we consider to asymptotic morphisms φ : A → B and ψ : B → C. By replacing φ and ψ with equivalent asymptotic morphisms we can assume that φ and ψ are continuous and homogeneous. Choose a dense, σ-compact *-subalgebra A′ of A, e.g. by taking A′ to be all *-polynomials in a dense subset of A. Now choose a sequence of compact, selfadjoint, symmetric sets K_{n} such that K_{n} ⊆ K_{n+1}, K_{n} + K_{n} ⊆ K_{n+1} and K_{n}K_{n} ⊆ K_{n+1}, ∪_{n}K_{n} = A′. The sets {φ(a + b) − φ(a) − φ(b)|a, b ∈ K_{n}}, {φ(ab) − φ(a)φ(b)|a, b ∈ K_{n}} and {φ(a^{∗})−φ(a)^{∗}|a ∈ K_{n}} are compact subsets of C_{0}(T, B) so we can pick an increasing sequence t_{n} ∈ T such that for t ≥ t_{n} and a, b ∈ K_{n} one has:

∥φ_{t}(a + b) − φ_{t}(a) − φ_{t}(b)∥ < 1/n
∥φ_{t}(ab) − φ_{t}(a)φ_{t}(b)∥ < 1/n
∥φ_{t}(a^{∗}) − φ_{t}(a)^{∗}∥ < 1/n
∥φ_{t}(a)∥ < ∥a∥ + 1/n

Set L_{n} = {φ_{t}(a)|a ∈ K_{n+2}, t ≤ t_{n+3}}. By ’fattening’ L_{n} we can find compact, selfadjoint, symmetric sets L′_{n} satisfying L_{n} ⊆ L′_{n}, L′_{n} ⊆ L′_{n+1}, L′_{n} + L′_{n} ⊆ L′_{n+1} and L′_{n}L′_{n} ⊆ L′_{n+1}. Choose, as before, an increasing sequence r_{n} ∈ T such that for r ≥ r_{n} and a, b ∈ L′_{n} one has:

∥ψ_{r}(a + b) − ψ_{r}(a) − ψ_{r}(b)∥ < 1/n
∥ψ_{r}(ab) − ψ_{r}(a)ψ_{r}(b)∥ < 1/n
∥ψ_{r}(a^{∗}) − ψ_{r}(a)^{∗}∥ < 1/n
∥ψ_{r}(a)∥ < ∥a∥ + 1/n

Choose now a continuous, increasing function r : T → T, such that r(t_{n}) ≥ r_{n}. Define (θ′_{t}) : A′ → C by θ′_{t}(a) = (ψ_{r(t)} ∘ φ_{t})(a). For a, b ∈ K_{n} and t_{n} ≤ t ≤ t_{n+1} we obtain φ_{t}(a), φ_{t}(b) ∈ L_{n−2}, φ_{t}(a + b), φ_{t}(a) + φ_{t}(b) ∈ L_{n−1} and φ_{t}(a + b) − φ_{t}(a) − φ_{t}(b) ∈ L_{n}. Therefore we obtain:

∥θ′_{t}(a + b) − θ′_{t}(a) − θ′_{t}(b)∥ ≤ ∥ψ_{r(t)}(φ_{t}(a + b)) − ψ_{r(t)}(φ_{t}(a) + φ_{t}(b))∥
+ ∥ψ_{r(t)}(φ_{t}(a) + φ_{t}(b)) − ψ_{r(t)}(φ_{t}(a)) − ψ_{r(t)}(φ_{t}(b))∥
< ∥ψ_{r(t)}(φ_{t}(a + b) − (φ_{t}(a) + φ_{t}(b))∥ + 1/n + 1/n
< ∥φ_{t}(a + b) − (φ_{t}(a) + φ_{t}(b))∥ + 1/n + 2/n
< 4/n

Similarly we get ∥θ′_{t}(ab)−θ_{t}(a)′θ′_{t}(b)∥ < 4/n, ∥θ′_{t}(a^{∗})−θ′_{t}(a)^{∗}∥ < 4/n and ∥θ′_{t}(a)∥ < ∥a∥ + 2/n. This shows that θ′ is an asymptotic morphism from the dense subset A′ to C with lim sup_{t} ∥θ′_{t}(a)∥ ≤ ∥a∥. This induces a *-homomorphism θ̃′ : A′ → C_{∞}, which by continuity extends to a *-homomorphism θ̃ : A → C_{∞}. Now let (θ_{t}) : A → C be a lift of θ̃. We define [[φ]] × [[ψ]] = [[θ]].

Theorem 1.23. The map × : [[A, B]] × [[B, C]] → [[A, C]] is a well defined associative product.

Proof. This is a bit tedious since there is quite a number of things to be checked. We leave it as an exercise to the diligent reader. The reader might seek some help in [16].

Remark 1.24. Note that this makes [[−, −]] into a bifunctor Asymp×Asymp → Set_{*}, the category of pointed sets. If C is stable or a suspension then for a fixed element [[φ]] ∈ [[A, B]] it is easy to see that the map [[φ]]× : [[B, C]] → [[A, C]] is a homomorphism since all group operations are performed on C.

However, even if both [[A, B]] and [[A, C]] are groups the map given by composition with a fixed element [[ψ]] ∈ [[B, C]] might not be a group homomorphism, unless the map ψ preserves the structure (e.g. both B and C being suspensions and ψ being a suspension map). In fact composition with [[ψ]] need not be a homomorphism even in the commutative case. Consider for instance the case where A = C(S^{1} ∨S^{1}), B = C = C(S^{1}). Then [[A, B]] = [[A, C]] = π_{1}(S^{1} ∨S^{1}), which is a free group on two generators. Choosing a degree 2 map S^{1} → S^{1}, the induced map is squaring, which is obviously not a group homomorphism. This counterexample is just due to the fact that the degree 2 map S^{1} → S^{1} is not a suspension. More elaborate counterexamples involving only abelian groups may also be obtained. This corrects a small mistake in [36], where it is claimed that the composition is bilinear, whenever this makes sense.

2. The relationship between asymptotic and *-homomorphisms

We will in this section examine under which conditions the map

c : [−, −] → [[−, −]]

is a bijection. We have already seen examples in Section 1 that this is not always the case. We will however show that this will be true under certain ‘niceness’ assumptions on the C^{∗}-algebra which we are mapping out of. Our treatment will roughly follow the one given in [3].

We first show how the an asymptotic morphism between commutative C^{∗}- algebras also can be seen to be approaching in the geometric sense of the word.

Lemma 2.1. Let Y be a locally compact space, and let φ be a positive, linear, contractive asymptotic morphism A → C_{0}(Y). For every open neighborhood V of Hom(A, ℂ) in F, where F is a weak* closed subset of the unit ball in A^{∗}, there exists a t_{0} such that f(t, y) = φ(·)(t)(y) ∈ V for all t ≥ t_{0} and y ∈ Y.

