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Pictures of KK-theory for real C*-algebras and almost commuting matrices

We give a systematic account of the various pictures of KK-theory for real C*-algebras, proving natural isomorphisms between the groups that arise from each picture. As part of this project, we develop the universal properties of KK-theory, and we use CRT-structures to prove that a natural transformation from F(A) to G(A) between homotopy equivalent, stable, half-exact functors defined on real C*-algebras is an isomorphism provided it is an isomorphism on the smaller class of C*-algebras. Finally, we develop E-theory for real C*-algebras and use that to obtain new negative results regarding the problem of approximating almost commuting real matrices by exactly commuting real matrices.


Introduction and Preliminaries
A real C * -algebra is a Banach * -algebra A over the real numbers such that x * x = x 2 holds for all x and such that every element of the form 1 + x * x is invertible in the unitization of A (see [32]). In this paper, we will adopt the term R * -algebra instead. Every R * -algebra is isometrically isomorphic to a closed * -algebra of bounded operators on a Hilbert space over R. In addition, every R * -algebra is isomorphic to the * -algebra of fixed elements of a C * -algebra with a conjugate linear involution.
In Kasparov's seminal paper [21] introducing KK-theory, he simultaneously considered both R *algebras and C * -algebras. Since then, many alternate but equivalent or closely related pictures of KK-theory have been introduced and developed by various authors ( [9], [10], [15], [16], [28], [34]). The ability to move among the various pictures has contributed immensely to the utility of KKtheory as a tool for solving problems. However, these authors have not following Kasparov's lead and the alternate pictures of KK-theory have been developed primarily in the complex case.
In recent years, substantial progress has been made in developing the tools to study R * -algebras including the development of united K-theory and the universal coefficient theorem in [2] and [3]. This has led to a classification of purely infinite simple R * -algebras (in [5]) and the classification of real forms of UHF-algebras that are stable over the CAR-algebra (in [33]).
Given the centrality of KK-theory for these projects, there has been a need to develop a systematic account of the various pictures of KK-theory for R * -algebras. In this paper, we will develop several of the alternate pictures of KK-theory in the context of R * -algebras and prove the appropriate equivalent theorems. In particular, in this paper we will consider the following pictures of KK-theory and prove appropriate equivalence theorems for each: the standard Kasparov bimodule picture of KK-theory, the Fredholm picture (both in Section 2), the universal property picture (Section 3), the extension picture (Section 4), and suspended E-theory using asymptotic morphisms (Section 5).
Furthermore, we prove a more general theorem which reduces the work required to replicate many of these equivalent theorems and promises to ease the way for similar projects in the future. Suppose that µ : F → G is a natural transformation between homotopy invariant, stable, half exact functors. We prove that if µ is an isomorphism for all C * -algebras, then it is an isomorphism for R *algebras. This is accomplished in Section 3 when we develop the universal properties of KK-theory and K-theory for R * -algebras.
In the last two sections, we will apply these ideas, using KK-theory to prove the existence of certain asymptotic morphisms, which in turn is used to obtain new results for the problem of approximating a set of almost commuting matrices over the field of real numbers. In particular, let the Halmos number be the largest integer d such that whenever d real self-adjoint matrices almost commute (pairwise) they can be approximated by d pairwise commuting matrices. More precisely, for all ǫ > 0 there should be a δ > 0 such that if Furthermore, the dependence of δ on ǫ must be uniform, independent of the dimension of the matrices H r . It is shown in [27] that in the context of real matrices, the statement is true for d = 2. We will show in Section 7 the statement is false for d = 5. Therefore, the Halmos number for real matrices is between 2 and 4, inclusive.

The Standard and Fredholm Pictures of KK-Theory
We take the following definition from Section 2.3 of [32] to be the standard definition of KK-theory for R * -algebras. It is essentially the same as that in [21] where it was simultaneously developed for both R * -algebras and C * -algebras.
Definition 2.1. Let A be a graded separable R * -algebra and B be an R * -algebra with σ-unital. are zero for all a ∈ A. By Proposition 2.3.3 of [32], degenerate bimodules are homotopic to trivial bimodules. The following theorem summarizes the principal properties of KK-theory for R * -algebras from Chapter 2 of [32].

Proposition 2.2. KK(A, B) is an abelian group for separable
A and σ-unital B. As a functor on separable R * -algebras (contravariant in the first argument and covariant in the second argument), it is homotopy invariant, stable, and has split exact sequences in both arguments. Furthermore, there is a natural associate pairing (the intersection product) We now turn to the Fredholm picture of KK-theory, which was developed in [15] with only the situation of C * -algebras in mind. However, the approach goes through the same for R * -algebras, as follows.
Definition 2.3. Let A and B be separable R * -algebras.
which is unique up to homotopy by Section 1.17 of [21].
is defined to be the set of equivalence classes of KK(A, B)-cycles.
The following lemma is the real version of Lemma 2.3 of [15]. Proof. For the first statement, we show that a cycle is homotopic to a degenerate cycle via the operator homotopy The functoriality is established as in [15] in Sections 2.4 through 2.7.
The next proposition establishes the isomorphism between the two pictures of KK-theory. First we review some preliminaries regarding graded R * -algebras and Hilbert modules.
