The Classification of Real

Purely Infinite Simple C*-Algebras

Jeffrey L. Boersema, Efren Ruiz, P. J. Stacey

Received: February 26, 2011

Communicated by Joachim Cuntz

Abstract. We classify real Kirchberg algebras using united Ktheory. Precisely, let A and B be real simple separable nuclear purely infinite C*-algebras that satisfy the universal coefficient theorem such that A_{ℂ} and B_{ℂ} are also simple. In the stable case, A and B are isomorphic if and only if K^{CRT}(A) ≅ K^{CRT}(B). In the unital case, A and B are isomorphic if and only if (K^{CRT}(A), [1_{A}]) ≅ (K^{CRT}(B), [1_{B}]). We also prove that the complexification of such a real C*-algebra is purely infinite, resolving a question left open from [43]. Thus the real C*-algebras classified here are exactly those real C*-algebras whose complexification falls under the classification result of Kirchberg [26] and Phillips [35]. As an application, we find all real forms of the complex Cuntz algebras O_{n} for 2 ≤ n ≤ ∞.

2010 Mathematics Subject Classification: 46L35, 46L80, 19K99 Keywords and Phrases: Real C*-algebras, K-theory, classification

1. Introduction

One of the highlights of the classification theory of simple amenable C*-algebras is the classification of purely infinite nuclear simple C*-algebras, obtained by Kirchberg and Phillips in [26] and [35]. This classification theorem relies in an essential way on the Universal Coefficient Theorem established by Rosenberg and Schochet in [40], where it was observed that “For reasons pointed out already by Atiyah, there can be no good Künneth Theorem or Universal Coefficient Theorem for the KKO groups of real C*-algebras; this explains why we deal only with complex C*-algebras”. Thus at the time of the Kirchberg and Phillips classification, the lack of a universal coefficient theorem was the primary barrier to extending the classification result to real C*-algebras. However, in [8], a new invariant called united K-theory was introduced for real C*-algebras and in [9] a universal coefficient theorem was proven for real C*-algebras using united K-theory. In the present paper, we take advantage of these developments to provide a complete classification of a class of real simple purely infinite C*-algebras in terms of united K-theory. The real C*-algebras that are classified are exactly those real C*-algebras for which the complexification is covered by the Kirchberg and Phillips theory. As an application of our classification we determine all the real forms of the complex Cuntz algebras O_{n} for 1 ≤ n ≤ ∞: there are two such forms when n is odd and one when n is even or infinite. The overall framework of the proof will be the same as that in the paper [35] and the underlying theory on which that paper was built. Furthermore, many of the proofs in the development leading to the main theorems of [35] carry over to the real case without significant change. In those cases, we will simply refer to the established proofs in the literature without reproducing them here. However there are many situations where the arguments in the real case require modification and we will then provide full proofs or full discussion of the necessary modifications. In Section 2, we describe the invariant of united K-theory and summarize its key properties. In Section 3 we then establish real analogues of some of the fundamental properties of purely infinite algebras, in the course of which we resolve a problem left hanging in [43] and [13] by showing that the complexification of a purely infinite simple real C*-algebra is also purely infinite (using the original definition for simple algebras). Following the complex case, as developed in [38], we then establish (in Theorem 5.2) criteria for two unital homomorphisms from the real Cuntz algebra O_{n}^{ℝ} (n even) to be approximately unitarily equivalent. Modifications of the complex arguments are required to establish some of the preliminary results: in Section 4 we modify the required results about exponential rank, noting that the close link between self-adjoint and skew-adjoint elements is absent in a real C*-algebra, and in Section 5 we modify the result from [15] establishing the Rokhlin property of the Bernoulli shift on the CAR-algebra. Our next step is to establish real analogues of Kirchberg’s tensor product theorems and his embedding theorem. This is achieved in Section 6 by using the relevant complex results and the embedding of ℂ into M_{2}(ℝ). In Sections 7, 8, 9 and 10, we closely follow [35] indicating how the results achieved for the complex case can be obtained in the real case. In particular, Section 7 contains a key result about uniqueness of homomorphisms from O_{∞}^{ℝ} to a real purely infinite C*-algebras. Section 8 contains the theory of asymptotic morphisms in the context of real C*-algebras and Section 9 culminates in a theorem identifying KK-theory to a group of asymptotic unitary equivalence classes of asymptotic morphisms as in Section 4 of [35]. To accomplish this, we make use of the axiomatic characterization of KK-theory for real C*-algebras established in [12]. This development culminates in Section 10, which contains the statements and proofs of our classification theorems, and in Section 11, which uses these results to describe the real forms of Cuntz algebras. The notation we use in these sections closely follows that in [35].

We will use the notation H^{ℝ} for a real Hilbert space; and B(H^{ℝ}) and K^{ℝ} for the real C*-algebras of bounded and compact operators H^{ℝ}. For the complex versions of these objects we will use H, B(H), and K. For a C*-algebra A, we will write M_{n}(A) for the matrix algebra over A; and M_{n} will stand for M_{n}(ℝ). Following standard convention, we will use O_{n} for the complex Cuntz algebras and O_{n}^{ℝ} for the real versions. The complexification of a real C*-algebra A will be denoted by A_{ℂ}. We will use Φ throughout to denote the conjugate linear automorphism of A_{ℂ} defined by a + ib ↦ a − ib (for a, b ∈ A). Note that A can be recovered from Φ as the fixed point set. Finally, a tensor product written as A ⊗ B will in most cases be the C*-algebra tensor product over ℝ, but should be understood to be a tensor product over ℂ if both A and B are known to be complex C*-algebras. Recall that if A and B are real C*-algebras, then (A ⊗ B)_{ℂ} ≅ A_{ℂ} ⊗ B_{ℂ}.

2. Preliminaries on United K-Theory

United K-theory was developed in the commutative context in [14] and subsequently extended to the context of real C*-algebras in [8]. United K-theory consists of the three separate K-theory modules as well as several natural transformations among them. In this section, we give the definition of united K-theory and summarize the features needed in this paper. Details are in [8], [9], [10].

Definition 2.1. Let A be a real C*-algebra. The united K-theory of A is given by

K^{CRT}(A) = {KO_{∗}(A), KU_{∗}(A), KT_{∗}(A), r, c, ε, ζ, ψ_{U}, ψ_{T} , γ, τ} .

In this definition, KO_{∗}(A) = K_{∗}(A) is the standard K-theory of a real C*- algebra, considered as a graded module over the ring K_{∗}(ℝ). This means in particular that there are operations

η_{O}: KO_{n}(A) → KO_{n+1}(A)
ξ: KO_{n}(A) → KO_{n+4}(A)
β_{O}: KO_{n}(A) → KO_{n+8}(A)

corresponding to multiplication by the elements of the same name in KO_{∗}(ℝ). The operation β_{O} is the periodicity isomorphism of real K-theory. The second item KU_{∗}(A) = K_{∗}(A_{ℂ}) is the K-theory of the complexification of A, having period 2. It is a module over K_{∗}(ℂ), which is to say that that there is an isomorphism of period 2 and the two remaining groups are independent with no operations between them. Finally, KT_{∗}(A) is the period 4 self-conjugate K-theory originally defined in the topological setting in [1]. In the non-commutative setting, it is more easily defined as KT_{∗}(A) = K_{∗}(T ⊗ A) in terms of the algebra T = {f ∈ C([0, 1], ℂ) | f(0) = \overline{f(1)}\} (see [8]). Self-conjugate K-theory is a module over the ring K_{∗}(T), giving operations

η_{O}: KT_{n}(A) → KT_{n+1}(A)
ω: KT_{n}(A) → KT_{n+3}(A)
β_{T}: KT_{n}(A) → KT_{n+4}(A) .

The rest of the information in united K-theory consists of operations

c_{n}: KO_{n}(A) ⟶ KU_{n}(A)  r_{n}: KU_{n}(A) ⟶ KO_{n}(A)
ε_{n}: KO_{n}(A) ⟶ KT_{n}(A)  ζ_{n}: KT_{n}(A) ⟶ KU_{n}(A)
(ψ_{U})_{n}: KU_{n}(A) ⟶ KU_{n}(A)  (ψ_{T})_{n}: KT_{n}(A) ⟶ KT_{n}(A)
γ_{n}: KU_{n}(A) ⟶ KT_{n−1}(A)  τ_{n}: KT_{n}(A) ⟶ KO_{n+1}(A)

among the three K-theory modules. For example, c is induced by the natural inclusion A → A_{ℂ}; r by the inclusion A_{ℂ} → M_{2}(A); and ψ_{U} by the involution Φ on A_{ℂ}. These operations are known to satisfy the following relations (see Proposition 1.7 of [8]):

rc = 2  ψ_{U}β_{U} = −β_{U}ψ_{U}  ξ = rβ2_{U}c
cr = 1 + ψ_{U}  ψ_{T} β_{T} = β_{T} ψ_{T}  ω = β_{T} γζ
r = τγ  εβ_{O} = β2_{T}ε  β_{T} ετ = ετβ_{T} + η_{T} β_{T}
c = ζε  ζβ_{T} = β2_{U}ζ  εrζ = 1 + ψ_{T}
(ψ_{U})^{2} = 1  γβ2_{U} = β_{T} γ  γcτ = 1 − ψ_{T}
(ψ_{T})^{2} = 1  τβ2_{T} = β_{O}τ  τ = −τψ_{T}
ψ_{T} ε = ε  γ = γψ_{U}  τβ_{T} ε = 0
ζγ = 0  η_{O} = τε  εξ = 2β_{T} ε
ζ = ψ_{U}ζ  η_{T} = γβ_{U}ζ  ξτ = 2τβ_{T} .

United K-theory takes values in the algebraic category of CRT-modules. A CRT-module consists of a triple (M^{O}, M^{U}, M^{T}) of graded modules, one over each of the rings K_{∗}(ℝ), K_{∗}(ℂ), and K_{∗}(T); as well as natural transformations c, r, ε, ζ, ψ_{U}, ψ_{T} , γ, τ that satisfy the above relations. For any real C*-algebra A, the CRT-module K^{CRT}(A) is acyclic, which means that the sequences

⋯ ⟶KO_{n}(A) ⟶^{η_{O}} KO_{n+1}(A) ⟶^{c} KU_{n+1}(A) ⟶^{rβ_{U}^{−1}} KO_{n−1}(A) ⟶ ⋯
⋯ ⟶KO_{n}(A) ⟶^{η_{O}^{2}} KO_{n+2}(A) ⟶^{ε} KT_{n+2}(A) ⟶^{τβ_{T}^{−1}} KO_{n−1}(A) ⟶ ⋯
⋯ ⟶KU_{n+1}(A) ⟶^{γ} KT_{n}(A) ⟶^{ζ} KU_{n}(A) ⟶^{1−ψ_{U}} KU_{n}(A) ⟶ ⋯

are exact. The important advantage of the full united K-theory over ordinary K-theory for a real C*-algebra A is that it yields both a Künneth formula (Theorem 4.2 of [8]) and a univeral coefficient theorem (Theorem 1.1 of [9]). For later reference, we now state two results that follow from those fundamental theorems.

Proposition 2.2. For any real C*-algebra A, (1) K^{CRT}(O_{2}^{ℝ} ⊗ A) = 0

(2) K^{CRT}(O_{∞}^{ℝ} ⊗ A) ≅ K^{CRT}(A).

Proof. By Table IV of [8], we have K^{CRT}(O_{2}^{ℝ}) = 0. Then (1) follows by the Künneth formula. The unital inclusion ℝ → O_{∞}^{ℝ} induces an isomorphism on united K-theory. This follows from Theorem 4 of [10] and the fact that the unital inclusion ℂ → O_{∞} induces an isomorphism on (complex) K-theory. Thus, Theorem 3.5 of [8] gives K^{CRT}(O_{∞}^{ℝ}) ⊗_{CRT} K^{CRT}(A) ≅ K^{CRT}(A) and Tor(K^{CRT}(O_{∞}^{ℝ}), K^{CRT}(A)) = 0. Then the isomorphism of (2) follows by the Main Theorem of [8]. □

Recall from [41] that the bootstrap class N is the smallest subcategory of complex, separable, nuclear C*-algebras that contains the separable type I C*- algebras; that is closed under the operations of taking inductive limits, stable isomorphisms, and crossed products by ℤ and ℝ; and that satisfies the two out of three rule for short exact sequences (i.e. if 0 → A → B → C → 0 is exact and two of A, B, C are in N, then the third is also in N).

Proposition 2.3 (Corollary 4.11 of [9]). Let A and B be real separable C*- algebras such that A_{ℂ} and B_{ℂ} are in N. Then A and B are KK-equivalent if and only if K^{CRT}(A) ≅ K^{CRT}(B).

This last result is the essential preliminary result for our classification of real purely infinite simple C*-algebras. We will also make use of Theorem 1 of [10], which states that every countable acyclic CRT-module can be realized as the united K-theory a real separable C*-algebra, indeed the C*-algebra can even be taken to be simple and purely infinite. We now describe a simpler variation of united K-theory that, by results from [23], contains as much information as the full version of united K-theory.

Definition 2.4. Let A be a real C*-algebra. Then

K^{CR}(A) = {KO_{∗}(A), KU_{∗}(A), r, c, ψ_{U}}

For any real C*-algebra, K^{CR}(A) is an acyclic CR-module, which means that the relations

rc = 2  ψ_{U}β_{U} = −β_{U}ψ_{U}  ξ = rβ2_{U}c
cr = 1 + ψ_{U}  ψ2_{U} = 1  ψ_{U}c = c

are satisfied and that the sequence

⋯ ⟶ KO_{n}(A) ⟶^{η_{O}} KO_{n+1}(A) ⟶^{c} KU_{n+1}(A) ⟶^{rβ_{U}^{−1}} KO_{n−1}(A) ⟶ ⋯

is exact.

Let Γ be the forgetful functor from the category CRT-modules to the category of CR-modules. It is immediate from Theorem 4.2.1 of [23] that Γ is injective (but not surjective) on the class of acyclic CRT-modules. Hence we have the following result.

Proposition 2.5. Let A and B be real C*-algebras. Then K^{CRT}(A) ≅ K^{CRT}(B) if and only if K^{CR}(A) ≅ K^{CR}(B).

Note, however, that the results of [10] do not extend to CR-modules. Not every countable acyclic CR-module can be realized as K^{CR}(A) for a real C*-algebra A.

3. Preliminaries on Real Simple Purely Infinite C*-Algebras

In this section, we provide some preliminaries on simple and purely infinite C*-algebras, including a theorem characterizing simple purely infinite real C*- algebras in terms of their complexification. One direction of this characterization was achieved in [43] and [13]. Let A be a real unital C*-algebra, let U(A) denote the group of unitary elements in A, and let U_{0}(A) denote the connected component of the identity in U(A). Note that if u is a unitary in a real C*-algebra, then the spectrum σ(u) ⊆ 𝕋 satisfies \overline{σ(u)} = σ(u) and the real C*-algebra generated by u is isomorphic to the algebra of complex-valued continuous functions f on σ(u) that satisfy \overline{f(z)} = f(\overline{z}). (If a is an element of A, then by definition the spectrum σ(a) is found by passing to A_{ℂ}.) We begin by making an explicit mention of a fairly well-known result about real simple C*-algebras.

