aa r X i v : . [ m a t h . OA ] A ug STABLE RANK OF C( X ) ⋊ Γ CHUN GUANG LI AND ZHUANG NIU
Dedicated to Professor George A. Elliott on the occasion of his 75th birthday
Abstract.
It is shown that, for an arbitrary free and minimal Z n -action on a compactHausdorff space X , the crossed product C*-algebra C( X ) ⋊ Z n always has stable rank one,i.e., invertible elements are dense. This generalizes a result of Alboiu and Lutley on Z -actions.In fact, for any free and minimal topological dynamical system ( X, Γ), where Γ is a count-able discrete amenable group, if it has the uniform Rokhlin property and Cuntz comparisonof open sets, then the crossed product C*-algebra C( X ) ⋊ Γ has stable rank one. Moreover,in this case, the C*-algebra C( X ) ⋊ Γ absorbs the Jiang-Su algebra tensorially if, and onlyif, it has strict comparison of positive elements.
Contents
1. Introduction 12. Notation and preliminaries 33. Some lemmas 94. Nilpotent elements, order zero maps, and limits of invertible elements 145. Property (D) and stable rank one 206. Non-invertible elements and zero divisors of C( X ) ⋊ Γ 267. Stable rank of C( X ) ⋊ Γ 308. Two remarks on Property (D) 51References 551.
Introduction
The topological stable rank of a unital C*-algebra A , denoted by tsr( A ), is introduced byRieffel in his seminal paper [21] as a topological version of the Bass stable rank of a ring:Denote by Lg n = { ( x , x , ..., x n ) ∈ A n : Ax + Ax + · · · + Ax n = A } . Then the topological stable rank of A , denote by tsr( A ), is the smallest n such that Lg n isdense in A n (if no such n exists, then the topological stable rank of A is ∞ ). It was shown in Date : August 11, 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Stable rank one, crossed product C*-algebras. [12] that the topological stable rank of a C*-algebra agrees with its Bass stable rank. Thus,we may just refer it as stable rank.The stable rank models dimension of a topological space: Consider the commutative C*-algebra C( X ), where X is a compact Hausdorff space. Its stable rank is ⌊ dim( X )2 ⌋ + 1, where ⌊·⌋ denotes the integer part. It is also shown in [30] that for any n ∈ { , , ..., ∞} , thereexists a simple unital separable C*-algebra A ( A actually can be chosen to be the limit ofan inductive sequence of homogeneous C*-algebras) such that tsr( A ) = n .The class of C*-algebras with stable rank one is particularly interesting. Any such C*-algebra A is stably finite, has cancellation of projections, and has the property that K ( A )is canonically isomorphic to U( A ) / U ( A ). It is also well known that a C*-algebra has stablerank one if, and only if, A = GL( A ), where GL( A ) denotes the group of invertible elementsof A .Many classes of simple C*-algebras have been shown to have stable rank one. For instance,any simple unital finite C*-algebra which absorbs a UHF algebra or, in general, absorbs theJiang-Su algebra Z , has stable rank one (see [22] and [24], respectively). These C*-algebrascertainly are well behaved from the perspective of the classification program.On the other hand, even beyond the classifiable C*-algebras, remarkably, it was shown byElliott, Ho, and Toms ([9], also see [13]) that any simple unital AH algebra with diagonalmaps (this class of C*-algebras contains the exotic AH algebras of [29] and [28], which cannotbe classified by the ordered K-groups together with the traces—the Elliott invariant), nomatter classifiable or not, always has stable rank one. This result actually is the mainmotivation of the current paper: Consider the C*-algebra of a minimal homeomorphism.In general, its behaviors are expected to be parallel to the behaviors of AH algebras withdiagonal maps (see, for instance, [15] and [10] on the classifiability and mean dimension),and it had been speculated for a while in the C*-algebra community that the C*-algebra ofa minimal homeomorphism should always has stable rank one. (Note that, as shown in [11],there exists a minimal homeomorphism of an infinite compact Hausdorff space such that thecorresponding C*-algebra is not classifiable by its Elliott invariant.)A considerable amount of work has been done concerning this question, and finally it issolved recently by Alboiu and Lutley in [1]. For the C*-algebra of a minimal homeomor-phism, one can consider the orbit-breaking subalgebra, which was introduced by Putnamfor Cantor dynamical systems ([20]) and then constructed for a general minimal homeomor-phism by Q. Lin ([14]). It was shown by Archey and Phillips ([2]) that if the orbit-breakingsubalgebra has stable rank one, then the transformation group C*-algebra must have stablerank one. If the minimal dynamical system has a Cantor factor (so that the orbit-breakingsubalgebra is an AH algebra with diagonal maps), with the result of [9], one has that thetransformation group C*-algebra has stable rank one (see [2]) (this result is also generalizedby Suzuki in [26] to the C*-algebra of a minimal almost finite groupoid). For a generalminimal homeomorphism ( Z -action), the orbit-breaking subalgebra might not be AH, butrather a unital inductive limit of subhomogeneous C*-algebras with diagonal maps (the DSHalgebras of [1]). Alboiu and Lutley show in [1] that any unital simple DSH algebra has stablerank one, and thus the C*-algebra of a minimal homeomorphism has stable rank one. TABLE RANK OF C( X ) ⋊ Γ 3
Beyond the case of Z -actions, however, it is not clear how to construct orbit-breakingsubalgebras in general. So, instead, one may consider the Uniform Rokhlin Property (URP)and Cuntz-comparison of Open Sets (COS) (see Definitions 2.19 and 2.21) for a topologicaldynamical system ( X, Γ), where Γ is a countable discrete amenable group. These two prop-erties are introduced in [17], and it is shown that, under the assumption of the (URP) and(COS), the radius of comparison of the crossed product C*-algebra C( X ) ⋊ Γ is dominatedby half of the mean dimension of ( X, Γ) ([17]), and the C*-algebra C( X ) ⋊ Γ is classifiedby its Elliott invariant if ( X, Γ) has mean dimension zero ([18]). Moreover, any free andminimal Z d -action have the (URP) and (COS) ([16]).In this paper, we still consider these two properties, and we show that if a free and minimalΓ-action has the (URP) and (COS), then the transformation group C*-algebra must havestable rank one (Theorem 7.8). Since any free and minimal Z d -actions have the (URP) and(COS), the C*-algebra C( X ) ⋊ Z d has stable rank one, no matter it is classifiable or not: Theorem (Corollary 7.9) . Let Z d act freely and minimally on a compact Hausdorff space X . Then tsr(C( X ) ⋊ Z d ) = 1 . As consequences of stable rank one, we obtain the following properties of the crossedproduct C*-algebra A = C( X ) ⋊ Z d , where ( X, Z d ) is free and minimal: • A has cancellation of projections, cancellation in Cuntz semigroup, and U( A ) / U ( A ) ∼ =K ( A ) (Corollary 7.11). • Approximately unitary equivalence classes of homomorphisms from an AI algebra to A is determined by the induced maps on Cuntz semigroups (Corollary 7.12). • Any strictly positive lower semicontinous affine function on T( A ) can be realized asthe rank function of some positive element of A ⊗ K (Corollary 7.13). • A absorbs the Jiang-Su algebra tensorially if, and only if, A has strict comparison ofpositive elements (Corollary 7.14). That is, A satisfies the Toms-Winter conjecture. • The real rank of A is either 0 or 1 (Corollary 7.15). Acknowledgements.
The research of the second named author is supported by an NSF grant(DMS-1800882). The result in this paper is obtained during the visit of the first namedauthor to the University of Wyoming in 2019-2020, which is supported by a CSC visitingscholar fellowship (No. 201906625028). The first named author thanks the Department ofMathematics and Statistics at the University of Wyoming for the hospitality. The research ofthe first named author is also partly supported by an NNSF grant of China (No. 11401088).2.
Notation and preliminaries
Topological Dynamical Systems.Definition 2.1.
Consider a topological dynamical system ( X, Γ), where X is a separablecompact Hausdorff space, Γ is a discrete group which acts on X from the right. The dynam-ical system ( X, Γ) is said to be minimal if
Y γ = Y, γ ∈ Γ CHUN GUANG LI AND ZHUANG NIU for some closed set Y ⊆ X implies Y = ∅ or Y = X ; and the dynamical system ( X, Γ) issaid to be free if xγ = x for some x ∈ X and γ ∈ Γ implies γ = e . Definition 2.2.
A Borel measure µ on X is invariant under the action σ if for any Borelset 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 E ⊆ Γ is said to be (
K, ε )-invariant if | EK ∆ E || E | < ε. The group Γ is amenable if there is a sequence (Γ n ) of finite subsets of Γ such that forany ( K, ε ), there is N such that Γ n is ( K, ε )-invariant for any n > N . The sequence (Γ n ) iscalled a Følner sequence.The K -interior of a finite set E ⊆ Γ is defined asint K ( E ) = { γ ∈ E : γK ⊆ E } , and the K -boundary of E is defined as ∂ K E := E \ int K ( E ) = { γ ∈ E : γγ ′ / ∈ E for some γ ′ ∈ K } . Note that | E \ int K ( E ) | ≤ | K | | EK \ E | ≤ | K | | EK ∆ E | , and hence for any ε >
0, if E is ( K, ε | K | )-invariant, then | E \ int K ( E ) || E | < ε. Remark . If a set E ⊆ Γ is ( F , ε )-invariant, then, for any γ ∈ Γ, the left translation γE is again ( F , ε )-invariant. Definition 2.5.
An (exact) tiling of a discrete group consists of • a finite collection S = { Γ , ..., Γ n } of finite subsets of Γ containing the unit e , calledthe shapes, • a finite collection C = { C ( S ) : S ∈ S} of disjoint subsets of Γ, called center sets,such that the left translations cS, c ∈ C ( S ) , S ∈ S form a partition of Γ. Remark . If Γ is amenable, it follows from [6] that for any finite set
F ⊆
Γ and any ε > F , ε )-invariant. TABLE RANK OF C( X ) ⋊ Γ 5
Crossed product C*-algebras.
Consider a topological dynamical system ( X, Γ).Then the group Γ acts (from the left) on the C*-algebra C( X ) by γ ( f ) = f ◦ 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, u γ u ∗ γ = u γ γ − , u e = 1 A , f ∈ C( X ) , γ, γ , γ ∈ Γ } . The C*-algebra A is nuclear if Γ is amenable (see, for instance, Corollary 7.18 of [31]). If,moreover, σ is minimal, the C*-algebra A is simple (Theorem 5.16 of [7] and Th´eor`eme 5.15of [33]), i.e., A has no non-trivial two-sided ideals.2.3. Cuntz-sub-equivalence and rank functions.Definition 2.7.
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. Example . Let f, g ∈ C( X ) be positive elements, and consider the open sets E := f − (0 , + ∞ ) and F := g − (0 , + ∞ ) . Then f - g if and only if E ⊆ F . That is, their Cuntz equivalence classes are determinedby their open supports.Throughout this paper, we use the following notation: Definition 2.9.
For any ε >
0, define the function f ε : [0 , → [0 ,
1] by f ε ( t ) = , t < ε/ , linear , ε/ ≤ t < ε, , t ≥ ε. Lemma 2.10 (Proposition 2.4(iv) of [23]) . Let A be a C*-algebra, and let a, b ∈ A be positive.If a - b , then for any δ > , there is ε > and r ∈ A such that f δ ( a ) = r ∗ f ε ( b ) r. In particular, denoted by v = f ε ( b ) r ∈ A , one has f δ ( a ) = v ∗ v and vv ∗ ∈ Her( b ) . Definition 2.11.
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 ) = µ τ (sp( a ) ∩ (0 , + ∞ )) , a ∈ A + , where µ τ is the Borel measure induced by τ on the spectrum of a . It is well known that if a - b , then d τ ( a ) ≤ d τ ( b ) , τ ∈ T( A ) . CHUN GUANG LI AND ZHUANG NIU
Example . Consider h ∈ C( X ) + and let µ be a Borel probability measure on X , where X is a compact Hausdorff space. Thend τ µ = µ ( f − (0 , + ∞ )) , where τ µ is the trace of C( X ) defined by τ µ ( f ) = Z f d µ, f ∈ C( X ) . If A = M n (C ( X )), where X is a locally compact Hausdorff space. Then, for any positiveelement 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 2.13.
For each open set E ⊆ X , pick a continuous function(2.1) ϕ E : X → [0 , E = ϕ − E ((0 , V ⊆ E such that ϕ E | V = 1. (In particular, k ϕ E k = 1.)For instance, one can pick ϕ E ( x ) = min { ε d ( x, X \ E ) , } , where d is a compatible metricon X and ε > ϕ E Aϕ E is independent of the choice of individualfunction ϕ E , where A is a C*-algebra containing C( X ). Therefore, one also denotes ϕ E Aϕ E by Her( E ) in the paper.2.4. Order zero maps and Rokhlin towers.Definition 2.14 (Order zero maps) . Let
A, B be C*-algebras. A linear map φ : A → B issaid to be order zero if a ⊥ b = ⇒ φ ( a ) ⊥ φ ( b ) , a, b ∈ A + . Let A be a C*-algebras and φ : M n ( C ) → A is a c.p. order zero map. Let C :=C ∗ ( φ (M n )) ⊆ A , and let h = φ (1 n ). Then h ∈ C ∩ C ′ , k h k = k φ k , and there is a ho-momorphism π φ : M n ( C ) → M ( C ) ∩ { h } ′ ⊆ A ∗∗ such that φ ( a ) = π φ ( a ) h, a ∈ M n ( C ) . Moreover, C ∼ = M n (C ((0 , . Definition 2.15.
Let φ : M n ( C ) → A be a c.p. order zero map, and let f ∈ C ((0 , k h k ]),where h = φ (1 n ). Then the mapM n ( C ) ∋ a π φ ( a ) f ( h ) ∈ A is again an order zero map, where π φ is as above. Denote this new order zero map by f ( φ ).An order zero map ψ : M n ( C ) → A is said to be extendable if there is a c.p. order zeromap ψ ′ : M n ( C ) → A such that φ = f δ ( ψ ′ ) for some δ > TABLE RANK OF C( X ) ⋊ Γ 7
The following is well known:
Lemma 2.16 ([32]) . Let v , v , ..., v n ∈ A , where A is a C*-algebra, such that • v ∗ v = v ∗ v = · · · = v ∗ n v n , • v ∗ v , v v ∗ , v v ∗ , ..., v n v ∗ n are mutually orthogonal, and • k v ∗ v k = 1 .Then there is an order zero map φ : M n +1 ( C ) → A such that k φ k = 1 , φ ( e , ) = v ∗ v , φ ( e i,i ) = v i v ∗ i , i = 1 , , ..., n. Definition 2.17.
A Rokhlin tower of a dynamical system ( X, Γ) is a pair ( B, Γ ), where B ⊆ X and Γ ⊆ Γ is finite, such that
Bγ, γ ∈ Γ , are mutually disjoint. It is an open tower if the base set B is open. Without loss of generality,one may assume that Γ contains the unit of Γ.One can naturally construct order-zero maps from Rokhlin towers, as follows:Let ( B, Γ ) be a tower, and pick a positive function e : X → [0 ,
1] such that e − ((0 , ⊆ B .Let γ , γ ∈ Γ and consider v := u ∗ γ e u γ . Then v ∗ v = u ∗ γ eu γ = e ◦ γ − and vv ∗ = u ∗ γ eu γ = e ◦ γ − . In general, if F , F ⊆ Γ are two disjoint sets with | F | = | F | . Pick a one-to-onecorrespondence θ : F → F , and consider v := X γ ∈ F u ∗ θ ( γ ) e u γ . Then v ∗ v = X γ ,γ ∈ F u ∗ γ e u θ ( γ ) u ∗ θ ( γ ) e u γ = X γ ,γ ∈ F u ∗ γ u θ ( γ ) ( e ◦ θ ( γ ) − )( e ◦ θ ( γ ) − ) u ∗ θ ( γ ) u γ = X γ ∈ F u ∗ γ eu γ , and the same calculation shows that vv ∗ = X γ ∈ F u ∗ θ ( γ ) eu θ ( γ ) = X γ ∈ F u ∗ γ eu γ . Now, suppose there are mutually disjoint setsΓ , , Γ , , ..., Γ ,n ⊆ Γ such that | Γ , | = | Γ , | = · · · = | Γ ,n | . CHUN GUANG LI AND ZHUANG NIU
Consider e := X γ ∈ Γ , u ∗ γ eu γ , ..., e n := X γ ∈ Γ ,n u ∗ γ eu γ . Then the above calculation shows that there are v , v , ..., v n − such that v ∗ v = v ∗ v = · · · = v ∗ n − v n − = e and v v ∗ = e , v v ∗ = e , ..., v n − v ∗ n − = e n . So, there is an order zero map φ : M n ( C ) → A such that φ ( e i,i ) = e i , i = 1 , ..., n. In summary, one has the following lemma:
Lemma 2.18.
