AF embeddability of crossed products of AT algebras by the integers and its application

MATUI Hiroki

August 2001

Abstract

We will show that the crossed products of unital simple real rank zero AT algebras by the integers are AF embeddable. This is a generalization of N. Brown’s AF embedding theorem. As an application, we will prove the AF embaddability of crossed product algebras arising from certain minimal dynamical systems induced by two commuting homeomorphisms.

1 Introduction

In [V2], Voiculescu raised the following question: for which ℤ^{2}-actions, on a metrizable compact set X, is the crossed product C(X)⋊_{α} ℤ^{2} embeddable into an AF algebra? In the ℤ-action case, Pimsner solved this AF embeddability problem in [Pi]. He showed that the crossed product C^{∗}-algebra arising from (X, T), where T is a homeomorphism on X, is AF embeddable if and only if T is pseudo-non-wandering, which means that there is no open subset U ⊂ X such that T (\overline{U}) ⊂ U and U \ T (\overline{U}) ≠ ∅. For general discrete group actions including ℤ^{2}, however, no general result has been known. In this paper we would like to give a partial answer for Voiculescu’s question.

Our setting is as follows. Let X be a compact metric space and T, S ∈ Homeo(X) be two commuting homeomorphisms. We denote the crossed product C^{∗}-algebra arising from (X, T, S) by C^{∗}(X, T, S). We say that (X, T, S) is a ℤ^{2}-minimal system, if {T^{n}S^{m}(x) ; n, m ∈ ℤ} is dense in X for every x ∈ X. This is evidently equivalent to non-existence of a non-trivial closed subset which is T -invariant and S-invariant. (A subset E is said to be T -invariant if T (E) = E.) When (X, T, S) is a ℤ^{2}-minimal system, there exists a probability measure supported on whole of X which is invariant under T and S. Hence, for every (n, m) ∈ ℤ^{2}, T^{n}S^{m} is pseudo-non-wandering. This fact seems to suggest the AF embeddability of the crossed product algebra C^{∗}(X, T, S). Actually we will prove that C^{∗}(X, T, S) is really AF embeddable if (X, T, S) is a ℤ^{2}-minimal system and satisfies a certain condition which will be introduced in Section 4. This condition is necessary for reducing the problem to the special case that T itself is a minimal homeomorphism. If X is the Cantor set and (X, T) is a minimal system, then the crossed product C^{∗}-algebra C^{∗}(X, T) is a simple unital AT algebra with real rank zero ([Pu]). Therefore the AF embaddability of C^{∗}(X, T, S) will be obtained as a corollary of the AF embaddability of crossed products of AT algebras by the integers, which is a generalization of [B, Corollary 4.10]. It should be also pointed out that quasidiagonality and stably finiteness follow immediately from AF embeddability.

Now we give an overview of each section below. In Section 2 we will generalize [KK, Theorem 3.1] to homomorphisms between simple unital AT algebras with real rank zero. It was proved in [E] that two homomorphisms between these algebras are approximately unitarily equivalent if and only if the induced homomorphisms on K-groups coincide. In the next section, however, we need a slight stronger equivalence relation, called asymptotically unitarily equivalence. By exactly the same way as in [KK], we will show that Ext and OrderExt groups can distinguish whether or not two homomorphisms are asymptotically unitarily equivalent. In Section 3 the generalization of [B, Corollary 4.10] will be proven. The strategy we will use is the same as in [B]. The only difference from the AF algebra case is that we have to take care of Bott elements in order to deduce the stability from the Rohlin property. The homotopy lemma of [BEEK] will play a very important role. As a corollary, we will also show that the crossed products of unital separable simple nuclear TAF algebras which satisfy UCT by the integers are AF embeddable. In Section 4 crossed product algebras arising from ℤ^{2}-minimal systems are examined. As mentioned before, we will prove that these algebras are AF embaddable under a certain assumption. We will construct a Cantor minimal system by using ordered Bratteli diagram ([HPS]) and consider a skew product extension. In Section 5 examples of ℤ^{2}-minimal systems will be given. It can be checked that our main theorem can be applied to many ℤ^{2}-minimal systems. In fact the author does not know a ℤ^{2}-minimal system for which our method does not work.

Acknowledgments.

The author would like to express his sincerest thanks to Professor IZUMI Masaki for his constant encouragement and helpful advice.

2 OrderExt and asymptotically unitarily equivalence

Let A and B be two unital C^{∗}-algebras. We denote the set of all unital homomorphisms from A to B by Hom(A, B). The unitary groups of A and B are denoted by U(A) and U(B). Two unital homomorphisms α, β ∈ Hom(A, B) are said to be approximately unitarily equivalent if there exists a sequence of unitaries {u_{n}}_{n∈ℕ} in U(B) such that Ad u_{n}α(a) converges to β(a) for all a ∈ A. If there exists a continuous map u : [0, ∞) → U(B) such that

lim_{t→∞} Ad u_{t}α(a) = β(a)

for every a, we say that α and β are asymptotically unitarily equivalent. In this section we will show that α and β are asymptotically unitarily equivalent if and only if an invariant η(α, β) defined later is zero in the OrderExt group. The reader may refer to [KK, Section 2] for the definition of OrderExt groups.

Suppose A is simple and two homomorphisms α, β ∈ Hom(A, B) are approximately unitarily equivalent. Define the mapping torus of α and β by

M_{α,β} = {x ∈ C([0, 1], B) ; x(0) = α(a) and x(1) = β(a) for some a ∈ A}.

From the exact sequence

0 → SB → M_{α,β} → A → 0,

we get two exact sequences of abelian groups:

0 → K_{1}(B) → K_{0}(M_{α,β}) → K_{0}(A) → 0

and

0 → K_{0}(B) → K_{1}(M_{α,β}) → K_{1}(A) → 0.

We write the set of all tracial states of B by T (B). For a unitary u ∈ M_{α,β} such that t ↦ u(t) is piecewise C^{1},

T (B) ∋ τ ↦ ∫_{0}^{1} τ(u̇(t)u(t)^{∗}) dt

gives an affine map from T (B) to ℝ. By exactly the same way as in [KK, Section 2], the map

R_{α,β} : K_{1}(M_{α,β}) → Aff(T (B)),

called the rotation map, is well defined. Therefore we obtain an element η(α, β) in

Ext(K_{0}(A), K_{1}(B)) ⊕ OrderExt(K_{1}(A), K_{0}(B)).

