aa r X i v : . [ m a t h . OA ] S e p Z -actions on AH algebras and Z -actions on AF algebras Hiroki MatuiGraduate School of ScienceChiba UniversityInage-ku, Chiba 263-8522, Japan
Abstract
We consider Z -actions (single automorphisms) on a unital simple AH algebra withreal rank zero and slow dimension growth and show that the uniform outerness impliesthe Rohlin property under some technical assumptions. Moreover, two Z -actions withthe Rohlin property on such a C ∗ -algebra are shown to be cocycle conjugate if they areasymptotically unitarily equivalent. We also prove that locally approximately innerand uniformly outer Z -actions on a unital simple AF algebra with a unique tracehave the Rohlin property and classify them up to cocycle conjugacy employing theOrderExt group as classification invariants. Classification of group actions is one of the most fundamental subjects in the theory ofoperator algebras. For AFD factors, a complete classification is known for actions ofcountable amenable groups. However, classification of automorphisms or group actions on C ∗ -algebras is still a far less developed subject, partly because of K -theoretical difficulties.For AF and AT algebras, A. Kishimoto [9, 10, 11, 12] showed the Rohlin property for acertain class of automorphisms and obtained a cocycle conjugacy result. Following thestrategy developed by Kishimoto, H. Nakamura [24] showed that aperiodic automorphismson unital Kirchberg algebras are completely classified by their KK -classes up to KK -trivial cocycle conjugacy. As for Z N -actions, Nakamura [23] introduced the notion of theRohlin property and classified product type actions of Z on UHF algebras. T. Katsuraand the author [8] gave a complete classification of uniformly outer Z -actions on UHFalgebras by using the Rohlin property. For Kirchberg algebras, M. Izumi and the author[7] classified a large class of Z -actions and also showed the uniqueness of Z N -actions onthe Cuntz algebras O and O ∞ . The present article is a continuation of these works.In the first half of this paper, we study single automorphisms (i.e. Z -actions) on aunital simple classifiable AH algebra. More precisely, we prove the Rohlin type theorem(Theorem 4.8) for an automorphism α of a unital simple AH algebra A with real rank zeroand slow dimension growth under the assumption that α r is approximately inner for some r ∈ N and A has finitely many extremal tracial states. Furthermore, we also prove that iftwo automorphisms α and β of a unital simple AH algebra with real rank zero and slowdimension growth have the Rohlin property and α ◦ β − is asymptotically inner, then thetwo Z -actions generated by α and β are cocycle conjugate (Theorem 4.9). These resultsare generalizations of Kishimoto’s work for AF and AT algebras ([9, 10, 11, 12]). For1he proofs, we need to improve some of the arguments in [11, 12] concerning projectionsand unitaries in central sequence algebras. As a byproduct, it will be also shown thatfor A as above, the central sequence algebra A ω satisfies Blackadar’s second fundamentalcomparability question (Proposition 3.8).In the latter half of the paper, we study Z -actions on a unital simple AF algebra A with a unique trace. We first show a Z -equivariant version of the Rohlin type theoremfor single automorphisms (Theorem 5.5), and as its corollary we obtain the Rohlin typetheorem for a Z -action ϕ : Z y A such that ϕ ( r, and ϕ (0 ,s ) are approximately innerfor some r, s ∈ N (Corollary 5.6). This is a generalization of [23, Theorem 3]. Next, byusing a Z -equivariant version of the Evans-Kishimoto intertwining argument [4], we classifyuniformly outer locally KK -trivial Z -actions on A up to KK -trivial cocycle conjugacy(Theorem 6.6). This is a generalization of [8, Theorem 6.5]. We remark that KK -trivialityof α ∈ Aut( A ) is equivalent to α being approximately inner, because A is assumed to beAF. For classification invariants, we employ the OrderExt group introduced in [13]. Thecrossed product of A by the first generator ϕ (1 , is known to be a unital simple AT algebrawith real rank zero. The second generator ϕ (0 , naturally extends to an automorphism ofthe crossed product and its OrderExt class gives the invariant of the Z -action ϕ . Howeverwe do not know the precise range of the invariant in general. We collect notations and terminologies relevant to this paper. For a Lipschitz continuousmap f between metric spaces, Lip( f ) denotes the Lipschitz constant of f . Let A be a unital C ∗ -algebra. For a, b ∈ A , we mean by [ a, b ] the commutator ab − ba . For a unitary u ∈ A ,the inner automorphism induced by u is written by Ad u . An automorphism α ∈ Aut( A )is called outer, when it is not inner. A single automorphism α is often identified with the Z -action induced by α . An automorphism α of A is said to be asymptotically inner, ifthere exists a continuous family of unitaries { u t } t ∈ [0 , ∞ ) in A such that α ( a ) = lim t →∞ Ad u t ( a )for all a ∈ A . When there exists a sequence of unitaries { u n } n ∈ N in A such that α ( a ) = lim n →∞ Ad u n ( a )for all a ∈ A , α is said to be approximately inner. The set of approximately innerautomorphisms of A is denoted by Inn( A ). Two automorphisms α and β are said tobe asymptotically (resp. approximately) unitarily equivalent if α ◦ β − is asymptotically(resp. approximately) inner. The set of tracial states on A is denoted by T ( A ). We meanby Aff( T ( A )) the space of R -valued affine continuous functions on T ( A ). The dimensionmap D A : K ( A ) → Aff( T ( A )) is defined by D A ([ p ])( τ ) = τ ( p ). For a homomorphism ρ between C ∗ -algebras, K ( ρ ) and K ( ρ ) mean the induced homomorphisms on K -groups.As for group actions on C ∗ -algebras, we freely use the terminology and notation de-scribed in [7, Definition 2.1].Let A be a separable C ∗ -algebra and let ω ∈ β N \ N be a free ultrafilter. We set c ( A ) = { ( a n ) ∈ ℓ ∞ ( N , A ) | lim n →∞ k a n k = 0 } , A ∞ = ℓ ∞ ( N , A ) /c ( A ) , ω ( A ) = { ( a n ) ∈ ℓ ∞ ( N , A ) | lim n → ω k a n k = 0 } , A ω = ℓ ∞ ( N , A ) /c ω ( A ) . We identify A with the C ∗ -subalgebra of A ∞ (resp. A ω ) consisting of equivalence classesof constant sequences. We let A ∞ = A ∞ ∩ A ′ , A ω = A ω ∩ A ′ and call them the central sequence algebras of A . A sequence ( x n ) n ∈ ℓ ∞ ( N , A ) is calleda central sequence if k [ a, x n ] k → n → ∞ for all a ∈ A . A central sequence is arepresentative of an element in A ∞ . An ω -central sequence is defined in a similar way.When α is an automorphism on A or an action of a discrete group on A , we can considerits natural extension on A ∞ , A ω , A ∞ and A ω . We denote it by the same symbol α .Next, we would like to recall the definition of uniform outerness introduced by Kishi-moto and the definition of the Rohlin property of Z N -actions introduced by Nakamura. Definition 2.1 ([9, Definition 1.2]) . An automorphism α of a unital C ∗ -algebra A is saidto be uniformly outer if for any a ∈ A , any non-zero projection p ∈ A and any ε > p , p , . . . , p n in A such that p = P p i and k p i aα ( p i ) k < ε for all i = 1 , , . . . , n .We say that an action α of a discrete group on A is uniformly outer if α g is uniformlyouter for every element g of the group other than the identity element.Let N be a natural number. Let ξ , ξ , . . . , ξ N be the canonical basis of Z N , that is, ξ i = (0 , , . . . , , . . . , , , where 1 is in the i -th component. For m = ( m , m , . . . , m N ) and n = ( n , n , . . . , n N ) in Z N , m ≤ n means m i ≤ n i for all i = 1 , , . . . , N . For m = ( m , m , . . . , m N ) ∈ N N , welet m Z N = { ( m n , m n , . . . , m N n N ) ∈ Z N | ( n , n , . . . , n N ) ∈ Z N } . For simplicity, we denote Z N /m Z N by Z m . Moreover, we may identify Z m = Z N /m Z N with { ( n , n , . . . , n N ) ∈ Z N | ≤ n i ≤ m i − ∀ i = 1 , , . . . , N } . Definition 2.2 ([23, Definition 1]) . Let ϕ be an action of Z N on a unital C ∗ -algebra A .Then ϕ is said to have the Rohlin property, if for any m ∈ N there exist R ∈ N and m (1) , m (2) , . . . , m ( R ) ∈ N N with m (1) , . . . , m ( R ) ≥ ( m, m, . . . , m ) satisfying the following:For any finite subset F of A and ε >
0, there exists a family of projections e ( r ) g ( r = 1 , , . . . , R, g ∈ Z m ( r ) )in A such that R X r =1 X g ∈ Z m ( r ) e ( r ) g = 1 , k [ a, e ( r ) g ] k < ε, k ϕ ξ i ( e ( r ) g ) − e ( r ) g + ξ i k < ε for any a ∈ F , r = 1 , , . . . , R , i = 1 , , . . . , N and g ∈ Z m ( r ) , where g + ξ i is understoodmodulo m ( r ) Z N . 3t is clear that if ϕ : Z N y A has the Rohlin property, then ϕ is uniformly outer.We recall the definition of tracial rank zero introduced by H. Lin. Definition 2.3 ([14, 15]) . A unital simple C ∗ -algebra A is said to have tracial rank zeroif for any finite subset F ⊂ A , any ε > x ∈ A thereexists a finite dimensional subalgebra B ⊂ A with p = 1 B satisfying the following.(1) k [ a, p ] k < ε for all a ∈ F .(2) The distance from pap to B is less than ε for all a ∈ F .(3) 1 − p is Murray-von Neumann equivalent to a projection in xAx .In [16], H. Lin gave a classification theorem for unital separable simple nuclear C ∗ -algebras with tracial rank zero which satisfy the UCT. ([16, Theorem 5.2]). Indeed, theclass of such C ∗ -algebras agrees with the class of all unital simple AH algebras with realrank zero and slow dimension growth. Lemma 3.1.