Let ( B, Γ ) be a Rokhlin tower, and let Γ , , Γ , , ..., Γ ,n ⊆ Γ be mutuallydisjoint sets such that | Γ , | = | Γ , | = · · · = | Γ ,n | . Let e : X → [0 , be a continuous function such that e − ((0 , ⊆ B . Set e := X γ ∈ Γ , u ∗ γ eu γ , ..., e n := X γ ∈ Γ ,n u ∗ γ eu γ . Then there is an order zero map φ : M n ( C ) → A such that φ ( e i,i ) = e i , i = 1 , ..., n. Uniform Rokhlin property and Cuntz comparison of open sets.Definition 2.19 ([17]) . A dynamical system ( X, Γ) is said to have the uniform Rokhlinproperty (URP) if for any finite set F , any ε >
0, there are open Rokhlin towers ( B , F ),..., ( B S , F S ) such that F , F , ..., F S are ( F , ε )-invariant, B s γ, γ ∈ F s , s = 1 , , ..., S are mutually disjoint, and µ ( X \ S G s =1 G γ ∈ F s B s γ ) < ε, µ ∈ M ( X, Γ) . Remark . If E ⊆ X is a closed set, then µ ( E ) < ε for all µ ∈ M ( X, Γ) if, and only if,the orbit capacity of E is at most ε . Definition 2.21 ([17]) . A topological dynamical system ( X, Γ) is said to have ( λ, m )-Cuntzcomparison of open sets, where λ ∈ (0 , + ∞ ) and m ∈ N , if, for any open sets E, F ⊆ X with µ ( E ) ≤ λµ ( F ) , µ ∈ M ( X, Γ) , one has ϕ E - ϕ F ⊕ · · · ⊕ ϕ F | {z } m in M ∞ (C( X ) ⋊ Γ) . A topological dynamical system is said to have Cuntz comparison of open sets (COS) ifit has ( λ, m )-Cuntz comparison of open sets for some λ and m . TABLE RANK OF C( X ) ⋊ Γ 9
It follows from Theorem 4.2 and Theorem 5.5 of [16] that
Theorem 2.22.
Any free and minimal dynamical system ( X, Z d ) has the (URP) and (COS). Some lemmas
In this section, let us develop some lemmas on dimension drop C*-algebras and order zeroc.p.c. maps with domain a matrix algebra. Let us start with a simple observation:
Lemma 3.1.
Let a, c be elements of a unital C*-algebra, and assume that ac = c and a ispositive. Then, f ( a ) c = f (1) c, f ∈ C([0 , k a k ]) . Proof. If f ( t ) = P nk =0 c k t k , then f ( a ) c = ( n X k =0 c k a k ) c = n X k =0 c k a k c = n X k =0 c k c = ( n X k =0 c k ) c = f (1) c. The general statement follows from the Weierstrass Theorem. (cid:3)
It is well known that the universal unital C*-algebra generated by v with respect torelations vv ∗ ⊥ v ∗ v and k vv ∗ k ≤ D = { f : [0 , → M ( C ) : f (0) ∈ C } , with v corresponding to [0 , ∋ t (cid:18) √ t (cid:19) . Using this identification, one has the following lemma:
Lemma 3.2.
Let A be a unital C*-algebra, and let v ∈ A . Consider a = vv ∗ and b = v ∗ v ,and assume that k a k ≤ and a ⊥ b . Define w = cos( π vv ∗ + v ∗ v )) + g ( vv ∗ ) v − g ( v ∗ v ) v ∗ , where g ( t ) = sin( π t ) √ t , t ∈ (0 , . Then w ∈ C ∗ { v, A } is a unitary such that (3.1) b ( w ∗ cw ) = ( w ∗ cw ) b = w ∗ cw, if ac = ca = c, and (3.2) a ( wcw ∗ ) = ( wcw ∗ ) a = wcw ∗ , if bc = cb = c. Proof.
Noting that vv ∗ + v ∗ v is central in C ∗ (1 , v ), vv ∗ ⊥ v ∗ v , and v ∗ h ( vv ∗ ) = h ( v ∗ v ) v ∗ , h ∈ C ((0 , , one has ww ∗ = (cos( π vv ∗ + v ∗ v )) + g ( vv ∗ ) v − g ( v ∗ v ) v ∗ )(cos( π vv ∗ + v ∗ v )) + v ∗ g ( vv ∗ ) − vg ( v ∗ v ))= cos ( π vv ∗ + v ∗ v )) + g ( vv ∗ ) vv ∗ + g ( v ∗ v ) v ∗ v = cos ( π vv ∗ + v ∗ v )) + sin ( π v ∗ v ) + sin ( π vv ∗ )= cos ( π vv ∗ + v ∗ v )) + sin ( π v ∗ v + vv ∗ ))= 1 . The same calculation shows that w ∗ w = 1, and hence w is a unitary.Let c ∈ A be an element satisfying ac = ca = c. Note that, for any f ∈ C([0 , f ( a ) c = f (1) c = cf ( a ) . Therefore, w ∗ cw = (cos( π vv ∗ + v ∗ v )) + v ∗ g ( vv ∗ ) − vg ( v ∗ v )) c (cos( π vv ∗ + v ∗ v )) + g ( vv ∗ ) v − g ( v ∗ v ) v ∗ )= (cos( π a )) + v ∗ g ( vv ∗ )) c (cos( π a )) + g ( vv ∗ ) v )= v ∗ g ( vv ∗ ) cg ( vv ∗ ) v = v ∗ cv, and hence ( w ∗ cw ) b = v ∗ cvb = v ∗ cvv ∗ v = v ∗ cav = v ∗ cv = w ∗ cw and b ( w ∗ cw ) = bv ∗ cv = v ∗ vv ∗ cv = v ∗ acv = v ∗ cv = w ∗ cw. A similar calculation shows that a ( wcw ∗ ) = ( wcw ∗ ) a = wcw ∗ if bc = cb = c . (cid:3) The following lemma is crucial in the proof of Proposition 5.4, in which it produces anelement that behaves as a lower triangular matrix.
Lemma 3.3.
Let A be a unital C*-algebra, and let v , v , ..., v n ∈ A be elements satisfying • v ∗ v = v ∗ v = · · · = v ∗ n v n , TABLE RANK OF C( X ) ⋊ Γ 11 • v ∗ v , v v ∗ , v v ∗ , ..., v n v ∗ n are mutually orthogonal, and • k v ∗ v k = 1 .Then there is a unitary w ∈ A satisfies the following properties: (1) If E , E , ..., E n ∈ A are mutually orthogonal positive elements such that [ v ∗ v , E i ] = 0 and ( v i v ∗ i ) E i = v i v ∗ i , i = 1 , , ..., n, then wE i ∈ ( E + E i + E i +1 ) A, ≤ i ≤ n − . (2) If D ⊆ A is a hereditary sub-C*-algebra such that v i v ∗ i ∈ D, i = 1 , , ..., n, and d ∈ D is an element such that [ d, v ∗ v ] = 0 and v ∗ n d = 0 , then wd ∈ DA. (3) If c ∈ A + satisfies ( v ∗ v ) c = c, then wc ∈ ( v v ∗ ) A. Proof.
Consider the universal algebra A generated by v , v , ..., v n with respect to the rela-tions: • v ∗ v = v ∗ v = · · · = v ∗ n v n , • v ∗ v , v v ∗ , v v ∗ , ..., v n v ∗ n are mutually orthogonal, • k v ∗ v k = 1.It is well known that A is isomorphic to the dimension drop C*-algebra(3.4) D n +1 := { f ∈ C([0 , , M n +1 ) : f (0) = 0 n +1 } ∼ = C ((0 , ⊗ M n +1 ( C )with v i ( t ) = √ t ⊗ e i, , t ∈ [0 , , i = 1 , , ..., n, where e i,j , i, j = 0 , , ..., n , are matrix units of M n +1 ( C ).With the identification (3.4), consider the unitary w ∈ D n +1 + C w ( t ) = n +1 , if t = 0 , n − k cos π n ( t − k − n ) 0 0 · · · sin π n ( t − k − n ) − sin π n ( t − k − n ) 0 0 · · · cos π n ( t − k − n ) − · · ·
0. . . . . . ... − , if t ∈ [ k − n , kn ].Then the image of w in A , which is still denoted by w , satisfies the properties of the lemma. Indeed, put w i,j = v i v ∗ j , i, j = 1 , ..., n, and w , = v ∗ v , w i, = v i , w ,j = v ∗ j , i, j = 1 , , ..., n. Then w = 1 + n X i =0 g i,i ( w i,i ) + n X i =1 g i,i − ( w i,i ) w i,i − + n − X i =0 g i,n ( w i,i ) w i,n , for some functions g i,j ∈ C ((0 , g , (1) = − . Let E , E , ..., E n be mutually orthogonal positive elements of A satisfying(3.6) w i,i E i = w i,i , i = 1 , , ..., n, and(3.7) [ w , , E i ] = 0 . Note that, by (3.6) and polar-decomposition, one has v ∗ j E i = v ′ j ( v j v ∗ j ) E i = v ′ j w j,j E i = v ′ j w j,j E j E i = 0 , j = i, ≤ i, j ≤ n, where v ′ j is a partial isometry in the enveloping von Neumann algebra. Then, for any1 ≤ i ≤ n − w i ,i E i = v i v ∗ i E i = 0 , i = i, ≤ i , i ≤ n and hence wE i = (1 + n X j =0 g j,j ( w j,j ) + n X j =1 g j,j − ( w j,j ) w j,j − + n − X j =0 g j,n ( w j,j ) w j,n ) E i = E i + g , ( w , ) E i + g i,i ( w i,i ) E i + g , ( w , ) w , E i + g i +1 ,i ( w i +1 ,i +1 ) w i +1 ,i E i . By (3.7), g , ( w , ) E i = E i g , ( w , ) ∈ E i A. By (3.6), g i,i ( w i,i ) E i = E i g i,i ( w i,i ) ∈ E i A,g , ( w , ) w , E i = E g , ( w , ) w , E i ∈ E A, and g i +1 ,i ( w i +1 ,i +1 ) w i +1 ,i E i = E i +1 g i +1 ,i ( w i +1 ,i +1 ) w i +1 ,i E i ∈ E i +1 A. Therefore, wE ∈ ( E + E i + E i +1 ) A, and this proves Property (1).For Property (2), let D be a hereditary sub-C*-algebra such that w i,i ∈ D, i = 1 , , ..., n, TABLE RANK OF C( X ) ⋊ Γ 13 and let d ∈ D be an element satisfying[ d, w , ] = 0 and w ,n d = 0 . Then wd = (1 + n X i =0 g i,i ( w i,i ) + n X i =1 g i,i − ( w i,i ) w i,i − + n − X i =0 g i,n ( w i,i ) w i,n ) d = d + g , ( w , ) d + n X i =1 g i,i ( w i,i ) d + n X i =1 g i,i − ( w i,i ) w i,i − d + n − X i =1 g i,n ( w i,i ) w i,n d ∈ DA.
For Property (3), let c ∈ A be a positive element such that w , c = c. Then, by Lemma 3.1 and (3.5), one has wc = (1 + n X i =0 g i,i ( w i,i ) + n X i =1 g i,i − ( w i,i ) w i,i − + n − X i =0 g i,n ( w i,i ) w i,n ) c = c + g , ( w , ) c + g , ( w , ) w , c = (1 + g , (1)) c + g , ( w , ) w , c = g , ( w , ) w , c ∈ w , A. This proves the lemma. (cid:3)
Lemma 3.4.
Let A be a unital C*-algebra, and let φ : M n ( C ) → A be an extendableorder zero c.p.c. map. Denote by e i = φ ( e i,i ) and h = φ (1 n ) . Then, for any permutation σ : { , , ..., n } → { , , ..., n } , there is a unitary u ∈ A such that [ u, h ] = 0 and u ∗ e i u = e σ ( i ) , i = 1 , , ..., n. Proof.
Since φ is extendable, there is an order zero map ˜ φ : M n ( C ) → A and δ > φ = f δ ( ˜ φ ). Note that C := C ∗ { ˜ φ (M n ( C )) } ∼ = { f : [0 , → M n ( C ) : f (0) = 0 n } ∼ = C ((0 , ⊗ M n ( C )with e i = f δ ⊗ e i,i , i = 1 , , ..., n , under the isomorphism. Denote by U ∈ M n ( C ) thepermutation unitary matrix such that U ∗ e i,i U = e σ ( i ) ,σ ( i ) , i = 1 , , ..., n. Since the unitary group of M n ( C ) is path connected, there is a continuous path of unitaries[0 , δ ∋ t U t ∈ M n ( C )such that U = 1 n and U δ/ = U . Set u : t u ( t ) = (cid:26) U t , t ∈ [0 , δ ] ,U, t ∈ [ δ , . Then the unitary u ∈ C + C A ⊆ A has the desired property. (cid:3) Nilpotent elements, order zero maps, and limits of invertible elements
An element a of a C*-algebra is said to be nilpotent if a n = 0 for some n ∈ N . It is wellknown that if a is nilpotent, then a + ε A is invertible for any non-zero ε ; in fact,(4.1) ( a + ε A ) − = 1 A ε − aε + a ε + · · · + ( − n − a n − ε n + · · · , where the infinite series is eventually zero, as a is nilpotent. Hence any nilpotent element isin the closure of invertible elements.The following lemma is a modified version of Lemma 4.2 of [2]. Lemma 4.1 (c.f. Lemma 4.2 of [2]) . Let A be a finite unital C*-algebra and let a ∈ A .Suppose there exist positive elements b , b , c , c such that (1) b + b = 1 A , (2) c + c = 1 A , (3) C ∗ { b , b , c , c } is commutative, (4) b c = c , (5) b c = b , (6) c b = 0 , (7) there are unitaries u, v ∈ A such that u ( b a ( b − c ) + b ab ) v and u ( b ab ) v are nilpotent, (4.2) v ( u ( b ab ) v ) n u ∈ b Ab , n = 1 , , ..., and c vub = 0(8) b a = 0 .Then a ∈ GL( A ) Proof.
The proof is the similar to that of Lemma 5.2 of [2], but with c = b = 0.Fix an arbitrary ε > A = c + ( b − c ) + b . Put a , = b ac a , = b a ( b − c ) a , = b ab . Then, by (1) and (8), a , + a , + a , = b a ( c + ( b − c ) + b ) = b a ( b + b ) = b a = ( b + b ) a = a. Note that a , = u − a ′ , v − , where a ′ , := ua , v = u ( b ab ) v TABLE RANK OF C( X ) ⋊ Γ 15 is nilpotent (by 7).Put t = u − ( a ′ , + ε A ) v − . Then t is invertible and(4.3) k t − a , k = (cid:13)(cid:13) εu − v − (cid:13)(cid:13) = ε. Using (4.1), write t − = v ( 1 A ε − a ′ , ε + ( a ′ , ) ε + · · · + ( − n − ( a ′ , ) n − ε n + · · · ) u = t + vuε , where, by (4.2), t := − va ′ , uε + v ( a ′ , ) uε + · · · + ( − n − v ( a ′ , ) n − uε n + · · · ∈ b Ab . (Note that the series above are eventually zero.) Therefore, together with c b = 0 and c vub = 0 , one has(4.4) a , t − a , = b ac ( t + vuε ) b ac = 0and(4.5) a , t − a , = b ac ( t + vuε ) b a ( b − c ) = 0 . Therefore, by (4.4), one has(4.6) ( a , + t ) t − (1 A − a , t − ) = a , t − − a , t − a , t − + 1 A − a , t − = 1 A and t − (1 A − a , t − )( a , + t ) = t − a , + 1 A − t − a , t − a , − t − a , = 1 A . In particular, the element ( a , + t ) is invertible with( a , + t ) − = t − (1 A − a , t − ) . Then, together with (4.5), one has t − (1 A − a , t − )( a , + a , + t ) = ( a , + t ) − ( a , + a , + t )(4.7) = 1 A + ( a , + t ) − a , = 1 A + t − (1 − a , t − ) a , = 1 A + t − a , . Consider t := 1 A + t − a , (4.8) = t − ( t + a , )= t − ( u − ( a ′ , + ε A ) v − + u − a ′ , v − )= t − u − ( a ′ , + a ′ , + ε A ) v − , where a ′ , = ua , v = u ( b a ( b − c )) v. Since a ′ , + a ′ , = u (( b ab ) + b a ( b − c )) v is nilpotent, one has that t is invertible.Put y = ( a , + t ) t . Since a , + t and t are invertible, one has y ∈ GL( A ) . Applying (4.6) in the third step, (4.7) in the fifth step, definition of t (see (4.8)) in the sixthstep, and (4.3) in the last step, one has k a − y k = k a , + a , + a , − ( a , + t ) t k≤ k a , − t k + k A ( a , + a , + t ) − ( a , + t ) t k≤ k a , − t k + (cid:13)(cid:13) ( a , + t ) t − (1 A − a , t − )( a , + a , + t ) − ( a , + t ) t (cid:13)(cid:13) ≤ k a , − t k + k a , + t k (cid:13)(cid:13) t − (1 A − a , t − )( a , + a , + t ) − t (cid:13)(cid:13) = k a , − t k + k a , + t k (cid:13)(cid:13) A + t − a , − t (cid:13)(cid:13) = k a , − t k = ε. Since ε is arbitrary, one has a ∈ GL( A ), as desired. (cid:3) Lemma 4.2.