If B is an R * -algebra, then the standard even grading on M 2 (B) is obtained by setting M 2 (B) (0) to be the set of diagonal matrices and M 2 (B) (1) the set of matrices with zero diagonal. The standard even grading on K R ⊗ B is obtained by choosing a * -isomorphism K R ⊗ B ∼ = M 2 (K R ⊗ B). This in turn induces a canonical (modulo a unitary automorphism) grading on M(K R ⊗ B).
Let H B be the Hilbert B-module consisting of all sequences , this grading coincides with the one described in the previous paragraph. Proof. We give A and B the trivial grading and we give M(K R ⊗ B) the standard even grading described above. For It is readily verified that α(x) is a Kasparov (A-B) bimodule. Furthermore, for x = (φ + , φ − , U ) and y = (ψ + , ψ − , V ) it is easy to see that α(x + y) and α(x) + α(y) are unitarily equivalent via a degree 0 unitary. We must show that α induces a well-defined homomoprhism Note first that α sends degenerate elements to degenerate elements. Next, suppose (φ + , φ − , U ) is a KK(A, IB)-cycle implementing a homotopy between x = (φ 0 + , φ 0 − , U 0 ) and y = (φ 1 is the map induced by the evaluation map ε t at t. Consider the Kasparov (A-IB)-bimodule Therefore α(x) and α(y) are homotopic. Hence α is well-defined.
To show that α is surjective, let y = (E, φ, T ) be a Kasparov (A,B)-bimodule. By Proposition 2.3.5 of [32], we may assume that E ∼ =ĤB and that T = T * . Thus, with respect to the graded isomorphism where φ + and φ − are * -homomorphisms from A to M(K R ⊗ B) and U is an element of M(K R ⊗ B).

The Universal Property of KK-Theory
Let F be a functor from the category C*R-Alg of separable R * -algebras to the cateogry Ab of abelian groups. We say that F is (i) homotopy invariant if (α 1 ) * = (α 2 ) * whenever α 1 and α 2 are homotopic * -homomorphisms on the level of R * -algebras.
defined via any rank one projection. (iii) split exact if any split exact sequence of separable R * -algebras (iv) half exact if any short exact sequence of separable R * -algebras In what follows we will see that if F is homotopy invariant and half exact, then it is split exact. Proof. Just as in the complex case (Theorem 4.1.13 of [19]), the map e KR : K R → K R ⊗ K R is homotopic to an isomorphism.
The following theorem is the version for R * -algebras of Theorem 3.7 of [15] and Theorem 22.3.1 of [1]. Furthermore, the pairing respects the intersection product on KK-theory in the sense that Proof. Let Φ ∈ KK(A, B). Using Theorem 2.5 we represent Φ with a KK(A, B) cycle and as in Lemma 3.6 of [15], we may assume that this cycle has the form (φ + , φ − , 1). We use the same construction as in Definitions 3.3 and 3.4 in [15]. In that setting F is assumed to be a functor from separable C * -algebras, but it goes through the same for functors from separable R * -algebras to any abelian category. This construction produces a homomorphism Φ * : F (A) → F (B) and we then define α(x ⊗ Φ) = Φ * (x). The proof of Theorems 3.7 and 3.5 of [15] carry over in the real case to show that α is natural, is well-defined, satisfies α(x ⊗ 1 A ) = x, and is unique. That α respects the Kasparov product follows from the uniqueness statement.
We also note the contravariant version of the result above. If F is a contravriant functor, otherwise satisfying the above hypotheses, then there is a pairing α : For any R * -algebra A, we define By iteration, S n A is defined for all n ∈ Z. Since SS −1 R is KK-equivalent to R, the formula S n S m A ≡ S n+m A holds up to KK-equivalence for all n, m ∈ Z. Then for any functor F on C*R-Alg and any integer n, we define F n (A) = F (S n (A)). Corollary 3.3. Let F be a functor from C*R-Alg to Ab that is homotopy invariant, stable, and split exact. Then F * (A) has the structure of a graded module over the ring K * (R). In particular, Proof. For all separable A and σ-unital B, the pairing of Proposition 2.2 gives KK * (A, B) the structure of a module over KK * (R, R). Taking A = B, we define a graded ring homomorphism β from K * (R) ∼ = KK * (R, R) to KK * (A, A) by multiplication by 1 A ∈ KK(A, A).
Then for any x ∈ F m (A) and y ∈ K n (R) we define x · y = α(x ⊗ β(y)) ∈ F n+m (A). The second statement follows from the KK-equivalence between R and S 8 R, and that between R and S −1 SR from Section 1.4 of [2].
It also follows that the pairing Theorem 3.2 extends to a well-defined graded pairing Let KK be the category whose objects are separable R * -algebras and the set of morphisms from A to B is KK(A, B). There is a canonical functor KK from C*R-Alg to KK that takes an object A to itself and which takes a * -homomorphism f : A → B to the corresponding element [f ] ∈ KK(A, B). Proposition 3.5. Let F be a functor from C*R-Alg to Ab that is homotopy invariant and half exact. Then for any short exact sequence there is a natural boundary map ∂ : F (SC) → F (A) that fits into a (half-infinite) long exact sequence Proof. Use the mapping cone construction as in Section 21.4 of [1]. Corollary 3.6. A functor F from C*R-Alg to Ab that is homotopy invariant and half exact is also split exact.
Proof. The splitting implies that g * is surjective. Thus in the sequence of Proposition 3.5, ∂ = 0 and f * is injective.