Let A be a unital separable approximately divisible C ∗ -algebra. Then forany m ∈ N , there exists a unital embedding of M m ⊕ M m +1 into A ∞ .Proof. Let l = m −
1. For any finite subset F ⊂ A and ε >
0, there exists a unital finitedimensional subalgebra B ⊂ A such that every central summand of B is at least l × l and k [ a, b ] k < ε for any a ∈ F and b ∈ B with k b k ≤
1. It is easy to find a unital subalgebra C of B such that C ∼ = M m ⊕ M m +1 , which completes the proof.The following is Lemma 3.6 of [11]. Lemma 3.2.
Let A be a unital simple AT algebra with real rank zero. For any finitesubset F ⊂ A and ε > , there exist a finite subset G ⊂ A , δ > and k ∈ N such thatthe following holds. If p, q ∈ A are projections satisfying k [ p ] ≤ [ q ] , k [ a, p ] k < δ and k [ a, q ] k < δ for all a ∈ G , then there exists a partial isometry v ∈ A such that v ∗ v = p , vv ∗ ≤ q and k [ a, v ] k < ε for all a ∈ F . We generalize the lemma above to AH algebras.
Lemma 3.3.
The above lemma also holds for any unital simple AH algebra with slowdimension growth and real rank zero.Proof.
Let A be a unital simple AH algebra with slow dimension growth and real rankzero and let Q be the UHF algebra such that K ( Q ) ∼ = Q . By the classification theorem([2, 1, 5]), A ⊗ Q is a unital simple AT algebra with real rank zero. Let F ⊂ A be afinite subset and ε >
0. Applying the lemma above to { a ⊗ | a ∈ F } ⊂ A ⊗ Q and ε/ >
0, we obtain a finite subset G ⊂ A ⊗ Q , δ > k ∈ N . We may assume G = { a ⊗ | a ∈ G } ∪ { ⊗ b | b ∈ G } , where G and G are finite subsets of A and Q ,respectively. We may further assume that G contains F and δ is less than ε/
2. We willprove that G , δ and 4 k meet the requirement.4uppose that p, q ∈ A are non-zero projections satisfying 4 k [ p ] ≤ [ q ], k [ a, p ] k < δ and k [ a, q ] k < δ for all a ∈ G . By Lemma 3.1, M ⊕ M embeds into A ∞ . Hence there exista projection r and a partial isometry s such that r ≤ q, s ∗ s = r, ss ∗ ≤ q − r, r ] > [ q ]and k [ a, s ] k < δ, k [ a, r ] k < δ ∀ a ∈ G . From 4 k [ p ] ≤ [ q ] < r ], we get k [ p ] < [ r ]. It follows that there exists a partial isometry u ∈ A ⊗ Q such that u ∗ u = p ⊗ uu ∗ ≤ r ⊗ k [ a ⊗ , u ] k < ε/ a ∈ F . Wemay assume that u belongs to some A ⊗ M m ⊂ A ⊗ Q . Put u = ( u i,j ) ≤ i,j ≤ m . Define w = ( w i,j ) ≤ i,j ≤ m +1 ∈ A ⊗ M m +1 by w i,j = u i,j if 1 ≤ i, j ≤ msu i, if i = m + 1 and j = m + 10 if i = m + 1 . It is not so hard to see that w ∗ w = p ⊗ ∈ A ⊗ M m +1 and ww ∗ ≤ q ⊗ ∈ A ⊗ M m +1 .Moreover, one has k [ a ⊗ , w ] k < ε for all a ∈ F . Let v = u ⊕ w ∈ A ⊗ ( M m ⊕ M m +1 ).Then v ∗ v = p ⊗ vv ∗ ≤ q ⊗ k [ a ⊗ , v ] k < ε for all a ∈ F .By [3] (see also [1, 5]), A is approximately divisible. By Lemma 3.1, there exists a unitalhomomorphism from M m ⊕ M m +1 to A ∞ , and so there exists a unital homomorphism π from A ⊗ ( M m ⊕ M m +1 ) to A ∞ such that π ( a ⊗
1) = a for a ∈ A . It follows that π ( v ) ∗ π ( v ) = p , π ( v ) π ( v ) ∗ ≤ q and k [ a, π ( v )] k < ε for all a ∈ F . Remark 3.4.
By using the lemma above and [27, Theorem 4.5], one can show the follow-ing. Let A be a unital simple AH algebra with real rank zero and slow dimension growth.Then for any α ∈ Aut( A ), there exists ˜ α ∈ Aut( A ) such that ˜ α has the Rohlin propertyin the sense of [12, Definition 4.1] and ˜ α is asymptotically unitarily equivalent to α .The following is a well-known fact. We have been unable to find a suitable referencein the literature, so we include a proof for completeness. Lemma 3.5.
Let A be a unital separable C ∗ -algebra and let ( p n ) n be a central sequenceof projections. For any extremal trace τ ∈ T ( A ) , one has lim n →∞ | τ ( ap n ) − τ ( a ) τ ( p n ) | = 0 for all a ∈ A .Proof. First, we deal with the case that there exists ε > τ ( p n ) ≥ ε for all n ∈ N . Consider a sequence of states ϕ n ( a ) = τ ( ap n ) τ ( p n )on A . Let ψ be an accumulation point of { ϕ n } n . Since ( p n ) n is a central sequence and τ ( p n ) is bounded from below, one can see that ψ is a trace. For any positive element5 ∈ A , it is easy to see ϕ n ( a ) ≤ ε − τ ( a ), and so ψ ( a ) ≤ ε − τ ( a ). Hence, ψ is equal to τ ,because τ is extremal. We have shown that any accumulation point of { ϕ n } is τ , whichimplies ϕ n converges to τ . Therefore, | τ ( ap n ) − τ ( a ) τ ( p n ) | goes to zero.Next, we consider the general case. Fix a ∈ A . Take ε > | τ ( ap n ) − τ ( a ) τ ( p n ) | is less than ε for sufficiently large n . We may assume k a k ≤
1. Put C = { n ∈ N | τ ( p n ) ≥ ε/ } . If n / ∈ C , then evidently | τ ( ap n ) − τ ( a ) τ ( p n ) | is less than ε . By applying the first partof the proof to ( p n ) n ∈ C , we have | τ ( ap n ) − τ ( a ) τ ( p n ) | < ε for sufficiently large n ∈ C ,thereby completing the proof. Lemma 3.6.