Proof. Assume that there exists a sequence (t_{i}, y_{i}) with t_{i} → ∞ such that f(t_{i}, y_{i}) ∉ V. Because F \ V is weak* compact there is a subnet such that f(t_{λ}, y_{λ}) → h in the weak* topology. It is obvious that h is a *-homomorphism, which contradicts the fact that V is an open neighborhood of Hom(A, ℂ).

Theorem 2.2. Let X, Y be compact spaces, and let M(X) denote the set of probability measures on X. For every continuous map f : T × Y → M(X) such that f_{t}(Y) approaches X in the weak* topology, defining φ_{t}(a)(y) = f(t, y)(a) yields an asymptotic morphism C(X) → C(Y). If X is metrizable every asymptotic morphism C(X) → C(Y) is obtained, up to equivalence, in this way.

Proof. We identify M(X) with the states on C(X), and X is identified with the character space of C(X), i.e. the pure states. To show that φ is an asymptotic morphism it suffices to show that φ is asymptotically multiplicative, because φ_{t} is linear and *-preserving for every t. Thus let a, b ∈ C(X), and ϵ > 0 be given. Let V = {μ ∈ S(C(X))| |μ(ab) − μ(a)μ(b)| < ϵ}. According to the hypothesis there exists a t_{0} ≥ 1 such that f_{t}(y) ∈ V for all t ≥ t_{0}, y ∈ Y. This implies that |φ_{t}(ab)(y) − φ_{t}(a)(y)φ_{t}(b)(y)| < ϵ for all t ≥ t_{0}, y ∈ Y, so φ is an asymptotic morphism.

Consider now any asymptotic morphism φ : C(X) → C(Y). The C^{∗}-algebra C(X) is separable and nuclear, so by replacing φ by an equivalent morphism, we can assume that φ is unital, positive and linear, cf. Theorem 1.9. Setting f(t, y)(a) = φ_{t}(a)(y) gives a continuous map T × Y → M(X). According to Lemma 2.1, for any neighborhood V of X there is a t_{0} such that f(t, y) ∈ V for all t ≥ t_{0}, y ∈ Y, which was to be shown.

We now turn to establishing conditions under which the canonical map c is bijective. We obtain that

Theorem 2.3. Let A be a separable C^{∗}-algebra, such that Hom(A, ℂ) is ANR, and let Y be a locally compact space. If φ : A → C_{0}(Y) is an asymptotic morphism there is an asymptotic morphism ψ equivalent to φ, such that ψ_{t} is a *-homomorphism for every t ∈ T.

Proof. Because A is separable and C_{b}(T, C_{0}(Y)) is nuclear, we can assume by the Choi-Effros theorem that φ is positive, linear and contractive. Define the continuous map f : T × Y → Pos(A) by f(t, y)(a) = φ_{t}(a)(y), where Pos(A) denotes the positive, contractive functionals on A. Due to the facts that Hom(A, ℂ) is closed in Pos(A), Pos(A) is normal (it is compact and Hausdorff) and Hom(A, ℂ) is ANR we can find a neighborhood U of Hom(A, ℂ) in Pos(A) and a retract r : U → Hom(A, ℂ). Because Pos(A) is a closed subset of the unit ball in A^{∗}, Lemma 2.1 shows that for some t_{0} ∈ T one has f(t, y) ∈ U for all t ≥ t_{0}, y ∈ Y. Now define a continuous map g : T ×Y → Hom(A, ℂ) by g(t, y) = r(f(t, y)) for t ≥ t_{0} and g(t, y) = r(f(t_{0}, y)) for t ≤ t_{0}. We now want to define ψ ∈ Hom(A, C_{b}(T, C_{0}(Y))). Set ψ(a)(t)(y) = r(f(t, y))(a). It is obvious that ψ ∈ Hom(A, C_{b}(T, C_{b}(Y))), because ∥ψ(a)∥ ≤ ∥a∥. If y → ∞ in Y^{+} we have f(t, y) → 0 in the weak* topology, and therefore r(f(t, y))(a) → 0 for every t ∈ T, a ∈ A. This shows that ψ ∈ Hom(A, C_{b}(T, C_{0}(Y))). We now show that ψ is equivalent to φ, e.g.

lim_{t→∞} sup_{y∈Y} ∥f(t, y)(a) − g(t, y)(a)∥ = 0

for all a ∈ A. Now A is separable so the topology on Pos(A) is induced by the metric d(σ, τ) = ∑_{k} 2^{−k}∥σ(a_{k}) − τ(a_{k})∥, where (a_{k}) is a dense subset of the unit ball of A. This gives the estimate ∥σ(a) − τ(a)∥ ≤ C_{a}d(σ, τ) for all a ∈ A, τ, σ ∈ Pos(A). It thus suffices to show that

lim_{t→∞} sup_{y∈Y} d(f(t, y), g(t, y)) = 0

So consider an arbitrary ϵ > 0. Set V = {σ ∈ U|d(σ, r(σ)) < ϵ}. According to lemma 2.1 for some t_{1} ≥ t_{0} we have f(t, y) ∈ V for all t ≥ t_{1}, y ∈ Y. This implies that d(f(t, y), g(t, y)) < ϵ for all t ≥ t_{1}, y ∈ Y, which shows the claim.

Corollary 2.4. Let A be a separable C^{∗}-algebra such that Hom(A, ℂ) is ANR, and let Y be a locally compact space. Then the canonical map [A, C_{0}(Y)] → [[A, C_{0}(Y)]] is bijective.

Proof. Let φ : A → C_{0}(Y) be an asymptotic morphism. According to Theorem 2.3, φ is equivalent to an asymptotic morphism ψ such that ψ_{t} is a *-homomorphism for every t. Hence it is in turn equivalent to ψ_{1}, cf. Remark 1.17. This shows the surjectivity.

Consider now *-homomorphisms φ, ψ : A → C_{0}(Y), that are homotopic as asymptotic morphisms, and let F : A → IC_{0}(Y) be an asymptotic morphism between these. Due to the fact that IC_{0}(Y) = C_{0}(I × Y) we can according to Theorem 2.3 find an asymptotic morphism G : A → IC_{0}(Y) consisting of *-homomorphisms, that is equivalent to F. Now since G is a retract of F for large t, cf. the proof of Theorem 2.3, there exists a t_{0} such that G_{t_{0}} : A → IC_{0}(Y) is a homotopy between φ and ψ as *-homomorphisms.

Remark 2.5. Theorem 2.3 and Corollary 2.4 can be generalized a little. In fact the statements also hold if Hom(A, ℂ) and C_{0}(Y) is replaced by respectively Hom(A, F) and C_{0}(Y) ⊗ F, where F is a finite dimensional C^{∗}-algebra. In the proofs of Lemma 2.1 and Theorem 2.3 the fact that we are mapping into ℂ is only used to conclude the compactness (and metrizability for A separable) of the unit ball of 𝔹(A, ℂ) = A^{∗} in the weak* topology. These properties also holds for the unit ball of 𝔹(A, F) = (A^{∗})^{N}, where N = dim(F).

