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Z-stability of crossed products by strongly outer actions II

We consider a crossed product of a unital simple separable nuclear stably finite Z-stable C*-algebra A by a strongly outer cocycle action of a discrete countable amenable group \Gamma. Under the assumption that A has finitely many extremal tracial states and \Gamma is elementary amenable, we show that the twisted crossed product C*-algebra is Z-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on Z are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite C*-algebras.


Introduction
This is a continuation of our previous paper [27] studying Z-stability of crossed product C * -algebras and classification of group actions on Z. Here Z denotes the Jiang-Su algebra introduced by X. Jiang and H. Su in [11], and a unital C * -algebra A is said to be Z-stable if A is isomorphic to A ⊗ Z. In Elliott's program to classify nuclear C * -algebras via Ktheoretic invariants (see [34] for an introduction to this subject), the Jiang-Su algebra Z has recently become to play a central role. In fact, all unital simple separable nuclear C * -algebras classified so far by their Elliott invariants are Z-stable. Also, Z-stability is known to be closely related to other regularity properties, such as finite nuclear dimension, strict comparison of positive elements and the property (SI) (see [35,46,28]). We refer the reader to [36,45] for several characterizations of Z.
In the first half of the paper, we study Z-stability of crossed products arising from strongly outer cocycle actions of discrete countable groups. Let A be a unital, simple, separable, nuclear, stably finite, Z-stable C * -algebra with finitely many extremal tracial states. Let (α, u) : Γ A be a cocycle action of a discrete group Γ (see Definition 2.1). We say that (α, u) is strongly outer when its extension to the weak closure in any tracial representation is outer (see Definition 2.5 for the precise definition). In this setting, we prove that if Γ is elementary amenable and (α, u) is strongly outer, then the twisted crossed product A ⋊ (α,u) Γ is Z-stable (Corollary 4.11). In our previous paper [27], we had to make extra hypotheses on the C * -algebra A and on the group Γ in order to deduce the weak Rohlin property (see Definition 2.5) from strong outerness. The hypothesis on A was necessary to replace central sequences in the sense of von Neumann algebras with central sequences in the sense of C * -algebras. In this paper, we have removed it by using amenability of nuclear C * -algebras proved by U. Haagerup [7] (see Proposition 3.5). As for the group Γ, we dealt with only Z N and finite groups in the previous paper. Now we extend our arguments to all elementary amenable groups by showing that they admit outer actions on the hyperfinite II 1 -factor with a 'nice' Rohlin property (see Definition 2.4 and Proposition 3.3). In order to prove Z-stability of crossed products, we need one more ingredient, namely the property (SI). In [28], we have already shown that Z-stability of A implies the property (SI) (Theorem 2.8). Furthermore, by using the weak Rohlin property of (α, u), we will prove that A satisfies an α-equivariant version of the property (SI) (Proposition 4.5). Together with an exact sequence of central sequence algebras (Theorem 4.3), this allows us to conclude that A ⋊ (α,u) Γ is Z-stable.
In Section 5, we will also show that if a unital separable nuclear C * -algebra A is not of type I, then the central sequence algebra has a subquotient which is isomorphic to a II 1 -factor (Theorem 5.1). This may be thought of as an analogue of Glimm's theorem.
In the latter half of the paper, we study cocycle actions of the Klein bottle group on UHF algebras and the Jiang-Su algebra. The Klein bottle group is the group generated by two elements a, b satisfying bab −1 = a −1 , and is isomorphic to the fundamental group of the Klein bottle. Classification of group actions is one of the most fundamental subjects in the theory of operator algebras. We briefly review classification results of automorphisms or group actions on simple stably finite C * -algebras known so far. For AF and AT algebras, A. Kishimoto [15,16,17] showed the Rohlin property for a certain class of automorphisms and obtained a cocycle conjugacy result. The first-named author [24] extended this result to unital simple AH algebras with real rank zero and slow dimension growth. The second-named author [38] proved that strongly outer Z-actions on Z are unique up to strong cocycle conjugacy. T. Katsura and the first-named author [14] gave a complete classification of strongly outer Z 2 -actions on UHF algebras by using the Rohlin property, and then this result was extended to a certain class of strongly outer Z 2 -actions on unital simple AF algebras [24]. The uniqueness of strongly outer Z N -actions on UHF algebras of infinite type was obtained in [26]. In our previous paper [27], we proved that strongly outer Z 2 -actions on Z are unique up to strong cocycle conjugacy.
In the present paper, we first show that strongly outer cocycle actions of the Klein bottle group on UHF algebras are unique up to cocycle conjugacy (Theorem 6.16). The proof is along the same lines as the proof of [24,Theorem 6.6], but we need to look at the OrderExt invariant carefully and to check that it always vanishes, unlike the case of Z 2 -actions. Then we establish a cohomology vanishing type result for cocycle actions of the Klein bottle group on the Jiang-Su algebra Z (Proposition 7.7). The Z-stability theorem proved in the first half of the paper plays an essential role. Finally, by using this cohomology vanishing, we obtain the uniqueness of strongly outer cocycle actions of the Klein bottle group on Z (Theorem 7.12). The uniqueness up to strong cocycle conjugacy of strongly outer actions is also given (Theorem 7.15). These are the first classification results of (cocycle) actions of non-abelian infinite groups on C * -algebras.

Notations
Let A be a C * -algebra. For a, b ∈ A, we mean by [a, b] the commutator ab − ba. The set of tracial states on A is denoted by T (A). For a ∈ A, we define τ (a * a) 1/2 If A is simple and T (A) is non-empty, then · 2 is a norm. For τ ∈ T (A), we let (π τ , H τ ) denote the GNS representation of A associated with τ . For τ ∈ T (A), the dimension function d τ associated with τ is given by d τ (a) = lim n→∞ τ (a 1/n ), for a positive element a ∈ A.
When A is unital, we mean by U (A) the set of all unitaries in A. For u ∈ U (A), the inner automorphism induced by u is written Ad u. An automorphism α ∈ Aut(A) is called outer, when it is not inner. For α ∈ Aut(A), the fixed point subalgebra {a ∈ A | α(a) = a} is written A α . We let T (A) α = {τ ∈ T (A) | τ • α = τ }. A single automorphism is often identified with a Z-action generated by it.
Let A be a separable C * -algebra and let ω ∈ βN \ N be a free ultrafilter. Set We identify A with the C * -subalgebra of A ∞ (resp. A ω ) consisting of equivalence classes of constant sequences. We let and call them the central sequence algebras of A. A sequence (x n ) n ∈ ℓ ∞ (N, A) is called a central sequence if [a, x n ] → 0 as n → ∞ for all a ∈ A. A central sequence is a representative of an element in A ∞ . An ω-central sequence is defined in a similar way. For τ ∈ T (A), we define τ ω ∈ T (A ω ) by where (x n ) n is a representative sequence of x ∈ A ω . When α is an automorphism of A, we can consider its natural extension on A ∞ , A ω , A ∞ and A ω . We denote it by the same symbol α. We let Z denote the Jiang-Su algebra introduced by X. Jiang and H. Su [11]. They proved that any unital endomorphisms of Z are approximately inner and that Z is isomorphic to the infinite tensor product of its replicas. In particular, Z is strongly self-absorbing, cf. [43]. When a C * -algebra A satisfies A ∼ = A ⊗ Z, we say that A absorbs Z tensorially, or A is Z-stable.

Group actions and cocycle conjugacy
We set up some terminologies for group actions. For a discrete group Γ, the neutral element is denoted by 1 ∈ Γ.
Definition 2.1. Let A be a unital C * -algebra and let Γ be a discrete group.
(2) A cocycle action (α, u) is said to be outer if α g is outer for every g ∈ Γ except for the neutral element.
(3) Two cocycle actions (α, u) : Γ A and (β, v) : Γ B are said to be cocycle conjugate if there exist a family of unitaries (w g ) g∈Γ in B and an isomorphism θ : A → B such that θ • α g • θ −1 = Ad w g • β g and θ(u(g, h)) = w g β g (w h )v(g, h)w * gh for every g, h ∈ Γ. Furthermore, when there exists a sequence (x n ) n of unitaries in B such that x n β g (x * n ) → w g as n → ∞ for every g ∈ Γ, we say that (α, u) and (β, v) are strongly cocycle conjugate.
(5) Let α : Γ A be an action. A family of unitaries (x g ) g∈Γ in A is called an α-cocycle if one has x g α g (x h ) = x gh for all g, h ∈ Γ.
A be a cocycle action of a discrete group Γ on a unital C * -algebra A. The twisted crossed product A ⋊ (α,u) Γ is the universal C * -algebra generated by A and a family of unitaries (λ α g ) g∈Γ satisfying λ α g λ α h = u(g, h)λ α gh and λ α g aλ α * g = α g (a) for all g, h ∈ Γ and a ∈ A.