Let A be a unital C*-algebra, and let φ : M n ( C ) → A be an extendable orderzero c.p.c map. Set h = φ (1 n ) and e i = φ ( e i,i ) , i = 1 , , ..., n, and set b = f δ ( h ) , b = 1 − b , c = f δ/ ( h ) , c = 1 − f δ/ ( h ) , for some δ ∈ (0 , .Then, for any m ∈ N with m ≤ n and for some ≤ i < i < · · · < i m ≤ n, there are unitaries u, v ∈ A satisfying the following properties: (1) c vub = 0 , and (2) if a ∈ c Ac satisfies e i ae j = 0 , if j − i > d, for some d ∈ N with d < m , and e i j a = ae i j = 0 , ≤ j ≤ m. Then uav is nilpotent. Moreover, if a ∈ b Ab , then v ( uav ) k u ∈ b Ab , k = 1 , , .... TABLE RANK OF C( X ) ⋊ Γ 17
Proof.
With the given m and i , i , ..., i m , define the permutation σ : { , , ..., n } → { , , ..., n } by stretching { , , ..., n − m } to { , , ..., n } \ { i , i , ..., i m } , and then moves { n − m + 1 , ..., n } to fill { i , i , ..., i m } . More precisely, write I = { ≤ i ≤ n − m : i < i } ,I = { ≤ i ≤ n − m : i ≤ i, i + 1 < i } , · · · I m − = { ≤ i ≤ n − m : i m − ≤ i + m − , i + m − < i m } ,I m = { ≤ i ≤ n − m : i m ≤ i + m − } , and note that { , , ..., n − m } = I ⊔ I ⊔ · · · ⊔ I m (some of I k , k = 1 , ..., m , might be empty).Then σ ( i ) = (cid:26) i + k, if i ∈ I k ,i i − n + m , n − m + 1 ≤ i ≤ n. Note that for any d ∈ N ,(4.9) σ ( j ) − σ ( i ) > d, if 1 ≤ i, j ≤ n − m and j − i > d. Since φ is extendable, by Lemma 3.4, there is a unitary w ∈ A such that[ w , h ] = 0 and w ∗ e i w = e σ ( i ) , i = 1 , , ..., n. By Lemma 3.4 again, there is a unitary w ∈ A satisfying[ w , h ] = 0and w ∗ e i w = (cid:26) e i + n − m , ≤ i ≤ m,e i − m , m + 1 ≤ i ≤ n. Then, the unitaries u := w w and v := w ∗ satisfy the property of the lemma.Indeed, since w and w commute with h and c b = 0, one has that c ( vu ) b = c ( w ∗ w w ) b = ( w ∗ w w )( c b ) = 0 . Let a ∈ c Ac satisfy e i ae j = 0 , if j − i > d for some natural number d < m , and e i j a = ae i j = 0 , ≤ j ≤ m. Consider the element w aw ∗ . Note that, for any n − m + 1 ≤ i ≤ n , σ ( i ) ∈ { i , i , ..., i m } . Hence(4.10) e i ( w aw ∗ ) = w ( e σ ( i ) a ) w ∗ = 0 , n − m + 1 ≤ i ≤ n and(4.11) ( w aw ∗ ) e i = w ( ae σ ( i ) ) w ∗ = 0 , n − m + 1 ≤ i ≤ n. Also note that, for any 1 ≤ i, j ≤ n − m satisfying j − i > d , by (4.9), one has that σ ( j ) − σ ( i ) > d , and hence e i ( w aw ∗ ) e j = w ( e σ ( i ) ae σ ( j ) ) w ∗ = 0 . Together with (4.10) and (4.11), one has(4.12) e i ( w aw ∗ ) e j = 0 , j − i > d, ≤ i, j ≤ n. Consider the element uav = w w aw ∗ , and note that for any 1 ≤ i, j ≤ n with j ≥ i , • if 1 ≤ i ≤ m , then n − m + 1 ≤ i + n − m ≤ n , and by (4.10), e i ( w w aw ∗ ) e j = w ( e i + n − m w aw ∗ ) e j = 0; • if m + 1 ≤ i ≤ n , then j − ( i − m ) ≥ m > d , and hence, by (4.12), e i ( w w aw ∗ ) e j = w ( e i − m w aw ∗ e j ) = 0 . That is,(4.13) e i ( uav ) e j = 0 , j ≥ i. Consider ˜ e i := f δ ( e i ) , i = 1 , , ..., n, and it follows from (4.13) that ˜ e i ( uav )˜ e j = 0 , j ≥ i. Since uav ∈ c Ac , one has uav = (˜ e + · · · + ˜ e n ) uav (˜ e + · · · + ˜ e n ) = n X i,j =1 ˜ e i ( uav )˜ e j = X i>j ˜ e i ( uav )˜ e j . That is, there is a decomposition uav = X i>j a (1) i,j , where a (1) i,j ∈ ˜ e i A ˜ e j . A direct calculation shows that( uav ) = X i>k>j a i,k a k,j = X i>k − a (2) i,j , where a (2) i,j ∈ ˜ e i A ˜ e j .Repeating n times, one has ( uav ) n = 0 . Hence uav is nilpotent.If, moreover, a ∈ b Ab , since u and v commute with h (and hence commute with b ), onehas v ( uav ) k u ∈ v ( u ( b Ab ) v ) k u ⊆ b Ab , k = 1 , , ..., as desired. (cid:3) TABLE RANK OF C( X ) ⋊ Γ 19
Proposition 4.3.
Let A be a unital C*-algebra, and let a ∈ A . If there exist an extendableorder zero c.p.c. map φ : M n ( C ) → A , natural numbers d < m and ≤ i < i < · · · < i m ≤ n , such that (1) (1 − h ) a = 0 , where h = φ (1 n ) , (2) e i j a = ae i j = 0 , j = 1 , , ..., m , where e i = φ ( e i,i ) , and (3) e i ae j = 0 , if j − i > d .Then a ∈ GL( A ) .Proof. Pick an arbitrary δ ∈ (0 , b = f δ ( h ) , c = f δ/ ( h ) , b = 1 − f δ ( h ) , and c = 1 − f δ/ ( h ) . Note that c b = b , b c = c , and c b = 0 . Since (1 − h ) a = 0, one has a = ha , and b a = f δ ( h ) a = f δ (1) a = a. Hence b a = 0 . Consider the elements b ab and b a ( b − c ) + b ab , and note that both of them are in c Ac . Also note that (since h commutes with e i , i =1 , , ..., n ) e i b ab e j = b e i ae j b = 0 , j − i > d and e i j ( b ab ) = b e i j ab = 0 = b ae i j b = ( b ab ) e i j , j = 1 , , ..., m ;and the same argument shows that e i ( b a ( b − c ) + b ab ) e j = 0 , j − i > d, and e i j ( b a ( b − c ) + b ab ) = 0 = ( b a ( b − c ) + b ab ) e i j , j = 1 , , ..., m. Let u, v ∈ A be the unitaries obtained by applying Lemma 4.2 to φ , δ , m and i , i , ..., i m .Then, by Lemma 4.2, one has c vub = 0 ,u ( b ab ) v and u ( b a ( b − c ) + b ab ) v are nilpotent, and v ( ub ab v ) n u ∈ b Ab , n = 1 , , .... Then, by Lemma 4.1, one has a ∈ GL( A ), as desired. (cid:3) Property (D) and stable rank one
Definition 5.1.
Let A be a unital C*-algebra. An element a ∈ A is said to be a D -operatorif there exists a nonzero positive element b ∈ A satisfying ba = ab = 0 , and there exists an order zero c.p.c. map φ : M pq ( C ) → A, where p, q ∈ N , and there exist r, l ∈ N such that, with e i := φ ( e i,i ) , i = 1 , , ..., pq,s k := e ( k − p +1 + · · · + e ( k − p + r , k = 1 , ..., q, and E k := e ( k − p +1 + · · · + e ( k − p + p , k = 1 , ..., q, one has(1) E k aE k = 0, k − k ≥ l , 1 ≤ k , k ≤ q ;(2) qr > ( l + 1) p ;(3) for each k = 1 , , ..., q , there are positive elements c k , d k with norm 1 such that(a) c k , d k ∈ bAb, (b) c k ⊥ s k and c k ⊥ d k , (c) c k E k = c k and d k E k = d k , and(d) s k - c k and 1 A − h - d k , where h := φ (1 pq ).Recall that Definition 5.2.
Let A be a C*-algebra, then, defineZD( A ) := { a ∈ A : d a = ad = 0 for some d , d ∈ A + \ { }} . And define
Definition 5.3.
The C*-algebra A is said to have Property (D) if for any a ∈ ZD( A ) and any ε >
0, there are unitaries u , u ∈ A and a D -operator a ′ ∈ A such that k u au − a ′ k < ε. It turns out that any D -operator is in the norm closure of invertible elements. Proposition 5.4.
Let a ∈ A be a D -operator of a unital C*-algebra A . Then a ∈ GL( A ) .Proof. Let a be a D -operator. Then there is b ∈ A + \ { } such that(5.1) ba = ab = 0;and there exists an order zero c.p.c. map φ ′ : M pq ( C ) → A, where p, q ∈ N , and there exist r, l ∈ N such that, with e ′ i := φ ′ ( e i,i ) , i = 1 , , ..., pq, TABLE RANK OF C( X ) ⋊ Γ 21 s k := e ′ ( k − p +1 + · · · + e ′ ( k − p + r , k = 1 , ..., q, and E k := e ′ ( k − p +1 + · · · + e ′ ( k − p + p , k = 1 , ..., q, one has(1) E k aE k = 0, k − k ≥ l , 1 ≤ k , k ≤ q ;(2) qr > ( l + 1) p ;(3) there are positive elements c k , d k , k = 1 , , ..., q , with norm 1 such that(a) c k , d k ∈ bAb, (b) c k ⊥ s k and c k ⊥ d k ,(c) c k E k = c k and d k E k = d k , and(d) s k - c k and 1 A − h ′ - d k , where h ′ := φ ′ (1 pq ).With above, one asserts that there exist an extendable order zero c.p.c. map φ : M pq ( C ) → A and unitaries u, w ∈ A such that, with h := φ (1 pq ) and e i := φ ( e i,i ) , i = 1 , , ..., pq, then • (1 A − h )( uw ∗ au ∗ ) = 0 , • e i ( uw ∗ au ∗ ) e j = 0 , j − i > ( l + 1) q, and • there are e i , ..., e i qr such that e i j ( uw ∗ au ∗ ) = ( uw ∗ au ∗ ) e i j = 0 , j = 1 , , ..., qr. It then follows from Proposition 4.3 (with d = ( l + 1) p and m = qr ) and Condition 2 that uw ∗ au ∗ ∈ GL( A ) . Since u, w are unitaries, one has that a ∈ GL( A ), and the proposition follows.Let us show the assertion. By Condition 3a and (5.1)(5.2) c k a = ac k = d k a = ad k = 0 , k = 1 , ..., q. Consider the positive element f ( h ′ ), and note that 1 A − f ( h ′ ) - A − h ′ . Then, byCondition 3d, one has 1 A − f ( h ′ ) - d k , k = 1 , , ..., q, and therefore, by Proposition 2.4(iv) of [23] (see Lemma 2.10), there are v , v , ..., v q ∈ A such that(5.3) v ∗ k v k = f (1 A − f ( h ′ )) and v k v ∗ k ∈ Her( d k ) ⊆ Her( b ) , k = 1 , ..., q. It follows from Condition 3c that f (1 A − f ( h ′ )) ⊥ d k ; then, using Condition 3c again, onehas f (1 A − f ( h ′ )) ⊥ v k v ∗ k and v k v ∗ k ∈ Her( E k ) , ≤ k ≤ q, and hence v ∗ v , v v ∗ , v v ∗ , ..., v q v ∗ q are mutually orthogonal. Applying Lemma 3.3 to v , v , ..., v q , one obtains the unitary w ∈ A which satisfies the properties of Lemma 3.3.By Condition 3d,(5.4) s k - c k , k = 1 , ..., q. Since s k , c k ∈ E k AE k , one may assume that the Cuntz sub-equivalences (5.4) hold in thehereditary sub-C*-algebra E k AE k . By Proposition 2.4(iv) of [23] (see Lemma 2.10), there is z k ∈ E k AE k such that(5.5) z ∗ k z k = f ( s k ) and z k z ∗ k ∈ Her( c k ) ⊆ Her( b ) . Note that, by Condition (3c),(5.6) z k ∈ f ( E k ) Af ( E k ) . By Condition 3b, z ∗ k z k = f ( s k ) ⊥ c k Ac k ∋ z k z ∗ k . Since f ( s k ) f ( s k ) = f ( s k ), applying Lemma 3.2 to v = z k , one has that, with(5.7) u k := cos( π z k z ∗ k + z ∗ k z k )) + z ∗ k g ( z k z ∗ k ) − z k g ( z ∗ k z k ) , where g ( t ) = sin( πt/ / √ t , t ∈ (0 , u k ∈ C ∗ ( z k ,
1) is a unitary such that u ∗ k f ( s k ) u k ∈ Her( c k ) ⊆ Her( b ) . By (5.6), one has that u k ∈ f ( E k ) Af ( E k ) + C A and hence u ∗ k E k u k ∈ E k AE k ,u ∗ k au k = a, a ∈ E k ′ AE k ′ , k = k ′ . (5.8) u ∗ k f ( E k ) u k ∈ f ( E k ) Af ( E k ) and [ u k , f ( E k )] = 0 . In particular, with u := q Y k =1 u k , one has(5.9) u ∗ E k u ∈ E k AE k , ≤ k ≤ q, (5.10) [ u, f ( E k )] = 0 , ≤ k ≤ q, and(5.11) u ∗ f ( s k ) u ∈ Her( c k ) ⊆ Her( b ) , ≤ k ≤ q. Consider the positive element 1 A − f ( h ′ ), and note that(1 A − f ( h ′ )) v ∗ v = (1 A − f ( h ′ )) f (1 A − f ( h ′ )) = 1 A − f ( h ′ ) . TABLE RANK OF C( X ) ⋊ Γ 23
