SYMMETRIES OF KIRCHBERG ALGEBRAS

DAVID J. BENSON, ALEX KUMJIAN, AND N. CHRISTOPHER PHILLIPS

Abstract. Let G_{0} and G_{1} be countable abelian groups. Let γ_{i} be an automorphism of G_{i} of order two. Then there exists a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem and with [1_{A}] = 0 in K_{0}(A), and an automorphism α ∈ Aut(A) of order two, such that K_{0}(A) ≅ G_{0}, such that K_{1}(A) ≅ G_{1}, and such that α_{∗}: K_{i}(A) → K_{i}(A) is γ_{i}. As a consequence, we prove that every ℤ_{2}-graded countable module over the representation ring R(ℤ_{2}) of ℤ_{2} is isomorphic to the equivariant K-theory K^{ℤ_{2}}(A) for some action of ℤ_{2} on a unital Kirchberg algebra A.

Along the way, we prove that every not necessarily finitely generated ℤ[ℤ_{2}]- module which is free as a ℤ-module has a direct sum decomposition with only three kinds of summands, namely ℤ[ℤ_{2}] itself and ℤ on which the nontrivial element of ℤ_{2} acts either trivially or by multiplication by −1.

1. Introduction

Following Definition 4.3.1 of Part 1 of [22], we use the term Kirchberg algebra for a purely infinite simple separable nuclear C*-algebra. In this paper, we prove that any order two automorphism of the K-theory of a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem, and with [1_{A}] = 0 in K_{0}(A), lifts to an order two automorphism of A.

Recall the classification theorem for unital Kirchberg algebras satisfying the Universal Coefficient Theorem ([14]; Theorem 4.2.4 of [19]): if A and B are such algebras, and if θ: K_{∗}(A) → K_{∗}(B) is a graded isomorphism such that θ([1_{A}]) = [1_{B}], then there is an isomorphism φ: A → B such that φ_{∗} = θ. The statement also holds with “isomorphism” replaced by “homomorphism” everywhere ([14]; in [19] see Theorem 4.1.3 and the proofs of Corollary 4.4.2 and Theorem 4.2.4). Moreover, every pair of countable abelian groups occurs as the K-theory of a unital Kirchberg algebra satisfying the Universal Coefficient Theorem, and the K_{0}-class of the identity can be arbitrary (Section 4.4 in Part 1 of [22] or Theorem 5.2 of [9]). Thus, the classification functor is surjective on isomorphism classes in this case.

George Elliott has asked to what extent this functor “splits”, in the same sense in which a surjection between abelian groups might (or might not) split. That is: How close can one come to constructing a functor F, from the values of the invariant, to Kirchberg algebras satisfying the Universal Coefficient Theorem, such that K_{∗}(F (G_{∗})) ≅ G_{∗}, etc.? One can’t do this exactly. (See Example 1.1 below.) We may then ask for some weaker sort of splitting, or we can try to eliminate the problem by reducing the number of morphisms in the categories. The question we consider here, of lifting finite order automorphisms of the K-theory to automorphisms of the algebra of the same order, is a special case of what one gets by restricting the morphisms to be isomorphisms.

Date: 7 Feb. 2003.

1991 Mathematics Subject Classification. Primary 20C10, 46L55; Secondary 19K99, 19L47, 46L40, 46L80.

Research partially supported by NSF grants DMS-9988110 (D. J. Benson) and DMS-0070776 (N. C. Phillips).

Our method is to find, for given countable abelian groups G_{i} and order two automorphisms γ_{i}, some separable nuclear C*-algebra A_{0} whose K-theory is G_{∗} and some order two automorphism α_{0} of A_{0} which lifts the γ_{i}. In fact, A_{0} will be type I. Then we apply the construction of [15] to get a Kirchberg algebra with the same properties. To construct A_{0}, we regard G_{∗} as a module over the group ring ℤ[ℤ_{2}]. (Throughout this paper, ℤ_{p} denotes ℤ/pℤ, not the p-adic integers.) Generalizing an old result for the finitely generated case (see Lemma 1 of [11]), we prove that every ℤ[ℤ_{2}]-module which is free as a ℤ-module has a direct sum decomposition with only three kinds of summands, namely ℤ[ℤ_{2}] itself and ℤ on which the nontrivial element of ℤ_{2} acts either trivially or by multiplication by −1. (Butler and Kovács have also obtained similar results, by slightly different methods [5]; a more general result is to appear in [4].) We produce A_{0} by combining this structure theorem with part of Schochet’s geometric realization technique from [24].

We want to explicitly point out that the K-theory of a C*-algebra A with an action of ℤ_{2}, regarded as a ℤ[ℤ_{2}]-module, is not the same as the ℤ_{2}-equivariant K-theory of A, regarded as a module over the representation ring R(ℤ_{2}) ≅ ℤ[ℤ_{2}]. It is, however, the same as the ℤ_{2}-equivariant K-theory of C^{∗}(ℤ_{2}, A), using the dual action. (Compare with Lemma 4.4.) Thus, as a corollary of our construction we obtain a realization theorem, Theorem 4.7, for R(ℤ_{2})-modules as the equivariant K-theory of actions of ℤ_{2} on Kirchberg algebras. There is also a version for type I C*-algebras.

The first author would like to thank the MSRI for its hospitality while this work was in progress. The second and third authors would like to thank the organizers of the conference “Aperiodic Order, Dynamical Systems, Operator Algebras, and Topology”, during which valuable discussions were held. The third author would like to acknowledge the hospitality of the Mathematisches Forschungsinstitut Oberwolfach, where Elliott’s raised his question and where Example 5.1 was constructed, and of the University of Georgia for its invitation for a visit in the spring of 2002.

This paper is organized as follows. In the rest of this section, we demonstrate the failure of the classification functor to split. In Section 2 we prove a structure theorem for ℤ[ℤ_{2}]-modules which are free as ℤ-modules. In Section 3 we prove the weak form of realization, using some type I C*-algebra with the right K-theory. In Section 4 we prove the main result, and also the realization theorems for ℤ_{2}- equivariant K-theory. Section 5 contains an explicit formula for the order two automorphism in the simplest nontrivial case of our theorem, namely the nontrivial automorphism of K_{∗}(M_{3}(O_{4})).

Here is the example which shows that the most obvious sort of splitting of the classification functor is not possible.

Example 1.1. There is no functor F, from the category of ℤ_{2}-graded abelian groups G_{∗} with distinguished element g in degree zero and graded homomorphisms preserving the distinguished elements, to the category of unital Kirchberg algebras satisfying the Universal Coefficient Theorem of [23] and unital homomorphisms, such that K_{∗}(F (G_{∗}, g)) ≅ G_{∗} with [1_{F (G_{∗},g)}] ↦ g for all G_{∗} and g, and such that the diagram

K_{∗}(F (G_{∗}, g)) ⟶^{F (θ)_{∗}} K_{∗}(F (H_{∗}, h))
↓  ↓
G_{∗}  ⟶^{θ}  H_{∗}

commutes whenever θ: G_{∗} → H_{∗} is a graded homomorphism such that θ(g) = h. Suppose we had such a functor. Take

G_{0} = 0, G_{1} = 0, H_{0} = ℤ, and H_{1} = 0.

Take g = 0 and h = 0. Let θ: G_{∗} → H_{∗} and ρ: H_{∗} → G_{∗} be the unique homomorphisms between these groups. Then ρ ∘ θ = id_{G_{∗}}. Therefore, using functoriality,

F (ρ) ∘ F (θ) = F (ρ ∘ θ) = F (id_{G_{∗}}) = id_{F (G_{∗})}.

In particular, F (ρ) is surjective. However, F (G_{∗}, g) (which is isomorphic to the Cuntz algebra O_{2} [6]) and F (H_{∗}, h) (which is isomorphic to a suitable corner of O_{∞}) are nonisomorphic simple C*-algebras, so there are no surjective homomorphisms from F (H_{∗}, h) to F (G_{∗}, g). Remark 1.2. The problem here has nothing to do with units, since the same argument works even if one considers stable Kirchberg algebras satisfying the Universal Coefficient Theorem. Moreover, one can even restrict, in either case, to the case K_{1} = 0.

