Minimal dynamics and Z-stable classification
aa r X i v : . [ m a t h . OA ] D ec MINIMAL DYNAMICS AND Z -STABLE CLASSIFICATION KAREN R. STRUNG AND WILHELM WINTER
Abstract.
Let X be an infinite compact metric space, α : X → X a minimalhomeomorphism, u the unitary implementing α in the transformation group C ∗ -algebra C ( X ) ⋊ α Z , and S a class of separable nuclear C ∗ -algebras thatcontains all unital hereditary C ∗ -subalgebras of C ∗ -algebras in S . Motivatedby the success of tracial approximation by finite dimensional C ∗ -algebras as anabstract characterization of classifiable C ∗ -algebras and the idea that classifi-cation results for C ∗ -algebras tensored with UHF algebras can be used to deriveclassification results up to tensoring with the Jiang–Su algebra Z , we provethat ( C ( X ) ⋊ α Z ) ⊗ M q ∞ is tracially approximately S if there exists a y ∈ X such that the C ∗ -subalgebra ( C ∗ ( C ( X ) , uC ( X \ { y } ))) ⊗ M q ∞ is traciallyapproximately S . If the class S consists of finite dimensional C ∗ -algebras, thiscan be used to deduce classification up to tensoring with Z for C ∗ -algebrasassociated to minimal dynamical systems where projections separate tracialstates. This is done without making any assumptions on the real rank or sta-ble rank of either C ( X ) ⋊ α Z or C ∗ ( C ( X ) , uC ( X \{ y } )), nor on the dimensionof X . The result is a key step in the classification of C ∗ -algebras associatedto uniquely ergodic minimal dynamical systems by their ordered K -groups. Italso sets the stage to provide further classification results for those C ∗ -algebrasof minimal dynamical systems where projections do not necessarily separatetraces. Introduction
The two subjects of C ∗ -algebras and dynamical systems have long been closeallies. On the one hand, dynamical systems provide a rich source of elegant andfundamental examples for C ∗ -algebra theory, see [5], [7] and [4], to name but afew. On the other hand, the techniques of C ∗ -algebras have been used to makeprogress in distinguishing dynamical systems, most notably in the work of Gior-dano, Putnam, and Skau on minimal Cantor systems [10]. The class of C ∗ -algebrasassociated to actions on infinite compact metric spaces has also played an inter-esting role in Elliott’s classification program for C ∗ -algebras. The classificationprogram was initiated when Elliott showed that the class of approximately finitedimensional (AF) C ∗ -algebras is classified by K -theory. Following this success, El-liott conjectured that simple separable nuclear C ∗ -algebras might be classified by acertain K -theoretic invariant, now called the Elliott invariant, in the sense that if A and B are two simple separable nuclear C ∗ -algebras with isomorphic invariants,then A and B are ∗ -isomorphic as C ∗ -algebras. Moreover, the isomorphism maybe chosen in such a way as to induce the isomorphism of Elliott invariants. Date : November 20, 2018.2000
Mathematics Subject Classification.
Key words and phrases. minimal homeomorphisms, classification, Z-stability.
Supported by:
EPSRC First Grant EP/G014019/1.
Evidence in support of the conjecture was successfully gathered by way of clas-sification results for various classes of C ∗ -algebras, including theorems for those C ∗ -algebras associated to minimal dynamical systems. Many of these results re-quire the presentation of the C ∗ -algebra as a direct limit, showing it is of a formknown to be classifiable. For example, the irrational rotation algebras, which arethe C ∗ -algebras associated to the dynamical system ( T , α ) where α is an irrationalrotation of the circle T , were shown by Elliott and Evans to be A T algebras withreal rank zero [7], and hence classifiable by their Elliott invariants. Similarly, thecrossed products associated to minimal homeomorphisms of the Cantor set wereshown, using results by Putnam, to be A T algebras with real rank zero and thusclassifiable [6].In the past decade, efforts have been made to avoid the use of specific directlimit presentations as a means of obtaining classification theorems and more ab-stract methods have been considered. To this end, Lin introduced the first notion oftracial approximation of C ∗ -algebras with his concept of tracially approximately fi-nite dimensional (TAF) C ∗ -algebras [15]. In the case of a simple unital C ∗ -algebra,these may be thought of as being approximated by finite dimensional C ∗ -algebrasin trace. Similarly, one may consider C ∗ -algebras that are tracially approximatelyinterval (TAI) algebras. These are C ∗ -algebras that can be approximated by in-terval algebras, that is, C ∗ -algebras of the form L ni =1 M m i ( C ( X i )) where X i is asingle point or X i = [0 , S C ∗ -algebras where S is an arbitrary class of C ∗ -algebras (seeDefinition 2.1 below). In particular, they look at which properties of the class S pass to the class TA S .The concept of tracial approximation has proven to be extremely useful in pro-viding classification results. One of the most notable results has been by Lin andPhillips in [20], where classification results for C ∗ -algebras with tracial rank zerowere successfully applied to many C ∗ -algebras of minimal dynamical systems. Let X be an infinite compact metric space, α : X → X a minimal homeomorphism and u the unitary implementing α in C ( X ) ⋊ α Z . Lin and Phillips proved that undercertain conditions, it is enough to verify that there exists a