It follows from Lemma 3.3(3) that(1 A − f ( h ′ )) w ∗ ∈ A ( v v ∗ ) ⊆ Ad and therefore, by (5.2),(5.12) (1 A − f ( h ′ )) w ∗ a = 0 . Since v ∗ v , E k ∈ C ∗ ( e ′ , ..., e ′ pq ), which is commutative, together with Condition 3c and(5.3), one has [ v ∗ v , E k ] = 0 and ( v k v ∗ k ) E k = v k v ∗ k , k = 1 , , ..., q. It then follows from Lemma 3.3(1) that E k w ∗ ∈ A ( E + E k + E k +1 ) , k = 1 , , ..., q − , and hence, by Condition 1, for any k − k ≥ l + 1 where 1 ≤ k , k ≤ q , one has that E k w ∗ aE k = 0 , and then, by (5.9), E k ( uw ∗ au ∗ ) E k = u ( u ∗ E k u ) w ∗ a ( u ∗ E k u ) u ∗ ∈ uE k AE k w ∗ aE k AE k u ∗ = { } . In particular,(5.13) f ( E k )( uw ∗ au ∗ ) f ( E k ) = 0 , k − k ≥ l + 1 . Consider c := q X k =1 c k . Note that c ∈ bAb. By Condition 3c, one has that ch ′ = c ; in particular, [ c, h ′ ] = 0 andhence, by (5.3), [ c, v ∗ v ] = 0 . Also note c ⊥ v q v ∗ q ∈ Her( d q ) . Then, it follows from Lemma3.3(2) (with c in the place of d and bAb in the place of D ) that(5.14) cw ∗ ∈ A ( bAb ) . By (5.11), for each k = 1 , , ..., q and i = 1 , ..., r , one has(5.15) u ∗ f ( e ′ ( k − p + i ) u ≤ u ∗ f ( s k ) u ∈ Her( c k ) ⊆ cAc ⊆ bAb. Then, together with (5.14),( u ∗ f ( e ′ ( k − p + i ) u ) w ∗ ∈ ( cAc ) w ∗ ⊆ cAcw ∗ ⊆ A ( bAb ) . Hence f ( e ′ n ( k − i )( uw ∗ au ∗ ) = u (( u ∗ f ( e ′ ( k − p + i ) u ) w ∗ ) au ∗ ∈ uA ( bAb ) au ∗ = { } ;and, on the other hand, by (5.15),( uw ∗ au ∗ ) f ( e ′ ( k − p + i ) = uw ∗ ( a ( u ∗ f ( e ′ ( k − p + i ) u )) u ∗ ∈ uw ∗ ( a ( bAb )) u ∗ = { } . That is, for any k = 1 , ..., q and i = 1 , ..., r ,(5.16) f ( e ′ ( k − p + i )( uw ∗ au ∗ ) = 0 and ( uw ∗ au ∗ ) f ( e ′ ( k − p + i ) = 0 . Let us show that(5.17) u ∗ (1 A − f ( h ′ )) uw ∗ a = 0 . By (5.10), one has u ∗ (1 A − f ( h ′ )) u (5.18) = 1 A − u ∗ q X k =1 p X i =1 f ( e ′ ( k − p + i ) u = (1 A − f ( h ′ )) + f ( h ′ ) − u ∗ q X k =1 p X i =1 f ( e ′ ( k − p + i ) u = (1 A − f ( h ′ )) + q X k =1 f ( E k ) − u ∗ q X k =1 p X i =1 f ( e ′ ( k − p + i ) u = (1 − f ( h ′ )) + u ∗ ( q X k =1 f ( E k ) − q X k =1 p X i =1 f ( e ′ ( k − p + i )) u = (1 A − f ( h ′ )) + q X k =1 p X i =1 u ∗ k ( f ( e ′ ( k − p + i ) − f ( e ′ ( k − p + i )) u k = (1 A − f ( h ′ )) + q X k =1 p X i =1 u ∗ k λ k,i u k , where λ k,i := f ( e ′ ( k − p + i ) − f ( e ′ ( k − p + i ) , k = 1 , ..., q, i = 1 , ..., p. Consider the elements ( u ∗ k λ k,i u k ) w ∗ a, k = 1 , ..., q, i = 1 , ..., p. If i = r + 1 , ..., p , then, by (5.5), z k λ k,i = λ k,i z k = 0;and hence, by (5.7), [ u k , λ k,i ] = 0 . Since λ k,i ⊆ Her(1 − f ( h ′ )), together with (5.12), one has( u ∗ k λ k,i u k ) w ∗ a = λ k,i w ∗ a = 0 . If i = 1 , ..., r , then, by (5.7), λ k,i u k w ∗ a = λ k,i cos( π z k z ∗ k + z ∗ k z k )) w ∗ a + λ k,i z ∗ k g ( z k z ∗ k ) w ∗ a − λ k,i z k g ( z ∗ k z k ) w ∗ a. By (5.5) and (5.14), λ k,i z ∗ k g ( z k z ∗ k ) w ∗ a ∈ λ k,i z ∗ k ( cAc ) w ∗ a = { } . Using (5.14) again, one has λ k,i z k = 0, and hence λ k,i z k g ( z ∗ k z k ) w ∗ a = 0 . Since [ λ k,i , z ∗ k z k ] = 0 (by (5.14)), λ k,i ( π z ∗ k z k ) n ∈ Her( λ k,i ) ⊆ Her(1 − f ( h ′ )) , n = 0 , , ..., TABLE RANK OF C( X ) ⋊ Γ 25 and therefore, by (5.12), λ k,i cos( π z k z ∗ k + z ∗ k z k )) w ∗ a = λ k,i ∞ X n =0 n )! ( π z k z ∗ k + z ∗ k z k )) n w ∗ a = ∞ X n =0 n )! λ k,i ( π z ∗ k z k ) n w ∗ a = 0 . This shows that λ k,i u k w ∗ a = 0 , k = 1 , ..., q, i = 1 , , ..., r, and hence(5.19) ( u ∗ k λ k,i u k ) w ∗ a = 0 , k = 1 , ..., q, i = 1 , ..., p. Then, together with by (5.18) and (5.12), u ∗ (1 A − f ( h ′ )) uw ∗ a = (1 A − f ( h ′ )) w ∗ a + q X k =1 p X i =1 u ∗ k λ k,i u k w ∗ a = 0 , and this proves (5.17).Therefore, with φ := f ( φ ′ ) and h := φ (1 pq ) , by (5.17), one that, (1 A − h )( uw ∗ au ∗ ) = 0 . Set e i := φ ( e i,i ) , i = 1 , , ..., pq. For any e i , e j , j − i > ( l + 1) p , there are 1 ≤ k , k ≤ q with k − k ≥ l + 1 such that e i ≤ f ( E k ) and e j ≤ f ( E k ) . Then it follows from 5.13 that e i ( uw ∗ au ∗ ) e j ∈ f ( E k ) Af ( E k ) uw ∗ au ∗ f ( E k ) Af ( E k ) = { } . Set { i , i , ..., i rq } = { ( k − p + i : k = 1 , ..., q, i = 1 , ..., r } . Then it follows from (5.16) that e i j ( u ( w ∗ a ) u ∗ ) = ( u ( w ∗ a ) u ∗ ) e i j = 0 , j = 1 , , ..., rq, as desired. Finally, it is clear that φ is extendable. This proves that φ satisfies the assertion,and hence the proposition follows. (cid:3) Theorem 5.5.
Let A be a unital C*-algebra which has Property (D). Then ZD( A ) ⊆ GL( A ) . In particular, if A is finite, then A = GL( A ) (in other words, tsr( A ) = 1 ). Proof.
Let a ∈ ZD( A ), and fix an arbitrary ε > A has theProperty (D), there exist unitaries u , u ∈ A and a D -operator a ′ ∈ A such that k u au − a ′ k < ε. By Proposition 5.4, one has that a ′ ∈ GL(A) . Since u , u are unitaries, it follows that u ∗ a ′ u ∗ ∈ GL( A ) , and hence dist( a, GL( A )) < ε. Since ε > a ∈ GL( A ) , and therefore(5.20) ZD( A ) ⊆ GL( A ) . If, moreover, the C*-algebra A is finite, by Proposition 3.2 of [22], A \ GL( A ) ⊆ ZD( A );and then, together with (5.20), A = GL( A ) ∪ ( A \ GL( A )) ⊆ GL( A ) ∪ ZD( A ) ⊆ GL( A ) ∪ GL( A ) = GL( A ) , as desired. (cid:3) Non-invertible elements and zero divisors of C( X ) ⋊ ΓIn the following two sections, let us show that the C*-algebra C( X ) ⋊ Γ has Property (D)if ( X, Γ) has the (URP) and (COS). Since C( X ) ⋊ Γ is finite, this shows that C( X ) ⋊ Γ hasstable rank one by Theorem 5.5.Recall that if B is a sub-C*-algebra of A , a conditional expectation from A to B is acompletely positive linear contraction E : A → B such that E ( b ) = b, E ( ba ) = b E ( a ) and E ( ab ) = E ( a ) b, b ∈ B, a ∈ A. If ( X, Γ) is a free dynamical system, and if E : C( X ) ⋊ Γ → C( X ) is a conditionalexpectation, where C( X ) ⋊ Γ is a crossed-product C*-algebra. Then E ( u γ ) = 0 , γ ∈ Γ \ { e } . Indeed, let γ ∈ Γ \ { e } and consider E ( u γ ) ∈ C( X ). Note that for all g ∈ C( X ), since u ∗ γ gu γ ∈ C( X ), one has g E ( u γ ) = E ( gu γ ) = E ( u γ u ∗ γ gu γ ) = E ( u γ )( u ∗ γ gu γ ) . Assume E ( u γ ) = 0. Then there is x ∈ X such that E ( u γ )( x ) = 0. Since ( X, Γ) is free, onehas x = x γ − . Pick g ∈ C( X ) such that g ( x ) = 1 and g ( x γ − ) = 0. Then g ( x ) E ( u γ )( x ) = E ( u γ )( x ) = 0 = E ( u γ )( x ) g ( x γ − ) = E ( u γ )( x )( u ∗ γ gu γ )( x ) , which is a contradiction.If Γ is amenable, then a conditional expectation E : C( X ) ⋊ Γ → C( X ) always exists, andis not only unique (see above) but also faithful (see, for instance, Proposition 4.1.9 of [4]). TABLE RANK OF C( X ) ⋊ Γ 27
Lemma 6.1.
Let ( X, Γ) be a free and minimal topological dynamical system, where Γ is acountable discrete group and X is a compact Hausdorff space. Denote by A = C( X ) ⋊ Γ acrossed-product C*-algebra, and assume there is a faithful conditional expectation E : A → C( X ) .Let a ∈ A such that ba = 0 for some non-zero positive element b . Then, for any ε > ,there is unitary u ∈ A and a (non-empty) open set E ⊆ X such that k ϕ E ua k < ε. Proof.
Since E is faithful, without loss of generality, one may assume k a k = 1 and k E ( b ) k = 1 . Pick ε ′ > k ca k < ε ′ for some positive element c with k c k ≤ ( k b k + 1) , then (cid:13)(cid:13)(cid:13) c a (cid:13)(cid:13)(cid:13) < ε/ ( k b k + 1) . Pick ε ′′ ∈ (0 ,
1) such that( k b k + ε ′′ ) ε ′′ < ε ′ , and pick b ′ ∈ C c (Γ , C( X )) such that k b − b ′ k < ε ′′ . Since k E ( b ) k = 1, one may assume that k E ( b ′ ) k = 1 . Write b ′ = X γ ∈ Γ f γ u γ for a finite set Γ ⊆ Γ with Γ = Γ − , where f γ ∈ C( X ). Since E ( b ′ ) = f e , one has that k f e k = 1 , and then there is x ∈ X such that | f e ( x ) | = 1 . Pick a neighbourhood U of x such that S γ ∈ Γ U γ = X, and pick an open set W = X suchthat [ γ ∈ Γ U γ ⊆ W. Therefore, there is a continuous function ϕ W : X → [0 ,
1] such that(6.1) ϕ − W ((0 , W and [ γ ∈ Γ U γ ⊆ ϕ − W (1) . Pick a continuous function ϕ U : X → [0 ,
1] so that(6.2) ϕ − U ((0 , U and ϕ U ( x ) = 1 . Note that b ′ ϕ U ( b ′ ) ∗ = ( X γ ∈ Γ f γ u γ ) ϕ U ( X γ ∈ Γ f γ u γ ) ∗ = ( X γ ∈ Γ f γ u γ ) ϕ U ( X γ ∈ Γ u ∗ γ f γ )= X γ,γ ′ ∈ Γ f γ ′ u γ ′ ϕ U u ∗ γ f γ = X γ,γ ′ ∈ Γ f γ ′ ( f γ ◦ ( γ ′ γ − ))( ϕ U ◦ γ ′ ) u γ ′ γ − . Hence, by (6.1),(6.3) ϕ W ( b ′ ϕ U ( b ′ ) ∗ ) = b ′ ϕ U ( b ′ ) ∗ . Also note that, by (6.2), E ( b ′ ϕ U ( b ′ ) ∗ )( x ) = X γ ∈ Γ | f γ ( x ) | ϕ U ( x γ ) ≥ | f e ( x ) | = 1;and in particular,(6.4) k E ( b ′ ϕ U ( b ′ ) ∗ ) k ≥ . Set b ′′ = 1 k E ( b ′ ϕ U ( b ′ ) ∗ ) k b ′ ϕ U ( b ′ ) ∗ . Note that E ( b ′′ ) = 1 (cid:13)(cid:13)(cid:13)P γ ∈ Γ | f γ | ϕ U (cid:13)(cid:13)(cid:13) X γ ∈ Γ | f γ | ϕ Uγ . So there is y ∈ X so that E ( b ′′ )( y ) = 1. By perturbing f γ , γ ∈ Γ , and ϕ U to be locallyconstant around y , there is an open neighbourhood V ∋ y such that(6.5) E ( b ′′ )( x ) = E ( b ′′ )( y ) = 1 , x ∈ V. Moreover, since ( X, Γ) is free, one may choose V small enough so that(6.6) V ∩ V γ = ∅ , γ ∈ Γ \ { e } , and since the action is minimal (so any orbit is dense), by choosing V even smaller, there is γ ∈ Γ such that(6.7)
V γ ∩ W = ∅ . Note that, by (6.4) and since b is positive, k b ′′ a k = (cid:13)(cid:13)(cid:13)(cid:13) k E ( b ′ ϕ U ( b ′ ) ∗ ) k b ′ ϕ U ( b ′ ) ∗ a (cid:13)(cid:13)(cid:13)(cid:13) = 1 k E ( b ′ ϕ U ( b ′ ) ∗ ) k k b ′ ϕ U ( b ′ ) ∗ a k≈ ( k b k + ε ′′ ) ε ′′ k E ( b ′ ϕ U ( b ′ ) ∗ ) k k b ′ ϕ U ba k = 0 . TABLE RANK OF C( X ) ⋊ Γ 29
That is,(6.8) k b ′′ a k ≤ ( k b k + ε ′′ ) ε ′′ < ε ′ . Since(6.9) k b ′′ k ≤ k b ′ k ≤ ( k b k + 1) , by the choice of ε ′ , one has(6.10) (cid:13)(cid:13)(cid:13) ( b ′′ ) a (cid:13)(cid:13)(cid:13) < ε k b k + 1 . Now, choose a continuous function h : X → [0 ,
1] such that h − ((0 , ⊆ V, and f − (1)contains a neighbourhood of y . Note that k h k = 1.By (6.6), hu γ h = 0 , γ ∈ Γ \ { e } , and hence, writing b ′′ = X γ ∈ Γ c γ u γ , together with (6.5), one has hb ′′ h = h ( X γ ∈ Γ c γ u γ ) h = X γ ∈ Γ c γ hu γ h = c e h = E ( b ′′ ) h = h . (6.11)By (6.7), one has u γ hu ∗ γ ⊥ ϕ W and hence, by (6.3),(6.12) u γ hu ∗ γ ⊥ b ′′ . Consider v := u γ h ( b ′′ ) . Then vv ∗ = u γ hb ′′ hu ∗ γ = u γ h u ∗ γ and v ∗ v = ( b ′′ ) h ( b ′′ ) . Pick an open set E such that(6.13) ϕ E vv ∗ = ϕ E ( u γ h u ∗ γ ) = ϕ E . (Such E exists because h is constantly 1 in a small neighbourhood of y .)By (6.12), vv ∗ ⊥ v ∗ v . By Lemma 3.2 and (6.13), there is a unitary u ∈ A such that( u ∗ ϕ E u )( b ′′ ) h ( b ′′ ) = ( u ∗ ϕ E u ) v ∗ v = u ∗ ϕ E u. Therefore, by (6.9), (6.10), k ϕ E ua k = k u ( u ∗ ϕ E u ) a k = (cid:13)(cid:13)(cid:13) u ( u ∗ ϕ E u )( b ′′ ) h ( b ′′ ) a (cid:13)(cid:13)(cid:13) ≤ ( k b k + 1) (cid:13)(cid:13)(cid:13) ( b ′′ ) a (cid:13)(cid:13)(cid:13) < ε, as desired. (cid:3) Proposition 6.2.
Let ( X, Γ) be a free and minimal topological dynamical system, where Γ isa countable discrete group and X is a compact Hausdorff space. Denote by A = C( X ) ⋊ Γ acrossed-product C*-algebra. Assume A is finite and there is a faithful conditional expectation E : A → C( X ) .Let a ∈ A be a non-invertible element. Then, for any ε > , there exist b ∈ C c (Γ , C( X )) ,a (non-empty) open set E ⊆ X , and unitaries u , u ∈ A such that k u au − b k < ε and ϕ E b = bϕ E = 0 . Proof.