2. Modules over the group ring of ℤ_{2}

In this section, we prove a structure theorem for modules over the group ring of ℤ_{2} which are free as ℤ-modules. We have learned that Butler and Kovác proved the main result of this section a little earlier than we did, using slightly different methods [5]. They use a sophisticated argument to reduce to the countably generated case first, which is avoided in our analysis. There is also a subsequent generalization to modules over the group ring of ℤ_{p} for an arbitrary prime p [4]. We have decided to retain our proof for completeness of exposition, and because it gives the result we need with a minimum of machinery.

For any unital ring R and any discrete group Γ, we let R[Γ] denote the ordinary algebraic group ring of Γ with coefficients in R. Although we are ultimately interested in the case R = ℤ, we will need the cases R = 𝔽_{2}, the field with two elements, and R = ℚ. Also, any ℤ[Γ]-module is automatically a ℤ-module (abelian group) by restriction of scalars.

The structure theorem is already known for the finitely generated case (see the first lemma), and the main point of this section is to remove the finite generation hypothesis. Lemma 2.1. Let N be a finitely generated ℤ[ℤ_{2}]-module which is free as a ℤmodule. Then N is a finite direct sum ⨁_{i∈I} N_{i} in which each N_{i} is isomorphic to one of the following three ℤ[ℤ_{2}]-modules:

• T_{1} = ℤ with the nontrivial element of ℤ_{2} acting trivially.

• T_{2} = ℤ with the nontrivial element of ℤ_{2} acting by multiplication by −1.

• T_{3} = ℤ[ℤ_{2}]. The multiplicities of T_{1}, T_{2} and T_{3} in such a direct sum decomposition are independent of the decomposition.

Proof. This is an immediate consequence of the canonical form for invertible elements a ∈ M_{n}(ℤ) with a^{2} = 1, given in Lemma 1 of [11], and the discussion following the proof of that lemma. For the matrix L of [11], the computation

L = ( \begin{matrix}1 & 0 \\ 1 & −1 \end{matrix} ) = ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} )^{−1} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} )

shows that it is similar over ℤ to the action of the nontrivial element of ℤ_{2} on ℤ[ℤ_{2}] in its usual basis.

Theorem 74.3 of [8] gives a structure theorem for finitely generated ℤ[ℤ_{p}]-modules which are free as a ℤ-modules, for arbitrary primes p. As described there, the direct sum decomposition is in general not unique. Lemma 2.2. Let M be a ℤ[ℤ_{2}]-module. Suppose M is free as a ℤ-module and M/2M is free as an 𝔽_{2}[ℤ_{2}]-module. Then M is free as a ℤ[ℤ_{2}]-module.

Proof. Theorem 6.1 of [2], with G = ℤ_{2} and H = {0}, shows that M is projective as a ℤ[ℤ_{2}]-module.

If M is finitely generated, apply Lemma 2.1. Since T_{1} and T_{2} are not projective as ℤ[ℤ_{2}]-modules, they can’t appear in the direct sum. Therefore M is free.

So suppose M is not finitely generated. A direct calculation shows that ℤ[ℤ_{2}] has no nontrivial idempotents. (This is true for any finite group in place of ℤ_{2}, by Corollary 8.1 of [26].) Therefore Corollary 4.5 of [1] applies, and shows that M is free.

Notation 2.3. For the rest of this section, we regard ℤ[ℤ_{2}] as a subring of ℚ[ℤ_{2}]. If M is a ℤ[ℤ_{2}]-module, then the corresponding ℚ[ℤ_{2}]-module is ℚ ⊗_{ℤ} M, and if M is free as a ℤ-module then we may clearly regard M as a submodule of ℚ⊗_{ℤ}M. We let s be the nontrivial element of ℤ_{2}, and use the same notation for the corresponding element of ℤ[ℤ_{2}]. Further set

e = \frac{1}{2}(1 + s) ∈ ℚ[ℤ_{2}] and f = \frac{1}{2}(1 − s) ∈ ℚ[ℤ_{2}],

which are idempotents with e + f = 1. Lemma 2.4. Let M be a ℤ[ℤ_{2}]-module which is free as a ℤ-module. Let m ∈ M, and assume (1 + s)m ∈ 2M. Then, following Notation 2.3,

m ∈ (M ∩ eM) + (M ∩ fM) ⊂ ℚ ⊗_{ℤ} M.

Proof. We have (1 − s)m = (1 + s)m − 2m ∈ 2M. Write (1 + s)m = 2m_{0} and (1 − s)m = 2m_{1} with m_{0}, m_{1} ∈ M. In ℚ ⊗_{ℤ} M we have m_{0} = \frac{1}{2}(1 + s) = em and m_{1} = fm. Then m_{0} ∈ M ∩ eM and m_{1} ∈ M ∩ fM. Moreover, 2m = 2(m_{0} + m_{1}), so m = m_{0} + m_{1}.

Lemma 2.5. Let M be a ℤ[ℤ_{2}]-module which is free as a ℤ-module. Following Notation 2.3, suppose that M ∩ eM = 2eM and M ∩ fM = 2fM. Then M is free as a ℤ[ℤ_{2}]-module.

Proof. By Lemma 2.2, it suffices to show that M/2M is free as an 𝔽_{2}[ℤ_{2}]-module.

Set N = (2eM + 2fM)/M. Consider 1 + s as a multiplication map on M/2M. We claim that

Ker(1 + s) = (1 + s)(M/2M) = N.

We consider Ker(1 + s) first. Suppose m ∈ M and m + 2M ∈ Ker(1 + s). Then (1+s)m ∈ 2M. Using Lemma 2.4 at the first step and the hypothesis at the second, we get

m ∈ (M ∩ eM) + (M ∩ fM) = 2eM + 2fM,

so m + 2M ∈ N. For the reverse, suppose m ∈ 2eM + 2fM. Write

m = (1 + s)m_{0} + (1 − s)m_{1}

with m_{0}, m_{1} ∈ M. Then

(1 + s)m = (1 + s)^{2}m_{0} + (1 + s)(1 − s)m_{1} = 2(1 + s)m_{0} ∈ 2M,

so m + 2M ∈ Ker(1 + s).

Now we consider the range of 1 + s. Since (1 + s)^{2} = 0 in 𝔽_{2}[ℤ_{2}], we get (1 + s)(M/2M) ⊂ Ker(1 + s) = N. For the reverse, let m ∈ 2eM + 2fM, and again write m = (1 + s)m_{0} + (1 − s)m_{1} with m_{0}, m_{1} ∈ M. Then

m + 2M = (1 + s)(m_{0} + m_{1}) − 2sm_{1} + 2M = (1 + s)(m_{0} + m_{1}) + 2M
∈ (1 + s)(M/2M).

This completes the proof of the claim.

Now N is a 𝔽_{2}-vector subspace of M/2M, so there exists a 𝔽_{2}-vector subspace V ⊂ M/2M such that V ⊕ N = M/2M. We claim that V ∩ sV = {0} and V + sV = M/2M. To prove the first, suppose v, w ∈ V and v = sw. Then (1 + s)v = (1 + s)sw = (1 + s)w, so that (1 + s)(v − w) = 0. Therefore v − w ∈ Ker(1 + s) ∩ V = N ∩ V, whence v = w. Now (1 + s)v = v + w = 2v = 0, so v ∈ (1 + s)(M/2M) ∩ V = N ∩ V, whence v = 0. For the second, let d ∈ M/2M. Using the first claim, we have N = (1 + s)(M/2M) = (1 + s)(V). So we can write d = x + (1 + s)w with w, x ∈ V. Then d = v + sw with v = x + sw and w both in V. This proves the claim.

Given the last claim, it is immediate that M/2M ≅ 𝔽_{2}[ℤ_{2}] ⊗_{𝔽_{2}} V, which is a free 𝔽_{2}[ℤ_{2}]-module.

We only need the next two lemmas for the prime 2. Since they hold for arbitrary primes, with the same proof, we may as well give them in that generality. Lemma 2.6. Let M be a free ℤ-module, and let p be a prime number. Let V be an 𝔽_{p}-vector subspace of M/pM. Then there exists a direct summand L of M such that the image of L under the map M → M/pM is exactly V.