Corollary 2.6. Let X be a locally compact, metrizable space and let Y be a locally compact space. If X is ANR then the canonical map [C_{0}(X), C_{0}(Y)] → [[C_{0}(X), C_{0}(Y)]] is bijective.

Corollary 2.7. Let X be a locally compact, metrizable space and let Y be a locally compact space. If X is ANR then the canonical map [C_{0}(X) ⊗ F_{1}, C_{0}(Y) ⊗ F_{2}] → [[C_{0}(X) ⊗ F_{1}, C_{0}(Y) ⊗ F_{2}]] is bijective where F_{1} and F_{2} are finite dimensional C^{∗}- algebras.

Sketch of Proof. By Corollary 2.4 and Remark 2.5 it is enough to prove that Hom(C(X) ⊗ F_{1}, F_{2}) is ANR. This is fairly routine point-set topology (cf. [13]).

Definition 2.8. Following Blackadar [12] we define semiprojectivity as follows. Let A be a separable C^{∗}-algebra, and let B be a separable C^{∗}-algebra with an ideal J such that J = ¯ ∪_{n}J_{n}, where J_{n} is an increasing sequence of ideals. We say that A is semiprojective if for all such B, J and all *-homomorphisms φ : A → B/J there exists an n and a *-homomorphism φ̃ : A → B/J_{n} such that the following diagram commutes:

[figure: B/J_{n}; A; φ̃; φ; B/J]

Remark 2.9. There exists another definition of semiprojectivity due to Effros and Kaminker, cf. [21].

Theorem 2.10. If A is semiprojective then the canonical map c : [A, B] → [[A, B]] is bijective for B arbitrary.

Proof. Let φ : A → B be an arbitrary morphism. This then in turn corresponds to a *-homomorphism A → B_{∞} = C_{b}(T, B)/C_{0}(T, B). Since C_{0}(T, B) = \overline{∪_{n}C_{0}([0; n[, B)} we can, by the semiprojectivity of A lift φ to a *-homomorphism A → C_{b}(T, B)/C_{0}([0; n[, B), which obviously can be lifted to A → C_{b}(T, B). Thus we can choose a representative of φ ∈ [[A, B]] which is a *-homomorphism by Remark 1.18, and thus in the image of c.

To prove injectivity let φ, ψ ∈ [A, B] and let F : A → (IB)_{∞} be a homotopy joining φ and ψ. By the semiprojectivity of A we can as before lift this to *- homomorphism F̃ : A → C_{b}(T, IB). Define a *-homomorphisms F̂ : A → IB by F̂(a) = F̃(a)(1), φ̂, ψ̂ : A → B by φ̂(a) = F̂(a)(0) and ψ̂(a) = F̂(a)(1). By construction, φ̂ and ψ̂ are homotopic as *-homomorphisms. To end the proof we want to see that φ and φ̂ are homotopic (the case ψ ≃ ψ̂ is analogous). First note that φ(a) = \underrightarrow{lim}_{t→∞}F̃(a)(t)(0), which we by an abuse of notation will write F̃(a)(∞)(0). Now note that G : A → IB given by G(a)(s) = F̃(a)(\frac{1}{s})(0) defines a continuous homotopy between φ and φ̂.

Remark 2.11. A good reference on semiprojectivity is [31]. From there we have the following results. A finitely presented universal C^{∗}-algebra is semiprojective iff the relations are stable. This implies that the C^{∗}-algebras C_{0}(ℝ), ℂ, C(S^{1}), qℂ = {f ∈ C_{0}(]0; 1], 𝕄_{2})|f(1) ∈ ℂ^{2}} and 𝕀_{n} = {f ∈ C_{0}(]0; 1], 𝕄_{n})|f(1) ∈ ℂI} are all semiprojective. However the C^{∗}-algebras C_{0}(ℝ^{2}) and C_{0}(I^{2} \ {(0, 0)}) are not semiprojective.

3. Cofibrations

Fibrations and cofibrations play a central role in topology, as the maps for which it is ‘homotopically acceptable’ to take respectively fibers and cofibers. Below we will give a definition of what is usually called cofibrations in C^{∗}-algebra theory. These will have the property that f : X → Y is a cofibration of topological spaces iff f^{∗} : C(Y) → C(X) is a cofibration of C^{∗}-algebras. In the language of closed model categories due to Quillen, (cf. [35, 20]) they constitute what would be called a fibration, since they have the same lifting properties. This inconsistency in the notation is due to the fact that the functor C(−) from Top to C^{∗} is contravariant.

Unfortunately, it is not possible to make the category of C^{∗}-algebras into a closed model category with the choice of these cofibrations as fibrations and (weak) equivalences being homotopy equivalences.

In this section we will examine the properties of these cofibrations and explore the relationship to properties of homotopy functors.

Definition 3.1. We say that a *-homomorphism q : A → B is a cofibration if it has the following homotopy lifting property: Given any C^{∗}-algebra C with a *- homomorphism p : C → B, a *-homomorphism p̃: C → A lifting p

[figure: A; q; C; p; p̃; B]

and a homotopy H : C → IB such that H(c, 0) = p(c) for all c then there exists a homotopy H̃ : C → IA such that H̃(c, 0) = p̃(c) and such that the following diagram

commutes [figure: IA; Iq; C; H; H̃; IB]

Proposition 3.2. A *-homomorphism q : A → B is a cofibration iff it has the homotopy lifting property with respect to I_{q}.

Proof. This is in analogy to the commutative case. Consider the diagram

[figure: A; q; IA; e_{0}; Iq; I_{q}; C; p; B; IB; e_{0}]

where the existence of the morphism I_{q} → IA follows by the assumed lifting property with respect to I_{q} and the morphism C → I_{q} comes from the construction of I_{q} as a pullback.

Remark 3.3. In terms of diagrams, this condition is the same as the condition for a map to be a fibration between topological spaces. However it is analogous to the topological cofibration since the functor from (compact, Hausdorff) spaces to C^{∗}-algebras is contravariant. That is, we have the following result: f : X → Y is a cofibration in the category of compact Hausdorff spaces iff C(Y) →^{f^{∗}} C(X) is a cofibration in the category of C^{∗}-algebras. This is seen by using the preceding proposition and considering the following two dual diagrams:

[figure: Y × 0; Y × I; C(Y); f^{∗}; IC(Y); If^{∗}; I_{f}; I_{f^{∗}}; X × 0; f; X × I; f×I; C(X); IC(X)]

where we have used the fact that C(I_{f}) = I_{f^{∗}} as described in [9].

Remark 3.4. Let q : A → B be a cofibration. We want to show that q is surjective. Let b be arbitrary and let h ∈ IB be a path from 0 to b.

[figure: A; q; IA; e_{0}; Iq; CB; 0; 0; B; IB; e_{0}]

The homotopy lifting property gives a h̃ ∈ IA such that Iq(h̃) = h. But then q(h̃(1)) = h(1) = b which shows that q is surjective.