point y ∈ X such thatthe C ∗ -subalgebra C ∗ ( C ( X ) , uC ( X \ { y } )) has tracial rank zero.In this paper, we generalize the results in [20] by following the strategy forclassification up to Z -stability as outlined in [35]. Here, Z denotes the Jiang–Sualgebra, introduced in [12]. There are many characterizations of this algebra, bothabstract and concrete, which single it out as a universal object playing a role asfundamental as that of the Cuntz algebra O ∞ , cf. [30] and [37]. Z -stability (i.e.,the property of absorbing Z tensorially) is an important structural property for C ∗ -algebras; it has recently been shown to be closely related to other topologicaland algebraic regularity properties, such as finite topological dimension and strictcomparison of positive elements. It is remarkable that all nuclear C ∗ -algebrasclassified so far by their Elliott invariants are in fact Z -stable.In [35], the second named author derived classification up to Z -stability fromclassification results up to UHF stability. The latter are usually much easier to es-tablish, since UHF stability ensures a wealth of projections. For any q ∈ N \ { } , let INIMAL DYNAMICS AND Z -STABLE CLASSIFICATION 3 M q ∞ denote the UHF algebra N ∞ n =1 M q . Our main theorem says that if S is a classof separable unital C ∗ -algebras such that the property of being a member of S passesto unital hereditary C ∗ -subalgebras, then ( C ( X ) ⋊ α Z ) ⊗ M q ∞ is TA S wheneverthere exists a y ∈ X such that the C ∗ -subalgebra C ∗ ( C ( X ) , uC ( X \ { y } )) ⊗ M q ∞ is TA S . We tensor with the UHF algebra to alleviate some of the restrictions on C ( X ) ⋊ α Z and C ∗ ( C ( X ) , uC ( X \ { y } )) that are found in [20] and to take advan-tage of results showing that classification results for C ∗ -algebras tensored with UHFalgebras can be used to derive classification results up to tensoring with the Jiang–Su algebra, as shown in [35], [16] and [19]. When S is the set of finite dimensional C ∗ -algebras, our result shows that the C ∗ -algebras associated to minimal dynam-ical systems of infinite compact metric spaces whose projections separate tracialstates are classified by their K -theory, up to tensoring with Z . Toms and the sec-ond named author show in [33] (see also [34]) that in the case where the base spacehas finite topological dimension the crossed product is Z -stable. Together with theresults of the present paper this completes the classification of C ∗ -algebras asso-ciated to uniquely ergodic minimal finite dimensional dynamical systems by theirordered K -groups. We are optimistic that our result will also help provide furtherclassification results for those C ∗ -algebras of minimal dynamical systems of infinitecompact metric spaces where projections do not necessarily separate traces.As an application, we obtain that the C ∗ -algebras associated to uniquely ergodicminimal homeomorphisms of odd spheres as considered by Connes in [4] are all iso-morphic. In the smooth case, this was already shown in [38], using the inductivelimit decomposition of [22]. The latter is technically very advanced; our methodprovides a somewhat easier path since the inductive limit structure of the subal-gebras C ∗ ( C ( X ) , uC ( X \ { y } )) can be established with much less effort. In thenot necessarily uniquely ergodic case, one expects the space of tracial states to bethe classifying invariant; building on our present results, this will be pursued in asubsequent article.The paper is organized as follows. In Section 2 we give the definition for a C ∗ -algebra that is TA S and outline the strategy for arriving at our main theoremin Section 3. In Section 4 we state and prove our main technical results and inSection 5 we apply these to derive classification results. We also discuss someexamples and outline the strategy for further results relating to the classificationof transformation group C ∗ -algebras.2. Preliminaries
We begin with a definition of what is meant by tracial approximation, cf. [13]and [8].
Definition 2.1.
Let S denote a class of separable unital C ∗ -algebras. Let A bea simple unital C ∗ -algebra. Then A is tracially approximately S (or TA S ) if thefollowing holds. For every finite subset F ⊂ A , every ǫ > , and every nonzeropositive element c ∈ A , there exists a projection p ∈ A and a unital C ∗ -subalgebra B ⊂ pAp with B = p and B ∈ S such that: (i) k pa − ap k < ǫ for all a ∈ F , (ii) dist( pap, B ) < ǫ for all a ∈ F , (iii) 1 A − p is Murray–von Neumann equivalent to a projection in cAc . KAREN R. STRUNG AND WILHELM WINTER
For a compact metric space X and a minimal homeomorphism α : X → X , put A = C ( X ) ⋊ α Z . We denote A { y } = C ∗ ( C ( X ) , uC ( X \ { y } )) .A { y } is a unital C ∗ -subalgebra of A , a generalization of those introduced by Putnamin [25]. This algebra carries much of the information contained in A while at thesame time is significantly more tractable. In particular, by Theorem 4.1(3) of [24],its K -group is isomorphic to that of A , and it can be written as an inductive limitof subhomogeneous algebras in a straightforward manner, see Section 3 of [21].There are natural bijections between the set of α -invariant probability measures on X , the set of tracial states on A and the set of tracial states on A { y } ([21], Theorem1.2).We recall the notion of strict comparison of positive elements for a C ∗ -algebra A . For two positive elements a, b ∈ A , write a . b if there exists r j ∈ A such thatlim j r j br ∗ j = a . For any C ∗ -algebra A , denote by T ( A ) the tracial state space of A .For τ ∈ T ( A ), define d τ ( a ) = lim n →∞ τ ( a /n ) ( a ∈ A + ) . If A is exact, then the set { d τ | τ ∈ T ( A ) } coincides with the set of normalizedlower semicontinuous dimension functions of A ([28] and [11]). If d τ ( a ) < d τ ( b )for all τ ∈ T ( A ) implies a . b , then we say that A has strict comparison (ofpositive elements) . It was recently shown by Toms ([32], Corollary 5.5) that if X is a compact smooth connected manifold and α : X → X a diffeomorphism, then C ( X ) ⋊ α Z has strict comparison. His result relies on the direct limit structureof C ( X ) ⋊ α Z given in [22]; such a structure is not known in the general case.However, because we have tensored with a UHF algebra, we are able to use resultsof Rørdam to circumvent this issue. For a compact metric space X and a minimalhomeomorphism α : X → X , let A = C ( X ) ⋊ α Z . Then, for any q ∈ N \ { } andany y ∈ X , the C ∗ -algebras A ⊗ M q ∞ and A { y } ⊗ M q ∞ have strict comparison byTheorem 5.2 of [28]. 3. Strategy
Our aim is to show that when A = C ( X ) ⋊ α Z is the C ∗ -algebra arising froma minimal dynamical system of an infinite compact metric space such that thesubalgebra A { y } ⊗ M q ∞ ⊂ A ⊗ M q ∞ is TA S , then A ⊗ M q ∞ is TA S . One cangenerally only expect such a passage from A { y } to A after tensoring with a UHFalgebra. We also need to assume the class S of Definition 2.1 to be closed withrespect to unital hereditary C ∗ -subalgebras.Our result (and method) is a generalization of the tracial rank zero case (with-out tensoring with a UHF algebra), which was obtained by Lin and Phillips in [20].A key lemma for their proof is showing that the finite dimensional C ∗ -subalgebrain the definition of tracial rank zero can be replaced by a sufficiently large simpleunital C ∗ -subalgebra of tracial rank zero. Since we are concerned with A ⊗ M q ∞ ,we show that for any simple unital C ∗ -algebra A , after tensoring with the UHFalgebra, A ⊗ M q ∞ is TA S when the subalgebra B ∈ S in Definition 2.1 is re-placed by a simple unital subalgebra of A ⊗ M q ∞ that is TA S . With the aim ofusing this result (Lemma 4.5 below), the key step is finding a suitable projection INIMAL DYNAMICS AND Z -STABLE CLASSIFICATION 5 p in the C ∗ -subalgebra A { y } ⊗ M q ∞ . Under the assumption that S is closed un-der taking unital hereditary C ∗ -subalgebras, the unital hereditary C ∗ -subalgebra B := p ( A { y } ⊗ M q ∞ ) p is also TA S ; this follows from Lemma 2.3 of [8]. The ideathen is to find a projection such that B will be nearly invariant under conjugationby u ⊗ M q ∞ and large enough for us to conclude that all of A ⊗ M q ∞ is TA S . Thisis achieved in Lemma 4.4.We impose no additional requirements on A { y } ⊗ M q ∞ in Lemma 4.5. Noticein particular, that by virtue of having the simple stably finite C ( X ) ⋊ α Z tensoredwith a UHF algebra, we do not require that the C ∗ -subalgebra A { y } has stable rankone, as is necessary for Lemma 4.2 of [20]; this follows from Corollary 6.6 of [27]. Inaddition, Lemma 4.2 of [20] requires the assumption that A { y } has real rank zero.This is used to pick out an initial small projection using Lemma 4.1 of [20]. Thisprojection is used to find a sequence of orthogonal projections, where conjugationby u ⊗ M q ∞ acts as a shift, which are then perturbed using Berg’s technique [1].When S is the set of finite dimensional C ∗ -algebras, then TA S is simply the classof C ∗ -algebras with tracial rank zero, in which case real rank zero is automaticby Theorem 3.4 of [15]. However if we are interested in the class of TAI algebras,for example, we can no longer assume this to be the case, as was shown in [17].Thus in an effort to allow the class S to be as general as possible, we remove theassumption that A ⊗ M q ∞ has real rank zero. Our Lemma 4.1 allows us to find aninitial small projection while avoiding the real rank zero restriction. We then finda projection p ∈ A { y } ⊗ M q ∞ satisfying the same conditions, (i) – (iii) of Lemma4.5. Here we do not get an initial sequence of projections all lying in A { y } ⊗ M q ∞ .However the projections lie close enough to this C ∗ -subalgebra that we are able topush them inside. After applying Berg’s technique, we once again end up outside A { y } ⊗ M q ∞ , but close enough to push the resulting loop inside, eventually endingup with the desired result. 4. Main Results
Lemma 4.1.
Let X be an infinite compact metric space and α : X → X a minimalhomeomorphism. Let y ∈ X , and set A { y } = C ∗ ( C ( X ) , uC ( X \ { y } )) , where u isthe unitary in C ( X ) ⋊ α Z implementing α . Let q ∈ N \ { } .Then for any η > and any open set V ⊂ X containing y , there exists an openset W ⊂ V with y ∈ W , functions g ∈ C ( W ) , g ∈ C ( V ) , ≤ g , g ≤ and aprojection q ∈ C ( V ) A { y } C ( V ) ⊗ M q ∞ such that g ( y ) = 1 , g | W = 1 , and k q ( g ⊗ − g ⊗ k ≤ η. Proof.
We claim that there is a non-zero projection in C ( V ) A { y } C ( V ) ⊗ M q ∞ .The set V is non-empty since y ∈ V . Thus C ( V ) is non-zero and hence wecan find a non-zero positive contraction in ( C ( V ) A { y } C ( V )) ⊗ M q ∞ , call it e .Since A { y } is simple by Proposition 2.5 of [20], so is A { y } ⊗ M q ∞ . Thus everytracial state τ ∈ T ( A { y } ⊗ M q ∞ ) is faithful, and in particular we have τ ( e ) > τ . Since A { y } ⊗ M q ∞ is unital, T ( A { y } ⊗ M q ∞ ) is compact. Thusmin τ ∈ T ( A { y } ⊗ M q ∞ ) τ ( e ) >