Let A be a unital simple separable C ∗ -algebra with tracial rank zero. Supposethat A has finitely many extremal traces. For any finite subset F ⊂ A and ε > , thereexist a finite subset G ⊂ A and δ > such that the following hold. If p, q ∈ A areprojections satisfying k [ a, p ] k < δ, k [ a, q ] k < δ ∀ a ∈ G and τ ( p ) + ε < τ ( q ) for all τ ∈ T ( A ) , then there exists a partial isometry v ∈ A such that v ∗ v ≤ p , vv ∗ ≤ q , k [ a, v ] k < ε ∀ a ∈ F and τ ( p − v ∗ v ) < ε for all τ ∈ T ( A ) .Proof. The proof is by contradiction. Suppose that the assertion does not hold for a finitesubset F ⊂ A and ε >
0. We would have central sequences of projections ( p n ) n and ( q n ) n such that τ ( p n ) + ε < τ ( q n ) ∀ τ ∈ T ( A ) , n ∈ N and any partial isometry does not meet the requirement.Use tracial rank zero to find a projection e ∈ A and a finite dimensional unital subal-gebra E ⊂ eAe such that the following are satisfied. • For any a ∈ F , k [ a, e ] k < ε/ • For any a ∈ F there exists b ∈ E such that k eae − b k < ε/ • τ (1 − e ) < ε for all τ ∈ T ( A ).Since ( p n ) n and ( q n ) n are central sequences of projections, we can find projections p ′ n and q ′ n in A ∩ E ′ such that k p n − p ′ n k → k q n − q ′ n k → n → ∞ . We would like to show that,for sufficiently large n , there exists a partial isometry v n ∈ eAe ∩ E ′ such that v ∗ n v n = ep ′ n and v n v ∗ n ≤ eq ′ n . Let { e , e , . . . , e m } be a family of minimal central projections in eAe ∩ E ′ such that e + e + · · · + e m = e . Clearly e i ( eAe ∩ E ′ ) is a unital simple C ∗ -algebra withtracial rank zero and the space of tracial states on e i ( eAe ∩ E ′ ) is naturally identified with T ( A ). By lemma 3.5, for sufficiently large n , one has τ ( e i p ′ n ) < τ ( e i q ′ n ) for all τ ∈ T ( A )and i = 1 , , . . . , m , because A has finitely many extremal traces. Hence [ e i p ′ n ] ≤ [ e i q ′ n ] in K ( e i ( eAe ∩ E ′ )). It follows that there exists a partial isometry v n ∈ eAe ∩ E ′ such that v ∗ n v n = ep ′ n and v n v ∗ n ≤ eq ′ n . Besides, τ ( p ′ n − v ∗ n v n ) = τ ( p ′ n (1 − e )) < ε and k [ a, v n ] k < ε for all a ∈ F . This is a contradiction. 6y using Lemma 3.6 and 3.3, we can show the following. Lemma 3.7.
Let A be a unital simple AH algebra with slow dimension growth and realrank zero. Suppose that A has finitely many extremal traces. For any finite subset F ⊂ A and ε > , there exist a finite subset G ⊂ A and δ > such that the following hold. If p, q ∈ A are projections satisfying k [ a, p ] k < δ, k [ a, q ] k < δ ∀ a ∈ G and τ ( p ) + ε < τ ( q ) for all τ ∈ T ( A ) , then there exists a partial isometry u ∈ A such that u ∗ u = p , uu ∗ ≤ q and k [ a, u ] k < ε for all a ∈ F .Proof. Notice that A has tracial rank zero by [15, Proposition 2.6]. Suppose that a finitesubset F ⊂ A and ε > F and ε/
2, we obtain afinite subset F ⊂ A , ε > k ∈ N . By applying Lemma 3.6 to F ∪ F ∪ F ∗ andmin { ε / , ε/k, ε/ } , we obtain a finite subset G ⊂ A , δ >
0. We would like to show that G ∪ F and min { δ, ε / } meet the requirement. Suppose that p, q ∈ A are projectionssatisfying k [ a, p ] k < min { δ, ε / } , k [ a, q ] k < min { δ, ε / } ∀ a ∈ G ∪ F and τ ( p ) + ε < τ ( q ) for all τ ∈ T ( A ). By Lemma 3.6, there exists a partial isometry v ∈ A such that v ∗ v ≤ p , vv ∗ ≤ q , k [ a, v ] k < min { ε / , ε/ } ∀ a ∈ F ∪ F ∪ F ∗ and τ ( p − v ∗ v ) < ε/k for all τ ∈ T ( A ). Let p ′ = p − v ∗ v and q ′ = q − vv ∗ . One has τ ( q ′ ) = τ ( q − vv ∗ ) = τ ( q ) − τ ( p ) + τ ( p − v ∗ v ) > ε, and so k [ p ′ ] ≤ [ q ′ ]. One also has k [ a, p ′ ] k < ε and k [ a, q ′ ] k < ε for all a ∈ F . By Lemma3.3, we obtain a partial isometry w ∈ A such that w ∗ w = p ′ , ww ∗ ≤ q ′ and k [ a, w ] k < ε/ a ∈ F . Put u = v + w . It is easy to see u ∗ u = p , uu ∗ ≤ q and k [ a, u ] k < ε for all a ∈ F .Any tracial state τ ∈ T ( A ) naturally extends to a tracial state on A ω and we write itby τ ω . Proposition 3.8.
Let A be a unital simple AH algebra with slow dimension growth and realrank zero. Suppose that A has finitely many extremal traces. If p, q ∈ A ω are projectionssatisfying τ ω ( p ) < τ ω ( q ) for all τ ∈ T ( A ) , then there exists v ∈ A ω such that v ∗ v = p and vv ∗ ≤ q . In particular, A ω satisfies Blackadar’s second fundamental comparabilityquestion.Proof. Let ( p n ) n and ( q n ) n be ω -central sequences of projections such thatlim n → ω τ ( p n ) < lim n → ω τ ( q n )for all τ ∈ T ( A ). Since A has finitely many extremal traces, there exists ε > C = { n ∈ N | τ ( p n ) + ε < τ ( q n ) for all τ ∈ T ( A ) } ∈ ω.
7e choose an increasing sequence { F m } ∞ m =1 of finite subsets of A whose union is densein A . By applying Lemma 3.7 to F m and ε/m , we obtain a finite subset G m ⊂ A and δ m >
0. We may assume that { G m } m is increasing and { δ m } m is decreasing. Put C m = { n ∈ C | k [ a, p n ] k < δ m and k [ a, q n ] k < δ m for all a ∈ G m } ∈ ω. For n ∈ C m \ C m +1 , by Lemma 3.7, there exists a partial isometry u n such that u ∗ n u n = p n , u n u ∗ n ≤ q n and k [ a, u n ] k < ε/m for all a ∈ F m . For n ∈ N \ C , we let u n = 0. Then ( u n ) n is a desired ω -central sequence of partial isometries.The following is Lemma 4.4 of [12]. Lemma 3.9.
Let A be a unital simple AT algebra with real rank zero. For any finite subset F ⊂ A and ε > , there exist a finite subset G ⊂ A and δ > such that the followingholds. If u : [0 , → A is a path of unitaries satisfying k [ a, u ( t )] k < δ for all a ∈ G and t ∈ [0 , , then there exists a path of unitaries v : [0 , → A satisfying v (0) = u (0) , v (1) = u (1) , k [ a, v ( t )] k < ε ∀ a ∈ F, t ∈ [0 , and Lip( v ) < π + 1 . We generalize the lemma above to AH algebras.
Lemma 3.10.