Remark 3.5. It is easily seen that the pullback of a cofibration is a cofibration and that the composition of cofibrations again is a cofibration.

Theorem 3.6. For a *-homomorphism p : A → B we have a commutative diagram

[figure: C_{p}; ι_{2}; A; ι_{1}; p; B; Cp̃; I_{p}; p̃; B]

where p̃ is a cofibration and the vertical arrows are homotopy equivalences. In particular we have that every *-homomorphism can be factored as a composition of an inclusion which is a homotopy equivalence and a cofibration.

Proof. We define p̃ by p̃(a ⊕ f) = f(0), and let ι_{1} and ι_{2} be the canonical inclusions given by ι_{1}(a) = a ⊕ (s ↦ p(a)) and ι_{2}(a ⊕ g) = a ⊕ (s ↦ p(a)) ⊕ g. It is clear that these maps are well defined and make the diagram commutative. It is also easily checked that ι_{1} and ι_{2} are homotopy equivalences. It remains to check that p̃ is a cofibration. Let f be a *-homomorphism f : C → B and let f̃ be a lifting C → I_{p} of f. Furthermore, let F : C → IB be a homotopy such that F_{0} = f. We want to construct a *-homomorphism F̃ : C → II_{p} such that F̃_{0} = f̃ and that (Ip̃)F̃ = F. We write F̃ = G ⊕ H where G : C → IA and H : C → IIB. Considering the constraint F̃_{0} = f̃ we put G(c)(t) = π_{A}(f̃(c)). This gives the following constraints on H in order for F̃(c) to be an element of II_{p} and F̃ to be a lifting of F: H_{0}(c)(t) = π_{IB}(f̃(c))(t), H_{s}(c)(1) = π_{IB}(f̃(c))(1) and H_{s}(c)(0) = F_{s}(c). This defines H(c) on three of the four sides of the unit square. By choosing an extension of H(c) to the unit square, in the same way for all c we obtain the desired homotopy F̃. One choice of H is given by:

H_{s}(c)(t) = \{ \begin{matrix}π_{IB}(f̃(c))(\frac{2t−s}{2−s}) & for s ≤ 2t \\ F (c)(s − 2t) & for s ≥ 2t \end{matrix}

Remark 3.7. Note suspension of A is exactly the mapping cone of the morphism 0 : 0 → A, i.e. the kernel of the map I_{0} → A. In fact choosing another factorization would still yield a kernel homotopy equivalent to ΣA.

Remark 3.8. The above Theorem 3.6 is one of the axioms in a closed model category (cf. [35, 20]). Such a structure is uniquely determined by giving the (weak) equivalences and fibrations. However it is not possible to impose such a structure on C^{∗} choosing as (weak) equivalences the homotopy equivalences and as fibrations the cofibrations. If these choices constituted a closed model category structure, the suspension functor should have an adjoint in the homotopy category, given by formally inverting the weak equivalences. This would especially imply that the suspension would commute with arbitrary products, i.e. that the natural map

Σ ∏_{I} A_{i} → ∏_{I} ΣA_{i}

would be a homotopy equivalence.

However taking I = ℕ, A_{i} = ℂ gives an example where this is not the case. This is because the induced map on K_{1} can be seen not to be surjective (ref or expansion????).

Definition 3.9. A short exact sequence 0 → J → A →^{p} B → 0 is called excisive if the canonical map i : J → C_{p} is a homotopy equivalence.

Proposition 3.10. If q : A → B is a cofibration then 0 → J → A →^{q} B → 0 is excisive.

Proof. We have a commutative diagram

[figure: A; q; I_{q}; π_{A}; e_{1}∘π_{IB}; B]

Thus we can lift the projection π_{IB} : I_{q} → IB to a *-homomorphism η : I_{q} → IA such that the following diagram commutes

[figure: IA; Iq; I_{q}; η; π_{IB}; IB]

Let κ denote the composition C_{q} ↪ I_{q} →^{η} IA →^{e_{0}} A, and note that this *- homomorphism actually maps into J. We claim that κ is a homotopy inverse of i : J → C_{q}. First note that the *-homomorphism F : J →^{i} C_{q} ↪ I_{q} →^{η} IA maps into IJ and is a homotopy between F_{0} = κ ∘ i and the identity. Next consider the map G : C_{q} → IA ⊕ ICB given by G(a ⊕ f) = η(a ⊕ f) ⊕ f̃ where f̃ is given by f̃(s, t) = f(min(s, t)). Since f̃(s, 1) = f(s) = Iq(η(a ⊕ f))(s) = q(η(a ⊕ f)(s)) we have that G actually maps into IC_{q}. Finally note that G_{1} = id_{C_{q}} and G_{0}(a ⊕ f) = η(a ⊕ f)(0) ⊕ 0 = i ∘ κ(a ⊕ f). Thus i is a homotopy equivalence.

Proposition 3.10 shows that for ‘nice’ surjections, i.e. cofibrations, the mapping cone coincides with the kernel, up to homotopy. Combining Proposition 3.10 and Theorem 3.6 shows that the mapping cone of an arbitrary morphism, up to homotopy equivalence, may be obtained by first turning the morphism into a cofibration and then taking the kernel of that morphism. The mapping cone of a morphism should thus be thought of as the ‘homotopy kernel’ of the morphism. The next theorem will shed further light on the relation to the topological case.

Theorem 3.11. Let Y be a compact Hausdorff space, and let i : X → Y be a cofibration. Then i^{∗} is a cofibration and 0 → C_{0}(Y \ X) → C(Y) →^{i^{∗}} C(X) → 0 is an excisive short exact sequence.

Proof. We have already seen that i^{∗} will be a cofibration, and therefore surjective. To identify the kernel note first that since i is a cofibration and Y is Hausdorff we have that X is homeomorphic to i(X) and i(X) is closed in Y. Thus the identification of X with its image in Y makes sense. The exactness of the theorem is now obvious, noting that the surjectivity of i^{∗} follows from Tietze’s extension theorem. Now since i^{∗} is a cofibration, the sequence is excisive by the preceding proposition. It can be also be seen directly:

C_{i^{∗}} = {f ∈ C(C_{i})|f([(0, x)]) = 0} ≃ {f ∈ C(Y/X)|f(X) = 0} = C_{0}(Y \ X)

where we have used the description of the mapping cone given in [9] and the fact that since i is a cofibration and i(X) is closed in Y the canonical map C_{i} → Y/X is a homotopy equivalence.

Definition 3.12. Let F be a (covariant) functor from C^{∗} to Set_{*}. We say that F is cofibration half exact if for any short exact sequence 0 → J → A →^{q} B → 0,

where q is a cofibration, we have an exact sequence F (J) → F (A) →^{F (q)} F (B) in Set_{*}.

We call F excisive if for any *-homomorphism p : A → B we have that F (i) : F (J) → F (C_{p}) is an isomorphism, where J is the kernel of p and i : J ↪ C_{p} is the canonical inclusion. We also make the analogous definitions for contravariant functors.