Remark that η(α, β) is zero if and only if the following three conditions are satisfied:

(i) The extensions of K_{i}(A) by K_{1−i}(B) described above are trivial.

(ii) The range of the rotation map R_{α,β} coincides with the range of the canonical map K_{0}(B) → Aff(T (B)).

(iii) The exact sequence 0 → Inf(K_{0}(B)) → ker R_{α,β} → K_{1}(A) → 0 is trivial,

where Inf(K_{0}(B)) denotes the infinitesimal subgroup

{x ∈ K_{0}(B) ; τ(x) = 0 for all τ ∈ T (B)}.

It is easy to see that η(α, β) + η(β, γ) equals η(α, γ) for approximately unitarily equivalent homomorphisms α, β and γ in Hom(A, B).

We would like to prove the following theorem.

Theorem 2.1. Let A and B be unital simple AT algebras with real rank zero. When α, β ∈ Hom(A, B) are approximately unitarily equivalent homomorphisms, the following are equivalent.

(i) α and β are asymptotically unitarily equivalent.

(ii) η(α, β) defined above is zero in Ext(K_{0}(A), K_{1}(B)) ⊕ OrderExt(K_{1}(A), K_{0}(B)).

Because the proof is the same as that of [KK, Theorem 3.1], we give only a rough sketch of the proof. In the next section we actually need the above theorem only for the case that B is an AF algebra and Inf(K_{0}(B)) = 0, and so the reader may consider that special case.

When u and v are almost commuting unitaries in a C^{∗}-algebra A, we denote the Bott element associated with u and v by B(u, v) ∈ K_{0}(A). We refer the reader to [BEEK] or [KK] for the Bott element.

At first we need to modify [KK, Lemma 3.4]. We say that the spectrum of a unitary u ∈ U(M_{n}) is ε-dense with multiplicity m in 𝕋, if for every t ∈ 𝕋 the unitary u has m eigenvalues including multiplicity in the ε-neighborhood of t.

Lemma 2.2. Let u be a unitary of C(𝕋, M_{n}). Suppose u(t) has n distinct eigenvalues and the spectrum is ε-dense with multiplicity m for all t ∈ 𝕋. Then, for every k with |k| ≤ m, there exists a unitary w ∈ C(𝕋, M_{n}) such that ||[u, w]|| < 2ε and the Bott element B(u, w) ∈ K_{0}(C(𝕋, M_{n})) ≅ ℤ is equal to k.

Proof. We identify C(𝕋, M_{n}) with a C^{∗}-subalgebra of C([0, 1], M_{n}). The unitary matrix u(0) has n distinct eigenvalues in 𝕋. We may assume u(0) is a diagonal matrix. Let w be a permutation unitary of M_{n} such that each eigenvalues of u(0) is permuted to the next eigenvalues in the counterclockwise order. Then, ||[u(0), w^{k}]|| < 2ε and B(u(0), w^{k}) = k for |k| < m. We can find a unitary v ∈ C([0, 1], M_{n}) such that v(t)^{∗}u(t)v(t) is a diagonal unitary, v(0) is one and v(1) equals w^{l} for some l. If we put w′(t) = v(t)w^{k}v(t)^{∗}, the unitary w′ is really in C(𝕋, M_{n}). It is easy to see that ||[u, w′]|| < 2ε and B(u, w′) = k.

Lemma 2.3. Let A be a unital simple AT algebra with real rank zero and z be a unitary in A with full spectrum. Then, for any ε > 0 there exists δ > 0 such that the following holds: If x ∈ K_{0}(A) satisfies |τ(x)| < δ for every τ ∈ T (A), then there exists w ∈ U(A) such that ||[z, w]|| < ε and B(z, w) = x.

Proof. Let (V, E) be the Bratteli diagram associated with K_{0}(A). We may assume that A is an inductive limit of C_{n}⊗C(𝕋) with injective connecting homomorphisms and the finite dimensional C^{∗}-algebra C_{n} is isomorphic to ⨁_{v∈V_{n}} M_{h(v)}, where h(v) is the number of paths from V_{0} to v.

By perturbing z a little bit, we may suppose that z is contained in C_{n} ⊗C(𝕋) for some n and the spectrum of z is ε-dense in 𝕋 with multiplicity one. Because A is simple, we may further assume that the spectrum of z(t) is ε-dense for every t ∈ V_{n} × 𝕋 ([DNNP, Proposition 2.1]). Let p_{v} be a minimal projection of M_{h(v)} ⊂ C_{n} ⊗ C(𝕋) and put p = ∑_{v∈V_{n}} p_{v}. Define

δ = inf{τ(p) ; τ ∈ T (B)}.

For m > n we denote the number of paths from V_{n} to v ∈ V_{m} by e_{v}. The K_{0} class of p is exactly (e_{v})_{v} ∈ ℤ^{V_{m}}.

Suppose x ∈ K_{0}(A) satisfies |τ(x)| < δ for all τ ∈ T (A). For sufficiently large m > n. there exists a representative (x_{v})_{v} ∈ ℤ^{V_{m}} of x such that |x_{v}| is less than e_{v} for every v ∈ V_{m}. Obviously the spectrum of z(t) for t ∈ V_{m} × 𝕋 is ε-dense with multiplicity e_{v}. By perturbing z a little bit in C_{m} ⊗ C(𝕋) again, we may assume that z(t) has h(v) distinct eigenvalues for every t ∈ {v} × 𝕋 and v ∈ V_{m} (see [E, Theorem 4.1 (i)] for example). By virtue of Lemma 2.2, we get a unitary w ∈ C_{m} ⊗ C(𝕋) such that

||[z, w]|| < 2ε, B(z, w) = (x_{v})_{v} ∈ K_{0}(C_{m} ⊗ C(𝕋)),

which completes the proof.

Lemma 2.4. Let A be a unital simple AT algebra with real rank zero. For every z ∈ U(A), x ∈ K_{1}(A) and ε > 0, there exists a unitary w ∈ A such that ||[z, w]|| < ε, B(z, w) = 0 and [w] = x.

