The Rohlin property for automorphisms of the Jiang–Su algebra

Journal of Functional Analysis 259 (2010) 453–476 www.elsevier.com/locate/jfa

The Rohlin property for automorphisms of the Jiang–Su algebra Yasuhiko Sato Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan Received 22 August 2009; accepted 7 April 2010 Available online 21 April 2010 Communicated by D. Voiculescu

Abstract For projectionless C ∗ -algebras absorbing the Jiang–Su algebra tensorially, we study a kind of the Rohlin property for automorphisms. We show that the crossed products obtained by automorphisms with this Rohlin property also absorb the Jiang–Su algebra tensorially under a mild technical condition on the C ∗ algebras. In particular, for the Jiang–Su algebra we show the uniqueness up to outer conjugacy of the automorphism with this Rohlin property. © 2010 Elsevier Inc. All rights reserved. Keywords: C ∗ -algebra; Automorphism; Jiang–Su algebra; Rohlin property

1. Introduction In the classification program established by Elliott, the Jiang–Su algebra Z is one of the most important C ∗ -algebras, see [5,14], and which has been investigated by many people [3,4,7,25, 29]. Toms and Winter proved that all approximately divisible C ∗ -algebras absorb the Jiang–Su algebra tensorially, i.e., A ∼ = A ⊗ Z [29]. Rørdam showed that the Cuntz semigroup of a Zabsorbing C ∗ -algebra is almost unperforated [25]. Recently, Winter has shown some criteria for the absorption of the Jiang–Su algebra [32]. For abstract characterizations of the Jiang–Su algebra in a streamlined way, we refer to the recent papers by Dadarlat, Rørdam, Toms, and Winter [4, 26,31]. H. Lin has shown the classification theorem for a large class of C ∗ -algebras consisting E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.04.006

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of limits of generalized dimension drop algebras when they absorb the Jiang–Su algebra tensorially [18]. In the case of von Neumann algebras Connes defined the Rohlin property for automorphisms, using a partition of unities consisting of projections, and classified automorphisms of the injective type II 1 factor up to outer conjugacy [2]. Kishimoto gave a method to prove the Rohlin property for automorphisms of AF-algebras for classifying automorphisms up to outer conjugacy, based on Elliott’s classification program [6,10,16,17]. For Kirchberg algebras, Nakamura completely classified automorphisms with the Rohlin property by their KK-classes up to outer conjugacy [21]. Recently, Matui has classified automorphisms of AH-algebras with real rank zero and slow dimension growth up to outer conjugacy [20]. For finite actions, Izumi defined the Rohlin property and has shown the classification theory [11,12]. Recently, Izumi, Katsura, and Matui showed classification results for Z2 -actions with the Rohlin property [13,15,19]. The aim of the paper is to introduce a kind of the Rohlin property for automorphisms of projectionless C ∗ -algebras and to give the two main theorems as follows. Definition 1.1. Let A be a unital C ∗ -algebra which has a unique tracial state τ and absorbs the Jiang–Su algebra Z tensorially, and α be an automorphism of A. We say that α has the weak Rohlin property, if for any k ∈ N there exist positive elements fn ∈ A1+ , n ∈ N such that (fn )n ∈ A∞ ,  j  α (fn ) n · (fn )n = 0, j = 1, 2, . . . , k − 1,   k−1  j τ 1− α (fn ) → 0. j =0

Here, we denote by A∞ the quotient ∞ (N, A)/c0 (A), and A∞ the central sequence algebra ∩ A . We extend a technical condition called property (SI) to C ∗ -algebras which do not necessarily have projections in Definition 3.3. Roughly speaking, property (SI) means that if two central sequence of positive elements are given such that one of them is infinitesimally small compared to the other in the sequence algebra, then in fact so in the central sequence algebra.

A∞

Theorem 1.2. Let A be a unital separable C ∗ -algebra which does not necessarily have projections, has a unique tracial state, and absorbs the Jiang–Su algebra tensorially. Suppose that A has property (SI) and α is an automorphism of A with the weak Rohlin property. Then A ×α Z also absorbs the Jiang–Su algebra tensorially. Theorem 1.3. Suppose that α and β are automorphisms of the Jiang–Su algebra with the weak Rohlin property. Then α and β are outer conjugate, i.e., there exist an automorphism δ of Z and a unitary u in Z such that α = Ad u ◦ δ ◦ β ◦ δ −1 . For a separable, nuclear C ∗ -algebra A absorbing the Jiang–Su algebra, Rørdam proved that A is purely infinite if and only if A is traceless in [25], and Nakamura proved that the aperiodicity for automorphisms of purely infinite C ∗ -algebras coincides with the Rohlin property in [21].

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If A is a projectionless C ∗ -algebra with a unique tracial state constructed in [28], the weak Rohlin property is equivalent to the aperiodicity of the automorphism in the GNS-representation associated with the tracial state. A similar definition for finite actions, which is called projection free tracial Rohlin property, has been defined in [1,22]. The first main theorem is an adaptation of the result showed by Hirshberg and Winter in [9] to projectionless C ∗ -algebras. The second main theorem is an adaptation of the result for UHF algebras showed by Kishimoto in [16] to the Jiang–Su algebra. The proofs of Theorem 1.2 and Theorem 1.3 will appear mainly in Lemma 4.3 and Corollary 5.6. As a specific example in the paper, we take the two-sided shift automorphism σ on the infinite tensor product n∈Z Z ∼ = Z of the Jiang–Su algebra. In Proposition 4.4 and Example 3.6, we will prove that σ has the weak Rohlin property and Z has the property (SI). So, as an application of Theorem 1.2, we obtain that: Corollary 1.4. 



Z ×σ Z ⊗ Z ∼ Z ×σ Z. = n∈Z

n∈Z

The paper is organized as follows: In Section 2, we recall the generators of the prime dimension drop algebras discovered by Rørdam and Winter in [26]. In Section 3, for projectionless cases we extend the technical property, which was called property (SI) in [27], to projectionless C ∗ -algebras. By this property, we can obtain the generators defined in Section 2. In Section 4, we prove Theorem 1.2. In Section 5, using the weak Rohlin property we show the stability for the automorphisms of the Jiang–Su algebra, and show Theorem 1.3. Concluding this section, we prepare some notations. When A is a C ∗ -algebra, we denote by Asa the set of self-adjoint elements of A, A1 the unit ball of A, A+ the positive cone of A, U (A) the unitary group of A, P (A) the set of projections of A, T (A) the tracial state space of A. We define an inner automorphism of A by Ad u(a) = uau∗ for u ∈ U (A) and a ∈ A. We denote (n) by Mn the C ∗ -algebra of n × n matrices with complex entries and ei,j the canonical matrix units (n) . We denote by (m, n) the greatest common divisor of m and n ∈ N. of Mn , and we set ei(n) = ei,i

2. The generators of prime dimension drop algebras The following argument was given by Rørdam and Winter in [25] and [26]. We would like to begin with some definitions about the generators of prime dimension drop algebras and show Proposition 2.1. We denote by I (k, k + 1), k ∈ N the prime dimension drop algebra 

  f ∈ C [0, 1] ⊗ Mk ⊗ Mk+1 ; f (0) ∈ Mk ⊗ 1k+1 , f (1) ∈ 1k ⊗ Mk+1 ,

and set the self-adjoint unitary u1 =

 i,j

(k)

(k)

ei,j ⊗ ej,i ∈ U (Mk ⊗ Mk ).

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(k)

Define non-unital ∗-homomorphisms ρ0 : Mk ⊗ Mk → Mk ⊗ Mk+1 by ρ0 (ei,j ⊗ el,m ) = (k)

(k+1)

ei,j ⊗ el,m , and ρ : C([0, 1]) ⊗ Mk ⊗ Mk → C([0, 1]) ⊗ Mk ⊗ Mk+1 by ρ(f )(t) = ρ0 (f (t)), t ∈ [0, 1]. Let u ∈ U (C([0, 1]) ⊗ Mk ⊗ Mk ) be such that u(0) = 1 and u(1) = u1 , and set v=

k 

(k)

(k+1)

e1,j ⊗ ej,k+1 ,

j =1 (k+1)

w(t) = ρ(u)(t) ⊕ cos1/2 (πt/2)1k ⊗ ek+1 ,   (k) cj (t) = w(t) e1,j ⊗ 1k+1 w ∗ (t), j = 1, 2, . . . , k, s(t) = sin(πt/2)w(t)v, (k)

(k+1)

Since cj (0) = e1,j ⊗ 1k+1 , cj (1) = 1k ⊗ e1,j cj , s ∈ I (k, k + 1). And we have that

t ∈ [0, 1]. (k+1)

, s(0) = 0, and s(1) = 1k ⊗ e1,k+1 , it follows that

 (k+1)  (k+1) w ∗ w(t) = ww ∗ (t) = 1k ⊗ 1k+1 − ek+1 ⊕ cos(πt/2)1k ⊗ ek+1 ,   (k) (k) ci cj∗ = ww ∗ w e1,i ej,1 ⊗ 1k+1 w ∗ = δi,j c12 , k 

 2 cj∗ cj = w ∗ w ,

j =1

s ∗ s(t) = sin2 (πt/2)1k ⊗ ek+1 , (k+1)

c1 s(t) = sin(πt/2)w



(k) e1

 k   (k)  ∗ (k+1) ⊗ 1k+1 w w(t) e1,i ⊗ ei,k+1 = s(t). i=1

From these computations, it follows that {cj }kj =1 ∪ {s} satisfies c1  0, k 

ci cj∗ = δi,j c12 ,

cj∗ cj + s ∗ s = 1,

c1 s = s.

j =1

To be convenient, we denote by Rk the above relations on generators of a unital C ∗ -algebra. Fix a separable infinite-dimensional Hilbert space H, and set Λ=

 k  1 cj j =1 ∪ s ⊂ B(H)1 ; satisfies Rk ⊂ 2B(H) .