Remark 3.13. A functor on topological spaces is traditionally called exact if it is cofibration half exact. Most of the well known topological homology functors will not be excisive in the topological definition analogous to the one above (even though they would satisfy the topological excision axiom). Consider for instance H_{1}(−; ℤ). Let W denote the Warsaw circle and let I be the subset 0×[−1; 1]. The topological quotient space W/I is homotopy equivalent to S^{1} so H_{1}(W/I; ℤ) = ℤ. On the other the mapping cone, C_{i}, of the inclusion will again be homotopy equivalent to W, so H_{1}(C_{i}; ℤ) = 0.

Theorem 3.14. For a functor F from C^{∗} to Set_{*} the following conditions are equivalent.

(1) F is cofibration half exact.

(2) For any cofibration q : A → B the sequence F (C_{q}) → F (A) →^{F (q)} F (B) is exact.

(3) For any *-homomorphism p : A → B the sequence F (C_{p}) → F (A) →^{F (p)} F (B) is exact.

Proof. By Proposition 3.10 (1) and (2) are equivalent, and obviously (3) implies (2). The implication (2) ⇒ (3) follows from Theorem 3.6.

Theorem 3.15. For a functor F from C^{∗} to Set_{*} we have that F is half exact iff F is cofibration half exact and excisive.

Proof. If F is cofibration half exact and excisive, then F is half exact by the equivalence of (1) and (3) in the preceding theorem. Suppose now that F is half exact. Then F is cofibration half exact by definition. Let p : A → B be any *-homomorphism. Define a C^{∗}-algebra by D = {g ∈ IA|g(0) ∈ J}, and a *- homomorphism η : D → C_{p} by η(g) = g(1) ⊕ (Ip)(g). We show that η is surjective. Consider an element a ⊕ h ∈ C_{f}. Because I is an exact functor we can find a lift g ∈ IA of h, and by adding a constant function in IJ we can assume that g(1) = a. Thus we have g ∈ D and η(g) = a ⊕ h. Furthermore it is easily checked that Ker η = C_{0}([0; 1[, J) ≅ CJ. Note that we can embed J in D as constant functions, and that this inclusion is a homotopy equivalence. By combining the established short exact sequence 0 → CJ → D →^{η} C_{p} → 0 with familiar short exact sequence 0 → J → C_{p} → CB → 0 we obtain the commutative diagram:

[figure: 0; 0; J; i; ≃; C_{p}; CB; 0; D; η; CJ; 0]

Applying F to this diagram gives

[figure: F (J); ≃; F (i); F (C_{p}); 0; F (D); F (η); 0]

Thus F (i) is an isomorphism.

4. The Puppe sequence

We now want to show that we can construct a Puppe sequence analogous to the topological Puppe sequence

X →^{f} Y → C_{f} → ΣX →^{−Σf} ΣY → ΣC_{f} → …

of a map f : X → Y.

Let p : A → B be a *-homomorphism. We have a short exact sequence 0 → ΣB →^{δ} C_{p} →^{k} A → 0, where δ is the canonical inclusion and and k is the canonical projection. Note that the *-homomorphism p∘k : C_{p} → B is null homotopic since it factors through the contractible C^{∗}-algebra CB by the pullback construction of C_{p}. We need the following essential lemma in the construction of the Puppe sequence.

Lemma 4.1. We have a homotopy commutative diagram of C^{∗}-algebras and *- homomorphisms

[figure: C_{k_{1}}; k_{2}; C_{k}; C_{p}; k; A; p; B; ΣA; ≃; −Σp; δ_{1}; ΣB; δ; ≃]

Proof. We have

C_{k} = {f ⊕ g ∈ C_{p} ⊕ CA|k(f) = g(1)}
= {a ⊕ h ⊕ g ∈ A ⊕ CB ⊕ CA|p(a) = h(1), a = g(1)}
≅ {h ⊕ g ∈ CB ⊕ CA|h(1) = p(g(1))}

We can thus make a canonical inclusion ι : ΣB ↪ C_{k} by h ↦ h ⊕ 0. It is easy to see that this map makes the right hand triangle commutative. We can construct a homotopy inverse η : C_{k} → ΣB by

η(h ⊕ g)(s) = \{ \begin{matrix}h(2s) & for 0 ≤ s ≤ \frac{1}{2} \\ p(g(2 − 2s)) & for \frac{1}{2} ≤ s ≤ 1 \end{matrix}

It is clear that η ∘ ι is homotopic to the identity. It is easily checked that

F_{t}(h ⊕ g)(s) = \{ \begin{matrix}h((2 − t)s) ⊕ g(ts) & for (2 − t)s ≤ 1 \\ p(g(2 − (2 − t)s)) ⊕ g(ts) & for (2 − t)s ≥ 1 \end{matrix}

is a homotopy between ι∘η and the identity on C_{k}. We urge the reader to visualize this homotopy. The upper left hand triangle is constructed in the same way as the right hand triangle so it now follows that this triangle is also commutative and that the inclusion ΣA ↪ C_{k_{1}} is a homotopy equivalence. We now proceed to the last triangle of the diagram. The *-homomorphism −Σp is (of course) defined by (−Σp)(f)(s) = p(f(1 − s)). We have that

(η ∘ δ_{1})(f)(s) = \{ \begin{matrix}0 & for 0 ≤ s ≤ \frac{1}{2} \\ p(f(2 − 2s)) & for \frac{1}{2} ≤ s ≤ 1 \end{matrix}

so η ∘ δ_{1} is homotopic to −Σp. This shows the lemma.

Theorem 4.2. We have a sequence of C^{∗}-algebras and *-homomorphisms

… → ΣC_{p} →^{−Σk} ΣA →^{−Σp} ΣB →^{δ} C_{p} →^{k} A →^{p} B

where the composition of two consecutive *-homomorphisms are null homotopic and which upon applying a cofibration half exact functor F gives an exact sequence

… → F (ΣC_{p}) →^{(−Σk)_{∗}} F (ΣA) →^{(−Σp)_{∗}} F (ΣB) →^{δ_{∗}} F (C_{p}) →^{k_{∗}} F (A) →^{p_{∗}} F (B)

If furthermore F is excisive or if p is a cofibration we get an exact sequence:

… → F (ΣJ) → F (ΣA) →^{(−Σp)_{∗}} F (ΣB) →^{δ_{∗}} F (J) → F (A) →^{p_{∗}} F (B)

Proof. We use Lemma 4.1. That the composition of two consecutive *-homomorphisms is null homotopic follows from Lemma 4.1 since in a mapping cone sequence C_{p} →^{k} A →^{p} B we have that p ∘ k is null homotopic since it factors through CB and all compositions in the sequence are up to homotopy of this form. Applying the functor F to the diagram in Lemma 4.1 gives a commutative diagram where the vertical arrows are isomorphisms and the rows are exact. This gives the first part of the theorem. The second part in an immediate corollary of the first.