Proposition 3.7. Let F be a functor from C*R-Alg to Ab that is homotopy invariant, stable, and half exact. Then for any short exact sequence there is a natural long exact sequence (with 24 distinct terms) Proof. From Corollary 3.6 and Corollary 3.3, F is periodic; so Proposition 3.5 gives the long exact sequence.
We say that a homotopy invariant, stable, half-exact functor F from C*R-Alg to the category Ab of abelian groups is an isomorphism.
Theorem 3.8. Let F be a functor from C*R-Alg to Ab that is homotopy invariant, stable, half exact and satisfies the dimension axiom. Then there is a natural transformation β : K n (A) → F n (A). If F is also continuous, then β is an isomorphism for all R * -algebras in the smallest class of separable R * -algebras which contains R and is closed under KK-equivalence, countable inductive limits, and the two-out-of-three rule for exact sequences.
Proof. Let z be a generator of F (R) ∼ = Z and for x ∈ K n (A) ∼ = KK(R, S n A) define a K * (R)-module homomorphism β : K * (A) → F * (A) by β(x) = α(z ⊗ x). Taking A = R, Theorem 3.2 yields that β(1 0 ) = z where 1 0 is the unit of the ring K * (R) = KK * (R, R). Therefore, β is an isomorphism for A = R. Then bootstrapping arguments show that β is an isomorphism for all R * -algebras in the class described.

From Section 2.1 of [3] we have distinguished elements
For any homotopy invariant, stable, split exact functor F on C*R-Alg, define the united F -theory of an R * -algebra A to be together with the collection of natural homomorphisms induced by the elements c, r, ε, ζ, ψ U , ψ T , γ, τ via the pairing of Theorem 3.2.
Proposition 3.9. Let F be a homotopy invariant, stable, split exact functor from C*R-Alg to Ab and let A be a separable R * -algebra.
hold among the operations {c n , r n , ε n , ζ n , (ψ U ) n , (ψ T ) n , γ n , τ n } on F CRT (A). But in the proof of Proposition 2.4 of [3], it is shown that these relations hold at the level of KK-elements. Therefore, using the associativity of the pairing of Theorem 3.2, the same relations hold among the operations of F CRT (A). Suppose now that F is also half-exact. To show that K CRT (A) is acyclic, we must show that the sequences 3) are exact. These can be derived from the short exact sequences Indeed, focusing on the first one for our argument, the short exact sequence gives rise to the long exact sequence where the homomorphisms are given by the multiplication of the elements [f ] ∈ KK 2 (R, R), [g] ∈ KK 0 (R, C), and ∂ ∈ KK −1 (C, R). In the case of the functor KK(B, −), it was shown in the proof of Proposition 2.4 of [3] that the resulting sequence has the form for all separable R * -algebras B. It then follows easily that the KK-element equalities hold. This proves that Sequence 3.1 is exact. Sequences 3.2 and 3.3 are shown to be exact the same way.
Theorem 3.10. Let F and G be homotopy invariant, stable, half exact functors from C*R-Alg to Ab with a natural transformation µ A : of acyclic CRT-modules which is, by hypothesis, an isomorphism on the complex part. Then the results in Section 2.3 of [6] imply that µ CRT A is an isomorphism.
Corollary 3.11. Any homotopy invariant, stable, half exact functor from C*R-Alg to Ab that vanishes on all C * -algebras vanishes on all R * -algebras.

Application: Isomorphism between KK −1 and Ext
In this section, we will use Theorem 3.10 to show that KK −1 (A, B) is naturally isomorphic to Ext(A, B) −1 for separable R * -algebras A and B. Let be an exact sequence of R * -algebras. As in the complex case, the associated mapping cone is defined to be and there are * -homomorphisms λ e : SA → C π , γ e : B → C π , and κ e : C π → E defined by λ e (f ) = (0, f ), γ e (b) = (ιb, 0), and κ e (e, f ) = e .
As in the complex case, there is an exact sequence 0 → SA Proof. Because of the section map σ, it follows that π * is surjective and that the boundary map vanishes in the sequence By the comments above, the sequence 0 → SA λe −→ C π κe −→ E → 0 induces the same long exact sequence on KK-theory. Thus the homomorphism KK n (SA, SA) Proof. As in the complex case, the conjugacy of the Busby invariants implies the corresponding extensions are equivalent. That is, we have the following commutative diagram: where we are identifying E i as a sub-algebra of M(B). Since the mapping cone construction is functorial, there exists a * -isomorphism α : C π1 → C π2 such that the diagram As in the complex case, we define the sum of two extensions in terms of the Busby invariants. Let B be stable and let π B : M(B) → Q(B) be the canonical projection. Then the sum of two extensions with Busby invariants τ i : A → Q(B) is the unique extension (up to strong equivalence) with Busby invariant τ 1 ⊕ τ 2 : A → Q(B). The * -homomorphism τ 1 ⊕ τ 2 is defined by fixing a * -isomorphism Proof. Let τ i be the Busby invariant corresponding to e i and τ 1 ⊕ τ 2 the Busby invariant of the sum e 1 ⊕ e 2 , as above. Let σ and ψ be as above also. Notice that σ is the Busby invariant of an extension of the form Since τ 1 ⊕ τ 2 = ψ • σ, there exist morphisms ψ and φ forming the following commutative diagram By the above paragraphs, the following diagrams The goal for the rest of this section is to produce an isomorphism between KK −1 (A, B) and the group Ext(A, B) −1 of invertible extensions. For r, s ∈ Z ≥0 , let Cℓ r,s denote the Clifford algebra Cℓ(R r+s , Q r,s ) where Q r,s is the quadratic form We will be using the fact that KK * (A⊗Cℓ p,q , B⊗Cℓ r,s ) depends, up to natural isomorphism, only on (p − q) − (r − s) (see Corollary 4.2.10 of [32]). In particular, for any trivially graded R * -algebras A and B, we have are elements of K R ⊗ B for all a ∈ A. Using an argument similar to that in the complex case, we may assume that F * = F = F −1 and E = H B (see Section 17.4 and 17.6 of [1]). Since Then p is a projection and π(p)(π • ψ(a)) = (π • ψ(a))π(p) for all a ∈ A. Define τ (E,φ,F ) : . This construction is well-defined and gives a natural transformation By Lemma 4.2, we also have a well-defined natural transformation . Furthermore, the diagram below commutes, showing that µ A,B is an isomorphism when B is a C * -algebra. It then follows by Theorem 3.10 that µ A,B is an isomorphism for all separable R * -algebras A and B.