For λ ∈ Λ, let cj,λ ∈ λ, j = 1, 2, . . . , k and sλ ∈ λ be generators corresponding to

cj , j = 1, 2, . . . , k, and s on the relations Rk , and define c˜j = λ∈Λ cj,λ , s˜ = λ∈Λ sλ ∈ B( λ∈Λ H). The set {c˜j }kj =1 ∪ {˜s } satisfies the relations Rk . Let C ∗ ({c˜j }kj =1 ∪ {˜s }) be the C ∗ -subalgebra

of B( λ∈Λ H) generated by {c˜j } ∪ {˜s }. Then, we can identify C ∗ ({c˜j }kj =1 ∪ {˜s }) with the universal C ∗ -algebra on a set of generators satisfying the relations Rk .

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Proposition 2.1. (See Proposition 5.1 in [26].) The universal C ∗ -algebra C ∗ ({c˜j }kj =1 ∪ {˜s }) is isomorphic to I (k, k + 1) with c˜j → cj and s˜ → s. Proof. First we show that C ∗ ({cj }j ∪ {s}) = I (k, k + 1). Since k 

cj∗ ss ∗ cj + s ∗ s(t) = sin2 (πt/2)(1k ⊗ 1k+1 ),

j =1

and 1k ⊗ 1k+1 ∈ C ∗ ({cj }j ∪ {s}), we have that C([0, 1]) ⊗ 1k ⊗ 1k+1 ⊂ C ∗ ({cj }j ∪ {s}). By a partition of unity argument on [0, 1], it suffices to show that C ∗ ({cj }j ∪ {s})(i) ∼ = Mk+i , i = 0, 1, (k) (k+1) ∗ ∼ and C ({cj }j ∪ {s})(t) = Mk ⊗ Mk+1 , t ∈ (0, 1). Since cj (0) = e1,j ⊗ 1k+1 , cj (1) = 1k ⊗ e1,j , (k+1) ∗ ∼ j = 1, 2, . . . , k, and s(1) = 1k ⊗ e , it follows that C ({cj }j ∪ {s})(i) = Mk+i , i = 0, 1. Since 1,k+1

 (k) (k+1)  scj s ∗ (t) = sin2 (πt/2) cos(πt/2)ρ(u) e1,1 ⊗ e1,j ρ(u)∗ (t), j = 1, 2, . . . , k,  (k) (k+1)  ρ(u)∗ (t), i, j = 1, 2, . . . , k, ⊗ e1,j scj s ∗ ci (t) = sin2 (πt/2) cos(πt/2)ρ(u) e1,i  (k) (k+1)  scj (t) = sin(πt/2) cos(πt/2)ρ(u)(t) e1,j ⊗ e1,k+1 , j = 1, 2, . . . , k, for t ∈ (0, 1), we have that C ∗ ({cj }j ∪ {s})(t) = Mk ⊗ Mk+1 for t ∈ (0, 1). Set A = C ∗ ({c˜j }kj =1 ∪ {˜s }). Let Φ : A → I (k, k + 1) be the ∗-homomorphism defined by Φ(c˜j ) = cj and Φ(˜s ) = s. It remains to show that Φ is injective. Let (π, H) be an irreducible representation of A. Because for any a ∈ A there exists an irreducible representation of A which preserves the norm of a (see [23, 4.3.10]), it suffices to show that there exists a representation ϕ of I (k, k + 1) on H such that ϕ(cj ) = π(c˜j ) and ϕ(s) = π(˜s ). Set b˜ =

k 

c˜j∗ s˜ s˜ ∗ c˜j + s˜ ∗ s˜ .

j =1

By the following computations, we see that b˜ is in the center of A. Since {c˜j }j ∪ {˜s } satisfies the relations Rk , in particular c˜12 = c˜j c˜j∗ , we have that b˜ c˜j = s˜ s˜ ∗ c˜j + s˜ ∗ s˜ c˜j , c˜j b˜ = s˜ s˜ ∗ c˜j + c˜j s˜ ∗ s˜ , s˜ ∗ s˜ c˜j = c˜j − c˜12 c˜j = c˜j s˜ ∗ s˜ , ˜ s = b˜ c˜1 s˜ = s˜ s˜ ∗ s˜ + s˜ ∗ s˜ 2 , b˜  s˜ b˜ = s˜ c˜j∗ s˜ s˜ ∗ c˜j + s˜ s˜ ∗ s˜ , s˜ ∗ s˜ 2 = c˜1 s˜ − c˜13 s˜ = s˜ − s˜ = 0, s˜ ∗ s˜ c˜j∗ s˜ s˜ ∗ c˜j = c˜j∗ s˜ s˜ ∗ c˜j − c˜j∗ c˜j c˜j∗ s˜ s˜ ∗ c˜j = 0. ˜ s˜ ] = 0. ˜ c˜j ] = 0 and [b, Hence [b,

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˜ Since 0  b˜  1, we have that sp(b) ˜ = [0, 1] and obSet c¯j = π(c˜j ), s¯ = π(˜s ), and b¯ = π(b). ¯ When β = 0 we have s¯ = 0. Thus {c¯j }j satisfies the relations for tain β ∈ [0, 1] such that β1 = b. (k) matrix units {e1,j } of Mk , and then H = Ck . Set V0 : I (k, k + 1) → Mk as the irreducible repre(k)

sentation at 0. Since V0 (cj ) = e1,j we obtain a unitary u0 ∈ U (Mk ) such that Ad u0 ◦ V0 (cj ) = c¯j and define ϕ = Ad u0 ◦ V0 . When β = 1, by the following computations, we see that c¯j∗ c¯j , j = 1, 2, . . . , n, and s¯ ∗ s¯ are  ∗ orthogonal projections. Since b¯ = 1, we have that c¯j (1 − s¯ s¯ ∗ )c¯j = 0. Then it follows that  2 4 2 ∗ ∗ 2 ∗ ∗ 2 ∗ 2 c¯1 = s¯ s¯ , c¯1 = c¯1 , (c¯j c¯j ) = c¯j c¯1 c¯j = c¯j c¯j , and (¯s s¯ ) = s¯ ∗ c¯12 s¯ = s¯ ∗ s¯ . From c¯j∗ c¯j + s¯ ∗ s¯ = 1 it follows that c¯j∗ c¯j , j = 1, 2, . . . , k and s¯ ∗ s¯ are mutually orthogonal projections. Hence {c¯j }j ∪ {¯s } (k+1)

(k+1) and satisfies the relations for matrix units {e1,j }k+1 j =1 of Mk+1 . Then we see that H = C can define ϕ : I (k, k + 1) → B(H) as the irreducible representation of I (k, k + 1) at t = 1 up to unitary equivalence. When 0 < β < 1, by the following computations, we see that

 −1 Ei,j = β(1 − β) c¯i∗ s¯ c¯j∗ c¯j s¯ ∗ c¯i ,  −1 Ej,k+1 = β(1 − β) c¯j∗ c¯j s¯ ∗ s¯ , ¯ s = s¯ s¯ ∗ s¯ and (1 − b)¯ ¯ s ∗ s¯ = i, j = 1, 2, . . . , k, are mutually orthogonal projections. Since b¯ ∗ ∗ (1 − s¯ s¯ )¯s s¯ , we have that

 i,j

Ei,j

 −2 2 = β(1 − β) c¯i∗ s¯ c¯j∗ c¯j s¯ ∗ c¯12 s¯ c¯j∗ c¯j s¯ ∗ c¯i Ei,j  −2 = β(1 − β) c¯i∗ s¯ s¯ ∗ s¯ c¯j∗ c¯j c¯j∗ c¯j s¯ ∗ c¯i   = β −1 (1 − β)−2 c¯i∗ s¯ 1 − s¯ ∗ s¯ c¯j∗ c¯j s¯ ∗ c¯i = Ei,j ,  −2  ∗ 2  ∗ 2 2 c¯j c¯j s¯ s¯ = β(1 − β) Ej,k+1   = β −1 (1 − β)−2 c¯j∗ c¯j 1 − s¯ ∗ s¯ s¯ ∗ s¯ = Ej,k+1 ,   k   −1    ∗   ∗ ∗ ∗ ∗ + Ej,k+1 = β(1 − β) c¯i s¯ 1 − s¯ s¯ s¯ c¯i + 1 − s¯ s¯ s¯ s¯ j

 =β

−1

i=1 k 

c¯i∗ s¯ s¯ ∗ c¯i

 ∗

+ s¯ s¯ = 1.

i=1

Set Fi,j = β −1 (1 − β)−1/2 s¯ c¯j s¯ ∗ c¯i ,  −1/2 s¯ c¯j , i, j = 1, 2, . . . , k. Fj,k+1 = β(1 − β) ∗ F =E , F∗ ∗ ∗ Then it follows that Fi,j i,j i,j j,k+1 Fj,k+1 = Ej,k+1 , Fi,j Fi,j = E1,1 , and Fj,k+1 Fj,k+1 = (k)

(k+1)

E1,1 . Thus {Fi,j }i,j ∪ {Fj,k+1 }j satisfies the same relations as matrix units {e1,i ⊗ e1,j }i,j ∪  (k) (k+1) ∗ F {e1,j ⊗ e1,k+1 }j of Mk ⊗ Mk+1 . It is not so hard to see that β 1/2 kj =1 F1,j j,k+1 = s¯ ,