0. Furthemore, d τ ( e ) > τ ( e ) so the previous observationsimply that min τ ∈ T ( A { y } ⊗ M q ∞ ) d τ ( e ) > KAREN R. STRUNG AND WILHELM WINTER
Since A { y } ⊗ M q ∞ has projections that are arbitrarily small in trace, there is aprojection p ∈ A { y } ⊗ M q ∞ satisfyingmax τ ∈ T ( A { y } ⊗ M q ∞ ) τ ( p ) < min τ ∈ T ( A { y } ⊗ M q ∞ ) d τ ( e ) . By the above, for the projection p and any τ ∈ T ( A { y } ⊗ M q ∞ ) we have d τ ( p ) = τ ( p ) < d τ ( e ), so by strict comparison there are x n ∈ A { y } ⊗ M q ∞ with x n ex ∗ n → p .Let a n = e / x ∗ n x n e / ∈ ( C ( V ) A { y } C ( V )) ⊗ M q ∞ . Then a n is self-adjoint and k a n − a n k → . Disregarding any a n such that k a n − a n k ≥ /
4, we obtain a sequence of projections b n satisfying k b n − a n k ≤ k a n − a n k → n , a projection b = b n contained in C ( V ) A { y } C ( V ) ⊗ M q ∞ , provingthe claim. Moreover, b is Murray–von Neumann equivalent to p , so min τ τ ( b ) =min τ τ ( p ).Let W be an open set contained in V such that y ∈ W and small enough so thatfor every function f ∈ C ( W ) with 0 ≤ f ≤ d τ ( f ⊗ M q ∞ ) ≤ min τ τ ( b )for every τ ∈ T ( A { y } ⊗ M q ∞ ). Choose g , g ∈ C ( W ) such that 0 ≤ g , g ≤ g ( y ) = 1 and g g = g . Then d τ ( g ⊗ < d τ ( b ) for every τ ∈ T ( A { y } ⊗ M q ∞ ), andso by the comparison of positive elements we have ( g ⊗ . b in A { y } ⊗ M q ∞ andhence also in C ( V ) A { y } C ( V ) ⊗ M q ∞ . Since A { y } ⊗ M q ∞ has stable rank one and C ( V ) A { y } C ( V ) ⊗ M q ∞ is a full hereditary C ∗ -subalgebra of A { y } ⊗ M q ∞ , it also hasstable rank one by Theorem 3.6 of [26] with Theorem 2.8 of [3]. By Proposition 2.4 of[28], for η/ > v in ( C ( V ) A { y } C ( V ) ⊗ M q ∞ ) + (the unitizationof C ( V ) A { y } C ( V ) ⊗ M q ∞ ) such that ( g ⊗ − η/ + ≤ vbv ∗ in ( C ( V ) A { y } C ( V ) ⊗ M q ∞ ) + , and hence in C ( V ) A { y } C ( V ) ⊗ M q ∞ . Put q = vbv ∗ . Then k q ( g ⊗ − ( g ⊗ k < k q ( g ⊗ − η/ + − ( g ⊗ k + η/ k ( g ⊗ − η/ + − ( g ⊗ k + η/ < η. (cid:3) We will use the previous lemma to choose an initial projection in A { y } ⊗ M q ∞ .However, since this projection actually only approximates the properties we wouldlike it to have, we require the following easy lemma that pushes orthogonal projec-tions into a C ∗ -subalgebra. The proof is straightforward and hence omitted. Lemma 4.2.
Given ǫ > and a positive integer n , there is a δ > with thefollowing property. Let A be a C ∗ -algebra, B a C ∗ -subalgebra of A . Suppose that p , . . . , p n are mutually orthogonal projections in A , the first k , ≤ k ≤ n , of whichare contained in B, and that a k +1 , . . . , a n are self adjoint elements of B such that k p i − a i k < min(1 / , δ ) , i = k + 1 , . . . , n. Then there are mutuallly orthogonal projections q , . . . , q n in B , where q i = p i for ≤ i ≤ k , and for k + 1 ≤ i ≤ n we have k q i − p i k < ǫ. Moreover, if A is unital then there are unitaries u i ∈ A such that q i = u i p i u ∗ i . INIMAL DYNAMICS AND Z -STABLE CLASSIFICATION 7 Lemma 4.3.
Let X be an infinite compact metric space with a minimal homeo-morphism α : X → X . Let A = C ∗ ( X ) ⋊ α Z and A { y } = C ∗ ( C ( X ) , uC ( X \ { y } )) .Then K ( A { y } ⊗ M q ∞ ) ∼ = K ( A ⊗ M q ∞ ) as ordered groups, with the isomorphism induced by the inclusion ι : A { y } → A .Proof. Since the K -group of the UHF algebra M q ∞ is { } , we have K ∗ ( B ) ⊗ K ∗ ( M q ∞ ) = ( K ( B ) ⊗ K ( M q ∞ )) ⊕ ( K ( B ) ⊗ K ( M q ∞ ))where B denotes A or A { y } .Let ι : A { y } → A be the inclusion map. We have the associated inclusion map ι ⊗ id M q ∞ : A { y } ⊗ M q ∞ → A ⊗ M q ∞ whence we get the group homomorphisms K ( ι ) : K ( A { y } ) → K ( A ) and K ( ι ⊗ id M q ∞ ) : K ( A { y } ⊗ M q ∞ ) → K ( A ⊗ M q ∞ ).Since K ∗ ( M q ∞ ) is torsion free, by the K¨unneth Theorem for Tensor Products, wehave homomorphisms α : K ∗ ( A { y } ) ⊗ K ∗ ( M q ∞ ) → K ∗ ( A { y } ⊗ M q ∞ )and α : K ∗ ( A ) ⊗ K ∗ ( M q ∞ ) → K ∗ ( A ⊗ M q ∞ ) , which are of degree 0, thus giving maps of the K groups, which we will also call α and α . We get the following diagram0 / / K ( A { y } ) ⊗ K ( M q ∞ ) K ( ι ) ⊗ id K Mq ∞ ) (cid:15) (cid:15) α / / K ( A { y } ⊗ M q ∞ ) / / K ( ι ⊗ id Mq ∞ ) (cid:15) (cid:15) / / K ( A ) ⊗ K ( M q ∞ ) α / / K ( A ⊗ M q ∞ ) / / K ( ι ) ⊗ id K ( M q ∞ ) is an isomorphism. We conclude that K ( ι ⊗ id M q ∞ ) is also anisomorphism.Clearly K ( ι ⊗ id M q ∞ )([1 ⊗ M q ∞ ]) = [1 ⊗ M q ∞ ], and thus preserves order units.It remains to show K ( ι ⊗ id M q ∞ ) preserves the order structure and hence is anorder isomorphism.Let η ∈ K ( A ⊗ M q ∞ ) + . Then there are projections p and q in M ∞ ( A { y } ⊗ M q ∞ )such that K ( ι ⊗ id M q ∞ )([ p ] − [ q ]) = η . We show that [ p ] − [ q ] ∈ K ( A { y } ⊗ M q ∞ ) + .Let τ ′ be the unique tracial state on M q ∞ . Then, by Theorem 1.2 (4) of [21], for σ ∈ T ( A { y } ⊗ M q ∞ ), we have σ = ( τ ◦ ι ) ⊗ τ ′ for some τ ∈ T ( A ). By simplicity of A ⊗ M q ∞ , we have K ( τ ⊗ τ ′ )( η ) >
0. It follows that ( τ ⊗ τ ′ )( ι ⊗ id M q ∞ )( p − q ) > τ ◦ ι ) ⊗ ( τ ′ ◦ id M q ∞ )( p ) > ( τ ◦ ι ) ⊗ ( τ ′ ◦ id M q ∞ )( q ) and finally, σ ( p ) > σ ( q ).Since A { y } ⊗ M q ∞ has strict comparison, we must have q . p , so [ p ] − [ q ] >
0, asdesired. (cid:3)
The following lemma generalizes Lemma 4.2 of [20], due to H. Lin and N. C. Phillips.