The weak Rohlin property
We recall the definitions of amenable groups and of elementary amenable groups (in the sense of M. M. Day [5]). The cardinality of a set F is written |F |. Definition 2.3. Let Γ be a countable discrete group.
(1) For a finite subset F ⊂ Γ and ε > 0, we say that a finite subset (2) The group Γ is said to be amenable if for any finite subset F ⊂ Γ and ε > 0 there exists an (F, ε)-invariant finite subset K ⊂ Γ.
(3) The class of elementary amenable groups is defined as the smallest family of groups containing all finite groups and all abelian groups, and closed under the processes of taking subgroups, quotients, group extensions and increasing unions.
For countable discrete amenable groups, we introduce the property (Q) as follows.
Definition 2.4. Let Γ be a countable discrete amenable group and let α : Γ R be an outer action on the AFD II 1 -factor R. We say that Γ has the property (Q) if the following holds: For any finite subset F ⊂ Γ and ε > 0, there exists an (F, ε)-invariant finite subset K ⊂ Γ and a sequence of projections (p n ) n in R such that The definition above does not depend on the choice of the outer action α : Γ R, because all outer actions of Γ on R are mutually cocycle conjugate ( [4,13,30,23]). In Proposition 3.3, it will be shown that any elementary amenable group has the property (Q).
Next we introduce the notions of strong outerness, the weak Rohlin property and the tracial Rohlin property for group actions on C * -algebras.
Definition 2.5 ([27, Definition 2.7]). Let A be a unital simple C * -algebra with T (A) non-empty and let Γ be a discrete countable amenable group.
(1) We say that an automorphism α ∈ Aut(A) is not weakly inner for τ ∈ T (A) α if the weak extension of α to an automorphism of π τ (A) ′′ is outer.
(2) A cocycle action (α, u) of Γ on A is said to be strongly outer if α g is not weakly inner for any τ ∈ T (A) αg and g ∈ Γ \ {e}.
(3) We say that a cocycle action (α, u) : Γ A has the weak Rohlin property if for any finite subset F ⊂ Γ and ε > 0 there exist an (F, ε)-invariant finite subset K ⊂ Γ and a central sequence (f n ) n in A such that 0 ≤ f n ≤ 1, When G = Z, in addition to the conditions above, we further impose the restriction that K is of the form {0, 1, . . . , k} for some k ∈ N.
(4) Let A be a unital simple C * -algebra with tracial rank zero. We say that a cocycle action (α, u) : Γ A has the tracial Rohlin property if the central sequence (f n ) n in the definition above can be chosen as a central sequence consisting of projections.
We remark that the definition of the tracial Rohlin property which we have introduced here is slightly stronger than that given in [31,32]

The property (SI)
We give the definition of the property (SI) which plays an important role in Section 4. The original definition given in [38] is slightly different from this version, but they are actually equivalent (see the discussion following [27,Definition 4.1]).
Definition 2.6 ([27, Definition 4.1]). We say that a separable C * -algebra A has the property (SI) when for any central sequences (x n ) n and (y n ) n in A satisfying 0 ≤ x n ≤ 1, 0 ≤ y n ≤ 1, Indeed, these two formulations are both equivalent to the following condition: for any finite subset F ⊂ A, ε > 0 and c > 0, there exist a finite subset G ⊂ A, δ > 0 and m ∈ N such that the following holds. If positive contractions x, y ∈ A satisfy τ (x) < δ and min then one can find s ∈ A such that [s, a] < ε ∀a ∈ F, s * s − x < ε and ys − s < ε.
In [27,Lemma 4.3], we proved that certain Z-stable C * -algebras have the property (SI). Recently we have generalized this result and obtained the following ( [28] (1) If A is Z-stable, then A has the property (SI).
(2) Assume further that A has finitely many extremal tracial states. If A has the property (SI), then A is Z-stable.
In Section 4, we will give a slightly different proof of (2) of Theorem 2.8 (Theorem 4.9).

Approximate representability of cocycle actions
In this subsection, we collect notations and terminologies which will be used in Section 6 and Section 7.
For a Lipschitz continuous function f , we denote its Lipschitz constant by Lip(f ). Let A be a unital C * -algebra. The collection of all continuous bounded affine maps from T (A) to R is denoted by Aff(T (A)). The dimension map For τ ∈ T (A), the de la Harpe-Skandalis determinant ( [8]) associated with τ is written We let log be the standard branch defined on the complement of the negative real axis. If u−1 < 2, then ∆ τ (u) = (2π √ −1) −1 τ (log u)+τ (K 0 (A)). We frequently use the following fact: when u, v ∈ U (A) satisfy u−1 + v−1 < 2, one has τ (log(uv)) = τ (log u)+τ (log v) for any τ ∈ T (A) (see [8,Lemma 1]).
For a homomorphism ϕ between C * -algebras, K 0 (ϕ) and K 1 (ϕ) are the induced homomorphisms on K-groups. Let A and B be unital C * -algebras. Two unital homomorphisms ϕ, ψ : A → B are said to be asymptotically unitarily equivalent if there exists a continuous family of unitaries for all a ∈ A. When there exists a sequence of unitaries (u n ) n∈N in B such that for all a ∈ A, ϕ and ψ are said to be approximately unitarily equivalent. If ϕ and ψ are approximately unitarily equivalent, then K i (ϕ) = K i (ψ) for i = 0, 1. An automorphism α ∈ Aut(A) is said to be asymptotically (resp. approximately) inner if α is asymptotically (resp. approximately) unitarily equivalent to the identity map. As usual, we denote by Inn(A) the set of all approximately inner automorphisms of A. Evidently Inn(A) is a normal subgroup of Aut(A).
Let ϕ ∈ Aut(A) be an automorphism of a unital C * -algebra A. The implementing unitary in the crossed product C * -algebra A⋊ ϕ Z is written λ ϕ . The set of all automorphisms ψ ∈ Aut(A ⋊ ϕ Z) satisfying ψ(A) = A and ψ(λ ϕ )λ ϕ * ∈ A is denoted by Aut T (A ⋊ ϕ Z) (similar definitions are found in [9, Section 2] and [26,Section 2]). An automorphism ψ ∈ Aut T (A ⋊ ϕ Z) is said to be T-asymptotically inner if there exists a continuous family of unitaries for all x ∈ A ⋊ ϕ Z. In an analogous way, one can define T-approximate innerness.
Next, we recall the notion of approximate (resp. asymptotical) representability of group actions. Definition 2.9 ([9, Definition 2.2]). Let Γ be a countable discrete group and let A be a unital C * -algebra. A cocycle action (α, u) : Γ A is said to be approximately representable if there exists a family of unitaries Asymptotical representability is defined in an analogous way.
It is routine to check that approximate (resp. asymptotical) representability is preserved under cocycle conjugacy.
The following proposition lists non-trivial examples of strongly outer, asymptotically representable actions on stably finite C * -algebras. Proposition 2.10. Let α : Γ A be a strongly outer action. Assume that one of the following conditions holds.
(1) Γ = Z, α g is asymptotically inner for every non-trivial g ∈ Z, and A is a unital simple AH algebra with real rank zero, slow dimension growth and finitely many extremal tracial states.
(2) Γ = Z N and A is a UHF algebra of infinite type.
(3) Γ = Z and A is the Jiang-Su algebra.