The above lemma also holds for any unital simple AH algebra with slowdimension growth and real rank zero, the Lipschitz constant being bounded by π .Proof. Let A be a unital simple AH algebra with slow dimension growth and real rankzero and let Q be the UHF algebra such that K ( Q ) ∼ = Q . By the classification theorem([2, 1, 5]), A ⊗ Q is a unital simple AT algebra with real rank zero. Let F ⊂ A be a finitesubset and ε >
0. We may assume that F is contained in the unit ball of A . Applyingthe lemma above to { a ⊗ | a ∈ F } ⊂ A ⊗ Q and ε/ >
0, we obtain a finite subset G ⊂ A ⊗ Q and δ >
0. We may assume G = { a ⊗ | a ∈ G } ∪ { ⊗ b | b ∈ G } , where G and G are finite subsets of A and Q , respectively. We will prove that G and δ meet therequirement.Suppose that u : [0 , → A is a path of unitaries satisfying k [ a, u ( t )] k < δ for all a ∈ G and t ∈ [0 , N ∈ N so that k u ( kN ) − u ( k +1 N ) k < ε/ k = 0 , , . . . , N − u k = u ( k/N ). By the lemma above, we can find continuous paths x : [0 , → A ⊗ Q , y k : [0 , → A ⊗ Q and z k : [0 , → A ⊗ Q for k = 1 , , . . . , N − x (0) = u ⊗ , x (1) = u N ⊗ , k [ a ⊗ , x ( t )] k < ε/ ∀ a ∈ F, t ∈ [0 , ,y k (0) = u ⊗ , y k (1) = u k ⊗ , k [ a ⊗ , y k ( t )] k < ε/ ∀ a ∈ F, t ∈ [0 , ,z k (0) = u k ⊗ , z k (1) = u N ⊗ , k [ a ⊗ , z k ( t )] k < ε/ ∀ a ∈ F, t ∈ [0 , , and Lip( x ) , Lip( y k ) , Lip( z k ) are less than 5 π + 1. We may assume that the ranges of x, y k , z k are contained in A ⊗ M n for some M n ⊂ Q .Put m = n ( N − v : [0 , → A ⊗ ( M m ⊕ M m +1 ) such that Lip( v ) < π , v (0) = u ⊗ v (1) = u N ⊗ k [ a ⊗ , v ( t )] k < ε for all a ∈ F and t ∈ [0 , x : [0 , → A ⊗ M m be the direct sum of N − x : [0 , → A ⊗ M n . Next, by using y , y , · · · , y N − , we can find a path˜ y : [0 , → A ⊗ M m +1 such that ˜ y (0) = u ⊗ , ˜ y (1) = diag( u , u , · · · , u | {z } n , u , · · · , u | {z } n , · · · , u N − , · · · , u N − | {z } n ) k [ a ⊗ , ˜ y ( t )] k < ε/ ∀ a ∈ F, t ∈ [0 , y ) < π + 1. Likewise, by using z , z , . . . , z N − , we can find a path ˜ z : [0 , → A ⊗ M m +1 such that˜ z (0) = diag( u , · · · , u | {z } n , u , · · · , u | {z } n , · · · , u N − , · · · , u N − | {z } n , u N )˜ z (1) = u N ⊗ , k [ a ⊗ , ˜ z ( t )] k < ε/ ∀ a ∈ F, t ∈ [0 , z ) < π + 1. Since k ˜ y (1) − ˜ z (0) k < ε/
2, if ε is sufficiently small, there exists apath w : [0 , → M m +1 such that w (0) = u ⊗ , w (1) = u N ⊗ , k [ a ⊗ , w ( t )] k < ε ∀ a ∈ F, t ∈ [0 , , and Lip( w ) < π . Then v = ˜ x ⊕ w is the desired path.By [3] (see also [1, 5]), A is approximately divisible. By Lemma 3.1, there exists a unitalhomomorphism from M m ⊕ M m +1 to A ∞ , and so there exists a unital homomorphism π from A ⊗ ( M m ⊕ M m +1 ) to A ∞ such that π ( a ⊗
1) = a for a ∈ A . It follows that the path˜ v : [0 , ∋ t π ( v ( t )) ∈ A ∞ satisfies˜ v (0) = u , ˜ v (1) = u N , k [ a, ˜ v ( t )] k < ε ∀ a ∈ F, t ∈ [0 , v ) < π , which completes the proof. In this section, we discuss the Rohlin property of automorphisms of AH algebras. For a ∈ A , we define k a k = sup τ ∈ T ( A ) τ ( a ∗ a ) / . If A is simple and T ( A ) is non-empty, then k·k is a norm. Proposition 4.1.
Let A be a unital simple separable C ∗ -algebra with tracial rank zeroand let Γ ⊂ Aut( A ) be a finite subset containing the identity. Suppose that there exists asequence of projections ( e n ) n in A satisfying the following property. (1) k γ ( e n ) γ ′ ( e n ) k → for any γ, γ ′ ∈ Γ such that γ = γ ′ . (2) k − P γ ∈ Γ γ ( e n ) k → . (3) For every a ∈ A , we have k [ a, e n ] k → . hen there exists a sequence of projections ( f n ) n in A satisfying the following. (1) k γ ( f n ) γ ′ ( f n ) k → for any γ, γ ′ ∈ Γ such that γ = γ ′ . (2) k e n − f n k → . (3) For every a ∈ A , we have k [ a, f n ] k → .Proof. This is almost the same as [19, Proposition 5.4]. In [19, Proposition 5.4], the finiteset Γ is assumed to be an orbit of a single automorphism γ of finite order. The proof,however, does not need this.The following is a variant of [9, Lemma 3.1] and [26, Theorem 2.17]. See [26, Definition2.1] for the definition of the tracial Rohlin property. Theorem 4.2.
Let A be a unital simple separable C ∗ -algebra with tracial rank zero. Sup-pose that A has finitely many extremal traces. Let α be an automorphism of A such that α m is uniformly outer for any m ∈ N . Then α has the tracial Rohlin property.Proof. Let { τ , . . . , τ d } be the set of extremal tracial states of A and let ( π i , H i ) be theGNS representation associated with τ i . It is well-known that π i ( A ) ′′ is a hyperfinite II -factor (see [26, Lemma 2.16]). Let ρ = L di =1 π i . Note that, for a bounded sequence( a n ) n in A , ρ ( a n ) converges to zero in the strong operator topology if and only if k a n k converges to zero. We regard A as a subalgebra of N = ρ ( A ) ′′ ∼ = L di =1 π i ( A ) ′′ and denotethe extension of the automorphism α to N by ¯ α . Let k be the minimum positive integersuch that τ i ◦ α k = τ i for all i = 1 , , . . . , d . In the same way as [9, Lemma 3.1], for any l ∈ N , one can find a sequence { f ( j )0 , . . . , f ( j ) kl − } of orthogonal families of projections in N such that P kl − i =0 f ( j ) i = 1, [ a, f ( j ) i ] → ∀ a ∈ A, ¯ α ( f ( j ) i ) − f ( j ) i +1 → ∀ i = 0 , , . . . , kl − j → ∞ , where f ( j ) kl = f ( j )0 . By [26, Lemma 2.15], wemay replace the projections f ( j ) i with projections of A . From the proposition above, wecan conclude that α has the tracial Rohlin property.The following is a well-known fact, but we include the proof for the reader’s conve-nience. Lemma 4.3.
Let A be a unital separable C ∗ -algebra and let α ∈ Inn( A ) . For any separablesubset C ⊂ A ∞ , there exists a unitary u ∈ A ∞ such that uxu ∗ = α ( x ) for all x ∈ C .Proof. We choose an increasing sequence { F n } n ∈ N of finite subsets of A whose union isdense in A . We can find a sequence of unitaries ( v n ) n in A such that k v n av ∗ n − α − ( a ) k < n − for all a ∈ F n , because α is approximately inner. We may assume that C is countable.Let C = { x , x , . . . } and let ( x i,j ) j be a representative of x i . There exists an increasingsequence ( m ( n )) n of natural numbers such that k [ v n , x i,j ] k < n − ∀ j ≥ m ( n )10or any i = 1 , , . . . , n , because ( x i,j ) j is a central sequence. Since α is in Inn( A ), one canfind a sequence of unitaries ( w n ) n in A such that k w n aw ∗ n − α ( a ) k < n − for all a in α − ( F n ) ∪ { x i,j | i = 1 , . . . , n, m ( n ) ≤ j < m ( n +1) } . For j ∈ N , find n ∈ N so that m ( n ) ≤ j < m ( n +1) and define a unitary u j by u j = w n v n .It is easy to see k [ u j , a ] k < /n ∀ a ∈ F n and k u j x i,j u ∗ j − α ( x i,j ) k < /n ∀ i = 1 , . . . , n, and so the proof is completed.We quote the following theorem by Lin and Osaka from [20]. See [20, Definition 2.4] forthe definition of the tracial cyclic Rohlin property. Since we need to discuss an ‘equivariantversion’ of this theorem later, we would like to include the proof briefly. Theorem 4.4 ([20, Theorem 3.4]) . Let A be a unital simple separable C ∗ -algebra andsuppose that the order on projections in A is determined by traces. Suppose that α ∈ Aut( A ) has the tracial Rohlin property. If α r is in Inn( A ) for some r ∈ N , then α has thetracial cyclic Rohlin property.Proof. Take m ∈ N and ε > l = r ( mr + 1). Since α has the tracialRohlin property and the order on projections is determined by traces, there exists a centralsequence of projections ( e n ) n such thatlim n →∞ k e n α i ( e n ) k = 0 ∀ i = 1 , , . . . , l − n →∞ τ (1 − ( e n + α ( e n ) + · · · + α l − ( e n )) = 0 ∀ τ ∈ T ( A ) . Let e ∈ A ∞ be the image of ( e n ) n and define ˜ e by˜ e = r − X i =0 α i ( mr +1) ( e ) . It follows from the lemma above that there exists a partial isometry v ∈ A ∞ such that v ∗ v = ˜ e and vv ∗ = α (˜ e ). The C ∗ -algebra C generated by v, α ( v ) , . . . , α mr − ( v ) is isomor-phic to M mr +1 and its unit is ˜ e + α (˜ e ) + · · · + α mr (˜ e ). The rest of the proof is exactly thesame as that of [9, Lemma 4.3] and we omit it. Remark 4.5.
