aa r X i v : . [ m a t h . OA ] A ug Z -STABILITY OF C( X ) ⋊ Γ ZHUANG NIU
Abstract.
Let ( X, Γ) be a free and minimal topological dynamical system, where X is aseparable compact Hausdorff space and Γ is a countable infinite discrete amenable group.It is shown that if ( X, Γ) has the Uniform Rokhlin Property and Cuntz comparison of opensets, then mdim( X, Γ) = 0 implies that (C( X ) ⋊ Γ) ⊗ Z ∼ = C( X ) ⋊ Γ, where mdim is themean dimension and Z is the Jiang-Su algebra. In particular, in this case, mdim( X, Γ) = 0implies that the C*-algebra C( X ) ⋊ Γ is classified by the Elliott invariant. Introduction
Let Γ be a discrete amenable group, and let (Ω , µ ) be a σ -finite standard measure space.Let (Ω , µ ) x Γ be a free and ergodic action with absolutely continuous finite invariantmeasure. By the classification of injective von Neumann algebras, it is well known that thevon Neumann II -factor L ∞ (Ω , µ ) ⋊ Γ is isomorphic to the unique hyperfinite II -factor R .Thus, all such crossed products L ∞ (Ω , µ ) ⋊ Γ are isomorphic.In the topological setting, consider a compact separable Hausdorff space X , and considera minimal and free action X x Γ. Then the crossed product C*-algebra C( X ) ⋊ Γ is simpleseparable unital nuclear and satisfies the UCT. Thus it is a very natural object for theElliott’s classification program of nuclear C*-algebras.Many efforts have been devoted to the classifiability of C( X ) ⋊ Γ (in term of the K-theoretical Elliott invariant); see, for instance, [23], [16] [15], [29], [26], [25], [32], etc. How-ever, as shown by Giol and Kerr in [7], there exist minimal and free actions X x Z suchthat the C*-algebras A = C( X ) ⋊ Z cannot be classified by the Elliott invariant, and theseC*-algebras do not absorb the Jiang-Su algebra Z tensorially (i.e., A ⊗ Z ≇ A ).The dynamical systems constructed in [7] have non-zero mean (topological) dimension;and in [5], it is shown that if a minimal and free Z -action has zero mean dimension (thisparticularly includes all strictly ergodic systems and all minimal dynamical systems withfinite topological entropy, see [17]), then the C*-algebra C( X ) ⋊ Z must be Z -absorbing andis classifiable (see [6] and [4]).In this note, one considers an arbitrary discrete amenable group Γ, and studies the Z -stability of C( X ) ⋊ Γ. Under the assumption that ( X, Γ) has the Uniform Rokhlin Property(URP) and Cuntz comparison of open sets (COS), which are introduced in [21], one hasthat mdim( X, Γ) = 0 implies that (C( X ) ⋊ Γ) ⊗ Z ∼ = C( X ) ⋊ Γ, where mdim is the meandimension. In particular, this implies that C( X ) ⋊ Γ is classified by its Elliott invariant.Recall
Date : August 11, 2020.The research is supported by an NSF grant (DMS-1800882).
Definition 1.1 (Definition 3.1 and Definition 4.1 of [21]) . A topological dynamical system( X, Γ), where Γ is a discrete amenable group, is said to have Uniform Rokhlin Property(URP) if for any ε > K ⊆ Γ, there exist closed sets B , B , ..., B S ⊆ X and ( K, ε )-invariant sets Γ , Γ , ..., Γ S ⊆ Γ such that B s γ, γ ∈ Γ s , s = 1 , ..., S, are mutually disjoint and ocap( X \ S G s =1 G γ ∈ Γ s B s γ ) < ε, where ocap denote the orbit capacity (see, for instance, Definition 5.1 of [18]).The dynamical system ( X, Γ) is said to have ( λ, m )-Cuntz-comparison of open sets, where λ ∈ (0 ,
1] and m ∈ N , if for any open sets E, F ⊆ X with µ ( E ) < λµ ( F ) , µ ∈ M ( X, Γ) , where M ( X, Γ) is the simplex of all invariant probability measures on X , then ϕ E - ϕ F ⊕ · · · ⊕ ϕ F | {z } m in C( X ) ⋊ Γ , where ϕ E and ϕ F are continuous functions supporting on E and F respectively.The dynamical system ( X, Γ) is said to have Cuntz comparison of open sets (COS) if ithas ( λ, m )-Cuntz-comparison on open sets for some λ and m .The properties of (URP) and (COS) seem to hold for all free and minimal actions by afinitely generated discrete amenable group: any free minimal Z d -action has the (URP) andhas ( , (2 ⌊√ d ⌋ +1) d +1)-Cuntz-comparison of open sets ([20]); any free and minimal Γ-actionhas the (URP) and has ( , X, Γ) is an extension of a Cantor system ([21]).In [21], it is shown that if ( X, Γ) has the (URP) and (COS), then the comparison radiusof the C*-algebra C( X ) ⋊ Γ is at most half of the mean dimension of ( X, Γ). In particular,if mdim( X, Γ) = 0, then the C*-algebra C( X ) ⋊ Γ has the strict comparison of positiveelements (see Definition 3.2), which, as a part of the Toms-Winter conjecture, to imply the Z -stability (this has been verified in the case that the C*-algebra has finitely many extremetracial states in [19], and then been generalized independently to the case that the set ofextreme tracial states is finite dimensional in [24], [13], and [28], and then to the case thatthe algebra has Uniform Property Gamma in [2]).Under the assumption that ( X, Γ) has the small boundary property (SBP) (which implieszero mean dimension, see [18], and is shown in [17] and [10] to be equivalent to zero meandimension in the case Γ = Z d ), Kerr and Szabo show in [12] (Theorem 9.4) that the C*-algebra C( X ) ⋊ Γ has the Uniform Property Gamma, and hence the strict comparison ofpositive elements implies Z -stability for C( X ) ⋊ Γ.In this note, one shows the following: -STABILITY OF C( X ) ⋊ Γ 3
Theorem (Theorem 4.9) . Let ( X, Γ) be a free and minimal dynamical system with the (URP)and (COS). If ( X, Γ) has mean dimension zero, then (C( X ) ⋊ Γ) ⊗ Z ∼ = C( X ) ⋊ Γ , where Z is the Jiang-Su algebra.In particular, let ( X , Γ ) and ( X , Γ ) be two free minimal dynamical systems with the(URP) and (COS), and zero mean dimension, then C( X ) ⋊ Γ ∼ = C( X ) ⋊ Γ if and only if Ell(C( X ) ⋊ Γ ) ∼ = Ell(C( X ) ⋊ Γ ) , where Ell( · ) = (K ( · ) , K +0 ( · ) , [1] , T( · ) , ρ, K ( · )) is the Elliott invariant. Moreover, these C*-algebras are inductive limits of unital subhomogeneous C*-algebras. As a consequence, the following crossed-product C*-algebras are Z -stable: Corollary (Corollary 4.10) . Let ( X, Γ) be a free and minimal dynamical system with meandimension zero. Assume that • either Γ = Z d for some d ≥ , or • ( X, Γ) is an extension of a Cantor system and Γ has subexponetial growth.Then, the C*-algebra C( X ) ⋊ Γ is classified by the Elliott invariant and is an inductive limitof unital subhomogeneous C*-algebras. Two approaches are included in this note: The first approach is more self-contained andmore C*-algebra oriented. It is to show that the C*-algebra C( X ) ⋊ Γ being considered istracially Z -stable; since C( X ) ⋊ Γ is nuclear, it follows from [19] and [11] that C( X ) ⋊ Γactually is Z -stable.In the second approach (Section 5), one proves the following dynamical system statement:mdim0 + URP ⇒ SBP . If, in addition, the system is assumed to have the (COS), it follows from [21] that the C*-algebra C( X ) ⋊ Γ has strict comparison of positive elements. Hence, with the SBP, the Z -stability of C( X ) ⋊ Γ also follows from the Theorem 9.4 and Corollary 9.5 of [12].2.