Lemma 4.4.
Let X be an infinite compact metric space, α : X → X a minimalhomeomorphism, y ∈ X and q ∈ N \ { } . Let A = C ( X ) ⋊ α Z and A { y } = C ∗ ( C ( X ) , uC ( X \ { y } )) , where u is the unitary implementing α in A . Then,for any finite subset F ⊂ A ⊗ M q ∞ and every ǫ > , there is a projection p in A { y } ⊗ M q ∞ such that KAREN R. STRUNG AND WILHELM WINTER (i) k pa − ap k < ǫ for all a ∈ F , (ii) dist( pap, p ( A { y } ⊗ M q ∞ ) p ) < ǫ for all a ∈ F , (iii) τ (1 A ⊗ M q ∞ − p ) < ǫ for all τ ∈ T ( A ⊗ M q ∞ ) .Proof. Let ǫ >
0. We first show that there exists a projection satisfying properties(i) – (iii) of the lemma when F is assumed to be of the form F = ( G ⊗ { M q ∞ } ) ∪ { u ⊗ M q ∞ } where G is a finite subset of C ( X ).Let N ∈ N such that π/ (2 N ) < ǫ/ δ > δ < ǫ/ g ∈ G we have k g ( x ) − g ( x ) k < ǫ/ d ( x , x ) < δ .Choose δ > δ < δ and such that d ( α − n ( x ) , α − n ( x )) < δ whenever d ( x , x ) < δ and 0 ≤ n ≤ N .Since α is minimal, there is an N > N + 1 such that d ( α N ( y ) , y ) < δ. Let R ∈ N be sufficiently large so that R > ( N + N + 1) / min(1 , ǫ ) . Minimality of α also implies that there is an open neighbourhood U of y suchthat α − N ( U ) , α − N +1 ( U ) , . . . , U, α ( U ) , . . . , α R ( U )are all disjoint. Making U smaller if necessary, we may assume that each α n ( U ), − N ≤ n ≤ R has diameter less than δ . To apply Berg’s technique, we only need U n for − N ≤ n ≤ N , however we require R to be larger in order to satisfy property(iii) of the lemma.Let λ = max {k g k | g ∈ G} , and choose0 < ǫ < min(1 / , ǫ/ (2( N + 3 N + 1)) , ǫ/ (32 N ( λ + ǫ/ , < ǫ < min( ǫ / , δ ǫ ,N / < η < min(2 ǫ, δ ǫ ,N + N +1 )where δ ǫ ,N + N +1 is given by Lemma 4.2 with respect to ǫ and N + N + 1 in placeof ǫ and n , respectively; similarly for δ ǫ ,N .Let f : X → [0 ,
1] be continuous with supp( f ) ⊂ U , and f | V = 1 for someopen set V ⊂ U containing y .By Lemma 4.1, there is an open set W ⊂ V containing y , functions g ∈ C ( W ), g ∈ C ( V ), 0 ≤ g , g ≤ q ∈ C ( V ) A { y } C ( V ) ⊗ M q ∞ suchthat g ( y ) = 1 , g | W = 1 and k q ( g ⊗ − g ⊗ k < η/ . Consequently, ( f ⊗ q = q = q ( f ⊗
1) and k q ( g ⊗ − g ⊗ k < η/ − N ≤ n ≤ N , set q n = ( u n ⊗ q ( u − n ⊗ , f n = u n f u − n = f ◦ α − n and U n = α n ( U ) . INIMAL DYNAMICS AND Z -STABLE CLASSIFICATION 9 Then supp( f n ) ⊂ U n and( f n ⊗ q n = (( u n f u − n ) ⊗ u n ⊗ q ( u − n ⊗
1) = ( u n ⊗ f ⊗ q ( u − n ⊗
1) = q n . Similarly, q n ( f n ⊗
1) = q n . Since the f n have disjoint support, it follows that theprojections q − N , . . . , q − , q , q , . . . , q N are mutually orthogonal.We claim that q − N , . . . , q − , q ∈ A { y } ⊗ M q ∞ and that there are self-adjoint c , . . . , c N ∈ A { y } ⊗ M q ∞ such that k q n − c n k < η for 1 ≤ n ≤ N .Let 1 ≤ n ≤ N and consider q − n . We have ( uf − n ⊗ ∈ A { y } ⊗ M q ∞ for1 ≤ n ≤ N since U − n ∩ U = ∅ . Let a n = f n u n ⊗
1. Then a n = f n u n ⊗ uu − f u u − f u u − · · · u n u − n f u n ) ⊗
1= ( uf − ⊗ uf − ⊗ · · · ( uf − n ⊗ ∈ A { y } ⊗ M q ∞ . From this it follows that q − n = ( u − n ⊗ q ( u n ⊗ u − n ⊗ f n ⊗ q ( f n ⊗ u n ⊗ a ∗ n q a n ∈ A { y } ⊗ M q ∞ . Note that q ( g ⊗