Proof. Let π: M → M/pM be the quotient map. Choose an 𝔽_{p}-vector subspace W of M/pM such that M/pM = V ⊕ W. Let σ: M/pM → W be the projection onto W obtained from this direct sum decomposition. Using an 𝔽_{p}-basis for W, find a free ℤ-module P_{0} with an isomorphism P_{0}/pP_{0} → W. Let κ_{0} : P_{0} → W be the composite of this isomorphism with the quotient map P_{0} → P_{0}/pP_{0}. Since M is a projective ℤ-module and κ_{0} is surjective, there exists a ℤ-module homomorphism φ_{0} : M → P_{0} making the following diagram commute:

M ⟶^{φ_{0}} P_{0}
π↓  ↓^{κ_{0}}
M/pM ⟶^{σ} W

The image φ_{0}(M) is a submodule of a free ℤ-module, hence free (Theorem 14.5 of [10]). Let (b_{i})_{i∈I_{0}} be a ℤ-basis for φ_{0}(M). Since κ_{0} ∘ φ_{0} = σ ∘ π is surjective, {κ_{0}(b_{i}): i ∈ I_{0}} spans W. Choose a subset I ⊂ I_{0} such that (κ_{0}(b_{i}))_{i∈I} is an 𝔽_{p}-basis for W. Set P = span_{ℤ}({b_{i}: i ∈ I}) ⊂ P_{0}, which is a free ℤ-module with basis (b_{i})_{i∈I}. For each j ∈ I_{0} \ I, choose α_{i,j} ∈ 𝔽_{p} for i ∈ I, all but finitely many of which are zero, such that κ_{0}(b_{j}) = ∑_{i∈I} α_{i,j}κ_{0}(b_{i}). For j ∈ I_{0} \ I and i ∈ I, choose β_{i,j} ∈ ℤ whose image in 𝔽_{p} is α_{i,j}, and such that if α_{i,j} = 0 then β_{i,j} = 0. For each j ∈ I_{0} \I, all but finitely many of the β_{i,j} are zero. Therefore there is a well defined surjective ℤ-module homomorphism μ: φ_{0}(M) → P such that p(b_{j}) = b_{j} for j ∈ I and μ(b_{j}) = ∑_{i∈I} β_{i,j}b_{i} for j ∈ I_{0} \ I. By construction, we have κ_{0} ∘ μ(b_{i}) = κ_{0}(b_{i}) for all i ∈ I. Therefore (κ_{0}|_{P}) ∘ μ = κ_{0}. Setting φ = μ ∘ φ_{0} and κ = κ_{0}|_{P}, we thus have a commutative diagram:

M ⟶^{φ} P
π↓  ↓^{κ}
M/pM ⟶^{σ} W

The ℤ-module P is still free, and in addition φ is surjective. Furthermore, Ker(κ) = pP, since a linear combination of (b_{i})_{i∈I} has image in W equal to zero if and only if all coefficients are even.

Set L = Ker(φ). Since P is projective, there is a ℤ-module homomorphism η: P → M such that φ ∘ η = id_{P}. Then M = L ⊕ η(P). We claim that π(L) = V. First, if m ∈ L then σ(π(m)) = κ(φ(m)) = 0, so π(m) ∈ Ker(σ) = V. Now suppose v ∈ V. Choose m_{0} ∈ M such that π(m_{0}) = v. Then κ(φ(m_{0})) = 0, whence φ(m_{0}) ∈ pP. Write φ(m_{0}) = px with x ∈ P. Then m = m_{0} − pη(x) ∈ Ker(φ) by a diagram chase, and π(m) = π(m_{0}) = v. This proves the claim, and the lemma.

Lemma 2.7. Let M be a free ℤ-module and let p be a prime number. Let N be a ℤsubmodule such that pM ⊂ N ⊂ M. Then there exist ℤ-submodules N_{0}, N_{1} ⊂ M such that M = N_{0} ⊕ N_{1} and N = pN_{0} ⊕ N_{1}.

Proof. Let π: M → M/pM be the quotient map. Using Lemma 2.6, write M = N_{0} ⊕ N_{1} for ℤ-submodules N_{0}, N_{1} ⊂ M, with π(N_{1}) = π(N). The result will now follow if we prove that N = pN_{0} + N_{1}.

We prove pN_{0} + N_{1} ⊂ N. Clearly pN_{0} ⊂ N. So let m ∈ N_{1}. Choose n ∈ N such that π(n) = π(m). Then π(m − n) = 0, so m − n ∈ pM ⊂ N. Thus m ∈ N. We have N_{1} ⊂ N.

Now we prove the reverse inclusion. Let n ∈ N. Write n = r + m_{1} with r ∈ N_{0} and m_{1} ∈ N_{1}. Then π(r) = π(n) − π(m_{1}). Because N_{1} ⊂ N, we get π(r) ∈ π(N_{0}) ∩ π(N). By considering a ℤ-basis for M made up of bases for N_{0} and N_{1}, we see that π(N_{0}) ∩ π(N) = {0}. Therefore π(r) = 0. So we can write r = pm_{0} with m_{0} ∈ M. Since M is torsion free and N_{0} is a summand, we get m_{0} ∈ N_{0}. Therefore n = pm_{0} + m_{1} ∈ pN_{0} + N_{1}.

The uniqueness statement in the following theorem will not be used in the rest of the paper, but we include it because it seems of independent interest. Theorem 2.8. Let M be a ℤ[ℤ_{2}]-module which is free as a ℤ-module. Then M is a direct sum ⨁_{i∈I} M_{i} in which each M_{i} is isomorphic to one of the three modules T_{1}, T_{2}, or T_{3} of Lemma 2.1. The multiplicities of T_{1}, T_{2} and T_{3} in such a direct sum decomposition are independent of the decomposition.

Proof. We use Notation 2.3.

We first prove uniqueness of the multiplicities. Suppose the ℤ[ℤ_{2}]-module M is isomorphic to the direct sum of n_{1} copies of T_{1}, n_{2} copies of T_{2}, and n_{3} copies of T_{3}. Then n_{1} is the 𝔽_{2}-vector space dimension of (M ∩eM)/2eM ⊂ eM/2eM, and similarly n_{2} = dim_{𝔽_{2}}((M ∩ fM)/2fM). Finally, one checks that n_{3} = dim_{𝔽_{2}}(eM/(M ∩ eM)), which is the same as dim_{𝔽_{2}}(fM/(M ∩ fM)).

Now we prove existence. We have

2eM ⊂ M ∩ eM ⊂ eM and 2fM ⊂ M ∩ fM ⊂ fM.

Apply Lemma 2.7 twice, to get ℤ-submodules R_{0}, R_{1} ⊂ eM and S_{0}, S_{1} ⊂ fM such that

eM = R_{0} ⊕ R_{1} and M ∩ eM = 2R_{0} ⊕ R_{1}

and

fM = S_{0} ⊕ S_{1} and M ∩ fM = 2S_{0} ⊕ S_{1}.

Since eM ∩ fM = {0} and e + f = 1, we can write

(M ∩ eM) ⊕ (M ∩ fM) ⊂ M ⊂ eM ⊕ fM.

Substituting from above, we have the inclusions of ℤ-module direct sums

2R_{0} ⊕ R_{1} ⊕ 2S_{0} ⊕ S_{1} ⊂ M ⊂ R_{0} ⊕ R_{1} ⊕ S_{0} ⊕ S_{1}.

Since R_{1}, S_{1} ⊂ M, we have the ℤ-module direct sum decomposition

M = R_{1} ⊕ S_{1} ⊕ N with N = M ∩ (R_{0} ⊕ S_{0}).

Now R_{1} ⊂ eM and sm = m for all m ∈ eM. So R_{1} is actually a ℤ[ℤ_{2}]-module. Since it is a ℤ-submodule of M, it is a free ℤ-module (Theorem 14.5 of [10]), and therefore as a ℤ[ℤ_{2}]-module it is a direct sum of ℤ[ℤ_{2}]-modules isomorphic to T_{1}. Similarly, from S_{1} ⊂ fM and sm = −m for all m ∈ fM, we see that S_{1} is a ℤ[ℤ_{2}]-module which is a direct sum of ℤ[ℤ_{2}]-modules isomorphic to T_{2}. We will complete the proof by showing that N is a ℤ[ℤ_{2}]-module, so that the direct sum decomposition above is really a ℤ[ℤ_{2}]-module direct sum decomposition, and that it is free, so that it is a direct sum of ℤ[ℤ_{2}]-modules isomorphic to T_{3}.