Notation and preliminaries
Topological Dynamical Systems.Definition 2.1.
A topological dynamical system ( X, Γ) consists of a separable compactHausdorff space X , a discrete group Γ, and a homomorphism Γ → Homeo( X ), whereHomeo( X ) is the group of homeomorphisms of X , acting on X from the right. In thispaper, we frequently omit the word topological, and just refer it as a dynamical system.The dynamical system ( X, Γ) is said to be free if xγ = x implies γ = e , where x ∈ X and γ ∈ Γ.A closed set Y ⊆ X is said to be invariant if Y γ = Y, γ ∈ Γ , ZHUANG NIU and the dynamical system ( X, Γ) is said to be minimal if ∅ and X are the only invariantclosed subsets. Definition 2.2.
A Borel measure µ on X is invariant if for any Borel set E ⊆ X , one has µ ( E ) = µ ( Eγ ) , γ ∈ Γ . Denote by M ( X, Γ) the set of all invariant Borel probability measures on X . It is a Choquetsimplex under the weak* topology. Definition 2.3.
Let Γ be a (countable) discrete group. Let K ⊆ Γ be a finite set and let δ >
0. Then a finite set F ⊆ Γ is said to be (
K, ε )-invariant if | F K ∆ F || F | < ε. The group Γ is amenable if there is a sequence (Γ n ) of finite subsets of Γ such that for any( K, ε ), the set Γ n is ( K, ε )-invariant if n is sufficiently large. The sequence (Γ n ) is called aFølner sequence.The K -interior of a finite set F ⊆ Γ is defined asint K ( F ) = { γ ∈ F : γK ⊆ F } . Note that | F \ int K ( F ) | ≤ | K | | F K \ F | ≤ | K | | F K ∆ F | , and hence for any ε >
0, if F is ( K, ε | K | )-invariant, then | F \ int K ( F ) || F | < ε. Definition 2.4 (see [18]) . Consider a topological dynamical system ( X, Γ), where Γ isamenable, and let E ⊆ X . The orbit capacity of E is defined byocap( E ) := lim n →∞ | Γ n | sup x ∈ X X γ ∈ Γ n χ E ( xγ ) , where (Γ n ) is a Følner sequence, and χ E is the characteristic function of E . The limit alwaysexists and is independent from the choice of the Følner sequence (Γ n ). Definition 2.5 (see [8] and [18]) . Let U be an open cover of X . Define D ( U ) = min { ord( V ) : V is an open cover of X and V (cid:22) U } , where ord( V ) = − x ∈ X X V ∈V χ V ( x ) , and V (cid:22) U means that, for any V ∈ V , there is U ∈ U with V ⊆ U .Consider a topological dynamical system ( X, Γ), where Γ is a discrete amenable group.The mean topological dimension is defined bymdim( X, Γ) := sup U lim n →∞ | Γ n | D ( _ γ ∈ Γ n γ − ( U )) , -STABILITY OF C( X ) ⋊ Γ 5 where U runs over all finite open covers of X , (Γ n ) is a Følner sequence (the limit is inde-pendent from the choice of (Γ n )), and α ∨ β denotes the open cover { U ∩ V : U ∈ α, V ∈ β } for any open covers α and β .2.2. Crossed product C*-algebras.
Consider a topological dynamical system ( X, Γ). The(full) crossed product C*-algebra A = C( X ) ⋊ Γ is defined to be the universal C*-algebraC* { f, u γ ; u γ f u ∗ γ = f ( · γ ) = f ◦ γ, u γ u ∗ γ = u γ γ − , u e = 1 , f ∈ C( X ) , γ, γ , γ ∈ Γ } . The C*-algebra A is nuclear (Corollary 7.18 of [30]) if Γ is amenable. If, moreover, ( X, Γ)is minimal and topologically free, the C*-algebra A is simple (Theorem 5.16 of [3] andTh´eor`eme 5.15 of [34]), i.e., A has no non-trivial two-sided ideals.3. The Cuntz semigroup of C( X ) ⋊ Γ Definition 3.1.
Let A be a C*-algebra, and let a, b ∈ A + . The element a is said to beCuntz sub-equivalent to b , denoted by a - b , if there are x i , y i , i = 1 , , ... , such thatlim i →∞ x i by i = a, and we say that a is Cuntz equivalent to b , denoted by a ∼ b , if a - b and b - a . Then theCuntz semigroup of A , denoted by W( A ), is defined as(M ∞ ( A )) + / ∼ with the addition [ a ] + [ b ] = (cid:20)(cid:18) a b (cid:19)(cid:21) , where (M ∞ ( A )) + := S ∞ n =1 M + n ( A ) and [ · ] denotes the equivalence class. Definition 3.2.
Let A be a C*-algebra, let T( A ) denote the set of all tracial states of A ,equipped with the topology of pointwise convergence. Note that if A is unital, the set T( A )is a Choquet simplex.Let a be a positive element of M ∞ ( A ) and τ ∈ T( A ); defined τ ( a ) = lim n →∞ τ ( a n ) , where τ is extended naturally to M ∞ ( A ). The functionT( A ) ∋ τ d τ ( a ) ∈ R + is the limit of an increasing sequence of strictly positive affine functions on T( A ), so it islower semicontinuous.It is well known that if a - b , thend τ ( a ) ≤ d τ ( b ) , τ ∈ T( A ) . If the C*-algebra A satisfies the property that for any positive elements a, b ∈ M ∞ ( A ) withd τ ( a ) < d τ ( b ) , τ ∈ T( A ) , ZHUANG NIU then a - b , the C*-algebra A is said to have the strict comparison of positive elements. Remark . Note that if A = M n (C ( X )), where X is a locally compact Hausdorff space,and τ be a trace of A . Then, for any positive element a ∈ M ∞ ( A ) ∼ = M ∞ (C ( X )) and any τ ∈ T( A ), one has τ ( a ) = Z X n Tr( a ( x )) dµ τ and d τ ( a ) = Z X n rank( a ( x )) dµ τ , where µ τ is the Borel measure on X induced by τ . Definition 3.4.
Let A be a unital C*-algebra. Denote by LAff b (T( A )) ++ the cone of allstrictly positive lower semicontinuous affine functions on T( A ), and denote by V( A ) thesemigroup of Murray-von Neumann equivalence classes of projections of S ∞ n =1 M n ( A ). ThenV( A ) ⊔ LAff b (T( A )) ++ form an ordered semigroup with addition of crossed terms to be( p + f ) = p ( τ ) + f ( τ ) ∈ LAff b (T( A )) ++ , p ∈ V( A ) , f ∈ LAff b (T( A )) ++ . Then the mapW( A ) ∋ [ a ] (cid:26) [ a ] ∈ V( A ) , if a is equivalent to a projection,( τ d τ ([ a ])) ∈ LAff b (T( A )) ++ , otherwise,is a representation of the Cuntz semigroup W( A ).The following is a version of Theorem 3.4 of [27] for the C*-algebra C( X ) ⋊ Γ. Proposition 3.5.
Let A = C( X ) ⋊ Γ , where ( X, Γ) is free, minimal, has the (URP) andzero mean dimension. Then, for any continuous affine function α : T( A ) → (0 , ∞ ) and any ε > , there is a positive element a ∈ M ∞ ( A ) such that | α ( τ ) − d τ ( a ) | < ε, τ ∈ T( A ) . Proof.