It remains now to prove that Θ A,B is surjective. Let τ : A → Q(B) be a * -homomorphism such that there exists a * -homomorphism τ −1 : A → Q(B) with τ ⊕ τ −1 is a trivial extension. Then there exists a * -homomorphism Hence, Θ A,B is surjective, which completes the proof.

Asymptotic Morphisms and E-theory
In this section, we will use Theorem 3.10 to show that KK(A, B) is naturally isomorphic to E(A, B) for separable R * -algebras A and B, when A is nuclear.
The following definition of an asymptotic morphism for R * -algebras is exactly analogous to that for C * -algebras. This definition also appears in Section 8 of [5].
Definition 5.1. Let A and B be R * -algebras. An asymptotic morphism from A to B is a family φ t , (t ∈ [1, ∞)) of maps from A to B with the following properties: (1) for all a ∈ A, t → φ t (a) is bounded and continuous; and (2) The set φ t is asymptotically * -linear and multiplicative, i.e. (a) lim for all a ∈ A. is a * -homomorphism from A to αB, which we denote by φ t . If φ t is equivalent to ψ t , then φ t = ψ t . Let φ : A → αB be a * -homomorphism and let ψ : A → βB be a lifting of φ. Set φ t = ev t • ψ. Then φ t is an asymptotic morphism from A to B such that φ t = φ.
From the above paragraph, we have the following proposition (compare to Remark 25.1.4 (a) of [1]).

Proposition 5.3.
There is a bijection from the set of * -homomorphism from A to αB to the set of equivalence classes of asymptotic morphisms from A to B.
Remark 5.4. Every * -homomorphism from A to B defines an asymptotic morphism as follows: Let φ : A → B be a * -homomorphism. The φ induces an asymptotic morphism φ t by φ t (a) = φ(a) for all t ∈ [1, ∞) and a ∈ A. We will denote this asymptotic morphism by φ .
Definition 5.5. Let φ t , ψ t : A → B be asymptotic morphisms. φ t is homotopic to ψ t if there exists an asymptotic morphism Φ t : A → IB such that for each t ∈ [1, ∞) and a ∈ A, ev [0,1] 0 (Φ t (a)) = φ t (a) and ev Proposition 5.6. Let φ t and ψ t be equivalent asymptotic morphisms from A to B. Then φ t and ψ t are homotopic.
for all a ∈ A and for all t, t ′ ∈ [1, ∞), we have that is a continuous function for all a ∈ A.
Proof. Note that there exists an asymptotic morphism Φ t : A → IB such that Φ t = Φ. Then ev • Φ t are homotopic. Hence, φ t and ψ t are homotopic. Proof. Note that the complexification I C of I is an ideal of the complexification A C of A and A C is separable since A is separable. Let τ be a conjugate-linear involution of A C such that A can be identified with {x ∈ A C : τ (x) = x}. Then I can be identified as {x ∈ I C : τ (x) = x}. Since I C is an ideal of a separable C * -algebra A C , there exists a continuous quasi-central approximate identity Then it is easy to verify that {u t } t∈[1,∞) is again a continuous quasi-central approximate identity of I C (hence of I) and is again quasi-central in A C (hence in A).
Proof. Note that, Therefore, Therefore, The lemma now follows from the Stone-Weierstrass Theorem and the above observations. We note that the existence of the cross-section σ is given by the Bartle-Graves selection theorem. x) is continuous and bounded. Note that if τ is another cross section of π, then σ(x) − τ (x) ∈ B for all x ∈ A. Then for each (f, x) ∈ SR × A, by Lemma 5.9, Therefore, θ t does not depend on the choice of cross section (up to equivalence of asymptotic morphisms). Let x ∈ A, then the function f → θ t (f, x) is linear for each t ∈ [1, ∞). Let f ∈ SR. Since are elements of B for all x, y ∈ A and λ ∈ R, by Lemma 5.9 for all x, y ∈ A and λ ∈ R. Since σ(xy) − σ(x)σ(y) ∈ B for all x, y ∈ A and (f g)(u t ) = f (u t )g(u t ) for all f, g ∈ SR and t ∈ [1, ∞) and since for all f, g ∈ SR and x, y ∈ A.