Without loss of generality, one may assume that k a k = 1. Let ε > a is not invertible and A is finite, by Proposition 3.2 of [22], there are a ′ ∈ A and nonzero b , b ∈ A + such that k a ′ k = 1 , k a − a ′ k < ε/ , and b a ′ = 0 = a ′ b . By Lemma 6.1, there are unitaries u , u ∈ A and open sets E ′ , F ′ ⊆ X such that k ϕ E ′ u a ′ k < ε/ k a ′ u ϕ F ′ k < ε/ . Since ( X, Γ) is minimal (so any orbit is dense), by passing to smaller open sets and changingthe unitary u , one may assume that E ′ = F ′ .Pick a ′′ ∈ C c (Γ , C( X )) such that k u a ′ u − a ′′ k < ε/ , and note that u au ≈ ε/ u a ′ u = ϕ E ′ u a ′ u ϕ E ′ + (1 − ϕ E ′ ) u a ′ u ϕ E ′ + ϕ E ′ u a ′ u (1 − ϕ E ′ ) + (1 − ϕ E ′ ) u a ′ u (1 − ϕ E ′ ) ≈ ε/ (1 − ϕ E ′ ) u a ′ u (1 − ϕ E ′ ) ≈ ε/ (1 − ϕ E ′ ) a ′′ (1 − ϕ E ′ )Pick an open set E ⊆ ϕ − E ′ (1) (so that ϕ E ϕ E ′ = ϕ E ), and define b = (1 − ϕ E ′ ) a ′′ (1 − ϕ E ′ ) . Then it is clear that k u au − b k < ε and ϕ E b = bϕ E = 0 , as desired. (cid:3) Stable rank of C( X ) ⋊ ΓIn this section, assuming ( X, Γ) has the (URP) and (COS), let us show that the element b obtained in Proposition 6.2 is an D -operator (Proposition 7.7). Hence the C*-algebraC( X ) ⋊ Γ has Property (D), and has stable rank one by Theorem 5.5.Let Γ be a discrete amenable group, and let Γ , Γ , ..., Γ T be finite subsets of Γ. Recallthat Γ is said to be tiled by Γ , Γ , ..., Γ T if there are sets group elements γ i,n , n = 1 , , ..., i = 1 , ..., T, TABLE RANK OF C( X ) ⋊ Γ 31 such that Γ = T G i =1 ∞ G n =1 γ i,n Γ i . Note that if Γ i is (right) ( F , ε )-invariant, then its (left) translation γ i,n Γ i is also ( F , ε )-invariant. Lemma 7.1.
Let Γ be an infinite amenable group, and let Γ , Γ , ..., Γ T ⊆ Γ be finite setswhich tile Γ . Let δ ∈ (0 , and let n ∈ N . Then, there is ( F , ε ) such that if F ⊆ Γ is ( F , ε ) -invariant, then there is H ⊆ F such that H is tiled by Γ , Γ , ..., Γ T with multiplicitiesdivided by n , and | H || F | > − δ. Proof.
Set K = Γ ∪ · · · ∪ Γ T , and choose δ ′ > − δ ′ )(1 − δ ′ > δ. Choose ( F , ε ) sufficiently large such that if F is ( F , ε )-invariant, then(7.1) | int K ( F ) || F | > − δ ′ . Since Γ is infinite, one may assume that ( F , ε ) large enough so that if F is ( F , ε )-invariant,then(7.2) | F | > n ( | Γ | + · · · + | Γ T | )(2 − δ ′ ) δ ′ . Then this ( F , ε ) satisfies the property of the lemma.Indeed, let F be an ( F , ε )-invariant set. Since Γ , ..., Γ T tile Γ, by (7.1), there is a set F ′ ⊆ F such that F ′ can be tiled by Γ , ..., Γ T (in fact, F ′ can be chosen as as the union ofthe tiles which intersect with int K F ) and(7.3) | F ′ || F | > − δ ′ . Write F ′ = ( m G i =1 γ ,i Γ ) ⊔ · · · ⊔ ( m T G i =1 γ T,i Γ T ) , where γ i,j ∈ Γ and m i , i = 1 , , ..., T , are non-negative integers. Note that m | Γ | + · · · + m T | Γ T | = | F ′ | . For each m i , i = 1 , , ..., T , consider r i which is the remainder of m i divided by n . Then,set H = ( m − r G i =1 γ ,m Γ ) ⊔ · · · ⊔ ( m T − r T G i =1 γ T,i Γ T ) . It is clear that H is tiled by Γ , Γ , ..., Γ T with multiplicities divided by n . Moreover, by(7.3) and (7.2), 1 − | H || F ′ | = r | Γ | + · · · + r T | Γ T || F ′ | < n | Γ | + · · · + n | Γ T || F ′ | < n ( | Γ | + · · · + | Γ T | )(2 − δ ′ ) 1 | F | < δ ′ , and hence, by (7.3) again, | H || F | > (1 − δ ′ )(1 − δ ′ > − δ, as desired. (cid:3) Lemma 7.2.
Let Γ be an infinite amenable group, and let Γ , Γ , ..., Γ T ⊆ Γ be finite setswhich tile Γ . Let δ ∈ (0 , , let n ∈ N , and let K ⊆ Γ be a finite set. Then, there exists ( F , ε ) such that if F , F , ..., F n are mutually disjoint ( F , ε ) -invariant sets and | F | = | F | = · · · = | F n | , then there are H ⊆ F , ..., H n ⊆ F n such that H i K ⊆ F i , i = 1 , , ..., n, each H i is tiled by Γ , Γ , ..., Γ T with multiplicities divided by n , | H | = | H | = · · · = | H n | , and | H i || F i | > − δ, i = 1 , , ..., n. Proof.
Apply Lemma 7.1 to δ T ) and n | Γ | | Γ | · · · | Γ T | , one obtains ( F ′ , ε ′ ). Choose ( F , ε ) such that if F is ( F , ε )-invariant, then int K F is ( F ′ , ε ′ )-invariant, and(7.4) | int K F || F | > − δ T ) . Then ( F , ε ) satisfies the property of the lemma.Indeed, let F , F , ..., F n be mutually disjoint ( F , ε )-invariant sets with | F | = | F | = · · · = | F n | . Consider the sets int K F , int K F , ..., int K F n . TABLE RANK OF C( X ) ⋊ Γ 33
Then each of them is ( F ′ , ε ′ )-invariant. Also note that(int K F i ) K ⊆ F i , (int K F i ) K ∩ (int K F j ) = ∅ , i, j = 1 , , ..., n, i = j, By Lemma 7.1, there are F ′ ⊆ int K F , F ′ ⊆ int K F , ..., F ′ n ⊆ int K F n , such that(7.5) | F ′ i || int K F i | > − δ T ) , i = 1 , , ..., n, and(7.6) F ′ i = ( m ( i )1 G j =1 γ ( i )1 ,j Γ ) ⊔ · · · ⊔ ( m ( i ) T G j =1 γ ( i ) T,j Γ T ) , i = 1 , , ..., n, and each m ( i ) t , i = 1 , , ..., n , t = 1 , , ..., T , is divided by n | Γ | | Γ | · · · | Γ T | .It follows from (7.4) and (7.5) that for each i = 1 , , ..., n , | F i | − | F ′ i | = ( | F i | − | int K ( F i ) | ) + ( | int K ( F i ) | − | F ′ i | )(7.7) < δ T ) | F i | + δ T ) | int K F i |≤ δ T ) | F i | + δ T ) | F i | = δ T | F i | . In the decomposition (7.6), if m ( i ) t < δ T | F i | , then set m ( i ) t = 0, and denote this possibly smaller new sets still by F ′ i . Then, by (7.7), withthe set new F ′ i , one has(7.8) 0 ≤ | F i | − | F ′ i | < δ T | F i | + T δ T | F i | = δ | F i | , i = 1 , , ..., n. Also note that if m ( i ) t = 0, then(7.9) m ( i ) t ≥ δ T | F i | ≥ δ | F i | . Set D = min {| F ′ | , | F ′ | , ..., | F ′ n |} . Since | F ′ | , | F ′ | , ..., | F ′ n | are divided by n | Γ | | Γ | · · · | Γ T | , there are non-negative integers d i , i = 1 , ..., n , such that | F ′ i | − D = d i | Γ | | Γ | · · · | Γ T | n, i = 1 , , ..., n. By (7.8) (and note that | F | = · · · = | F n | ),(7.10) D | F i | > | F i | − δ | F i || F i | = 1 − δ, i = 1 , , ..., n, and so | F ′ i | − D ≤ | F i | − D ≤ δ | F i | , i = 1 , , ..., n. For each i = 1 , , ..., n , consider { t , t , ..., t S } = n t = 1 , , ..., T : m ( i ) t = 0 o . Then, there are 0 ≤ c ( i ) t , ..., c ( i ) t S ≤ d i | Γ | | Γ | · · · | Γ T | ≤ δ | F i | n such that d i | Γ | | Γ | · · · | Γ T | = c ( i ) t | Γ t | + · · · + c ( i ) t S | Γ t S | . (Actually, one can choose c ( i ) t = d i | Γ | | Γ | · · · | Γ T | / | Γ t | and c ( i ) t s = 0, s = 2 , ..., S .) Notethat, by (7.9), δ | F i | n ≤ m ( i ) t s n , s = 1 , , ..., S. For each t / ∈ { t , ..., t S } , set c ( i ) t = 0. Then, one has that(7.11) 0 ≤ c ( i ) t ≤ d i | Γ | | Γ | · · · | Γ T | ≤ δ | F i | n ≤ m ( i ) t n , t = 1 , , ..., T, and d i | Γ | | Γ | · · · | Γ T | = c ( i )1 | Γ | + · · · + c ( i ) T | Γ T | . Put H i = ( m ( i )1 − c ( i )1 n G j =1 γ ( i )1 ,j Γ ) ⊔ · · · ⊔ ( m ( i ) T − c ( i ) T n G j =1 γ ( i ) T,j Γ T ) , i = 1 , , ..., n. (Note that, by (7.11), m ( i ) t − c ( i ) t n ≥ m ( i ) t is divisible by n , it is clear that each H i is tiled by Γ , ..., Γ T with multiplicities divisible by n . Since H ⊆ int K F , H ⊆ int K F , ..., H n ⊆ int K F n , one has H K ⊆ F , H K ⊆ F , ..., H n K ⊆ F n , Also note that | H | = | H | = · · · = | H n | = D, and hence, by (7.10), | H i || F i | > − δ, i = 1 , , ..., n, as desired. (cid:3) Lemma 7.3.
Let Γ be an infinite amenable group, and let ( X, Γ) be a minimal dynamicalsystem with the (URP). Let λ > be arbitrary, and let O , , ..., O ,M , O , , ..., O ,M ⊆ X bemutually disjoint non-empty open sets together with { κ , (= e ) , κ , , ..., κ ,M , κ , (= e ) , κ , , ..., κ ,M } ⊆ Γ TABLE RANK OF C( X ) ⋊ Γ 35 such that O i,m = O i, κ i,m , i = 0 , , m = 1 , ..., M. Put δ := min { µ ( O i,m ) : i = 0 , , m = 1 , ..., M, µ ∈ M ( X, Γ) } , and let K ⊆ Γ be a symmetric finite set. (Since ( X, Γ) is minimal, one has that δ > .)Then, there are ( F , ε ) , and n ∈ N ( n > ) such that if ( B, F ) is a tower of ( X, Γ) with F being ( F , ε ) -invariant, then there is an order zero c.p.c. map φ : M n ( C ) → A, where A = C( X ) ⋊ Γ , such that if h = φ (1) and e i = φ ( e i,i ) , i = 1 , , ..., n , and b k := e n ( k − + · · · + e n ( k − n , k = 1 , , ..., n, then e i ∈ C( X ) and if denote by E i = e − i ((0 , , i = 1 , , ..., n , then (1) n G i =1 E i ⊆ G γ ∈ F Bγ, and µ ( G γ ∈ F Bγ \ n G i =1 E i ) < λ δ n µ ( G γ ∈ F Bγ ) , µ ∈ M ( X, Γ) . (2) for each k = 1 , , ..., n , there are mutually disjoint open sets O k , , ..., O k ,M and O k , , ..., O k ,M such that (a) O k ,m ⊆ O ,m ∩ F nkj = n ( k − E j and O k ,m ⊆ O ,m ∩ F nkj = n ( k − E j , m = 1 , , ..., M , (b) O ki,m = O ki, κ i,m , i = 0 , , m = 1 , , ..., M , (c) µ ( O k , ) , µ ( O k , ) > δ n µ ( F n i =1 E i ) , µ ∈ M ( X, Γ) . (3) b k ⊥ u γ b k u ∗ γ , γ ∈ K, k = k , ≤ k , k ≤ n, where u γ ∈ A is the canonicalunitary of γ .Moreover, n can be chosen arbitrarily large.Proof. Choose n ∈ N such that(7.12) 0 < n − < δ , λ δ n < , and 3 n < δ . Pick ( F ′ , ε ′ ) such that if a finite set Γ ⊆ Γ is ( F ′ , ε ′ )-invariant, then(7.13) 1 | Γ | |{ γ ∈ Γ : xγ ∈ O i,m }| > δ , x ∈ X, i = 0 , , m = 0 , , ..., M, and(7.14) (cid:12)(cid:12)(cid:12) ∂ K M Γ (cid:12)(cid:12)(cid:12) | Γ | < δ , where K := { κ , , κ , , ..., κ ,M , κ , , κ , , ..., κ ,M } . By Theorem 4.3 of [6], there are ( F ′ , ε ′ )-invariant finite setsΓ , Γ , ..., Γ T ⊆ Γwhich tile Γ. Applying Lemma 7.2 to λδ/ n , n , and K with respect to the finite sets Γ ,...,Γ T , one obtains ( F ′′ , ε ′′ ).By Theorem 4.3 of [6] again, there are ( F ′′ , ε ′′ )-invariant finite setsΓ ′ , Γ ′ , ..., Γ ′ T ′ ⊆ Γwhich tile Γ. Applying Lemma 7.1 to λδ/ n and n with respect to the finite sets Γ ′ , Γ ′ , ...,Γ ′ T ′ , one obtains ( F , ε ). Since Γ is infinite, one may assume that ( F , ε ) is sufficiently largesuch that if F is ( F , ε )-invariant, then | F | > n .Then, ( F , ε ) satisfies the property of the lemma.Indeed, let ( B, F ) be a tower such that F is ( F , ε )-invariant. Then, by Lemma 7.1, thereis a finite set R ⊆ F such that(7.15) | R || F | < λ δ n and F \ R can be tiled by Γ ′ , ..., Γ ′ T ′ with multiplicities divided by n . By a grouping of thetilings, one has F \ R = Γ ′′ ⊔ Γ ′′ ⊔ · · · ⊔ Γ ′′ n , where Γ ′′ i , i = 1 , ..., n , are mutually disjoint ( F ′′ , ε ′′ )-invariant set and | Γ ′′ | = | Γ ′′ | = · · · = | Γ ′′ n | . By Lemma 7.2 and the choice of ( F ′′ , ε ′′ ), there are finite sets Γ ′′′ i ⊆ Γ ′′ i such thatΓ ′′′ i K ⊆ Γ ′′ i , i = 1 , , ..., n, | Γ ′′′ i || Γ ′′ i | > − λ δ n , i = 1 , , ..., n, | Γ ′′′ | = | Γ ′′′ | = · · · = | Γ ′′′ n | , and each Γ ′′′ i is tiled by Γ , ..., Γ T with multiplicities divided by n . Since Γ ′′ i , i = 1 , ..., n , aremutually disjoint, one hasΓ ′′′ i K ∩ Γ ′′′ j = ∅ , i, j = 1 , , ..., n, i = j. Write R = ( F \ R ) \ (Γ ′′′ ⊔ Γ ′′′ ⊔ · · · ⊔ Γ ′′′ n ) , and one has F \ ( R ∪ R ) = Γ ′′′ ⊔ Γ ′′′ ⊔ · · · ⊔ Γ ′′′ n , TABLE RANK OF C( X ) ⋊ Γ 37 and(7.16) | R || F | ≤ λ δ n . Note that, by (7.12), (7.15), and (7.16), one has(7.17) 2 | F \ ( R ∪ R ) | > | F | . Then, inside each Γ ′′′ i , since it is tiled by Γ , ..., Γ T (which are ( F ′ , ε ′ )-invariant) withmultiplicities divided by n , after a grouping, one hasΓ ′′′ i = Γ i, ⊔ · · · ⊔ Γ i,n , where Γ i,j is ( F ′ , ε ′ )-invariant with | Γ i, | = | Γ i, | = · · · = | Γ i,n | , i = 1 , , ..., n. In summary, one obtains the decomposition F \ ( R ∪ R ) = (Γ , ⊔ · · · ⊔ Γ ,n ) ⊔ · · · ⊔ (Γ n, ⊔ · · · ⊔ Γ n,n )with properties(1)(7.18) | Γ i ,j | = | Γ i ,j | , ≤ i , i , j , j ≤ n, (2) each Γ i,j is ( F ′ , ε ′ )-invariant,(3)(7.19) (Γ i, ⊔ · · · ⊔ Γ i,n ) K ⊆ F, i = 1 , , ..., n, and(4) if i = j , then(7.20) (Γ i, ⊔ · · · ⊔ Γ i,n ) K ∩ (Γ j, ⊔ · · · ⊔ Γ j,n ) = ∅ . Set e = ϕ B , and set e γ = u ∗ γ eu γ , γ ∈ F. For each 1 ≤ i ≤ n , write i = n ( k −