1) = q ( g g ⊗ g | W = 1 and g ∈ C ( W ). Thus k ( g g ⊗ − q ( g ⊗ k = k ( g ⊗ − q ( g ⊗ g ⊗ k < η/ . Also, g f = g since f | V = 1 and g ∈ C ( V ). Hence k ( q − g ⊗ f ⊗ − g ⊗ − ( q − g ⊗ k = k q ( f ⊗ − q ( g ⊗ − ( g f ) ⊗ g g ) ⊗ − q + g ⊗ k = k ( g g ⊗ − q ( g ⊗ k < η/ . Since f ( y ) = 1 = g ( y ), we have that u ( f − g ) ⊗ ∈ A { y } ⊗ M q ∞ . Set c = ( u ( f − g ) ⊗ q − g ⊗ u ( f − g ) ⊗ ∗ + ( ug u ∗ ⊗ . Then c is a self-adjoint element in A { y } ⊗ M q ∞ and k q − c k = k ( u ⊗ q ( u ∗ ⊗ − ( u ( f − g ) ⊗ q − g ⊗ u ( f − g ) ⊗ ∗ − ( u ⊗ g ⊗ u ∗ ⊗ k = k q − (( f − g ) ⊗ q − g ⊗ f − g ) ⊗ − g ⊗ k≤ k ( q − g ⊗ − ( q − g ⊗ f − g ) ⊗ k + k ( q − g ⊗ f − g ) ⊗ − (( f − g ) ⊗ q − g ⊗ f − g ) ⊗ k≤ k ( q − g ⊗ − ( q − g ⊗ f − g ) ⊗ k + k ( q − g ⊗ − (( f − g ) ⊗ q − g ⊗ k k (( f − g ) ⊗ k < η. For 2 ≤ n ≤ N , define c n = (( uf n − · · · uf ) ⊗ c (( uf n − · · · uf ) ⊗ ∗ . The c n are self-adjoint elements in A { y } ⊗ M q ∞ since f n − , . . . , f all vanish at y .Furthermore, k q n − c n k = k ( u n ⊗ q ( u − n ⊗ − c n k = (cid:13)(cid:13) (( u n f n − ) ⊗ q (( f n − u − n ) ⊗ − c n (cid:13)(cid:13) = (cid:13)(cid:13) (( u n f n − u − ) ⊗ q (( uf n − u − n ) ⊗ − c n (cid:13)(cid:13) = k (( uf n − · · · uf ) ⊗ q (( uf n − · · · uf ) ⊗ ∗ − c n k≤ k q − c k < η. This proves the claim.We now apply Lemma 4.2 to obtain projections p , . . . , p N in A { y } ⊗ M q ∞ suchthat q − N , . . . , q − , q , p , . . . , p N are mutually orthogonal, and, for 1 ≤ n ≤ N , we have k p n − q n k < ǫ and unitaries y n such that p n = y n q n y ∗ n . Since p n ∼ q n and q n ∼ q , we have [ p N ] = [ q ] in K ( A ⊗ M q ∞ ). Since K ( ι ⊗ id M q ∞ ) : K ( A { y } ⊗ M q ∞ ) → K ( A ⊗ M q ∞ ) is an isomorphism by Lemma 4.3,we also have [ p N ] = [ q ] in K ( A { y } ⊗ M q ∞ ). Moreover, simplicity of A { y } ([20],Proposition 2.5) implies A { y } ⊗ M q ∞ has stable rank one by Corollary 6.6 of [27].Thus projections in matrix algebras over A { y } ⊗ M q ∞ satisfy cancellation, and thereis a partial isometry w ∈ A { y } ⊗ M q ∞ such that w ∗ w = q and ww ∗ = p N .For t ∈ R , set v ( t ) = cos( πt/ q + p N ) + sin( πt/ w − w ∗ ) . Then v ( t ) is a unitary in the corner ( q + p N )( A { y } ⊗ M q ∞ )( q + p N ). The matrixof v ( t ) with respect to the obvious decomposition is (cid:18) cos( πt/ − sin( πt/ πt/
2) cos( πt/ (cid:19) . For 0 ≤ k ≤ N , define w k = ( u − k ⊗ v ( k/N )( u k ⊗ . Also, let w ′ k = ( a k + b k ) ∗ v ( k/N )( a k + b k )where a k = ( f k u k ) ⊗ uf − · · · uf − k ) ⊗ b k = ( f kN u k ) ⊗ uf N − . . . uf N − k ) ⊗ . Both a k and b k are in A { y } ⊗ M q ∞ , hence w ′ k ∈ A { y } ⊗ M q ∞ . INIMAL DYNAMICS AND Z -STABLE CLASSIFICATION 11 We show what w k is close to w ′ k . Define x k = ( u − k ⊗ q + q N ) v ( k/N )( q + q N )( u k ⊗ . We have that k w k − x k k = k ( u − k ⊗ q + p N ) v ( k/N )( q + p N )( u k ⊗ − ( u − k ⊗ q + q N ) v ( k/N )( q + q N )( u k ⊗ k = k q v ( k/N ) p N − q v ( k/N ) q N + p N v ( k/N ) q − q N v ( k/N ) q + p N v ( k/N ) p N − q N v ( k/N ) q N k≤ k p N − q N k < ǫ . Also, k w ′ k − ( a k + b k ) ∗ ( q + q N ) v ( k/N )( q + q N )( a k + b k ) k≤ k a k + b k k k ( q + p N ) v ( k/N )( q + p N ) − ( q + q N ) v ( k/N )( q + q N ) k≤ k p N − q N k < ǫ . But ( a k + b k ) ∗ ( q + q N ) v ( k/N )( q + q N )( a k + b k )= ( f k u k + f kN u k ) ⊗ ∗ ( q + q N ) v ( k/N )( q + q N )( f k u k + f kN u k ) ⊗ u − k ⊗ q + q N ) v ( k/N )( q + q N )( u k ⊗ x k . Thus k w k − w ′ k k < ǫ . Comparing w k to w k +1 conjugated by u ⊗ k ( u ⊗ w k +1 ( u − ⊗ − w k k = k v (( k + 1) /N ) − v ( k/N ) k ≤ π/ (2 N ) < ǫ/ ≤ k < N . Define projections e = q , e n = p n for 1 ≤ n < N − N and