Proof. We use the same notation as in the proof of Lemma 2.3. We may assume that z is contained in a basic building block C_{n} ⊗ C(𝕋) and all eigenvalues of z(t) are distinct for every t ∈ V_{n} × 𝕋. Suppose (x_{v})_{v} ∈ ℤ^{V_{n}} is a representative of x. For every v ∈ V_{n} we can find continuous maps λ : [0, 1] → 𝕋 and p : [0, 1] → M_{h(v)} such that p(t) is a rank one projection and z(t)p(t) = p(t)z(t) = λ(t)p(t). Define w_{v}(t) = e^{2πix_{v}t}p(t) + (1 − p(t)). Then, w_{v} is a unitary of M_{h(v)} ⊗ C(𝕋) and commutes with z. It is clear that the K_{1}-class of the direct sum of w_{v}’s equals x.

We can prove Theorem 2.1 by using the above lemmas. Proof of Theorem 2.1. The implication (i)⇒(ii) is obvious (see [KK]). Let us prove the other implication. As in Lemma 2.3, let (V, E) be the Bratteli diagram associated with K_{0}(A) and suppose that A is the inductive limit of C_{n} ⊗ C(𝕋) with injective connecting homomorphisms, where C_{n} is isomorphic to ⨁_{v∈V_{n}} M_{h(v)}. We denote the generating unitary of the center of C_{n} ⊗ C(𝕋) by z_{n}. Let p_{v} be a minimal rank one projection of M_{h(v)} ⊂ C_{n} for each v ∈ V_{n} and define z_{v} = z_{n}p_{v} + (1 − p_{v}). We may further assume that z_{n} equals z_{n+1}a + z_{n+1}^{∗}b + c for some partial isometries a, b, c ∈ C_{n+1} for every n.

Suppose {ε_{n}}_{n} is a decreasing sequence of positive real numbers. By applying Lemma 2.3 to β(p_{v})Bβ(p_{v}), β(z_{v}p_{v}) for v ∈ V_{n} and ε_{n}/2, we get a small number δ_{v}. Let δ_{n} be the minimum of ε_{n}(4π)^{−1} and these δ_{v}’s for v ∈ V_{n}. Because α is approximately unitarily equivalent to β, there exists u_{n} ∈ U(B) such that Ad u_{n}α|C_{n} = β|C_{n} and

|| Ad u_{n}α(z_{n}) − β(z_{n})|| < sin 2πδ.

For v ∈ V_{n} we define

h_{v} = \frac{1}{2πi} log β(z_{v})u_{n}α(z_{v})u_{n}^{∗},

then h_{v} is in β(p_{v})Bβ(p_{v}) and ||h_{v}|| < δ. In the same way as in [KK], we can see that the affine function T (B) ∋ τ ↦ τ(h_{v}) is in the range of the rotation map R_{α,β}. From the assumption η(α, β) = 0, there exists x_{v} ∈ K_{0}(β(p_{v})Bβ(p_{v})) such that τ(h_{v}) = τ(x_{v}) for every τ ∈ T (β(p_{v})Bβ(p_{v})). Therefore, by Lemma 2.3, we obtain a unitary w_{v} ∈ β(p_{v})Bβ(p_{v}) such that ||[β(z_{v}p_{v}), w_{v})]|| < ε_{n}/2 and B(β(z_{v}p_{v}), w_{v}) = x_{v}. By taking a direct sum of h(v) copies of w_{v}, we get a partial unitary of B commuting with β(M_{h(v)}). Let w_{n} be the direct sum of these partial unitaries. Then, w_{n} is a unitary of B ∩ β(C_{n})′. When we replace u_{n} by w_{n}u_{n}, we can get the following:

Ad u_{n}α|C_{n} = β|C_{n},
|| Ad u_{n}α(z_{n}) − β(z_{n})|| < ε_{n}

and τ(h_{v}) = 0 for all τ ∈ T (B), where h_{v} is equal to

\frac{1}{2πi} log β(z_{v})u_{n}α(z_{v})u_{n}^{∗}.

Thus we have shown [KK, Lemma 3.3] in our situation.

Lemma 3.5, 3.6 and 3.7 of [KK] are valid with no modification. If one uses Lemma 2.4 above, it is not so hard to see that [KK, Lemma 3.9] is also valid. Therefore we have chosen the above u_{n}’s so that the Bott element B(u_{n+1}^{∗}u_{n}, α(z_{v})) is zero for all v ∈ V_{n} and [u_{n+1}^{∗}u_{n}α(p_{v})] is also zero in K_{1}(α(p_{v})Bα(p_{v})). Note that the argument in this paragraph is not necessary, when B is an AF algebra and Inf(K_{0}(B)) = 0.

Now we can apply the homotopy lemma [BEEK, Theorem 8.1] in a similar fashion to [KK] and conclude that α and β are asymptotically unitarily equivalent. □

3 Crossed products of AT algebras by the integers

The aim of this section is to generalize N. Brown’s AF embedding theorem [B, Corollary 4.10]. Let us begin with the definition of the universal UHF algebra U and an automorphism σ ∈ Aut(U) with the Rohlin property ([B, Example 2.2]). The universal UHF algebra is the infinite tensor product C^{∗}-algebra ⨂_{n∈ℕ} M_{n}. Let u_{n} ∈ M_{n} be the unitary matrix satisfying Ad u_{n}(e_{i,j}) = e_{i+1,j+1}, where {e_{i,j}}_{i,j} is a system of matrix units in M_{n} and the addition is understood modulo n. The infinite product automorphism σ = ⊗_{n} Ad u_{n} has a special property, so called the Rohlin property.

In order to explain this property, we have to recall the idea of central sequence C^{∗}-algebras. Let A be a C^{∗}-algebra. Define a C^{∗}-algebra and its ideal by

ℓ^{∞}(ℕ, A) = {(a_{n})_{n∈ℕ} ; a_{n} ∈ A and sup_{n} ||a_{n}|| < ∞}

and

c_{0}(ℕ, A) = {(a_{n})_{n} ∈ ℓ^{∞}(ℕ, A) ; lim a_{n} = 0} .

Let A^{∞} = ℓ^{∞}(ℕ, A)/c_{0}(ℕ, A) be the quotient C^{∗}-algebra. The central sequence C^{∗}-algebra A_{∞} is defined by A_{∞} = A^{∞} ∩ A′, where the original C^{∗}-algebra A is embedded into A^{∞} as constant sequences.