By Corollary 3.10 of [1], there is a positive element a ′ ∈ A such that α ( τ ) = τ ( a ′ ) , τ ∈ T( A ) . Since the action is minimal, the algebra A is simple, and hence there is δ ∈ (0 ,
1) such that(3.1) τ ( a ′ ) > δ, τ ∈ T( A ) . Also pick
M > k a ′ k so τ ( a ′ ) < M, τ ∈ T( A ) . Let ε ∈ (0 , ) be arbitrary. By Theorem 3.8 of [21], for any finite subset F ⊆ A and any ε ′ ∈ (0 , ε ) ( F and ε ′ will be fixed in the next paragraph), there exist a ′′ ∈ A , a finite set F ′ ⊆ A , h ∈ C( X ) + , and a sub-C*-algebra C ⊆ A with C ∼ = L Ss =1 M n s (C ( Z s )) and closedsets [ Z s ] ⊆ Z s such that(1) for any f ∈ F , there is f ′ ∈ F ′ such that k f − f ′ k < ε ′ ,(2) k a ′ − a ′′ k < ε ′ , k ha ′′ − a ′′ h k < ε ′ , k hf ′ − f ′ h k < ε ′ , f ′ ∈ F ′ ,(3) h ∈ C , ha ′′ h ∈ C , hf ′ h ∈ C , f ′ ∈ F ′ ,(4) k h k ≤ τ (1 − h ) < ε ′ , τ ∈ T ( A ),(5) µ ( X \ h − (1)) < ε ′ M +1 , µ ∈ M ( X, Γ), -STABILITY OF C( X ) ⋊ Γ 7 (6) under the isomorphism C ∼ = L Ss =1 M n s (C ( Z s )), the element h has the form h = S M s =1 diag { h s, , ..., h s,n s } , where h s,i : Z s → [0 , n s |{ ≤ i ≤ n s : h s,i ( x ) = 1 }| > − ε ′ , x ∈ [ Z s ] , s = 1 , ..., S, (7) dim([ Z s ]) n s < ε ′ δ, s = 1 , , ..., S, (8) each n s is sufficiently large such that the interval (2 n s δε + 1 , n s ε −
1) contains atleast one integer.Put a ′ = (1 − h ) a ′′ (1 − h ) and a ′ = h a ′′ h . One asserts that with F sufficiently large and ε ′ sufficiently small, one has(3.2) M > τ ( π ( a ′ )) > δ, τ ∈ T( π ( C )) , where π is the standard quotient map from C ∼ = L Ss =1 M n (C ( Z s )) to L Ss =1 M n (C ([ Z s ])).Then, fix this pair of ( F , ε ′ ).Indeed, suppose the contrary, there then exist a sequence of finite subset F ′ i ⊆ A withdense union and a sequence of positive numbers ε i decreasing to 0, sub-C*-algebras C i ⊆ A , a ′′ i ∈ A , positive elements h i ∈ C i such that • k a ′ − a ′′ i k < ε i , • (cid:13)(cid:13)(cid:13) h i f ′ − f ′ h i (cid:13)(cid:13)(cid:13) < ε i , f ′ ∈ F ′ i , • h i a ′′ i h i ∈ C i , h i ∈ C i , and h i f ′ h i ∈ C i , f ′ ∈ F ′ i , so that h i a ′′ i h i ∈ C i , h i a ′′ i h i ∈ C i , h i f ′ h i ∈ C i , and h i f ′ h i ∈ C i , f ′ ∈ F ′ i , • there exists τ i ∈ T( π ( C i )) such that τ i ( π i ( h a ′′ i h i )) ≤ δ or τ i ( π i ( h a ′′ i h i )) ≥ M, where π i is the standard quotient map from C i ∼ = L Ss =1 M n (C ( Z s )) to L Ss =1 M n (C ([ Z s ])), • τ ( π ( h i )) > − ε i , for any τ ∈ T( π ( C i )) (this follows from 6).Consider the linear functional ρ i : A ∋ a τ i ( π ( h i ah i )) ∈ C , and note that k ρ i k = ρ i (1 A ) = τ i ( π ( h i )) > − ε i . Also note that for, any a, b ∈ F ′ i , ρ i ( ab ) = τ i ( π ( h i abh i )) ≈ ε i τ i ( π ( h i ah i h i bh i )) = τ i ( π ( h i bh i h i ah i )) ≈ ε i τ i ( π ( h i bah i )) = ρ i ( ba ) . ZHUANG NIU
Thus, any accumulation point of { ρ i } , say ρ ∞ , is actually a tracial state. However, ρ ∞ ( a ′ ) = lim i →∞ τ i ( π i ( h a ′′ i h i )) ≤ δ or ρ ∞ ( a ′ ) = lim i →∞ τ i ( π i ( h a ′′ i h i )) ≥ M, which contradicts to (3.1). This proves the assertion.Denote by Z the (abstract) disjoint union of Z s , s = 1 , ..., S , and denote by [ Z ] the(abstract) disjoint union of [ Z s ], s = 1 , ..., S . Consider π ( a ′ ) ∈ π ( C ), and consider thecontinuous function [ Z ] ∋ x Tr( π ( a ′ )( x )) ∈ (0 , + ∞ ) . For each s = 1 , , ..., S , by 8, one picks an integer∆ s ∈ (2 n s δε + 1 , n s ε − . Define f : [ Z ] ∋ x
7→ ⌈
Tr( π ( a ′ )( x )) ⌉ + ∆ s , if x ∈ [ Z s ]and g : [ Z ] ∋ x
7→ ⌊
Tr( π ( a ′ )( x )) ⌋ − ∆ s , if x ∈ [ Z s ]where ⌊ t ⌋ = max { k ∈ Z : k ≤ t } and ⌈ t ⌉ = min { k ∈ Z : k ≥ t } . Note that by (3.2), for any x ∈ [ Z s ], s = 1 , ..., S , one has ⌊ Tr( π ( a ′ )( x )) ⌋ − ∆ s ≥ n s tr( π ( a ′ )( x )) − n s δε − > n s δ − n s δε − > . That is, the function g is a positive. Also note that for any x ∈ [ Z s ], s = 1 , ..., S , f ( x ) ≤ max {⌈ Tr( π ( a ′ )( y )) ⌉ + ∆ s : y ∈ [ Z s ] }≤ max { Tr( π ( a ′ )( y )) + 4 n s ε : y ∈ [ Z s ] }≤ n s max { tr( π ( a ′ )( y )) + 4 ε : y ∈ [ Z s ] }≤ n s ( M + 1) . Therefore f and g satisfy(a) g is positive upper semicontinuous and f is lower semicontinuous,(b) 0 < g ( x ) < Tr( π ( a ′ )( x )) < f ( x ) ≤ n s ( M + 1), x ∈ [ Z s ], and(c) 4dim([ Z s ]) < n s δε < s − < f ( x ) − g ( x ) ≤ s + 2 < εn s , x ∈ [ Z s ].It then follows from Proposition 2.9 of [27] that there is a positive element a ′′′ ∈ M ∞ ( π ( C ))such that g ( x ) < rank( a ′′′ ( x )) < f ( x ) , x ∈ [ Z s ] . -STABILITY OF C( X ) ⋊ Γ 9