To prove that N is a ℤ[ℤ_{2}]-module, we simply observe that R_{0} and S_{0} are ℤ[ℤ_{2}]- modules, for the same reason that R_{1} and S_{1} are. For freeness, we know as above that N is a free ℤ-module because it is a ℤ-submodule of M. By Lemma 2.5, it now suffices to prove that N ∩eN = 2eN and N ∩fN = 2fN. That 2eN ⊂ N ∩eN and 2fN ⊂ N ∩ fN is clear, so we prove the reverse inclusions.

Let m ∈ N ∩ eN. Use m ∈ N to write m = a + b with a ∈ R_{0} and b ∈ S_{0}. Use m ∈ eN to write m = e(x + y) with x ∈ R_{0} and y ∈ S_{0}. Note that ex = x and ey = 0, and combine the two expressions for m to get x − a = b. Since x − a ∈ eM and b ∈ fM, we get b = 0. Therefore m = a ∈ R_{0}. From m ∈ M ∩ eM = 2R_{0} ⊕ R_{1} we get m ∈ 2R_{0}. Now R_{0} ⊂ eM implies eR_{0} = R_{0}, so R_{0} ⊂ N implies R_{0} ⊂ eN. Therefore m ∈ 2eN, as desired.

Now suppose m ∈ N ∩ fN. Write m = a + b = f(x + y) with a, x ∈ R_{0} and b, y ∈ S_{0}. From fx = 0 and fy = y, we get y − b = a ∈ eM ∩ fM = {0}. So m = b ∈ S_{0}. From m ∈ M ∩ fM = 2S_{0} ⊕ S_{1} we now get m ∈ 2S_{0}. Since S_{0} ⊂ N ∩ fM ⊂ fN, it follows that m ∈ 2fN, as desired. This completes the proof.

3. Geometric resolution

In this section, we prove that for every order two automorphism γ of a countable ℤ_{2}-graded abelian group G_{∗}, there is some separable C*-algebra A with an order two automorphism α ∈ Aut(A), such that there is a graded isomorphism μ: G_{∗} → K_{∗}(A) which identifies γ with α_{∗}. We borrow the method of geometric realization of resolutions from Schochet’s proof of the Künneth formula for C*-algebras [24]. As it turns out, the C*-algebra A we produce will be of type I and nonunital.

Although we do not state this version formally, the same proofs show that if G_{∗} is not countable, we can still produce an automorphism of a nonseparable type I C*-algebra satisfying the remaining conditions.

For any C*-algebra A, we let A^{+} denote its usual unitization, in which we add a new unit even if A is already unital.

Lemma 3.1. Let C = ⨁_{n∈ℕ}[C_{0}(ℝ) ⊕ C_{0}(ℝ)], and let γ ∈ Aut(C) be the order two automorphism given by

γ((f_{1}, g_{1}), (f_{2}, g_{2}), …) = ((g_{1}, f_{1}), (g_{2}, f_{2}), …).

Let M be a subgroup of K_{1}(C) which is invariant under γ_{∗}. Then there exists a separable Hilbert space H with an action of ℤ_{2}, a separable commutative C*-algebra B with an automorphism β of order two, and a ℤ_{2}-equivariant homomorphism φ: B → K(H) ⊗ C, where ℤ_{2} acts on K(H) by conjugation, such that K_{0}(B) = 0, such that φ_{∗}: K_{1}(B) → K_{1}(C) is injective, and such that the image of φ_{∗} is exactly M.

Proof. The K-theory K_{∗}(C) is a countable abelian group with an automorphism γ_{∗} of order two, and hence in an obvious way a ℤ[ℤ_{2}]-module. As ℤ[ℤ_{2}]-modules, K_{1}(C) is a countable direct sum of copies of ℤ[ℤ_{2}] and K_{0}(C) = 0. Moreover, the hypothesis on M is just that it is a ℤ[ℤ_{2}]-submodule. It is a free ℤ-module because subgroups of free abelian groups are free (Theorem 14.5 of [10]), and it is clearly countably generated, so by Theorem 2.8 there is an isomorphism M ≅ ⨁_{i∈I} M_{i} with I finite or countable, and with each M_{i} isomorphic to one of the three modules T_{j} of Lemma 2.1. We regard each M_{i} as a ℤ[ℤ_{2}]-submodule of K_{1}(C).

For each i ∈ I, we construct a separable Hilbert space H_{i} with an action of ℤ_{2}, a separable commutative C*-algebra B_{i} with an automorphism β_{i} of order two, and a ℤ_{2}-equivariant homomorphism φ_{i}: B_{i} → K(H_{i}) ⊗ C, such that K_{0}(B_{i}) = 0, such that (φ_{i})_{∗}: K_{1}(B_{i}) → K_{1}(C) is injective, and such that the image of (φ_{i})_{∗} is exactly M_{i}. There are three cases, but we begin by introducing notation common to all three. Define u: ℝ → S^{1} by

u(t) = exp (πi (1 + t(1 + t^{2})^{−1/2})) .

Then u is a unitary in the unitization C_{0}(ℝ)^{+} such that [u] generates K_{1}(C_{0}(ℝ)), and u−1 generates C_{0}(ℝ) as a C*-algebra. In particular, if E is any C*-algebra and v ∈ E^{+} is any unitary such that v − 1 ∈ E, then there is a unique homomorphism ψ: C_{0}(ℝ) → E such that ψ(u − 1) = v − 1. Further let s be the nontrivial element of ℤ_{2}, and use the same notation for the corresponding element of ℤ[ℤ_{2}]. We will also write elements of K_{1}(C) as sequences

(k_{1} + l_{1}s, k_{2} + l_{2}s, …) ∈ ⨁_{n∈ℕ} ℤ[ℤ_{2}],

where, with all nontrivial entries being in the n-th position, (0, …, 0, 1, 0, …) corresponds to the K_{1}-class of the unitary

1 + ((0, 0), …, (0, 0), (u − 1, 0), (0, 0), …) ∈ C^{+}

and (0, …, 0, s, 0, …) corresponds to the K_{1}-class of the unitary

1 + ((0, 0), …, (0, 0), (0, u − 1), (0, 0), …) ∈ C^{+}.

The first case is M_{i} ≅ T_{1}. Let m be a generator of M_{i} as a ℤ-module. Then sm = m. Therefore m has the form

m = (k_{1} + k_{1}s, k_{2} + k_{2}s, …),

with all but finitely many of the k_{j} equal to zero. Take H_{i} = ℂ with the trivial action of ℤ_{2}, take B_{i} = C_{0}(ℝ), take β_{i} = id_{B_{i}}, and take φ_{i} to be the homomorphism determined by

φ_{i}(u − 1) = ((u^{k_{1}} − 1, u^{k_{1}} − 1), (u^{k_{2}} − 1, u^{k_{2}} − 1), …).

The required properties are immediate.

Next, suppose M_{i} ≅ T_{2}. Let m be a generator of M_{i} as a ℤ-module. Then sm = −m. Therefore m has the form

m = (k_{1} − k_{1}s, k_{2} − k_{2}s, …),

with all but finitely many of the k_{j} equal to zero. Take H_{i} = ℂ with the trivial action of ℤ_{2}, and take B_{i} = C_{0}(ℝ). Take β_{i}: C_{0}(ℝ) → C_{0}(ℝ) to be the homomorphism β_{i}(f)(t) = f(−t), which is the unique homomorphism such that β_{i}(u − 1) = u^{∗} − 1. Take φ_{i} to be the homomorphism determined by

φ_{i}(u − 1) = ((u^{k_{1}} − 1, u^{−k_{1}} − 1), (u^{k_{2}} − 1, u^{−k_{2}} − 1), …).

Equivariance follows from

φ_{i}(u^{∗} − 1) = ((u^{−k_{1}} − 1, u^{k_{1}} − 1), (u^{−k_{2}} − 1, u^{k_{2}} − 1), …),

and the rest of the required properties are immediate.