Since f (u t )σ(x) ≤ f σ(x) for all f ∈ SR and x ∈ A and since θ t is independent of the choice of σ, we may choose a cross section of π such that σ(x) = x for all x ∈ A. Therefore, θ t defines a * -homomorphism from ψ : By construction, lim t→∞ (φ e t (f ⊗ a) − f (u t )σ(a)) = 0, for all f ∈ SR and a ∈ A. Also, by the above observation, τ is another cross section of π, then the induce * -homomorphism from SR ⊗ A to αB is ψ. Thus, the induced asymptotic morphism is equivalent to φ e t . Given an asymptotic morphism φ t : A → B and an element γ ∈ SA, the composition φ t • γ is an element of SB for all t ∈ [1, ∞). We let Sφ t denote the resulting map SA → SB and using the compactness of the circle, is it easy to show that Sφ t is an asymptotic morphism. Furthermore it is easy to show if two asymptotic morphisms φ t and ψ t are homotopic, then Sφ t and Sψ t are also homotopic. Thus, every asymptotic morphism from A to B induces a unique element of E (A, B). It also follows that there is a well-defined map Σ : E(A, B) → E(SA, SB), which can easily be shown to be a group homomorphism.
In fact, the map Σ is a special case of the tensor product construction for asymptotic morphisms described in Lemma II.B.β.5 of [7], which carries over to the case of R * -algebras. Given two asymptotic morphisms φ t : A → C and ψ t : B → D, there is a tensor product asymptotic morphism We also will need to make use of the associative product structure on E-theory described in Proposition II.B.β.4 of [7], which also carries over to the case of R * -algebras. Given two asymptotic morphisms φ t : A → B and ψ t : B → C, there is a composition asymptotic morphism ψ • φ t : A → C, defined uniquely up to homotopy. In the special case that ψ or φ is an actual * -homomorphism, then this product is a literal composition. In the general case, a reparametrization is necessary to construct an asymptotic morphism from the composition (see Lemma II.B.β.3). The resulting product induces a natural homomorphism E(A, B) ⊗ E(B, C) → E(A, C).

Theorem 5.13. E(A, B) is a bivariant functor from separable R * -algebras to abelian groups. In both arguments, it is homotopy invariant, stable, half exact, and has a degree 8 periodicity isomorphism.
Proof. The homotopy invariance is immediate and the stability follows from Proposition 3. We postpone the proof of half-exactness until after a series of lemmas.
For any elements x ∈ E(A, B) and z ∈ KK(B, C), the pairing described in the proof above gives an element in E(A, C) which we will denote by x ⊗ α z. Taking 1 A ∈ E(A, A) we obtain a homomorphism ǫ : KK(A, B) → E (A, B). For x ∈ E(A, B) and y ∈ E(B, C), we let x ⊗ E y denote the product in E(A, C). Similarly for z ∈ KK(A, B) and w ∈ KK(B, C), we let z ⊗ KK w denote the product in KK(A, C).
Proof. First we show that ǫ(x ⊗ KK y) = ǫ(x) ⊗ α y. Indeed, using Theorem 3.2 we have Secondly, we prove that z ⊗ α y = z ⊗ E ǫ(y) for z ∈ E (A, B). Indeed, this clearly holds for y = 1 B ∈ KK(B, B) so it holds in general by the uniqueness statement of Theorem 3.2.
From these facts, we complete the proof with the calculation Proof. Let α ∈ E(R, S 8 R) and β ∈ E(S 8 R, R) be Bott elements arising from the corresponding Bott elements in KK-theory via ǫ. Then it follows from Lemma 5.14 that α ⊗ E β = 1 R and β ⊗ α = 1 S 8 R . It follows that the map z → E(A, B). We call this isomorphism Θ. Now consider the homomorphism Σ 8 : E(A, B) → E (S 8 A, S 8 B). From the comments following Definition 5.12, this homomorphism can be expressed as z → 1 S 8 R ⊗ z. Then for z ∈ E(A, B) the composition is given by It follows that Σ : E(A, B) → E(SA, SB) is injective.
Since Θ • Σ 8 = id E(A,B) and since Θ is already known to be an isomorphism, it also follows that Replacing A and B with suspensions of appropriate degrees, we obtain that Σ : Finally, consider the diagram where the vertical maps and the lower map are all known to be isomorphisms. This diagram commutes modulo a homomorphism induced by the rearrangement of the order of the suspension factors of S 9 A and S 9 B. Since the rearrangement of the factors corresponds to an even permutation, it is homotopic to the identity and induces an identity homomorphism on E-theory. It follows that the Σ : E(A, B) → E(SA, SB) is an isomorphism.