Corollary 4.3. Every half exact functor F is split exact.

Proof. That F is split exact follows from the long exact sequence in the last part of the previous theorem and Theorem 3.15.

Proposition 4.4. Let (φ_{t}) : A → C and (ψ_{t}) : B → C and suppose that the commutator [φ_{t}(a), ψ_{t}(b)] → 0 for t → ∞ for all a ∈ A and b ∈ B. Then there exists a (up to equivalence) unique asymptotic morphism θ : A ⊗_{max} B → C such that θ_{t}(a ⊗ b) − φ_{t}(a)ψ_{t}(b) → 0 for t → ∞. Particularly for asymptotic morphisms (φ_{t}) : A → C and (ψ_{t}) : B → D we get an asymptotic morphism φ⊗ψ : A⊗_{max}B → C ⊗ D. The tensor product passes to homotopy classes.

Proof. The asymptotic morphisms induces two commuting *-homomorphisms φ̃ : A → C_{∞} and ψ̃ : B → C_{∞} The thus obtain a *-homomorphism A ⊗_{max} B → C_{∞} i.e. an asymptotic morphism (θ_{t}) : A ⊗_{max} B → C. The last part is an obvious corollary of the first part in the case where C and D are unital since we can embed C and D as commuting subalgebras of C ⊗_{max} D. In the general case we just add units and observe that C ⊗_{max} D ↪ C^{+} ⊗_{max} D^{+} and that θ in this case lands us in C ⊗_{max} D. To check that the tensor product passes to homotopy classes consider two homotopies A → IC and B → ID. The tenor product of these gives an asymptotic morphism A ⊗_{max} B → I^{2}(C ⊗_{max} D). Composition with the diagonal map d : I → I^{2} thus gives an asymptotic morphism A ⊗_{max} B → I(C ⊗_{max} D). This is the desired homotopy.

Remark 4.5. Note that the map [[A, C]] × [[B, D]] → [[A ⊗_{max} B, C ⊗_{max} D]] is linear in each variable when it makes sense i.e. it is linear in the first variable if C is stable or a suspension and linear in the second variable when D is stable or a suspension.

Remark 4.6. As a special case of the above proposition we can define the suspension of asymptotic morphisms. Given a asymptotic morphism φ : A → B, the suspension Σφ : ΣA → ΣB is given by Σφ_{t}(f)(s) = φ_{t}(f(s)).

Definition 4.7. We define the class of stable asymptotic morphisms of degree q ∈ ℤ between two C^{∗}-algebras A and B as \{\{A, B\}\}_{q} = \underrightarrow{lim}_{n}[[Σ^{n+q}A, Σ^{n}B]], where the map [[Σ^{n+q}A, Σ^{n}B]] → [[Σ^{n+q+1}A, Σ^{n+1}B]] is given by suspension. This can be done for arbitrary C^{∗}-algebras A and B. To get a composition product we have to restrict to separable C^{∗}-algebras. We therefore define the category Asymp^{S} with objects separable C^{∗}-algebras and morphisms the stable asymptotic morphisms of any degree. As a subcategory we have Asymp_{0}^{S} with the same objects and as morphisms the stable asymptotic morphisms of degree 0. Note that we have functorial isomorphisms {{Σ−, −}}_{q−1} = {{−, −}}_{q} = {{−, Σ−}}_{q+1}. We denote {{−, −}}_{0} simply by {{−, −}}.

Remark 4.8. The preceding definition is an analogue of the ’classical’ definitions for homotopy classes of *-homomorphisms (cf. [36]), where one defines {A, B}_{q} = \underrightarrow{lim}_{n}[Σ^{n+q}A, Σ^{n}B]. If X, Y are compact Hausdorff topological spaces we have that {C(Y), C(X)}_{q} = [Σ^{∞}X, Σ^{∞}Y]_{−q}, Σ^{∞}X being the suspension spectrum of X.

The following proposition also holds in this setting, and the proof is similar.

Proposition 4.9. {{A, B}}_{q} is an abelian group for all A and B. For separable C^{∗}-algebras the composition product {{A, B}}_{p} × {{B, C}}_{q} → {{A, C}}_{p+q} is bilinear.

Proof. That {{A, B}}_{q} is an abelian group follows since for n ≥ 2, [[Σ^{n+q}A, Σ^{n}B]] is an abelian group and the suspension map is a group homomorphism. Linearity in the second variable follows from the corresponding linearity on the level of [[−, −]]. Linearity in the first variable follows since our representative φ_{t} of {{φ_{t}}} ∈ {{B, C}}_{q} is a suspension.

Proposition 4.10. Let C be a C^{∗}-algebra. The functors [C, −], [[C, −]] : C^{∗} → Set_{*}, {C, −}_{q}, {{C, −}}_{q} : C^{∗} → Ab are cofibration half exact.

Proof. We give the proof for [[C, −]] from which the statement for {{C, −}}_{q} follows since suspensions commute with mapping cones (i.e. Σ^{n}C_{f} = C_{Σ^{n}f} ) and that \underrightarrow{lim} is an exact functor. We have p_{∗} ∘ k_{∗} = 0 since p ∘ k is null homotopic. Let [[φ_{t}]] ∈ [[C, A]] such that p ∘ φ_{t} is null homotopic as an asymptotic morphism. Choose a homotopy F_{t} : C → IB between the zero map and p ∘ φ_{t}. But then φ_{t} ⊕ F_{t} defines an element in [[C, C_{p}]] that maps to [[φ_{t}]] under k_{∗}. The proof for [C, −] and {C, −}_{q} are analogous.

Proposition 4.11. Let C be a C^{∗}-algebra. The functors {−, C}_{q}, {{−, C}}_{q} : C^{∗} → Ab are cofibration half exact.

Proof. Let p : A → B be a cofibration. Consider the following diagram:

[figure: [[B, C]]; p^{∗}; Σ; [[A, C]]; [[C_{p}, ΣC]]; δ^{∗}; [[ΣB, ΣC]]]

We first show that the kernel of p^{∗} is mapped into the image of δ^{∗} by Σ. Suppose we have [[φ]] ∈ [[B, C]] such that p^{∗}([[φ]]) = 0. Since φ_{t}(0) ∈ C_{0}(T, C) we may assume that φ_{t}(0) = 0. Choose a homotopy F_{t} : A → IC such that F_{t}(a)(0) = φ_{t}(p(a)) and F_{t}(a)(1) = 0. By replacing F_{t}(·) by F_{t}(·) − F_{t}(0) we may also assume that F_{t}(0) = 0. Define an asymptotic morphism ψ_{t} : C_{p} → ΣC by

ψ_{t}(a ⊕ f)(s) = \{ \begin{matrix}φ_{t}(f(2s)) & for 0 ≤ s ≤ \frac{1}{2} \\ F_{t}(a)(2s − 1) & for \frac{1}{2} ≤ s ≤ 1 \end{matrix}

It is easy to see that ψ maps into ΣC and actually is an asymptotic morphism. Furthermore we have