Finally, suppose M_{i} ≅ T_{3}. Let m ∈ K_{1}(C) be the image of 1 ∈ T_{3} under the isomorphism T_{3} → M_{i}. Write m = (k_{1} + l_{1}s, k_{2} + l_{2}s, …). Then the image of s is

sm = (l_{1} + k_{1}s, l_{2} + k_{2}s, …),

and these two elements are linearly independent over ℤ. Take H_{i} = ℂ^{2}, with s(ξ_{1}, ξ_{2}) = (ξ_{2}, ξ_{1}). Take B_{i} = C_{0}(ℝ) ⊕ C_{0}(ℝ), with β_{i}(f_{1}, f_{2}) = (f_{2}, f_{1}). To define

φ_{i}: C_{0}(ℝ) ⊕ C_{0}(ℝ) → L(ℂ^{2}) ⊗ C,

let p_{1}, p_{2} ∈ L(ℂ^{2}) be the projections on the first and second coordinates, and let ψ_{1}, ψ_{2} : C_{0}(ℝ) → C be the unique homomorphisms such that

ψ_{1}(u − 1) = ((u^{k_{1}} − 1, u^{l_{1}} − 1), (u^{k_{2}} − 1, u^{l_{2}} − 1), …)

and

ψ_{2}(u − 1) = ((u^{l_{1}} − 1, u^{k_{1}} − 1), (u^{l_{2}} − 1, u^{k_{2}} − 1), …).

Then define

φ_{i}(f_{1}, f_{2}) = p_{1} ⊗ ψ_{1}(f_{1}) + p_{2} ⊗ ψ_{2}(f_{2}).

This is a homomorphism because p_{1} and p_{2} are orthogonal. Equivariance follows from the fact that the action of s exchanges p_{1} and p_{2}, and the formulas for m and sm. That (ψ_{i})_{∗} is injective follows from the fact that m and sm are linearly independent over ℤ, and the remaining required properties are clear.

Now let H be the Hilbert direct sum H = ⨁_{i∈I} H_{i}, and let B be the C*-algebra direct sum B = ⨁_{i∈I} B_{i}. Give both the action of ℤ_{2} coming from the actions on the summands. Define φ: B → K(H) ⊗ C by taking φ((b_{i})_{i∈I}) to be the block diagonal element ⨁_{i∈I} φ_{i}(b_{i}). This element is in fact in K(H) ⊗ C because for every ε > 0, we have ∥b_{i}∥ < ε for all but finitely many i. Then φ is the required homomorphism.

Lemma 3.2. Let G be a countable abelian group, and let ν: G → G be an automorphism of G of order two. Then there exist a separable type I C*-algebra A such that K_{1}(A) = 0, an automorphism α ∈ Aut(A) of order two, and an isomorphism μ: G → K_{0}(A), such that μ ∘ ν = α_{∗} ∘ μ.

Proof. The group G is a ℤ[ℤ_{2}]-module in an obvious way, and it is countable by assumption. Set N = ⨁_{n∈ℕ} ℤ[ℤ_{2}], and choose a surjective ℤ[ℤ_{2}]-module homomorphism τ: N → G. Set M = Ker(τ). Let C be as in Lemma 3.1, and identify N with K_{1}(C). Apply Lemma 3.1 with this M, obtaining a ℤ_{2}-equivariant homomorphism of separable type I C*-algebras φ: B → K(H) ⊗ C such that (by abuse of notation) K_{1}(B) = M, K_{1}(K(H) ⊗ C) = N, and φ_{∗} is the inclusion. To simplify the notation, set D = K(H) ⊗ C.

Let A be the mapping cone

A = {(f, b) ∈ C([0, 1], D) ⊕ B: f(0) = 0 and f(1) = φ(b)}.

Since φ is equivariant, this algebra has an obvious action of ℤ_{2}. (The action is trivial on [0, 1].) Moreover, with SD = C_{0}((0, 1), D) being the usual suspension of D, there is an equivariant short exact sequence

0 ⟶ SD ⟶ A ⟶ B ⟶ 0.

By naturality the corresponding six term exact sequence in K-theory is actually a sequence of ℤ[ℤ_{2}]-modules. Since K_{0}(B) = 0 and K_{0}(D) = 0, this sequence reduces to

0 ⟶ K_{1}(A) ⟶ K_{1}(B) ⟶ K_{0}(SD) ⟶ K_{0}(A) ⟶ 0.

Using the Bott periodicity isomorphism K_{0}(SD) ≅ K_{1}(D), and appropriately identifying the maps following Theorem 3.5 of [25], this sequence can be naturally identified with

0 ⟶ K_{1}(A) ⟶ K_{1}(B) ⟶^{φ_{∗}} K_{1}(D) ⟶ K_{0}(A) ⟶ 0.

By our construction and by naturality of the sequence, we therefore have the sequence of ℤ[ℤ_{2}]-modules

0 ⟶ K_{1}(A) ⟶ M ⟶^{τ} N ⟶ K_{0}(A) ⟶ 0,

from which it follows that K_{1}(A) = 0 and that K_{0}(A) ≅ N/M ≅ G as ℤ[ℤ_{2}]- modules.

Proposition 3.3. Let G_{0} and G_{1} be countable abelian groups. Let γ_{i} be an automorphism of G_{i} of order two. Then there exist a separable type I C*-algebra A, an automorphism α ∈ Aut(A) of order two, and a graded isomorphism μ: G_{∗} → K_{∗}(A), such that the diagram

G_{∗} ⟶^{γ_{∗}} G_{∗}
μ↓  ↓^{μ}
K_{∗}(A) ⟶^{α_{∗}} K_{∗}(A)

commutes.

Proof. Apply Lemma 3.2 to G_{0} and γ_{0}, obtaining a C*-algebra A_{0} with an order two automorphism α_{0}. Apply Lemma 3.2 to G_{1} and γ_{1}, obtaining a C*-algebra A_{1} with an order two automorphism α_{1}. Take A = A_{0} ⊕ SA_{1} and α = α_{0} ⊕ Sα_{1}. This C*-algebra and automorphism satisfy the conclusions by Bott periodicity.

4. From type I to purely infinite simple

For countable abelian groups G_{0} and G_{1} and order two automorphisms γ_{i} of G_{i}, we can now produce a separable type I C*-algebra A with K-theory G_{0} ⊕ G_{1}, and an order two automorphism which induces γ_{i} on K-theory. We want to use the construction of [15] to produce a unital Kirchberg algebra. However, A is not unital, and this construction will then not produce a unital C*-algebra. Unitizing A changes the K-theory. To remedy this problem, we introduce in the next proposition a kind of unitization functor which does not change the K-theory. Our functor preserves nuclearity and the Universal Coefficient Theorem, but not type I (or simplicity or finiteness). Although we will not use this fact, it also seems interesting to observe that the functor sends exact C*-algebras to exact C*-algebras . Proposition 4.1. There is a functor F from the category of all C*-algebras and C*- algebra homomorphisms to the subcategory of all unital C*-algebras and unital C*- algebra homomorphisms, and a natural transformation η from the identity functor to F, such that:

(1) For every C*-algebra A, the homomorphism η_{A}: A → F (A) is a KKequivalence, and in particular is an isomorphism on K-theory.

(2) If A is separable, or nuclear, or exact, or satisfies the Universal Coefficient Theorem of [23], then F (A) also has the same property.

Proof. Choose and fix a nonzero projection e ∈ O_{∞} such that [e] = 0 in K_{0}(O_{∞}), a projection q ≤ e such that [q] = [1_{O_{∞}}] in K_{0}(O_{∞}), and a unital subalgebra D ⊂ eO_{∞}e such that D ≅ O_{2}. For any C*-algebra A, let A^{+} be the unitization (as in Section 3), let π_{A}: A^{+} → ℂ be the canonical map, and define

F (A) = {b ∈ eO_{∞}e ⊗ A^{+} : (id_{eO_{∞}e} ⊗ π_{A})(b) ∈ D}.

Let ι_{A}: A → A^{+} be the inclusion, and define η_{A}: A → F (A) by η_{A}(a) = q ⊗ ι_{A}(a). This element is in F (A) because (id_{eO_{∞}e} ⊗ π_{A})(q ⊗ ι_{A}(a)) = 0.

Since the map λ ↦ λq from ℂ to eO_{∞}e is a KK-equivalence, so is the map a ↦ q ⊗ a from A to eO_{∞}e. (See Example 19.1.2(c) of [3].) Moreover, there is a split short exact sequence

0 ⟶ eO_{∞}e ⊗ A ⟶ F (A) ⟶ O_{2} ⟶ 0.

Since O_{2} is KK-equivalent to the zero C*-algebra, it follows that, for every separable C*-algebra B, the inclusion φ: eO_{∞}e ⊗ A → F (A) induces isomorphisms

φ_{B}^{∗}: KK(F (A), B) → KK(eO_{∞}e ⊗ A, B)

and

φ_{∗}^{B}: KK(B, eO_{∞}e ⊗ A) → KK(B, F (A)).