The weak Rohlin property
In this section we prove Theorem 3.6 and Theorem 3.7. See Definition 2.4 for the definition of the property (Q). (2) Any finite group has the property (Q).
(3) If Γ = n Γ n is an increasing union of countable discrete amenable groups Γ n with the property (Q), then Γ also has the property (Q).
(4) If 1 → N → Γ → H → 1 is a short exact sequence of countable discrete amenable groups and N, H have the property (Q), then Γ also has the property (Q).
(1) follows from [4] and (2) follows from [13]. (3) is obvious. (5) is an immediate consequence of the other assertions, because any countable abelian group is an increasing union of finitely generated abelian groups.
It remains for us to show (4). Let N and H be countable discrete amenable groups with the property (Q) and let 1 → N → Γ → H → 1 be a short exact sequence. Let π : Γ → H be the quotient map and let σ : H → Γ be its right inverse. Let α : Γ R be an outer action of Γ on the AFD II 1 -factor R and let β : H R be an outer action of H on R. Suppose that we are given a finite subset F ⊂ Γ and ε > 0. Without loss of generality, we may assume that there exist finite subsets F 1 ⊂ N and F 2 ⊂ H such that F = F 1 ∪ σ(F 2 ). Since H has the property (Q), we can find an (F 2 , ε/2)-invariant finite subset K 2 ⊂ H and a sequence of projections (q n ) n in R satisfying Define a finite subsetF 1 ⊂ N bỹ Since N has the property (Q), we can find an (F 1 , ε/2)-invariant finite subset K 1 ⊂ N and a sequence of projections (p n ) n in R satisfying Also, for any t ∈ F 2 , h ∈ K 2 ∩ s∈F 2 s −1 K 2 and g ∈ K 1 ∩ s∈F 1 s −1 K 1 , one has Define an outer action γ : Γ R⊗R by γ s = α s ⊗ β π(s) for s ∈ Γ. It is easy to see which completes the proof.
Remark 3.2. Let N be a countable discrete amenable group with the property (Q) and let 1 → N → Γ → Z → 1 be a short exact sequence. Thus, Γ is isomorphic to a semidirect product N ⋊ Z. Letting a ∈ Z denote a generator, we write Γ = {ga i | g ∈ N, i ∈ Z}. From the lemma above, Γ has the property (Q). Moreover, its proof tells us that the 'invariant' set in Γ can be chosen of the form {ga i | g ∈ K, 0 ≤ i ≤ k−1}, where K ⊂ N is an 'invariant' set. More precisely, we get the following. Let α : Γ R be an outer action of Γ on the hyperfinite II 1 -factor R. Then, for any finite subset F ⊂ N , ε > 0 and k ∈ N, there exists an (F, ε)-invariant finite subset K ⊂ N and a sequence (p n ) n of projections in R such that Proof. This follows from the lemma above and [2, Proposition 2.2 (b)].
In order to state and prove Proposition 3.5 below, we need to recall U. Haagerup's result [7,Theorem 3.1], which says that any nuclear C * -algebra has a virtual diagonal in the sense of [12]. In particular, any nuclear C * -algebra is amenable. Here we quote the following theorem from [6], which is an interpretation of [7,Theorem 3.1]. See also [19,Lemma 4.2]. . Let A be a unital nuclear C * -algebra. For any finite subset F ⊂ A and ε > 0, there exist w 1 , w 2 , . . . , w m ∈ A such that The following is a variant of [39, Lemma 2.1]. We include the proof for completeness.
Proposition 3.5. Let A be a unital nuclear C * -algebra. For any finite subset F ⊂ A and ε > 0, there exist a finite subset G ⊂ A and δ > 0 such that the following hold. If x ∈ A is a positive element such that then there exists a positive element y ∈ A such that y ≤ 1, x − y 2 < ε and [y, a] < ε ∀a ∈ F.
Proof. By applying the theorem above to F ⊂ A and ε > 0, we get Furthermore, for any a ∈ F , We are now ready to show the equivalence of strong outerness and the weak Rohlin property (see Definition 2.5 for the definitions) for cocycle actions of groups with the property (Q).
Theorem 3.6. Let A be a unital, simple, separable, nuclear, stably finite, infinite dimensional C * -algebra with finitely many extremal tracial states and let Γ be a countable discrete amenable group with the property (Q). For a cocycle action (α, u) of Γ on A, the following conditions are equivalent.
Proof. The implication (2)⇒(1) was proved in [27,Remark 2.8]. We show the other implication. Suppose that (α, u) : Γ A is strongly outer. Let E be the set of extremal tracial states on A. For τ ∈ E, it is well-known that π τ (A) ′′ is the AFD II 1 -factor. Set π = τ ∈E π τ and M = π(A) ′′ = τ ∈E π τ (A) ′′ . We identify A with π(A) and omit π. The cocycle action (α, u) on A naturally extends to the cocycle action (ᾱ, u) on M . Note that, for a bounded sequence (x n ) n in M , x n converges to zero in the strong operator topology if and only if x n 2 converges to zero.
Suppose that we are given a finite subset F ⊂ Γ and ε > 0. Let R be the AFD II 1 -factor and let γ : Γ R be an outer action. Since Γ has the property (Q), there exist an (F, ε)-invariant finite subset K ⊂ Γ and a sequence of projections (p n ) n in R such that First, we would like to show that there exists a sequence of projections (q n ) n in M such that Put Γ e = {g ∈ Γ |ᾱ g (e) = e}. Then Γ e is a subgroup of Γ and Γ = h∈He hΓ e holds for every e ∈ E 0 . Let (ᾱ e , u e ) be the restriction of (ᾱ, u) to Γ e and M e, i.e.ᾱ e g = α g |M e, u e (g, h) = u(g, h)e for g, h ∈ Γ e . Since (α, u) is strongly outer, the cocycle action (ᾱ e , u e ) of Γ e on M e is outer. Likewise, we let γ e denote the restriction of γ : Γ R to Γ e . It follows from [30, Chapter 1] (see also [23,Theorem 2.3]) that (ᾱ e , u e ) is cocycle conjugate to (ᾱ e ⊗ γ e , u e ⊗ 1). Hence, there exist an isomorphism π e : M e⊗R → M e and a family of unitaries (v e g ) g∈Γe in M e such that We define an isomorphism π : M⊗R → M by Let q n = π(1 ⊗ p n ). We would like to show ᾱ g (q n ) − π(1 ⊗ γ g (p n )) 2 → 0 for every g ∈ Γ as n → ∞. Once this is done, it is evident that (q n ) n is the desired sequence of projections. We have Take g ∈ Γ. For each h ∈ H e there exists k ∈ H e such that gh ∈ kΓ e , and the map h → k is a bijection. Then one can verify By Kaplansky's density theorem, we may replace (q n ) n with a sequence (x n ) n of positive contractions in A. Thanks to Proposition 3.5, we can further replace (x n ) n with a sequence (y n ) n of positive contractions in A satisfying [a, y n ] → 0, α g (y n )α h (y n ) 2 → 0 and τ (y n ) → 1/|K| for all a ∈ A, g, h ∈ K with g = h and τ ∈ T (A) as n → ∞. Finally, by the same way as [27, Proposition 3.3], we obtain a sequence (z n ) n of positive contractions in A satisfying for all a ∈ A, g, h ∈ K with g = h and τ ∈ T (A) as n → ∞. Consequently (α, u) has the weak Rohlin property.
For a C * -algebra A with tracial rank zero, we can prove that strong outerness is equivalent to the tracial Rohlin property in the sense of Definition 2.5 (4).
Theorem 3.7. Let A be a unital simple separable C * -algebra with tracial rank zero and with finitely many extremal tracial states. Let Γ be a countable discrete amenable group with the property (Q). For a cocycle action (α, u) of Γ on A, the following conditions are equivalent.
Proof. The implication (2)⇒(1) can be shown in a similar fashion to [27, Remark 2.8]. We prove (1)⇒ (2). Suppose that (α, u) : Γ A is strongly outer. Define M and (ᾱ, u) : Γ M in the same way as Theorem 3.6. It is well-known that M is isomorphic to a finite direct sum of the hyperfinite II 1 -factor (see [31,Lemma 2.16], [27, Remark 2.6 (8)]). Exactly the same argument as Theorem 3.6 shows that there exists a sequence of projections (q n ) n in M such that for all x ∈ M , g, h ∈ K with g = h and τ ∈ T (M ) as n → ∞. By [31, Lemma 2.15], we may replace q n with projections e n in A, that is, there exists a sequence of projections for all a ∈ A, g, h ∈ K with g = h and τ ∈ T (A) as n → ∞. Thus (α, u) has the tracial Rohlin property.

Z-stability of crossed products
In this section, we prove Theorem 4.9. We also prove that the fixed point subalgebra of the central sequence algebra has strict comparison (Theorem 4.8).