Extend a ′′′ to an element of M n ( C ) ⊆ M n ( A ) and denote it by a . One then has that forany x ∈ [ Z ], (cid:12)(cid:12)(cid:12)(cid:12) n ( x ) rank( a ( x )) − tr( a ′ ( x )) (cid:12)(cid:12)(cid:12)(cid:12) (3.3) = (cid:12)(cid:12)(cid:12)(cid:12) n ( x ) rank( a ′′′ ( x )) − tr( a ′ ( x )) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) n ( x ) Tr( a ′ ( x )) − tr( a ′ ( x )) (cid:12)(cid:12)(cid:12)(cid:12) + 1 n ( x ) ( f ( x ) − g ( x )) ≤ ε. Note that the element a can be chosen so that for any x ∈ Z s \ [ Z s ], s = 1 , ..., S ,rank( a ( x )) ≤ max { f ( x ) : x ∈ [ Z s ] } ≤ n s ( M + 1) . Now, let τ ∈ T( A ) be arbitray, and let µ τ denote the Borel measure on Z induced by therestriction of τ to C . Note that 1 − ε < k µ τ k ≤ τ ( h ) ≥ − ε ′ > − ε ), and alsonote that µ τ ( Z \ [ Z ]) ≤ d τ (˜ c − h ) ≤ d τ (1 A − h ) < µ ( X \ h − (1)) < εM + 1 , where ˜ c ≥ h is some strict positive element of C ⊆ A , and µ is the invariant measure on X corresponding to τ ( µ is not µ τ ). Therefore, Z Z \ [ Z ] n ( x ) rank( a ( x )) dµ τ ≤ Z Z \ [ Z ] ( M + 1) dµ τ < ε, and (by 3.2) Z Z \ [ Z ] tr( a ′ ( x )) dµ τ ≤ Z Z \ [ Z ] k a ′ k dµ τ ≤ Z Z \ [ Z ] k a k dµ τ ≤ Z Z \ [ Z ] M dµ τ < ε, where n ( x ) = n s if x ∈ Z s . In particular (cid:12)(cid:12)(cid:12)(cid:12)Z Z \ [ Z ] n ( x ) rank( a ( x )) dµ τ − Z Z \ [ Z ] tr( a ′ ( x )) dµ τ (cid:12)(cid:12)(cid:12)(cid:12) < ε. By (3.3), d τ ( a ) = Z Z n ( x ) rank( a ( x )) dµ τ = Z [ Z ] n ( x ) rank( a ( x )) dµ τ + Z Z \ [ Z ] n ( x ) rank( a ( x )) dµ τ ≈ ε Z [ Z ] n ( x ) rank( a ( x )) dµ τ + Z Z \ [ Z ] tr( a ′ ( x )) dµ τ ≈ ε Z [ Z ] tr( a ′ ( x )) dµ τ + Z Z \ [ Z ] tr( a ′ ( x )) dµ τ = Z Z tr( a ′ ( x )) dµ τ = τ ( a ′ ) ≈ ε τ ( a ′ ) . Since ε is arbitrary, this proves the desired conclusion. (cid:3) Corollary 3.6.
Let ( X, Γ) be a free and minimal dynamical system with the (URP), andassume that A = C( X ) ⋊ Γ has Cuntz comparison of open sets. If ( X, Γ) has mean dimensionzero, then, (3.4) W( A ) ∼ = V( A ) ⊔ LAff b (T( A )) ++ . In particular, for any positive element a ∈ M ∞ ( A ) , and any k ∈ N , there is x ∈ W( A ) suchthat (3.5) kx ≤ [ a ] ≤ ( k + 1) x. In other words, A is -almost divisible (hence tracially -divisible)(see Definition 3.5(i) of [31] ).Proof. By Theorem 4.8 of [21], the C*-algebra A has strict comparison of positive elements.Then (3.4) follows from Proposition 3.5 and the proof of Theorem 5.3 of [1].Let a ∈ M ∞ ( A ) be a non-zero positive element, and pick δ > τ ( a ) > δ, τ ∈ T( A ) . Since A is simple and non-elementary, there is a non-zero positive element c ∈ A such thatsp( c ) = [0 ,
1] and d τ ( c ) < δk < k d τ ( a ) , τ ∈ T( A ) . Consider [ a ] + [ c ], and note that [ a ] + [ c ] ∈ LAff b (T( A )) ++ . By (3.4), there is x ∈ M ∞ ( A ),which is not Cuntz equivalent to a projection, such thatd τ ( x ) = 1 k + 1 d τ ([ a ] + [ c ]) , τ ∈ T( A ) . Then, for any τ ∈ T( A ), k (d τ ( x )) = kk + 1 d τ ([ a ] + [ c ])= kk + 1 d τ ([ a ]) + kk + 1 d τ ( c ) < kk + 1 d τ ( a ) + 1 k + 1 d τ ( a ) = d τ ( a ) , and ( k + 1)(d τ ( x )) = d([ a ] + [ c ])( τ ) > d τ ( a ) . Together with (3.4), this proves (3.5). (cid:3)
Remark . Note that a straightforward argument shows that there is m such that for any k ∈ N , there is x ∈ W( A ) such that kx ≤ [1 A ] ≤ m ( k + 1) x, whenever ( X, Γ) has the (URP) and Cuntz-comparison of open sets, even without meandimension zero. Then, as a natural question, is C( X ) ⋊ Γ always tracially m -divisible for -STABILITY OF C( X ) ⋊ Γ 11 some m ∈ N if ( X, Γ) has the (URP) and Cuntz-comparison of open sets, but without anyassumptions on mean dimension?4.
Approximate central order zero maps from M k ( C ) to C( X ) ⋊ Γ and the Z -stability of C( X ) ⋊ ΓOne considers the Z -stability of C( X ) ⋊ Γ in this section. First, one has the followinglemma which is enssentailly Theorem 3.8 of [21], stating that the C*-algebra A = C( X ) ⋊ Γcan be (weakly) tracially approximated by homogeneous C*-algebras, but with an extraconclusion that there is an element h in the homogeneous sub-C*-algebra, which is approxi-mately central in A , large in trace, and is orthogonal to the elements with smaller trace inthe decomposition obtained from the tracial approximation. Lemma 4.1.