The following was shown by Lin in [17, Theorem 3.4]. Let A be a unitalsimple separable C ∗ -algebra with tracial rank zero. Suppose that α ∈ Aut( A ) has thetracial cyclic Rohlin property and that there exists r ∈ N such that K ( α r ) | G = id G forsome subgroup G ⊂ K ( A ) for which D A ( G ) is dense in D A ( K ( A )). Then A ⋊ α Z hastracial rank zero. 11y using Lemma 3.3, we can show the following. Lemma 4.6.
Let A be a unital simple AH algebra with slow dimension growth and realrank zero. Suppose that α ∈ Aut( A ) has the tracial cyclic Rohlin property and that thereexists r ∈ N such that τ ◦ α r = τ for any τ ∈ T ( A ) . Then, for any m ∈ N , there existprojections e, f ∈ A ∞ and a partial isometry v ∈ A ∞ such that v ∗ v = f, vv ∗ ≤ e, f + mr X i =0 α i ( e ) = 1 and α mr +1 ( e ) = e .Proof. Suppose that we are given m ∈ N . Let l = r ( mr + 1). Since α has the tracial cyclicRohlin property, we can find central sequences of projections ( e n ) n and ( f n ) n such that f n + l − X i =0 α i ( e n ) → , e n − α l ( e n ) → , sup τ ∈ T ( A ) τ ( f n ) → n → ∞ . There exists a central sequence of projections (˜ e n ) n such thatlim n →∞ ˜ e n − r − X i =0 α i ( mr +1) ( e n ) = 0 . Then f n + mr X i =0 α i (˜ e n ) → , ˜ e n − α mr +1 (˜ e n ) → n → ∞ . It is also easy to see τ ( α (˜ e n )) = τ (˜ e n ) for all τ ∈ T ( A ), and so τ (˜ e n ) goes to( mr + 1) − for all τ ∈ T ( A ). Therefore, by Lemma 3.3, one can find a central sequence ofpartial isometries ( v n ) n such that v ∗ n v n = f n and v n v ∗ n ≤ ˜ e n for sufficiently large n , whichcompletes the proof.By using the lemma above, we can show the following theorem. Theorem 4.7.
Let A be a unital simple AH algebra with slow dimension growth and realrank zero. Suppose that α ∈ Aut( A ) has the tracial cyclic Rohlin property. If there exists r ∈ N such that τ ◦ α r = τ for any τ ∈ T ( A ) , then α has the Rohlin property.Proof. Suppose that we are given M ∈ N . Choose a natural number m ∈ N so that m ≥ M and m ≡ r ). Let k, l be sufficiently large natural numbers satisfying k ≡ l ≡ r ). By the lemma above, we can find projections e, f ∈ A ∞ and a partial isometry v ∈ A ∞ such that v ∗ v = f, vv ∗ ≤ e, f + klm − X i =0 α i ( e ) = 1and α klm ( e ) = e . Define ˜ e, w ∈ A ∞ by˜ e = k − X i =0 α ilm ( e ) and w = 1 √ k k − X i =0 α ilm ( v ) . e is a projection and w is a partial isometry satisfying f + lm − X i =0 α i (˜ e ) = 1 , α lm (˜ e ) = ˜ e and w ∗ w = f, ww ∗ ≤ ˜ e, k α lm ( w ) − w k ≤ √ k . Let D be the C ∗ -algebra generated by w, α ( w ) , . . . , α lm − ( w ). Then D is isomorphic to M lm +1 and the unit 1 D of D is equal to f + ww ∗ + · · · + α lm − ( ww ∗ ). From the spectralproperty of α restricted to D , if k and l are sufficiently large, we can obtain projections p , . . . , p m − , q , . . . , q m of D such that m − X i =1 p i + m X i =1 q i = 1 D , α ( p i ) ≈ p i +1 , α ( q i ) ≈ q i +1 , where p m = p and q m +1 = q . We define projections p ′ i in A ∞ by p ′ i = p i + l − X j =0 α i + jm (˜ e − ww ∗ ) . Then the projections p ′ , . . . , p ′ m − , q , . . . , q m meet the requirement. See [9, 10] for details.Combining the theorems above, we obtain the following theorem which is a general-ization of [11, Theorem 2.1]. Theorem 4.8.
Let A be a unital simple AH algebra with slow dimension growth and realrank zero and let α ∈ Aut( A ) . Suppose that A has finitely many extremal traces and that α r is approximately inner for some r ∈ N . Then the following are equivalent. (1) α has the Rohlin property. (2) α m is uniformly outer for any m ∈ N . We can also generalize [12, Theorem 5.1] by using Lemma 3.10 instead of [12, Lemma4.4].
Theorem 4.9.
Let A be a unital simple AH algebra with slow dimension growth and realrank zero. If α, β ∈ Aut( A ) have the Rohlin property and α is asymptotically unitarilyequivalent to β , then there exist µ ∈ Inn( A ) and a unitary u ∈ A such that Ad u ◦ α = µ ◦ β ◦ µ − . The proof is similar to that of [12, Theorem 5.1] and we omit it.As an application of the theorems above, we can show the following, which will beused in Section 6. 13 emma 4.10.
Let A be a unital simple AF algebra with finitely many extremal traces andlet α be an approximately inner automorphism of A such that α m is uniformly outer forall m ∈ N . For any finite subset F ⊂ A and ε > , there exist a finite subset G ⊂ A and δ > satisfying the following. If u : [0 , → A is a path of unitaries such that k [ a, u ( t )] k < δ and k u ( t ) − α ( u ( t )) k < δ ∀ a ∈ G, t ∈ [0 , , then there exists a path of unitaries v : [0 , → A such that v (0) = u (0) , v (1) = u (1) , k [ a, v ( t )] k < ε, k v ( t ) − α ( v ( t )) k < ε, ∀ a ∈ F, t ∈ [0 , and Lip( v ) < π .Proof. Let x n be a unitary of M n ( C ) such that Sp( x n ) = { ω k | k = 0 , , . . . , n − } , where ω = exp(2 π √− /n ). One can find an increasing sequence { A n } ∞ n =1 of unital finite dimen-sional subalgebras of A such that S n A n is dense in A and there exists a unital embedding π n : M n ⊕ M n +1 → A n +1 ∩ A ′ n . Let y n = π n ( x n ⊕ x n +1 ). Define an automorphism σ of A by σ = lim n →∞ Ad( y y . . . y n ). Then σ is approximately inner and σ m is uniformly outerfor all m ∈ N .We would like to show that the assertion holds for σ . Suppose that we are given F ⊂ A 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 A n . Applying [8, Lemma 4.2] to ε/
2, we obtain a positivereal number δ >
0. We may assume δ is less than min { , ε } . Choose a finite subset G ⊂ A and δ > z : [0 , → A is a path of unitaries such that k [ a, z ( t )] k < δ for all a ∈ G and t ∈ [0 , z : [0 , → A ∩ A ′ n suchthat k z ( t ) − ˜ z ( t ) k < δ /
6. Let δ = min { δ / , δ } . Suppose that u : [0 , → A is a path ofunitaries such that k [ a, u ( t )] k < δ and k u ( t ) − σ ( u ( t )) k < δ ∀ a ∈ G, t ∈ [0 , . By the choice of δ , we can find ˜ u : [0 , → A ∩ A ′ n such that k u ( t ) − ˜ u ( t ) k < δ /
6. Wemay assume that there exists m > n such that the range of ˜ u is contained in A m . Put y = y n y n +1 . . . y m − ∈ A m ∩ A ′ n . Then k [ y, ˜ u ( t )] k = k ˜ u ( t ) − σ (˜ u ( t )) k < k u ( t ) − σ ( u ( t )) k + δ / < δ + δ / ≤ δ / t ∈ [0 , k [ y, ˜ u ( t )˜ u (0) ∗ ] k is less than δ . It follows from [8, Lemma 4.2]that one can find a path of unitaries w : [0 , → A m ∩ A ′ n such that w (0) = 1 , w (1) = ˜ u (1)˜ u (0) ∗ , Lip( w ) ≤ π + ε and k [ y, w ( t )] k < ε/ t ∈ [0 , yw ( t ) y ∗ is equal to σ ( w ( t )). Byperturbing w ( t ) u (0) a little bit, the required v : [0 , → A is obtained.Suppose that α is an approximately inner automorphism of A such that α m is uniformlyouter for all m ∈ N . By Theorem 4.8 and Theorem 4.9, there exist µ ∈ Aut( A ) and aunitary u ∈ A such that Ad u ◦ α = µ ◦ σ ◦ µ − . Moreover, one can choose u arbitrarilyclose to 1, because A is AF (see [6, 12]). Therefore the assertion also holds for α . Remark 4.11.