Fixed point subalgebras of central sequence algebras
Throughout this subsection, we let A be a unital, simple, separable, nuclear, stably finite, infinite dimensional C * -algebra with finitely many extremal tracial states. We also fix a countable discrete amenable group Γ and a cocycle action (α, u) : Γ A with the weak Rohlin property. Note that we allow Γ to be trivial.
Let E be the set of extremal tracial states on A. Set π = τ ∈E π τ and M = π(A) ′′ = τ ∈E π τ (A) ′′ . It is well-known that π τ (A) ′′ is the AFD II 1 -factor R for τ ∈ E. We identify A with π(A) and omit π. It is clear that T (A) is identified with T (M ). We let be the central sequence algebra of R in the sense of von Neumann algebra, where R is identified with the subalgebra consisting of equivalence classes of constant sequences. It is also well-known that R is a II 1 -factor (see [41,Theorem XIV.4.18] for example). Similarly, we define M by Clearly M is isomorphic to τ ∈E R. Put J = Ker ρ. For an ω-central sequence (x n ) n in A, its image in A ω belongs to J if and only if x n 2 → 0 as n → ω. In other words, for x ∈ A ω , x belongs to J if and only if τ ω (x * x) = 0 for any τ ∈ T (A) (see Section 2.1 for the definition of τ ω ). The action α globally preserves J, and so the fixed point subalgebra J α is well-defined. Throughout this subsection, we keep the above notations.
The following lemma is folklore among experts. For the reader's convenience, we include a proof here.
R be an outer cocycle action of Γ on the AFD II 1 -factor R. Then R γ is a II 1 -factor.
Hence there exists a unital homomorphism ϕ : R → R γ (or one may use [30,Lemma 8.3], which says that R γ is of type II 1 ). Let τ be the unique trace on R. Suppose that p is a projection belonging to the center of R γ . Take a projection q ∈ R such that τ (ϕ(q)) = τ (p). There exists a unitary x ∈ R such that xpx * = ϕ(q), because R is a II 1 -factor. Then (xγ g (x * )) g is a γ-cocycle in R∩{ϕ(q)} ′ . By the cohomology vanishing theorem [30, Proposition 7.2], we can find a unitary y ∈ R ∩ {ϕ(q)} ′ such that xγ g (x * ) = yγ g (y * ) for all g ∈ Γ. Therefore y * xpx * y = ϕ(q) and y * x ∈ R γ . Since p is in the center of R γ , we obtain p = ϕ(q). This means p = 0 or p = 1, and whence R γ is a II 1 -factor.
M e i be the restriction of (ᾱ, u) to Γ i and M e i . It is not so hard to see Since (ᾱ (i) , u (i) ) : Γ i M e i is an outer cocycle action of a countable discrete amenable group on the AFD II 1 -factor, by the lemma above, the fixed point algebra (Me i )ᾱ (i) is a II 1 -factor, which completes the proof.
(2) The restriction of ρ to (A ω ) α induces the following short exact sequence: sequence of x consisting of positive contractions. By Kaplansky's density theorem, we may assume that x n is in A. We choose an increasing sequence (F m ) m of finite subsets of A whose union is dense in A. Applying Proposition 3.5 to F m and 1/m, we get a finite subset G m ⊂ A and δ m > 0. Define Then Ω m belongs to ω because x is in M. For each n ∈ Ω m \Ω m+1 , by means of Proposition 3.5, we obtain a positive contraction y n ∈ A such that x n − y n 2 < 1/m and [y n , a] < 1/m ∀a ∈ F m .
Put y n = 0 for n ∈ N \ Ω 1 . It is easy to see that (y n ) n is an ω-central sequence. Let y ∈ A ω be the image of (y n ) n . Since x n − y n 2 → 0 as n → ω, we have ρ(y) = x which completes the proof.
Take a positive element x ∈ Mᾱ. We would like to prove that ρ((A ω ) α ) contains x. Since ρ is surjective, there exists a ∈ A ω such that ρ(a) = x. Choose a finite subset F ⊂ Γ, ε > 0 and m ∈ N arbitrarily. Choose δ > 0 so that Since (α, u) has the weak Rohlin property, there exists an (F, δ)-invariant finite subset K ⊂ Γ and a positive contraction f ∈ A ω such that τ ω (f ) = |K| −1 and α g (f )α h (f ) = 0 for all τ ∈ T (A) and g, h ∈ K with g = h. Evidently (ρ(α g (f ))) g∈K forms a partition of unity consisting of projections in M. By the standard reindexation trick, we may assume that f commutes with a. Define a function ℓ : K → N by It is easy to see that c − α g (c) is less than 1/m for all g ∈ F . Also, we have Since F , ε and m were arbitrary, the standard trick on central sequences implies that there exists a positive contraction c ∈ (A ω ) α such that a − c ∈ J, which completes the proof. Proof. Take x ∈ J α . Let (x n ) n be a representative sequence of x. We choose an increasing sequence (F m ) m of finite subsets of A whose union is dense in Put y n = x * n x n + x n x * n . It is easy to see that (y n ) n is an ω-central sequence of positive elements satisfying τ (y n ) → 0 and α g (y n ) − y n → 0 as n → ω for any τ ∈ T (A) and g ∈ Γ.
Let Ω m be the set of all natural numbers n ≥ m such that Then (e n ) n is an ω-central sequence satisfying τ (e n ) → 0 and α g (e n ) − e n → 0 as n → ω for all τ ∈ T (A) and g ∈ Γ. Thus, the image e ∈ A ω of (e n ) n is in J α . Moreover, for every n ∈ Ω m \ Ω m+1 , we have Hence x = ex. Likewise, we obtain x = xe, which completes the proof.
In the following proposition, we consider an equivariant version of the property (SI). Notice that we need not assume that A has finitely many extremal tracial states for this proposition. See Definition 2.6 for the definition of the property (SI).
there exists an ω-central sequence (s n ) n in A such that lim n→ω s * n s n − x n = 0, lim n→ω y n s n − s n = 0, Proof. The assertion is actually contained in the proof of [27, Theorem 4.7], but we include a proof here for completeness. We first note that, by Remark 2.7, the assertion is equivalent to the property (SI) itself, when Γ is trivial. Let us consider the general case. Let (x n ) n and (y n ) n be as in the statement. Put Take a finite subset F ⊂ Γ and ε > 0 arbitrarily. Since (α, u) : Γ A has the weak Rohlin property, we can find an (F, ε)-invariant finite subset K ⊂ Γ and a central sequence (f l ) l of positive contractions satisfying We claim that there exists an ω-central sequence (ỹ n ) n of positive contractions in A such that To this end, it suffices to show that for any finite subset G ⊂ A, δ > 0 and m ∈ N, there exists a sequence (ỹ n ) n of positive contractions in A such thatỹ Setỹ n = y We also have lim for all a ∈ G. Thus, the proof of the claim is completed. Because A has the property (SI) (see also Remark 2.7), there exists an ω-central sequence (r n ) n in A such that r * n r n − x n → 0 andỹ n r n − r n → 0 as n → ω. Define s n ∈ A by Then it is easy to see s * n s n − x n → 0 and y n s n − s n → 0 as n → ω. Besides, for any h ∈ F , Since F ⊂ Γ and ε > 0 were arbitrary, the proof is completed.
Proof. Let x ∈ J α be a positive contraction. It suffices to show τ (x) = 0 for all τ ∈ T ((A ω ) α ). By Lemma 4.4, we can find a positive contraction e 1 ∈ J α such that xe 1 = x. By Proposition 4.5, there exists s 1 ∈ J α such that s * 1 s 1 = x and (1 − e 1 )s 1 = s 1 . Applying Lemma 4.4 to the positive contraction x + s 1 s * 1 , we obtain a positive contraction e 2 ∈ J α such that ( Lemma 4.7. Suppose that A has the property (SI).
(3) If A has a unique tracial state, then so does (A ω ) α . Proof.
The following proposition says that (A ω ) α has strict comparison in a certain sense. We remark that the condition , thanks to the lemma above.
Proof. For a subset X ⊂ R, we let 1 X be the characteristic function of X. Notice that we keep the identification between T (A), T (M ) and T (M). By τ • ρ = τ ω , we have where the support projection 1 (0,∞) (ρ(a)) is well-defined because Mᾱ is a von Neumann algebra. Similarly we get  1]) is the identity map. By the usual fast reindexation trick, we may and do assume that the range ofφ commutes with b. Let p ∈ M k be a rank one projection. Since ρ(b) commutes with elements of ϕ(M k ), we have and for all τ ∈ T (M). As T (M) has finitely many extremal points, there exists ε > 0 such that min Then Thus, a − rbr * is a positive element of J α . On the other hand, one has This, together with (4.1), implies Therefore, by appealing to Proposition 4.5, we can find s ∈ (A ω ) α such that Clearly rbt * = 0, and so we get

Z-stability of crossed products
In this subsection, we state direct consequences of the results obtained in Section 3 and Section 4.1. Throughout this subsection, we let A be a unital, simple, separable, nuclear, stably finite, infinite dimensional C * -algebra with finitely many extremal tracial states. See Definition 2.1 and Definition 2.2 for the definitions of (strong) cocycle conjugacy and twisted crossed products. We note that the following theorem contains Theorem 2.8 (2) as a special case (namely Γ = {1}). Theorem 4.9. Suppose that A has the property (SI). Let (α, u) : Γ A be a cocycle action of a countable discrete amenable group Γ with the weak Rohlin property. Then there exists a unital homomorphism from Z to (A ω ) α , and hence (α, u) : Γ A is strongly cocycle conjugate to (α⊗id, u⊗1) : Γ A⊗Z. In particular, the twisted crossed product A⋊ (α,u) Γ is Z-stable.
Proof. We first prove the existence of a unital homomorphism from Z to (A ω ) α . As shown in [11], Z is an inductive limit of the prime dimension drop algebras I(k, k+1)'s. Hence, in the light of [44, Proposition 2.2], it suffices to construct a unital homomorphism from I(k, k+1) to (A ω ) α . Let (e i,j ) i,j be a system of matrix units for M k . As in the proof of Proposition 4.8, we can find a homomorphismφ : C 0 ((0, 1]) ⊗ M k → (A ω ) α such that τ ω (φ(ι m ⊗ e 1,1 )) = 1/k for every τ ∈ T (A) and m ∈ N, where ι ∈ C 0 ((0, 1]) is the identity map. Clearlỹ By Lemma 4.5, there exists s ∈ (A ω ) α such that It follows from [38, Proposition 2.1] that there exists a unital homomorphism from I(k, k+1) to (A ω ) α . Once we get a unital embedding Z → (A ω ) α , strong cocycle conjugacy between (α, u) and (α ⊗ id, u ⊗ 1) follows by essentially the same method as [34, Theorem 7.2.2]. Then we can conclude that the twisted crossed product A ⋊ (α,u) Γ is Z-stable.
Corollary 4.10. Let Γ be a countable discrete amenable group with the property (Q). Suppose that A has the property (SI). Let (α, u) : Γ A be a strongly outer cocycle action. Then there exists a unital homomorphism from Z to (A ω ) α , and hence (α, u) is strongly cocycle conjugate to (α ⊗ id, u ⊗ 1) : Γ A ⊗ Z. In particular, the twisted crossed product A ⋊ (α,u) Γ is Z-stable.
Proof. By Theorem 3.6, the cocycle action (α, u) has the weak Rohlin property. Then the conclusion follows from the theorem above. A be a strongly outer cocycle action. Then there exists a unital homomorphism from Z to (A ω ) α , and hence (α, u) is strongly cocycle conjugate to (α ⊗ id, u ⊗ 1) : Γ A ⊗ Z. In particular, the twisted crossed product A ⋊ (α,u) Γ is Z-stable.
Proof. By Proposition 3.3, Γ has the property (Q). Then the conclusion follows from the corollary above.
Remark 4.12. In the corollaries above, the assumption of the property (SI) can be replaced with Z-stability of A, thanks to Theorem 2.8 (1).