Definition 3.1. A real C*-algebra A is c-simple if A_{ℂ} is simple.

Proposition 3.2. A simple real C*-algebra A is either c-simple or is isomorphic to a simple complex C*-algebra.

Proof. Let I be a proper ideal in A_{ℂ}. Then J = A ∩ I ∩ Φ(I) = 0 and so I ∩ Φ(I) = 0. Furthermore, I + Φ(I) = A_{ℂ}. It then follows that the map x ↦ x + Φ(x) is an isomorphism from the complex C*-algebra I onto A. □

As the structure of simple complex C*-algebras is comparatively wellunderstood, our primary interest lies in c-simple C*-algebras. As in the complex case, we will use the tilde ∼ to denote the relation of Murrayvon Neumann equivalence of projections. A projection is said to be infinite if it is Murray-von Neumann equivalent to a proper subprojection of itself. The following definition of purely infinite is from [43]. Bearing in mind subsequent developments, such as [27] and [28], a different definition should be made in the non-simple case. However the focus in this paper is on simple algebras, for which the definition below is appropriate.

Definition 3.3. Let A be a real simple C*-algebra.

(1) A subalgebra B is a regular hereditary subalgebra of A if there is an element x ∈ A_{+} such that B = \overline{xAx}.

(2) A is purely infinite if every regular hereditary subalgebra of A contains an infinite projection.

Proposition 3.4. Let A be a separable simple purely infinite real C*-algebra. Then either A is unital or there is a real unital simple purely infinite C*-algebra A_{0} such that A ≅ K^{ℝ} ⊗ A_{0}.

Proof. As in Section 27.5 of [2]. □

Proposition 3.5. Let A be a simple purely infinite C*-algebra and let p be a projection in A. Then pAp and A are stably isomorphic.

Proof. In the complex case, this result follows from Corollary 2.6 of [16]. The proof of that result and the proofs of the preliminary lemmas of Section 2 of [16] work the same in the real case. □

For the rest of this section, f_{ε} will denote the real-valued function such that f_{ε}(t) = 0 for t ≤ ε/2, f_{ε}(t) = 1 for t ≥ ε, and f_{ε}(t) is linear on [ε/2, ε].

Lemma 3.6. For any real C*-algebra A, the following are equivalent. (1) For any non-zero a, b ∈ A there exist x, y ∈ A with a = xby.

(2) For any non-zero positive a, b ∈ A there exists x ∈ A with a = xbx^{∗}.

Proof. (2) ⇒ (1). Let 0 ≠ a, b ∈ A. As in the complex case, described in 1.4.5 of [33], there exists u ∈ A with a = u(a^{∗}a)^{1/4}. Let x ∈ A with (a^{∗}a)^{1/4} = xbb^{∗}x^{∗} and observe that a = (ux)b(b^{∗}x^{∗}).

(1) ⇒ (2). This uses the argument for the complex case, from Lemma 1.7 and Proposition 1.10 of [18]. If a, b ∈ A are positive and non-zero and ε is chosen so that f_{ε}(b) ≠ 0 then a = (zz^{∗}zk)b(zz^{∗}zk)^{∗}, where x, y are chosen so that a^{1/6} = xf_{ε}(b)y, k ≥ 0 is chosen so that f_{ε/2}(b) = kbk and z = x(f_{ε}(b)yy^{∗}f_{ε}(b))^{1/2}. □

Lemma 3.7. Let A be a real C*-algebra such that for all non-zero elements a, b there exist x, y with a = xby. Suppose that A contains a non-zero projection and let c be a non-zero positive element such that \overline{cAc} ≠ A. Then \overline{cAc} contains an infinite projection.

Proof. The argument from (vii) ⇒ (i) of Theorem 2.2 of [31] applies to the real case to show that for any non-trivial projection p and positive element x there is a Murray-von Neumann equivalence between p and a subprojection of x. We will repeatedly use this fact. In the unital case, this shows that the unit 1 is Murray-van Neumann equivalent to a projection of \overline{cAc}, which is necessarily infinite. Now suppose that A has no unit but has a non-zero projection p. Applying the fact above to a non-zero positive element d in (1−p)A(1−p) gives a projection q such that p ∼ q and p ⊥ q. Now apply the fact again using the projection p + q and the positive element p to show that p + q is infinite. Finally, apply the same fact using the projection p + q and the positive element c to show that p + q is Murray-von Neumann equivalent to a projection in \overline{cAc}. □

Lemma 3.8. Let A be a real simple C*-algebra. Then the following are equivalent:

(1) A is purely infinite,

(2) A is not isomorphic to ℝ, ℂ, or ℍ and for each pair of non-zero elements a, b ∈ A there exist x, y ∈ A such that a = xby,

(3) A is not isomorphic to ℝ, ℂ, or ℍ and for each pair of non-zero positive

elements a, b ∈ A there exists x ∈ A such that a = xbx^{∗}. Furthermore, if these conditions are satisfied, then for all ε > 0 the element x in (3) can be chosen to satisfy ∥x∥ ≤ (∥a∥/∥b∥)^{1/2} + ε.

Proof. As the result is well-known in the complex case, we may assume by Theorem 3.1 that A is c-simple. By Lemma 3.6, (2) and (3) are equivalent. For (1) ⇒ (2), let a, b be non-zero elements of A, identified with e_{11}(K^{ℝ} ⊗ A)e_{11}. We are assuming A_{ℂ} is simple, so Theorem 2.4 of [17] applied to the unital algebra pAp implies that K ⊗ pA_{ℂ}p is algebraically simple. Then by Proposition 3.5, K⊗A_{ℂ} is algebraically simple, whence K^{ℝ}⊗A is. The argument from (ii) ⇒ (xi) of Theorem 2.2 of [31], then produces x, y ∈ K^{ℝ} ⊗ A with a = xby, so a = (e_{11}xe_{11})b(e_{11}ye_{11}). For (2) ⇒ (1), we use a simplified argument based on the proof of Theorem 1.2 of [31]. Note first that if a nonzero projection can be found in A then Lemma 3.7 gives the result. (In particular, this takes care of the unital case.) Let a, d be non-zero positive elements of A with da = ad = a (for a positive element x with norm 1 take a = f_{1/2}(x) and d = f_{1/4}(x)). Then let s, t ∈ A with d = sat and let y = (as^{∗}sa)^{1/2}t. An easy argument shows that |y||y^{∗}| = |y^{∗}| hence f_{1/2}(|y|)f_{1/8}(|y^{∗}|) = f_{1/8}(|y^{∗}|). Unless f_{1/4}(|y|) is a projection, Lemma 4.2 of [7] gives a scaling element t ∈ A. In this case, p_{n} = f_{n} + f_{n}^{1/2}tf_{n}^{1/2} + f_{n}^{1/2}t^{∗}f_{n}^{1/2} (where f_{n} = t^{n}(t^{∗})^{n} − t^{n+1}(t^{∗})^{n+1} for n ≥ 2) is a projection by Theorem 3.1 of [7], . The final condition holds as in Lemma 2.4 of [28]. □

Theorem 3.9. A real c-simple C*-algebra A is purely infinite if and only if A_{ℂ} is purely infinite.

Proof. From Theorem 3.3 of [43] we know that A is purely infinite if A_{ℂ} is. For the converse, suppose A is purely infinite, let ω be a free ultrafilter on ℕ and let A_{ω} be the corresponding ultrapower algebra, defined in Definition 6.2.2 of [39]. Note that the proofs of Proposition 6.2.6 of [39] and the preliminary Lemma 6.2.3 carry over directly to the real case (using Lemma 3.8). Therefore A_{ω} is simple and purely infinite. Suppose that D is a dimension function, as defined in Definition I.1.2 of [5], on the complexification (A_{ω})_{ℂ} ≅ (A_{ℂ})_{ω}. For each positive non-zero a, b in A_{ω} there exist x, y ∈ A_{ω} with b = xax^{∗} and a = yby^{∗} so D(a) = D(b). For any infinite projection p ∈ A_{ω}, there exists a projection q < p with D(p) = D(q) + D(p − q) = D(p) + D(p), so D(a) = D(p) = 0 for each positive a ∈ A_{ω}. Then for each positive a ∈ (A_{ω})_{ℂ}, we have 0 ≤ D(a) ≤ D(a + Φ(a)) = 0. So there is no dimension function on (A_{ω})_{ℂ} and therefore, by Theorem II.2.2 of [5], no 2-quasitrace. Therefore A_{ℂ} is weakly purely infinite by Theorem 4.8 of [28]. By Corollary 4.16 of [28] it is therefore purely infinite. □

Corollary 3.10. (1) If A and B are stably isomorphic real C*-algebras,

and if A is purely infinite and c-simple then so is B.

(2) Any inductive limit of real purely infinite c-simple C*-algebras is again purely infinite and c-simple.

(3) If A and B are purely infinite and c-simple, then so is A ⊗_{min} B.

Proof. These results follow immediately from Theorem 3.9 and the same results in the complex case (see Proposition 4.1.8 of [39]). □

We now work toward showing that the K_{0} and K_{1} groups of a real purely infinite algebra can be described in a similar way to the complex case. The next two lemmas provide the required modification of Lemma 1.7 of [19].

Lemma 3.11. Let A be a real c-simple purely infinite unital C*-algebra and let u ∈ U(A) and let λ ∈ σ(u). For any ε > 0 there exists v ∈ U(A) such that ∥u − v∥ < ε and

(1) if λ = λ^{∗} then v = v_{0} + λp where p is a non-zero projection in A and v_{0} ∈ U(p^{⊥}Ap^{⊥}).

(2) if λ ≠ λ^{∗} then v = v_{0} + λp_{1} + λ^{∗}p_{2} where p_{1} and p_{2} are orthogonal non-zero orthogonal projections in A_{ℂ} satisfying Φ(p_{1}) = p_{2} and v_{0} ∈ U((p_{1} + p_{2})^{⊥}A(p_{1} + p_{2})^{⊥}).

Proof. First assume that λ = λ^{∗}. Let h be a positive function on σ(u) such that supp(h) ⊂ N_{ε_{0}}(λ) and h(z^{∗}) = h(z) for all z ∈ σ(u). Then h(u) ∈ A and let p be a non-zero projection in \overline{h(u)Ah(u)}. As in the proof of Lemma 1.7 of [19], we have ∥u − (p^{⊥}up^{⊥} + λp)∥ ≤ 3ε_{0}. Then the polar decomposition of (p^{⊥}up^{⊥} + λp) yields a unitary v of the required form that, if ε_{0} is sufficiently small, will satisfy ∥u − v∥ < ε. Now assume λ ≠ λ^{∗}. Choose ε_{0} small enough so that N_{ε_{0}}(λ) ∩ N_{ε_{0}}(λ^{∗}) = ∅. Let h_{1} be a positive function on σ(u) such that supp(h_{1}) ⊂ N_{ε_{0}}(λ). By Theorem 3.9, A_{ℂ} is purely infinite so there is a non-zero projection p_{1} in B = \overline{h_{1}(u)A_{ℂ}h_{1}(u)}. Define p_{2} = Φ(p_{1}) ∈ Φ(B) and p = p_{1} + p_{2}. Now Φ(h_{1}(u)) = h_{2}(u) where h_{2} is the continuous function on σ(u) defined by h_{2}(z) = h_{1}(z^{∗}). Since supp(h_{2}) ⊂ N_{ε_{0}}(λ^{∗}), we have h_{1}(u)h_{2}(u) = 0. Thus p_{1} and p_{2} are orthogonal projections and p ∈ A. As in Lemma 1.7 of [19], we have ∥up_{1} − λp_{1}∥ ≤ ε_{0} and ∥up_{2} − λ^{∗}p_{2}∥ ≤ ε_{0} from which it follows that ∥u − (p^{⊥}up^{⊥} + λp_{1} + λ^{∗}p_{2})∥ ≤ 8ε_{0}. The required unitary v is obtained by taking the polar decomposition of p^{⊥}up^{⊥} + λp_{1} + λ^{∗}p_{2} in A. □

Lemma 3.12. Let A and u be as above. Then there is a projection p in A and a unitary v in U(p^{⊥}Ap^{⊥}) such that u ∼ v + p.

Proof. If 1 ∈ σ(u) then using Lemma 3.11, approximate u by an element of the form v + p. If the approximation is close enough, then the two unitaries will be in the same path component. If λ ∈ σ(u) where λ ≠ λ^{∗}, use Lemma 3.11 to approximate u by v +λp_{1} +λ^{∗}p_{2}. Then we can easily find a path from λp_{1} +λ^{∗}p_{2} to p_{1} +p_{2} in (p_{1} +p_{2})A(p_{1} +p_{2}). The only possibility left is u = −1. In that case, find two orthogonal projections q_{1} and q_{2} and a partial isometry s such that ss^{∗} = q_{1} and s^{∗}s = q_{2}. Let p = q_{1} + q_{2}. The projection p can be rotated to −p within the 2 × 2 matrix algebra generated by q_{1}, q_{2} and s. Hence the unitary −1 = −(p^{⊥}) + −p can be connected to the unitary −(p^{⊥}) + p. □

Proposition 3.13. Let A be a c-simple purely infinite real C*-algebra. Then (1) K_{0}(A) = {[p] | p is a non-zero projection in A}

(2) K_{1}(A) = U(A)/U_{0}(A) (for A unital).

Proof. In the complex case, these results are proven in Section 1 of [19]. The proofs of those results as well as the proofs of the preliminary lemmas carry over to the real case, with two modifications. The first is to the proof of Lemma 1.7 of [19], which we already addressed with the proof of Lemma 3.12 above. Secondly, in the proof of Lemma 1.1 of [19] the author uses an element of the form

w̃ = w + w^{∗} + (1 − w^{∗}w − ww^{∗}), (where w^{2} = 0)

that is a unitary lying in the finite dimensional C*-algebra generated by w. In the complex case it follows that w̃ ∈ U_{0}(A), whereas in the real case unitary groups of finite dimensional C*-algebras are not connected in general. However, if instead we take w̃ = w − w^{∗} + (1 − w^{∗}w − ww^{∗}) then w̃ is in the connected component of the identity, as it corresponds to a matrix of the form ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). The proof of Lemma 1.1 of [19] can be completed without change using this alternative w̃. □

We note that part (1) of Proposition 3.13 appeared as Proposition 11 of [10].

4. Exponential Rank

Definition 4.1. An element a in a real C*-algebra A is skew-adjoint if a^{∗} = −a. The set of skew-adjoint elements is denoted by A_{sk}.

If a is skew-adjoint, then σ(a) = −σ(a) ⊆ iℝ and the real unital C*-algebra generated by a is isomorphic to

{f ∈ C(σ(a), ℂ) | f(it)^{∗} = f(−it)} .

Furthermore, if a is a skew-adjoint element in a real unital C*-algebra A, then exp(a) is a unitary in A.

Lemma 4.2. Let A and B be unital real C*-algebras.

(1) U_{0}(A) = {∏_{i=1}^{n} exp(k_{i}) | k_{i} ∈ A_{sk}, n ∈ ℕ}.