1) + j , where 1 ≤ j ≤ n , and set e i = X γ ∈ Γ k,j e γ By (7.18), it follows from Lemma 2.18 that there is a order zero map φ : M n ( C ) → A such that φ ( e i,i ) = e i , i = 1 , , ..., n . Denote by E i := e − i ((0 , i = n ( k −
1) + j with 1 ≤ j ≤ n , one has E i = G γ ∈ Γ k,j Bγ, and it is clear that(7.21) n G i =1 E i ⊆ G γ ∈ F Bγ, and, by (7.15) and (7.16), one has µ ( G γ ∈ F Bγ \ n G k =1 E k ) = µ ( G γ ∈ R ∪ R Bγ ) < λ δ n µ ( G γ ∈ F Bγ ) , µ ∈ M ( X, Γ) . This proves Property 1.Consider b k := e n ( k − + · · · + e nk , k = 1 , ..., n. Note that, with Γ k := Γ k, ⊔ · · · ⊔ Γ k,n , one has b k = X γ ∈ Γ k e γ , and hence, if Γ k γ ⊆ F for a group element γ , then u ∗ γ b k u γ = X γ ′ ∈ Γ k u ∗ γ e γ ′ u γ = X γ ′ ∈ Γ k e γ ′ γ = X γ ′ ∈ Γ k γ e γ ′ Thus, by (7.19), (7.20) and the assumption that K = K − , one has that for any k = k , b k ⊥ u γ b k u ∗ γ , γ ∈ K. This proves Property 3.Also consider C := C ∗ { u γ f : γ ∈ F, f ∈ C ( B ) } ⊆ A, the C*-algebra of the tower ( B, F ). Note that, by Lemma 3.12 of [17], C ∼ = M | F | (C ( B )) , and under this isomorphism, for any g ∈ C ( F γ ∈ F Bγ ) ⊆ C( X ), one has g ∈ C and g ( x X γ ∈ F g ( xγ ) e γ,γ ) . In particular, since for each i = 0 , k = 1 , , ..., n and m = 1 , ..., M , b k ϕ O i,m ∈ C ( F γ ∈ F Bγ ),one has b k ϕ O i,m ∈ C. TABLE RANK OF C( X ) ⋊ Γ 39
Noting that Γ i,j are ( F ′ , ε ′ )-invariant, by (7.13) and (7.17), regarding b k ϕ O i,m as an elementof C ∼ = M | F | (C ( B )) , one has that for any x ∈ B ,rank( b k ϕ O i,m ( x )) = (cid:12)(cid:12)(cid:8) γ ∈ F : ( b k ϕ O i,m )( xγ ) > (cid:9)(cid:12)(cid:12) = |{ γ ∈ Γ k, ⊔ Γ k, ⊔ · · · ⊔ Γ k,n : xγ ∈ O i,m }|≥ δ | Γ k, | + · · · + | Γ k,n | )= δ n | Γ k, | > δ n | F | ;then, for any µ ∈ M ( X, Γ), by (7.21) in the last step, µ ( O i,m ∩ nk G j = n ( k − E j ) = µ ( (cid:8) x ∈ X : ( b k ϕ O i,m )( x ) > (cid:9) )(7.22) = Z B rank( b k ϕ O i,m ( x ))d µ ≥ Z B δ n | F | d µ = δ n | F | µ ( B ) > δ n µ ( n G j =1 E j ) . Now, for each i = 0 , k = 1 , , ..., n , let us construct open sets O ki,m , m = 1 , , ..., M .Note that (recall Γ k = Γ k, ⊔ · · · ⊔ Γ k,n ) O i,m ∩ nk [ j = n ( k − E j = O i,m ∩ G γ ∈ Γ k Bγ, i = 0 , , m = 1 , , ..., M. Consider Γ ◦ k := Γ k \ (Γ k, ∪ Γ k, ∪ Γ k, ) = Γ k, ⊔ Γ k, ⊔ · · · ⊔ Γ k,n , and define O k , := O , ∩ G γ ∈ int KM (Γ ◦ k ) Bγ and O k ,m := O k , κ ,m , m = 1 , , ..., M, and O k , := O , ∩ G γ ∈ int KM (Γ k ) Bγ and O k ,m := O k , κ ,m , m = 1 , , ..., M. Then it is clear that O k ,m ⊆ O ,m ∩ nk [ i = n ( k − E i and O k ,m ⊆ O ,m ∩ nk [ j = n ( k − E j , m = 1 , , ..., M, Since Γ i,j are ( F ′ , ε ′ )-invariant, the sets Γ ◦ k is also ( F ′ , ε ′ )-invariant. By (7.14), one has (cid:12)(cid:12)(cid:12) ∂ K M (Γ ◦ k ) (cid:12)(cid:12)(cid:12) | Γ ◦ k | < δ , and therefore, together with (7.12) and (7.22), for any µ ∈ M ( X, Γ) and i = 0 , µ ( O ki, ) ≥ µ ( O i, ∩ G γ ∈ int KM (Γ ◦ k ) Bγ ) ≥ µ ( O i, ∩ G γ ∈ Γ ◦ k Bγ ) − µ ( G γ ∈ ∂ KM (Γ ◦ k ) Bγ ) ≥ µ ( O i, ∩ G γ ∈ Γ ◦ k Bγ ) − δ | Γ ◦ k | µ ( B ) > µ ( O i, ∩ nk G j = n ( k − E j ) − δ n µ ( n G j =1 E j ) ≥ µ ( O i, ∩ nk G j = n ( k − E j ) − n µ ( n G j =1 E j ) − δ n µ ( n G j =1 E j ) ≥ δ n µ ( n G j =1 E j ) − δ n µ ( n G j =1 E j )= δ n µ ( n G j =1 E j ) . This proves Property 2, as desired. (cid:3)
Lemma 7.4.
Let Γ be an infinite discrete amenable group, and let ( X, Γ) be a minimaltopological dynamical system with the (URP). Let λ > be arbitrary, and let O , , ..., O ,M , O , , ..., O ,M ⊆ X be mutually disjoint non-empty open sets together with { κ , (= e ) , κ , , ..., κ ,M , κ , (= e ) , κ , , ..., κ ,M } ⊆ Γ such that O i,m = O i, κ i,m , i = 0 , , m = 1 , ..., M. Let K ⊆ Γ be a symmetric finite set.Then there exist n ∈ N ( n > ) and an order zero c.p.c. map φ : M n ( C ) → A, where A = C( X ) ⋊ Γ , such that if h := φ (1) and e i := φ ( e i,i ) , i = 1 , , ..., n , and b k := e n ( k − + · · · + e n ( k − n , k = 1 , , ..., n, then e i ∈ C( X ) TABLE RANK OF C( X ) ⋊ Γ 41 and if denote by E i = e − ((0 , , i = 1 , , ..., n , one has (1) for each k = 1 , , ..., n , there are mutually disjoint open sets O k , , ..., O k ,M and O k , , ..., O k ,M such that (a) O k ,m ⊆ O ,m ∩ F ni =4 E n ( k − i and O k ,m ⊆ O ,m ∩ F ni =1 E n ( k − i , m = 1 , , ..., M , (b) O ki,m = O ki, κ i,m , i = 0 , , m = 1 , , ..., M , (c) λµ ( O k , ) > n and λµ ( O k , ) > µ ( X \ n G i =1 E i ) , µ ∈ M ( X, Γ) , (2) b k ⊥ u γ b k u ∗ γ , γ ∈ K, k = k , ≤ k , k ≤ n, where u γ ∈ A is the canonical unitary of γ .Proof. Applying Lemma 7.3 with respect to O , , O , , ..., O ,M and O , , O , , ..., O ,M , and δ := min { µ ( O i,m ) : i = 0 , , m = 1 , ..., M, µ ∈ M ( X, Γ) } > , one obtains ( F ′ , ε ′ ) and n . Since n can be chosen arbitrarily large, one may assume that(7.23) 3 n < λ δ n < . Since ( X, Γ) is assumed to have the (URP), there exist open towers( B , F ) , ..., ( B S , F S )such that each F s , s = 1 , ..., S , is ( F ′ , ε ′ )-invariant and(7.24) µ ( X \ S G s =1 G γ ∈ F s B s γ ) < λ δ n , µ ∈ M ( X, Γ) . For each tower ( B s , F s ), since F s is ( F ′ , ε ′ )-invariant, by Lemma 7.3, there is an order zeroc.p.c. map φ s : M n ( C ) → A, where A = C( X ) ⋊ Γ, such that if h s := φ s (1) and e ( s ) i := φ s ( e i,i ) , i = 1 , , ..., n , and b s,k := e ( s ) n ( k − + · · · + e ( s ) n ( k − n , k = 1 , , ..., n, then e ( s ) i ∈ C( X )and if denote by E s,i = ( e ( s ) i ) − ((0 , , i = 1 , , ..., n , then (1) n G i =1 E s,i ⊆ G γ ∈ F s B s γ, and(7.25) µ ( G γ ∈ F s B s γ \ n G i =1 E s,i ) < λ δ n µ ( G γ ∈ F s B s γ ) , µ ∈ M ( X, Γ) . (2) for each k = 1 , , ..., n , there are open sets O k,s , , ..., O k,s ,M and O k,s , , ..., O k,s ,M such that(a) O k,s ,m ⊆ O ,m ∩ F nkj = n ( k − E s,j and O k,s ,m ⊆ O ,m ∩ F nkj = n ( k − E s,j , m = 1 , , ..., M ,(b) O k,si,m = O k,si, κ i,m , i = 0 , m = 1 , , ..., M ,(c)(7.26) µ ( O k,s , ) , µ ( O k,s , ) > δ n µ ( n G i =1 E s,i ) , µ ∈ M ( X, Γ) . (3) b s,k ⊥ u γ b s,k u ∗ γ , γ ∈ K, k = k , ≤ k , k ≤ n. Then, the order zero c.p.c. map(7.27) φ := S X s =1 φ s is the desired map.Indeed, it follows (7.27) that h = φ (1) = S X s =1 φ s (1) = h + · · · + h S ,e i = φ ( e i,i ) = S X s =1 φ s ( e i,i ) = e (1) i + · · · + e ( S ) i , i = 1 , , ..., n . In particular, b k = b ,i + · · · + b S,i , i = 1 , , ..., n and E i = e − i ((0 , E ,i ⊔ · · · ⊔ E S,i , i = 1 , , ..., n . For each i = 0 , k = 1 , , ..., n , and m = 1 , , ..., M , set O ki,m = S G s =1 O k,si,m . By Conditions 2a and 2b, it is clear that O k ,m ⊆ O ,m ∩ n G j =4 E n ( k − j and O ki,m ⊆ O i,m ∩ n G j =1 E n ( k − j and O ki,m = O ki, κ m , i = 0 , , m = 1 , , ..., M. TABLE RANK OF C( X ) ⋊ Γ 43
By (7.24), (7.25) and (7.23), for any µ ∈ M ( X, Γ), µ ( X \ n G i =1 E i ) = µ ( X \ S G s =1 n G i =1 E s,i )(7.28) = µ ( X \ S G s =1 G γ ∈ F s B s γ ) + S X s =1 µ ( G γ ∈ F s B s γ \ n G i =1 E s,i ) < λ δ n + S X s =1 λ δ n µ ( G γ ∈ F s B s γ ) < λ δ n + λ δ n < λ δ n < . and, then by (7.26) and (7.28) λµ ( O ki, ) = λ S X s =1 µ ( O k,si, ) > λ δ n S X s =1 µ ( n G i =1 E s,i ) > λ δ n (1 −
14 ) = λ δ n> µ ( X \ n G i =1 E i ) . Also note that, by (7.23), λµ ( O k , ) > λ δ n > n . This verifies Property 1.Property 2 follows from Condition 3 straightforwardly. This proves the lemma. (cid:3)
Next, let us perturb further the order zero map φ obtained by Lemma 7.4. First, we havethe following simple observation. Lemma 7.5.
Let X be compact Hausdorff space, and let T be a compact set of probabilityBorel measures. (1) If O ⊆ X is an open set and λ, δ > satisfy λµ ( O ) > δ, µ ∈ T, then there is a closed set D ⊆ O such that λµ ( D ) > δ, µ ∈ T. (2) If O ⊆ X is an open set and C ⊆ X is closed set satisfying λµ ( O ) > µ ( C ) , µ ∈ T, for some λ > , then there exist a closed set D ⊆ O and an open set F ⊇ C suchthat λµ ( D ) > µ ( F ) , µ ∈ T. Proof.
Let us prove the second statement only. The first statement can be shown with asimilar argument.For any µ ∈ T , pick continuous functions f µ , g µ : X → [0 ,
1] such that f µ | X \ O = 0, g µ | C = 1, and λτ µ ( f µ ) > τ µ ( g µ ) + δ µ for some δ µ >
0, where τ µ ( f ) := R f dµ . Then, pick a open neighborhood N µ of µ such that λ | τ µ ( f µ ) − τ µ ′ ( f µ ) | < δ µ | τ µ ( g µ ) − τ µ ′ ( g µ ) | < δ µ , τ ′ ∈ N µ , and a straightforward calculation shows λτ µ ′ ( f µ ) > τ µ ′ ( g µ ) + δ µ , µ ′ ∈ N µ . Since T is compact, there is a finite open cover of T consists of N µ , ..., N µ n , where µ , ..., µ n ∈ T . With f := max { f µ , ..., f µ n } , g := min { g µ , ..., g µ n } , and δ := 12 min { δ µ , ..., δ µ n } , one has f | X \ O = 0 , g | C = 1 , and λτ µ ( f ) > τ µ ( g ) + δ, µ ∈ T. Then, with a sufficiently small ε >
0, the closed set D := f − ([ ε, F := g − ((1 − ε, (cid:3) Lemma 7.6.