A remark on central sequence algebras
In this section, we show that for any unital separable nuclear C * -algebra A which is not of type I, the central sequence algebra A ω has a subquotient which is isomorphic to a II 1 -factor (Theorem 5.1). This may be regarded as a variant of Glimm's theorem, which says that if a separable C * -algebra is not of type I, then it has a subquotient isomorphic to a UHF algebra. Let M be a von Neumann algebra with separable predual M * . We recall the definition of the central sequence algebra of M in the sense of von Neumann algebra from [3, Section II] (see also [41,Chapter XIV §4]).
For a normal positive functional ϕ ∈ M * , we let Theorem 5.1. Let A be a unital separable nuclear C * -algebra which is not of type I. Then there exists a subquotient of A ω = A ω ∩ A ′ which is isomorphic to a II 1 -factor.
Proof. Since A is not of type I, by Glimm's theorem, there exist a type II factor M with separable predual and a unital homomorphism π : A → M such that π(A) is * -strongly dense in M . Moreover, M is hyperfinite, because A is assumed to be nuclear. Let C, I and M be as above. We let B ⊂ ℓ ∞ (N, A) be the set of all ω-central sequences (a n ) n such that the sequence (π(a n )) n in M is strongly ω-central. It is routine to check that B is a unital C * -subalgebra of ℓ ∞ (N, A) and that B contains c ω (A). Let π : ℓ ∞ (N, A) → ℓ ∞ (N, M ) be the homomorphism which is naturally induced by π. Theñ π(B) is contained in C andπ(c ω (A)) is contained in I. Henceπ|B induces a unital homomorphism from B/c ω (A) ⊂ A ω to M. As mentioned above, M is known to be a II 1 -factor. Therefore, it remains for us to show that the homomorphism B/c ω (A) → M is surjective. The proof is essentially the same as [39, Lemma 2.1], but we include it for completeness.
Take a strongly ω-central sequence (x n ) n ∈ C arbitrarily. We assume x n ≤ 1. Let ψ ∈ M * be a normal positive faithful functional. Then · # ψ is a norm and the * -strong topology agrees with the topology induced by the norm · # ψ on any bounded subset of M (see [40,Proposition III.5.3] for example). Hence it suffices to show that there exists an ω-central sequence (a n ) n in A such that x n − π(a n ) # ψ → 0 as n → ω. By Kaplansky's density theorem (see [40,Theorem II.4.8] for example), we can find (b n ) n ∈ ℓ ∞ (N, A) such that b n ≤ 1 and x n − π(b n ) # ψ → 0 as n → ∞. We choose an increasing sequence (F m ) m of finite subsets of A whose union is dense in A. Since A is nuclear, we can apply Theorem 3.4 to F m and 1/m, and get a finite subset G m ⊂ A. Define As (x n ) n is strongly ω-central, (π(b n )) n is also strongly ω-central. Therefore, by [41, Lemma XIV.3.4 (i)], [π(b n ), π(w)] tends to zero in the * -strong topology as n → ω for every w ∈ A. It follows that Ω m belongs to ω. For each n ∈ Ω m \ Ω m+1 , we define a n ∈ A by a n = w∈Gm wb n w * .
By the choice of G m , we have [a n , c] < 1/m for all c ∈ F m , that is, (a n ) n is an ω-central sequence. Besides, and so x n − π(a n ) # ψ → 0 as n → ω.
Remark 5.2. In the theorem above, suppose that M is the hyperfinite II 1 -factor. Then, by [41, Lemma XIV.3.4 (ii)], we can see that for any ω-central sequence (a n ) n in A, the sequence (π(a n )) n in M is strongly ω-central. Thus, B above equals A ω = A ω ∩ A ′ , and whence there exists a unital homomorphism from A ω to the II 1 -factor M.

Actions of the Klein bottle group I
In this section, we would like to prove that all strongly outer cocycle actions of the Klein bottle group on a UHF algebra are mutually cocycle conjugate (Theorem 6.16). We note that some of the ideas and techniques used in this section are borrowed from [10].

The OrderExt invariant
In this subsection, we would like to collect some basic facts on the OrderExt group for later use. The OrderExt invariant was introduced by A. Kishimoto and A. Kumjian [18] in order to determine when an approximately inner automorphism of a unital simple AT algebra with real rank zero is asymptotically inner. In the later subsections, we use this invariant for classification of certain group actions on AF algebras. We begin with the definition of the OrderExt group ([18, Section 2]). Let G 0 , G 1 , F be abelian groups and let D : G 0 → F be a homomorphism. When is exact, R is in Hom(E ξ , F ) and R • ι = D, the pair (ξ, R) is called an order-extension. Two order-extensions (ξ, R) and (ξ ′ , R ′ ) are equivalent if there exists an isomorphism θ : E ξ → E ξ ′ such that R = R ′ • θ and is commutative. Then OrderExt(G 1 , G 0 , D) consists of equivalence classes of all orderextensions. As shown in [18], OrderExt(G 1 , G 0 , D) is equipped with an abelian group structure. The map sending (ξ, R) to ξ induces a homomorphism from OrderExt(G 1 , G 0 , D) onto Ext(G 1 , G 0 ). When (ξ, R) is an order-extension, there exists r ∈ Hom(G 1 , F/ Im D) such that the following diagram is commutative: Clearly r depends only on the OrderExt class of (ξ, R) and the map (ξ, R) → r gives rise to a homomorphism. We denote it by ρ : OrderExt(G 1 , G 0 , D) → Hom(G 1 , F/ Im D). Lemma 6.1. Let G 0 , G 1 , F be abelian groups and let D : G 0 → F be a homomorphism. If D is injective, then the homomorphism ρ : OrderExt(G 1 , G 0 , D) → Hom(G 1 , F/ Im D) is an isomorphism.
Proof. Suppose that r ∈ Hom(G 1 , F/ Im D) is given. Define ξ ∈ Ext(G 1 , G 0 ) by We would like to recall the main result of [18]. Let B be a unital simple AT algebra with real rank zero. As described in [18], there exist natural homomorphisms The following is the main result of [18].

Theorem 6.2 ([18, Theorem 4.4]).
Suppose that B is a unital simple AT algebra with real rank zero. Then the homomorphism is surjective and its kernel equals the set of all asymptotically inner automorphisms of B.
In the rest of this subsection, we let A be a unital simple AF algebra and let ϕ ∈ Inn(A) be an approximately inner automorphism such that ϕ n is not weakly inner for every n ∈ N. By the Pimsner-Voiculescu exact sequence, (K 0 (A⋊ ϕ Z), K 1 (A⋊ ϕ Z)) are naturally identified with (K 0 (A), K 0 (A)) (see [24,Section 6]). The space of tracial states T (A ⋊ ϕ Z) is also identified with T (A). These identifications induce the natural isomorphism From now on we will freely identify these two OrderExt groups. Lemma 6.3. For any approximately inner automorphism ψ ∈ Aut T (A ⋊ ϕ Z), one has η 1 (ψ) = 0.
Proof. This follows from the proof of [24, Lemma 6.3].
From Theorem 6.2 and the lemma above, we obtain the following. Proposition 6.4. Suppose that A ⋊ ϕ Z is a unital simple AT algebra with real rank zero. Let ψ ∈ Aut T (A ⋊ ϕ Z) be an approximately inner automorphism. Ifη 0 (ψ) = 0, then ψ is asymptotically inner.
Lemma 6.5. Let ρ : OrderExt(K 0 (A), K 0 (A), D A ) → Hom(K 0 (A), Aff(T (A))/ Im D A ) be the homomorphism described above. For every t ∈ R and x ∈ K 0 (A), one has Proof. Let t ∈ R and let p ∈ A be a projection. It suffices to show the equation for x = [p]. Under the identification between K 0 (A) and Then f is a unitary in the mapping torus ofφ t and We conclude this subsection by the following lemma, which will be used in Section 6.3.