(2) If α: A → B is unital and surjective, then α(U_{0}(A)) = U_{0}(B).

Proof. Suppose first that u is a unitary element in A with ∥u − 1∥ < 2. Then −1 ∉ σ(u). We define a continuous function f: 𝕋 \ {0} → i(−π, π) by f(exp(it)) = it for t ∈ (−π, π). Then f(u) is in the real C*-algebra generated by u, is skew-adjoint, and satisfies exp(f(u)) = u. More, generally, if u ∈ U_{0}(A) then there exists a chain u = u_{0}, u_{1}, u_{2}, …, u_{n} = 1 with ∥u_{i−1} − u_{i}∥ < 2 for all i ∈ {1, 2, …, n}. Then applying the previous paragraph we have u_{i−1}u_{i}^{∗} = exp(k_{i}) for all i with 1 ≤ i ≤ n. Then u = ∏_{i=1}^{n} exp(k_{i}). Conversely, if {k_{i}}_{i=1}^{n} is any collection of skew-adjoint elements, then u(t) = ∏_{i=1}^{n} exp(tk_{i}) for 0 ≤ t ≤ 1 is a continuous path of unitaries from 1_{A} to ∏_{i=1}^{n} exp(k_{i}). This proves (1). For (2), the inclusion α(U_{0}(A)) ⊆ U_{0}(B) is immediate. Let u ∈ U_{0}(B). Then u = ∏_{i=1}^{n} exp(k_{i}) for some skew-adjoint elements k_{i} ∈ B. Let l_{i} ∈ A be elements such that α(l_{i}) = k_{i}. We may assume that l_{i} is skew-adjoint for all i, by replacing with \frac{1}{2}(l_{i} − l_{i}^{∗}) if necessary. Then u = α (∏_{i=1}^{n} exp(l_{i})). □

Let E = {exp(k) | k ∈ A_{sk}} and let E^{n} be the set of all products of at most n elements of E. Thus U_{0}(A) = ∪_{n=1}^{∞}E^{n}. The argument in the proof above also implies that the set E^{n+1} contains the topological closure of E^{n} so that we have the an increasing sequence

E ⊆ \overline{E} ⊆ E^{2} ⊆ \overline{(E)^{2}} ⊆ ⋯ ⊆ (E)^{n} ⊆ \overline{(E^{n})} ⊆ (E)^{n+1} ⊆ …

similar to that in [37], motivating the following definition. Definition 4.3.

(1) The exponential rank of A, written cer(A), is equal to the integer n if E^{n} is the smallest set in this sequence to be equal to U_{0}(A) and is equal to the symbol n + ε if \overline{E^{n}} is the smallest set to be equal to U_{0}(A). If E^{n} ≠ U_{0}(A) for all n then cer(A) = ∞.

(2) The exponential length of A, written cel(A), is equal to the smallest number 0 < cel(A) ≤ ∞ such that every unitary u in U_{0}(A) can be written in the form

u = exp(k_{1}) exp(k_{2}) ⋯ exp(k_{n})

where k_{i} ∈ A_{sk} and

∥k_{1}∥ + ∥k_{2}∥ + ⋯ + ∥k_{n}∥ ≤ cel(A) .

With these definitions, the proofs of Section 2 of [37] can be applied with minimal modification to prove the following results.

Lemma 4.4. Let A be a real unital C*-algebra and let n be a positive integer. (1) If cel(A) < nπ then cer(A) ≤ n.

(2) If cel(A) ≤ nπ then cer(A) ≤ n + ε.

Lemma 4.5. Let A be a real unital C*-algebra. If every unitary u ∈ U_{0}(A) can be connected to the identity by a rectifiable path of length no more than M, then cel(A) ≤ M.

Definition 4.6. A real C*-algebra A has real skew rank zero if the elements of A_{sk} with finite spectrum are dense in A_{sk}.

In the case of a complex C*-algebra A there is a bicontinuous bijection A_{sa} → A_{sk} given by multiplication by i, showing that A has skew rank zero if and only it has real rank zero. However, in the case of real C*-algebras things are more subtle. For example the condition of being skew-rank zero is not equivalent (in the unital case) to the condition that the invertible elements of A_{sk} are dense. Indeed, all finite dimensional real C*-algebras have real skew rank zero, but the invertibles of (M_{n})_{sk} are dense only if n is even.

Proposition 4.7. Let A be a real unital c-simple purely infinite C*-algebra satisfying [1] ∈ 2K_{0}(A). Then the invertibles of A_{sk} are dense in A_{sk} and A has real skew rank zero.

Proof. Let A be a real purely infinite C*-algebra such that [1] ∈ 2K_{0}(A). Let a ∈ A_{sk} and let ε > 0 be given. Define functions g: iℝ → ℝ and f: iℝ → iℝ by

g(it) = max\{ε − |t|, 0\}  and  f(it) = \begin{cases}i(t + ε) & t ≤ −ε \\ 0 & |t| < ε \\ i(t − ε) & t ≥ ε . \end{cases}

Then g(a) ∈ A_{+} and f(a) ∈ A_{sk}. Since A is purely infinite, there is a projection p ∈ \overline{g(a)Ag(a)} with 2[p] = [1] ∈ K_{0}(A). Then [1 − p] = [p] so there is a partial isometry s such that s^{∗}s = 1 − p and ss^{∗} = p. Since f(a)g(a) = 0 we have f(a) = (1 − p)f(a)(1 − p). Let b = f(a) + ε(s − s^{∗}). In matrix form under the decomposition indicated by the projection sum 1 = (1 − p) + p we have

b = (\begin{matrix}f(a) & −ε \\ ε & 0 \end{matrix})

whence b is invertible. This proves the first statement. For the second statement, again let a ∈ A_{sk} and let ε > 0 be given. By the first part of the theorem, we may assume that a is invertible, hence σ(a) ⊂ iℝ \ {0}. Write a = a_{1} + a_{2} where the elements a_{i} ∈ A_{ℂ} satisfy σ(a_{1}) ⊂ i(0, ∞) and σ(a_{2}) ⊂ i(−∞, 0). Note also that Φ(a_{1}) = a_{2}. Since A_{ℂ} is simple and purely infinite it has real rank zero, so there exists b_{1} ∈ (A_{ℂ})_{sk} such that σ(b_{1}) is a finite subset of iℝ^{+} and ∥a_{1} − b_{1}∥ < ε/2. Let b_{2} = Φ(b_{1}) and let b = b_{1} + b_{2}. Then b is a skew-adjoint element of A with finite spectrum and ∥a − b∥ < ε. □

Lemma 4.8. Let A be a real c-simple unital C*-algebra such that [1] ∈ 2K_{0}(A). Let u ∈ U(A) be a unitary such that σ(u) ≠ S^{1}. Then for every ε > 0 there is a unitary v with finite spectrum such that ∥u − v∥ < ε.

Proof. If −1 ∉ σ(u), then there is a continuous function f: σ(u) → i[−π, π] that is a right inverse to the function it ↦ exp(it) and that satisfies f(z^{∗}) = f(z)^{∗}. Then f(u) ∈ A_{sk} can be approximated within δ by a skew adjoint element b with finite spectrum by Proposition 4.7. For an appropriate choice of δ, this implies that exp(b) ∈ U(A) approximates u within ε. Similarly, if 1 ∉ σ(u), then there is a continuous function f: σ(u) → i[−π, π] that is a right inverse to the function it ↦ − exp(−it) and that satisfies f(z^{∗}) = f(z)^{∗}. In the general case, suppose that λ ∉ σ(u) for some λ ∈ S^{1}. Let σ_{1} = {w ∈ σ(u) | Re(w) > Re(λ)} and let σ_{2} = {w ∈ σ(u) | Re(w) < Re(λ)}. Then σ = σ_{1} ∪ σ_{2}. Let u_{i} = u_{i}E_{u}(σ_{i}), where E_{u}(σ_{i}) denotes the spectral projection of u associated with the clopen subset σ_{i} of σ. Then 1 ∉ σ(u_{2}) and −1 ∉ σ(u_{1}). Using the results from the first two paragraphs, let v_{i} be a unitary that approximates u_{i} in E_{u}(σ_{i})AE_{u}(σ_{i}) within ε. Then since u = u_{1} + u_{2} we have that v = v_{1} + v_{2} is a unitary that approximates u within ε. □

Lemma 4.9. Let A be a real unital simple purely infinite C*-algebra let u ∈ U(A) and let {λ_{1}, …, λ_{n}} be a subset of σ(u) that is closed under conjugation. For any ε > 0 there exist v ∈ U(A) and orthogonal projections p_{1}, …, p_{n} ∈ A_{ℂ} such that ∥u − v∥ < ε and v = v_{0} + λ_{1}p_{1} + ⋯ + λ_{n}p_{n} with v_{0} ∈ U((p_{1} + ⋯ + p_{n})^{⊥}A(p_{1} + ⋯ + p_{n})^{⊥}). Furthermore, the elements ∑_{i=1}^{n} p_{i} and ∑_{i=1}^{n} λ_{i}p_{i} are both in A.

Proof. Use the constructions of Lemma 3.11 above as in the proof of Lemma 6 of [34]. □

Lemma 4.10. Let A be a real unital C*-algebra and let u ∈ U(A). For any ε > 0 there exists an h ∈ M_{2}(A)_{sk} such that ∥u ⊕ u^{∗} − exp(h)∥ < ε.

Proof. As in the proof of Corollary 5 of [34], there exists a continuous path v(t) of unitaries in M_{2}(A) with v(0) = 1 and v(π/2) = u ⊕ u^{∗} such that −1 ∉ σ(v(t)) for 0 ≤ t < π/2. Thus we can find a t close enough to π/2 such that ∥u ⊕ u^{∗} − v(t)∥ < ε and v(t) = exp(h) for a skew-adjoint h. □

Lemma 4.11. Let A be a real unital c-simple purely infinite C*-algebra such that [1] ∈ 2K_{0}(A). Let e_{1}, e_{2}, e_{3}, e_{4} be nonzero orthogonal projections in A that sum to 1. Let a be a partial isometry such that a^{∗}a = e_{2} and aa^{∗} = e_{3}. Let u ∈ U(e_{1}Ae_{1}) and v ∈ U(e_{2}Ae_{2}) be unitaries with σ(u) = S^{1}. Then for all ε > 0 there is a unitary z ∈ U(A) and a unitary w ∈ U(e_{4}Ae_{4}) with finite spectrum such that

∥z^{∗}(u + 1 − e_{1})z − (u + v + av^{∗}a^{∗} + w)∥ < ε .

Proof. This proof closely follows that of Lemma 7 of [34]. By Lemma 4.10 there is a unitary in (e_{2} + e_{3})A(e_{2} + e_{3}) that is arbitrarily close to v + av^{∗}a^{∗} and that has the form exp h for h ∈ A_{sk}. This in turn can be approximated by a unitary that has finite spectrum by Proposition 4.7. The general form of such a unitary is

∑_{k=1}^{n}(λ_{k}q_{k1} + λ_{k}^{∗}q_{k2}) + 1q_{01} + (−1)q_{02}

where λ_{k}^{∗} ≠ λ_{k}, the nonzero projections q_{ki} ∈ A_{ℂ} satisfy Φ(q_{k1}) = q_{k2} for 1 ≤ k ≤ n, and the (possibly zero) projections q_{0i} are in A. Furthermore, the q_{ki} are orthogonal and sum to e_{2} + e_{3}. Without loss of generality, we assume that v + av^{∗}a^{∗} has this form. With an obvious choice of coefficients λ_{ki} we can write this as

v + av^{∗}a^{∗} = ∑_{k=0}^{n} ∑_{i=1}^{2} λ_{ki}q_{ki} = ∑ λ_{ki}q_{ki} .

(Henceforth in this proof will use an undecorated ∑ to represent a double sum indexed as ∑_{k=0}^{n} ∑_{i=1}^{2}.) Now we replace u by a nearby element of the form given by Lemma 4.9. Specifically, there are orthogonal projections p_{ki} ∈ e_{1}A_{ℂ}e_{1} and, setting p = e_{1} − ∑ p_{ki} ∈ A, there is a unitary u_{0} ∈ pAp such that

u = u_{0} + ∑ λ_{ki}p_{ki}

(where the projection p_{0i} = 0 if and only if q_{0i} = 0). For each k ∈ {1, …, n} let c_{k1} ∈ A_{ℂ} be a partial isometry such that c_{k1}^{∗}c_{k1} = p_{k1} and c_{k1}c_{k1}^{∗} < p_{k1}. Then c_{k2} = Φ(c_{k1}) satisfies c_{k2}^{∗}c_{k2} = p_{k2} and c_{k2}c_{k2}^{∗} < p_{k2} and c_{k} = c_{k1}+c_{k2} ∈ A satisfies c_{k}^{∗}c_{k} = p_{k1}+p_{k2} and c_{k}c_{k}^{∗} < p_{k1}+p_{k2}. For k = 0 we obtain partial isometries c_{0k} ∈ A such that c_{0i}^{∗}c_{0i} = p_{0i} and c_{ki}c_{ki}^{∗} < p_{ki}. Then c = p + ∑ c_{ki} ∈ A satisfies

c^{∗}c = e_{1},  cc^{∗} = e_{1} − ∑(p_{ki} − c_{ki}c_{ki}^{∗}),  and
cuc^{∗} = u_{0} + ∑ λ_{ki}c_{ki}c_{ki}^{∗} .

Similarly we can find a collection of partial isometries d_{ki} with domain projection q_{ki} and range projection a subprojection of p_{ki} − c_{ki}^{∗}c_{ki} that also satisfy Φ(d_{k1}) = d_{k2} for k ≠ 0 and Φ(d_{ki}) = d_{ki} for k = 0. Then the partial isometry d = ∑ d_{ki} ∈ A satisfies

d^{∗}d = e_{2} + e_{3},  dd^{∗} ≤ ∑(p_{ki} − c_{ki}c_{ki}^{∗}),  and
d (∑ λ_{ki}q_{ki}) d^{∗} = ∑ λ_{ki}d_{ki}d_{ki}^{∗} .

Now, choose a partial isometry b such that

b^{∗}b < e_{4}, bb^{∗} = ∑(p_{ki} − c_{ki}c_{ki}^{∗} − d_{ki}d_{ki}^{∗})

and define

w_{0} = ∑ λ_{ki}b^{∗}(p_{ki} − c_{ki}c_{ki}^{∗} − d_{ki}d_{ki}^{∗})b .

Then z_{0} = b + c + d is a partial isometry with z_{0}^{∗}z_{0} = e_{1} + e_{2} + e_{3} + b^{∗}b and z_{0}z_{0}^{∗} = e_{1}. So in K_{0}(A) we have [e_{1}] = [e_{1} + e_{2} + e_{3} + b^{∗}b], which implies [1 − e_{1}] = [e_{4} − b^{∗}b]. By Proposition 11 of [10], there is a partial isometry z_{1} ∈ A such that z_{1}z_{1}^{∗} = 1 − e_{1} and z_{1}^{∗}z_{1} = e_{4} − b^{∗}b. Then w = w_{0} + e_{4} − b^{∗}b is a unitary with finite spectrum in e_{4}Ae_{4} and z = z_{0} + z_{1} is a unitary in A that satisfies z^{∗}(u + 1 − e_{1})z = u + ∑ λ_{ki}q_{ki} + w. □

Theorem 4.12. Let A be a real unital c-simple purely infinite C*-algebra such that [1] ∈ 2K_{0}(A). For every u ∈ U_{0}(A) and every ε > 0 there is a unitary v with finite spectum such that ∥u − v∥ < ε.