Completion of proof of Theorem 5.13. Finally, to prove half-exactness let 0 → J ι → A q → B → 0 be an extension of separable R * -algebras and D be an R * -algebra. Suppose h is an asymptotic morphism from SD ⊗ K R to SA ⊗ K R such that [q • h] = [0]. Lemma 25.5.12 of [1] and its proof carry over to the real case, so there exists an asymptotic morphism k from the vertical maps are isomorphisms, so there exists an asymptotic morphism g from SD ⊗ K R to Half-exactness in the first argument is proved in a similar way. is an isomorphism when B is a C * -algebra. In the diagram we use KK C (−, −) and E C (−, −) to denote the versions of these functors on C * -algebras (for example E C (−, −) consists of homotopy classes of asymptotic morphisms that are asymptotically linear over C). Since A C is nuclear, the top horizontal homomorphism is an isomorphism by Theorem 25.6.3 of [1]. The left vertical homomorphism is an isomorphism by Lemma 4.3 of [3]. The right vertical homomorphism is defined by restriction-a complex asymptotic morphism defined on SA C ⊗ K restricts to a real asymptotic morphism defined on SA ⊗ K R -and it is an isomorphism since every real asymptotic morphism on SA⊗ K R can be extended uniquely to a complex asymptotic morphism on SA C ⊗ K. The square commutes by Theorem 3.7 of [15], since the two directions around the square give natural transformations KK C (A C , −) → E(A, −) of functors defined on separable C *algebras, each of which sends 1 A ∈ KK C (A C , A C ) to the asymptotic morphism represented by the inclusion of A into A C . Therefore, the bottom row is an isomorphism as desired.
Finally, we present below the real analog of Theorem 5.8 of [28], showing that E-theory is a special case of KK-theory. That we can obtain this result without reproducing the extensive technicalities of the proof in [28] shows the utility of Theorem 3.10.
Since Proof. Let ε ′ = γ •ε where ε is the isomorphism of Theorem 5.16. We know that ε ′ is an isomorphism in the complex case by Theorem 5.8 of [28]. So it suffices by Theorem 3.10 to show that ε ′ fits in a commutative square in the same way that ε does in the proof of Theorem 5.16. The square in question can be factored into two squares as follows: The first square is just a specialization of the square in Theorem 5.16. The second square commutes since the horizontal maps are homomorphisms that arise from stabilization, from long exact sequences, and from suspensions; all of which commute with the vertical restriction map.

Application: asymptotic morphisms on spheres
The goal of this section is to prove the following theorem, using the results of the previous sections, the universal coefficient theorem for real C * -algebras, and a united K-theory analysis.
Theorem 6.1. There exists asymptotic morphisms φ t that induce non-trivial homomorphisms on K-theory of the following forms: (1) S d R → K R where d > 0 and d ≡ 0, 4, 6, 7 (mod 8), (2) S d R → K R ⊗ H where d > 0 and d ≡ 0, 2, 3, 4 (mod 8) Furthermore, if n ∈ N does not satisfy the specified condition in each case, then there does not exist a corresponding asymptotic morphism that is non-trivial on K-theory.
Before we prove Theorem 6.1 we need to establish some relationships between asymptotic morphisms and mapping cones which are well-known for C * -algebras. Note that if φ : B → C is a * -homomorphism and ψ t : A → B is an asymptotic morphism, then φ • ψ t : A → C is an asymptotic morphism.
where φ e t is the asymptotic morphism associated to e. Proof. Let σ : A → E be a cross section of π such that σ(x) = x . Let {u t } t∈[1,∞) be an approximate identity of B that is quasi-central in E. For f ∈ SR and x ∈ A and s ∈ [0, 1], By the continuity of continuous functional calculus, we have that ψ t (f, x) is continous and φ t (f, x) is continuous. Since and For all f ∈ SR and x ∈ A, it is clear that t → ψ t (f, x) is continuous and bounded in ICA and that t → φ t (f, x) is the same in IE. Hence, the map t → (φ t (f, x), ψ t (f, x)) ∈ IC π , is continuous and bounded on [1, ∞).
Define Ψ : SR ⊗ A → βIC π by Ψ(f, x)(t) = (φ t (f, x), ψ t (f, x)). Let b ∈ B, x ∈ E, and t ∈ [1, ∞). Then Set f t ∈ I E by f t (s) = (1 − s)u t + s1 E . Then by the above paragraph, {f t } t∈[1,∞) is a family of positive elements in I E with f t ≤ 1 such that for all x ∈ E, where we identify b and x in I E as constant functions. Therefore, for all f, g ∈ SR, for all x, y ∈ A, and for all λ ∈ R, using Lemma 5.9 as in the proof of Proposition 5.10.
By the definition of ψ t , for all f, g ∈ SR, for all x, y ∈ A, and for all λ ∈ R, It is clear from the definition of both φ t and ψ t that φ t (·, x) and ψ t (·, x) are linear functions. From the above observations, there exists a * -homomorphism Φ : SR ⊗ A → αIC π such that if Φ is a set-theoretical lifting of Φ and Φ t = ev t • Φ is the associated asymptotic morphism, then for each f ∈ SR and x ∈ A, Hence, for all f ∈ SR and x ∈ A, Thus, ev • Φ t is homotopic to γ e • φ e t and ev • Φ t is homotopic to λ e . Since the asymptotic morphism Φ t gives a homotopy between ev Let B be an R * -algebra. Then we have the following exact sequence of R * -algebras The above exact sequence induces a long exact sequence in K-theory for each n ∈ Z. Since β 0 B is contractible, K * (β 0 B) = 0. Hence, (ρ B ) * is an isomorphism. Definition 6.3. Let A and B be R * -algebras and let φ t : A → B be an asymptotic morphism. Let φ t * denote the composition (ev The next three lemmas are well-known in the context of C * -algebras. We are not able to find a reference for R * -algebras so for the convenience of the reader we provided the proof here. Lemma 6.4. Let A and B be R * -algebras and let φ t , ψ t : A → B be asymptotic morphisms. If φ t is homotopic to ψ t , then φ t * = ψ t * . Proof. Let Φ t : A → C([0, 1], B) be an asymptotic morphism such that ev for all s ∈ [0, 1] and since for each f ∈ βIB, the function Note that ev A similar computation shows that (ev which implies that Lemma 6.5. Let A and B be R * -algebras and let φ : A → B be a * -homomorphism. Then φ * = φ * : K * (A) → K * (B).