Let Γ be an infinite group, and let ( X, Γ) be a minimal topological dynamicalsystem with the (URP). Let λ > be arbitrary, and let O , , ..., O ,M , O , , ..., O ,M ⊆ X bemutually disjoint non-empty open sets together with { κ , (= e ) , κ , , ..., κ ,M , κ , (= e ) , κ , , ..., κ ,M } ⊆ Γ such that O i,m = O i, κ i,m , i = 0 , , m = 1 , ..., M. Let K ⊆ Γ be a symmetric finite set.Then there is an order zero c.p.c. map φ : M n ( C ) → A for some n > such that if h := φ (1) and e i := φ ( e i,i ) , i = 1 , , ..., n , and b k := e n ( k − + · · · + e n ( k − n , k = 1 , , ..., n, then e i ∈ C( X ) and (1) for each k = 1 , , ..., n , there are mutually orthogonal positive functions c k, , ..., c k,M , d k, , ..., d k,M ∈ C( X ) such that TABLE RANK OF C( X ) ⋊ Γ 45 (a) c k,m ∈ Her( O ,m ) and d k,m ∈ Her( O ,m ) , m = 1 , , ..., M , (b) c k,m ⊥ ( e ( k − n +1 + e ( k − n +2 + e ( k − n +3 ) , m = 1 , , ..., M , (c) c k,m b k = c k,m and d k,m b k = d k,m , m = 1 , , ..., M, (d) c k,m = u ∗ κ m c k, u κ m and d k,m = u ∗ κ m d k, u κ m , m = 1 , , ..., M, and (e) λ d τ ( c k, ) > n and λ d τ ( d k, ) > d τ (1 − h ) , τ ∈ T( A ) , (2) b k ⊥ u γ b k u ∗ γ , γ ∈ K, k = k , ≤ k , k ≤ n, where u γ ∈ A is the canonical unitary of γ .Proof. It follows form Lemma 7.4 that there exist n ∈ N ( n >
3) and an order zero c.p.c. map φ ′ : M n ( C ) → A such that if h ′ := φ ′ (1) and e ′ i := φ ′ ( e i,i ) , i = 1 , , ..., n , and b ′ k := e ′ n ( k − + · · · + e ′ n ( k − n , k = 1 , , ..., n, then e ′ i ∈ C( X )and if denote by E i = ( e ′ i ) − ((0 , , i = 1 , , ..., n , then(1) for each k = 1 , , ..., n , there are mutually disjoint open sets O k , , ..., O k ,M and O k , , ..., O k ,M such that(a) O k ,m ⊆ O ,m ∩ F ni =4 E n ( k − i and O k ,m ⊆ O ,m ∩ F ni =1 E n ( k − i , m = 1 , , ..., M ,(b) O ki,m = O ki, κ i,m , i = 0 , m = 1 , , ..., M ,(c) λµ ( O k , ) > n and λµ ( O k , ) > µ ( X \ n G i =1 E i ) , µ ∈ M ( X, Γ) , (2) b ′ k ⊥ u γ b ′ k u ∗ γ , γ ∈ K, k = k , ≤ k , k ≤ n. Since M ( X, Γ) is compact, O k , and O k , are open, and X \ F n i =1 E i is closed, by Condition(1c) and Lemma 7.5, there are closed sets D ki, ⊆ O ki, , i = 0 ,
1, and an open set U ⊇ X \ F n i =1 E i such that(7.29) λµ ( D k , ) > n and λµ ( D k , ) > µ ( U ) , µ ∈ M ( X, Γ) . For any ε >
0, define(7.30) V ε := int( f ε ( h ′ ) − ( { } )) = { x ∈ X : h ′ ( x ) > ε } , and consider the open sets(7.31) W ki,ε := ( O ki, ∩ V ε ) ∩ ( O ki, ∩ V ε ) κ − i, ∩ · · · ∩ ( O ki,M ∩ V ε ) κ − i,M , which increases to O ki, as ε → i = 0 ,
1. Since D ki, is compact, there is a sufficiently small ε > W ki,ε ⊇ D ki, , i = 0 , . Pick a such ε , and one may also assume that U ⊇ { x ∈ X : h ′ ( x ) < ε } , and then note U ⊇ { x ∈ X : h ′ ( x ) < ε } = (1 − f ε ( h ′ )) − ((0 , . Then, together with (7.29),(7.32) λµ ( W k ,ε ) > λµ ( D k , ) > n , µ ∈ M ( X, Γ) , and(7.33) λµ ( W k ,ε ) > λµ ( D k , ) > µ ( U ) ≥ µ ((1 − f ε ( h ′ )) − ((0 , , µ ∈ M ( X, Γ) , It also follows from (7.31) that(7.34) W ki,ε κ i,m ⊆ V ε , i = 0 , , m = 1 , , ..., M. Set c k, = ϕ W k ,ε and c k,m = u ∗ κ m c k, u κ m , m = 2 , , ..., M, and d k, = ϕ W k ,ε and c k,m = u ∗ κ m c k, u κ m , m = 2 , , ..., M. Note that, by (7.31), c k,m ∈ Her( O k ,m ) and d k,m ∈ Her( O k ,m ) , m = 1 , , ..., M. It follows from (7.34) and (7.30) that c k,m f ε ( b ′ k ) = c k,m f ε ( h ′ ) = c k,m and d k,m f ε ( b ′ k ) = d k,m f ε ( h ′ ) = d k,m , and it follows from (7.32) and (7.33) that for any τ ∈ T( A ), λ d τ ( c k, ) = λµ τ ( W k ,ε ) > n . and λ d τ ( d k, ) = λµ τ ( W k ,ε ) > µ τ ((1 − f ε ( h ′ )) − ((0 , τ (1 − f ε ( h ′ )) . Then φ := f ε ( φ ′ ) : M n ( C ) → A is the desired order-zero map.Indeed, noting that h = φ (1) = f ε ( h ′ ) , the existence of c k,m , d k,m , k = 1 , ..., n , m = 1 , ..., M , and Property 1 are verified above.Consider any b k , b k with k = k , 1 ≤ k , k ≤ n . Note that b k = f ε ( b ′ k ) ∈ Her( b ′ k ) and u γ b k u ∗ γ = f ε ( u γ b ′ k u ∗ γ ) ∈ Her( u γ b ′ k u ∗ γ ) , γ ∈ K, TABLE RANK OF C( X ) ⋊ Γ 47 and therefore, it follows from Condition 2 that b k ⊥ u γ b k u ∗ γ , γ ∈ K. This verified Property 2, as desired. (cid:3)
We are now ready for the main results of the paper.
Proposition 7.7.
Let Γ be an infinite countable discrete amenable group, and let ( X, Γ) be aminimal free topological dynamical system with the (URP) and (COS). Then the C*-algebra C( X ) ⋊ Γ has Property (D).Proof. Let a ∈ ZD( A ) and let ε > u , u ∈ A , a ′ ∈ C c (Γ , C( X )) and a non-empty open set E ⊆ X such that k u au − a ′ k < ε and ϕ E a ′ = a ′ ϕ E = 0 . In the following, let us verify that a ′ is actually a D -operator. Since ε is arbitary, this showsthat A has Property (D).Note that, since ( X, Γ) has the (COS), the sub-C*-algebra C( X ) has the ( λ, M )-Cuntzcomparison insider A for some λ ∈ (0 , + ∞ ) and M ∈ N . Fix λ and M .Write(7.35) a ′ = X γ ∈ K f γ u γ , where f γ ∈ C( X ) and K ⊆ Γ is a symmetric finite set.Consider the open set E . Since ( X, Γ) is minimal, all orbits are dense, and hence thereexist non-empty mutually orthogonal open sets O , , ..., O ,M , O , , ..., O ,M ⊆ E and(7.36) { κ , (= e ) , κ , , ..., κ ,M , κ , (= e ) , κ , , ..., κ ,M } ⊆ Γsuch that O i,m = O i, κ i,m , i = 0 , , m = 1 , ..., M. Since ( X, Γ) has the (URP), it follows from Lemma 7.6 that there is an order zeroc.p.c. map φ : M n ( C ) → A for some n > h := φ (1) , e i := φ ( e i,i ) , i = 1 , , ..., n ,s k := e ( k − p +1 + · · · + e ( k − p +3 , k = 1 , ..., n, and E k := e n ( k − + · · · + e n ( k − n , k = 1 , , ..., n, then(7.37) e i ∈ C( X )and (1) for each k = 1 , , ..., n , there are mutually orthogonal positive functions c k, , ..., c k,M , d k, , ..., d k,M ∈ C( X )such that(a) c k,m ∈ Her( O ,m ) and d k,m ∈ Her( O ,m ), m = 1 , , ..., M ,(b) c k,m ⊥ s k , m = 1 , , ..., M ,(c) c k,m E k = c k,m and d k,m E k = d k,m , m = 1 , , ..., M, (d) c k,m = u ∗ κ m c k, u κ m and d k,m = u ∗ κ m d k, u κ m , m = 1 , , ..., M, and(e) λ d τ ( c k, ) > n and λ d τ ( d k, ) > d τ (1 − h ), τ ∈ T( A ) , (2) E k ⊥ u γ E k u ∗ γ , γ ∈ K, k = k , ≤ k , k ≤ n, where u γ ∈ A is the canonical unitary of γ .Let us verify that the order zero map φ satisfies Definition 5.1 with p = q = n , l = 1, and r = 3. (With the given p, q, l, r , it is straightforward to verify that 2 of Definition 5.1 holds.)Note that, by Equations (7.35), (7.37), and Condition 2, for any k = k , 1 ≤ k , k ≤ n , E k a ′ E k = E k ( X γ ∈ K f γ u γ ) E k = X γ ∈ K E k f γ u γ E k = X γ ∈ K f γ E k u γ E k = X γ ∈ K f γ ( E k u γ E k u ∗ γ ) u γ = 0 . In particular, this verifies 1 of Definition 5.1.Set c k = c k, + · · · + c k,M and d k = d k, + · · · + d k,M , k = 1 , ..., n. Then, 3a 3b 3c of Definition 5.1 follows directly from Conditions 1a, 1b, and 1c above.As for 3d of Definition 5.1, note that it follows from Condition 1e above thatd τ ( s k ) ≤ n < λ d τ ( c k, ) and d τ (1 − h ) < λ d τ ( d k, ) , τ ∈ T( A ) . Since ( X, Γ) has ( λ, M )-Cuntz comparison of open sets and c k, , d k, , h, s k ∈ C( X ), one has s k - c k, ⊕ · · · ⊕ c k, | {z } M and 1 − h - d k, ⊕ · · · ⊕ d k, | {z } M . By Condition 1d above, the positive elements c k,m , m = 1 , ..., M , are mutually orthogonaland mutually Cuntz equivalent, and the positive elements d k,m , m = 1 , ..., M are mutuallyorthogonal and mutually Cuntz equivalent. One then has c k ∼ c k, ⊕ · · · ⊕ c k, | {z } M and d k ∼ d k, ⊕ · · · ⊕ d k, | {z } M , and hence s k - c k and 1 − h - d k . This shows that a ′ is a D -operator, as desired. (cid:3) TABLE RANK OF C( X ) ⋊ Γ 49
Theorem 7.8.
Let Γ be a countable discrete amenable group, and let ( X, Γ) be a free andminimal topological dynamical system with the (URP) and (COS). Then tsr(C( X ) ⋊ Γ) = 1 .Proof. If | Γ | < ∞ , since ( X, Γ) is minimal, the space X must consist of finitely many pointsand C( X ) ⋊ Γ ∼ = M | Γ | ( C ). In particular, it has stable rank one.If | Γ | = ∞ , then it follows from Proposition 7.7 that C( X ) ⋊ Γ has Property (D). SinceC( X ) ⋊ Γ is finite, it follows from Theorem 5.5 that tsr(C( X ) ⋊ Γ) = 1, as desired. (cid:3)
Corollary 7.9.
Let ( X, Z d ) be a free and minimal topological dynamical system. Then tsr(C( X ) ⋊ Z d ) = 1 . Proof.
By Theorem 4.2 and Theorem 5.5 of [16], any free and minimal dynamical system( X, Z d ) has the (URP) and (COS). It then follows from Theorem 7.8 that tsr(C( X ) ⋊ Z d ) =1 . (cid:3) Remark . Without simplicity, the C*-algebra C( X ) ⋊ Γ might not have stable rank onein general, even if X is the Cantor set, Γ = Z , and ( X, Z ) has finitely many minimal closedinvariant subsets (see, [19] or [3]). Corollary 7.11.
Let ( X, Z d ) be a free and minimal dynamical system, and set A = C( X ) ⋊Z d . Then (1) A has cancellation of projections, i.e., if p, q ∈ A ⊗ K are two projections such that p ⊕ r ∼ q ⊕ r for some projections r ∈ A ⊗ K , then p ∼ q . (2) A has cancellation in Cuntz semigroup: let x, y ∈ W( A ) such that x +[ c ] ≤ y +[( c − ε ) + ] for some positive element c ∈ M ∞ ( A ) , then x ≤ y . (3) The canonical map U( A ) / U ( A ) → K ( A ) is an isomorphism. That is, any unitaryof A ⊗ K is homotopic to a unitary of A , and if a unitary u of A is connected to theidentity with a path of unitaries of ^ A ⊗ K , then u can be connected to the identity bya path of unitaries of A .Proof. Statements 1 and 3 are well known fact for C*-algebras with stable rank one ([21]).Statement 2 follows from Theorem 4.3 of [25] (an earlier version were obtained in [8]). (cid:3)
By Theorem 4.1 of [5], the Cuntz semigroup classifies homomorphisms from an inductivelimit of interval algebras (AI algebra) to a C*-algebra A with stable rank one. Therefore wehave the following corollary. Corollary 7.12.
Let ( X, Z d ) be a free and minimal dynamical system. Let φ , φ : I → A = C( X ) ⋊ Z d be two homomorphisms, where I is an AI algebra. Then φ and φ areapproximately unitarily equivalent if, and only if, [ φ ] = [ φ ] on the Cuntz semigroups. The next corollary follows from [27]:
Corollary 7.13.
Let ( X, Γ) be a free and minimal dynamical system with the (URP) and(COS). Then for every f ∈ LAff(T( A )) ++ , where A = C( X ) ⋊ Γ , there exists a ∈ ( A ⊗ K ) + such that d τ ( a ) = f ( τ ) , τ ∈ T( A ) . Moreover, if A has strict comparison of positive elements, then the Cuntz semigroup of A is almost divisible (see [27] ). In this case, there are canonical order-isomorphisms Cu( A ) ∼ = V ( A ) ⊔ LAff(T( A )) ++ ∼ = Cu(A ⊗ Z ) . In particular, the statements above hold for
Γ = Z d .Proof. This follows directly from Theorem 8.11 and Corollary 8.12 of [27]. (cid:3)
In fact, if C( X ) ⋊Z d has strict comparison of positive elements, then it actually is Jiang-Sustable: Corollary 7.14.
Let ( X, Γ) be a free and minimal dynamical system with the (URP) and(COS), and denote by A = C( X ) ⋊ Γ . Then A ∼ = A ⊗Z if, and only if, A has strict comparisonof positive elements (that is, it satisfies the Toms-Winter conjecture). In particular, thestatement holds for Γ = Z d .Proof. One only need to show the “if” part. Since A is assumed to have strict comparisonof positive elements, by Corollary 7.13, one has that Cu( A ) ∼ = V ( A ) ⊔ LAff(T( A )) ++ andhence it is tracially 0-divisible (see Corollary 2.6 of [18] and its proof). It then follows fromProposition 3.8 of [18] and the strict comparison assumption that A is tracially Z -stable.Since A is nuclear, one has that A ∼ = A ⊗ Z , as desired. (cid:3) Since the real rank of a C*-algebra A is at most 2 · tsr( A ) −
1, one has the followingestimate:
Corollary 7.15.
Let ( X, Z d ) be a free and minimal dynamical system. The real rank of C( X ) ⋊ Z d is either or .Remark . Consider a simple unital AH algebra A with diagonal maps. It is known that if A has real rank zero (or just projections separate traces), then A is classifiable ([15]). Doesthe same statement hold for the crossed-product C*-algebras C( X ) ⋊ Z (or C( X ) ⋊ Γ ingeneral)? That is, if C( X ) ⋊ Z (or C( X ) ⋊ Γ, in general) has real rank zero, does C( X ) ⋊ Z (or C( X ) ⋊ Γ, in general) absorb the Jiang-Su algebra Z tensorially? What if one onlyassumes that projections separate traces instead of real rank zero?Let Γ be a countable discrete group with sub-exponential growth, and let ( X, Γ) be a freeand minimal dynamical system. Assume that ( X, Γ) is an extension of a minimal Γ-actionon a Cantor set. Then it was shown in [26] that the C*-algebra C( X ) ⋊ Γ has stable rankone. Note that, by Corollary 3.8 and Corollary 8.11 of [17], the dynamical system ( X, Γ) hasthe (URP) and (COS), and therefore this result also can follows from Theorem 7.8.