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 ∗ F ∗ ∗ ∗ ∗ β kj =1 F1,j i,j = s¯ s¯ c¯i , and β c¯i = s¯ s¯ c¯i + s¯ s¯ c¯i . Then we have that C ({Fi,j } ∪ {Fj,k+1 }) = ∗ k(k+1) C ({c¯j }j ∪ {¯s }) and H = C . Let Vβ be the irreducible representation of I (k, k + 1) at ˜ = β and there exists a unitary uβ such that t ∈ (0, 1) with sin2 (πt/2) = β. Then Vβ ◦ Φ(b)   Fi,j = β −1 (1 − β)−1/2 Ad uβ ◦ Vβ scj s ∗ ci ,  −1/2 Ad uβ ◦ Vβ (scj ). Fj,k+1 = β(1 − β) Hence, we have that Ad uβ ◦ Vβ (cj ) = c¯j and Ad uβ ◦ Vβ (s) = s¯ and obtain ϕ = Ad uβ ◦ Vβ . This completes the proof. 2 3. Property (SI) for projectionless C ∗ -algebras Definition 3.1. Let A be a unital C ∗ -algebra and τ ∈ T (A). We recall the dimension function dτ and define d τ : A1+ → R+ by   dτ (f ) = lim τ (1/n + f )−1 f , n→∞   d τ (f ) = lim τ f n , f ∈ A1+ . n→∞

Lemma 3.2. For fn ∈ A1+ , n ∈ N with (fn )n ∈ A∞ and an increasing sequence mn ∈ N, n ∈ N with mn  ∞, it follows that: (1) If limn→∞ maxτ ∈T (A) τ (fn ) = 0 then there exist f˜n ∈ A1+ , n ∈ N such that (f˜n )n = (fn )n and limn→∞ maxτ ∈T (A) dτ (f˜n ) = 0. (2) There exist f˜n ∈ A1+ , n ∈ N such that (f˜n )n = (fn )n and lim infn→∞ minτ ∈T (A) d τ (f˜n )  lim infn minτ τ (fnmn ). (3) If A absorbs Z tensorially, then there exist fn(i) ∈ A1+ , i = 0, 1, n ∈ N such that (fn(i) )n ∈ A∞ , fn(0) fn(1) = 0, (fn(i) )n  (fn )n , i = 0, 1, and lim infn→∞ minτ ∈T (A) d τ (fn(i) )  lim infn minτ τ (fnmn )/2. Proof. (1) Let εn > 0 be such that εn  0 and maxτ ∈T (A) τ (fn )  εn2 . Set  gε (t) =

(1 − ε)−1 (t − ε),

ε  t  1,

0,

0  t  ε,

and f˜n = gεn (fn ). Then we have that f˜n − fn   εn and εn limm→∞ (1/m + f˜n )−1 f˜n  fn , which implies that dτ (f˜n )  εn , for any τ ∈ T (A). (2) Let εn > 0 be such that εn  0, and (1 − εn )mn → 0. Set  gε (t) =

(1 − ε)−1 t, 0  t  1 − ε, 1, 1 − ε  t  1,

and f˜n = gεn (fn ). Then we have that f˜n − fn   εn and fnmn = fnmn (liml→∞ f˜nl + χ([0, 1 − εn ))(fn ))  liml→∞ f˜nl + (1 − εn )mn (where χ(S) means the characteristic function of S), which implies that τ (fnmn )  d τ (f˜n ) + (1 − εn )mn , for any τ ∈ T (A).

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 (3) Set c = lim infn→∞ minτ ∈T (A) τ (fnmn ). Since A ∼ = A ⊗ n∈N Z, we obtain ln ∈ N and  ln f n ∈ (A ⊗ j =1 Z)1+ such that ln  ∞ and mn f n − fn  → 0, which implies that (f n )n ∈ A∞ n and lim infn→∞ minτ ∈T (A) τ (f m n ) = c. By an argument as in the proof of (2), we obtain f˜n ∈ ln 1 Z)+ , n ∈ N such that (f˜n )n = (f n ) = (fn ) and lim infn→∞ minτ ∈T (A) d τ (f˜n )  c. (A ⊗ (i)

l=1

(0) (1)

(i)

1 , i = 0, 1, n ∈ N be such that g g Let gn ∈ Z+ n n = 0, lim infn d τZ (gn ) = 1/2, i = 0, 1, where  n +1 (i) (i) τZ means the unique tracial state of Z. Set fn = f˜n ⊗ gn ∈ A ⊗ ll=1 Z. Since ln  ∞, it p (i) (i) p (i) p follows that (fn )n ∈ A∞ , and since τ ((fn ) ) = τ (f˜n ⊗ 1)τZ ((gn ) ), p ∈ N, τ ∈ T (A), it (i) (i) follows that lim infn minτ d τ (fn ) = lim infn minτ d τ (f˜n )d τZ (gn )  c/2, i = 0, 1. 2

In [27], we have defined a technical condition, called property (SI), for C ∗ -algebra with nontrivial projections. For C ∗ -algebras which do not necessarily have projections, we generalize this technical condition in the following. Definition 3.3. We say that A has the property (SI), when for any en and fn ∈ A1+ , n ∈ N satisfying the following conditions: (en )n , (fn )n ∈ A∞ , lim max τ (en ) = 0,

n→∞ τ ∈T (A)

  lim inf min τ fnn > 0, n→∞ τ ∈T (A)

there exist sn ∈ A1 , n ∈ N, such that (sn )n ∈ A∞ and  ∗  sn sn = (en ),

(fn sn ) = (sn ).

Example 3.4. Any UHF algebra has the property (SI). 1 , n ∈ N satisfy the conditions in the Proof. Let B be a UHF algebra, and let en and fn ∈ B+ property (SI). Let Bn , n ∈ N be an increasing sequence of matrix subalgebras of B such that  ( n∈N Bn ) = B and 1Bn = 1B . For any Bn , we denote by Φn : B → Bn ∩ B the conditional expectations [27, Section 1]. By (en )n , (fn )n ∈ B∞ , we obtain a slow increasing sequence mn ∈ N, n ∈ N such that mn  ∞, mn  n, (Φmn (en ))n = (en )n , and

  lim mn Φmn (fn ) − fn  = 0,

n→∞

∩ B )1 such that and we obtain a fast increasing sequence ln , n ∈ N, and en , f n ∈ (Bm ln + n mn < ln , (en )n = (Φmn (en ))n , and limn→∞ mn f n − Φmn (fn ) = 0. Then we have that mn mn mn n lim τ (en ) = 0 and lim f m n −fn  = 0, which implies that lim inf τ (f n ) = lim inf τ (fn ) > 0. 1 By Lemma 3.2(1), we obtain e˜n ∈ (Bmn ∩ Bln )+ such that (e˜n )n = (en )n and lim dτ (e˜n ) = 0. By ∩ B )1 such that (f˜ ) = (f ) and Lemma 3.2(2), we obtain f˜n ∈ (Bm ln + n n n n n

  n > 0. lim inf d τ (f˜n )  lim inf τ f m n n→∞

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Taking a large N ∈ N, we have that dTrn (e˜n ) = dτ (e˜n ) < d τ (f˜n ) = d Trn (f˜n ),

n  N,

∩ B . Then, we obtain s ∈ (B ∩ B )1 such that where Trn is the normalized trace of Bm ln n ln mn n sn∗ sn = e˜n , f˜n sn = sn , hence we have that (sn )n ∈ B∞ , (sn∗ sn )n = (en )n , and (fn sn )n = (sn )n . 2

The following proposition is motivated by Lemma 3.3 in [20]. Combining this proposition and the above example we conclude Example 3.6. Proposition 3.5. Let A be a unital C ∗ -algebra absorbing the Jiang–Su algebra Z tensorially. If A ⊗ B has the property (SI) for any UHF algebra B, then A also has the property (SI). Example 3.6. The Jiang–Su algebra has the property (SI). In order to prove the above proposition, we define the projectionless C ∗ -algebra Zk for k ∈ N \ {1} by    Zk = f ∈ C [0, 1] ⊗ Mk ∞ ⊗ M(k+1)∞ ;

f (0) ∈ Mk ∞ ⊗ 1(k+1)∞ , f (1) ∈ 1k ∞ ⊗ M(k+1)∞ .

This projectionless C ∗ -algebra Zk was introduced by Rørdam and Winter in [26,30] as a mediator between C ∗ -algebras absorbing UHF algebras and C ∗ -algebras absorbing the Jiang–Su algebra. Proof of Proposition 3.5. Suppose that en and fn ∈ A1+ , n ∈ N satisfy the conditions in the property (SI). Let k be a natural number with k  2, B (i) the UHF algebra of rank (k + i)∞ , and Φ (i) the canonical unital embeddings of A ⊗ B (i) into A ⊗ B (0) ⊗ B (1) , i = 0, 1. By Lemma 3.2 there exist fn(i) ∈ A1+ , i = 0, 1, n ∈ N, such that (fn(i) )n ∈ A∞ , n

(fn(0) )n (fn(1) )n = 0, (fn(i) )n  (fn )n , and lim infn→∞ minτ ∈T (A) τ (fn(i) )  (i) lim infn minτ d τ (fn )  lim infn minτ τ (fnn )/2 > 0, i = 0, 1. 1 Applying the property (SI) of A ⊗ B (i) to en ⊗ 1B (i) and fn(i) ⊗ 1B (i) ∈ A ⊗ B (i) + we obtain (i)

1

(i)

sn ∈ A ⊗ B (i) , i = 0, 1, n ∈ N such that (sn )n ∈ (A ⊗ B (i) )∞ ,  (i) ∗ (i)  sn sn n = (en ⊗ 1B (i) ),

  (i)   fn ⊗ 1B (i) · sn(i) n = sn(i) .