Proof. With the lemmas that we have developed, the proof is now the same as that of the unital case of Theorem 1 and Corollary 2 of [34], except that wherever there is an element of the form exp(ih) where h is self-adjoint, we use exp(k) where k is skew-adjoint. □

As in the complex case, we have the following corollary concerning exponential length.

Corollary 4.13. Let A be a real unital c-simple purely infinite C*-algebra such that [1] ∈ 2K_{0}(A). Then cel(A) ≤ 4.

Proof. By Theorem 4.12, every unitary u ∈ U_{0}(A) can be approximated within ε by a unitary v with finite spectrum. For ε sufficiently small, ∥v^{∗}u − 1∥ < ε implies there exists a skew-adjoint k_{2} such that v^{∗}u = exp(k_{2}) with ∥k_{2}∥ ≤ 4 − π. As v has finite spectrum, there exists a skew-adjoint k_{1} such that v = exp(k_{1}) and ∥k_{1}∥ ≤ π. Then u = exp(k_{1}) exp(k_{2}) and ∥k_{1}∥+∥k_{2}∥ ≤ 4. □

5. Homomorphisms from O_{n}^{ℝ}

The following theorem gives the real version of the Rokhlin property of the Bernoulli shift, established in [15] and summarized in [39]. Let M_{2^{∞}} = lim_{k→∞} M_{2^{k}} be the real CAR algebra and let ℍ be the real C*-algebra of quaternions.

Proposition 5.1. Let σ be the one-sided Bernoulli shift on M_{2^{∞}}. For each ε > 0 and for each r ∈ ℕ there exist k ∈ ℕ and projections e_{0}, e_{1}, …, e_{2^{r}} = e_{0} ∈ M_{2^{k}} such that ∑_{j=1}^{2^{r}} e_{j} = 1 and ∥σ(e_{j})−e_{j+1}∥ < ε for all j = 0, 1, 2, …, 2^{r} −1.

Proof. Let A_{k} = M_{2^{k}} and let A = M_{2^{∞}}. Using the notation of Proposition 5.1.3 of [39], let S denote the unilateral shift on ℓ^{2}(ℕ, ℂ), let ω_{k} = exp(2πi/2^{k}) for each k ≥ 0 and, given δ > 0, let

f_{0} = \frac{1}{\sqrt{n_{0}}} (1, 1, …, 1, 0, 0, …) ∈ ℓ^{2}(ℕ, ℝ)

be a unit vector with ∥Sf_{0} − f_{0}∥ < δ and let

f_{1} = \frac{1}{\sqrt{n_{1}}} (0, 0, … 0, 1, −1, 1, −1, …, −1, 0, 0, …)

be a unit vector in ℓ^{2}(ℕ, ℝ), orthogonal to f_{0}, with ∥Sf_{1} + f_{1}∥ < δ. Then, for r ∈ ℕ, let f_{2}, …, f_{r} ∈ ℓ^{2}(ℕ, ℂ) be defined by

f_{j} = \frac{1}{\sqrt{n_{j}}} (0, 0, … 0, 1, ω_{j}, ω_{j}^{2}, …, ω_{j}^{n_{j}−1}, 0, 0, …)

where there are sufficiently many initial zeros to make f_{j} orthogonal to its predecessors and where n_{j} is chosen so that

⟨f_{j}, \overline{f_{j}}⟩ = 1 + ω_{j}^{2} + … ω_{j}^{2(n_{j}−1)} = 0 and ∥Sf_{j} − ω_{j}f_{j}∥ < δ. If f_{j} = g_{j} + ih_{j} with g_{j}, h_{j} ∈ ℓ^{2}(ℕ, ℝ) then, from the orthogonality of f_{j} and \overline{f_{j}}, ∥g_{j}∥ = ∥h_{j}∥ = 1/\sqrt{2}. Let a : ℓ^{2}(ℕ, ℂ) → A_{ℂ} be the map described in [15] and [39] satisfying the canonical anticommutation relations and observe that a maps ℓ^{2}(ℕ, ℝ) into A. Let v_{1} = w_{1} = a(f_{1})(a(f_{0}) + a(f_{0})^{∗}) and, for each 2 ≤ j ≤ r let v_{2j−2} = a(f_{j})(a(f_{0}) + a(f_{0})^{∗}), v_{2j−1} = a(\overline{f_{j}})(a(f_{0}) + a(f_{0})^{∗}), w_{2j−2} = a(\sqrt{2}g_{j})(a(f_{0}) + a(f_{0})^{∗}) = (v_{2j−2} + v_{2j−1})/\sqrt{2} and w_{2j−1} = a(\sqrt{2}h_{j})(a(f_{0}) + a(f_{0})^{∗}) = −i(v_{2j−2} − v_{2j−1})/\sqrt{2}. Note that \{w_{1}, w_{2}, …, w_{2r−1}\} ⊂ A_{k} for all sufficiently large k. It is noted in the proof of Proposition 4.1 of [15] that the elements v_{i} for 1 ≤ i ≤ 2r − 1 satisfy the relations v_{i}v_{j} + v_{j}v_{i} = 0 and v_{i}v_{j}^{∗} + v_{j}^{∗}v_{i} = δ_{ij}1. It follows from this that the elements w_{i} for 1 ≤ i ≤ 2r − 1 satisfy the same relations. Therefore, using the matrix units described in the proof of Proposition 4.1 of [15], the real C*-algebra B generated by w_{1}, …, w_{2r−1} is isomorphic to M_{2^{2r−1}}. Slightly varying the proof of Proposition 4.1 of [15], let β be the automorphism of the complexification of B determined by β(v_{1}) = −v_{1}, β(v_{2j}) = ω_{j}v_{2j} and β(v_{2j+1}) = \overline{ω_{j}}v_{2j+1} for each 1 ≤ j ≤ r−1. Note that β(w_{2j}) = \frac{1}{2}(ω_{j} +\overline{ω_{j}})w_{2j} + \frac{i}{2}(ω_{j} − \overline{ω_{j}})w_{2j+1} and β(w_{2j+1}) = −\frac{i}{2}(ω_{j} − \overline{ω_{j}})w_{2j} + \frac{1}{2}(ω_{j} + \overline{ω_{j}})w_{2j+1}, so that β leaves the real algebra B invariant. Identifying B with M_{2^{2r−1}}, there is an orthogonal matrix W implementing β. By standard linear algebra, described for example in Section 81 of [22], W is orthogonally conjugate to an orthogonal matrix consisting of diagonal elements ±1 and diagonal 2×2 rotation matrices, determined by the eigenvalues of W. As in [39], on the complexification of B, identified with M_{2^{2r−1}}(ℂ), β is implemented by a diagonal unitary with entries 1, ω_{r}, ω_{r}^{2}, …, ω_{r}^{2^{r}−1}, each repeated 2^{r−1} times. (The unitary arises as the tensor product of one diagonal unitary with entries 1, ω_{r}, ω_{r}^{2}, …, ω_{r}^{2^{r}−1} and another with entries 1, \overline{ω_{r}}, \overline{ω_{r}}^{2}, …, \overline{ω_{r}}^{2^{r−1}−1}.) On B ≅ M_{2^{2r−1}} the orthogonal matrix W implementing β is therefore conjugate to an orthogonal matrix with 2 × 2 diagonal blocks diag(1, −1), R, R^{2}, …, R^{2^{r−1}−1}, each repeated 2^{r−1} times, where

R = ( \begin{matrix}cos(π/2^{r−1}) & − sin(π/2^{r−1}) \\ sin(π/2^{r−1}) & cos(π/2^{r−1}) \end{matrix} ) .

The cyclic shift on M_{2^{r}} is implemented by the unitary

V = (\begin{matrix}0 & 0 & 0 & … & 0 & 1 \\ 1 & 0 & 0 & … & 0 & 0 \\ 0 & 1 & 0 & … & 0 & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ & ⋮ \\ 0 & 0 & 0 & … & 1 & 0 \end{matrix}) ,

which is orthogonally conjugate to diag (diag(1, −1), R, R^{2}, …, R^{2^{r−1}−1}). It follows that the orthogonal element W implementing β on B is orthogonally conjugate to a direct sum of 2^{r−1} copies of V and thus that β is conjugate to a direct sum of 2^{r−1} cyclic shifts. It follows that there are 2^{r} orthogonal projections e_{0}, e_{1}, …, e_{2^{r}} = e_{0} in B (each of rank 2^{r−1}) that are cyclically permuted by β. As in the proof of Proposition 4.1 of [15], a suitable choice of δ at the start of the proof ensures that ∥σ(e_{j}) − β(e_{j})∥ < ε for each j and therefore the projections e_{0}, e_{1}, …, e_{2^{r}} = e_{0} have the required properties. □

Theorem 5.2. Let D be a real unital C*-algebra satisfying

(i) the canonical homomorphism U(D)/U_{0}(D) → K_{1}(D) is an isomorphism, and (ii) cel(D) < ∞.

Let n be an even integer, let ϕ, ψ be unital homomorphisms from O_{n}^{ℝ} to D, let λ be the endomorphism of D defined by λ(a) = ∑_{j=1}^{n} ϕ(s_{j})aϕ(s_{j})^{∗} and let u ∈ U(D) be defined by u = ∑_{j=1}^{n} ψ(s_{j})ϕ(s_{j})^{∗}, where s_{1}, …, s_{n} are the canonical generators of O_{n}^{ℝ}. Then the following are equivalent:

(1) u ∈ \overline{\{vλ(v)^{∗} | v ∈ U(D)\}},

(2) [u] ∈ (n − 1)K_{1}(D)

(3) [ϕ] = [ψ] ∈ KK_{0}(O_{n}^{ℝ}, D),

(4) ϕ and ψ are approximately unitarily equivalent. In particular, these statements are equivalent if D is a real unital purely infinite c-simple C*-algebra.

Proof of Theorem 5.2. The proof of the equivalence of the four statements, assuming (i) and (ii), is similar to that of the complex case in Sections 3 and 4 of [38], modified only by the use of unitaries of the form exp(h) with h ∈ A_{sk} in the proof of the real version of Lemma 4.6 of [38]. We note that in the proof of the real version of Lemma 3.7 of [38], the required result from [19] holds, as was observed already in the proof of Proposition 3.13 above. Suppose D is a real unital purely infinite c-simple C*-algebra. Then condition (i) holds for D by Proposition 3.13. Since K_{0}(O_{n}^{ℝ}) = ℤ_{n−1} and n is even, we have [1_{O_{n}^{ℝ}}] ∈ 2K_{0}(O_{n}^{ℝ}). Using the unital homomorphism ϕ (or ψ) we obtain [1_{D}] ∈ 2K_{0}(D). Then condition (ii) holds by Corollary 4.13. □

Corollary 5.3.

(1) Let A be a real unital purely infinite c-simple C*-algebra. Any two unital homomorphisms ϕ, ψ: O_{2}^{ℝ} → A are approximately unitarily equivalent.

(2) Any inductive limit of the form O_{2}^{ℝ} → O_{2}^{ℝ} → O_{2}^{ℝ} → …, with unital connecting homomorphisms, is isomorphic to O_{2}^{ℝ}.

(3) O_{2}^{ℝ} ⊗ O_{2}^{ℝ} ≅ O_{2}^{ℝ}.

(4) ⨂_{n=1}^{∞} O_{2}^{ℝ} ≅ O_{2}^{ℝ}.
(5) O_{2}^{ℝ} ⊗ M_{2^{∞}} ≅ O_{2}^{ℝ}.
(6) O_{2}^{ℝ} ⊗ ℍ ≅ O_{2}^{ℝ}.

Proof. We know that K^{CRT}(O_{2}^{ℝ}) = 0 from Section 5 of [8] so the universal coefficient theorem (Theorem 4.1 of [9]) implies that KK_{0}(O_{2}^{ℝ}, D) = 0. Then part (1) follows immediately from Theorem 5.2.

Parts (2) and (3) can be proven in the same way as in the complex case. See Corollary 5.1.5 and Theorem 5.2.1 in [39]. Then part (4) follows from parts (2) and (3). There is an isomorphism O_{2}^{ℝ} ≅ M_{2}(O_{2}^{ℝ}), established as in the complex case: if s_{1} and s_{2} are generators of O_{2}^{ℝ} satisfying the canonical relations s_{i}^{∗}s_{j} = δ_{ij}1_{O_{2}^{ℝ}} and ∑_{i=1}^{2} s_{i}s_{i}^{∗} = 1, then

S_{1} = ( \begin{matrix}s_{1} & s_{2} \\ 0 & 0 \end{matrix} )  and  S_{2} = ( \begin{matrix}0 & 0 \\ s_{1} & s_{2} \end{matrix} )

satisfy the same relations and generate M_{2}(O_{2}^{ℝ}). Using that isomorphism, part (5) follows from part (2). Finally, part (6) follows from (5) and the formula M_{2^{∞}} ⊗ ℍ ≅ M_{2^{∞}}, which follows from Theorem 10.1 of [21] or from Theorem 4.8 of [42]). □

6. Tensor Product Theorems

In this section, we reproduce for real C*-algebras some standard results regarding tensor products with O_{2}^{ℝ} and O_{∞}^{ℝ}. Definition 6.1.

(1) A real (resp. complex) C*-algebra A is amenable if for all ε > 0 and all finite subsets F ⊂ A, there is a finite dimensional real (resp. complex) C*-algebra B and contractive completely positive linear maps ϕ: A → B and ψ: B → A such that

∥ψ ∘ ϕ(a) − a∥ < ε for all a ∈ F.

(2) A real (resp. complex) C*-algebra A is nuclear if for all real (resp. complex) C*-algebras B the algebraic tensor product A ⊗_{ℝ} B (resp. A ⊗_{ℂ} B) has a unique C*-norm.

(3) A real (resp. complex) C*-algebra A is exact if the tensor product functor B ↦ A ⊗_{min} B is exact. Here the tensor product is over ℝ (resp. ℂ) and B can be any real (resp. complex) C*-algebra.

Lemma 6.2. Let A be a real C*-algebra. Then (1) A is amenable if and only if A_{ℂ} is amenable.

(2) A is nuclear if and only if A_{ℂ} is nuclear.

(3) A is exact if and only if A_{ℂ} is exact. Consequently, A is amenable if and only if it is nuclear; and in this case it is also exact.