Cocycle actions of Z 2 on UHF algebras
In this subsection, we would like to summarize the results obtained in [24, Section 6] and generalize them to cocycle actions of Z 2 . Throughout this subsection, we write Z 2 = a, b | bab −1 = a . For a cocycle action (α, u) : Z 2 A, we puť A be a strongly outer and locally KK-trivial (i.e. K 0 (α g ) = id) cocycle action of Z 2 on a unital simple AF algebra A. The automorphism α b ∈ Aut(A) extends toα b ∈ Aut T (A ⋊ αa Z) by lettingα b (x) = α b (x) for all x ∈ A andα b (λ αa ) =ǔλ αa . As mentioned in the previous subsection, by the Pimsner-Voiculescu exact sequence, (K 0 (A ⋊ αa Z), K 1 (A ⋊ αa Z)) are naturally identified with (K 0 (A), K 0 (A)). The space of tracial states T (A ⋊ αa Z) is also identified with T (A). Under these identifications, we regardη 0 (α b ) as an element in OrderExt(K 0 (A), K 0 (A), D A ), and define our invariant c(α, u) ∈ OrderExt(K 0 (A), K 0 (A), D A ) by  Proof. There exists a family of unitaries (w g ) g∈Z 2 in A such that θ • α g • θ −1 = Ad w g • β g and θ(u(g, h)) = w g β g (w h )v(g, h)w * gh hold for every g, h ∈ Z 2 . Then θ extends to an isomorphism from A ⋊ αa Z to A ⋊ βa Z by letting θ(λ αa ) = w a λ βa . Define an isomorphismθ :
In the same way as [24, Theorem 6.6], we can prove the following. Theorem 6.9. Let (α, u), (β, v) : Z 2 A be strongly outer, locally KK-trivial cocycle actions of Z 2 on a unital simple AF algebra A with finitely many extremal tracial states. The following are equivalent.
Proof. We give only sketchy arguments as the proof of [24, Theorem 6.6] applies almost verbatim. (2)⇒(1) was shown in the lemma above without assuming that A has finitely many extremal traces. Let us consider the other implication (1)⇒(2). It is well-known (see [24,Section 4] for example) that A ⋊ αa Z and A ⋊ βa Z are unital simple AT algebras with real rank zero. By [24,Theorem 4.8,4.9], we may assume that there exists w ∈ U (A) such that β a = Ad w • α a . Define an isomorphism θ : A ⋊ βa Z → A ⋊ αa Z by θ(x) = x for x ∈ A and θ(λ βa ) = wλ αa . We can observẽ It follows from Proposition 6.4 thatα b and θ •β b • θ −1 are asymptotically unitarily equivalent. Then [24, Theorem 6.1] applies and yields an approximately inner automorphism µ ∈ Aut T (A ⋊ αa Z) and t ∈ U (A) such that µ|A is in Inn(A) and (Actually, in [24, Theorem 6.1], A is assumed to have a unique trace. We can weaken this hypothesis by using Theorem 3.7 instead of [24,Lemma 5.1]. See also Theorem 6.14.) Let s = µ(λ αa )(λ αa ) * ∈ U (A). One can deduce Moreover, we can verify which implies that (α, u) and (β, v) are cocycle conjugate.
Let us consider the case that A is a UHF algebra. As in [14,Section 5], for every prime number p, we define In [14], it was shown that there exists a bijective correspondence between the set of (strong) cocycle conjugacy classes of strongly outer Z 2 -actions on A and the abelian group As mentioned in [24,Remark 6.7], it is easy to see that this group is isomorphic to It is also known that this group is isomorphic to the fundamental group π 1 (Aut(A)) ( [42]).
The following theorem is a strengthened version of the main result of [14] (see also [27,Theorem 8.6]). Theorem 6.10. Assume that A is a UHF algebra. There exist bijective correspondences between the following three sets.
(1) Cocycle conjugacy classes of strongly outer cocycle actions of Z 2 on A.
We will use the following lemma in the next subsection. Lemma 6.11. Let A be a UHF algebra and let ϕ be an automorphism of A such that ϕ n is not weakly inner for every n ∈ N. For any c ∈ OrderExt(K 0 (A), Proof. From the theorem above, there exists a strongly outer cocycle action (α, u) : Z 2 A such that c(α, u) = c. By [15,Theorem 1.3], we may assume that there exists w ∈ U (A) such that α a = Ad w • ϕ. Define an isomorphism θ : A ⋊ αa Z → A ⋊ ϕ Z by θ(x) = x for x ∈ A and θ(λ αa ) = wλ ϕ . Then θ •α b • θ −1 is in Aut T (A ⋊ ϕ Z) and

Actions of the Klein bottle group on UHF algebras
Throughout this subsection, we let Γ = a, b | bab −1 = a −1 . The group Γ is isomorphic to the fundamental group of the Klein bottle and is called the Klein bottle group. In this subsection, we will show that all strongly outer cocycle actions of Γ on a UHF algebra are cocycle conjugate to each other.
First, we would like to show the existence of strongly outer, approximately representable Γ-actions on the Jiang-Su algebra Z. See Definition 2.9 for the definition of approximate representability. Lemma 6.12. There exists a homomorphism π : Γ → U (C([0, 1]) ⊗ M 2 ) such that π(g) is not a scalar multiple of 1 for any g ∈ Γ \ {1} and π(g)(0) = 1 for all g ∈ Γ.
Proposition 6.13. There exists a strongly outer, approximately representable action α : Γ Z. In particular, for any unital Z-stable C * -algebra A, there exists a strongly outer, approximately representable action α : Γ A.
Proof. Since the C * -algebra is embeddable to Z, by the lemma above, there exists a homomorphism π : Γ → U (Z) such that Ad π(g) = id for all g ∈ Γ \ {1}. Define an action α of Γ on the infinite tensor product of Z by α g = k∈N Ad π(g).
It is clear that α is strongly outer and approximately representable. For any unital C *algebra A, the action id ⊗α : Γ A⊗Z is strongly outer and approximately representable.
The following is a Rohlin type theorem for Γ-actions on unital simple AF algebras. The proof is almost the same as [29,Theorem 3] and [24,Theorem 5.5]. In addition, by the proof of Lemma 3.1 (4), we may assume that K is of the form for some m 1 , m 2 ∈ N. Since α r a is approximately inner, when m 1 is large enough, we can construct a central sequence of projections (f ′ n ) n in A such that Next we would like to establish a Γ-action version of [24, Theorem 6.1] (or [9, Theorem 4.11]). We note that the idea of the proof is taken from [10].
Theorem 6.15. Let A be a unital simple AF algebra with finitely many extremal tracial states and let ϕ ∈ Inn(A). For i = 1, 2, assume that ψ i ∈ Aut(A ⋊ ϕ Z) satisfies the following conditions.
(3) ψ r i |A is approximately inner for some r ∈ N.
Assume further that ψ 1 • ψ −1 2 is asymptotically inner. Then there exists an approximately inner automorphism µ ∈ Aut T (A ⋊ ϕ Z) and a unitary v ∈ U (A) such that µ|A is also approximately inner and Proof. We would like to apply the Evans-Kishimoto intertwining argument to ψ 1 and ψ 2 . It is easy to see that . By Proposition 2.10 (1), (the Z-action generated by) ϕ is asymptotically representable. Then [9,Theorem 4.8] implies that ψ 1 •ψ −1 2 is T-asymptotically inner. For each i = 1, 2, the cocycle action of Γ generated by ϕ and ψ i |A is strongly outer. It follows from the theorem above that we can find Rohlin projections for ψ i in the fixed point algebra We are now ready to prove the uniqueness of strongly outer cocycle actions of the Klein bottle group on UHF algebras. Theorem 6.16. Let (α, u) and (β, v) be strongly outer cocycle actions of Γ on a UHF algebra A. Then (α, u) and (β, v) are cocycle conjugate.
Proof. This immediately follows from the theorem above and Proposition 6.13. Remark 6.18. Let (α, u) : Γ A be a strongly outer cocycle action on a UHF algebra A. By [27, Corollary 5.9], A ⋊ (α,u) Γ ∼ = (A ⋊ αa Z) ⋊α b Z is a unital simple AH algebra with real rank zero and slow dimension growth. By means of the Pimsner-Voiculescu exact sequence, we get Note that ζ A (2) = ∞ if and only if A absorbs the UHF algebra of type 2 ∞ tensorially.

Actions of the Klein bottle group II
Throughout this section, we let Γ denote the Klein bottle group a, b | bab −1 = a −1 . In this section we will prove that all strongly outer cocycle actions of Γ on the Jiang-Su algebra are cocycle conjugate to each other (Theorem 7.12).