∗

Note that (Φ (i) (sn(i) sn(i) ))n = (en ⊗1B (0) ⊗B (1) )n , (fn ⊗1B (0) ⊗B (1) )n ·(Φ (i) (sn(i) ))n = (Φ (i) (sn(i) ))n , (0) (1) and (Φ (0) (sn ))∗n (Φ (1) (sn ))n = 0. 1 Define sn ∈ A ⊗ Zk , n ∈ N by     sn (t) = cos(πt/2)Φ (0) sn(0) + sin(πt/2)Φ (1) sn(1) , Since

t ∈ [0, 1].

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 ∗  ∗   ∗    sn sn (t) n = cos2 (πt/2)Φ (0) sn(0) sn(0) + sin2 (πt/2)Φ (1) sn(1) sn(1)   ∗    ∗   + cos · sin(πt/2) Φ (0) sn(0) Φ (1) sn(1) + Φ (1) sn(1) Φ (0) sn(0) n = (en ⊗ 1B (0) ⊗B (1) )n ,

t ∈ [0, 1],

(fn ⊗ 1B (0) ⊗B (1) )n (sn (t)) = (sn (t)), t ∈ [0, 1], and Lip(sn ) = π , n ∈ N, it follows that  ∗  (fn ⊗ 1Zk )n (sn )n = (sn )n . sn sn n = (en ⊗ 1Zk )n ,  Set ι : A∞ → (A ⊗ Zk )∞ by ι((an )n ) = (an ⊗ 1Zk )n . Since A ∼ = A ⊗ n∈N Z and Zk ⊂unital Z, for any finite subset F ⊂ A∞ , we obtain a unital embedding ΦF : (A ⊗ Zk )∞ → A∞ such that ΦF ◦ ι(x) = x, x ∈ F . Define s = Φ{(en ),(fn )} ((sn )n ) ∈ A∞ , then we conclude that s ∗ s = (en ) and (fn )s = s. 2 4. Z absorption of Crossed products In this section we prove Theorem 1.2. We denote by Aα the fixed point algebra of α ∈ Aut(A) and by α∞ the automorphism of A∞ induced by α. In the following Lemma 4.1, mimicking Theorem 4.4 in [9], we use the weak Rohlin property to obtain a set of elements in (A∞ )α∞ which satisfies the same relations as {cj }kj =1 in Rk . After that, applying the property (SI) and the weak Rohlin property, we obtain the generators of prime dimension drop algebras in (A∞ )α∞ . Lemma 4.1. Let A be a unital separable C ∗ -algebra which has a unique tracial state τ and absorbs the Jiang–Su algebra tensorially. Suppose that α ∈ Aut(A) has the weak Rohlin property. Then for any k ∈ N there exist cj,n ∈ A, j = 1, 2, . . . , k, n ∈ N such that (cj,n )n ∈ (A∞ )1α∞ , (c1,n )n  0, (ci,n )n (cj,n )∗ = δi,j (c1,n )2 ,   k       n    (c1,n )n  = 1, lim τ c1,n = 1/k (cj,n )∗n (cj,n ) = 1, 1 − n→∞   j =1

(which implies limn→∞ τ (1A −

k

∗ j =1 cj,n cj,n ) = 0).

Proof. Let Φm , m ∈ N be the unital embeddings of Z into A ⊗ , Φm (x) = 1A ⊗ 1mi=1 Z ⊗ x ⊗ 1∞ i=m+2 Z



m∈N Z

∼ = A defined by

x ∈ Z,

and Φ be the unital embedding of Z into A∞ defined by Φ(x) = (Φm (x))m , x ∈ Z. Note that τ (Φm (x)) = τZ (x), m ∈ N, x ∈ Z. 2 (j ) In the definition of cj ∈ I (k, k + 1) in Section 2, replacing cos with cos1/n we obtain cn ∈ I (k, k + 1) ⊂ Z, j = 1, 2, . . . , k such that (j ) ∗

2

cn(1)  0, cn(i) cn = δi,j cn(1) ,   k     (1)    n ∗ (j )   (j )  c  τZ cn(1)  1/k. cn cn  = 1, 1 − n n = 1,   j =1

(1) n

Let εn > 0, n ∈ N be such that εn  0 and τZ (cn

) > 1/k − εn , n ∈ N.

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463

Let kn and ln ∈ N, n ∈ N be such that ln  ∞ and ln2 < kn . Since α ∈ Aut(A) has the (n) weak Rohlin property and A and Z are separable C ∗ -algebras, we obtain f (n) = (fm )m ∈  j ∞ (n) (A ∪ j ∈Z α∞ (Φ(Z))) ∩ A such that f  = 1,  p  α∞ f (n) f (n) = 0,

   n and τ fr(n) > 1/ 2(kn + ln ) + 1 − εm , (n)

for all p with 0 < |p|  2(kn + ln ) and for all r  m. Note that any subsequence of (fm )m (n) satisfies the above conditions. Then, taking a subsequence of (fm )m , we may suppose that    (1) n  (n) n   n  n  τ Φm c ·f − τZ c(1) τ f (n)  < εm . n

m

n

m

For p ∈ Z, define ap,n  0 by ⎧ ⎨ 1 − (|p| − kn )/ ln , kn < |p|  kn + ln , |p|  kn , ap,n = 1, ⎩ 0, kn + ln < |p|, and define completely positive maps ϕn : Z → A∞ , by 

ϕn (x) =

 p  p  ap,n α∞ Φ(x) α∞ f (n) .

|p|kn +ln

Then we have that       α∞ ϕn (x) − ϕn (x) =  



 |p|kn +ln

 p  p  (ap,n − ap−1,n ) · α∞ Φ(x) α∞ f (n)  

= x/ ln , x ∈ Z,   (i)   (j ) ∗ p   (j ) ∗  p  (n) 2 2 α∞ f = ap,n α∞ Φ cn(i) cn ϕn cn ϕn cn |p|kn +ln

 2 = δi,j ϕn cn(1) , (1)

ϕn (cn ) = 1, and 1 − (j ) cn,m

n ∈ N,

k

(j ) ∗ (j ) j =1 ϕn (cn ) ϕn (cn ) = 1.

(j )

(j )

Let ∈ A1 , j = 1, 2, . . . , k, m ∈ N be components of ϕn (cn ) (i.e., (cn,m )m = (j ) (1) ϕn (cn ) ∈ A∞ ) with cn,m  0, then we have that  (1) n  lim inf τ cn,m = lim inf m→∞

m



n     n   n ap,n τ α p Φm cn(1) α p fm(n)

|p|kn +ln

  n n > (2kn + 1) lim inf τ Φm cn(1) · fm(n) m

 n  n = (2kn + 1) lim inf τZ cn(1) τ fm(n) m

(2kn + 1) (1/k − εn ) →n→∞ 1/k. > (2(kn + ln ) + 1)

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 Let Fn be an increasing sequence of finite subsets of A1 with n Fn = A1 . By the above conditions, we obtain an increasing sequence mn ∈ N, n ∈ N such that  (j )   cn,m , x  < ε, x ∈ Fn , n   (j )  (j )  α cn,m − cn,m  n n < 1/ ln + εn ,   (i) (j ) ∗ (1) 2  c < εn , i, j = 1, 2, . . . , k, n,mn cn,mn − δi,j cn,mn   k     (1)  ∗ (j )   (j ) c  > 1 − εn , 1 − c c  n,mn n,mn  > 1 − εn , n,mn   j =1

 (1) n  > τ cn,m n

2kn + 1 (1/k − εn ). 2(kn + ln ) + 1

(j )

Define cj,n = cn,mn , j = 1, 2, . . . , k, then we have that (cj,n )n ∈ (A∞ )α∞ ,

  (c1,n )n  = 1,

(ci,n )n (cj,n )∗ = δi,j (c1,n )2 , (c1,n )n  0,   k     n    ∗ lim τ c1,n = 1/k. (cj,n )n (cj,n ) = 1, 1 − n→∞  

2

j =1

By the technique in the proof of Lemma 4.6 in [17] we obtain a generator s in (A∞ )α∞ satisfying the relations in Rk together with {(cj,n )n } above. In the proof of the following proposition, x ≈ε y means x − y < ε. Proposition 4.2. Let A be a unital separable C ∗ -algebra which has a unique tracial state τ , absorbs the Jiang–Su algebra Z tensorially, and has the property (SI). Suppose that α ∈ Aut(A) has the weak Rohlin property. Then for any k ∈ N there exists a set of norm-one elements {cj }kj =1 ∪{s} in (A∞ )α∞ satisfying Rk . (j )

(j )

Proof. By Lemma 4.1 we obtain cm ∈ A1 , j = 1, 2, . . . , k, m ∈ N such that (cm )m ∈  (j ) (j ) (j ) (j ) (1) (i) (1) (A∞ )α∞ , (cm )m  0, (cm )m (cm )∗ = δi,j (cm )2 , (cm )m  = 1, 1 − kj =1 (cm )∗ (cm ) = 1,  ∗ m (j ) (j ) (1) limm→∞ τ (cm ) = 1/k, and limm→∞ τ (1 − cm cm ) = 0, where τ is a unique tracial state (1) m of A. Let εm > 0, m ∈ N be such that εm  0 and τ (cm )  1/k − εm . (l) Because of the weak Rohlin property of α ∈ Aut(A) we obtain fm ∈ A1+ , l, m ∈ N, such that (l) (fm )m ∈ A∞ and    p  α∞ fm(l) m fm(l) = 0, p = 1, 2, . . . , l − 1,  (l) (1)   f , c  < εm , r  m, r m  (l) m  > 1/ l − εm , r  m. τ fr