Let ( X, Γ) be a free dynamical system with the (URP). Then, for any finitelymany elements f , f , ..., f n ∈ C( X ) ⋊ Γ and any ε > , there exist a C*-algebra C ⊆ C( X ) ⋊ Γ with C ∼ = L Ss =1 M k s (C ( U s )) for some k s ∈ N and locally compact Hausdorff space U s , s = 1 , ..., S , positive functions h ∈ C( X ) ∩ C , and f (0)1 , f (1)1 , f (0)2 , f (1)2 , ..., f (0) n , f (1) n ∈ C( X ) ⋊ Γ such that (1) (cid:13)(cid:13)(cid:13) f i − ( f (0) i + f (1) i ) (cid:13)(cid:13)(cid:13) < ε , ≤ i ≤ n , (2) f (1) i ∈ C , ≤ i ≤ n , (3) (cid:13)(cid:13)(cid:13) f (0) i h (cid:13)(cid:13)(cid:13) = 0 , ≤ i ≤ n , (4) (cid:13)(cid:13)(cid:13) [ f (1) i , h ] (cid:13)(cid:13)(cid:13) < ε , ≤ i ≤ n , (5) τ (1 − h ) < ε , τ ∈ T( A ) .Proof. The proof is similar to that of Theorem 3.8 of [21], but without dealing with meandimension.Denote by A the crossed product C*-algebra C( X ) ⋊ Γ. Without loss of generality, onemay assume f i = X γ ∈N f i,γ u γ for some finite set N ⊆
Γ with e ∈ N = N − , and some f i,γ ∈ C( X ). Denote by M = max { , k f i,γ k : i = 1 , ..., n, γ ∈ N } . For the given ε >
0, choose ε ∈ (0 , ε ) such that if a positive element a ∈ A with k a k ≤ k af i − f i a k < ε , ≤ i ≤ n, then (cid:13)(cid:13)(cid:13) a f i − f i a (cid:13)(cid:13)(cid:13) < ε , ≤ i ≤ n. Pick a natural number
L > M |N | ε , and pick a sufficiently large finite set K ⊆ Γ and a sufficiently small positive number δ sothat if a finite set Γ ⊆ Γ is (
K, δ )-invariant, then(4.1) | Γ \ int N L +1 (Γ ) || Γ | < ε . Since ( X, Γ) has the (URP), there exist closed sets B , B , ..., B S ⊂ X and ( K, δ )-invariantsets Γ , Γ , ..., Γ S ⊆ Γ such that B s γ, γ ∈ Γ s , s = 1 , ..., S, are mutually disjoint and ocap( X \ S G s =1 G γ ∈ Γ s B s γ ) < ε . Pick two open sets U s , V s ⊆ X , s = 1 , , ..., S , satisfying U s ⊇ V s ⊇ B s , U s ⊇ V s , and U s γ, γ ∈ Γ s , s = 1 , ..., S, are mutually disjoint.Consider the sub-C*-algebra(4.2) C := C ∗ { u ∗ γ f : f ∈ C ( U s ) , γ ∈ Γ s , s = 1 , , ..., S } ⊆ C( X ) ⋊ Γ , which, by Lemma 3.11 of [21], is isomorphic to S M s =1 M | Γ s | (C ( U s )) . For each s = 1 , , ..., S , pick continuous functions χ U s , χ V s : X → [0 ,
1] such that(4.3) χ U s | V s = 1 , χ V s | B s = 1 , χ U s | X \ U s = 0 , and χ V s | X \ V s = 0 . Note that χ U s , χ V s ∈ C , and(4.4) χ U s f, χ V s f ∈ C, f ∈ C( X ) . For each Γ s , s = 1 , , ..., S , define the subsets Γ s,L +1 = int N L +1 (Γ s ) , Γ s,L = int N L (Γ s ) \ int N L +1 (Γ s ) , Γ s,L − = int N L − (Γ s ) \ int N L (Γ s ) , ... ... ...Γ s, = Γ s \ int N (Γ s ) . Then, for any γ ∈ N , one has(4.5) Γ s,l γ ⊆ Γ s,l − ∪ Γ s,l ∪ Γ s,l +1 , ≤ l ≤ L. Indeed, pick an arbitrary γ ′ ∈ Γ s,l . By the construction, one has(4.6) γ ′ N l ⊆ Γ s but γ ′ N l +1 * Γ s . -STABILITY OF C( X ) ⋊ Γ 13
Therefore γ ′ γ N l − ⊆ γ ′ N l ⊆ Γ s and hence γ ′ γ ∈ int N l − Γ s (since e ∈ N l − ).Thus, to show (4.5), one only has to show that γ ′ γ / ∈ int N l +2 Γ s . Suppose γ ′ γ N l +2 ⊆ Γ s .Since N is symmetric, one has γ − ∈ N ; hence N l +1 ⊆ γ N l +2 and γ ′ N l +1 ⊆ γ ′ γ N l +2 ⊆ Γ s , which contradicts (4.6).Also note that(4.7) Γ s,L +1 γ ⊆ Γ s,L +1 ∪ Γ s,L . For each γ ∈ Γ s , define ℓ ( γ ) = l, if γ ∈ Γ s,l . By (4.5) and (4.7), the function ℓ satisfies(4.8) | ℓ ( γ ′ γ ) − ℓ ( γ ) | ≤ , γ ′ ∈ N , γ ∈ Γ s, ∪ · · · ∪ Γ s,L +1 . Define h U = S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L ( χ U s ◦ γ − ) = S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L u ∗ γ χ U s u γ ∈ C( X ) ∩ C, and h V = S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L ( χ V s ◦ γ − ) = S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L u ∗ γ χ V s u γ ∈ C( X ) ∩ C. Note that h U h V = ( S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L u ∗ γ χ U s u γ )( S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L u ∗ γ χ V s u γ )= S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L u ∗ γ χ U s χ V s u γ = S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L u ∗ γ χ V s u γ = h V , and hence(4.9) (1 − h U ) h V = 0 . By (4.3) (and (4.1)),ocap( X \ h − V (1)) ≤ max { | Γ s \ int N L +1 (Γ s ) || Γ s | : s = 1 , ..., S } +ocap( X \ S G s =1 G γ ∈ Γ s B s γ ) ≤ ε ε < ε, and therefore τ (1 − h V ) < ε, τ ∈ T( A ) . Note that, by the construction of C (see (4.2)), χ U s u γ ∈ C, γ ∈ Γ s . Hence, for each γ ′ ∈ N , since γγ ′ ∈ Γ s , γ ∈ Γ s,l , l = 1 , , ..., L + 1, one has h U u γ ′ = S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L u ∗ γ χ U s u γγ ′ = S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L ( u ∗ γ χ U s )( χ U s u γγ ′ ) ∈ C, and therefore, h U u γ h U ∈ C, γ ∈ N . For any f ∈ C( X ), by (4.4), one has h U f = S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L u ∗ γ χ U s u γ f = S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L u ∗ γ χ U s ( u γ f u ∗ γ ) u γ ∈ C, and therefore h U f i h U ∈ C, ≤ i ≤ n. Note that, for each γ ′ ∈ N , by (4.8), (cid:13)(cid:13) u ∗ γ ′ h U u γ ′ − h U (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L χ U s ◦ ( γ ′ γ ) − − S X s =1 L +1 X l =1 X γ ∈ Γ s,l l − L χ U s ◦ γ − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = max { (cid:12)(cid:12)(cid:12)(cid:12) ℓ ( γ ′ γ ) − L − ℓ ( γ ) − L (cid:12)(cid:12)(cid:12)(cid:12) : γ ∈ Γ s \ Γ s, , s = 1 , , ..., S } < L < ε M |N | , and hence(4.10) k h U f i − f i h U k < ε , i = 1 , , ..., n. The same argument also shows that(4.11) k h V f i − f i h V k < ε < ε, i = 1 , , ..., n. -STABILITY OF C( X ) ⋊ Γ 15
It follows from (4.10) and the choice of ε that (cid:13)(cid:13)(cid:13) h U f i − f i h U (cid:13)(cid:13)(cid:13) < ε (cid:13)(cid:13)(cid:13) (1 − h U ) f i − f i (1 − h U ) (cid:13)(cid:13)(cid:13) < ε , i = 1 , , ..., n, and hence (cid:13)(cid:13)(cid:13) f i − ((1 − h U ) f i (1 − h U ) + h U f i h U ) (cid:13)(cid:13)(cid:13) < ε, ≤ i ≤ n. Put f (0) i = (1 − h U ) f i (1 − h U ) and f (1) i = h U f i h U . By (4.9), f (0) i h V = 0 , i = 1 , ..., n. One also has, by (4.11), f (1) i h V = h U f i h U h V = h U f i h V h U ≈ ε h U h V f i h U = h V h U f i h U = h V f (1) i . Thus (cid:13)(cid:13)(cid:13) f (1) i h V − h V f (1) i (cid:13)(cid:13)(cid:13) < ε, i = 1 , ..., n. Then the element h := h V satisfies the lemma. (cid:3) Recall
Definition 4.2 ([33]) . Let A , B be C*-algebras, and let ϕ : A → B be a completely positivecontractive linear map (c.p.c map). ϕ is said to be order zero if a ⊥ b = ⇒ ϕ ( a ) ⊥ ϕ ( b ) , a, b ∈ A. Definition 4.3 ([11]) . A unital C*-algebra A is said to be tracially Z -stable if for any finiteset F ⊆ A , any ε >
0, and any non-zero positive element a ∈ A , there is a c.p.c. order zeromap ϕ : M ( C ) → A such that(1) k [ ϕ ( a ) , f ] k < ε , a ∈ M ( C ), k a k ≤ f ∈ F ,(2) 1 A − ϕ (1 ) - a .Based on [19], for nuclear C*-algebras, the tracial Z -stability is shown to be equivalent tothe Z -stability in [11]: Theorem 4.4.