Proof. Part (1) can be found in Proposition 3 of [25] and the preceding text. We claim that there is a one-to-one correspondence between C*-norms on the algebraic tensor product A ⊗_{ℝ} B and those on A_{ℂ} ⊗_{ℂ} B_{ℂ}. Let γ be a C*-norm on A ⊗_{ℝ} B, and let A ⊗_{γ} B be the real C*-algebra obtained by completion. Then the complexification (A ⊗_{γ} B)_{ℂ} has a unique C*-norm extending that on A ⊗_{ℝ} B. Thus every C*-norm on the algebraic tensor product A ⊗_{ℝ} B extends uniquely to a C*-norm on A_{ℂ} ⊗_{ℂ} B_{ℂ}. Part (2) follows immediately from this claim.

It also follows that the restriction of the minimal C*-norm on A_{ℂ}⊗_{ℂ}B_{ℂ} gives the minimal C*-norm on A⊗_{ℝ} B. This fact, plus the fact that the complexification functor A ↦ A_{ℂ} is exact, implies (3). The final statement then follows from the corresponding statement for complex C*-algebras. See Theorem 6.1.3 of [39] and Theorem 6.5.2 of [32]. □

Proposition 6.3. Let A be a real separable C*-algebra A. Then A is exact if and only if there is an injective homomorphism ι: A → O_{2}^{ℝ}. If A is unital then ι can be chosen to be unital.

Proof. Suppose that A is exact. Then A_{ℂ} is separable and exact. Thus, by Theorem 6.3.11 of [39], there is an injective homomorphism ι_{ℂ}: A_{ℂ} → O_{2} (which is unital if A_{ℂ} is unital). Then we can take ι to be the composition

A ↪ A_{ℂ} ⟶^{ι_{ℂ}} O_{2} ↪ M_{2}(O_{2}^{ℝ}) ≅ O_{2}^{ℝ} .

Conversely, if there is an injective homomorphism ι: A → O_{2}^{ℝ} then the complexification yields an injective homomorphism from A_{ℂ} to O_{2}. By Theorem 6.3.11 of [39] this implies that A_{ℂ} is exact, hence A is exact. □

Lemma 6.4. Let A be a real purely infinite c-simple nuclear unital C*-algebra. Then all unital endomorphisms on A ⊗ O_{2}^{ℝ} are approximately unitarily equivalent.

Proof. In the complex case, this result is found as Theorem 6.3.8 of [39]. We will use that result to prove the real version. By Corollary 5.3, Part (5) it suffices to show that any unital homomorphism

γ: A ⊗ O_{2}^{ℝ} ⊗ M_{2^{∞}} → A ⊗ O_{2}^{ℝ} ⊗ M_{2^{∞}}

is approximately unitarily equivalent to the identity. We write A′ = A ⊗ O_{2}^{ℝ} and let

α_{ℓ,k}: A′ ⊗ M_{2^{k}} ↪ A′ ⊗ M_{2^{ℓ}} for k < ℓ
α_{k}: A′ ⊗ M_{2^{k}} ↪ A′ ⊗ M_{2^{∞}}

be the canonical injections. Then we use the commutative diagram

\xymatrix{A′ \ar[r] \ar[d]^{c} & … \ar[r] & A′ ⊗ M_{2^{k}} \ar[r]^{α_{k+1,k}} \ar[d]^{c} & A′ ⊗ M_{2^{k+1}} \ar[r] \ar[d]^{c} & … \ar[r] & A′ ⊗ M_{2^{∞}} \ar[d]^{c} \\ A′_{ℂ} \ar[r] & … \ar[r] & (A′ ⊗ M_{2^{k}})_{ℂ} \ar[r]^{α_{k+1,k}} & (A′ ⊗ M_{2^{k+1}})_{ℂ} \ar[r] & … \ar[r] & (A′ ⊗ M_{2^{∞}})_{ℂ}}

By Theorem 6.3.8 of [39], there is a sequence of unitaries u_{n} ∈ (A′ ⊗ M_{2^{∞}})_{ℂ} such that

∥u_{n}au_{n}^{∗} − γ(a)∥ → 0 for all a ∈ A′ ⊗ M_{2^{∞}} .

For each n find an integer k(n) and a unitary v_{n} ∈ (A′ ⊗ M_{2^{k(n)}})_{ℂ} such that ∥α_{k(n)}(v_{n}) − u_{n}∥ < 1/n. Let w_{n} = r(v_{n}) ∈ A′ ⊗ M_{2^{k(n)+1}}, where r is induced by the realification map M_{2^{k(n)}} ⊗ ℂ → M_{2^{k(n)+1}}. We may assume that the sequence {k(n)}_{n=1}^{∞} is increasing. Let a ∈ A′ ⊗ M_{2^{∞}} be given such that ∥a∥ = 1 and let ε > 0. Then find an integer N large enough so that, for all n ≥ N,

• ∥α_{k(n)}(v_{n}) − u_{n}∥ < ε,
• ∥u_{n}au_{n}^{∗} − γ(a)∥ < ε,
• there exist a_{n}, b_{n} ∈ A′ ⊗ M_{2^{k(n)}} such that
∥a − α_{k(n)}(a_{n})∥ < ε  and  ∥γ(a) − α_{k(n)}(b_{n})∥ < ε .

Then a calculation shows that, for all n ≥ N,

∥v_{n}a_{n}v_{n}^{∗} − b_{n}∥ = ∥α_{k(n)}(v_{n})α_{k(n)}(a_{n})α_{k(n)}(v_{n})^{∗} − α_{k(n)}(b_{n})∥ < 5ε.

Now for any element x ∈ A′ ⊗ M_{2^{k(n)}} we have

α_{k(n)+1}rc(x) = α_{k(n)+1}α_{k(n)+1,k(n)}(x) = α_{k(n)}(x) .

It follows that

∥α_{k(n)+1}(w_{n})aα_{k(n)+1}(w_{n})^{∗} − γ(a)∥
<∥α_{k(n)+1}(w_{n})α_{k(n)}(a_{n})α_{k(n)+1}(w_{n})^{∗} − α_{k(n)}(b_{n})∥ + 2ε
=∥α_{k(n)+1}r(v_{n})α_{k(n)+1}rc(a_{n})α_{k(n)+1}r(v_{n})^{∗} − α_{k(n)+1}rc(b_{n})∥ + 2ε
=∥v_{n}c(a_{n})v_{n}^{∗} − c(b_{n})∥ + 2ε
=∥v_{n}a_{n}v_{n}^{∗} − b_{n}∥ + 2ε < 7ε.
□

Theorem 6.5. Let A be a real C*-algebra. Then A is c-simple, separable, unital, and nuclear if and only if A ⊗ O_{2}^{ℝ} ≅ O_{2}^{ℝ}

Proof. Suppose that A is c-simple, separable, unital, and nuclear. There is a unital homomorphim γ: O_{2}^{ℝ} → A⊗O_{2}^{ℝ} given by x ↦ 1⊗x and there is a unital homomorphism κ: A ⊗ O_{2}^{ℝ} → O_{2}^{ℝ} by Lemma 6.2 and Proposition 6.3. Then by Theorem 5.2 we have κ ∘ γ ≈_{u} 1_{O_{2}^{ℝ}} and by Lemma 6.4 we have γ ∘ κ ≈_{u} 1_{A⊗O_{2}^{ℝ}}. Therefore, by (the real analog of) Corollary 2.3.4 of [39], A ⊗ O_{2}^{ℝ} ≅ O_{2}^{ℝ}. Conversely, if the isomorphism A ⊗ O_{2}^{ℝ} ≅ O_{2}^{ℝ} holds for a real C*-algebra A, then we have A_{ℂ} ⊗ O_{2} ≅ O_{2} which implies by Theorem 7.1.2 of [39] that A_{ℂ} is simple, separable, unital, and nuclear. Therefore A is c-simple, separable, unital, and nuclear. □

We note that the hypothesis above requiring that A be c-simple cannot be relaxed, as the result does not hold for A = O_{2} (considered as a real C*- algebra). Theorem 6.6.

(1) Any two unital homomorphisms ϕ, ψ from O_{∞}^{ℝ} into a real, unital, purely infinite, nuclear, c-simple C*-algebra A are approximately unitarily equivalent.

(2) Let A be a real c-simple, separable, and nuclear C*-algebra. Then A is isomorphic to A ⊗ O_{∞}^{ℝ} if and only if A is purely infinite.

(3) O_{∞}^{ℝ} ≅ ⨂_{n=1}^{∞} O_{∞}^{ℝ}.

Proof. As in Section 7.2 of [39]. □

Corollary 6.7. Let A and B be real, c-simple, separable, nuclear C*-algebras. If A or B is purely infinite, then A ⊗ B is purely infinite.

Proof. From part (2) of Theorem 6.6. □

7. Homomorphisms from O_{∞}^{ℝ}

The goal of this section is to prove the following theorem, analogous to Proposition 2.2.7 of [35].

Theorem 7.1. Let D be a real unital purely infinite simple C*-algebra, and let ϕ, ψ: O_{∞}^{ℝ} → D be unital homomorphisms. Then ϕ is asymptotically unitarily equivalent to ψ.

The proof of Theorem 7.1 will be the same as that in [35]. However, there are a couple of background topics that need to be addressed in the context of real C*-algebras. We begin with a discussion of approximately divisible real C*-algebras, following [6]. It is sufficient to consider only separable unital C*-algebras. Also, we skirt the general topic of completely noncommutative C*-algebras by taking into account Definition 2.6 of [6] and the subsequent comment.

Definition 7.2. A separable unital real C*-algebra A is approximately divisible if for all x_{1}, x_{2}, …, x_{n} ∈ A and ε > 0, there is a unital subalgebra B isomorphic to M_{2}, M_{3}, or M_{2} ⊕ M_{3} such that ∥x_{i}y − yx_{i}∥ < ε for all i = 1, 2, …, n and all y in the unit ball of B.

The following theorem is the real version of Corollary 2.1.6 of [35].

Lemma 7.3. The tensor product O_{∞}^{ℝ} ⊗ D is approximately divisible for any real separable unital C*-algebra D. In particular, every c-simple, separable, nuclear, purely infinite, unital real C*-algebra is approximately divisible.

Proof. Let A = O_{∞}^{ℝ} ⊗ D. Using the isomorphism O_{∞}^{ℝ} ≅ ⨂_{n=1}^{∞} O_{∞}^{ℝ} of Theorem 6.6 we obtain a sequence of mutually commuting unital homomophisms ϕ_{n}: O_{∞}^{ℝ} → A such that ∥ϕ_{n}(a)b − bϕ_{n}(a)∥ → 0 for all a ∈ O_{∞}^{ℝ} and all b ∈ A. Choose a unital map γ: M_{2} ⊕ M_{3} → O_{∞}^{ℝ} and let ψ_{n} = ϕ_{n} ∘ γ. Then for large enough n, the subalgebra B = ψ_{n}(M_{2} ⊕ M_{3}) works. The second statement follows from part (2) of Theorem 6.6. □

Lemma 7.4. Let p and q be full projections in M_{∞}(A) where A is a real, separable, unital, approximately divisible C*-algebra. Then p ∼ q if and only if [p] = [q] in K_{0}(A).

Proof. The proof is the same as the proof of (the first part of) Proposition 3.10 in [6] in complex case. That proof relies on a progression of results from Section 2 of [6] which can all be proven in the real case in the same way with one minor caveat. The proof of Proposition 2.1 of [6] (which in that paper was left to the reader) relies on the fact that a complex C*-algebra is spanned by its unitaries. While this fact is not true in general for real C*-algebras, it can easily be shown to be true for finite dimensional real C*-algebras, which is the relevant case. The proof of Proposition 3.10 in [6] also relies on Theorem 3.1.4 of [3], which is a ring-theoretic result stated in enough generality to apply to real C*-algebras.

□

We remark that a more direct proof of Lemma 7.4 can be achieved in the special case (which is sufficient for our purposes) that A = O_{∞}^{ℝ} ⊗ D where D is separable and unital. In that case, we write A = ⨂_{i=1}^{∞} O_{∞}^{ℝ} ⊗ D and let A_{n} = ⨂_{i=1}^{n} O_{∞}^{ℝ} ⊗ D be the unital subalgebra of A consisting of the first n factors in the tensor product. Then for each n and each k, it is easy to find a unital subalgebra B_{n} ⊂ A′_{n} ∩ A that is isomorphic to M_{2^{k}} ⊕ M_{3^{k}}. Thus we achieve the result of Corollary 2.10 of [6] without having to recheck all the earlier material of Section 2 of [6] in the real case.

Lemma 7.5. Let D be a unital real C*-algebra and let p, q be any two full projections in K^{ℝ} ⊗ O_{∞}^{ℝ} ⊗ D. Then p is Murray-von Neumann equivalent to a subprojection of q. Furthermore, p is homotopic to q if and only if they represent the same class in K_{0}(K^{ℝ} ⊗ O_{∞}^{ℝ} ⊗ D) ≅ K_{0}(D).

Proof. With our Lemmas 7.3 and 7.4, as well as Theorem 3.6 of [11], the proof is the same as that of Lemma 2.1.8 of [35]. □

Proof of Theorem 7.1. With these preliminary definitions and results, the proof is the same as the proof of Proposition 2.2.7 of [35] including all of the lemmas and intermediate results in Sections 2.1 and 2.2 of [35]. We note that in [35], the proofs of Propositions 2.1.9 and 2.1.10 (having to do with exact stability of the relations defining O_{m}^{ℝ} and E_{m}(δ)) are referred back to the proofs of parts (1) and (2) of Lemma 1.3 of [30]. The proof given there for part (2) produces isometries w_{j} that live in the real algebra E_{n}(δ). Therefore the homomorphisms ϕ_{δ}^{(m)} constructed in the complex case restrict to homomorphisms between the real algebras. The same will be true for the analogous proof of part (1). We also note that the proofs for the real versions of Lemmas 2.2.1 and 2.2.3 of [35] rely on our Theorem 5.2 which is only established for n even. Hence for real C*-algebras, we need to take m to be even in Lemma 2.2.1 and n to be even in Lemma 2.2.3. This is however, sufficient for all subsequent arguments. □

8. Asymptotic Morphisms

We appropriate the following definition of an asymptotic morphism from Section 25.1 of [4]. The other definitions in this section and the next are adapted from [35].

Definition 8.1. Let A and B be real C*-algebras. An asymptotic morphism ϕ from A to B is a family {ϕ_{t}}_{t∈[0,∞)} of maps ϕ_{t}: A → B such that

(1) the map t ↦ ϕ_{t}(a) is continuous for each a ∈ A, and (2) for all a, b ∈ A and all λ ∈ ℝ, the following functions vanish in norm

as t → ∞:

(a) ϕ_{t}(a + b) − ϕ_{t}(a) − ϕ_{t}(b),

(b) ϕ_{t}(λa) − λϕ_{t}(a),

(c) ϕ_{t}(ab) − ϕ_{t}(a)ϕ_{t}(b),

(d) ϕ_{t}(a^{∗}) − ϕ_{t}(a)^{∗}.

We say that two asymptotic morphisms ϕ_{t} and ψ_{t} from A to B are equivalent if ∥ϕ_{t}(a) − ψ_{t}(a)∥ vanishes as t → ∞ for all a ∈ A. We say that ϕ_{t} and ψ_{t} are homotopic if there is an asymptotic morphism Φ_{t} from A to C([0, 1], B) such that Φ_{t}(a)(0) = ϕ_{t}(a) and Φ_{t}(a)(1) = ψ_{t}(a) for all a ∈ A. Equivalent asymptotic morphisms are homotopy equivalent (see Remark 25.1.2 of [4]). We leave the easy proof of the next lemma to the reader.