It now follows from the Yoneda Lemma that φ is a KK-equivalence (see Section III.2 of [16]), but we give the easy direct argument here. In terms of the Kasparov product, these isomorphisms have the formulas φ_{B}^{∗}(γ) = [φ] · γ and φ_{∗}^{B}(γ) = γ · [φ]. (See Proposition 18.7.2 of [3].) Choose

α ∈ KK(F (A), eO_{∞}e ⊗ A) and β ∈ KK(eO_{∞}e ⊗ A, F (A))

such that

φ_{eO_{∞}e⊗A}^{∗}(α) = [id_{eO_{∞}e⊗A}] and φ_{∗}^{F (A)}(β) = [id_{F (A)}].

Then

[φ] · α = [id_{eO_{∞}e⊗A}] and β · [φ] = [id_{F (A)}],

from which it follows that φ is also a KK-equivalence. So η_{A} is a KK-equivalence. This proves the property (1).

It is immediate from the KK-equivalence of (1) that F (A) satisfies the Universal Coefficient Theorem if A does. That F preserves separability and nuclearity is immediate from the split exact sequence above, since eO_{∞}e and O_{2} are separable and nuclear. The same argument works for exactness: the minimal tensor product of exact C*-algebras is exact by Proposition 7.1(iii) of [13], and extensions with completely positive splittings preserve exactness by Proposition 7.1(vi) of [13].

Next, we need an equivariant version of the construction of [15]. Proposition 4.2. Let α: Γ → Aut(A) be an action of a countable discrete group Γ on a separable unital C*-algebra A. Then there exists a separable unital purely infinite simple C*-algebra B, an action β: Γ → Aut(B), and an equivariant unital homomorphism φ: A → B, such that φ is a KK-equivalence. Moreover, if A is nuclear then so is B, and if A satisfies the Universal Coefficient Theorem then so does B.

Proof. Choose an injective unital representation π_{0} : A → L(H_{0}) of A on a separable Hilbert space H_{0} such that π_{0}(A) ∩ K(H_{0}) = {0}. Let H = l^{2}(Γ, H_{0}), and let u: Γ → U(H) and π: A → L(H) be the components of the regular covariant representation of (Γ, A) associated to π_{0}, as in 7.7.1 in [17]. Then u_{γ}π(a)u_{γ}^{∗} = π(α_{γ}(a)) for all a ∈ A and γ ∈ Γ. (See [17].) Moreover, π(A) ∩ K(H) = {0}.

Now we follow the proof of Proposition 2.1 of [15]. Let E = H ⊗_{ℂ} A be the Hilbert A-bimodule defined there, and let φ: A → L(E) be as there. Each γ ∈ Γ induces a isometric ℂ-linear map v_{γ}: E → E given by v_{γ}(ξ ⊗ a) = u_{γ}ξ ⊗ α_{γ}(a) for ξ ∈ H and a ∈ A. These maps have the following properties:

(1) v_{γ}(ξa) = v_{γ}(ξ)α_{γ}(a) for γ ∈ Γ, ξ ∈ E, and a ∈ A.

(2) ⟨v_{γ}ξ, v_{γ}η⟩ = α_{γ}(⟨ξ, η⟩) for γ ∈ Γ and ξ, η ∈ E.
(3) ∥v_{γ}ξ∥ = ∥ξ∥ for γ ∈ Γ and ξ ∈ E.
(4) v_{γ^{−1}} = v_{γ}^{−1} for γ ∈ Γ.
(5) v_{γ}φ(a)v_{γ}^{−1} = φ(α_{γ}(a)) for γ ∈ Γ and a ∈ A.

Caution: v_{γ} is not a right A-module homomorphism. Nevertheless, these properties imply that b ↦ v_{γ}bv_{γ}^{−1} is a *-automorphism of the C*-algebra L(E) of all adjointable right A-module morphisms of E; moreover, φ is equivariant.

Next, let E_{+} = ⨁_{n=0}^{∞} E^{⊗n} be the Fock space of E, as after Definition 1.3 of [15] or as at the beginning of Section 1 of [20]. We let γ ∈ Γ act on E^{⊗n} via v_{γ}^{⊗n} (via α_{γ} on E^{⊗0} = A), and on E_{+} via the direct sum w_{γ} of these actions. This action is still isometric, and we again have the analogs of Properties (1)–(5) above. The role of φ is now played by the homomorphism φ_{+} : A → L(E_{+}) of the discussion after Definition 1.3 of [15]. Following the construction there, the Toeplitz algebra T_{E} is by definition the C*-subalgebra of L(E_{+}) generated by the operators T_{ξ} defined there, for ξ ∈ E. One checks that T_{v_{γ}ξ} = w_{γ}T_{ξ}w_{γ}^{−1} for γ ∈ Γ and ξ ∈ E. Therefore Γ acts on T_{E} via *-automorphisms. Since π(A) ∩ K(H) = {0}, Lemma 2.1 of [15] and Corollary 3.14 and Theorem 3.13 of [20] show that the canonical map from T_{E} to the Cuntz-Pimsner algebra O_{E} is an isomorphism. Therefore we have an action of Γ on O_{E}, and φ_{+} : A → O_{E} is equivariant.

Theorem 2.8 of [15] shows that O_{E} is purely infinite and simple. Separability is clear. The proof of Theorem 3.1 of [15] shows that if A is nuclear, then so is O_{E}. Corollary 4.5 of [20] shows that φ_{+} is a KK-equivalence, from which it is immediate that if A satisfies the Universal Coefficient Theorem then so does O_{E}.

Theorem 4.3. Let G_{0} and G_{1} be countable abelian groups. Let γ_{i} be an automorphism of G_{i} of order two. Then there exist a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem and with [1_{A}] = 0 in K_{0}(A), an automorphism α ∈ Aut(A) of order two, and a graded isomorphism μ: G_{∗} → K_{∗}(A), such that the diagram

G_{∗} ⟶^{γ_{∗}} G_{∗}
μ↓  ↓^{μ}
K_{∗}(A) ⟶^{α_{∗}} K_{∗}(A)

commutes.

Proof. Use Proposition 3.3 to find a separable type I C*-algebra A_{0}, an automorphism α_{0} ∈ Aut(A_{0}) of order two, and a graded isomorphism μ_{0} : G_{∗} → K_{∗}(A_{0}), such that μ_{0} ∘ γ = (α_{0})_{∗} ∘ μ_{0}.

Let F be the functor of Proposition 4.1, and let η: A_{0} → F (A_{0}) be the natural transformation from there. By Part (2) of Proposition 4.1, the algebra F (A_{0}) is separable, unital, nuclear, and satisfies the Universal Coefficient Theorem; also, η is a KK-equivalence. Since F is a functor, F (α_{0}) is an order two automorphism of F (A_{0}), and naturality implies that η ∘ α_{0} = F (α_{0}) ∘ η.

Now use Proposition 4.2 to find a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem, an automorphism α ∈ Aut(A) of order two, and a ℤ_{2}-equivariant homomorphism φ: F (A_{0}) → A which is a KK-equivalence. Then φ ∘ η: A_{0} → A is ℤ_{2}-equivariant and is a KK-equivalence. The theorem is therefore proved by taking μ = φ_{∗} ∘ η_{∗} ∘ μ_{0}.

It should be easy to arrange to have [1_{A}] correspond to any element in G of the form η + γ_{0}(η), but we don’t know how to get arbitrary γ_{0}-invariant elements of G_{0}.

We now turn to the problem of realizing R(ℤ_{2})-modules as equivariant K-theory for actions on C*-algebras. The main point is contained in the next lemma, which we state in greater generality.

Before stating it, recall (Section 2.2 of [18]) that if a compact group G acts on a C*-algebra A, then the equivariant K-theory K_{∗}^{G}(A) is, in a canonical way, a module over the representation ring R(G) of G. Further recall that when G is abelian, the representation ring R(G) is just ℤ[Ĝ]; in particular, if G is a countable abelian group, then R(Ĝ) is canonically isomorphic to ℤ[G]. Lemma 4.4. Let G be a countable abelian group, and let α: G → Aut(A) be an action of G on a C*-algebra A. Regard K_{∗}(A) as a ℤ[G]-module via the action of G on A. Regard the equivariant K-theory KĜ ; ∗(C^{∗}(G, A, α)) of the dual action α̂: Ĝ → Aut(C^{∗}(G, A, α)) as a ℤ[G]-module as discussed above. Then KĜ ; ∗(C^{∗}(G, A, α)) ≅ K_{∗}(A) as ℤ[G]-modules.