Corollary 7.17 (c.f. Main Theorem of [26]) . Let Γ be a countable discrete group with sub-exponential growth, let ( X, Γ) be a free and minimal dynamical system. Assume that ( X, Γ) is an extension of a Γ -action on the Cantor set. Then tsr(C( X ) ⋊ Γ) = 1 . TABLE RANK OF C( X ) ⋊ Γ 51 Two remarks on Property (D)
In the final section, let us remark that simple Z -stable C*-algebras and simple AH-algebraswith diagonal maps all have Property (D). These C*-algebras (if finite for the case of Z -stableC*-algebras) are known to have stable rank one (see [24] and [9]).8.1. Z -stable C*-algebras. Let A be a unital simple exact C*-algebra such that A ∼ = A ⊗ Z , where Z is the Jiang-Su algebra. Note that A has strict comparison of positiveelements (we include purely infinite C*-algebras, which have empty tracial simplices).Let a ∈ ZD( A ) with k a k = 1 and let ε > d , d ∈ A + such that k d k = k d k = 1 and d a = ad = 0 . By regarding A as A ⊗ Z ⊗ Z ⊗ Z , one obtains ˜ a, ˜ d , ˜ d ∈ A ⊗ Z ⊗ ⊗ u ∈ A ⊗ Z ⊗ Z ⊗ k uau ∗ − ˜ a k < ε , (cid:13)(cid:13)(cid:13) ud u ∗ − ˜ d (cid:13)(cid:13)(cid:13) < ε , (cid:13)(cid:13)(cid:13) ud u ∗ − ˜ d (cid:13)(cid:13)(cid:13) < ε , and then (cid:13)(cid:13)(cid:13) ˜ d ˜ a (cid:13)(cid:13)(cid:13) < ε (cid:13)(cid:13)(cid:13) ˜ a ˜ d (cid:13)(cid:13)(cid:13) < ε . With a small perturbation of ˜ d and ˜ d , one may assume that there are positive elements d ′ , d ′ ∈ A ⊗ Z ⊗ ⊗ k d ′ k = k d ′ k = 1 such that d ′ ˜ d = d ′ and d ′ ˜ d = d ′ Note that(1 − ˜ d )˜ a (1 − ˜ d ) ≈ ε ˜ a and d ′ (1 − ˜ d )˜ a (1 − ˜ d ) = (1 − ˜ d )˜ a (1 − ˜ d ) d ′ = 0 . Pick two orthogonal nonzero positive elements s , s ∈ ⊗ ⊗Z ⊗
1, and consider the positiveelements ˜ d s and ˜ d s . Since s , s commute with ˜ d , ˜ d , one has that˜ d s ⊥ ˜ d s and ( ˜ d s )((1 − d ′ )˜ a (1 − ˜ d )) = ((1 − ˜ d )˜ a (1 − ˜ d ))( d ′ s ) = 0 . Since A ⊗ Z ⊗ Z ⊗ v ∈ A ⊗ Z ⊗ Z ⊗ k v k = 1 such that vv ∗ ∈ Her( d s ) and v ∗ v ∈ Her( d s ) , and moreover, with the polar decomposition and a further perturbation, one may assumethat there is a positive element b such that k b k = 1 and ( vv ∗ ) b = b (hence b ∈ Her( d s )). Itthen follows from Lemma 3.2 that there is a unitary w ∈ A ⊗ Z ⊗ Z ⊗ wbw ∗ ∈ Her( d s ) . Thus, with a ′ := (1 − ˜ d )˜ a (1 − ˜ d ) w, one has k uau ∗ w − a ′ k < ε and ba ′ = 0 = a ′ b. Let us show that a ′ is a D -operator, and thus A has Property (D). Since A is not of type I, there are positive elements c, d ∈ b ( A ⊗ Z ⊗ Z ⊗ b such that c ⊥ d and k c k = k d k = 1. Since A is simple, there is δ > τ ( c ) , τ ( d ) > δ, τ ∈ T( A ) . Now, consider the embedding φ ′ : Z → ⊗ ⊗ ⊗ Z , and note that(8.1) [ a ′ , φ ′ ( a )] = 0 and [ b, φ ′ ( a )] = 0 , a ∈ Z . Pick n ∈ N sufficiently large such that ( n − δ > , and pick a standard embedding ι : M n (C ((0 , → Z n ,n +1 → Z . Denote by φ ′′ the order zero map induced by the homomorphism φ ′ ◦ ι , and choose ε ′ > τ ( f ε ′ ( φ ′′ (1))) > − δ/ n, τ ∈ T( A ) . For each k = 1 , , ..., n , define c k = c · f ε ′ ( φ ′′ )( e ( k − n +4 + · · · + e ( k − n + n ) , and d k = d · f ε ′ ( φ ′′ )( e ( k − n +1 + · · · + e ( k − n + n ) . A straightforward calculation (using (8.1)) shows that c k , d k ∈ b ( A ⊗ Z ⊗ Z ⊗ Z ) b , c k ⊥ d k and c k E k = c k and d k E k = d k , where E k := f ε ′ ( φ ′′ )( e ( k − n +1 + · · · + e ( k − n + n ).Note that, for any τ ∈ T( A ),d τ ( c k ) > δ · n − n · n − δ n > n · n − n > n and d τ ( d k ) > δ · n · n − δ n > δ · n · n − n > δ n > d τ (1 − f ε ( φ ′′ (1))) . Since A has strict comparison of positive elements, one has that s k - c k and 1 − f ε ( φ ′′ (1)) - d k . This shows that a ′ is a D -operator (with φ = f ε ′ ( φ ′′ ), p = q = n , r = 3, and l = 1 inDefinition 5.1). TABLE RANK OF C( X ) ⋊ Γ 53
AH algebras with diagonal maps.
Recall that an AH algebra with diagonal mapsis the limit of a unital inductive sequence ( A n , ψ n ), where A n = h n M i =1 M k n,i (C( X n,i ))for some compact metrizable space X n,i , and if D n := M i { diag { f , f , ..., f k n,i } : f k ∈ C( X n,i ) } ⊆ M i M k n,i (C( X n,i )) = A n , then ψ n ( D n ) ⊆ D n +1 . Let A be simple AH algebra with diagonal maps. It then follows from Theorem 3.4 of [9]that A has Property (D), and we leave the details to readers. Alternatively, let us proposethe following approach which is in the similar line of our approach to the crossed productC*-algebra C( X ) ⋊ Γ: Consider D := lim −→ D n ⊆ lim −→ A n = A. Then the commutative sub-C*-algebra D actually behaves like the sub-C*-algebra C( X ) ofC( X ) ⋊ Γ.Let a ∈ A satisfy k a k = 1 and d a = ad = 0 for some nonzero positive elements d , d ,and let ε > a, ˜ d , ˜ d ∈ A with norm one such that k a − ˜ a k < ε (cid:13)(cid:13)(cid:13) ˜ d ˜ a (cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13) ˜ a ˜ d (cid:13)(cid:13)(cid:13) < ε , where ˜ d , ˜ d are positive. Since ˜ d , ˜ d has norm one, there is x ∈ X ,i for some i such that (cid:13)(cid:13)(cid:13) ˜ d ( x ) (cid:13)(cid:13)(cid:13) = 1 and (cid:13)(cid:13)(cid:13) ˜ d ( x ) (cid:13)(cid:13)(cid:13) = 1 . Since ˜ d , ˜ d are positive, by conjugating some constant unitary matrices, one may assumethat ˜ d and ˜ d are diagonal matrices at x . Hence, by cutting the diagonal entry whichhas value 1 at x , one can find a positive element h ∈ D which is constantly 1 on a smallneighborhood of x such that h ˜ d ≈ ε h and h ˜ d ≈ ε h. Then a straightforward calculation shows that k h ˜ a k < ε k ˜ ah k < ε . Since h is constantly 1 on a neighborhood of x , there is a positive element b ∈ D n withnorm 1 such that bh = b . Then a ′ := (1 − h )˜ a (1 − h )and b satisfy k a − a ′ k < ε and ba ′ = a ′ b = 0 . Now, let us show that a ′ is a D -operator, and thus A has Property (D). Choose positive orthogonal functions c, d ∈ D such that c, d ∈ bD b . Set δ = min { τ ( c ) , τ ( d ); τ ∈ T( A ) } > . By another telescoping if necessary, one may assume that(8.2) 3 k ,i < δ < min { rank( c ( x )) , rank( d ( x )) } k ,i , x ∈ X ,i . Choose l ∈ N such that 12 l − < δ . Set K := max { k , , ..., k ,h } + 1. Consider A , and to simplify notation, rewrite A = L Ss =1 M k s (C( X s )). With a telescoping of the inductive sequence if necessary, one has that,insider each direct summand of A ,(1) the element a ′ is a matrix of continuous functions with(8.3) a ′ i,j = 0 , if | i − j | ≥ K, (2) write c = S M s =1 diag { c ( s )1 , ..., c ( s ) k s } and d = S M s =1 diag { d ( s )1 , ..., d ( s ) k s } , where c ( s ) i , d ( s ) i ∈ C( X s ), then, by (8.2), for any L = 1 , ..., k s ,(8.4) δ − KL ) < (cid:12)(cid:12)(cid:12) { i ≤ i ≤ i + L − c ( s ) i ( x ) = 0 } (cid:12)(cid:12)(cid:12) L , ≤ i < k s − L, x ∈ X s , and(8.5) δ − KL ) < (cid:12)(cid:12)(cid:12) { i ≤ i ≤ i + L − d ( s ) i ( x ) = 0 } (cid:12)(cid:12)(cid:12) L , ≤ i < k s − L, x ∈ X s , (3) with k s = m s l + r s , 0 ≤ r s < l , one has(8.6) m s l > K, Km s < , and 4 r s m s l − K < δ . Then, for each s = 1 , ..., S , consider M k s ( C ) ⊆ M k s (C( X s )), and consider p ( s ) i := diag { l m s z }| { m s , ..., m s , m s | {z } im s , m s , ..., m s , r s } , i = 1 , ..., l . Note that p ( s )1 , p ( s )2 , ..., p ( s ) l ⊆ M k s ( C ) have the same rank and are mutually orthogonal. There-fore, there is a homomorphism φ s : M l ( C ) ∋ e i,i p ( s ) i ∈ M k s ( C ) ⊆ A . Define φ := M s φ s : M l ( C ) → M s M k s ( C ) ⊆ A , TABLE RANK OF C( X ) ⋊ Γ 55 and set e i = φ ( e i,i ), i = 1 , , ..., l , s k = e l ( k − + · · · + e l ( k − , E k = e l ( k − + · · · + e lk , k = 1 , ..., l , and h = φ (1).Then, since m s l > K , by (8.3), E k a ′ E k = 0 , k − k ≥ . For each k = 1 , , ..., l , consider c k := M s diag { l m s z }| { m s , ..., m s | {z } l ( k − m s , m s , ..., m s | {z } m s , c ( s )( l ( k − m s +1 , ..., c ( s ) lkm s | {z } lm s , m s , ..., m s , r s } , and d k := M s diag { l m s z }| { m s , ..., m s | {z } l ( k − m s , d ( s ) l ( k − m s +1 , ..., d ( s ) lkm s | {z } lm s , m s , ..., m s , r s } . Then it is clear that c k ⊥ s k , c k ⊥ d k , c k E k = c k and d k E k = d k .Note that, by (8.4), (8.5), and (8.6),14 rank( c k ( x )) > · δ l − m s − K ) > m s = rank( s k ( x ))and 14 rank( d k ( x )) > · δ lm s − K ) > r s = rank((1 − h )( x )) . Since s k , c k , d k and 1 − h are diagonal elements, by Theorem 7.8 of [17], one has that s k - c k and 1 − h - d k . Therefore, a ′ is a D -operator (with p = q = l , l = 2 and r = 4 in Definition 5.1). References [1] M. Alboiu and J. Lutley. The stable rank of diagonal ASH algebras and crossed products by minimalhomeomorphisms. 05 2020. URL: https://arxiv.org/pdf/2005.00148.pdf , arXiv:2005.00148 .[2] D. Archey and N. C. Phillips. Permanence of stable rank one for centrally large subalgebras andcrossed products by minimal homeomorphisms. J. Operator Theory , 83(2):353–389, 2020. URL: doi:10.7900/jot.2018oct10.2236 .[3] S. Bezuglyi, Z. Niu, and W. Sun. C*-algebras of a Cantor system with finitely many minimal subsets:structures, K-theories, and the index map.
Ergodic Theory Dynam. Systems , in press, 2020.[4] N. P. Brown and N. Ozawa.
C*-algebras and finite-dimensional approximations , volume 88 of
Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2008. URL: https://mathscinet.ams.org/mathscinet-getitem?mr=2391387 , doi:10.1090/gsm/088 .[5] A. Ciuperca and G. A. Elliott. A remark on invariants for C*-algebras of stable rank one. Int. Math.Res. Not. IMRN , (5):Art. ID rnm 158, 33, 2008. URL: http://dx.doi.org/10.1093/imrn/rnm158 , doi:10.1093/imrn/rnm158 .[6] T. Downarowicz, D. Huczek, and G. Zhang. Tilings of amenable groups. J. Reine Angew.Math. , 747:277–298, 2019. URL: https://mathscinet.ams.org/mathscinet-getitem?mr=3905135 , doi:10.1515/crelle-2016-0025 . [7] E. G. Effros and F. Hahn. Locally compact transformation groups and C*- algebras . Memoirs of theAmerican Mathematical Society, No. 75. American Mathematical Society, Providence, R.I., 1967.[8] G. A. Elliott. Hilbert modules over a C*-algebra of stable rank one.
C. R. Math. Acad. Sci. Soc. R.Can. , 29(2):48–51, 2007. URL: https://mathscinet.ams.org/mathscinet-getitem?mr=2367726 .[9] G. A. Elliott, T. M. Ho, and A. S. Toms. A class of simple C*-algebras with stable rankone.
J. Funct. Anal. , 256(2):307–322, 2009. URL: http://dx.doi.org/10.1016/j.jfa.2008.08.001 , doi:10.1016/j.jfa.2008.08.001 .[10] G. A. Elliott and Z. Niu. The C ∗ -algebra of a minimal homeomorphism of zero mean dimension. DukeMath. J. , 166(18):3569–3594, 2017. doi:10.1215/00127094-2017-0033 .[11] J. Giol and D. Kerr. Subshifts and perforation.
J. Reine Angew. Math. , 639:107–119, 2010. URL: http://dx.doi.org/10.1515/CRELLE.2010.012 , doi:10.1515/CRELLE.2010.012 .[12] R. H. Herman and L. N. Vaserstein. The stable range of C*-algebras. Invent. Math. , 77(3):553–555, 1984.URL: https://mathscinet.ams.org/mathscinet-getitem?mr=759256 , doi:10.1007/BF01388839 .[13] T. M. Ho. On inductive limits of homogeneous C*-algebras with diagonal morphisms be-tween the building blocks, 2006. Thesis (Ph.D.)–University of Toronto (Canada). URL: https://mathscinet.ams.org/mathscinet-getitem?mr=2709772 .[14] Q. Lin. Analytic structure of the transformation group C*-algebra associated with minimal dynamicalsystems. Preprint .[15] Z. Niu. Mean dimension and AH-algebras with diagonal maps.
J. Funct. Anal. , 266(8):4938–4994, 2014.URL: http://dx.doi.org/10.1016/j.jfa.2014.02.010 , doi:10.1016/j.jfa.2014.02.010 .[16] Z. Niu. Comparison radius and mean topological dimension: Z d -actions. arXiv:1906.09171 , 2019.[17] Z. Niu. Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets,and subhomogeneous C*-algebras. arXiv:1906.09172 , 2019.[18] Z. Niu. Z -stability of C( X ) ⋊ Γ. Preprint , 2019.[19] Y. T. Poon. Stable rank of some crossed product C*-algebras.
Proc. Amer. Math. Soc. , 105(4):868–875,1989. URL: http://dx.doi.org/10.2307/2047045 , doi:10.2307/2047045 .[20] I. F. Putnam. The C*-algebras associated with minimal homeomor-phisms of the Cantor set. Pacific J. Math. , 136(2):329–353, 1989. URL: http://projecteuclid.org/getRecord?id=euclid.pjm/1102650733 .[21] Marc A. Rieffel. Dimension and stable rank in the K-theory of C*-algebras.
Proc. Lon-don Math. Soc. (3) , 46(2):301–333, 1983. URL: http://dx.doi.org/10.1112/plms/s3-46.2.301 , doi:10.1112/plms/s3-46.2.301 .[22] M. Rørdam. On the structure of simple C*-algebras tensored with a UHF algebra. J. Funct. Anal. ,100:1–17, 1991.[23] M. Rørdam. On the structure of simple C*-algebras tensored with a UHF-algebra. II.
J.Funct. Anal. , 107(2):255–269, 1992. URL: http://dx.doi.org/10.1016/0022-1236(92)90106-S , doi:10.1016/0022-1236(92)90106-S .[24] M. Rørdam. The stable and the real rank of Z -absorbing C*-algebras. Internat. J.Math. , 15(10):1065–1084, 2004. URL: http://dx.doi.org/10.1142/S0129167X04002661 , doi:10.1142/S0129167X04002661 .[25] M. Rørdam and W. Winter. The Jiang-Su algebra revisited. J. Reine Angew. Math. , 642:129–155, 2010.[26] Y. Suzuki. Almost finiteness for general etale groupoids and its applications to stable rank of crossedproducts.
Int. Math. Res. Not. IMRN , 2018.[27] H. Thiel. Ranks of operators in simple C*-algebras with stable rank one.
Comm. Math. Phys. , to appear.[28] A. S. Toms. On the classification problem for nuclear C*-algebras.
Ann. of Math.(2) , 167(3):1029–1044, 2008. URL: http://dx.doi.org/10.4007/annals.2008.167.1029 , doi:10.4007/annals.2008.167.1029 .[29] J. Villadsen. Simple C*-algebras with perforation. J. Funct. Anal. , 154(1):110–116, 1998. URL: http://dx.doi.org/10.1006/jfan.1997.3168 , doi:10.1006/jfan.1997.3168 . TABLE RANK OF C( X ) ⋊ Γ 57 [30] J. Villadsen. On the stable rank of simple C*-algebras.
J. Amer. Math. Soc. ,12(4):1091–1102, 1999. URL: http://dx.doi.org/10.1090/S0894-0347-99-00314-8 , doi:10.1090/S0894-0347-99-00314-8 .[31] D. P. Williams. Crossed products of C*-algebras . Mathematical Surveys and Monographs, Volume 134.American Mathematical Society, Providence, RI, 2007. URL: http://dx.doi.org/10.1090/surv/134 , doi:10.1090/surv/134 .[32] W. Winter. Covering dimension for nuclear C*-algebras. II. Trans. Amer. Math. Soc. ,361(8):4143–4167, 2009. URL: https://mathscinet.ams.org/mathscinet-getitem?mr=2500882 , doi:10.1090/S0002-9947-09-04602-9 .[33] G. Zeller-Meier. Produits crois´es d’une C*-alg`ebre par un groupe d’automorphismes. J. Math. PuresAppl. (9) , 47:101–239, 1968.
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