Lemma 8.2. If A and B are real C*-algebras and ϕ is an asymptotic morphism from A to B, then there is an asymptotic morphism ϕ_{ℂ}: A_{ℂ} → B_{ℂ} defined by (ϕ_{ℂ})_{t}(a + ib) = ϕ_{t}(a) + iϕ_{t}(b).

It can be proven, then, from the same result in the complex case, that for any asymptotic morphism ϕ we have lim sup_{t→∞} ∥ϕ_{t}(a)∥ ≤ ∥a∥ for all a ∈ A (see Proposition 25.1.3 of [4]). Thus, an asymptotic morphism {ϕ_{t}} gives rise to a unique homomorphism

ϕ: A → C_{b}([0, ∞), B)/C_{0}([0, ∞), B)

defined in the natural way; and every such homomorphism represents an asymptotic morphism, unique up to equivalence.

Lemma 8.3. Let A be separable and nuclear. Every asymptotic morphism from A to B is equivalent to one that is completely positive and contractive. Furthermore, if ϕ and ψ are homopic completely positive and contractive asymptotic morphisms from A to B, then in fact there is a homotopy from ϕ to ψ consisting of completely positive and contractive asymptotic morphisms.

Proof. Let ϕ be an asymptotic morphism from A to B. Then by Proposition 1.1.5 of [35], the complexification ϕ_{ℂ} is equivalent to an asymptotic morphism ψ that is completely positive and contractive. The map α: B_{ℂ} → B defined by α(a + ib) = a is completely positive and contractive. Then the restriction of α∘ψ to A is a completely positive, contractive asymptotic morphism from A to B and is equivalent to ϕ. The same construction can be applied to a homotopy to prove the second statement. □

Definition 8.4. Let ϕ and ψ be asymptotic morphisms from A to K^{ℝ} ⊗ D. We define an asymptotic morphism ϕ ⊕ ψ, also from A to K^{ℝ} ⊗ D, as follows. Choose an isomorphism δ: M_{2}(K^{ℝ}) → K^{ℝ} and define

(ϕ ⊕ ψ)_{t}(a) = (δ ⊗ 1_{D}) (\begin{matrix}ϕ_{t}(a) & 0 \\ 0 & ψ_{t}(a) \end{matrix}) .

Lemma 8.5. The asymptotic morphism ϕ ⊕ ψ is well defined up to unitary equivalence, as well as up to homotopy.

Proof. As in the complex case every automorphism of K^{ℝ} is implemented by a unitary in U(B(H^{ℝ})) (the proof in, for example, Lemma V.6.1 of [20] works in the real case). Furthermore, by [36], U(B(H^{ℝ})) is path connected. (In fact, by Theorem 3 of [29], it is contractible.) □

Definition 8.6. Let ϕ: A → B be an asymptotic morphism of real C*-algebras and let p ∈ A be a projection. A tail projection for ϕ(p) is a continuous path p_{t} of projections for t ∈ [0, ∞) such that lim_{t→∞} ∥ϕ_{t}(p) − p_{t}∥ = 0. We say that ϕ is full if there is a full projection p ∈ A such that ϕ(p) has a full tail projection.

Definition 8.7. Let A and B be real C*-algebras. Two asymptotic morphisms ϕ and ψ from A and B are asymptotically unitarily equivalent if there is a continuous family of unitary elements u_{t} ∈ B̃ such that lim_{t→∞} ∥u_{t}ϕ_{t}(a)u_{t}^{∗} − ψ_{t}(a)∥ = 0 for all a ∈ A.

With these definitions, all the results of Sections 1.2 and 1.3 of [35] hold for real C*-algebras.

Definition 8.8. Let A and D be real C*-algebras. An asymptotic morphism ϕ: A → D has a standard factorization through O_{∞}^{ℝ}⊗A if there is an asymptotic morphism ψ: O_{∞}^{ℝ} ⊗ A → D such that the asymptotic morphisms ϕ(a) and ψ(1 ⊗ a) (both from A to D) are asymptotically unitarily equivalent. Similarly, ϕ is asymptotically trivially factorizable if there is an asymptotic morphism ψ: O_{2}^{ℝ} ⊗ A → D such that ϕ(a) and ψ(1 ⊗ a) are asymptotically unitarily equivalent.

Theorem 8.9 (Theorem 2.3.7 of [35]). Let A be a separable, nuclear, unital, and c-simple. Let D_{0} be a unital C*-algebra, and let D = O_{∞}^{ℝ} ⊗ D_{0}. Then two full asymptotic morphisms from A to K^{ℝ} ⊗ D are asymptotically unitarily equivalent if and only they are homotopic.

Proof. The proof of Theorem 2.3.7 in [35] as well as the proofs of all of the preceeding lemmas in Section 2.3 of [35] can be proven in the real case with the same proofs, with some extra attention paid to the issue of connectedness of unitary groups. In a few places Phillips uses the fact that the unitary group of O_{2} is connected. It is also true that O_{2}^{ℝ} is connected since K_{1}(O_{2}^{ℝ}) ≅ 0. However, on page 85 of [35], Phillips also uses the fact that the unitary group of a corner algebra of O_{∞} is connected. The corresponding statement in the real case is not true since K_{1}(O_{∞}^{ℝ}) ≅ ℤ_{2}. We will show how to adjust the proof so that it works in the real case. At this point in the proof we are (using Phillips’ notation) trying to find a path of partial isometries from w_{n} + f_{n+2} to v_{n+1} + w_{n+1} (these are partial isometries from f_{n+1} + f_{n+2} to f_{n+2} + e). If the unitaries (w_{n} + f_{n+2})^{∗}(v_{n+1} + w_{n+1}) and f_{n+1} + f_{n+2} are not in the same connected component of (f_{n+1} + f_{n+2})O_{∞}^{ℝ}(f_{n+1} + f_{n+2}), then this can be changed by by multiplying w_{n+1} on the right by a suitable unitary in f_{n+2}O_{∞}^{ℝ}f_{n+2}. Thus by re-choosing the w_{n}’s inductively, we can be sure that there is an appropriate path of partial isometries at each step. □

9. Groups of Asymptotic Morphisms

Definition 9.1. Let A be a real, separable, nuclear, unital, c-simple C*-algebra and let D be unital. We define E_{A}(D) to be the the set of homotopy classes of full asymptotic morphisms from A to K^{ℝ} ⊗ O_{∞}^{ℝ} ⊗ D. That is,

E_{A}(D) = [[A, K^{ℝ} ⊗ O_{∞}^{ℝ} ⊗ D]]_{+} .

More generally, for D unital or not, we define

Ẽ_{A}(D) = ker (E_{A}(D^{+}) → E_{A}(ℝ)) .

Proposition 9.2. Let A be real, separable, nuclear, unital, and c-simple. Then Ẽ_{A}(−) is a functor from the category of separable real C*-algebras with homotopy classes of asymptotic morphisms to abelian groups, that is homotopy invariant, stable, half exact, and split exact.

Proof. In the complex case, these results are proven in Section 3.1 of [35]. In the real case, they are proven the same way. Note that split exactness follows from homotopy invariance and half exactness by Corollary 3.5 of [12]. □

Lemma 9.3. Let A and B be C*-algebras (real or complex). Let ϕ: A → B be an asymptotic morphism. If p, q are projections in A with p ≤ q, then there are tail projections p_{t} (for ϕ(p)) and q_{t} (for ϕ(q)) in B with p_{t} ≤ q_{t} for all t.

Proof. Let p̃_{t} and q_{t} be arbitrary tail projections corresponding to ϕ(p) and ϕ(q), respectively (these exist as in Remark 1.2.2 of [35]). One can easily show that

lim_{t→∞} ∥p̃_{t} − q_{t}p̃_{t}q_{t}∥ = 0 .

For each t, the element q_{t}p̃_{t}q_{t} is a self adjoint and asymptotically idempotent element of q_{t}Bq_{t}. Therefore, there is a continuous path of projections p_{t} ∈ q_{t}Bq_{t} such that

lim_{t→∞} ∥q_{t}p̃_{t}q_{t} − p_{t}∥ = 0 .

The tail projections p_{t} and q_{t} have the desired properties. □

We note that if A and D are complex C*-algebras there are two groups one might consider: we let Ẽ_{A}^{ℂ}(D) denote the functor of [35] that is based on complex asymptotic morphisms. On the other hand, according to the notation established in Definition 9.1, the asymptotic morphisms comprising Ẽ_{A}(D) are only required to be asymptotically linear over ℝ (thus the complex structures of A and D are forgotten). The following theorem relates the two groups.

Proposition 9.4. If A is a real C*-algebra satisfying the hypotheses of Definition 9.1 and D is a complex unital C*-algebra, then there is a isomorphism

Ẽ_{A}(D) ≅ Ẽ_{A_{ℂ}}^{ℂ}(D)

which is natural with respect to complex homomorphisms.

Proof. We show that for a real unital C*-algebra A and a complex C*-algebra B, there is a bijection

[[A, B]]_{+} ≅ [[A_{ℂ}, B]]_{+}^{ℂ}

of equivalence classes of full asymptotic morphisms. Given a complex asymptotic morphism ϕ from A_{ℂ} to B, then we let Γ(ϕ) be the restriction of ϕ to A. If ϕ is full, then we claim that Γ(ϕ) is full. Since ϕ is full, there is a full projection p ∈ A_{ℂ} and a full tail projection r_{t} ∈ B such that ∥ϕ_{t}(p) − r_{t}∥ → 0. Applying Lemma 9.3 to p ≤ 1 we obtain tail projections p_{t} and q_{t} for p and 1, respectively, such that p_{t} ≤ q_{t} for all t. Since the tail projections p_{t} and r_{t} are asymptotically equal, it must be that p_{t} are full projections. It follows that q_{t} are also full projections; and since they are tail projections for the full projection 1_{A} in A, it follows that Γ(ϕ) is full. Given a real asymptotic morphism ψ from A to B, then

Δ(ψ)_{t}(a + ib) = ψ_{t}(a) + iψ_{t}(b)

defines a complex asymptotic morphism from A_{ℂ} to B. Suppose that ψ is full. Let p be a full projection in A and let q_{t} ∈ B be a full tail projection for ψ(p). Then clearly p is full in A_{ℂ} and q_{t} is a full tail projection for Δ(ψ(p)). Hence Δ(ψ) is full. It is immediate that Δ is a two-sided inverse for Γ. Furthermore, in the case that B is stable, it is easy to see that Γ preserves the semigroup operation of Definition 8.4. Therefore, under the hypotheses of the theorem, there is an group isomorphism Ẽ_{A}(D) ≅ Ẽ_{A_{ℂ}}^{ℂ}(D). □

Proposition 9.5. Let A be a separable, nuclear, c-simple unital, real C*- algebra. Let B be a separable real C*-algebra. Then there is a natural isomophism KK(A, B) ≅ Ẽ_{A}(B).

The proof in the complex case takes place in Section 3.2 of [35]. Rather than reconstructing all of the arguments in the real case, we give a proof that uses results from [12] to reduce the real case to the complex case.

Proof of Proposition 9.5. Fix A satisfying the hypotheses above. Let e be a rank one projection in K^{ℝ} and let ι_{A}: A → K^{ℝ}⊗O_{∞}^{ℝ}⊗A be the homomorphism defined by ι_{A}(a) = e ⊗ 1 ⊗ a. Let [[ι_{A}]] be the induced element of Ẽ_{A}(A). Let [1_{A}] ∈ KK(A, A) be the class of the identity. By Corollary 3.3 of [12], there is a unique natural transformation α from KK(A, −) to Ẽ_{A}(−) such that α([1_{A}]) = [[ι_{A}]]. We will show that

α: KK(A, B) → Ẽ_{A}(B)

is an isomorphism for all separable real C*-algebras B. By Theorem 3.9 of [12] it suffices to show that α is an isomorphism when B is complex. In the complex case we have the element [1_{A_{ℂ}}] ∈ KK^{ℂ}(A_{ℂ}, A_{ℂ}) and the homomorphism

(ι_{A})_{ℂ}: A_{ℂ} → K^{ℝ} ⊗ O_{∞}^{ℝ} ⊗ A_{ℂ} ≅ K ⊗ O_{∞} ⊗ A_{ℂ} .

By Theorem 3.7 of [24] there is a unique natural transformation α^{ℂ} from KK^{ℂ}(A_{ℂ}, −) to Ẽ_{A_{ℂ}}^{ℂ}(−) such that α^{ℂ}([1_{A_{ℂ}}]) = [[ι_{A}^{ℂ}]]. A special case of Theorem 3.2.6 of [35] shows that α^{ℂ} is an isomorphism for all separable complex C*-algebras B. Consider the following diagram for a complex C*-algebra B,

\xymatrix{KK^{ℂ}(A_{ℂ}, B) \ar[r]^{α^{ℂ}} \ar[d]^{ν} & Ẽ_{A_{ℂ}}^{ℂ}(B) \ar[d]^{μ} \\ KK(A, B) \ar[r]^{α} & Ẽ_{A}(B)}

where μ is the isomorphism of Proposition 9.4 above and ν is the isomorphism of Lemma 4.3 of [9]. To complete the proof, we only need to show that the diagram commutes. Since the homomorphism α^{ℂ} is characterized by the value of α^{ℂ}([1_{A_{ℂ}}]) it suffices to consider the case B = A_{ℂ} as in the diagram

\xymatrix{KK^{ℂ}(A_{ℂ}, A_{ℂ}) \ar[r]^{α^{ℂ}} \ar[d]^{ν} & Ẽ_{A_{ℂ}}^{ℂ}(A_{ℂ}) \ar[d]^{μ} \\ KK(A, A_{ℂ}) \ar[r]^{α} & Ẽ_{A}(A_{ℂ})}

and to show that α^{ℂ}([1_{A_{ℂ}}]) = (μ^{−1} ∘ α ∘ ν)([1_{A_{ℂ}}]) or, equivalently, (μ ∘ α^{ℂ})([1_{A_{ℂ}}]) = (α ∘ ν)([1_{A_{ℂ}}]). From the construction of ν in the proof of Lemma 4.3 of [9] it is apparent that ν([1_{A_{ℂ}}]) = [c_{A}] = (c_{A})_{∗}([1_{A}]) where c_{A}: A → A_{ℂ} is the real C*-algebra homomorphism induced by the unital inclusion c: ℝ ↪ ℂ. Thus

(α ∘ ν)([1_{A_{ℂ}}]) = α((c_{A})_{∗}([1_{A}])) = (c_{A})_{∗}(α([1_{A}])) = (c_{A})_{∗}([[ι_{A}]]) = [[c_{A}]] .

On the other hand, it is apparent from the construction of μ in the proof of Proposition 9.4 above that μ([[ι_{A}^{ℂ}]]) = [[c_{A}]]. Thus

(μ ∘ α^{ℂ})([1_{A_{ℂ}}]) = μ(α^{ℂ}([1_{A_{ℂ}}])) = μ([[ι_{A}^{ℂ}]]) = [[c_{A}]] .
□

The following is the real version of Theorems 4.1.1 and 4.1.3 of [35].