Proof. Note that φ : A → βB which maps a to the constant function φ(a) is a * -homomorphism and φ = ρ B • φ t . Therefore, Lemma 6.6. Suppose φ : B → C is a * -homomorphism and ψ t : As an aside, we note that it is possible to prove a more general statement: the formula φ t • ψ t * = φ t * • ψ t * holds for asymptotic morphisms φ t : B → C and ψ t : A → B.  (1) and (2) follow as well as the converse statements.
We are now ready to prove Theorem 6.1.
Proof of Theorem 6.1. For d ≥ 1, where d is not in one of the required congruence classes, Theorem 6.7 implies that no asymptotic morphism can exist that induces a non-trivial homomorphism on K-theory.  C π ). Furthermore, the homomorphism of CRT -modules (λ e ) * : K CRT (A) → K CRT (C π ) is nonzero. This is because K CRT (A) is a free CRTmodule so the Universal Coefficient Theorem implies that is an isomorphism. Even more, this implies that the homomorphism on K-theory (λ e ) * : K * (A) → K * (C π ) is nonzero, by Theorem 4 of [4].

Almost commuting matrices
We can use all this machinery to find novel examples of almost commuting real symmetric matrices. Our approach is to use commutative R * -algebras and create asymptotic morphisms out of these. On the one hand these carry K-theory data that can distinguish them from actual * -homomorphisms. On the other, in the image the relations that make the R * -algebra commutative get turned into the property of being almost commuting.
Let F denote either R or C or H. We generalize a question of Halmos [12] to ask about d almost commuting self-adjoint matrices over F. Lin [23] showed that in the complex case the anwer is "yes" for d = 2 while it was known much earlier [35] that the answer is "no" in the complex case for d = 3.
For the quaternionic case, the result is the same: yes for d = 2 [27] and no for d = 3 [14]. These leaves the real case, arguably the most important case. We know the answer is "yes" for d = 2 ( [27]). We will show that the answer is "no" for d = 5, leaving open the cases d = 3, 4. The proof techniques used for a negative result for d = 3 in the complex and quaternionic cases rely on the fact that K −2 (H) = 0 and K −2 (C) = 0 and so will not work for F = R since K −2 (R) = 0. However, since K −4 (R) is nontrivial, we will see that these methods will apply for d = 5.
We start by connecting this problem to a problem couched in the theory of R * -algebras. For any sequence B n of R * -algebras, let π be the quotient map from the product ∞ n=1 B n to its quotient by the sum ∞ n=1 B n / ∞ n=1 B n . Problem 7.2. Does every * -homomorphism of the form n=1 is a sequence of integers, lift to a * -homomorphism such that ψ = π • ψ? Theorem 7.3. For a fixed positive integer d and division algebra F, if the answer to Problem 7.1 is "yes" then the answer to Problem 7.2 is also "yes".
We prove Theorem 7.3 below, following two lemmas. Then, the rest of the section is devoted to showing that the answer to Problem 7.2 is "no" when d = 5, using a K-theoretic obstruction. We start with a lemma stating the universal properties of C (S d−1 , R). This is a standard result that can be proven using the techniques of Chapter 3 of [24]. Lemma 7.5. For each ε > 0 there exists a δ > 0 such that the following holds. Let k 1 , . . . , k d be commuting self-adjoint elements in a unital R * -algebra A that satisfy Proof. Assume that 0 < δ < .5 and that the elements k i satisfy Since the elements k i commute, we can treat them as real-valued functions in C(X, R) for some compact space . Thus for any ε > 0, we can find a δ small enough to ensure Proof of Theorem 7.3. Suppose that the answer to Problem 7.1 is "yes" for some d and F, and let be a * -homomorphism. Then taking the images of the coordinate functions in C(S d−1 , F) we find that there exist sequences of matrices H in ∈ M m(n) (F) (i ∈ {1, . . . , d}, n ∈ N) that are asymptotically (as n → ∞) self-adjoint contractions, that satisfy d i=1 H 2 in = 1 asymptotically, and such that H in and H jn asymptotically commute for each i, j ∈ {1, . . . , d}. In fact, we may assume that each H in is exactly a self-adjoint contraction. This is achieved by replacing H in by 1 2 (H in + H * in ) and then by f (H in ) where f ∈ C(R, R) is defined by We may also assume without loss of generality that Since lim n→∞ L in − H in = 0 it follows by the uniqueness statement of Lemma 7.4 that π •ψ = ψ. Lemma 7.6. Suppose F is a division algebra and d ≥ 1. Any * -homorphism Proof. It suffices to show that any * -homomorphism ϕ : S d R → M m(n) (F) is homotopic to the zero map. As the codomain is finite dimensional, this ϕ must factor through C(X, R) where X is finite. Thus ϕ is a sum of point evaluations, each of which is homotopic to 0.
induced by the collection of projection maps In [11], there is a thorough analysis of the circumstances under which Θ is surjective and injective on K 0 and K 1 for C * -algebras B i . The following lemma states that the situation for R * -algebras partially reduces to that for C * -algebras.