Homotopy of unitaries
In this subsection we establish Lemma 7.1 and Lemma 7.3, which will be used in the next subsection to obtain certain cohomology vanishing type results.
The following lemma is a variant of [24,Lemma 4.10] or [27,Lemma 6.3], and is essentially contained in the proof of [38,Theorem 5.3]. But we include a proof here for completeness.
Lemma 7.1. Let B be a UHF algebra with a unique trace τ and let ϕ be an automorphism of B such that ϕ n is not weakly inner for all n ∈ N. For any finite subset F ⊂ B and ε > 0, there exist a finite subset G ⊂ B and δ > 0 such that the following holds. If then we can find a continuous map z : In addition, if F = ∅, then G = ∅ is possible.
Proof. The case F = G = ∅ is contained in [27,Lemma 8.1], and so we treat the general case here. Let (d n ) n be a sequence of natural numbers such that d n < d n+1 and We regard B n = M d 1 ⊗ · · · ⊗ M dn as a subalgebra of B. Let y n ∈ B n ∩ B ′ n−1 be a unitary whose spectrum is {ζ k n | k = 1, 2, . . . , d n }, where ζ n = exp(2π √ −1/d n ). Define an automorphism σ of B by σ = lim n→∞ Ad(y 1 y 2 . . . y n ). Then σ m is not weakly inner for all m ∈ N. By [15, Theorem 1.3, 1.4], the Z-actions generated by ϕ and σ are strongly cocycle conjugate. Hence it suffices to show the assertion for σ.
Suppose that we are given F ⊂ B and ε > 0. Without loss of generality, we may assume that there exists n ∈ N such that F is contained in the unit ball of B n . Applying [14,Lemma 4.2] to ε/2, we obtain a positive real number δ 1 > 0. We may assume δ 1 is less than min{2, ε}. Choose a finite subset G ⊂ B and δ 2 > 0 so that if u ∈ U (B) satisfies [u, c] < δ 2 for all c ∈ G, then there exists u 0 ∈ U (B ∩ B ′ n ) such that u − u 0 < δ 1 /4.
In order to prove Lemma 7.3, we need the following lemma. The proof uses [21].
Lemma 7.2. Let C be a unital simple AT algebra with a unique trace τ . Assume that K i (C) is isomorphic to a non-zero subgroup of Q for each i = 0, 1. Let w ∈ U (C) be such that [w] = 0. Suppose that (x n ) n is a central sequence of unitaries in C such that [x n ] = 0 and τ (log(x n wx * n w * )) = 0 for all sufficiently large n. Then there exists a central sequence (z n ) n of unitaries in C([0, 1], C) such that z n (0) = 1, z n (1) = x n and Lip(z n ) < 7 for all sufficiently large n.
• Let p ∈ C be a projection. Then for all sufficiently large n, the K 1 -class of x n p+1−p is zero.
• Let u ∈ C be a unitary. Then for all sufficiently large n, Bott(u, x n ) equals zero.
Once this is done, one can construct (z n ) n as follows. Let F ⊂ C be a finite subset and let ε > 0. By applying [21,Corollary 17.9] (see also [25,Theorem 5.4]), we get a finite subset G ⊂ C, δ > 0, a finite subset P of projections in C and a finite subset U of unitaries in C. Notice that we do not need to work with the entire K-group K(C) of C, because K i (C) are torsion free. For all sufficiently large n, [c, x n ] < δ, [x n p + 1 − p] = 0 and Bott(u, x n ) = 0 hold for all c ∈ G, p ∈ P and u ∈ U . It follows from [21,Corollary 17.9] that there exists a continuous map z n : [0, 1] → U (C) such that z n (0) = 1, z n (1) = x n , Lip(z n ) < 2π + ε and [c, z n (t)] < ε for all c ∈ F and t ∈ [0, 1]. Since F and ε were arbitrary, the proof is completed. Lemma 7.3. Let B be a UHF algebra with a unique trace τ and let (α, u) : Γ B be a strongly outer cocycle action. Suppose that B absorbs the UHF algebra of type 2 ∞ tensorially. For any finite subset F ⊂ B and ε > 0, there exist a finite subset G ⊂ B and δ > 0 such that the following holds. If x ∈ U (B) satisfies then we can find a continuous map z : Proof. The proof is by contradiction. Assume that there exist a finite subset F ⊂ B and ε > 0 such that the assertion does not hold. Then we would have a central sequence (x n ) n of unitaries in B satisfying lim n→∞ α g (x n ) − x n = 0 ∀g ∈ Γ and τ (log(x n α b (x * n ))) = 0, and such that there does not exist a unitary z ∈ C([0, 1], B) as described in the statement. Let C = B ⋊ (α,u) Γ be the twisted crossed product C * -algebra. Since B absorbs the UHF algebra of type 2 ∞ , by Remark 6.18, C is a unital simple AT algebra with real rank zero and K 0 (C) We regard B as a subalgebra of C and think of τ as the unique tracial state on C. Then the sequence (x n ) n consisting of unitaries in B is a central sequence in C such that [x n ] = 0 in K 1 (C) and τ (log(x n λ α b x * n (λ α b ) * ))) = τ (log(x n α b (x * n ))) = 0. It follows from the lemma above that there exists a central sequence (z n ) n of unitaries in C([0, 1], C) such that z n (0) = 1, z n (1) = x n and Lip(z n ) < 7 for all sufficiently large n. By Corollary 6.17, (α, u) : Γ B is approximately representable. Therefore, in the same way as [26,Lemma 3.3], we may further assume that z n (t) is in B for all n ∈ N and t ∈ [0, 1]. When n is sufficiently large, we have [z n (t), c] < ε, α a (z n (t)) − z n (t) < ε and α b (z n (t)) − z n (t) < ε for all c ∈ F and t ∈ [0, 1], which is a contradiction.