As explained in Section 2 of [B], the automorphism σ ∈ Aut(U) has the Rohlin property in the following sense: for every natural number N there exists a family of projections E_{0}, E_{1}, …, E_{N} ∈ U_{∞} such that ∑_{i} E_{i} = 1 and σ(E_{i}) = E_{i+1} (with addition modulo N). This family of projections is sometimes called the Rohlin tower. Moreover, for any unital C^{∗}-algebra B and automorphism β ∈ Aut(B), it is easy to see that the automorphism β ⊗ σ on B ⊗ U also satisfies the same property. In general the Rohlin property does not require that the number of the Rohlin tower is one, and the height of the tower may not be exactly equal to the given natural number N (see [B, Definition 2.1]). In this paper, however, we will adopt the stronger version described above as definition of the Rohlin property for convenience.

At first we need a technical lemma.

Lemma 3.1. Let A be a unital simple AT algebra with real rank zero. For every finite subset F of A and ε > 0, there exist a finite subset G of the unit ball of A and δ > 0 such that the following holds: If φ : A → B is a unital embedding to an AF algebra B and u : [0, 1] → U(B) is a continuous path with u(0) = 1 satisfying

||[u(t), φ(a)|| < δ

for all t ∈ [0, 1] and a ∈ G, then there exists a continuous path w : [0, 1] → U(B) with length less than 6π such that w(0) = 1, w(1) = u(1) and

||[w(t), φ(a)]|| < ε

for all t ∈ [0, 1] and a ∈ F.

Proof. Use Lemma 7.1 or Theorem 8.1 of [BEEK]. See also [K, Lemma 4.4].

The following proposition is the key to Theorem 3.3.

Proposition 3.2. Let A be a unital simple AT algebra with real rank zero and α be an automorphism of A. Let B be a unital simple AF algebra and φ : A → B be an embedding. Suppose there exists an automorphism β ∈ Aut(B) such that φα and βφ are asymptotically unitarily equivalent. Then, the crossed product C^{∗}-algebra A ⋊_{α} ℤ is AF embeddable.

Proof. Since we can replace (B, β) with (B ⊗ U, β ⊗ σ), we may assume that β has the Rohlin property.

Let F_{1} ⊂ F_{2} ⊂ F_{3} ⊂ … be an increasing sequence of finite subsets of the unit ball of A, and suppose the union ⋃ F_{n} is dense in the unit ball of A. Let k_{1} < k_{2} < k_{3} < … be a sequence of natural numbers which increase sufficiently rapidly and ε_{1} > ε_{2} > ε_{3} > … be a sequence of positive real numbers which decrease sufficiently rapidly. By applying Lemma 3.1 to ⋃_{i=0}^{k_{n}−1} α^{−i}(F_{n}) and ε_{n}, we get a finite subset G_{n} of the unit ball of A and δ′_{n} > 0. We may assume F_{n} ⊂ G_{n}. Let δ_{1} > δ_{2} > δ_{3} > … be a decreasing sequence of positive real numbers such that k_{n}^{2}δ_{n} < δ′_{n} and (k_{n}^{2} + k_{n})δ_{n} < ε_{n}.

Since φα and βφ are asymptotically unitarily equivalent, by perturbing β with an inner automorphism, we may assume the following: There exists a continuous map u : [0, ∞) → U(B) with u(0) = 1 such that

φα(a) = lim_{t→∞} Ad u(t)βφ(a)

holds for all a ∈ A and

||φα(a) − Ad u(t)βφ(a)|| < \frac{δ_{1}}{2}

for a ∈ ⋃_{i=0}^{k_{1}−1} α^{−i}(G_{1}) and t ∈ [0, ∞).

For sufficiently large s > 0 we have

||φα(a) − Ad u(t)βφ(a)|| < \frac{δ_{2}}{2}

for all t ∈ [s, ∞) and a ∈ ⋃_{i=0}^{k_{2}−1} α^{−i}(G_{2}). For t ∈ [0, s] we define

v_{0}(t) = 1, v_{1}(t) = u(t), v_{2}(t) = u(t)β(u(t)),
v_{3}(t) = u(t)β(u(t))β^{2}(u(t)), ….

Then, for a ∈ G_{1} we get

φα(a) ∼_{δ_{1}/2} βφ(a) = βφαα^{−1}(a) ∼_{δ_{1}/2} β^{2}φα^{−1}(a) = …,

and so

||φα(a) − β^{k}φα^{−k+1}(a)|| < \frac{kδ_{1}}{2}

is obtained for all a ∈ G_{1} and k ≤ k_{1}. Hence we have

||[v_{k}(t), φα(a)]|| ≤ ∑_{j=0}^{k−1} ||[β^{j}(u(t)), φα(a)]||

< ∑_{j=0}^{k−1} (||[β^{j}(u(t)), β^{j}φα^{−j+1}(a)]|| + jδ_{1})
< k_{1}(δ_{1} + (k_{1} − 1)δ_{1}) = k_{1}^{2}δ_{1} < δ′_{1},

for all t ∈ [0, s], a ∈ G_{1} and k ≤ k_{1}. Especially, we can apply Lemma 3.1 to the case k = k_{1}, and get a continuous map w : [0, 1] → U(B) satisfying w(0) = 1, w(1) = v_{k_{1}}(s) and

||w(l) − w(l′)|| < 6π|l − l′|
||[w(l), φα(a)]|| < ε_{1}

for l ∈ [0, 1] and a ∈ ⋃_{i=0}^{k_{1}−1} α^{−i}(F_{1}). Take a Rohlin tower E_{1}, E_{2}, …, E_{k_{1}} ∈ B_{∞} and define

V = ∑_{j=1}^{k_{1}} v_{j−1}(s)β^{j−1} (w (\frac{j − 1}{k_{1} − 1})) E_{j} ∈ B^{∞}.