Theorem 9.6. Let A be a real separable unital nuclear c-simple C*-algebra and let D be a separable unital C*-algebra. Then the following groups are naturally isomorphic, via the obvious maps.

(1) KK(A, D)

(2) The set of asymptotic unitary equivalence classes of full homomorphisms from A to K^{ℝ} ⊗ O_{∞}^{ℝ} ⊗ D.

(3) The set of homotopy classes of full homomorphisms from A to K^{ℝ} ⊗ O_{∞}^{ℝ} ⊗ D.

(4) The set of asymptotic unitary equivalence classes of full homomorphisms from K^{ℝ} ⊗ O_{∞}^{ℝ} ⊗ A to K^{ℝ} ⊗ O_{∞}^{ℝ} ⊗ D.

(5) The set of homotopy classes of full homomorphisms from K^{ℝ} ⊗O_{∞}^{ℝ} ⊗A to K^{ℝ} ⊗ O_{∞}^{ℝ} ⊗ D.

Proof. The proof of the isomorphism of (1), (2), and (3) is the same as the proof of Theorem 4.1.1 in [35]. The proof of the isomorphism of (1), (4), and (5) relies on Lemma 9.7 below (which is the real version of Lemma 4.1.2 of [35]). Once that lemma is established, the proof of the isomorphism of (1), (4), and (5) is the same as the proof of Theorem 4.1.3 of [35]. □

Lemma 9.7. Let A be separable, nuclear, unital, and c-simple; let D_{0} be separable and unital; and let D = O_{∞}^{ℝ} ⊗ D_{0}. Let t ↦ ϕ_{t}, for t ∈ [0, ∞), be a continuous path of full homomorphisms from K^{ℝ} ⊗ A to K^{ℝ} ⊗ D, and let ψ: K^{ℝ} ⊗ A → K^{ℝ} ⊗ D be a full homomorphism. Assume that [ϕ_{0}] = [ψ] in KK_{0}(A, D). Then there is an asymptotic unitary equivalence from ϕ to ψ that consists of unitaries in U_{0}((K^{ℝ} ⊗ D)^{+}).

The proof will be essentially the same as the proof of Lemma 4.1.2 of [35]. However, that proof has an error in the third paragraph. The element w_{t} introduced there does not seem to be a unitary as purported. Also, the order of the product in the definition of z_{t} seems wrong. Fortunately, there is an easy fix and most of the proof can be left as it is. For clarity and completeness we present the entire proof, but the only significant difference is the unitary w in the third paragraph and following. In places where the proof does not change (such as the entire first and second paragraphs, and most of the final paragraph), we use exactly the same language as in [35], except for the references to previous results in the present paper.

Proof of Lemma 9.7. Let {e_{ij}} be a system of matrix units for K^{ℝ}. Identify A with the subalgebra e_{11} ⊗ A of K^{ℝ} ⊗ A. Define ψ_{t}^{(0)} and ψ^{(0)} to be the restrictions of ϕ_{t} and ψ to A. Then [ϕ_{0}^{(0)}] = [ψ^{(0)}] in KK_{0}(A, D). It follows from (the equivalence of (1) and (3) of) Theorem 9.6 that ϕ_{0}^{(0)} is homotopic to ψ^{(0)}. Therefore ϕ_{0}^{(0)} and ψ^{(0)} are homotopic as asymptotic morphisms, and Theorem 8.9 provides an asymptotic unitary equivalence t ↦ u_{t} in U((K^{ℝ} ⊗ D)^{+}) from ϕ^{(0)} to ψ^{(0)}. Let c ∈ U((K^{ℝ} ⊗ D)^{+}) be a unitary with cψ^{(0)}(1) = ψ^{(0)}(1)c = ψ^{(0)}(1) and such that c is homotopic to u_{0}^{−1}. Then c commutes with every ψ^{(0)}(a). Replacing u_{t} by cu_{t}, we obtain an asymptotic unitary equivalence, which we again call t ↦ u_{t}, from ϕ^{(0)} to ψ^{(0)} which is in U_{0}(K^{ℝ} ⊗ D)^{+}). Define \overline{e}_{ij} = e_{ij} ⊗ 1. Then in particular u_{t}ϕ_{t}(\overline{e}_{11})u_{t}^{∗} → ψ(\overline{e}_{11}) as t → ∞. Therefore there is a continuous path t → z_{t}^{(1)} ∈ U_{0}((K^{ℝ}⊗D)^{+}) such that z_{t}^{(1)} → 1 and z_{t}^{(1)}u_{t}ϕ_{t}(\overline{e}_{11})u_{t}^{∗}(z_{t}^{(1)})^{∗} = ψ(\overline{e}_{11}) for all t. We still have z_{t}^{(1)}u_{t}ϕ_{t}(e_{11} ⊗ a)u_{t}^{∗}(z_{t}^{(1)})^{∗} → ψ(e_{11} ⊗ a) for a ∈ A. For convenience, set f_{ijt}^{(1)} = z_{t}^{(1)}u_{t}ϕ_{t}(\overline{e}_{ij})u_{t}^{∗}(z_{t}^{(1)})^{∗} and set g_{ij} = ψ(\overline{e}_{ij}). For each fixed t, the f_{ijt}^{(1)} are matrix units for K^{ℝ} as are the g_{ij}. Also, we have f_{11t}^{(1)} = g_{11}. The projections f_{11t}^{(1)}+f_{22t}^{(1)} and g_{11}+g_{22} represent the same element of K_{0}(D) so (using Lemma 7.4) there is a continuous path of partial isometries x_{t}^{(1)} in K^{ℝ}⊗D such that x_{t}^{(1)}(x_{t}^{(1)})^{∗} = 1−g_{11} −g_{22} and (x_{t}^{(1)})^{∗}x_{t}^{(1)} = 1−f_{11t}^{(1)} −f_{22t}^{(1)}. Set w_{t}^{(1)} = g_{11}+g_{21}f_{12t}^{(1)}+x_{t}^{(1)} ∈ U((K^{ℝ}⊗D)^{+}). Then one checks that w_{t}^{(1)}f_{ijt}^{(1)}(w_{t}^{(1)})^{∗} = g_{ij} for all t and for 1 ≤ i, j ≤ 2. Choose c^{(1)} ∈ U((K^{ℝ} ⊗ D)^{+}) with

c^{(1)}(g_{11} + g_{22}) = (g_{11} + g_{22})c^{(1)} = g_{11} + g_{22}
and c^{(1)}w_{1}^{(1)} ∈ U_{0}((K^{ℝ} ⊗ D)^{+}) .

Set z^{(2)} = c^{(1)}w_{t}^{(1)} for t ≥ 1 and extend z_{t}^{(2)} continuously over [0, 1] through unitaries so that z_{0}^{(2)} = 1, retaining the property that z_{t}^{(2)}g_{11} = g_{11}z_{t}^{(2)} = g_{11}. This gives z_{t}^{(2)} = 1 for t = 0, z_{t}^{(2)}g_{11} = g_{11}z_{t}^{(2)} = g_{11} for all t, and

z_{t}^{(2)}z_{t}^{(1)}u_{t}ϕ_{t}(\overline{e}_{ij})u_{t}^{∗}(z_{t}^{(1)})^{∗}(z_{t}^{(2)})^{∗} = ψ(\overline{e}_{ij})

for t ≥ 1 and 1 ≤ i, j ≤ 2. Set p^{(m)} = ∑_{k=1}^{m} g_{kk} for all postive integers m. For the induction step, assume that we have continuous paths unitaries z_{t}^{(1)}, z_{t}^{(2)}, …, z_{t}^{(n)} defined on [0, ∞) such that

• z_{t}^{(n)} = 1 for 0 ≤ t ≤ n − 2,

• z_{t}^{(n)}p^{(n−1)} = p^{(n−1)}z_{t}^{(n)} = p^{(n−1)} for all t ≥ 0,

• z_{t}^{(n)} ⋯ z_{t}^{(1)}u_{t}ϕ_{t}(\overline{e}_{ij})u_{t}^{∗}(z_{t}^{(1)})^{∗} ⋯ (z_{t}^{(n)})^{∗} = ψ(\overline{e}_{ij}) for t ≥ n − 1 and 1 ≤

i, j ≤ n.

We must construct a z_{t}^{(n+1)} with the corresponding properties. Initially, working with t ∈ [n, ∞), set

f_{ijt}^{(n)} = z_{t}^{(n)} … z_{t}^{(1)}u_{t}ϕ_{t}(\overline{e}_{ij})u_{t}^{∗}(z_{t}^{(1)})^{∗} … (z_{t}^{(n)})^{∗}

and let x_{t}^{(n)} be a continuous path of partial isometries such that x_{t}^{(n)}(x_{t}^{(n)})^{∗} = 1 − ∑_{k=1}^{n} g_{kk} = 1 − p^{(n)} and (x_{t}^{(n)})^{∗}x_{t}^{(n)} = 1 − ∑_{k=1}^{n} f_{kkt}. Set w_{t}^{(n)} = p^{(n)} + g_{(n+1) 1}f_{1 (n+1) t}^{(n)} + x_{t}^{(n)}. This continuous path of unitaries satisfies w_{t}^{(n)}p^{(n)} = p^{(n)}w_{t}^{(n)} = p^{(n)} and w_{t}^{(n)}f_{ijt}^{(n)}(w_{t}^{(n)})^{∗} = g_{ij} for all t ≥ n and all 1 ≤ i, j ≤ n + 1. As above, we can find a unitary c^{(n)} such that z_{t}^{(n+1)} = c^{(n)}w_{t}^{(n)} is in the connected component of the identity and c^{(n)}p^{(n+1)} = p^{(n+1)}c^{(n)} = c^{(n)}. Then extend z_{t}^{(n+1)} so that it is defined for all t ≥ 0 and z_{t}^{(n+1)} = 1 for 0 ≤ t ≤ n−1. Check that this z^{(n+1)} satisfies the corresponding properties listed above. Now define

z_{t} = ( lim_{n→∞} z_{t}^{(n)} ⋯ z_{t}^{(2)}z_{t}^{(1)}) u_{t} .

In a neighborhood of each t, all but finitely many of the z_{t}^{(k)} are equal to 1, so this limit of products yields a continuous path of unitaries of U_{0}((K^{ℝ} ⊗ D)^{+}). Moreover, z_{t}ϕ_{t}(\overline{e}_{ij})z_{t}^{∗} = ψ(\overline{e}_{ij}) whenever t ≥ i, j, so that lim_{t→∞} z_{t}ϕ_{t}(\overline{e}_{ij})z_{t}^{∗} = ψ(\overline{e}_{ij}) for all i and j, while

lim_{t→∞} z_{t}ϕ_{t}(e_{11} ⊗ a)z_{t}^{∗} = lim_{t→∞} z_{t}^{(1)}u_{t}ϕ_{t}(e_{11} ⊗ a)u_{t}^{∗}(z_{t}^{(1)})^{∗} = ψ(e_{11} ⊗ a)

for all a ∈ A. Since the \overline{e}_{ij} and e_{11} ⊗ a generate K^{ℝ} ⊗ A, this shows that t ↦ z_{t} is an asymptotic unitary equivalence. □

10. Classification of Real Kirchberg Algebras

We now present our main classification theorems for real Kirchberg algebras, analogous to the results of Section 4.2 of [35].

Theorem 10.1. Let A and B be unital separable nuclear purely infinite c-simple C*-algebras.

(1) Let η be an invertible element in KK(A, B). Then there is an isomorphism ϕ: K^{ℝ} ⊗ A → K^{ℝ} ⊗ B such that [ϕ] = η.

(2) Let η be an invertible element in KK(A, B) such that [1_{A}] × η = [1_{B}]. Then there is an isomorphism ϕ: A → B such that [ϕ] = η.

Proof. As in the proofs of Theorem 4.2.1 and Corollary 4.2.2 of [35]. □

Theorem 10.2. Let A and B be unital separable nuclear purely infinite c-simple C*-algebras that satisfy the universal coefficient theorem.

(1) The stable C*-algebras K^{ℝ} ⊗ A and K^{ℝ} ⊗ B are isomorphic if and only if K^{CRT}(A) and K^{CRT}(B) are isomorphic CRT-modules.

(2) The unital C*-algebras A and B are isomorphic if and only if the invariants (K^{CRT}(A), [1_{A}]) and (K^{CRT}(B), [1_{B}]) are isomorphic.

(3) The stable C*-algebras K^{ℝ} ⊗ A and K^{ℝ} ⊗ B are isomorphic if and only if K^{CR}(A) and K^{CR}(B) are isomorphic CR-modules.

(4) The unital C*-algebras A and B are isomorphic if and only if the invariants (K^{CR}(A), [1_{A}]) and (K^{CR}(B), [1_{B}]) are isomorphic.

Proof. Parts (1) and (2) are proven as in the proof of Theorem 4.2.4 of [35], using Proposition 2.3. Parts (3) and (4) then follow by Proposition 2.5. □

Corollary 10.3.

(1) The functor A ↦ K^{CRT}(A) is a bijection from isomorphism classes of real stable separable nuclear purely infinite c-simple C*-algebras that satisfy the universal coefficient theorem to isomorphism classes of countable acyclic CRT-modules.

(2) The functor A ↦ (K^{CRT}(A), [1_{A}]) is a bijection from isomorphism classes of real unital separable nuclear purely infinite c-simple C*- algebras that satisfy the universal coefficient theorem to isomorphism classes of countable acyclic CRT-modules M with distinguished element m ∈ M_{0}^{O}.

Proof. Combine Theorem 10.2 above with Theorem 1 of [10]. □ Definition 10.4.

(1) Let A be a complex C*-algebra. A real form of A is a real C*-algebra B such that B_{ℂ} ≅ A.

(2) Let G_{∗} = (G_{0}, G_{1}) be a pair of groups. A real form of G_{∗} is an acyclic CRT-module such that M_{∗}^{U} ≅ G_{∗}.

(3) Let G_{∗} = (G_{0}, G_{1}, g) be a pair of groups with a distinguished element g ∈ G_{0}. A real form of G_{∗} is a pair (M, m) where M is an acyclic CRT-module and m is a distinguished element of M_{0}^{O} such that (M_{0}^{U}, M_{1}^{U}, c(m)) ≅ (G_{0}, G_{1}, g).

Corollary 10.5. Let A be a complex unital separable nuclear purely infinite simple C*-algebra satisfying the universal coefficient theorem.

(1) The functor B ↦ K^{CRT}(B) is a bijection from isomorphism classes of real forms of K^{ℝ} ⊗ A to isomorphism classes of real forms of K_{∗}(A).

(2) The functor B ↦ (K^{CRT}(B), [1_{B}]) is a bijection from isomorphism classes of real forms of A to isomorphism classes of real forms of (K_{∗}(A), [1_{A}]).