Cohomology vanishing
In this subsection we prove Proposition 7.7, which is a cohomology vanishing type result for cocycle actions of Γ on Z. We also prove that all strongly outer actions of Γ on a certain UHF algebra are strongly cocycle conjugate (Theorem 7.9).
In addition, if F = ∅ and u(g, h) = 1 for all g, h ∈ Γ, then G = ∅ is possible.
Proof. The proof is by contradiction. Assume that there exist a finite subset F ⊂ B and ε > 0 such that the assertion does not hold. Then we would have a central sequence (x n ) n of unitaries in B satisfying α a (x n ) − x n → 0 as n → ∞ and τ (log(x n α a (x * n ))) = 0, and such that there does not exist a unitary y ∈ B as described in the statement. Let x ∈ U ((B ∞ ) αa ) be the image of (x n ) n . Choose m ∈ N so that 4/m < ε. Define x i,n ∈ U (B) for i = 0, 1, 2, . . . by x 0,n = 1 and x i+1,n = x n α b (x i,n ). It is easy to see that (x i,n ) n is also a central sequence and α a (x i,n ) − x i,n → 0 as n → ∞. Putǔ = u(b, a)u(a −1 , b) * u(a −1 , a) (we need u(a −1 , a) because α a −1 is not necessarily equal to α −1 a ). For each i, when n is sufficiently large, where the last equality follows since (α b (x i,n )) n is central andǔ is homotopic to 1. Therefore, for each i, we have τ (log(x i,n α a (x * i,n ))) = 0 for all sufficiently large n. Let x i ∈ U ((B ∞ ) αa ) be the image of (x i,n ) n . By Lemma 7.1, there exist paths of unitaries z 0 , z 1 ∈ C([0, 1], (B ∞ ) αa ) such that z 0 (1) = z 1 (1) = 1, z 0 (0) = x m , z 1 (0) = x m+1 , Lip(z 0 ) ≤ 4 and Lip(z 1 ) ≤ 4. By Theorem 6.14, there exist projections e, f ∈ (B ∞ ) αa such that We may further assume that the projections e, f commute with It is easy to check x − yα b (y * ) ≤ 4/m < ε. This is a contradiction. We consider the case F = ∅ and u(g, h) = 1 for all g, h ∈ Γ. In this case we would have a unitary x in (B ∞ ) αa , andǔ defined above is equal to 1. By replacing (B ∞ ) αa with (B ∞ ) αa , the same proof works. Lemma 7.5. Let B be a UHF algebra of infinite type with a unique trace τ and let (α, u) : Γ B be a strongly outer cocycle action. For any finite subset F ⊂ B, ε > 0 and r ∈ τ (K 0 (B)) with |r| < 1/2, there exists z ∈ U (B) such that Proof. Suppose that we are given r ∈ τ (K 0 (B)) with |r| < 1/2. Let σ = lim Ad(y 1 y 2 . . . y n ) be the automorphism of . . constructed in the proof of Lemma 7.1. Since B is of infinite type, we may further impose the condition that d n divides d n+1 .
We regard M dn as a subalgebra of B. When n is sufficiently large, there exists a unitary z n ∈ M dn such that z n y n z * n y * n = e 2π √ −1r . Hence we have τ (log(z n σ(z * n ))) = 2π √ −1r and z n − σ(z n ) = |e 2π √ −1r − 1|. Moreover (z n ) n is a central sequence in B. Take a strongly outer action γ : Γ Z. Define β : Γ Z ⊗ B by β a = γ a ⊗ id and Z ⊗ B be a strongly outer cocycle action. By Theorem 6.16, (α, u) is cocycle conjugate to β. Thus, there exist θ ∈ Aut(Z ⊗ B) and a family of unitaries (w g ) g∈Γ in Z ⊗ B such that Put z ′′ n = θ −1 (z ′ n ). Then (z ′′ n ) n is again a central sequence in Z ⊗ B. It is straightforward to see lim n→∞ z ′′ n − α a (z ′′ n ) = 0 and lim In addition, for sufficiently large n, because (z ′ n ) n is central and w b is homotopic to 1. The proof is completed.
Lemma 7.6. Let (α, u) : Γ B be a strongly outer cocycle action on a UHF algebra B of infinite type with a unique trace τ . For any finite subset F ⊂ B and ε > 0, there exist a finite subset G ⊂ B and δ > 0 such that the following holds. If x ∈ U (B) satisfies then there exists y ∈ U (B) such that [y, c] < ε ∀c ∈ F, α a (y) − y < ε, x − yα b (y * ) < ε and τ (log(y * xα b (y))) = 0.
Thus yz does the job.
The following is a cohomology vanishing type result for cocycle actions of the Klein bottle group Γ on the Jiang-Su algebra Z. We will use this proposition in the next subsection to prove the uniqueness of strongly outer cocycle actions of Γ on Z. In the proof, Corollary 4.11 plays a central role. We remark that the basic idea of the proof is the same as that of [27,Lemma 7.5,7.6] or [38,Theorem 5.3].
Proposition 7.7. Let (α, u) : Γ Z be a strongly outer cocycle action and let τ denote the unique trace on Z. For any finite subset F ⊂ Z and ε > 0, there exist a finite subset G ⊂ Z and δ > 0 such that the following holds. If x ∈ U (Z) satisfies Proof. According to [36,Proposition 3.3], the Jiang-Su algebra Z contains a unital subalgebra isomorphic to where B 0 and B 1 are the UHF algebras of type 2 ∞ and 3 ∞ , respectively. We identify B 0 and B 1 with B 0 ⊗ C and C ⊗ B 1 . Set B = B 0 ⊗ B 1 and denote the unique trace on Z ⊗ B by τ . Clearly (α ⊗ id, u ⊗ 1) : Γ Z ⊗ B j and (α ⊗ id, u ⊗ 1) : Γ Z ⊗ B are strongly outer.
As a byproduct of Lemma 7.4, we get the following.
Proposition 7.8. Let α : Γ B be a strongly outer action on a UHF algebra B with a unique trace τ . For an α-cocycle (x g ) g∈Γ in B, the following conditions are equivalent.
(2) There exists a sequence (w n ) n of unitaries in B such that w n α g (w * n ) → x g as n → ∞ for every g ∈ Γ.
Proof. We show (2)⇒(1). Suppose that there exists a unitary w ∈ B such that x g − wα g (w * ) < 1 for each g = a, b, a −1 . Let x ′ g = w * x g α g (w). Then (x ′ g ) g is also an αcocycle, and so we have Thus τ (log x ′ a ) = 0. Hence ∆ τ (x a ) = ∆ τ (x ′ a ) = 0. Let us consider the other implication. Take ε > 0 arbitrarily. By applying Lemma 7.4 to α : Γ B with the empty set and ε/3 in place of F and ε, we obtain δ > 0. Since the Z-action generated by α a has the Rohlin property, there exists a unitary u ∈ B such that x a − uα a (u * ) < min{δ/4, ε/3}. .
In the setting of the proposition above, the map g → ∆ τ (x g ) is a homomorphism from Γ to R/τ (K 0 (B)). Therefore we always have 2∆ τ (x a ) = 0 because of bab −1 = a −1 . Moreover, if K 0 (B) is 2-divisible (or equivalently, if B absorbs the UHF algebra of type 2 ∞ tensorially), then we always have ∆ τ (x a ) = 0. Theorem 7.9. Let B be a UHF algebra. If B absorbs the UHF algebra of type 2 ∞ tensorially or B does not contain a unital copy of M 2 , then all strongly outer actions of Γ on B are strongly cocycle conjugate.
Proof. By Theorem 6.16, all strongly outer actions of Γ on B are cocycle conjugate. If B absorbs the UHF algebra of type 2 ∞ tensorially, then the conclusion follows from Proposition 7.8 and the observation above. Suppose that B does not contain a unital copy of M 2 . Then {c ∈ R/τ (K 0 (B)) | 2c = 0} is isomorphic to Z/2Z and 1/2 + τ (K 0 (B)) is the generator. Let α : Γ B be a strongly outer action and let (x g ) g be an α-cocycle in B such that ∆ τ (x a ) = 0. By the observation above, ∆ τ (x a ) equals 1/2 + τ (K 0 (B)). For g = a n b m ∈ Γ, set ζ g = (−1) n ∈ C ⊂ B. It is easy to see that (ζ g x g ) g is an α-cocycle satisfying Ad(ζ g x g ) = Ad x g and ∆ τ (ζ a x a ) = 0. Hence the conclusion follows.

Actions of the Klein bottle group on the Jiang-Su algebra
In this subsection we prove that all strongly outer cocycle actions of Γ on Z are mutually cocycle conjugate (Theorem 7.12). We also show that all strongly outer actions of Γ on Z are strongly cocycle conjugate to each other (Theorem 7.15). The following is an analogue of Theorem 6.15 for the Jiang-Su algebra.
Assume further that ψ 1 • ψ −1 2 is approximately inner. Then there exists an approximately inner automorphism µ ∈ Aut T (Z ⋊ ϕ Z) and a unitary v ∈ U (Z) such that Proof. We denote the unique trace on Z by τ . It is easy to see that ψ 1 • ψ −1 2 is in Aut T (Z ⋊ ϕ Z). By Proposition 2.10 (3), (the Z-action generated by) ϕ is asymptotically representable. Then [9,Theorem 4.8] implies that ψ 1 • ψ −1 2 is T-approximately inner, that is, for any finite subset F ⊂ Z ⋊ ϕ Z and ε > 0, there exists a unitary x ∈ Z such that ψ 1 (c) − xψ 2 (c)x * < ε for all c ∈ F , and we may also assume ∆ τ (x) = 0 by replacing x with its suitable scalar multiple. Moreover, for each i = 1, 2, the cocycle action of Γ on Z generated by ϕ and ψ i |Z is strongly outer. Hence, by virtue of Proposition 7.7, we have the following for each i = 1, 2: if a sequence (x n ) n of unitaries in Z is a central sequence in Z ⋊ ϕ Z and ∆ τ (x n ) = 0, then there exists a sequence (y n ) n of unitaries in Z such that (y n ) n is a central sequence in Z ⋊ ϕ Z and satisfies x n − y n ψ i (y * n ) → 0 as n → ∞. Then the Evans-Kishimoto intertwining argument shows the statement (see [17,Theorem 5.1] for example).
Corollary 7.13. Any strongly outer cocycle action of Γ on the Jiang-Su algebra Z is approximately representable.
Proof. This immediately follows from the theorem above and Proposition 6.13.
Finally, we would like to prove that all strongly outer actions of Γ on Z are mutually strongly cocycle conjugate (Theorem 7.15).
Proposition 7.14. Let α : Γ Z be a strongly outer action. For two α-cocycles (x g ) g and (y g ) g in Z, the following conditions are equivalent.
(2) There exists a sequence (w n ) n of unitaries in Z such that w n x g α g (w * n ) → y g as n → ∞ for every g ∈ Γ.
Let α ⊗ : Γ Z ⊗ be the two-sided infinite tensor product of α : Γ Z, that is, First, we claim that the assertion holds for α ⊗ . Let (x g ) g be an α ⊗ -cocycle in Z ⊗ . Let ζ g ∈ C ⊂ Z ⊗ be the unitary satisfying ∆ τ (x g ) = ∆ τ (ζ g ). Then (ζ g ) g is clearly an α ⊗cocycle, and so it suffices to show that there exists a sequence (w n ) n of unitaries in Z such that w n x g α g (w * n ) → ζ g as n → ∞. Note that g → ζ g is a homomorphism. In particular, ζ a = 1 or ζ a = −1.
As in the proof of the proposition above, let ζ g ∈ C ⊂ Z be the unitary satisfying ∆ τ (x g ) = ∆ τ (ζ g ). Then (ζ −1 g x g ) g is also a β-cocycle. Hence, by replacing x g with ζ −1 g x g , we may assume ∆ τ (x g ) = 0. It follows from the proposition above that there exists (w n ) n in U (Z) such that w n β g (w * n ) → x g as n → ∞ for all g ∈ Γ, which implies that α is strongly cocycle conjugate to β.