In the proof of the lemma above, it is easily seen that (the Z -action gener-ated by) σ is asymptotically representable ([7, Definition 2.2]). Hence the automorphism α stated in the lemma above is also asymptotically representable. Besides, it is not sohard to see that the crossed product C ∗ -algebra A ⋊ σ Z is a unital simple AT algebra withreal rank zero. Therefore, A ⋊ α Z is a unital simple AT algebra with real rank zero, too.14 The Rohlin property of Z -actions on AF algebras In this section, we would like to show that certain Z -actions on an AF algebra have theRohlin property. This is a generalization of Nakamura’s theorem [23, Theorem 3].Throughout this section, we keep the following setting. Let A be a unital simpleseparable C ∗ -algebra with tracial rank zero and suppose that A has a unique tracial state τ . Suppose that automorphisms α, β ∈ Aut( A ) and a unitary w ∈ A satisfy β ◦ α = Ad w ◦ α ◦ β and α m ◦ β n is uniformly outer for all ( m, n ) ∈ Z \ { (0 , } . We remark that α and β induce a Z -action on A ∞ . Lemma 5.1.
For any m , m ∈ N , there exists a central sequence of projections ( e n ) n in A such that lim n →∞ τ ( e n ) = 1 m m and lim n →∞ k β j ( α i ( e n )) β l ( α k ( e n )) k = 0 for all ( i, j ) = ( k, l ) in { ( i, j ) | ≤ i ≤ m − , ≤ j ≤ m − } .Proof. Set I = { ( i, j ) | ≤ i ≤ m − , ≤ j ≤ m − } . Let ( π τ , H τ ) be the GNSrepresentation associated with τ . It is well-known that π τ ( A ) ′′ is a hyperfinite II -factor(see [26, Lemma 2.16]). For ( i, j ) ∈ Z , we put ϕ ( i,j ) = α i ◦ β j . Then ϕ : Z → Aut( A ) is acocycle action of Z . We denote its extension to π τ ( A ) ′′ by ¯ ϕ . Since α m ◦ β n is uniformlyouter for all ( m, n ) ∈ Z \ { (0 , } , ¯ ϕ is an outer cocycle action of Z on π τ ( A ) ′′ . It followsfrom [25] that there exists a sequence of projections ( e n ) n in π τ ( A ) ′′ such that X ( i,j ) ∈ I β j ( α i ( e n )) → , [ x, e n ] → ∀ x ∈ π τ ( A ) ′′ and β j ( α i ( e n )) β l ( α k ( e n )) → ∀ ( i, j ) , ( k, l ) ∈ I with ( i, j ) = ( k, l )in the strong operator topology as n → ∞ . By [26, Lemma 2.15], we may replace e n withprojections in A . Applying Proposition 4.1 to Γ = { β j ◦ α i | ( i, j ) ∈ I } , we obtain theconclusion.Let ( e n ) n be the projections as in the lemma above. If α r is in Inn( A ) for some r ∈ N and m is large enough, then we can construct a central sequence of projections ( e ′ n ) n in A such that lim n →∞ k e ′ n ( e n + α ( e n ) + · · · + α m − ( e n )) − e ′ n k = 0 ,α ( e ′ n ) ≈ e ′ n and τ ( e ′ n ) ≈ m τ ( e n ) , by using the arguments in [9, Lemma 3.1] (see also [23, Lemma 6]). Consequently, we getthe following. 15 emma 5.2. If α r is in Inn( A ) for some r ∈ N , then for any m ∈ N , there exists a centralsequence of projections ( e n ) n in A such that lim n →∞ τ ( e n ) = 1 m , lim n →∞ k e n − α ( e n ) k = 0 and lim n →∞ k e n β j ( e n ) k = 0 for all j = 1 , , . . . , m − . Our next task is to achieve the cyclicity condition β m ( e n ) ≈ e n . Lemma 5.3.
Suppose that A is AF. Suppose that a projection e ∈ A ∞ and a partialisometry u ∈ A ∞ satisfy e = α ( e ) and e = u ∗ u = uu ∗ . Then there exists a partialisometry w ∈ A ∞ such that w ∗ w = ww ∗ = e and u = w ∗ α ( w ) .Proof. Theorem 4.8 tells us that α possesses the Rohlin property. We can modify thestandard argument deducing stability from the Rohlin property (see [6, 4]) and apply itto the unitary u + (1 − e ). We leave the details to the readers. Lemma 5.4.
Suppose that either of the following holds. (1) A is AF, α r is approximately inner for some r ∈ N and β s is approximately innerfor some s ∈ N . (2) α r is approximately inner for some r ∈ N and there exist a natural number s ∈ N and a sequence of unitaries ( u n ) n in A such that lim n →∞ k u n − α ( u n ) k = 0 and lim n →∞ k u n au ∗ n − β s ( a ) k = 0 ∀ a ∈ A. Then for any m ∈ N , there exists a central sequence of projections ( e n ) n such that lim n →∞ τ ( e n ) = 1 m , lim n →∞ k e n − α ( e n ) k = 0 , lim n →∞ k e n β j ( e n ) k = 0 for all j = 1 , , . . . , m − and lim n →∞ k e n − β m ( e n ) k = 0 . Proof.
Choose a large natural number l such that l ≡ s ). By using Lemma 5.2and the assumption that β s is in Inn( A ) for some s ∈ N , one can find a projection e ∈ A ∞ and a partial isometry v ∈ A ∞ such that e = α ( e ) , v ∗ v = e, vv ∗ = β ( e ) and eβ j ( e ) = 0 ∀ j = 1 , , . . . , l − n →∞ τ ( e n ) = l − , where ( e n ) n is a representative sequence of e consisting of projections. Note that β j ( e ) isfixed by α , because e is a central sequence. In the case (2), clearly we may further assume v = α ( v ). In the case (1), the lemma above applies to v ∗ α ( v ) and yields w ∈ A ∞ satisfying w ∗ w = e , ww ∗ = e and v ∗ α ( v ) = w ∗ α ( w ). By replacing v with vw ∗ , we get v = α ( v ), too.Then the conclusion follows from exactly the same argument as [9, Lemma 4.3].16 heorem 5.5. Suppose that the conclusion of Lemma 5.4 holds. Then for any m ∈ N ,there exist projections e and f in A ∞ such that α ( e ) = e, α ( f ) = f, β m ( e ) = e, β m +1 ( f ) = f and m − X i =0 β i ( e ) + m X j =0 β j ( f ) = 1 . Proof.
Let ( e n ) n be the central sequence of projections obtained in Lemma 5.4. Define f n = 1 − m − X j =0 β j ( e n ) . There exists a sequence of unitaries ( u n ) n in A such that u n → n → ∞ and u n α ( e n ) u ∗ n = e n for sufficiently large n . The Z -action on e n Ae n generated by Ad u n ◦ α isuniformly outer, and so it has the tracial Rohlin property by Theorem 4.2 (or [26, Theorem2.17]). It follows that, for any k ∈ N , there exists a central sequence of projections (˜ e n ) n such that˜ e n ≤ e n , lim n →∞ τ (˜ e n ) = 1 /mk, and lim n →∞ k ˜ e n α i (˜ e n ) k = 0 ∀ i = 1 , , . . . , k − . Let e, f, ˜ e ∈ A ∞ be the images of ( e n ) n , ( f n ) n , (˜ e n ) n , respectively. By Lemma 3.3, thereexists a partial isometry v such that v ∗ v = f and vv ∗ ≤ ˜ e . We define a partial isometry˜ v ∈ A ∞ by ˜ v = 1 √ k k − X i =0 α i ( v ) . Then one has ˜ v ∗ ˜ v = f, ˜ v ˜ v ∗ ≤ e and k ˜ v − α (˜ v ) k < / √ k. By a standard trick on central sequences, we may assume α (˜ v ) = ˜ v . Thus, we haveobtained the α -invariant version of the conclusion of Lemma 4.6. We can complete theproof by the same argument as in Theorem 4.7.The following is a generalization of [23, Theorem 3]. Corollary 5.6.
Let ϕ : Z y A be a Z -action on a unital simple AF algebra A withunique trace. When ϕ ( r, and ϕ (0 ,s ) are approximately inner for some r, s ∈ N , thefollowing are equivalent. (1) ϕ has the Rohlin property. (2) ϕ is uniformly outer.Proof. This immediately follows from Theorem 5.5 and [23, Remark 2] (see also [22, Re-mark 2.2]).The next corollary also follows from Theorem 5.5 immediately, because condition (2) ofLemma 5.4 is satisfied in this case. See [7, Definition 2.2] for the definition of approximaterepresentability. 17 orollary 5.7.