Let A be a simple separable unital nuclear C*-algebra. Then A ∼ = A ⊗ Z ifand only if A is tracially Z -stable, where Z is the Jiang-Su algebra.Remark . In general, there are non-nuclear C*-algebras which are tracially Z -stable butnot Z -stable (see [22]).The following two lemmas are simple observations. Lemma 4.6.
Let A be a unital C*-algebra, and let τ be a tracial state of A . Assume a, b ∈ A are positive elements with norm at most and τ (1 − a ) < ε and τ (1 − b ) < ε, then τ ( ab ) > − ε. Proof.
It follows from the assumption that1 − ε < τ ( a ) and − ε < τ ( b − . Also note that0 ≤ τ ((1 − a ) (1 − b )(1 − a ) ) = τ ((1 − a )(1 − b )) = τ (1 − a − b + ab ) , and so τ ( a + b − ≤ τ ( ab ) . Then 1 − ε = (1 − ε ) − ε < τ ( a ) + τ ( b −
1) = τ ( a + b − ≤ τ ( ab ) , as desired. (cid:3) Lemma 4.7.
For any ε > , if ϕ : M k ( C ) → A is a c.p.c. order zero map with τ (1 A − φ (1 k )) < ε, τ ∈ T( A ) , then there is a c.p.c. order zero map ϕ ′ : M k ( C ) → A such that k ϕ ′ − ϕ k < √ ε and d τ (1 A − ϕ ′ (1 k )) < √ ε, τ ∈ T( A ) . Proof.
Since ϕ has order zero, it follows from Theorem 1.2 of [33] that there is h ∈ M (C*( ϕ (M k ))) ∩ (C*( ϕ (M k ))) ′ and a unital homomorphism˜ ϕ : M k ( C ) → M (C*( ϕ (M k ))) ∩ ( h ) ′ such that ϕ ( a ) = ˜ ϕ ( a ) h. Note that h = ϕ (1 k ).Let τ ∈ T( A ) be arbitrary, and denote by µ τ the probability measure induced by τ onsp( h ) ⊆ [0 , τ (1 A − h ) < ε , one has1 − ε < Z (0 , t d µ τ = Z (0 , −√ ε ] t d µ τ + Z (1 −√ ε, t d µ τ ≤ (1 − √ ε ) µ τ ([0 , − √ ε ]) + (1 − µ τ ([0 , − √ ε ])) , and hence µ τ ([0 , − √ ε ]) < √ ε. Set f ( t ) = min { t −√ ε , } . Consider f ( h ) and the c.p.c. order zero map ϕ ′ := ˜ ϕ ( a ) f ( h ) , a ∈ M k ( C ) . Note that k h − f ( h ) k < √ ε ; one has that k ϕ − ϕ ′ k < ε. -STABILITY OF C( X ) ⋊ Γ 17
On the other hand, for any τ ∈ T( A ), one hasd τ (1 − ϕ ′ (1 k )) = d τ (1 − f ( h )) = µ τ ([0 , − √ ε ]) < √ ε, as desired. (cid:3) Proposition 4.8.
Let ( X, Γ) be a free and minimal dynamical system with the (URP).If C( X ) ⋊ Γ is tracially m -almost divisible for some m ∈ N , then, for any finite set { f , f , ..., f n } ⊆ C( X ) ⋊ Γ , any ε > , and any k ∈ N , there is a c.p.c. order zero map φ : M k ( C ) → C( X ) ⋊ Γ such that (1) k [ φ ( a ) , f i ] k < ε , a ∈ M( C ) with k a k = 1 and ≤ i ≤ n , and (2) d τ (1 A − φ (1 k )) < ε , τ ∈ T( A ) .Proof. Denote by A = C( X ) ⋊ Γ. By Lemma 4.7, it is enough to show that for any given ε > { f , f , ..., f n } ⊆ A , there is a c.p.c. order-zero map φ : M k ( C ) → A such that(1) k [ φ ( a ) , f i ] k < ε , a ∈ M k ( C ) with k a k = 1 and 1 ≤ i ≤ n , and(2) τ (1 A − φ (1 k )) < ε , τ ∈ T( A ).Since order zero maps from M k ( C ) are weakly stable (see Proposition 2.5 of [14]), one isable to pick δ > ρ : M k ( C ) → A satisfies a ⊥ b ⇒ k ρ ( a ) ρ ( b ) k < δ, a, b ∈ M k ( C ) , k a k = k b k = 1 , there is a c.p.c order zero map θ : M k ( C ) → A such that k ρ ( a ) − θ ( a ) k < ε , a ∈ M k ( C ) , k a k = 1 . By Lemma 4.1, there are positive elements f , f (1)1 , f , f (1)2 , ..., f n , f (1) n ∈ A , a C*-algebra B ⊆ A with B ∼ = L Ss =1 M k s (C ( Z s )) for some locally compact Hausdorff spaces Z s , s =1 , ..., S , a positive element h ∈ A with norm 1 such that(4.12) (cid:13)(cid:13)(cid:13) f i − ( f i + f (1) i ) (cid:13)(cid:13)(cid:13) < ε , ≤ i ≤ n, (4.13) h ∈ B and f (1) i ∈ B, ≤ i ≤ n, (4.14) (cid:13)(cid:13)(cid:13) f (0) i h (cid:13)(cid:13)(cid:13) < ε , ≤ i ≤ n, (4.15) (cid:13)(cid:13)(cid:13) [ f (1) i , h ] (cid:13)(cid:13)(cid:13) < ε , ≤ i ≤ n, and(4.16) τ (1 − h ) < ε , τ ∈ T( A ) . Consider the unitization ˜ B = B + C A , and note that˜ B ∼ = { f ∈ C( {∞} ∪ S G s =1 Z s , S M s =1 M k s ( C )) : f ( ∞ ) ∈ C } . Since the space {∞} ∪ F Ss =1 Z s is an inverse limit of finite dimensional CW-complexes, witha small perturbation of f (1)1 , f (1)2 , ..., f (1) n , and h , one may assume that ˜ B (and B ) has finitenuclear dimension.Since A is assumed to be tracially m -divisible, applying Lemma 5.11 of [31] to ˜ B and using(4.13), one obtains a c.p.c. order zero map ϕ : M k ( C ) → A such that(4.17) (cid:13)(cid:13)(cid:13) [ ϕ ( a ) , f (1) i ] (cid:13)(cid:13)(cid:13) < ε , ≤ i ≤ n, a ∈ M k ( C ) , k a k = 1 , (4.18) k [ ϕ ( a ) , h ] k < δ, a ∈ M k ( C ) , k a k = 1 , and(4.19) τ (1 A − ϕ (1 k )) < ε , τ ∈ T( A ) . Consider the c.p.c. map M k ( C ) ∋ a h ϕ ( a ) h ∈ A. Then, for any elements a, b ∈ M k ( C ) with a ⊥ b and k a k = k b k = 1, one has (by (4.18))( h ϕ ( a ) h )( h ϕ ( b ) h ) = h ϕ ( a ) hϕ ( b ) h ≈ δ h ϕ ( a ) ϕ ( b ) h = 0 , and hence, by the choice of δ , there exists a c.p.c order zero map φ : M k ( C ) → A such that(4.20) (cid:13)(cid:13)(cid:13) φ ( a ) − h ϕ ( a ) h (cid:13)(cid:13)(cid:13) < ε , a ∈ M k ( C ) , k a k = 1 . Then, for any a ∈ M k ( C ) with k a k = 1 and any 1 ≤ i ≤ n , one has k [ φ ( a ) , f i ] k < (cid:13)(cid:13)(cid:13) [ h ϕ ( a ) h , f i ] (cid:13)(cid:13)(cid:13) + ε < (cid:13)(cid:13)(cid:13) [ h ϕ ( a ) h , f (0) i + f (1) i ] (cid:13)(cid:13)(cid:13) + 3 ε (cid:13)(cid:13)(cid:13) [ h ϕ ( a ) h , f (0) i ] (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) [ h ϕ ( a ) h , f (1) i ] (cid:13)(cid:13)(cid:13) + 3 ε < ε ε ε ε (by (4.14), (4.15) and (4.17)).Moreover, applying Lemma 4.6 with (4.16) and (4.19), together with (4.20), one has τ ( φ (1 k )) ≈ ε τ ( h ϕ (1 k ) h ) = τ ( hϕ (1 k )) > − ε , τ ∈ T( A ) , as desired. (cid:3) Theorem 4.9.