Proof. Let B = C^{∗}(Ĝ, C^{∗}(G, A, α), α̂) be the second crossed product, with second dual action β: G → Aut(B). Regard K_{∗}(B) as a ℤ[G]-module via this action of G. Proposition 2.7.10 of [18], applied to the action α̂ of Ĝ on C^{∗}(G, A, α), shows that KĜ ; 0(C^{∗}(G, A, α)) ≅ K_{0}(B) as ℤ[G]-modules. Applying this result to the suspension SA, we obtain the same isomorphism for K_{1} as well. By Takai duality [27], there is an isomorphism B ≅ A ⊗ K(l^{2}(G)) which intertwines the action β with the tensor product of α and an inner action on K(l^{2}(G)). It is therefore immediate that K_{∗}(B) ≅ K_{∗}(A) as ℤ[G]-modules.

We immediately get a realization theorem using type I C*-algebras. Theorem 4.5. Let G_{∗} be a countable ℤ_{2}-graded R(ℤ_{2})-module. Then there exists a separable type I C*-algebra A, and an action α of ℤ_{2} on A, such that K_{∗}^{ℤ_{2}}(A) ≅ G_{∗} as R(ℤ_{2})-modules.

Proof. Identify R(ℤ_{2}) with ℤ[ℤ̂_{2}] as discussed before Lemma 4.4. Since ℤ̂_{2} ≅ ℤ_{2}, we can use Proposition 3.3 to find a separable type I C*-algebra B and an action β: ℤ̂_{2} → Aut(B) such that K_{∗}(B) ≅ G_{∗} as R(ℤ_{2})-modules. Let A = C^{∗}(ℤ̂_{2}, B, β), equipped with the dual action α = β̂ of ℤ_{2}. Lemma 4.4 implies that K_{∗}^{ℤ_{2}}(A) ≅ G_{∗} as R(ℤ_{2})-modules. Clearly A is separable, and Theorem 4.1 of [21] implies that A is type I.

For the realization theorem using Kirchberg algebras, we need another lemma. Lemma 4.6. Let

A_{0} ⟶^{ρ_{1}} A_{1} ⟶^{ρ_{2}} A_{2} ⟶^{ρ_{3}} ⋯

be a direct system of C*-algebras in which each A_{n} is a finite direct sum of Kirchberg algebras satisfying the Universal Coefficient Theorem. Suppose that for each n the partial map determined by ρ_{n} from any summand of A_{n−1} to any other summand of A_{n} is nonzero. Then the direct limit of this system is again a Kirchberg algebra satisfying the Universal Coefficient Theorem.

Proof. Let A = lim_{⟶} A_{n}. Clearly A is separable and nuclear. The condition that all the partial maps be nonzero, together with the simplicity of all the summands, guarantees that the algebraic direct limit of the A_{n} is simple. A standard argument now shows that A is simple. It is easy to check, using standard direct limit methods, that A has real rank zero and that every nonzero projection in A is properly infinite. (For the second statement, use the fact that every projection in A is equivalent to a projection in the algebraic direct limit.) Therefore A is purely infinite simple. Finally, A satisfies the Universal Coefficient Theorem because the class of such algebras is closed under direct limits. This is essentially Proposition 2.3(b) of [23], in view of Theorem 4.1 of [23] and its converse (the converse being trivial).

Theorem 4.7. Let G_{∗} be a countable ℤ_{2}-graded R(ℤ_{2})-module. Then there exists a unital Kirchberg algebra A, and an action α of ℤ_{2} on A, such that K_{∗}^{ℤ_{2}}(A) ≅ G_{∗} as R(ℤ_{2})-modules, and such that C^{∗}(ℤ_{2}, A, α) is a Kirchberg algebra.

Proof. Identify R(ℤ_{2}) with ℤ[ℤ̂_{2}] as discussed before Lemma 4.4. Let τ ∈ ℤ̂_{2} be the nontrivial element.

First suppose that the action of τ on the R(ℤ_{2})-module G_{∗} is nontrivial. Since ℤ̂_{2} ≅ ℤ_{2}, we can use Theorem 4.3 to find a unital Kirchberg algebra B and an action β: ℤ̂_{2} → Aut(B) such that K_{∗}(B) ≅ G_{∗} as R(ℤ_{2})-modules. Let A = C^{∗}(ℤ̂_{2}, B, β), equipped with the dual action α = β̂ of ℤ_{2}. Lemma 4.4 implies that K_{∗}^{ℤ_{2}}(A) ≅ G_{∗} as R(ℤ_{2})-modules. Moreover, β_{τ} is outer (because it is nontrivial on K-theory), so Corollary 4.4 of [12] implies that A is a Kirchberg algebra. Moreover C^{∗}(ℤ_{2}, A, α) ≅ M_{2} ⊗ B is also a Kirchberg algebra.

Now suppose that the action of τ on the R(ℤ_{2})-module G_{∗} is trivial. Choose a unital Kirchberg algebra A_{0} satisfying the Universal Coefficient Theorem such that K_{∗}(A_{0}) ≅ G_{∗} as abelian groups. Let α^{(0)} : ℤ_{2} → Aut(A_{0} ⊕ A_{0}) be the action such that α_{τ}^{(0)}(a, b) = (b, a). We identify the crossed product C^{∗} (ℤ_{2}, A_{0} ⊕ A_{0}, α^{(0)}). Let u ∈ C^{∗} (ℤ_{2}, A_{0} ⊕ A_{0}, α^{(0)}) be the canonical unitary of order two. Then there is an isomorphism σ: C^{∗} (ℤ_{2}, A_{0} ⊕ A_{0}, α^{(0)}) → M_{2}(A_{0}), determined by

σ(a, b) = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )

for a, b ∈ A_{0}, and

σ(u) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

The dual action is inner, so K_{∗}^{ℤ_{2}}(A_{0} ⊕ A_{0}) ≅ G_{∗} as R(ℤ_{2})-modules.

The algebra A_{0}⊕A_{0} is not a Kirchberg algebra. To remedy this, choose a nonzero projection q ∈ A_{0} such that [q] = 0 in K_{0}(A_{0}). Set p = 1 − q, so that [p] = [1_{A_{0}}] in K_{0}(A_{0}). Choose ([14]; or Theorem 4.1.3 and the proofs of Corollary 4.4.2 and Theorem 4.2.4 of [19]) unital homomorphisms φ: A_{0} → pA_{0}p and ψ: A_{0} → qA_{0}q such that

A_{0} ⟶^{φ} pA_{0}p ⟶ A_{0}

is the identity on K-theory and ψ_{∗} = 0. Define a unital equivariant homomorphism ρ: A_{0} ⊕ A_{0} → A_{0} ⊕ A_{0} by φ(a, b) = (φ(a) + ψ(b), φ(b) + ψ(a)) for a, b ∈ A_{0}. Let A be the direct limit of the system

A_{0} ⊕ A_{0} ⟶^{ρ} A_{0} ⊕ A_{0} ⟶^{ρ} A_{0} ⊕ A_{0} ⟶^{ρ} · · · .

Then A is a unital Kirchberg algebra by Lemma 4.6. Let α: ℤ_{2} → Aut(A) be the direct limit action.

We next show that K_{∗}^{ℤ_{2}}(A) ≅ G_{∗} as R(ℤ_{2})-modules. We do this by proving that ρ_{∗}: K_{∗}^{ℤ_{2}}(A_{0} ⊕ A_{0}) → K_{∗}^{ℤ_{2}}(A_{0} ⊕ A_{0}) is an isomorphism and using Proposition 2.5.4 of [18] on the direct system and its suspension. We claim that the corresponding homomorphism

\overline{ρ}: C^{∗} (ℤ_{2}, A_{0} ⊕ A_{0}, α^{(0)}) → C^{∗} (ℤ_{2}, A_{0} ⊕ A_{0}, α^{(0)})

is, under the identification of the crossed product above, given by

\overline{ρ}(x) = (id_{M_{2}} ⊗ φ)(x) + u(id_{M_{2}} ⊗ ψ)(x)u^{∗}.

It is enough to check this on the image of A_{0} ⊕ A_{0}, namely the diagonal matrices, and on u; this is easy. It is immediate from the choice of φ and ψ that this map is the identity on K-theory.