Obviously V is a unitary. Moreover we obtain

||V β(V)^{∗} − u(s)|| < \frac{6π}{k_{1} − 1}

and

||[V, φα(a)]|| < (k_{1}^{2} + k_{1})δ_{1} + ε_{1} < 2ε_{1}

for all a ∈ F_{1}. Therefore there exists a unitary v_{1} in B such that the above two inequalities hold for v_{1} instead of V. We define w_{1} = v_{1}^{∗}u(s)β(v_{1}), φ_{1} = Ad v_{1}^{∗}φ, β_{1} = Ad w_{1}β and u_{1}(t) = v_{1}^{∗}u(t + s)u(s)^{∗}v_{1}. Then we can check that

φ_{1}α(a) = lim_{t→∞} Ad u_{1}(t)β_{1}φ_{1}(a)

holds for all a ∈ A and

||φ_{1}α(a) − Ad u_{1}(t)β_{1}φ_{1}(a)|| < \frac{δ_{2}}{2}

for all t ∈ [0, ∞) and a ∈ ⋃_{i=0}^{k_{2}−1} α^{−i}(G_{2}).

By repeating this procedure, we get

φ_{n} = Ad v_{n}v_{n−1} … v_{2}v_{1}φ

and

β_{n} = Ad w_{n}w_{n−1} … w_{2}w_{1}β

for each n. Since we have

||[v_{n}, φα(a)]|| < 2ε_{n}

for a ∈ F_{n}, there exists φ′ ∈ Hom(A, B) such that φ′(a) = lim_{n} φ_{n}(a) for every a ∈ A. Because ||w_{n} − 1|| is less than 6π(k_{n} − 1)^{−1},

w = lim_{n→∞} w_{n}w_{n−1} … w_{2}w_{1}

exists. Moreover, for all a ∈ F_{n}, we have

||φ_{n}α(a) − β_{n}φ_{n}(a)|| < \frac{δ_{n}}{2} ,

which implies φ′α = Ad wβφ′. Since we have embedded (A, α) to (B, Ad wβ) covariantly, the conclusion follows from [B, Corollary 4.10] (or from [V1, Theorem 3.6], when the original β is approximately inner.)

Theorem 3.3. When A is a unital simple AT algebra with real rank zero and α is an automorphism of A, the crossed product C^{∗}-algebra A ⋊_{α} ℤ is AF embeddable.

Proof. Let τ ∈ T (B) be an α-invariant tracial state. When we define D = τ(K_{0}(A)) and D^{+} = D ∩ ℝ^{+}, the triple (D, D^{+}, 1) is a simple dimension group. We can find a unital simple AF algebra B and φ ∈ Hom(A, B) such that K_{0}-group of B is isomorphic to (D, D^{+}, 1) and φ_{∗} is equal to τ on K_{0}(A). From Elliott’s theorem [E, Theorem 7.4] two homomorphisms φ and φα are approximately unitarily equivalent.

Since T (B) consists of one point, the range of the rotation map R_{φα,φ} can be identified with a countable subgroup of ℝ containing D ≅ K_{0}(B). Let D′ be this countable group and D′^{+} be D′ ∩ ℝ^{+}. There exists a unital simple AF algebra B′ whose K_{0}-group is isomorphic to (D′, D′^{+}, 1). We can find ψ ∈ Hom(B, B′) which induces the canonical inclusion from D to D′. Evidently the range of the rotation map R_{ψφα,ψφ} is equal to D′, which also implies that the extension of K_{1}(A) by K_{0}(B′) is trivial because Inf(K_{0}(B)) = 0. Hence we get η(ψφα, ψφ) = 0. Then, Theorem 2.1 tells us that two homomorphisms ψφα and ψφ are asymptotically unitarily equivalent. From Proposition 3.2, we get the desired AF embedding.

By using the recent classification result, we obtain the following corollary.

Corollary 3.4. Let A be a unital separable simple nuclear TAF algebra which satisfies UCT. When α is an automorphism of A, the crossed product C^{∗}-algebra A ⋊_{α} ℤ is AF embeddable.

Proof. Notice that A is a unital simple real rank zero AH algebra with slow dimension growth (see [L] and [EG]). Since A ⊗ U is also such a kind of C^{∗}-algebra and its K-groups are torsion free, the classification theorems in the above papers imply that A ⊗ U is a unital simple AT algebra with real rank zero. Because we can apply Theorem 3.3 to (A ⊗ U, α ⊗ id), the crossed product A ⋊_{α} ℤ is embedded into an AF algebra.

4 C^{∗}-algebras arising from ℤ^{2}-minimal systems

We will discuss the AF embeddability of the crossed product C^{∗}-algebra C^{∗}(X, T, S) arising from a ℤ^{2}-minimal system (X, T, S) in this section. The next proposition is our starting point.

Proposition 4.1. Let T be a minimal homeomorphism on the Cantor set X and S be a homeomorphism which commutes with T. Then the crossed product C^{∗}-algebra C^{∗}(X, T, S) is AF embeddable.

Proof. The homeomorphism S induces the automorphism α of C^{∗}(X, T) and C^{∗}(X, T, S) is isomorphic to C^{∗}(X, T) ⋊_{α} ℤ. Thanks to [Pu, Theorem 2.1] the crossed product C^{∗}-algebra C^{∗}(X, T) is a unital simple AT algebra with real rank zero (see also [HPS, Section 8]). We get the conclusion from Theorem 3.3.

Let T be a homeomorphism on a compact Hausdorff space X. We introduce the following condition called (#) for T:

(#) For every T -invariant non-empty open subset U there exists a T -invariant non-empty open subset V with \overline{V} ⊂ U.

The following lemma is an easy observation.

Lemma 4.2. (i) If (X, T) is a minimal system and Y is a compact Hausdorff space, then T × id ∈ Homeo(X × Y) satisfies condition (#).

(ii) If (X, T) is a dynamical system and there exists a T -invariant metric on X, then T satisfies condition (#).

(iii) If (T_{s})_{s∈ℝ} is a minimal flow on X, then T_{s} satisfies condition (#) for every s ∈ ℝ.

Proof. (i) Every T ×id-invariant open set takes the form of X ×U for an open set U of Y. Since there exists a non-empty open set V with \overline{V} ⊂ U, we can check condition (#).

(ii) Let d(·, ·) be a T -invariant metric and U be a T -invariant open set. Take x ∈ U and let W be an open ball with the center x and the radius d(x, U^{c})/2. When we define V = ⋃_{n∈ℤ} T^{n}(W ∩ U), we can see that d(U^{c}, V) = d(U^{c}, W ∩ U) is not less than d(x, U^{c})/2. Hence \overline{V} is contained in U.