Let ( X, Γ) be a free and minimal dynamical system with the (URP) and(COS). If ( X, Γ) has mean dimension zero, then (C( X ) ⋊ Γ) ⊗ Z ∼ = C( X ) ⋊ Γ .In particular, let ( X , Γ ) and ( X , Γ ) be two free minimal dynamical systems with the(URP), Cuntz comparison of open sets, and zero mean dimension, then C( X ) ⋊ Γ ∼ = C( X ) ⋊ Γ if and only if Ell(C( X ) ⋊ Γ ) ∼ = Ell(C( X ) ⋊ Γ ) , where Ell( · ) = (K ( · ) , K +0 ( · ) , [1] , T( · ) , ρ, K ( · )) is the Elliott invariant. Moreover, these C*-algebras are inductive limits of unital subhomogeneous C*-algebras. -STABILITY OF C( X ) ⋊ Γ 19
Proof.
It follows from Corollary 3.6 that C( X ) ⋊ Γ is tracially 0-divisible. It follows fromTheorem 4.8 of [21] that C( X ) ⋊ Γ has strict comparison of positive elements. Together withProposition 4.8, one has that C( X ) ⋊ Γ is tracially Z -stable. Since C( X ) ⋊ Γ is nuclear, itis Z -stable as desired. (cid:3) Corollary 4.10.
Let ( X, Γ) be a free and minimal dynamical system with mean dimensionzero. Assume that • either Γ = Z d for some d ≥ , or • ( X, Γ) is an extension of a Cantor system and Γ has subexponetial growth.Then, the C*-algebra C( X ) ⋊ Γ is classified by the Elliott invariant and is an inductive limitof unital subhomogeneous C*-algebras.Proof. It follows from [21] and [20] that the dynamical systems being considered have the(URP) and (COS). The statement then follows from Theorem 4.9. (cid:3) An alternative approach: mdim0 + URP ⇒ SBPIn this section, one considers the zero mean dimension together with the (URP), andshows that these two conditions actually implies that the dynamical system has the smallboundary property (SBP). Together with [12] and [21], this gives another proof of Theorem4.9.
Theorem 5.1.
Let ( X, Γ) be a free topological dynamical system with the (URP). If mdim( X, Γ) = 0 , then ( X, Γ) has the small boundary property.Proof. It follows from Lemma 5.5 and Theorem 5.3 of [10] that, in order to show that ( X, Γ)has the (SBP), it is enough to show that for any continuous function f : X → R and any ε >
0, there is a continuous function g : X → R such that(1) k f − g k ∞ < ε , and(2) ocap( { x ∈ X : g ( x ) = 0 } ) < ε .Let f : X → R and ε > U to be a finite open cover of X such that | f ( x ) − f ( y ) | < ε , x, y ∈ U, U ∈ U . Since mdim( X, Γ) = 0, there is (
K, ε ′ ), where K ⊆ Γ is a finite set and ε ′ >
0, such thatif Γ ⊆ Γ is (
K, ε ′ )-invariant, there is an open cover V such that(1) V refines W γ ∈ Γ U γ , and(2) ord( V ) < ε | Γ | .Since ( X, Γ) has the (URP), there are closed sets B , B , ..., B S and ( K, ε ′ )-invariant setsΓ , Γ , ..., Γ S ⊆ Γ such that B s γ, γ ∈ Γ s , ≤ s ≤ S are mutually disjoint and ocap( X \ S G s =1 G γ ∈ Γ s B s γ ) < ε . Pick a small neighborhood U s of each B s , s = 1 , , ..., S , such that U s γ, γ ∈ Γ s , ≤ s ≤ S, are still mutually disjoint. Note thatocap( X \ S G s =1 G γ ∈ Γ s U s γ ) ≤ ocap( X \ S G s =1 G γ ∈ Γ s B s γ ) < ε . For each s = 1 , , ..., S , since Γ s is ( K, ε ′ )-invariant, there is an open cover V of X suchthat(1) V refines W γ ∈ Γ U γ , and(2) ord( V ) < ε | Γ s | .Then, consider the collection of open sets V s := { V ∩ U s : V ∈ V} . Note that V s covers B s and for any V ∈ V s and any γ ∈ Γ s , there is U ∈ U such that V γ ⊆ U. For V s , pick continuous functions φ ( s ) V : X → [0 , , V ∈ V s such that ( φ ( s ) V ) − ((0 , ⊆ V, X V ∈V s φ ( s ) V ( x ) ≤ , x ∈ X, and X V ∈V s φ ( s ) V ( x ) = 1 , x ∈ B s . For V s , also consider the simplicial complex ∆ s spanned by [ V ], V ∈ V s , with[ V ] , [ V ] , ..., [ V d ]span a simplex if and only if V ∩ V ∩ · · · ∩ V d = ∅ . Note that(5.1) dim(∆ s ) = ord( V s ) ≤ ord( V ) ≤ ε | Γ s | . Define the map η s : X ∋ x X V ∈V s φ ( s ) V ( x )[ V ] ∈ C∆ s , where C∆ s is the cone over ∆ s . Note that η s ( B s ) ⊆ ∆ s . -STABILITY OF C( X ) ⋊ Γ 21
For each V ∈ V s , pick a point x ∗ V ∈ V , and define˜ f = f (1 − S X s =1 X γ ∈ Γ s X V ∈V s ( φ ( s ) V ◦ γ − )) + S X s =1 X γ ∈ Γ s X V ∈V s f ( x ∗ V γ )( φ ( s ) V ◦ γ − ) . Then, for any x ∈ X , (cid:12)(cid:12)(cid:12) f ( x ) − ˜ f ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − ( f ( x )(1 − S X s =1 X γ ∈ Γ s X V ∈V s φ ( s ) V ( xγ − )) + S X s =1 X γ ∈ Γ s X V ∈V s f ( x ∗ V γ ) φ ( s ) V ( xγ − )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x )(1 − S X s =1 X γ ∈ Γ s X V ∈V s φ ( s ) V ( xγ − ) + S X s =1 X γ ∈ Γ s X V ∈V s φ ( s ) V ( xγ − )) − ( f ( x )(1 − S X s =1 X γ ∈ Γ s X V ∈V s φ ( s ) V ( xγ − )) + S X s =1 X γ ∈ Γ s X V ∈V s f ( x ∗ V γ ) φ ( s ) V ( xγ − )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S X s =1 X γ ∈ Γ s X V ∈V s ( f ( x ) − f ( x ∗ V γ )) φ ( s ) V ( xγ − )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ S X s =1 X γ ∈ Γ s X V ∈V s | f ( x ) − f ( x ∗ V γ ) | φ ( s ) V ( xγ − )) < ε . That is,(5.2) (cid:13)(cid:13)(cid:13) f − ˜ f (cid:13)(cid:13)(cid:13) < ε . Define piecewise linear function F s : C∆ s → R | Γ s | by F s ([ V ]) = M γ ∈ Γ s f ( x ∗ V γ ) ∈ R | Γ s | . Then ˜ f = f (1 − S X s =1 X γ ∈ Γ s X V ∈V s ( φ ( s ) V ◦ γ − )) + S X s =1 X γ ∈ Γ s π s,γ ◦ F s ◦ η s ◦ γ − , where π s,γ is the projection of R | Γ s | to the γ -coordinate.By Lemma 5.7 of [10], there is a linear map ˜ F s : C∆ s → R | Γ s | such that (cid:13)(cid:13)(cid:13) F s − ˜ F s (cid:13)(cid:13)(cid:13) ∞ < ε (cid:12)(cid:12)(cid:12) { γ ∈ Γ s : π s,γ ( ˜ F s ( x )) = 0 } (cid:12)(cid:12)(cid:12) ≤ dim∆ s , x ∈ C∆ s . Put g = f (1 − S X s =1 X γ ∈ Γ s X V ∈V s ( φ ( s ) V ◦ γ − )) + S X s =1 X γ ∈ Γ s π s,γ ◦ ˜ F s ◦ η s ◦ γ − , and then, for any x ∈ X , (cid:12)(cid:12)(cid:12) ˜ f ( x ) − g ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S X s =1 X γ ∈ Γ s π s,γ ◦ F s ◦ η s ( xγ − ) − S X s =1 X γ ∈ Γ s π s,γ ◦ ˜ F s ◦ η s ( xγ − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S X s =1 X γ ∈ Γ s ( π s,γ ◦ F s ◦ η s ( xγ − ) − π s,γ ◦ ˜ F s ◦ η s ( xγ − )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . If x / ∈ F Ss =1 F γ ∈ Γ s U s γ , then η s ( xγ − ) = , γ ∈ Γ s , s = 1 , ..., S. Hence π s,γ ◦ F s ◦ η s ( xγ − ) = π s,γ ◦ ˜ F s ◦ η s ( xγ − ) = 0 , γ ∈ Γ s , s = 1 , ..., S, and(5.4) ˜ f ( x ) = g ( x ) . If x ∈ F Ss =1 F γ ∈ Γ s U s γ , then there exist s ∈ { , ..., S } and γ ∈ Γ s such that x is only in U s γ . Then η s ( xγ − ) = , γ ∈ Γ s , s = s , and η s ( xγ − ) = , γ = γ . Hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S X s =1 X γ ∈ Γ s ( π s,γ ◦ F s ◦ η s ( xγ − ) − π s,γ ◦ ˜ F s ◦ η s ( xγ − )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) π s ,γ ◦ F s ◦ η s ( xγ − ) − π s ,γ ◦ ˜ F s ◦ η s ( xγ − ) (cid:12)(cid:12)(cid:12) < ε , and (cid:12)(cid:12)(cid:12) ˜ f ( x ) − g ( x ) (cid:12)(cid:12)(cid:12) < ε . Together with (5.4), one has (cid:13)(cid:13)(cid:13) ˜ f − g (cid:13)(cid:13)(cid:13) < ε k f − g k < ε < ε. -STABILITY OF C( X ) ⋊ Γ 23
Let us estimate ocap( { x ∈ X : g ( x ) = 0 } ). Fist, note that for an arbitrary x ∈ B s , where s ∈ { , , ..., S } , by (5.3),(5.5) { γ ∈ Γ s : g ( xγ ) = 0 } = { γ ∈ Γ s : S X s =1 X γ ∈ Γ s π s,γ ( ˜ F s ( η s ( x ))) = 0 } ≤ dim∆ s . Let Γ ⊆ Γ be a finite set which is sufficiently invariant such that(5.6) (cid:12)(cid:12)(cid:12) int S Ss =1 (Γ si ) − Γ (cid:12)(cid:12)(cid:12) | Γ | > − ε , and(5.7) 1 | Γ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { γ ∈ Γ : xγ ∈ X \ S G s =1 G γ ∈ Γ s B s γ } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε , x ∈ X. Denote by K = S [ s =1 Γ s , and note that int S Ss =1 Γ − s Γ = Γ ∩ (Γ K )Let x ∈ X be arbitrary, and consider the orbit x Γ. The partition X = ( X \ S G c =1 G γ ∈ Γ s B s γ ) ⊔ S G c =1 G γ ∈ Γ s B s γ induces a partition of x Γ; since the action is free, this induces a partition of Γ:Γ = Λ ⊔ ∞ G i =1 c i Γ s ( i ) , where s ( i ) ∈ { , , ..., S } for each i = 1 , , ... ,Λ = { γ ∈ Γ : xγ ∈ X \ S G s =1 G γ ∈ Γ s B s γ } , and c i ∈ Γ, i = 1 , , ... , satisfying xc i ∈ B s ( i ) . Restrict this partition to Γ , one hasΓ = (Γ ∩ Λ) ∪ G c i Γ s ( i ) * Γ (Γ ∩ ( c i Γ s ( i ) )) ∪ G c i Γ s ( i ) ⊆ Γ c i Γ s ( i ) . A straightforward calculation shows that if γ ∈ Γ ∩ ( c i Γ s ( i ) ) and c i Γ s ( i ) * Γ , then γ / ∈ int (Γ s ( i ) ) − Γ . Therefore G c i Γ s ( i ) * Γ (Γ ∩ c i Γ s ( i ) ) ⊆ Γ \ int S Ss =1 (Γ si ) − (Γ ) =: ∂ S Ss =1 (Γ si ) − Γ , and, by (5.5), (5.6), (5.7), and (5.1),1Γ |{ γ ∈ Γ : g ( xγ ) = 0 }|≤ | Γ ∩ Λ || Γ | + (cid:12)(cid:12)(cid:12) ∂ S Ss =1 (Γ si ) − Γ (cid:12)(cid:12)(cid:12) | Γ | + 1 | Γ | X c i Γ s ( i ) ⊆ Γ dim∆ s ( i ) ≤ ε ε P c i Γ s ( i ) ⊆ Γ dim∆ s ( i ) P c i Γ s ( i ) ⊆ Γ (cid:12)(cid:12) Γ s ( i ) (cid:12)(cid:12) ≤ ε ε ε, as desired. (cid:3) Remark . Note that if Γ = Z d , it follows from Theorem 1.10.1 and Theorem 1.10.3 of [9]that TRP + mdim0 ⇔ SBP , where TRP stands for the Topological Rokhlin Property in the sense of 1.9 of [9] (edim( X, Z d ) ≤ l densely for some l ∈ N is actually not needed in Theorem 1.10.3). It is easy to see that URPimplies TRP. Therefore, in this case, the statement of Theorem 5.1 is covered by Theorem1.10.3 of [9]. It was also proved later in [10] (Corollary 5.4) thatmdim0 ⇔ SBPfor any Z d -actions with marker property.With the Uniform Property Gamma and [2], Kerr and Szabo has the following: Theorem 5.3 (Corollary 9.5 of [12]) . Assume that ( X, Γ) has the small boundary property.Then, C( X ) ⋊ Γ has the strict comparison if and only if it is Z -stable. Thus, together with Theorem 5.1 and Theorem 4.8 of [21], one has the following:
Alternative proof of Theorem 4.9.
Since ( X, Γ) is assumed to have the (URP), by Theorem5.1, it has the (SBP) since it has mean dimension zero. Therefore, by Theorem 5.3, inorder to prove the theorem, it is enough to show that C ( X ) ⋊ Γ has the strict comparisonof positive elements. But since ( X, Γ) has the (COS) and zero mean dimension, the strictcomparison of C( X ) ⋊ Γ follows from Theorem 4.8 of [21]. (cid:3)
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