Proof. If B is a real form of K^{ℝ} ⊗ A, then B is necessarily stable separable nuclear purely infinite and c-simple. Then KU_{∗}(B) = K_{∗}(B_{ℂ}) ≅ K_{∗}(A), so K^{CRT}(B) is a real form of K_{∗}(A). Conversely, suppose M is a real form of K_{∗}(A). Since K_{∗}(A) is countable, the exact sequences of Section 2.3 of [14] imply that M is countable. Then by Corollary 10.3, M ≅ K^{CRT}(B) for some real stable separable nuclear purely infinite c-simple C*-algebra satisfying the universal coefficient theorem. Since K_{∗}(B_{ℂ}) ≅ K_{∗}(A), it follows from Theorem 4.2.4 of [35] that B_{ℂ} ≅ A hence B is a real form of A. Furthermore, Corollary 10.3 also implies that B is unique up to isomorphism. In the unital case, suppose that B is a real form of A. As there is a isomorphism B_{ℂ} ≅ A and the unit of B_{ℂ} is c(1_{B}), there is an isomorphism ϕ: KU_{∗}(B) → K_{∗}(A) such that ϕ_{∗}(c([1_{B}])) = [1_{A}]. Thus (K^{CRT}(B), [1_{B}]) is a real form of (K_{∗}(A), [1_{A}]). Conversely, if (M, m) is a real form of K_{∗}(A), then let B be a real unital separable nuclear purely infinite c-simple C*-algebra such that (K^{CRT}(B), [1_{B}]) ≅ (M, m). Again, Theorem 4.2.4 of [35], implies that B is a real form of A. □

11. Real Forms of Cuntz Algebras

In this section, we use Corollary 10.5 to give a complete description of all real forms of the complex Cuntz algebras O_{n} for n ∈ {2, …, ∞}. The natural real form of O_{n} is the real Cuntz algebra O_{n}^{ℝ}, but we will find that there are others when n is odd. For reference, we show in Table 1 the groups making up K^{CRT}(O_{n}^{ℝ}). In the case of n = ∞ this arises from the isomorphism K^{CRT}(ℝ) ≅ K^{CRT}(O_{∞}^{ℝ}) of Proposition 2.2; while for finite n, these CRT-modules were computed in Section 5.1 of [8].

Table 1

K^{CRT}(O_{∞}^{ℝ})

0  1  2  3  4  5  6  7  8
KO_{∗} ℤ  ℤ_{2}  ℤ_{2}  0  ℤ  0  0  0  ℤ
KU_{∗} ℤ  0  ℤ  0  ℤ  0  ℤ  0  ℤ
KT_{∗} ℤ  ℤ_{2}  0  ℤ  ℤ  ℤ_{2}  0  ℤ  ℤ
K^{CRT}(O_{n}^{ℝ}) for n even
0  1  2  3  4  5  6  7  8
KO_{∗} ℤ_{n−1} 0  0  0 ℤ_{n−1} 0  0  0 ℤ_{n−1}
KU_{∗} ℤ_{n−1} 0 ℤ_{n−1} 0 ℤ_{n−1} 0 ℤ_{n−1} 0 ℤ_{n−1}
KT_{∗} ℤ_{n−1} 0  0 ℤ_{n−1} ℤ_{n−1} 0  0 ℤ_{n−1} ℤ_{n−1}
K^{CRT}(O_{n}^{ℝ}) for n − 1 ≡ 2 (mod 4)
0  1  2  3  4  5  6  7  8
KO_{∗} ℤ_{n−1} ℤ_{2}  ℤ_{4}  ℤ_{2} ℤ_{n−1} 0  0  0 ℤ_{n−1}
KU_{∗} ℤ_{n−1} 0 ℤ_{n−1} 0 ℤ_{n−1} 0 ℤ_{n−1} 0 ℤ_{n−1}
KT_{∗} ℤ_{n−1} ℤ_{2}  ℤ_{2} ℤ_{n−1} ℤ_{n−1} ℤ_{2}  ℤ_{2} ℤ_{n−1} ℤ_{n−1}
K^{CRT}(O_{n}^{ℝ}) for n − 1 ≡ 0 (mod 4)
0  1  2  3  4  5  6  7  8
KO_{∗} ℤ_{n−1} ℤ_{2}  ℤ_{2}^{2}  ℤ_{2} ℤ_{n−1} 0  0  0 ℤ_{n−1}
KU_{∗} ℤ_{n−1} 0 ℤ_{n−1} 0 ℤ_{n−1} 0 ℤ_{n−1} 0 ℤ_{n−1}
KT_{∗} ℤ_{n−1} ℤ_{2}  ℤ_{2} ℤ_{n−1} ℤ_{n−1} ℤ_{2}  ℤ_{2} ℤ_{n−1} ℤ_{n−1}

Theorem 11.1. (1) For n even or n = ∞, there is up to isomorphism only

one real form of O_{n}: the real Cuntz algebra O_{n}^{ℝ}.

(2) For n odd, there are up to isomorphism two real forms of O_{n}: the real Cuntz algebra O_{n}^{ℝ} and an exotic real form E_{n}.

Proof. First check that for odd integers n, n ≥ 3, the groups and operations shown in Table 2 form an acyclic CRT-module. Using Corollary 10.3 (that is, Theorem 1 of [10]), let E_{n} be the unique real unital separable nuclear csimple purely infinite C*-algebra satisfying the universal coefficient theorem with united K-theory as shown in Table 2 and such that [1_{E_{n}}] corresponds to a generator of the group in the real part in degree 0. By Corollary 10.5, the problem of classifying real forms of O_{n} (for n ∈ {2, 3, …, ∞}) reduces to the algebraic problem of classifying real forms of (K_{∗}(O_{n}), [1_{O_{n}}]). Suppose that (M, m) is such a real form. For n even (respectively n = ∞) we will show that (M, m) is isomorphic to (K^{CRT}(O_{n}^{ℝ}), [1_{O_{n}^{ℝ}}]) (respectively (K^{CRT}(O_{∞}^{ℝ}), [1_{O_{∞}^{ℝ}}])). For n odd we will show that (M, m) is either isomorphic to (K^{CRT}(O_{n}^{ℝ}), [1_{O_{n}^{ℝ}}]) or to (K^{CRT}(E_{n}), [1_{E_{n}}]). Furthermore, by

Table 2. K^{CRT}(E_{n}), for n odd and n ≥ 3.

0  1  2  3  4  5  6  7  8
KO_{∗} ℤ_{2(n−1)} ℤ_{2} ℤ_{2}  0 ℤ_{(n−1)/2} 0 ℤ_{2}  ℤ_{2} ℤ_{2(n−1)}
KU_{∗} ℤ_{n−1} 0 ℤ_{n−1} 0  ℤ_{n−1}  0 ℤ_{n−1} 0  ℤ_{n−1}
KT_{∗} ℤ_{n−1} ℤ_{2} ℤ_{2} ℤ_{n−1} ℤ_{n−1} ℤ_{2} ℤ_{2} ℤ_{n−1} ℤ_{n−1}
c_{∗}  1  0  0  0  2  0 \frac{n−1}{2}  0  1
r_{∗}  2  0  1  0  1  0  0  0  2
ε_{∗}  1  1  0  0  2  0  1  \frac{n−1}{2}  1
ζ_{∗}  1  0 \frac{n−1}{2}  0  1  0 \frac{n−1}{2}  0  1
(ψ_{U})_{∗}  1  0 −1  0  1  0 −1  0  1
(ψ_{T})_{∗}  1  1  1  −1  1  1  1  −1  1
γ_{∗}  1  0  1  0  1  0  1  0  1
τ_{∗}  1  1  0  1  0  0  1  2  1

Proposition 2.5 it suffices to restrict our attention to the CR-module consisting of the real and complex parts of M. Since (M, m) is a real form of (K_{∗}(O_{n}), [1_{O_{n}}]) we know that M_{0}^{U} ≅ ℤ_{n−1} (respectively M_{0}^{U} ≅ ℤ when n = ∞), M_{1}^{U} = 0, and m ∈ M_{0}^{O}. We further suppose that c_{0}(m) ∈ M_{0}^{U} is a generator (corresponding to the class of the unit in K_{0}(O_{n})). We will compute the real part of M (and the behavior of the operations η_{O}, ξ, r, c, ψ_{U}) using the long exact sequence

⋯ → M_{n}^{O} ⟶^{η_{O}} M_{n+1}^{O} ⟶^{c} M_{n+1}^{U} ⟶^{rβ_{U}^{−1}} M_{n−1}^{O} → …

and the CRT-relations described in Section 2. Since M_{k}^{U} = 0 for k odd it follows that (η_{O})_{k} is injective for k odd and surjective for k even. Furthermore, our hypothesis that c_{0}(m) generates M_{0}^{U} implies that c_{0} is surjective, which implies that r_{−2} = 0 and that (η_{O})_{−2} is injective. Thus (η_{O})_{−2} : M_{−2}^{O} → M_{−1}^{O} is an isomorphism and η3_{O}: M_{−3}^{O} → M_{0}^{O} is injective. Then the relations η3_{O} = 0 and 2η_{O} = 0 imply that M_{−3}^{O} = 0 and that M_{−2}^{O} consists only of 2-torsion. Suppose first that M_{−2}^{O} ≅ M_{−1}^{O} = 0. Then using the long exact sequence above and the relation rc = 2, the rest of the groups of M^{O} can be easily computed; except that in the case that n is odd we encounter an extension problem wherein M_{2}^{O} is either isomorphic to ℤ_{4} or to ℤ_{2} ⊕ ℤ_{2}. In that case, the same argument as in the computation of K^{CRT}(O_{n}^{ℝ}) in Section 5.1 of [8] shows that M_{2}^{O} ≅ ℤ_{4} exactly when n − 1 ≡ 0 (mod 4) and M_{2}^{O} ≅ ℤ_{2} ⊕ ℤ_{2} exactly when n−1 ≡ 2 (mod 4). Thus we find that the real and complex parts of M (as well as the operations η_{O} ξ, r, c, ψ_{U}) are isomorphic to the real and complex parts of K^{CRT}(O_{n}^{ℝ}) (respectively K^{CRT}(O_{∞}^{ℝ})).

For the remaining case, suppose that M_{−2}^{O} ≠ 0. Since this leads to the exotic CRT-module K^{CRT}(E_{n}), we will include all of the details of this computation. Since c_{0} is surjective, the relation ψ_{U}c = c implies that (ψ_{U})_{0} = 1. Then the relation β_{U}ψ_{U} = −ψ_{U}β_{U} implies that ψ_{U} = 1 in degrees congruent to 0 (mod 4) and ψ_{U} = −1 in degrees congruent to 2 (mod 4). From M_{−3}^{O} = 0 it follows that c_{−2} is injective. But the only non-trivial 2torsion subgroup of M_{−2}^{U} is isomorphic to ℤ_{2}, and that occurs only when n is finite and odd. Thus M_{−2}^{O} ≅ M_{−1}^{O} ≅ ℤ_{2} and the complexification map c_{−2} : ℤ_{2} → ℤ_{n−1} is multiplication by (n − 1)/2 (in terms of chosen generators). The map r_{−4} is surjective and has kernel equal to ((n − 1)/2)ℤ_{n−1} ≅ ℤ_{2} so M_{−4}^{O} ≅ ℤ_{(n−1)/2}. The relation c_{−4}r_{−4} = 1 + (ψ_{U})_{−4} = 2 implies that the map c_{−4} : ℤ_{(n−1)/2} → ℤ_{n−1} is multiplication by 2. Continuing to work our way down, the fact that c_{−4} is injective implies that M_{−5}^{O} = 0. The fact that the image of c_{−4} is 2ℤ_{n−1} implies that M_{−6}^{O} ≅ ℤ_{2} and r_{−6} is surjective. The relation c_{−6}r_{−6} = 1 + (ψ_{U})_{−6} = 0 implies that c_{−6} = 0 from which we see that η_{−7} is an isomorphism. Thus M_{−7}^{O} ≅ ℤ_{2}. Finally, we compute M_{−8}^{O} ≅ M_{0}^{O}. The exact sequence indicates that it is an extension of ℤ_{2} by ℤ_{n−1}. We will prove that it is isomorphic to ℤ_{2(n−1)}. If not, then M_{0}^{O} ≅ ℤ_{2} ⊕ ℤ_{n−1} and we can arrange the direct sum decomposition so that η_{−1} = ( \begin{matrix}1 \\ 0 \end{matrix} ) and c_{0} = (^{0} ^{1} ). Then the relation rc = 2 implies that r_{0} = ( \begin{matrix}0 \\ 2 \end{matrix} ). But then there is no isomorphism from M_{0}^{O}/image(r_{0}) ≅ ℤ_{2} ⊕ ℤ_{2} to M_{1}^{O} ≅ ℤ_{2} as required by the long exact sequence. Thus in the case that M_{−2}^{O} ≠ 0 it must be that n is odd and it must be that the real and complex parts of M are isomorphic to the real and complex parts of K^{CRT}(E_{n}) as in Table 2, completing the proof. □

We remark that the above result can instead be obtained using the analysis of acyclic CRT-modules in [23]. Indeed, let M be an acyclic CRT-module such that M_{0}^{U} is isomorphic to ℤ_{k−1} or ℤ, M_{1}^{U} = 0, and c_{0} : M_{0}^{O} → M_{0}^{U} is surjective (hence (ψ_{U})_{0} = 1). By Lemma 8.3.1, Proposition 8.3.2, and Theorem 8.3.3 of [23], there are isomorphisms

h_{k}(M) := ker(1 − (ψ_{U})_{k})/image(1 + (ψ_{U})_{k}) ≅ η_{O}M_{k}^{O} ⊕ η_{O}M_{k+4}^{O}

and, furthermore, M is determined up to isomorphism by M^{U}, ψ_{U}, and the resulting decompositions of h_{k}(M) for k = 0 and k = 2. Using (ψ_{U})_{0} = 1 and (ψ_{U})_{2} = −1, we obtain

(h_{0}(M), h_{2}(M)) = \begin{cases}(ℤ_{2}, 0) & if M_{0}^{U} = ℤ \\ (0, 0) & if M_{0}^{U} = ℤ_{n−1} with n even \\ (ℤ_{2}, ℤ_{2}) & if M_{0}^{U} = ℤ_{n−1} with n odd. \end{cases}

The resulting possibilities for M are realized by the united K-theory of O_{∞}^{ℝ} and ℍ ⊗ O_{∞}^{ℝ} in the first case; by that of O_{n}^{ℝ} in the second case; and by that of O_{n}^{ℝ}, ℍ ⊗ O_{n}^{ℝ}, E_{n}, and ℍ ⊗ E_{n} in the third case. The assumption that c_{0} is surjective reduces the possibilities to the united K-theory of O_{∞}^{ℝ}, O_{n}^{ℝ}, or E_{n}.

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Jeffrey L. Boersema Department of Mathematics Seattle University Seattle, WA USA boersema@seattleu.edu Jeffrey L. Boersema Department of Mathematics University of Hawaii Hilo, HI USA ruize@hawaii.edu

P. J. Stacey

Department of Mathematics LaTrobe University Melbourne, VIC Australia P.Stacey@latrobe.edu.au 656

Documenta Mathematica 16 (2011)