(iii) It suffices to show that T_{1} satisfies condition (#). Let F be a minimal T_{1}-invariant closed subset. If T_{s}(F) ∩ F is not empty, from the minimality of F, we have T_{s}(F) = F. When we define G = {s ∈ ℝ ; T_{s}(F) = F }, it is easily seen that G is a closed subgroup of ℝ containing one. When G equals ℝ, one has F = X and condition (#) is obviously satisfied. Suppose G = n^{−1}ℤ for n ∈ ℕ. Then, ⋃_{s∈[0,n^{−1})} T_{s}(F) gives a disjoint partition of X. We can prove that T_{1} satisfies condition (#) in a similar fashion to (i).

Let (X, T, S) and (Y, T′, S′) be two ℤ^{2}-minimal systems. A continuous map π : Y → X is called a factor map, when it satisfies T π = πT′ and Sπ = πS′. The factor map π induces a canonical embedding of C^{∗}(X, T, S) to C^{∗}(Y, T′, S′).

Lemma 4.3. Let π : Y → X be a factor map between ℤ^{2}-minimal systems (X, T, S) and (Y, T′, S′). Suppose T satisfies condition (#). If either of the following holds, then T′ also satisfies condition (#).

(i) The factor map π is almost one-to-one, that is, there exists y ∈ Y with π^{−1}(π(y)) = {y}.

(ii) The factor map π is a local homeomorphism, that is, each point y ∈ Y has a neighborhood U such that π(U) is open and π|U is a homeomorphism.

Proof. (i) Suppose U ⊂ Y is a T′-invariant non-empty open subset. Since (Y, T′, S′) is a ℤ^{2}- minimal system, there exists y ∈ U with π^{−1}(π(y)) = {y}. We can find a neighborhood W_{0} of π(y) such that π^{−1}(W_{0}) is contained in U. Define W = ⋃_{n∈ℤ} T (W_{0}). By the assumption, there exists a T -invariant non-empty open set V with \overline{V} ⊂ W. Then, we have π^{−1}(\overline{V}) ⊂ π^{−1}(W) ⊂ U, which implies condition (#).

(ii) Notice that π^{−1}(x) is a finite set and its cardinality does not depend on x. Let U be a T′-invariant open set and define

k = max {# (U ∩ π^{−1}(x)) ; x ∈ X} .

Then, W = {y ∈ U; #(U ∩ π^{−1}π(y)) = k} is a T′-invariant open subset of U. There exists a T -invariant open set V ⊂ X with \overline{V} ⊂ π(W), because T satisfies condition (#). We would like to show that the T′-invariant open set V′ = W ∩ π^{−1}(V) satisfies \overline{V′} ⊂ W. Suppose that a sequence \{y_{n}\}_{n} ⊂ V′ converges to y ∉ W. Since y is contained in π^{−1}(\overline{V}), there exist k distinct preimages of π(y) in W. Then, one can see that there exist k + 1 preimages of π(y_{n}) in W for sufficiently large n. This contradicts the assumption.

Let us prove the main theorem. In order to do that, we need a series of lemmas.

Lemma 4.4. When (X, T) is a dynamical system, the following are equivalent.

(i) T satisfies condition (#).

(ii) T^{n} satisfies condition (#) for all n ∈ ℕ.

(iii) T^{n} satisfies condition (#) for some n ∈ ℕ.

Proof. (i)⇒(ii). The proof goes by induction. Assume that the assertion has been shown for all natural numbers less than n. Let U ⊂ X be a T^{n}-invariant open subset. Take the minimum natural number k such that T^{k}(U)∩U is not empty. If k = n, W = U ∪T (U)∪⋯∪T^{n}(U) is a T - invariant open set and we get a T -invariant open set V with \overline{V} ⊂ W. Then U ∩V is a T^{n}-invariant open set and \overline{U ∩ V} ⊂ (W \U)^{c}∩\overline{V} ⊂ U. Suppose k < n. When U′ = U ∩T^{k}(U)∩⋯∩T^{(n−1)k}(U) is not empty, by the induction step, we can find a T^{l}-invariant open set V with \overline{V} ⊂ U′, where l is the greatest common divisor of k and n. Since V is also T^{n}-invariant, we have the conclusion. We may assume U′ is empty. Let m be the natural number such that U′′ = U ∩T^{k}(U)∩⋯∩T^{mk}(U) is not empty and U ∩ T^{k}(U) ∩ ⋯ ∩ T^{(m+1)k}(U) is empty. Clearly U′′ is a T^{n}-invariant open subset of U and U′′ ∩ T^{i}(U′′) = ∅ for i = 1, 2, …, k − 1, k. By repeating this argument for U′′, we will complete the proof.

(ii)⇒(iii) is obvious.

(iii)⇒(i). Suppose U is a T -invariant non-empty open subset. From the assumption there exists a T^{n}-invariant open subset V with \overline{V} ⊂ U. Then V′ = V ∪ T (V) ∪ … T^{n−1}(U) is a T -invariant open subset with \overline{V′} ⊂ U.

Lemma 4.5. Let (X, T, S) be a ℤ^{2}-minimal system and suppose T satisfies condition (#). Let a, b and c be three natural numbers and U be an open subset of X. If a and b are relatively prime, there exists a sequence of homeomorphisms ϕ_{0}, ϕ_{1}, …, ϕ_{l} ∈ Homeo(X) with ϕ_{0} = id such that ϕ_{i}^{−1}ϕ_{i+1} equals either of T^{a}, T^{b}, and T^{c}S for all i = 0, 1, …, l − 1 and X = ⋃_{i=0}^{l} ϕ_{i}(U).

Proof. Since ⋃_{k∈ℤ} T^{ka}(U) is T^{a}-invariant and open, with the aid of Lemma 4.4 we obtain a T^{a}-invariant open subset V with \overline{V} ⊂ ⋃ T^{ka}(U). Because \overline{V} is compact, there exists n such that

U ∪ T^{a}(U) ∪ ⋯ ∪ T^{na}(U) ⊃ V.

Hence we have

⋃_{k=0}^{a−1} T^{kb+kna}(U ∪ T^{a}(U) ∪ ⋯ ∪ T^{na}(U)) ⊃ ⋃_{k=0}^{a−1} T^{kb}(V).