ψ_{t}(δ(f))(s) = \{ \begin{matrix}φ_{t}(f(2s)) & for 0 ≤ s ≤ \frac{1}{2} \\ 0 & for \frac{1}{2} ≤ s ≤ 1 \end{matrix}

Thus ψ ∘ δ is homotopic to Σφ so that δ^{∗}([[ψ]]) = [[Σφ]]. This immediately

gives us an exact sequence {{C_{p}, C}}_{q−1} →^{δ^{∗}} {{B, C}}_{q} →^{p^{∗}} {{A, C}}_{q}. Apply-

ing this to the *-homomorphism k : C_{p} → A we find that {{C_{k}, C}}_{q−1} →^{δ_{1}^{∗}}

{{A, C}}_{q} →^{k^{∗}} {{C_{p}, C}}_{q} is exact. By Lemma 4.1 we now get an exact sequence

{{ΣB, C}}_{q−1} →^{(−Σp)^{∗}} {{A, C}}_{q} →^{k^{∗}} {{C_{p}, C}}_{q}. Since flipping loops induces multi-

plication by −1 on {{−, −}} we are done.

The proof in the case of {−, C}_{q} can be copied almost verbatim.

Assumption: Since we will be using some theorems whose proofs involve the use of quasicentral approximate units, we will in the rest of this section assume that all C^{∗}-algebras are separable.

The following result was discovered by Connes-Higson [17, 16, 15] and independently by Dădărlat [3], where a detailed proof is given.

Theorem 4.12. For a morphism p : A → B between separable C^{∗}-algebras, the canonical map Σi : ΣJ → ΣC_{p} is an equivalence in Asymp.

Sketch of Proof. To any short exact sequence 0 → J → A →^{p} B → 0 it is possible to construct an asymptotic morphism ΣB → J. Applying this to the short exact sequence 0 → ΣJ → CA → C_{p} one can check that one obtains an inverse to Σi : ΣJ → ΣC_{p} in Asymp.

Corollary 4.13. The functors [[C, Σ−]] : C^{∗} → Grp, {{C, −}}, {{−, C}} : C^{∗} → Ab are half exact.

Proof. This is immediate from Theorem 4.12 and Theorem 3.15.

Corollary 4.14. We have long exact sequences

… → {{C, J}}_{q} → {{C, A}}_{q} →^{(Σp)_{∗}} {{C, B}}_{q} →^{δ_{∗}} {{C, J}}_{q−1} → {{C, A}}_{q−1} →^{p_{∗}} {{C, B}}_{q−1} → …

and

… → {{B, C}}_{q} →^{p^{∗}} {{A, C}}_{q} → {{J, C}}_{q} →^{δ^{∗}} {{B, C}}_{q+1} →^{p^{∗}} {{A, C}}_{q+1} → {{J, C}}_{q+1} → …

Proof. This follows directly from Theorem 4.2 and the previous corollary.

We have the following result from [6, Prop. 3.2]

Proposition 4.15. Let 0 → J → A →^{p} B → 0 be a split exact sequence of separable C^{∗}-algebras. Then the canonical inclusion i : I → C_{p} is an equivalence in Asymp.

This shows that split exact sequences are ‘excisive’ in Asymp. Arbitrary sequences only become ‘excisive’ in Asymp after suspension (cf. Theorem 4.12).

Consider the exact sequence

0 → C_{0}(S^{1} \ pt) → C_{0}(W \ pt) → C_{0}(I \ pt) → 0.

Here W denotes the Warsaw circle. This sequence is split exact since we can choose a splitting induced from projecting W onto I. The Puppe sequence now gives us a surjection

[[C, C_{0}(S^{1} \ pt)]] → [[C, C_{0}(W \ pt)]].

for all separable C^{∗}-algebras C, since C_{0}(I \pt) is contractible. Setting C = C_{0}(W \ pt) we get an asymptotic morphism φ_{t} : C_{0}(W \pt) → C_{0}(S^{1} \pt) such that i∘φ_{t} ≃ id_{C_{0}(W \pt)} as asymptotic morphisms. Applying K_{1} to the split exact sequence above reveals that K_{1}(C_{0}(W \ pt)) = K_{1}(C_{0}(S^{1} \ pt)) = ℤ. Thus K_{1}(i)K_{1}(φ_{t}) = K_{1}(i ∘ φ_{t}) = id_{ℤ}. Thus K_{1}(φ_{t} ∘ i) = id_{ℤ} (e.g. since End ℤ is abelian). By Corollary 2.6 we have that φ ∘ i is homotopic to a *-homomorphism, which in turn is induced by a pointed map S^{1} → S^{1}. This must be a degree 1 map since it induces the identity on topological K-theory (to see this e.g. appeal to the Atiyah-Hirzebruch spectral sequence or check the definitions). Thus φ_{t} ∘ i is homotopic to the identity so C_{0}(S^{1} \ pt) and C_{0}(W \ pt) are equivalent as objects in Asymp. This also holds after adding units, so we get that the Warsaw circle is equivalent to S^{1} in Asymp.

Note that this example also shows that split exact sequences are not in general excisive (that is excisive in C^{∗}) since the Warsaw circle is not homotopy equivalent to S^{1}.

Remark 4.16. It could be interesting to consider the ‘stable’ category of C^{∗}- algebras i.e. the category with objects being suspension spectra of C^{∗}-algebras and the corresponding stable maps (either *-homomorphisms or asymptotic morphisms). In this category the suspension would have an adjoint, just given by Σ^{−1}, and furthermore one would suspect that fibrations and cofibrations would coincide. Thus this category should be a stable closed model category.

Indeed the homotopy category of this category is easily seen to be a triangulated category (cf. e.g. [32]).

It could be interesting to explore how far the homotopy category is being a stable homotopy category of Hovey, Palmeri and Strickland [25]. Under what conditions do we get Brown representability? What if we restrict to the subcategory of (matrix) stable C^{∗}-algebras? Does the category have a closed symmetric monoidal category structure (when do internal function objects exist)?

5. The category E

We will in this final section give the definition of E-theory and prove the existence of its two six-term exact sequences. We will furthermore state some additional results and discuss the relation to K- and KK-theory. Most of these, however, will be stated without proof.

Notation: From now on all C^{∗}-algebras will be assumed to be separable.

Definition 5.1. We define the category E as the category with objects separable C^{∗}-algebras and morphisms E(A, B) = [[ΣA ⊗ 𝕂, ΣB ⊗ 𝕂]].