Let ϕ : Z y A be an approximately representable Z -action on a unitalsimple AH algebra A with real rank zero and slow dimension growth. Suppose that A hasa unique trace. Then the following are equivalent. (1) ϕ has the Rohlin property. (2) ϕ is uniformly outer. Z -actions on AF algebras In this section, we will show a classification result of a certain class of Z -actions onunital simple AF algebras. We freely use the terminology and notation introduced in [7,Definition 2.1]. For an automorphism α of a C ∗ -algebra A , we write the crossed product C ∗ -algebra A ⋊ α Z by C ∗ ( A, α ) and the implementing unitary by λ α . The mapping torus M ( A, α ) is defined by M ( A, α ) = { f ∈ C ([0 , , A ) | α ( f (0)) = f (1) } . When A is an AF algebra, ‘ KK -triviality’ of α ∈ Aut( A ) is equivalent to K ( α ) = id, andalso equivalent to α being approximately inner.The following theorem is a Z -equivariant version of Theorem 4.9. Let A be a unitalsimple AF algebra with unique trace and let α ∈ Inn( A ). Let Aut T ( C ∗ ( A, α )) denote theset of all automorphisms of C ∗ ( A, α ) commuting with the dual action ˆ α . For i = 1 ,
2, wesuppose that an automorphism β i ∈ Aut( A ) and a unitary w i ∈ A are given and satisfy β i ◦ α = Ad w i ◦ α ◦ β i . Then β i extends to ˜ β i ∈ Aut T ( C ∗ ( A, α )) by setting ˜ β i ( λ α ) = w i λ α . Suppose further that α m ◦ β ni is uniformly outer for all ( m, n ) ∈ Z \ { (0 , } and that β s i i is approximately innerfor some s i ∈ N . Theorem 6.1.
In the setting above, if ˜ β and ˜ β are asymptotically unitarily equivalent,then there exist an approximately inner automorphism µ ∈ Aut T ( C ∗ ( A, α )) and a unitary v ∈ A such that µ | A is also approximately inner and µ ◦ ˜ β ◦ µ − = Ad v ◦ ˜ β . Proof.
We can apply the argument of [24, Theorem 5] to ˜ β and ˜ β in a similar fashionto [7, Theorem 4.11]. By Remark 4.11, (the Z -action generated by) α is asymptoticallyrepresentable. Then [7, Theorem 4.8] implies that ˜ β and ˜ β are T -asymptotically unitarilyequivalent. Moreover, by Theorem 5.5, we can find Rohlin projections for ˜ β i in the fixedpoint algebra ( A ∞ ) α . Hence, by using Lemma 4.10 instead of [24, Theorem 7], the usualintertwining argument shows the statement.Let us recall the OrderExt invariant introduced in [13]. Let G , G , F be abelian groupsand let D : G → F be a homomorphism. When ξ : 0 −−−−→ G ι −−−−→ E ξ q −−−−→ G −−−−→ 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 ξ : 0 −−−−→ G −−−−→ E ξ −−−−→ G −−−−→ (cid:13)(cid:13)(cid:13) y θ (cid:13)(cid:13)(cid:13) ξ ′ : 0 −−−−→ G −−−−→ E ξ ′ −−−−→ G −−−−→ G , G , D ) consists of equivalence classes of all order-extensions. As shown in [13], OrderExt( G , G , D ) is equipped with an abelian groupstructure. The map sending ( ξ, R ) to ξ induces a homomorphism from OrderExt( G , G , D )onto Ext( G , G ).Let B be a unital C ∗ -algebra with T ( B ) non-empty. We denote by Aut ( B ) the set ofall automorphisms γ of B such that K ( γ ) = K ( γ ) = id and τ ◦ γ = τ for all τ ∈ T ( B ).When B is a unital simple AT algebra with real rank zero, Aut ( B ) equals Inn( B ). Let D B : K ( B ) → Aff( T ( B )) denote the dimension map defined by D B ([ p ])( τ ) = τ ( p ). Asdescribed in [13], there exist natural homomorphisms˜ η : Aut ( B ) → OrderExt( K ( B ) , K ( B ) , D B )and η : Aut ( B ) → Ext( K ( B ) , K ( B )) . The following is the main result of [13]. See [21, 18] for further developments.
Theorem 6.2 ([13, Theorem 4.4]) . Suppose that B is a unital simple AT algebra with realrank zero. Then the homomorphism ˜ η ⊕ η : Inn( B ) → OrderExt( K ( B ) , K ( B ) , D B ) ⊕ Ext( K ( B ) , K ( B )) is surjective and its kernel equals the set of all asymptotically inner automorphisms of B . By using this OrderExt invariant, we introduce an invariant of certain Z -actions asfollows. Let A be a unital simple AF algebra and let ϕ : Z y A be an action of Z on A . Suppose that ϕ is uniformly outer and locally KK -trivial (i.e. locally approximatelyinner). We write B = C ∗ ( A, ϕ (1 , ). Then ϕ (0 , extends to ˜ ϕ (0 , ∈ Aut( B ) by setting˜ ϕ (0 , ( λ ϕ (1 , ) = λ ϕ (1 , . Let ι : A → B = C ∗ ( A, ϕ (1 , ) be the canonical inclusion. One cancheck the following immediately. • K ( ι ) is an isomorphism from K ( A ) to K ( B ). • The connecting map ∂ : K ( B ) → K ( A ) in the Pimsner-Voiculescu exact sequenceis an isomorphism and ∂ − ([ p ]) = [ λ ϕ (1 , ι ( p ) + ι ( v (1 − p ))] for any projection p ∈ A ,where v is a unitary of A satisfying vpv ∗ = ϕ (1 , ( p ). • The map ι ∗ : T ( B ) → T ( A ) sending τ to τ ◦ ι is an isomorphism and satisfies D B ( K ( ι )( x ))( τ ) = D A ( x )( ι ∗ ( τ )) for x ∈ K ( A ) and τ ∈ T ( B ).19rom these properties, we can obtain a natural isomorphism ζ ϕ (1 , : OrderExt( K ( B ) , K ( B ) , D B ) → OrderExt( K ( A ) , K ( A ) , D A ) . In addition, it is easy to see K ( ˜ ϕ (0 , ) = K ( ˜ ϕ (0 , ) = id and τ ◦ ˜ ϕ (0 , = τ for all τ ∈ T ( B ),that is, ˜ ϕ (0 , belongs to Aut ( B ). Lemma 6.3.
In the setting above, η ( ˜ ϕ (0 , ) ∈ Ext( K ( B ) , K ( B )) is zero.Proof. There exists a natural commutative diagram0 −−−−→ C ((0 , , B ) −−−−→ M ( B, ˜ ϕ (0 , ) −−−−→ B −−−−→ x x x ι −−−−→ C ((0 , , A ) −−−−→ M ( A, ϕ (0 , ) −−−−→ A −−−−→ , where the horizontal sequences are exact. From the naturality of the six-term exactsequence, we obtain the commutative diagram0 −−−−→ K ( B ) −−−−→ K ( M ( B, ˜ ϕ (0 , )) −−−−→ K ( B ) −−−−→ x x x K ( ι ) −−−−→ K ( A ) −−−−→ K ( M ( A, ϕ (0 , )) −−−−→ K ( A ) −−−−→ , where the horizontal sequences are exact. Since K ( A ) is zero and K ( ι ) is an isomorphism,we can conclude η ( ˜ ϕ (0 , ) = 0. Definition 6.4.
In the setting above, we define our invariant [ ϕ ] by[ ϕ ] = ζ ϕ (1 , (˜ η ( ˜ ϕ (0 , )) ∈ OrderExt( K ( A ) , K ( A ) , D A ) . Proposition 6.5.