Since V′ = ⋃_{k=0}^{a−1} T^{kb}(V) is a T -invariant open subset, by the minimality, we can find a natural number m such that V′ ∪ S(V′) ∪ ⋯ ∪ S^{m}(V′) = X. Therefore we get

⋃_{j=0}^{m} T^{cj}S^{j} ( ⋃_{k=0}^{a−1} T^{kb+kna}(U ∪ T^{a}(U) ∪ ⋯ ∪ T^{na}(U))) = X,

which completes the proof.

The following lemma will be used to reduce the problem to Proposition 4.1. We refer the reader to [HPS] or [M, Section 2] for ordered Bratteli diagrams.

Lemma 4.6. Let (X, T, S) be a ℤ^{2}-minimal system and suppose T satisfies condition (#). Then there exist a Cantor minimal system (Y, ψ) and a continuous function f : Y → ℤ such that γ(x, y) = (T S^{f(y)}(x), ψ(y)) is a minimal homeomorphism on X × Y.

Proof. We will construct a properly ordered simple Bratteli diagram B = (V, E). We denote the Bratteli-Vershik system of B by (Y, ψ) and the unique minimal infinite path by y_{0} ∈ Y. Define V_{0} = {v_{0}} and V_{n} = {a_{n}, b_{n}, c_{n}, d_{n}} for every n ∈ ℕ. We connect the vertex v_{0} to each vertex of V_{1} by a single edge. These four edges in E_{1} give a partition of Y, namely Y = U_{a} ∪ U_{b} ∪ U_{c} ∪ U_{d}. We put a ℤ-valued continuous function f by f|U_{a} = f|U_{b} = 0, f|U_{c} = 1 and f|U_{d} = −1. Inductively we will define the partially ordered edge set E_{n} for n ≥ 2 so that the homeomorphism γ determined by f becomes minimal on X × Y. We write the projections from X × Y to X and Y by π_{X} and π_{Y}. Let {U_{n}}_{n∈ℕ} be an open basis of X. We write the number of paths from v to v′ by E(v, v′). Suppose that the partially ordered edge sets E_{1}, E_{2}, …, E_{n} have been already defined so that the following properties are satisfied.

(i) Two natural numbers E(v_{0}, a_{n}) and E(v_{0}, b_{n}) are relatively prime.

(ii) E(c_{1}, a_{n}) = E(d_{1}, a_{n}).

(iii) E(c_{1}, b_{n}) = E(d_{1}, b_{n}).

(iv) E(c_{1}, c_{n}) − E(d_{1}, c_{n}) = 1.

(v) E(c_{1}, d_{n}) − E(d_{1}, d_{n}) = −1.

(vi) Every maximal edges and minimal edges in E_{n} goes through a_{n−1} and every pair of vertices in V_{n−1} and V_{n} is connected.

(vii) For x ∈ X there exists a non-negative integer k less than E(v_{0}, a_{n}) such that π_{X}γ^{k}(x, y_{0}) is contained in U_{n−1} and the initial n − 2 edges of π_{Y} γ^{k}(x, y_{0}) = ψ^{k}(y_{0}) coincide with y_{0}.

We would like to define the partially ordered edge set E_{n+1}. By applying Lemma 4.5 to (X, T^{−1}, S^{−1}), U_{n} and three natural numbers E(v_{0}, a_{n}), E(v_{0}, b_{n}) and E(v_{0}, c_{n}), we obtain a sequence of homeomorphisms ϕ_{0}, ϕ_{1}, …, ϕ_{l} with ϕ_{0} = id. At first, let us consider r^{−1}(a_{n+1}). For i = 1, 2, …, l we determine the source vertex of i-th edge e_{i} in r^{−1}(a_{n+1}) by the homeomorphism ϕ_{i−1}ϕ_{i}^{−1}, that is, we define

s(e_{i}) = \begin{cases}a_{n} & ϕ_{i−1}ϕ_{i}^{−1} = T^{E(v_{0},a_{n})} \\ b_{n} & ϕ_{i−1}ϕ_{i}^{−1} = T^{E(v_{0},b_{n})} \\ c_{n} & ϕ_{i−1}ϕ_{i}^{−1} = ST^{E(v_{0},c_{n})}. \end{cases}

This definition for n = 1 has ambiguity because E(v_{0}.a_{1}) = E(v_{0}, b_{1}) = 1. But we may choose a_{1} or b_{1} freely when n = 1. Lemma 4.5 and the definition above tell us that the property (vii) in the n + 1-st step is satisfied. By adding more edges greater than e_{l} to r^{−1}(a_{n+1}), we can achieve the property (ii) and (vi) in the n + 1-st step. For the other edge sets r^{−1}(b_{n+1}), r^{−1}(c_{n+1}) and r^{−1}(c_{n+1}) we have much flexibility. It is easy to achieve the other properties and make the Bratteli diagram simple. By repeating this procedure, we get a properly ordered simple Bratteli diagram B = (V, E).

From the construction we can show that the closure of the γ-orbit {γ^{n}(x, y_{0}); n ∈ ℤ} contains X × {y_{0}} for every x ∈ X. Let x, x′ ∈ X be two points. When U is an open neighborhood of x′, there exists a sufficiently large number n such that x′ ∈ U_{n} ⊂ U. From the property (vii), we can find a natural number k so that π_{X}γ^{k}(x, y_{0}) is contained in U_{n} and ψ^{k}(y_{0}) is very close to y_{0}. Hence the closure of {γ^{n}(x, y_{0}); n ∈ ℤ} contains (x′, y_{0}).

Let us check that the minimality of γ. Take (x, y) ∈ X × Y arbitrarily. We would like to show that the closure of the γ-orbit of (x, y) is X × Y. Since (Y, ψ) is minimal and X is compact, we can find a sequence of integers {m_{n}}_{n} such that γ^{m_{n}}(x, y) converges to a point in X × {y_{0}}. Hence the closure of the γ-orbit of (x, y) contains the whole of X × {y_{0}}. By using the minimality of (Y, ψ) again, we get the conclusion.

Although the following lemma may be well-known, the author would like to present the proof for the reader’s convenience.

Lemma 4.7. Let (X, T, S) be a ℤ^{2}-minimal system and suppose X is not a finite set. Then there exist a ℤ^{2}-minimal system (Y, T′, S′) and an almost one-to-one factor map π : Y → X such that Y is the Cantor set.