Y. Sato / Journal of Functional Analysis 259 (2010) 453–476

465

(l)

Note that any subsequence of (fm )m satisfies the above conditions. Since τ is the unique (l) (1) (l) tracial state, taking a subsequence of (fm )m we may suppose that τ ((cm fm )m ) ≈εm (1) m (l) m (1) m (l) m τ (cm fm ) ≈εm τ (cm )τ (fm )  1/(kl) − 2εm . Set (l) (1) = cm gm

1/2 (l) (1) 1/2 f m cm

∈ A1+ ,

m ∈ N, (l) m

(l)

then we have that (gm )m ∈ A∞ 1+ , l ∈ N and lim infm→∞ τ (gm )  1/(kl). (l)

(l)

(l) ∗ (l)

By the property (SI) of A, we obtain sm ∈ A1 , m ∈ N, such that (sm )m ∈ A∞ , (sm sm ) =  (j ) ∗ (j ) (l) (l) (l) (l) (l) (1) (l) (l) (1 − kj =1 cm cm ), and (gm sm ) = (sm ). Remark that (gm ) = (fm )(cm )  (fm ), (gm )  (1)

(l)

(l)

(l)

(l)

(1)

−1/2

(l)

(cm ), (fm )(sm ) = (sm ), and (cm )m (sm ) = (sm ). Let Ln ∈ N be such that 2Ln (Ln  ∞) and define −1/2

(Ln ) sm = Ln

L n −1

< εn

 (L )  n . α p sm

p=0

Then we have that   (L )   α s n − s (Ln )   2Ln−1/2 < εn , m

m

m ∈ N.

(Ln ) (Ln ) Let mn ∈ N, n ∈ N be an increasing sequence with mn  ∞ such that (sm n )n ∈ A∞ , smn   ∗ (Ln ) (Ln ) (Ln ) (Ln ) − sm  < εn /Ln , α p (fm(Ln n ) )fm(Ln n )  < εn /Ln , sm smn − (1 − 1 + εn , fm(Ln n ) sm n n n k (j ) ∗ (j ) (j ) (j ) p j =1 cmn cmn )  εn , α (cmn ) − cmn  < εn /(2k), j = 1, 2, . . . , k, p = 1, 2, . . . , Ln − 1, and (1) (L )

(L )

(L )

cmn smnn − smnn  < εn /Ln , and set sn = smnn . Then we have that α∞ ((sn )n ) = (sn ) ∈ A∞ , sn∗ sn

≈2εn L−1 n

 L −1 n 

 L −1  n   (L ) ∗ (L )   (L ) (L )  q α smn n fmn n α fmn n smn n p

p=0

≈εn L−1 n

L n −1

q=0

 (L ) ∗ (L ) 2 (L )  n f n s n α p sm mn mn n

p=0

 (L ) ∗ (L )  n s n α p sm mn n   k   (j ) ∗ (j ) −1 p Ln cm n cm n α 1−

≈εn L−1 n ≈εn



j =1

≈εn 1 −

k 

(j ) ∗ (j )

cm n cm n ,

j =1 (1)

(j )

(sn )n  = 1, and (cmn )(sn ) = (sn ). Hence we conclude that {(cmn )n }kj =1 ∪ {(sn )n } ⊂ (A∞ )1α∞ and they satisfy the relations Rk . 2

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Proof of Theorem 1.2. Applying Proposition 2.2 in [29] to (A ×α Z)∞ it suffices to show the following lemma. Lemma 4.3. Let A be a unital separable C ∗ -algebra which does not necessarily have projections, has a unique tracial state, and absorbs the Jiang–Su algebra tensorially. Suppose that A has the property (SI) and α ∈ Aut(A) has the weak Rohlin property. Then for any k ∈ N there exists a unital ∗-homomorphism Φk from I (k, k + 1) to (A∞ )α∞ . Proof. By Proposition 4.2 we obtain a set of norm-one generators {cj }kj =1 ∪ {s} in (A∞ )α∞ satisfying Rk . Then, by Proposition 2.1, we conclude the above lemma. 2 At the end of this section, we see the following proposition as an example of automorphisms with the weak Rohlin property, hence we conclude Corollary 1.4. Proposition 4.4. The two-sided shift automorphism σ on



n∈Z Z

has the weak Rohlin property.

 Proof. Identify n∈Z Z with Z. Let (π, H) be the GNS-representation of Z associated with the unique tracial state τZ and α the weak extension of α ∈ Aut(Z) on π(Z) . Because of Theorem 1.2 in [28], we have shown that the weak Rohlin property is equivalent to the aperiodicity in the GNS-representation associated with the unique tracial state. Thenit suffices to show that σ k = Ad V for any V ∈ U (π(Z)) and any k ∈ N. In particular, since ki=1 Z ∼ = Z, it suffices to show that σ = Ad V for any V ∈ U (π(Z) ). Assume that there exists Vσ ∈ U (π(Z) ) such that Ad Vσ = σ . Note that σ k (Vσ ) = Vσ for any k ∈ N. However we see that π(Z) σ = C1, this is a contradiction. N and Indeed,  V ∈ U (π(Z) ) with σ (V ) = V and any ε > 0, we obtain ∗N ∈ N for any v ∈ −N Z ⊂ n∈Z Z (= Z) such that V − π(v)2 < ε, where x2 := τZ (x x)1/2 , then σ k (V ) − π ◦ σ k (v)2 < ε for all k ∈ N. Hence, for any a ∈ Z 1 , it follows that [V , π(a)]2 < 2ε. Since ε is arbitrary, we conclude that V ∈ π(Z) ∩ π(Z) = C1. 2 5. Stability for automorphisms In this section, using the weak Rohlin property, we show the stability for automorphisms of the Jiang–Su algebra Theorem 5.3 and prove Theorem 1.3. First, we recall the generalized determinant introduced by P. de la Harpe and G. Skandalis (see [8,15,24]). Let A be a unital C ∗ -algebra with a unique tracial state τ . For any piecewise differentiable path ξ : [0, 1] → U (A), we define  τ (ξ ) = 

1 √ 2π −1

1

  τ ξ˙ (t)ξ ∗ (t) dt ∈ R.

0

τ (ξ ) ∈ τ (K0 (A)). For any u ∈ U0 (A), there exists a pieceWhen ξ(0) = ξ(1) = 1 we have that  wise differentiable path ξu : [0, 1] → U (A) such that ξu (0) = 1, ξu (1) = u. The generalized determinant τ associated with the tracial state τ is the map from U0 (A) to R/τ (K0 (A)) de (ξu ) + τ (K0 (A)). Note that τ is a group homomorphism. fined by τ (u) =  Mimicking the proof of Lemma 6.2 in [15] we prove the following proposition. Hereinafter, we let log be the standard branch defined on the complement of the negative real axis.

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467

Proposition 5.1. Let B be the UHF algebra of rank k ∞ , where k ∈ N \ {1}, τ the unique tracial state of B, β ∈ Aut(B), and un ∈ U (B), n ∈ N with (un )n ∈ B∞ . Suppose that β ∈ Aut(B) has the Rohlin property and τ (un ) = 0,

for any n ∈ N.

Then there exist vn ∈ U (B), n ∈ N such that (vn )n ∈ B∞ ,   vn β(vn )∗ n = (un )n ,

  τ ◦ log vn β(vn )∗ u∗n = 0,

for any n ∈ N.

The following lemma was essentially proved in [8]. Lemma 5.2. Let A be a unital C ∗ -algebra with a unique tracial state τ . (1) For u1 , u2 ∈ U (A) with, ui − 1 < 1/2, i = 1, 2 it follows that τ ◦ log(u1 u2 ) = τ ◦ log(u1 ) + τ ◦ log(u2 ). (2) For u1 , u2 , and v ∈ U (A) with u1 − u2  < 1/2 and v − 1 < 1/4, it follows that     τ ◦ log u1 vu∗2 v ∗ = τ ◦ log u1 u∗2 . √ Proof. (1) Let hi√∈ Asa be such that exp(2π −1hi√ ) = ui , i = 1, 2, √ and h3 ∈ Asa be such that exp(2π −1h ) = u u . Set u(t) = exp(2π −1th ) · exp(2π −1th2 ), w(t) = 3 1 2 1 √ exp(2π −1th3 ), t ∈ [0, 1]. Since 1 − u(t) < 1, 1 − w(t) < 1, and 1 − w ∗ u(t) < 2, t ∈ [0, 1], we can define h ∈ C([0, 1]) ⊗ Asa by h(t) = log(w ∗ u(t)), t ∈ [0, 1], then u and w are homotopic, by H (s, t) = w(t) exp((1 − s)h(t)) with fixed endpoints H (s, 0) = 1 and H (s, 1) = w(1). Hence, we have that √

1

τ ◦ log(u1 u2 ) = 2π −1τ (h3 ) = √



 τ ww ˙ (t) dt = ∗

0

1

 ∗  τ uu ˙ (t) dt

0

= 2π −1τ (h1 + h2 ) = τ ◦ log(u1 ) + τ ◦ log(u2 ). (2) Set U1 = v ∗ u1 vu∗1 , U2 = u1 u∗2 , then it follows that Ui − 1 < 1/2, i = 1, 2. Applying (1), since τ ◦ log(U1 ) = τ ◦ log(v ∗ ) + τ ◦ log(u1 vu∗1 ) = 0 we have that τ ◦ log(U1 U2 ) = τ ◦ log(U1 ) + τ ◦ log(U2 ) = τ ◦ log(U2 ). 2 Proof of Proposition 5.1. Because β ∈ Aut(B) has the Rohlin property in [16], there exist vn ∈ U (B), n ∈ N such that (vn )n ∈ B∞ , and   ∗  vn β vn n = (un ). By the assumption and

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Y. Sato / Journal of Functional Analysis 259 (2010) 453–476

 ∗     1 √ τ ◦ log vn β vn u∗n + τ K0 (B) 2π −1    ∗   ∗  = τ vn β vn u∗n = τ vn β vn − τ (un ) = −τ (un ), we have that 1 √

2π −1

  ∗    τ ◦ log vn β vn u∗n ∈ τ K0 (B) ,

n ∈ N.