Let ϕ, ψ : Z y A be uniformly outer, locally KK -trivial Z -actionson a unital simple AF algebra A . If ϕ and ψ are KK -trivially cocycle conjugate, then [ ϕ ] = [ ψ ] .Proof. For µ ∈ Inn( A ), it is straightforward to see that the Z -action µ ◦ ϕ ◦ µ − has thesame invariant as ϕ . Hence, it suffices to show [ ϕ ] = [ ϕ u ] for any ϕ -cocycle { u n } n ∈ Z .Define an isomorphism π from C ∗ ( A, ϕ (1 , ) to C ∗ ( A, ϕ u (1 , ) by π ( λ ϕ (1 , ) = u ∗ (1 , λ ϕ u (1 , and π ( a ) = a ∀ a ∈ A, where A is identified with subalgebras of the crossed products. For γ ∈ Aut( C ∗ ( A, ϕ u (1 , )),one can check ζ ϕ (1 , (˜ η ( π − ◦ γ ◦ π )) = ζ ϕ u (1 , (˜ η ( γ )) ∈ OrderExt( K ( A ) , K ( A ) , D A ) , where ˜ η in the left hand side is defined for C ∗ ( A, ϕ (1 , ) and ˜ η in the right hand side isdefined for C ∗ ( A, ϕ u (1 , ). We also have( π − ◦ ˜ ϕ u (0 , ◦ π )( a ) = π − ( ˜ ϕ u (0 , ( a )) = ˜ ϕ u (0 , ( a ) = (Ad u (0 , ◦ ˜ ϕ (0 , )( a ) ∀ a ∈ A π − ◦ ˜ ϕ u (0 , ◦ π )( λ ϕ (1 , ) = ( π − ◦ ˜ ϕ u (0 , )( u ∗ (1 , λ ϕ u (1 , )= π − ( ˜ ϕ u (0 , ( u ∗ (1 , ) λ ϕ u (1 , )= ϕ u (0 , ( u ∗ (1 , ) u (1 , λ ϕ (1 , = u (0 , ϕ (0 , ( u ∗ (1 , ) u ∗ (0 , u (1 , λ ϕ (1 , = u (0 , ϕ (1 , ( u ∗ (0 , ) u ∗ (1 , u (1 , λ ϕ (1 , = u (0 , ϕ (1 , ( u ∗ (0 , ) λ ϕ (1 , = u (0 , λ ϕ (1 , u ∗ (0 , = (Ad u (0 , ◦ ˜ ϕ (0 , )( λ ϕ (1 , ) . Thus π − ◦ ˜ ϕ u (0 , ◦ π = Ad u (0 , ◦ ˜ ϕ (0 , . Since inner automorphisms are contained in thekernel of ˜ η , we obtain ζ ϕ (1 , (˜ η ( ˜ ϕ (0 , )) = ζ ϕ (1 , (˜ η (Ad u (0 , ◦ ˜ ϕ (0 , ))= ζ ϕ (1 , (˜ η ( π − ◦ ˜ ϕ u (0 , ◦ π ))= ζ ϕ u (1 , (˜ η ( ˜ ϕ u (0 , )) , which completes the proof. Theorem 6.6.
Let ϕ, ψ : Z y A be uniformly outer, locally KK -trivial Z -actions on aunital simple AF algebra A with unique trace. The following are equivalent. (1) [ ϕ ] = [ ψ ] . (2) ϕ and ψ are KK -trivially cocycle conjugate.Proof. (2) ⇒ (1) was shown in the proposition above without assuming that A has a uniquetrace. Let us consider the other implication (1) ⇒ (2). By Theorem 4.8 and Theorem 4.9, wemay assume that there exists a unitary u ∈ A such that ψ (1 , = Ad u ◦ ϕ (1 , . By Theorem4.8 and Remark 4.11 (or Remark 4.5), the crossed product C ∗ -algebra C ∗ ( A, ϕ (1 , ) is aunital simple AT algebra with real rank zero.Clearly ϕ (0 , extends to ˜ ϕ (0 , ∈ Aut( C ∗ ( A, ϕ (1 , )) by˜ ϕ (0 , ( a ) = a ∀ a ∈ A and ˜ ϕ (0 , ( λ ϕ (1 , ) = λ ϕ (1 , . Since ψ (0 , ◦ ϕ (1 , = Ad( ψ (0 , ( u ∗ ) u ) ◦ ϕ (1 , ◦ ψ (0 , , we can extend ψ (0 , to ω ∈ Aut( C ∗ ( A, ϕ (1 , )) by ω ( a ) = a ∀ a ∈ A and ω ( λ ϕ (1 , ) = ψ (0 , ( u ∗ ) uλ ϕ (1 , . In order to apply Theorem 6.1 to ˜ ϕ (0 , and ω , we would like to check that these au-tomorphisms are asymptotically unitarily equivalent. There exists an isomorphism π : C ∗ ( A, ϕ (1 , ) → C ∗ ( A, ψ (1 , ) defined by π ( a ) = a ∀ a ∈ A and π ( λ ϕ (1 , ) = u ∗ λ ψ (1 , .
21s mentioned in the proof of Proposition 6.5, for any γ ∈ Aut( C ∗ ( A, ϕ (1 , )), one has ζ ϕ (1 , (˜ η ( γ )) = ζ ψ (1 , (˜ η ( π ◦ γ ◦ π − )) . Moreover it is easy to see that π ◦ ω ◦ π − is equal to ˜ ψ (0 , , which is defined by˜ ψ (0 , ( a ) = a ∀ a ∈ A and ˜ ψ (0 , ( λ ψ (1 , ) = λ ψ (1 , . It follows that ζ ϕ (1 , (˜ η ( ω )) = ζ ψ (1 , (˜ η ( π ◦ ω ◦ π − ))= ζ ψ (1 , (˜ η ( ˜ ψ (0 , )) = [ ψ ] = [ ϕ ] = ζ ϕ (1 , (˜ η ( ˜ ϕ (0 , )) , and so ˜ η ( ω ) = ˜ η ( ˜ ϕ (0 , ). By Lemma 6.3, η ( ω ) = η ( ˜ ϕ (0 , ) = 0. Therefore, by Theorem6.2, ˜ ϕ (0 , and ω are asymptotically unitarily equivalent.Then, Theorem 6.1 applies and yields an approximately inner automorphism µ ∈ Aut T ( C ∗ ( A, ϕ (1 , )) and a unitary v ∈ A such that µ | A is in Inn( A ) and µ ◦ ω ◦ µ − = Ad v ◦ ˜ ϕ (0 , . (6.1)By restricting this equality to A , we get( µ | A ) ◦ ψ (0 , ◦ ( µ | A ) − = Ad v ◦ ϕ (0 , . (6.2)Let z ∈ A be the unitary satisfying µ ( λ ϕ (1 , ) = zλ ϕ (1 , . Then( µ | A ) ◦ ψ (1 , ◦ ( µ | A ) − = ( µ | A ) ◦ Ad u ◦ ϕ (1 , ◦ ( µ | A ) − = Ad µ ( u ) z ◦ ϕ (1 , . (6.3)From (6.1), one can see that( µ ◦ ω ◦ µ − )( λ ϕ (1 , ) = ( µ ◦ ω )( µ − ( z ∗ ) λ ϕ (1 , )= µ ( ψ (0 , ( µ − ( z ∗ )) ψ (0 , ( u ∗ ) uλ ϕ (1 , )= (Ad v ◦ ϕ (0 , )( z ∗ µ ( u ∗ )) µ ( u ) zλ ϕ (1 , = vϕ (0 , ( z ∗ µ ( u ∗ )) v ∗ µ ( u ) zλ ϕ (1 , is equal to (Ad v ◦ ˜ ϕ (0 , )( λ ϕ (1 , ) = vλ ϕ (1 , v ∗ = vϕ (1 , ( v ∗ ) λ ϕ (1 , . Hence one obtains vϕ (0 , ( µ ( u ) z ) = µ ( u ) zϕ (1 , ( v ) . (6.4)It follows from (6.2), (6.3), (6.4) that ψ and ϕ are KK -trivially cocycle conjugate.22 emark 6.7. We do not know the precise range of our invariant which takes its val-ues in OrderExt. At least, the following observation shows that the range does notexhaust OrderExt. Let ϕ : Z y A be a locally KK -trivial and uniformly outer Z -action on a unital simple AF algebra. Suppose that ( ξ, R ) is a representative of [ ϕ ] ∈ OrderExt( K ( A ) , K ( A ) , D A ). Since ξ : 0 −−−−→ K ( A ) ι −−−−→ E ξ q −−−−→ K ( A ) −−−−→ R : E ξ → Aff( T ( A )) satisfies R ◦ ι = D A , there exists a homomorphism R : K ( A ) → Aff( K ( A )) / Im D A such that R ( q ( x )) = R ( x ) + D A ( K ( A )) for any x ∈ E ξ . It is easy to see R ([1 A ]) = 0, because the implementing unitary λ ϕ (1 , is fixedby ˜ ϕ (0 , . Thus, [ ϕ ] belongs to the subgroup { [( ξ, R )] ∈ OrderExt( K ( A ) , K ( A ) , D A ) | R ([1 A ]) = 0 } . When A is a UHF algebra, one can see that this subgroup coincides with the range of theinvariant introduced in [8]. Therefore, Theorem 6.6 yields a new proof of [8, Theorem 6.5]. References [1] M. Dadarlat,
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