Proof. Let \{U_{n}\}_{n∈ℕ} be an open basis of X. Define U_{0,n} = \overline{U_{n}} and U_{1,n} = U_{n}^{c}. It is clear that

X′ = {y ∈ {0, 1}^{ℕ×ℤ×ℤ} ; ⋂_{n,k,l} T^{k}S^{l}(U_{y(n,k,l),n}) is not empty}

is a closed subset of {0, 1}^{ℕ×ℤ×ℤ}. Obviously there exists a continuous surjection π : X′ → X. The map π is one-to-one at least on the points of

⋂_{n,k,l} T^{k}S^{l}(U_{n} ∪ \overline{U_{n}}^{c}),

which is not empty by the Baire category theorem. We denote the subshifts on the second and third coordinates of {0, 1}^{ℕ×ℤ×ℤ} by T′ and S′. By Zorn’s lemma we can find a minimal T′-invariant and S′-invariant non-empty closed subset Y in X′. Then (Y, T′, S′) is a ℤ^{2}-minimal system and π : Y → X gives an almost one-to-one factor map. Since Y is totally disconnected and X is not a finite set, we can see that Y is the Cantor set.

Now we are ready to prove the main theorem.

Theorem 4.8. Let (X, T, S) be a ℤ^{2}-minimal system. If T^{n}S^{m} satisfies condition (#) for some (n, m) ∈ ℤ^{2} \ 0, then the crossed product C^{∗}-algebra C^{∗}(X, T, S) is AF embeddable.

Proof. We may assume that n and m are relatively prime because of Lemma 4.4. By replacing generators of the ℤ^{2}-action, we may further assume that T satisfies condition (#). From Lemma 4.3 (i) and 4.7, it suffices to consider the case that X is the Cantor set.

By Lemma 4.6 we can find a Cantor minimal system (Y, ψ), a continuous function f : Y → ℤ and a minimal homeomorphism γ on X × Y determined by f. Since we can apply Proposition 4.1 to the Cantor minimal system (X × Y, γ) and the homeomorphism τ = S × id, the crossed product C^{∗}-algebra C^{∗}(X × Y, γ, τ) is AF embeddable. We denote the implementing unitaries corresponding to γ and τ by u and v. Then, for any g ∈ C(X × Y) we have ugu^{∗} = gγ^{−1} and vgv^{∗} = gτ^{−1}. Define a unitary w ∈ C(X × Y, γ, τ) by

w = ∑_{n∈ℤ}uv^{−n}1_{X×f^{−1}(n)},

where 1_{E} denotes the characteristic function on E and the sum is actually a finite sum. It is easily seen that

w(g ⊗ 1_{Y})w^{∗} = gT^{−1} ⊗ 1_{Y} , v(g ⊗ 1_{Y})v^{∗} = gS^{−1} ⊗ 1_{Y}

for every g ∈ C(X), and so a unital homomorphism π : C^{∗}(X, T, S) → C^{∗}(X × Y, γ, τ) is obtained. Because there are faithful conditional expectations E : C^{∗}(X, T, S) → C(X) and F : C^{∗}(X × Y, γ, τ) → C(X × Y) such that πE = F π, we can show that π is an embedding.

5 Examples

At the final section we would like to give examples of ℤ^{2}-minimal systems to which Theorem 4.8 can be applied.

(1) Many examples of minimal ℝ-flows on compact connected manifolds are known ([FH]). Let α : ℝ × X → X be such a minimal flow. If s and t are rationally independent real numbers, then T = α_{t} and S = α_{s} induce a ℤ^{2}-minimal system on X. From Lemma 4.2 (iii) we can see that C^{∗}(X, T, S) is AF embeddable.

(2) We would like to consider Toeplitz sequences over ℤ^{2}. The reader may refer to [W] for the usual ℤ case. Define

RF (ℤ^{2}) = {H ⊂ ℤ^{2} ; H is a subgroup of ℤ^{2} with ℤ^{2}/H finite }.

Let A be a finite set. For ξ : ℤ^{2} → A and H ∈ RF (ℤ^{2}) we denote by Per(ξ, H) the set of all p ∈ ℤ^{2} such that ξ(p) = ξ(p + q) for all q ∈ H. We call ξ a Toeplitz sequence over ℤ^{2}, when

ℤ^{2} = ⋃_{H∈RF (ℤ^{2})}Per(ξ, H).

Note that this definition of Toeplitz sequences is valid not only for ℤ^{2} or ℤ but also for all residually finite groups.

Let T and S be the subshift over ℤ^{2}, that is, for η ∈ A^{ℤ^{2}} we define T (η)(n, m) = η(n + 1, m) and S(η)(n, m) = η(n, m + 1). Let X be the closure of {T^{n}S^{m}(ξ); n, m ∈ ℤ} for a Toeplitz sequence ξ. Then, by the same way as the ℤ case, one can prove that (X, T, S) is a ℤ^{2}- minimal system. We call H ∈ RF (ℤ^{2}) a essential period of ξ, if Per(ξ, H) is not empty and Per(ξ, H′) ⊊ Per(ξ, H) holds for all H′ ∈ RF (ℤ^{2}) with H ⊊ H′. For H ∈ RF (ℤ^{2}), let T_{H} and S_{H} be the translations on ℤ^{2}/H by (1, 0) and (0, 1) of ℤ^{2}. Let H be an essential period of ξ. For p ∈ ℤ^{2}, we can see that the closure of

{T^{n}S^{m}(ξ) ∈ X ; (n, m) ∈ H + p}

is contained in the closed set

U_{H,p} = {η ∈ X ; η(q − p) = ξ(q) for all q ∈ Per(ξ, H)}.

When p runs over representatives of all cosets in ℤ^{2}/H, it can be checked that U_{H,p}’s give a disjoint partition of X. Therefore U_{H,p} is a clopen subset of X and there exists a factor map π_{H} : (X, T, S) → (ℤ^{2}/H, T_{H}, S_{H}). By considering all essential periods, we obtain a factor map π from (X, T, S) to a ℤ^{2}-minimal system induced by two rotations over a compact abelian group and π is one-to-one at ξ. From Lemma 4.2 (ii) and Lemma 4.3 (i), it follows that T ∈ Homeo(X) satisfies condition (#). Hence C^{∗}(X, T, S) is AF embeddable.

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e-mail; matui@kusm.kyoto-u.ac.jp Department of Mathematics, Kyoto University, Kitasirakawa-Oiwaketyô, Sakyôku, Kyôto, 606-8502, Japan.