Since B is the UHF algebra of rank k ∞ , we obtain ln ∈ N and mn ∈ Z such that (mn , k) = 1 and k −ln mn = −

  ∗    1 √ τ ◦ log vn β vn u∗n ∈ τ K0 (B) . 2π −1

√ Set λn = exp(2π −1k −ln mn ), then we have that λn → 1, by (vn β(vn )∗ u∗n )n = 1. By the Rohlin property of β ∈ Aut(B), there exist pn ∈ P (B) and zn ∈ U (B), n ∈ N such that (pn )n ∈ B∞ , (zn )n = 1B∞ , and ln −1 k

(Ad zn ◦ β)j (pn ) = 1B .

j =0

Define vn =

ln −1 k

  √ exp 2π −1j k −ln mn · (Ad zn ◦ β)j (pn ),

j =0

vn = vn v n ∈ U (B),

n ∈ N.

Taking a subsequence of (pn )n and (zn )n , we may suppose that (v n )n ∈ B∞ . Then it follows that (vn )n ∈ B∞ . By the definition of v n we have that v n Ad zn ◦ β(v n )∗ = λn and         ∗  vn β(vn )∗ u∗n n = vn Ad zn ◦ β vn∗ u∗n n = λn vn β vn u∗n n = 1. And, by Lemma 5.2, we have that     τ ◦ log vn β(vn )∗ u∗n = τ ◦ log vn Ad zn ◦ β(vn )∗ u∗n   ∗  = τ ◦ log v n Ad zn ◦ β(v n )∗ Ad zn ◦ β vn u∗n vn √   ∗  = 2π −1k −ln mn + τ ◦ log vn β vn u∗n = 0, n ∈ N.

2

Theorem 5.3. Suppose that α ∈ Aut(Z) has the weak Rohlin property and un ∈ U (Z), n ∈ N satisfy (un )n ∈ Z∞ and that τZ (un ) = 0,

for any n ∈ N.

Y. Sato / Journal of Functional Analysis 259 (2010) 453–476

469

Then there exist vn ∈ U (Z), n ∈ N such that (vn )n ∈ Z∞ and   vn α(vn )∗ n = (un )n . The following lemma is a direct adaptation of Proposition 4.6 in [15]. Lemma 5.4. For any c > 0 there exists c > 0 such that the following holds. Let B be a UHF algebra, τ the unique tracial state of B. Suppose that  un ∈ U (C([0, 1]) ⊗ B), n ∈ N satisfy that ( un )n ∈ (C([0, 1]) ⊗ B)∞ , ( un (i))n = 1, i = 0, 1, τ (  un ) = 0,

  τ ◦ log  un (i) = 0,

i = 0, 1, n ∈ N,

and Lip( un ) < c, n ∈ N. Then there exist yn ∈ U (C([0, 1]2 ) ⊗ B), n ∈ N such that (yn )n ∈ 2 (C([0, 1] ) ⊗ B)∞ , yn (0, t) = 1B , yn (1, t) =  un (t), t ∈ [0, 1],     yn (s, i) = exp log  un (i) s , i = 0, 1, s ∈ [0, 1], and Lip(yn ) < c , n ∈ N. Proof. Set ∂E = {(s, t) ∈ [0, 1]2 ; {s, t} ∩ {0, 1} = φ}. By Proposition 4.6 in [15], for c > 0, we obtain c > 0 satisfying that: for any AF-algebra A and for any z ∈ U (C(∂E) ⊗ A) with z(0, 0) = 1, Lip(z) < c, and [z]1 = 0 ∈ K1 (C(∂(E)) ⊗ A), there exists  z ∈ U (C([0, 1]2 ) ⊗ A) such that  z|∂E = z and Lip( z ) < c . Suppose that  un ∈ U (C([0, 1]) ⊗ B) satisfies the conditions in the lemma. Define Un ∈ U (C(∂E) ⊗ B) by ⎧ ⎨ 1, un (t), Un (s, t) =  ⎩ exp(log( un (i))s),

s = 0, s = 1, t = i, i = 0, 1.

Then we have that Lip(Un ) < c for any n ∈ N. By the assumption, regarding Un ∈ U (C(T) ⊗ B),  τ (Un ) = 0 in τ (K0 (B)). we have that [Un ]1 =  Let Bn , n ∈ N be an increasing sequence of matrix subalgebras of B with 1Bn = 1B and  Bn = B. Since (Un )n ∈ U ((C(∂E) ⊗ B)∞ ), slightly modifying Un , we obtain an increasing ∩ B)) such that m  ∞, (U ) = (U ) , sequence mn ∈ N, n ∈ N and Un ∈ U (C(∂E) ⊗ (Bm n n n n n n Un (0, 0) = 1, and Lip(Un ) < c. Since Bmn ∩ B has the unique tracial state τ |Bm n ∩B , it foln ∈ τ  τB (Un ) =   τB (Un ) = 0, then we obtain U (Un ) =  lows that [Un ]K1 (Bm n ∩B) =  Bmn ∩B 2 n |∂E = Un and Lip(U n ) < c . Then we have U (C([0, 1] ) ⊗ (Bmn ∩ B)), n ∈ N such that U 2   n on ∂E, we that (Un )n ∈ (C([0, 1] ) ⊗ B)∞ . Since Un |∂E = Un , n ∈ N, slightly modifying U 2  obtain yn ∈ U (C([0, 1] ) ⊗ B), n ∈ N and ε > 0 such that (yn )n = (Un )n , yn |∂E = Un , and Lip(yn ) < c + ε for any n ∈ N. 2 As in the proof of Proposition 2.2 in [29], unital ∗-homomorphisms from I (k, k + 1) to (A∞ )α∞ obtained in Lemma 4.3 implies the following lemma.

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Lemma 5.5. Let A be a unital separable C ∗ -algebra which does not necessarily have projections, has a unique tracial state, and absorbs the Jiang–Su algebra tensorially. Suppose that A has the property (SI) and α is an automorphism of A with the weak Rohlin property. Then there exists a unital embedding of Z into (A∞ )α∞ . Proof of Theorem 5.3. Let Zk , B (i) , and Φ (i) , i = 0, 1, be the projectionless C ∗ -algebra, the UHF algebras, and the unital embeddings in the proof of Proposition 3.5. Because of Z ⊗ B (i) ∼ = B (i) , the Rohlin property of α ⊗ idB (i) ∈ Aut(Z ⊗ B (i) ), and τZ⊗B (i) (un ⊗ 1B (i) ) = 0,

   in R/τ K0 B (i) , n ∈ N,

applying Proposition 5.1, we obtain Vn (i) ∈ U (Z ⊗ B (i) ), i = 0, 1, n ∈ N such that (Vn (i) )n ∈ (Z ⊗ B (i) )∞ and  (i)  ∗  Vn α ⊗ idB (i) Vn (i) n = (un ⊗ 1B (i) )n , i = 0, 1,  ∗   τZ ⊗B (i) ◦ log Vn(i) α ⊗ idB (i) Vn(i) u∗n ⊗ 1B (i) = 0, n ∈ N. By the following argument, we obtain a path of unitaries  vn in Z ⊗ Zk with endpoints (i) Φ (i) (Vn ) ∈ U (Z ⊗ B (0) ⊗ B (1) ), i = 0, 1 which satisfies  vn α ⊗ idZk ( vn∗ ) ≈ un ⊗ 1Zk . Set  ∗   Un,1 = Φ (0) Vn(0) Φ (1) Vn(1) , Wn = Un,1 α ⊗ idB (0) ⊗B (1) (Un,1 )∗ ,

n ∈ N.

Then it follows that (Un,1 )n ∈ (Z ⊗ B (0) ⊗ B (1) )∞ , (Wn )n = 1(Z ⊗B (0) ⊗B (1) )∞ , and, by (1) in Proposition 5.2, τZ ⊗B (0) ⊗B (1) ◦ log(Wn )   ∗   = τ ◦ log Φ (1) Vn(1) α ⊗ idB (1) Vn(1) u∗n ⊗ 1B (1)   ∗ ∗  · Φ (0) Vn(0) α ⊗ idB (0) Vn(0) u∗n ⊗ 1B (0)    ∗    = (−1)1−i τ ◦ log Φ (i) Vn(i) α ⊗ idB (i) Vn(i) u∗n ⊗ 1B (i) = 0, i=0,1

for any n ∈ N. n ∈ U (C([0, 1]) ⊗ Z ⊗ B (0) ⊗ B (1) ), n ∈ N Since (Un,1 )n ∈ (Z ⊗ B (0) ⊗ B (1) )∞ , there exist U n (1) = Un,1 , (U n )n ∈ (C([0, 1]) ⊗ Z ⊗ B (0) ⊗ B (1) )∞ , and Lip(U n ) < n (0) = 1, U such that U (j ) (0) (1)  π + ε for some ε > 0. Define Tn ∈ U (C([0, 1]) ⊗ Z ⊗ B ⊗ B ), j , n ∈ N by (j ) n idC([0,1]) ⊗ α j ⊗ idB (0) ⊗B (1) (U n )∗ , Tn = U (0) and Tn = 1. Note that

 (j −1) ∗ (j ) Tn id ⊗ α ⊗ id Tn = Tn(1) ,

j, n ∈ N.