Lemma 7.7. Let {B i } i∈N be a sequence of R * -algebras and let (B i ) C be their respective complexifications. Then the map

is an isomorphism if and only if
is an isomorphism.
Proof. Since K CRT is a CRT-module homomorphism, this follows immediately from the results of Section 2.3 of [6].
We shall say that a family of R * -algebras {B i } i∈N has property Θ if Θ CRT is an isomorphism. We shall say further that an R * -algebra B has property Θ if {B i } i∈N has property Θ where B i = B for all i ∈ N. In particular F ⊗ K R ∼ = K F has property Θ.
We note that if B has property Θ, then the * -homomorphism and the resulting map∆ into the quotient are both injective on K-theory and united K-theory.
Recall that we have βB = C b ([1, ∞), B) and αB = C b ([1, ∞), B)/C 0 ([1, ∞), B). We can evaluate an element of βB at the natural numbers, and get a surjection which passes to a * -homomorphismδ Lemma 7.8. Suppose A is a separable R * -algebra and B is a separable R * -algebra with property Θ.
is an asympotic morphism, then there is a * -homomorphism In particular, if φ t * is non-zero, then so is ψ * .
Proof. Define ψ =δ N • φ t . Consider the following diagram. The square commutes exactly. The triangle on the top commutes at the level of K-theory, as can be seen using the fact that δ 1 is homotopic to δ k for any k ∈ N and identifying the K-theory of the product with the product of the K-theory.
Using the relation φ t * = δ 1 • (ρ B ) −1 * • φ t * , an easy calculation shows that∆ * • φ t * = ψ * Lemma 7.9. Suppose B is a separable R * -algebra, and A is a finitely generated R * -subalgebra of Then there is a sequence m(1) < m(2) < on natural numbers so that A ⊆ More generally, for a finite set of elements in A, we can use a single sequence of projections as above to show that We now put these results together for our main theorem. Proof. Let d be as above. By Theorem 6.1, there exists an asymptotic morphism φ t : S d R → K F that induces a non-zero map on K-theory. Then by Lemma 7.8, we obtain a * -homomorphism of the form that is non-zero on K-theory. By Lemma 7.9, we have that this * -homomorphism factors through a * -homomorphism of the form it follows that φ * is also nonzero. Now φ cannot be lifted to a * -homomorphism ψ with values in The result above for d = 3, F = H, replicates results from [27]. Considering this case more closely, we can clarify the K-theory behind the invariant for 3D topological insulators considered in [14]. We consider H as sitting inside its complexification, so is to be regarded as the set of See [26]. The first several homotopy groups for Sp(n) stablize early. The connection with K-theory is that K 1 (C(S 3 , H)) is given by homotopy classes of maps from the three sphere into the symplectic unitary matrices of various sizes. Let x 1 , . . . , x 4 be the coordinate functions in R 4 restricted to the unit sphere S 3 . From [36, §10.6] we learn that is a generator of π 3 (Sp(1)) and so gives us a generator for K −3 (C(S 3 , R)) ∼ = K 1 (C(S 3 , H)) ∼ = Z/2.
is easy enough to work out, since it is entirely governed by the determinants of real othogonal matrices. It is a subgroup of When d = 3, the induced map ϕ * on K-theory is such that the generator of is sent to the class of 1, 1, . . .
This means that the Z 2 -index [14] can be non-trivial for arbitrarily small commutators. Now we switch to considering M N (H) as a real part of M 2N (C) with respect to the dual operation This is evidence that the phenomena of 3D systems with topologically protected states as witnessed in [14] would persist in larger system size.

The ten-fold way
Some of the motivation of this work is to explain index studies of topological insulators. The symmetries present in topological insulators are essential in their behavior. Physicists think of timereversal (TR) symmetry in terms of commutating relations with anti-unitary operators, but it is equivalent to think in terms of anti-multiplicative involutions.
For single particle systems, this picture suffices. There can be no TR symmetry, where the relevant algebra for a finite model is the C * -algebra M N (C). When TR symmetry is present, the TR symmetry comes in two forms, and leading to either the transpose or the dual operation. We are looking at Dyson's "three-fold" way, or real, complex or quaternionic matrices.
In more complex systems, one needs to track additional symmetries, in particular particle-hole symmetry. The picture here, in terms of algebras, is that there are two anti-multiplicative involutions τ and γ and the Hamiltonian H ∈ M N (C) now satisfies There are more general symmetry groups to consider, acting by unitary and anti-unitary operators. See [13]. For now, we consider just the symmetries beyond being self-adjoint.
In physics, it is expected that these two "τ -operations" will commute. As one can have either symmetry non-existant, or equivalent to the transpose, or equivalent to the dual operation, it seems that we will get nine symmetry classes. Instead, we find ten as the combined symmetry of τ followed by γ can be preserved, or broken, when each is individually broken. So condensed matter physics is now investigated in terms of the ten-fold way. See [18] for a nice introduction to the matrix classes that arise.
It is easy to check that when τ and γ commute, α(a) = (a τ ) γ defines an order-two automorphism. If we are to follow the ten-fold way, we need to be working with graded real C * -algebras. It was in the context of graded real C * -algebras KK-theory was introduced. In work on the standard model [8], Connes considered real structures on spectral triples. It seems likely that progress in graded real noncommutative geometry will have application in the study of topological insulators.