Theorem 5.2 (Bott periodicity). For any stable half exact functor F : Sep-C^{∗} → Ab we have that the functor F Σ^{2} is naturally equivalent to F. This equivalence is also natural in F i.e. if we have a natural transformation α : F → F′ between two stable half exacts functors we get a commutative diagram:

[figure: F; α; ≃; F Σ^{2}; αΣ^{2}; F′; ≃; F′Σ^{2}]

Proof. This result is due to Cuntz [18]. We can easily obtain a proof from our previous results. Note first that since F is half exact it is additive and split exact cf. Corollary 4.3. Furthermore to every short exact sequence 0 → J → A → B → 0 we have an exact sequence

… → F (ΣJ) → F (ΣA) →^{(Σp)_{∗}} F (ΣB) →^{δ_{∗}} F (J) → F (A) →^{p_{∗}} F (B)

The proof of Bott periodicity in [9] can now be copied verbatim. The last part of the theorem follows easily from the construction of the isomorphism.

Theorem 5.3. For all A, B we have that E(A, B) is an abelian group. The functor E(−, −) is stable and half exact in each variable. The functor Σ is self-adjoint equivalence i.e. the suspension Σ : E(−, −) → E(Σ−, Σ−) is an isomorphism and we have the functorial isomorphisms E(−, Σ−) ≃ E(Σ−, −) and E(Σ^{2}−, −) ≃ E(−, Σ^{2}−) ≃ E(−, −). Furthermore we have a natural equivalence E(−, −) ≃ {{− ⊗ 𝕂, − ⊗ 𝕂}}.

Proof. By construction E(−, −) is stable in both variables. Let A an arbitrary separable C^{∗}-algebra. By Corollary 4.13 we have that E(A, −) is half exact. We thus have that E(A, −) ≃ E(A, Σ^{2}−) by Bott periodicity. This equivalence is also natural in A as is seen from the last part of the Bott periodicity theorem. Thus Σ^{2} is a right adjoint to the identity I in the category E. Since adjoints are unique (cf. Mac Lane [26, Corollary IV.1.1]) we get that Σ^{2} is naturally equivalent to I in E, in particular we get a natural isomorphism E(−.−) ≃ E(Σ^{2}−, −). That Σ^{2} is naturally equivalent to I means by definition that Σ is an equivalence of categories. By Mac Lane [26, Theorem IV.4.1] it follows that Σ is self-adjoint and that the map Σ : E(−, −) → E(Σ−, Σ−) is an isomorphism. Therefore E(A, B) = {{A ⊗ 𝕂, B ⊗ 𝕂}} and thus E(−, B) is half exact.

Remark 5.4. The definition of E-theory can be simplified somewhat. Note that the functor tensoring with 𝕂 gives rise to an idempotent monad on the category Asymp in a natural way. Hence we have a natural equivalence

[[−, − ⊗ 𝕂]] ≃ [[− ⊗ 𝕂, − ⊗ 𝕂]],

since for a general idempotent monad < T, η, μ > we have a commutative diagram:

[figure: Hom(T x, T^{2}y); ≃; μ_{∗}; Hom(x, T y); Hom(T x, T y); η^{∗}; Hom(x, T y)]

So we actually have E(−, −) = [[Σ−, Σ−⊗𝕂]], i.e. we can leave out the compacts in the first variable, and thus if B is stable E(A, B) = {{A, B}}.

Sometimes it is possible to entirely leave out the suspensions. Following [6] we say that a C^{∗}-algebra A is homotopy symmetric if a ↦ a ⊗ e_{11} is invertible in [[A, A ⊗ 𝕂]]. For instance Σℂ and qℂ are homotopy symmetric. If A is homotopy symmetric then A⊗B will also be homotopy symmetric. If A is stable the following conditions are equivalent [6]:

• A is homotopy symmetric.

• [[A, B]] is a group for all stable B.

• E(A, B) = [[A, B]] for all stable B.

Corollary 5.5. For any short exact sequence 0 → J → A →^{p} B → 0 and any separable C^{∗}-algebra C we have cyclic six-term exact sequences:

[figure: E(C, J); E(C, A); p_{∗}; E(C, B); δ; E(C, ΣB); δ; E(C, ΣA); (Σp)_{∗}; E(C, ΣJ); E(B, C); p^{∗}; E(A, C); E(J, C); δ; E(ΣJ, C); δ; E(ΣA, C); E(ΣB, C); (Σp)^{∗}]

Proof. Since E(−, −) is half exact in both variables, this follows immediately using Bott periodicity and Theorem 4.2.

Proposition 5.6. We have a isomorphisms E(ℂ, −) ≃ K_{0}(−) and thus E(Σℂ, −) ≃ K_{1}(−). These are functorial in the sense that they coincide with the maps induced on K-theory by asymptotic morphisms, cf. Remark 1.19.

Sketch of Proof. Let φ_{t} : Σℂ ⊗ 𝕂 → ΣA ⊗ 𝕂 be an asymptotic morphism. On K-theory this induces a map ℤ → K_{1}(ΣA) ≃ K_{0}(A), which evaluated in 1 gives an element in K_{0}(A). We claim that this map induces an isomorphism. We’ll give the flavor of things by proving surjectivity in the case where A is unital. Let p ∈ 𝕄_{n}(A) be a projection, and let ℂ → 𝕄_{n}(A) be the corresponding *-homomorphism sending 1 ↦ p. Tensoring with 𝕂 and suspending gives an element in E(ℂ, A) mapping to p ∈ K_{0}(A). Since K_{0}(A) consists of differences of projections in matrices over A, we are done.

Remark 5.7. The above proposition shows that in the category E, K-theory becomes a representable functor. Likewise K-theory with coefficients becomes realizable as E(𝕀_{n}, −) = K_{0}(−; ℤ/n). It would be interesting to have a classification of the representable functors—do non-representable functors exist?

Remark 5.8. Consider the short exact sequence 0 → 𝕂 → T_{0} → Σℂ → 0 used in the proof of Bott periodicity. Applying the construction mentioned in the sketch of the proof of Theorem 4.12 we get an asymptotic morphism Σ^{2}ℂ → 𝕂, inducing an isomorphism in K-theory. This map is called the Bott map, and its inverse in E-theory is given by a *-homomorphism Σℂ → Σ^{3}ℂ ⊗ 𝕂. Thus we get a very concrete description of the periodicity isomorphisms.

We end this section by stating the following characterization of E-theory due to Connes and Higson [17, 16], generalizing earlier work of Higson [24].

Theorem 5.9. Let F be a functor Sep-C^{∗} → 𝔸, where 𝔸 is an additive category. Suppose that F is homotopy invariant and stable. Furthermore assume that F is half exact in the sense that the functors Hom(F (A), F (−)) and Hom(F (−), F (B)) from Sep-C^{∗} to Ab are half exact for all A, B ∈ Sep-C^{∗}.

Then there exists a unique functor F̂ such that the following diagram commutes:

[figure: E; F̂; _{Sep-C_{∗}}; F; 𝔸]

Remark 5.10. The above theorem gives us an abstract description of E-theory. Although this is not that useful for concrete calculations, it is essential for others, e.g. proving the concordance of E-theory and KK-theory in ‘good’ cases (e.g. E(A, B) = KK(A, B) for (K-) nuclear A).

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[41] Edwin H. Spanier. Algebraic Topology. McGraw-Hill Ltd., 1966.

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