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n )n ∈ (C([0, 1]) ⊗ Z ⊗ B (0) ⊗ B (1) )∞ , (Wn )n = 1, τ ◦ log(Wn ) = 0, and Lip(U n ) < π + ε, By (U (j ) (j ) (j ) (0) (1)   we have that (Tn )n ∈ (C([0, 1]) ⊗ Z ⊗ B ⊗ B )∞ , (Tn (1))n = 1, τ ◦ log(Tn (1)) = (j ) j τ ◦ log(Wn ) = 0, and Lip(Tn ) < 2(π + ε), j ∈ N. Then, by Lemma 5.4, we obtain a constant (j ) (j ) c > 0 and yn ∈ U (C([0, 1]2 ) ⊗ Z ⊗ B (0) ⊗ B (1) ), j ∈ N such that (yn )n ∈ (C([0, 1]2 ) ⊗ Z ⊗ B (0) ⊗ B (1) )∞ , (j ) (j ) (j ) yn (0, t) = 1, yn (1, t) = Tn (t), t ∈ [0, 1],    (j ) (j ) (j ) yn (s, 1) = exp log Tn (1)s , yn (s, 0) = 1, s ∈ [0, 1], (j )

and Lip(yn ) < c, n ∈ N. (l) (l) By the Rohlin property of α ⊗ idB (0) ⊗B (1) we obtain pm ∈ P (Z ⊗ B (0) ⊗ B (1) ) and zm ∈ (l) (l) U (Z ⊗ B (0) ⊗ B (1) ), l, m ∈ N such that (pm )m ∈ (Z ⊗ B (0) ⊗ B (1) )∞ , (zm )m = 1, and l −1 k

 j  (l)  (l) Ad zm = 1. ◦ α ⊗ id pm

j =0 (j )

(j )

Set yn (s)(t) = yn (s, t), s, t ∈ [0, 1], j , n ∈ N,   (l) (l) = idC([0,1]) ⊗ Ad zm ◦ α ⊗ idB (0) ⊗B (1) ,  αm l,m,n W =

l −1 k

l, m ∈ N,

∗  (l) j   (j )  (l) j −k l  (k l )  (l) yn j/k l 1C([0,1]) ⊗ pm . αm ·  αm Tn · 

j =0 (l) j (l) Since (( αm ) (1C([0,1]) ⊗ pm ))m ∈ (C([0, 1]) ⊗ Z ⊗ B (0) ⊗ B (1) )∞ , j = 0, 1, . . . , k l − 1 are l,m,n ∈  )m is a unitary and obtain W mutually orthogonal projections, we have that (W l,m,n (0) (1)   U (C([0, 1]) ⊗ Z ⊗ B ⊗ B ), l, m, n ∈ N such that (Wl,m,n )m = (Wl,m,n )m . By the definition  , we have that of W l,m,n

      (W l,m,n )m · idC([0,1]) ⊗ α ⊗ id(W l,m,n )∗ − Tn(1)  < c/k l , m m

l, n ∈ N.

l

(j ) (l) j −k l (k ) (l) j (l) αm ) (yn (j/k l )))n , and ( αm ) (1C([0,1]) ⊗ pm )m ∈ (C([0, 1]) ⊗ Z ⊗ Since (Tn )n , (( B (0) ⊗ B (1) )∞ , and 1 − Wn  → 0, we obtain a slow increasing sequence ln , n ∈ N and a fast increasing sequence mn ∈ N, n ∈ N such that ln  ∞, mn  ∞,

    ln ,mn ,n )n ∈ C [0, 1] ⊗ Z ⊗ B (0) ⊗ B (1) , (W ∞   2ln ∗    Wln ,mn ,n id ⊗ α ⊗ id(Wln ,mn ,n ) − Tn(1)  < c/k ln . k 1 − Wn  → 0, Set     l∗ ,m ,n U n = W n ∈ U C [0, 1] ⊗ Z ⊗ B (0) ⊗ B (1) , V n n n )n ∈ (C([0, 1]) ⊗ Z ⊗ B (0) ⊗ B (1) )∞ and Then it follows that (V

n ∈ N.

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  ∗  n idC([0,1]) ⊗ α ⊗ idB (0) ⊗B (1) V n V n    ∗ (1) l ,m ,n Tn · id ⊗ α ⊗ id(W ln ,mn ,n ) · W l∗ ,m ,n = Ad W = 1. n n n n n (k ln )

Since 1 − Tn (1)  j 1 − Wn  → 0, and 1 − yn that (j )

l ,m ,n (1) W n n

=

ln −1 k

(k ln )

(i/k ln , 1)  1 − Tn

(1), it follows

 (l ) j −k ln  (k ln )  ∗  (l ) j  (l )  (j ) yn j/k ln , 1  αmnn pmnn αmnn Tn (1) 

j =0

≈δn 1, where δn = 2k 2ln 1 − Wn , n ∈ N, and then we have that        n (1) = (Un,1 )n = Φ (0) Vn(0) ∗ Φ (1) Vn(1) . V n n Define   n (t), n (t) = Φ (0) Vn(0) V V

t ∈ [0, 1].

n )n ∈ (C([0, 1]) ⊗ Z ⊗ B (0) ⊗ B (1) )∞ Then we have that (V      n (i) = Φ (i) Vn(i) , V n n

i = 0, 1,

and   n idC([0,1]) ⊗ α ⊗ idB (0) ⊗B (1) (V n )∗ V n  (0)  ∗   (0) = 1C([0,1]) ⊗ Φ Vn α ⊗ idB (0) Vn(0) n = (1C([0,1]) ⊗ un ⊗ 1B (0) ⊗B (1) )n . n at the end points, we obtain  n )n , Slightly modifying V vn ∈ U (Z ⊗ Zk ) such that ( vn )n = (V    vn (i) = Φ (i) Vn(i) ,

i = 0, 1,

   vn α ⊗ idZk ( vn )∗ n = (un ⊗ 1Zk )n .

vn ∈ Z ⊗ Zk and satisfies (vn α(vn∗ ))n = Finally, we obtain (vn )n ∈ Z∞ which corresponds to  (un )n , by the following. By Lemma 5.5 and Zk ⊂unital Z, we obtain a unital embedding Ψ : Z ⊗ Zk → Z ∞ such that α∞ ◦ Ψ = Ψ ◦ α ⊗ idZk and Ψ (a ⊗ 1Zk ) = a ∈ Z ⊂ Z ∞ , a ∈ Z. Let sequence of finite subsets of Z 1 and εn > 0, n ∈ N a decreasing Fn ⊂ Z 1 , n ∈ N be an  increasing 1 sequence such that Fn = Z , εn  0,   [ vn , x ⊗ 1Zk ] < εn ,

x ∈ Fn ,    vn α ⊗ idZk ( vn )∗ − un ⊗ 1Zk  < εn . It follows that [Ψ ( vn ), x] = Ψ ([ vn , x ⊗ 1Zk ]) < εn , x ∈ Fn and

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473

    vn )∗ = Ψ  vn α ⊗ idZk ( Ψ ( vn )α∞ Ψ ( vn )∗ ≈εn Ψ (un ⊗ 1Zk ) = un . Denote by vn,p ∈ U (Z), p ∈ N components of Ψ ( vn ) ∈ U (Z ∞ ), then we obtain an increasing sequence pn ∈ N, n ∈ N such that (vn,pn )n ∈ Z∞ ,

  vn,pn α(vn,pn )∗ n = (un )n .

Define vn = vn,pn . This completes the proof.

2

Corollary 5.6. Suppose that α ∈ Aut(Z) has the weak Rohlin property. For any finite subset F of Z 1 and ε > 0, there exist a finite subset G of Z 1 and δ > 0 satisfying that: for any u ∈ U (Z) with [u, y] < δ, y ∈ G, there exist v ∈ U (Z) and λ ∈ T such that   vα(v)∗ − λu < ε,

  [v, x] < ε,

x ∈ F.

√ Proof. For un ∈ U (Z), n ∈ N with (un )n ∈ Z∞ , set λn = exp(−2π −1τZ (un )) ∈ T. Since τZ (λn un ) = 0 ∈ R/τ (K0 (Z)), by the above theorem we obtain vn ∈ U (Z), n ∈ N such that (vn )n ∈ Z∞ and   vn α(vn )∗ n = (λn un )n . Assume that there exist a finite subset F of Z 1 and ε > 0 satisfying that: For any finite subset G of Z 1 and δ > 0 there exists u ∈ U (Z) with [u, y] < δ, y ∈ G such that if v ∈ U (Z) and λ ∈ T satisfy vα(v)∗ − λu < ε then [v, x]  ε for some x ∈ F . This contradicts the above statement. 2 Proof of Theorem 1.3. By using the stability of the above form instead of Proposition 4.3 in [17] and by the Evans–Kishimoto intertwining argument in the proof of Theorem 5.1 in [17] we can give the proof. The details are as follows. Let ε > 0 and let {xn }n∈N be a dense sequence in Z 1 . We shall construct inductively finite subsets Fn , Gn of Z 1 , un , vn ∈ U (Z), and δn > 0, n ∈ N satisfying the following conditions: Set F0 = G0 = {1Z }, u0 = v0 = 1Z , α−1 = α, β0 = β, δ0 = 1, α2n+1 = Ad u2n+1 ◦ α2n−1 ,

β2n+2 = Ad u2n+2 ◦ β2n ,

n ∈ N ∪ {0}.