It remains to show that C^{∗}(ℤ_{2}, A, α) is a Kirchberg algebra. This algebra is the direct limit of the system of crossed products

C^{∗}(ℤ_{2}, A_{0}⊕A_{0}, α^{(0)}) ⟶^{\overline{ρ}} C^{∗}(ℤ_{2}, A_{0}⊕A_{0}, α^{(0)}) ⟶^{\overline{ρ}} C^{∗}(ℤ_{2}, A_{0}⊕A_{0}, α^{(0)}) ⟶^{\overline{ρ}} ⋯,

which, by the computation at the beginning of this case, can be rewritten as

M_{2}(A_{0}) ⟶ M_{2}(A_{0}) ⟶ M_{2}(A_{0}) ⟶ · · · .

The direct limit of this system is a Kirchberg algebra by Lemma 4.6.

Remark 4.8. When the action of the nontrivial element of ℤ̂_{2} is nontrivial, we do not know whether the C*-algebra in Theorem 4.7 satisfies the Universal Coefficient Theorem.

5. An explicit example

In this section, we write down a formula for an automorphism in the smallest nontrivial case of Theorem 4.3. This is the case K_{0}(A) ≅ ℤ_{3}, K_{1}(A) = 0, and γ_{0} is the unique nontrivial automorphism of ℤ_{3}. Since γ_{0} fixes only the identity element of ℤ_{3}, the automorphism will exist only when [1_{A}] = 0 in K_{0}(A). Let O_{4} be the Cuntz algebra [6], with K-theory as computed in [7]. Then the unital Kirchberg algebra A with this K-theory and satisfying the Universal Coefficient Theorem is M_{3} ⊗ O_{4}.

Example 5.1. We give an explicit formula, in terms of the standard generators, for an order two automorphism φ of A = M_{3} ⊗ O_{4} which is nontrivial on K_{0}(A).

Let s_{1}, s_{2}, s_{3}, and s_{4} be the standard generating isometries of O_{4}, and define p_{m} = s_{m}s_{m}^{∗}. Further let e_{j,k}, for 1 ≤ j, k ≤ 3, be the standard matrix units of M_{3}, satisfying e_{j,k}e_{k,l} = e_{j,l} etc. Then φ is determined by the formulas:

e_{1,1} ⊗ 1 ↦ f_{1,1} = (e_{2,2} + e_{3,3}) ⊗ 1
e_{2,2} ⊗ 1 ↦ f_{2,2} = e_{1,1} ⊗ (p_{1} + p_{2})
e_{3,3} ⊗ 1 ↦ f_{3,3} = e_{1,1} ⊗ (p_{3} + p_{4})
e_{1,2} ⊗ 1 ↦ f_{1,2} = e_{2,1} ⊗ s_{1}^{∗} + e_{3,1} ⊗ s_{2}^{∗}
e_{1,3} ⊗ 1 ↦ f_{1,3} = e_{2,1} ⊗ s_{3}^{∗} + e_{3,1} ⊗ s_{4}^{∗}
e_{1,1} ⊗ s_{1} ↦ v_{1} = e_{2,2} ⊗ s_{1} + e_{2,3} ⊗ s_{2}
e_{1,1} ⊗ s_{2} ↦ v_{2} = e_{2,2} ⊗ s_{3} + e_{2,3} ⊗ s_{4}
e_{1,1} ⊗ s_{3} ↦ v_{3} = e_{3,2} ⊗ s_{1} + e_{3,3} ⊗ s_{2}
e_{1,1} ⊗ s_{4} ↦ v_{4} = e_{3,2} ⊗ s_{3} + e_{3,3} ⊗ s_{4}

One must check two things: that φ is a homomorphism, and that φ^{2} = id_{A}. For the first, one checks the following relations:

• f_{1,1}, f_{2,2}, and f_{3,3} are orthogonal projections which sum to 1.
• f_{1,j}f_{1,j}^{∗} = f_{1,1} and f_{1,j}^{∗}f_{1,j} = f_{j,j} for j = 2, 3.
• v_{m}^{∗}v_{m} = f_{1,1} for 1 ≤ m ≤ 4 and ∑_{m=1}^{4} v_{m}v_{m}^{∗} = f_{1,1}.

The details of the computation are somewhat long, and are omitted.

To prove that φ^{2} = id_{A}, one checks that φ(φ(a)) = a for each of the generators used above. Again, we omit the details.

References

[1] H. Bass, Big projective modules are free, Illinois J. Math. 7(1963), 24–31.

[2] D. J. Benson and K. R. Goodearl, Periodic flat modules, and flat modules for finite groups, Pacific J. Math. 196(2000), 45–67.

[3] B. Blackadar, K-Theory for Operator Algebras, MSRI Publication Series 5, Springer-Verlag, New York, Heidelberg, Berlin, Tokyo, 1986.

[4] M. C. R. Butler, J. M. Campbell, and L. G. Kovács, Infinite rank integral representations of groups and orders of finite lattice type, in preparation.

[5] M. C. R. Butler and L. G. Kovács, Large lattices over classical orders of finite lattice type, draft preprint.

[6] J. Cuntz, Simple C*-algebras generated by isometries, Commun. Math. Phys. 57(1977), 173– 185.

[7] J. Cuntz, K-theory for certain C*-algebras, Ann. Math. 113(1981), 181–197.

[8] C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience Publishers, New York, London, Sydney, 1962.

[9] G. A. Elliott and M. Rørdam, Classification of certain infinite simple C*-algebras II , Comment. Math. Helv. 70(1995), 615–638.

[10] L. Fuchs, Infinite Abelian Groups. Vol. I , Academic Press, New York, London, 1970.

[11] L. K. Hua and I. Reiner, Automorphisms of the unimodular group, Trans. Amer. Math. Soc. 71(1951), 331–348.

[12] J. A. Jeong and H. Osaka, Extremally rich C*-crossed products and the cancellation property, J. Austral. Math. Soc. (Series A) 64(1998), 285–301.

[13] E. Kirchberg, Commutants of unitaries in UHF algebras and functorial properties of exactness, J. reine angew. Math. 452(1994), 39–77.

[14] E. Kirchberg, The classification of purely infinite C*-algebras using Kasparov’s theory, preliminary preprint (3rd draft).

[15] A. Kumjian, On certain Cuntz-Pimsner algebras, preprint.

[16] S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag Graduate Texts in Math. no. 5, Springer-Verlag, New York, Heidelberg, Berlin, 1971.

[17] G. K. Pedersen, C*-Algebras and their Automorphism Groups, Academic Press, London, New York, San Francisco, 1979.

[18] N. C. Phillips, Equivariant K-Theory and Freeness of Group Actions on C*-Algebras, Springer-Verlag Lecture Notes in Math. no. 1274, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987.

[19] N. C. Phillips, A classification theorem for nuclear purely infinite simple C*-algebras, Doc. Math. 5(2000), 49–114 (electronic).

[20] M. Pimsner, A class of C*-algebras generalizing both Cuntz-Krieger algebras and crossed products by ℤ, pages 189–212 in: Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Commun. vol. 12, Amer. Math. Soc., Providence, RI, 1997.

[21] M. A. Rieffel, Actions of finite groups on C*-algebras, Math. Scand. 47(1980), 157–176.

[22] M. Rørdam and E. Størmer, Classification of nuclear C*-algebras. Entropy in operator algebras, Springer-Verlag, Berlin, 2002.

[23] J. Rosenberg and C. Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J. 55(1987), 431–474.

[24] C. Schochet, Topological methods for C*-algebras II: geometric resolutions and the Künneth formula, Pacific J. Math. 98(1982), 443–458.

[25] C. Schochet, Topological methods for C*-algebras III: axiomatic homology, Pacific J. Math. 114(1984), 399–445.

[26] R. G. Swan, Induced representations and projective modules, Ann. of Math. 71(1960), 552– 578.

[27] H. Takai, On a duality for crossed products of C*-algebras, J. Funct. Anal. 19(1975), 25–39.

Department of Mathematics, University of Georgia, Athens GA 30602-7403, USA.

E-mail address: djb@byrd.math.uga.edu

Department of Mathematics, University of Nevada, Reno NV 89557-0045, USA. E-mail address: alex@unr.edu

Department of Mathematics, University of Oregon, Eugene OR 97403-1222, USA.

E-mail address: ncp@darkwing.uoregon.edu