Define ∗ w2n = u2n β2n−2 (v2n )v2n ,

∗ w2n+1 = u2n+1 α2n−1 (v2n+1 )v2n+1 ,

for n ∈ N, and inductively define   w2n , = w2n Ad v2n w2n−2

  w2n+1 , = w2n+1 Ad v2n+1 w2n−1

for n ∈ N, where w0 = 1, w1 = w1 . The conditions indexed by n ∈ N ∪ {0} are given by

474

(1) (2) (3) (4) (5) (6) (7)

Y. Sato / Journal of Functional Analysis 259 (2010) 453–476 Fn+1 ⊃ {xi }n+1 i=1 ∪ {vn } ∪ {wn }, Fn+1 ⊃ Fn , Gn+1 ⊃ Fn+1 ∪ Gn ,  Ad u2n+1 ◦ α2n−1 (x) − β2n (x) < 2−1 δ2n+1 , x ∈ G2n+1 ,  Ad u2n+2 ◦ β2n (x) − α2n+1 (x) < 2−1 δ2n+2 , x ∈ G2n+2 , v2n+1 α2n−1 (v2n+1 )∗ − u2n+1  < 2−2n−1 ε, [v2n+1 , x] < 2−2n−1 ε, x ∈ F2n , v2n+2 β2n (v2n+2 )∗ − u2n+2  < 2−2n−2 ε, [v2n+2 , x] < 2−2n−2 ε, x ∈ F2n+1 , δ2n+1  2−1 δ2n , and if u ∈ U (Z) satisfies that [u, y] < δ2n+1 for any y ∈ β2n (G2n+1 ), then there exist v ∈ U (Z) and λ ∈ T such that

  vβ2n (v)∗ − λu < 2−2n−2 ε,

  [v, x] < 2−2n−2 ,

for any x ∈ F2n+1 ,

(8) δ2n+2  2−1 δ2n+1 , and if u ∈ U (Z) satisfies that [u, y] < δ2n+2 for any y ∈ α2n+1 (G2n+2 ), then there exist v ∈ U (Z) and λ ∈ T such that   vα2n+1 (v)∗ − λu < 2−2n−3 ε,

  [v, x] < 2−2n−3 ,

for any x ∈ F2n+2 .

First, we construct F1 satisfying (1) for n = 0. Assuming that we have constructed Fn , Gn , un , vn , δn , n  2k, and F2k+1 satisfying (1) for n  2k, (2) for n  2k − 1, and (3)–(8) for n  k − 1, we proceed as follows: Since β2k has the weak Rohlin property, by Corollary 5.6, we obtain a finite subset G2k+1 and δ2k+1 > 0 satisfying (2) for n = 2k and (7) for n = k. Because any automorphism of the Jiang–Su algebra is approximately inner [14], we obtain u2k+1 ∈ U (Z) satisfying (3) for n = k. When we obtain G1 , δ1 , and u1 , take the same argument for k = 0. By (4) for n = k − 1, (3) for n = k, G2k ⊂ G2k+1 , and δ2k > δ2k+1 (when k > 0), we have that   [u2k+1 , y] < 2−1 (δ2k + δ2k+1 ) < δ2k , for any y ∈ α2k−1 (G2k ). Then by (8) for n = k − 1 we obtain v2k+1 ∈ U (Z) and λ2k+1 ∈ T such that   v2k+1 α2k−1 (v2k+1 )∗ − λ2k+1 u2k+1  < 2−2k−1 ε,

  [v2k+1 , x] < 2−2k−1 ,

for x ∈ F2k . When we obtain v1 , because for F = φ we may assume G = φ in Corollary 5.6, we obtain v1 ∈ U (Z) and λ1 ∈ T such that v1 α(v1 )∗ − λ1 u1  < 2−1 ε. Since Ad u2k+1 = Ad λ2k+1 u2k+1 , replacing u2k+1 we can obtain the ones which satisfy (3) and (5) for n = k (for u1 and v1 , we take k = 0). Let F2k+2 satisfy (1) for n = 2k + 1. Similarly, by the weak Rohlin property of α2k+1 and Corollary 5.6, we obtain a finite subset G2k+2 and δ2k+2 > 0 satisfying (2) for n = 2k + 1 and (8) for n = k. By approximately innerness of automorphisms of the Jiang–Su algebra, (3) for n = k, and (7) for n = k, we obtain unitaries u2k+2 and v2k+2 satisfying (4) and (6) for n = k. Finally we obtain a finite subset F2k+3 satisfying (1) for n = 2k + 2. This completes the induction. Set σ2n = Ad(v2n v2n−2 · · · v2 ) and σ2n+1 = Ad(v2n+1 v2n−1 · · · v1 ). From (1), (5),  and (6) it follows that [v2n+i , v2n−2+i ] < 2−(2n+i) for any n ∈ N and i = 0, 1. Then, since n∈N F2n is dense in Z 1 , we can define automorphisms of Z by  σ0 = lim σ2n ,

 σ1 = lim σ2n+1 .

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Indeed for x ∈ F2m and n > m it follows that σ2n+2 (x) − σ2n (x) < (2n + 1) · 2−2n−2 ε, then σ2n (x), n ∈ N is a Cauchy sequence. Similarly σ2n+1 (x), n ∈ N is also a Cauchy sequence. Thus −1 −1 σ1 . Since σ2n+2+i (x) − σ2n+i (x) < 2−2n−2−i , x ∈ F2m , we can define ∗-homomorphisms  σ0 ,  −1 , i = 0, 1, on Z. It is not so hard to n > m, we also define ∗-homomorphisms  σi−1 := lim σ2n+i −1 −1 σi = idZ =  σi ◦  σi , i = 0, 1. see that  σi ◦  By (1), (5) and (6), we see that w2n+i − 1 < 2−2n−i ε and [v2n+i , w2n−2+i ] < 2−2n−i ε, then w2n+i , n ∈ N, i = 0, 1 converge to w˜ i ∈ U (Z), i = 0, 1 such that w˜ i − 1 < ε. By (3), we have that for x ∈ F2n+1  Ad w

2n+1

 −1 −1 ◦ σ2n+1 ◦ α ◦ σ2n+1 (x) − Ad w2n ◦ σ2n ◦ β ◦ σ2n (x) < 2−1 δ2n+1 .

Since δn → 0, we conclude that Ad w˜ 1 ◦  σ1 ◦ α ◦  σ1−1 = Ad w˜ 0 ◦  σ0 ◦ β ◦  σ0−1 . This completes the proof.

2

Acknowledgments The author would like to thank his advisor Akitaka Kishimoto for many valuable discussions, and he also would like to thank Hiroki Matui for the preprint [20] and useful advices, as well as Wilhelm Winter and the reviewer for suggesting the interesting example Corollary 1.4. References [1] D. Archey, Crossed product C ∗ -algebras by finite group actions with the projection free tracial Rokhlin property, arXiv:0902.3324. [2] A. Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. Ec. Norm. Super. (4) 8 (1975) 383–419. [3] M. Dadarlat, N.C. Phillips, A.S. Toms, A direct proof of Z-stability for AH algebras of bounded topological dimension, arXiv:0806.2855. [4] M. Dadarlat, W. Winter, On the K-theory of strongly self-absorbing C ∗ -algebras, Math. Scand. 104 (1) (2009) 95–107. [5] G.A. Elliott, A.S. Toms, Regularity properties in the classification program for separable amenable C ∗ -algebras, Bull. Amer. Math. Soc. 45 (2) (2008) 229–245. [6] D.E. Evans, A. Kishimoto, Trace scaling automorphisms of certain stable AF algebras, Hokkaido Math. J. 26 (1997) 211–224. [7] G. Gong, X. Jiang, H. Su, Obstructions to Z-stability for unital simple C ∗ -algebras, Canad. Math. Bull. 43 (4) (2000) 418–426. [8] P. de la Harpe, G. Skandalis, Déterminant associé à une trace sur une algéebre de Banach, Ann. Inst. Fourier (Grenoble) 34 (1984) 241–260. [9] I. Hirshberg, W. Winter, Rokhlin actions and self-absorbing C ∗ -algebras, Pacific J. Math. 233 (1) (2007) 125–143. [10] M. Izumi, The Rohlin property for automorphisms of C ∗ -algebras, in: Mathematical Physics in Mathematics and Physics, Siena, 2000, in: Fields Inst. Commun., vol. 30, Amer. Math. Soc., Providence, RI, 2001, pp. 191–206. [11] M. Izumi, Finite group actions on C ∗ -algebras with the Rohlin property. I, Duke Math. J. 122 (2004) 233–280. [12] M. Izumi, Finite group actions on C ∗ -algebras with the Rohlin property. II, Adv. Math. 184 (2004) 119–160. [13] M. Izumi, H. Matui, Z2 -actions on Kirchberg algebras, preprint, arXiv:0902.0194. [14] X. Jiang, H. Su, On a simple unital projectionless C ∗ -algebra, Amer. J. Math. 121 (2) (1999) 359–413. [15] T. Katsura, H. Matui, Classification of uniformly outer actions of Z2 on UHF algebras, Adv. Math. 218 (2008) 940–968, arXiv:0708.4073. [16] A. Kishimoto, The Rohlin property for automorphisms of UHF algebras, J. Reine Angew. Math. 465 (1995) 183– 196.

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