e n = w N − n q n − N w ∗ N − n for N − N ≤ n ≤ N. Also define d n = q n for 0 ≤ n < N − N and d n = x N − n q n − N x ∗ N − n for N − N ≤ n ≤ N. Note that this gives d N = x q x ∗ = ( q + q N ) v (0) q v (0) ∗ ( q + q N ) = q = d andand e N = v (0) q v (0) ∗ = q = e . We also have that the x ∗ k x l = 0 when k = l . Thisfollows from the fact that, if k = l , then q − k , q N − k , q − l and q N − l , 0 ≤ k = l ≤ N ,are mutually orthogonal and( q + q N )( u k ⊗ u − l ⊗ q + q N )= ( u k ⊗ q − k + q N − k )( q − l + q N − l )( u − l ⊗ . Also, if 0 < m < N − N and N − N ≤ n ≤ N then q m ( u − ( N − n ) ⊗ q + q N ) = q m ( q − ( N − n ) + q n )( u − ( N − n ) ⊗
1) = 0 , and similarly ( q + q N )( u N − n ⊗ q m = 0 . From this it follows that d m d n = 0 for 0 ≤ m = n ≤ N .For 1 ≤ n ≤ N − N − k e n − d n k = k p n − q n k < ǫ and k e − d k = 0 . If N − N ≤ n ≤ N , then k e n − d n k = k w N − n q − ( N − n ) w ∗ N − n − x N − n q − ( N − n ) x ∗ N − n k = k w N − n q − ( N − n ) w ∗ N − n − w N − n q − ( N − n ) x ∗ N − n + w N − n q − ( N − n ) x ∗ N − n − x N − n q − ( N − n ) x ∗ N − n k≤ k w ∗ N − n − x ∗ N − n k + k w N − n − x N − n k < ǫ + 4 ǫ = 8 ǫ . We now show that conjugating the d n by u ⊗ ≤ n ≤ N − N − u ⊗ d n − ( u ⊗ ∗ = d n since d n = q n .If n = N − N , then d N − N = x N q − N x ∗ N = ( u − N ⊗ q + q N ) v (1)( q + q N ) q ( q + q N ) v ( − q + q N )( u N ⊗ u − N ⊗ q + q N ) p N ( q + q N )( u N ⊗ u − N ⊗ q N p N q N ( u N ⊗ . Thus k ( u ⊗ d N − N − ( u ∗ ⊗ − d N − N k = k q N − N − ( u − N ⊗ q N p N q N ( u N ⊗ k = k ( u − N ⊗ q N ( u N ⊗ − ( u − N ⊗ q N p N q N ( u N ⊗ k≤ k q N − p N k < ǫ . INIMAL DYNAMICS AND Z -STABLE CLASSIFICATION 13 When N − N < n ≤ N , first consider what happens to the e n using theestimation made on the w k above. We have k ( u ⊗ e n − ( u ∗ ⊗ − e n k = k ( u ⊗ w N − ( n − ( u ∗ ⊗ q n − N ( u ⊗ w ∗ N − ( n − ( u ∗ ⊗ − w N − n q n − N w ∗ N − n k≤ k ( u ⊗ w N − ( n − ( u ∗ ⊗ q n − N ( u ⊗ w ∗ N − ( n − ( u ∗ ⊗ − ( u ⊗ w N − ( n − ( u ∗ ⊗ q n − N w ∗ N − n k + k ( u ⊗ w N − ( n − ( u ∗ ⊗ q n − N w ∗ N − n − w N − n q n − N w ∗ N − n k≤ k ( u ⊗ w ∗ N − ( n − ( u ∗ ⊗ − w ∗ N − n k + k ( u ⊗ w N − ( n − ( u ∗ ⊗ − w N − n k < ǫ/ . From this we have k ( u ⊗ d n − ( u ∗ ⊗ − d n k < k e n − − d n − k + k e n − d n k + ǫ/ < ǫ + ǫ/ . Now we use the fact that the w k are almost in A { y } ⊗ M q ∞ to find projectionsin A { y } ⊗ M q ∞ that lie close to the e n and hence also close to the d n . When0 ≤ n ≤ N − N −
1, we have e n = p n ∈ A { y } ⊗ M q ∞ . Also, since e N = q , we onlyneed to find projections when N − N ≤ n ≤ N −
1. In this case, we have k e n − w ′ N − n q − ( N − n ) ( w ′ N − n ) ∗ k = k w N − n q − ( N − n ) w ∗ N − n − w ′ N − n q − ( N − n ) ( w ′ N − n ) ∗ k < ǫ . Since w ′ N − n q − ( N − n ) ( w ′ N − n ) ∗ ∈ A { y } ⊗ M q ∞ , by Lemma 4.2 we find orthogonalprojections r n ∈ A { y } ⊗ M q ∞ with k r n − e n k < ǫ and r n = z n e n z ∗ n for unitaries z n ∈ A ⊗ M q ∞ . This also implies that k r n − d n k < ǫ + 8 ǫ < ǫ . For 1 ≤ n ≤ N − N − r n = e n = p n and put r N = e N = q . Then set r = N X n =1 r n and p = 1 − r. We verify that the projection p ∈ A { y } ⊗ M q ∞ satisfies properties (i) – (iii) of thelemma.Let d = P Nn =1 d n . Note that d − ( u ⊗ d ( u ⊗ ∗ = N X n = N − N (( u ⊗ d n − ( u ⊗ ∗ − d n ) . For N − N ≤ m = n ≤ N , we have(( u ⊗ d n − ( u ⊗ ∗ − d n )(( u ⊗ d m − ( u ⊗ ∗ − d m )= ( u ⊗ d n − d m − ( u ⊗ ∗ − ( u ⊗ d n − ( u ⊗ ∗ d m − d n ( u ⊗ d m − ( u ⊗ ∗ + d n d m = − ( u ⊗ x N − ( n − q n − − N x ∗ N − ( n − ( u ⊗ ∗ x N − m q m − N x ∗ N − m − x N − n q n − N x ∗ N − n ( u ⊗ x N − ( m − q m − − N x ∗ N − ( m − ( u ⊗ ∗ = − ( u ⊗ x N − ( n − q n − − N ( u ⊗ ∗ x ∗ N − n x N − m q m − N x ∗ N − m − x N − n q n − N x ∗ N − n x N − m ( u ⊗ q m − − N x ∗ N − ( m − ( u ⊗ ∗ = 0 . Thus the terms in the sum are mutually orthogonal with norm at most 16 ǫ + ǫ/ k d − ( u ⊗ d ( u ⊗ ∗ k < ǫ + ǫ/ . Now k p − ( u ⊗ p ( u ⊗ ∗ k = k (( u ⊗ r ( u ∗ ⊗ − r ) − (( u ⊗ d ( u ∗ ⊗ − d ) + (( u ⊗ d ( u ∗ ⊗ − d ) k≤ k r − d k + 16 ǫ + ǫ/ < N − N − X n =1 k p n − q n k + N − X m = N − N k r m − d m k + 16 ǫ + ǫ/ < N − N − ǫ + 4 N ǫ + 16 ǫ + ǫ/ < ( N − N − ǫ + 4 N ǫ + 2 ǫ + ǫ/ < ǫ. Since g ( y ) = 1 it follows that u (1 − g ) ⊗ ∈ A { y } ⊗ M q ∞ . Thus we also havethat p ( u ⊗ − g ) ⊗ − q ) p ∈ A { y } ⊗ M q ∞ . Note that p ≤ − q . Usingthis and the fact that k g ⊗ − ( g ⊗ q k < η/ < ǫ , it follows that k p ( u ⊗ p − p ( u ⊗ − g ) ⊗ − q ) p k = k p ( u ⊗ p − p ( u ⊗ p + p ( u ⊗ g ⊗ − q ) p k≤ k p ( u ⊗ g ⊗ p − p ( u ⊗ g ⊗ q p k < ǫ. This proves (i) and (ii) for the element u ⊗ ∈ F .Now consider g ⊗ ∈ F , where g ∈ C ( X ).Since d ( α N ( y ) , y ) < δ , we have d ( α n ( y ) , α n − N ( y )) < δ for N − N ≤ n ≤ N . Itfollows that U n − N ∪ U n has diameter less than 2 δ + δ ≤ δ . The function g ∈ G varies by at most ǫ/ δ , and since the sets U , U , . . . , U N − N − , U N − N ∪ U − N , U N − N +1 ∪ U − N +1 , . . . , U N ∪ U are open and pairwise disjoint, there is ˜ g ∈ C ( X ) which is constant on each ofthese sets and satisfies k g − ˜ g k < ǫ/
4. Let the values of ˜ g on these sets be λ on U through to λ N on U N ∪ U . INIMAL DYNAMICS AND Z -STABLE CLASSIFICATION 15 For 0 ≤ n ≤ N − N − k ( f n ⊗ r n − r n k = k ( f n ⊗ r n − ( f n ⊗ q n + q n − r n k≤ k q n − p n k < ǫ . Thus k (˜ g ⊗ r n − λ n · r n k ≤ k (˜ g ⊗ r n − (˜ g ⊗ f n ⊗ r n k + k (˜ g ⊗ f n ⊗ r n − λ n · r n k < k ˜ g k ǫ . For N − N ≤ n ≤ N , we have that ( f n − N + f n ) x N − n = x N − n , since we may write x N − n = ( q n − N + q n )( u n − N ⊗ v (( N − n ) /N )( u N − n ⊗ q n − N + q n ). Similarly, x ∗ N − n ( f n − N + f n ) = x ∗ N − n . Thus ( f n − N + f n ) d n = d n = d n ( f n − N + f n ). It followsthat k ( f n − N + f n ) r n − r n k = k ( f n − N + f n ) r n − ( f n − N + f n ) d n + d n − r n k < ǫ . Thus, similar to the above, k (˜ g ⊗ r n − λ n · r n k < k ˜ g k ǫ .Hence k ( g ⊗ p − p ( g ⊗ k < k (˜ g ⊗ p − p (˜ g ⊗ k + ǫ/ ≤ N X n =1 k (˜ g ⊗ r n − λ n · r n + λ n · r n − r n (˜ g ⊗ k + ǫ/ ≤ N (8 k ˜ g k ǫ ) + ǫ/ < ǫ. This shows property (i) of the lemma for g ⊗ g ∈ G . The second condition isimmediate since g ⊗ A { y } ⊗ M q ∞ .It remains to verify the third condition.Since the sets α − N ( U ) , α − N +1 ( U ) , . . . , U, α ( U ) , . . . , α R ( U ) are all disjoint and R > ( N + N + 1) / min(1 , ǫ ), it follows that R X n = − N u n f u − n = f − N + · · · + f + · · · + f R ≤ τ ( f ) ≤ τ (1) / ( R + N + 1) < ǫ/ ( N + N + 1) for every τ ∈ T ( A ). Sinceany τ ∈ T ( A ⊗ M q ∞ ) is of the form τ = τ ⊗ τ for τ ∈ T ( A ) and τ the uniquetracial state on M q ∞ , we have τ ( q ) ≤ τ ( f ⊗
1) = τ ( f ) < ǫ/ ( N + N + 1). For1 ≤ n ≤ N − N − r n is just q conjugated by a unitary so τ ( r n ) = τ ( q ). For N − N ≤ n ≤ N , r n = z n e n z ∗ n = z n w N − n q − ( N − n ) w ∗ N − n z ∗ n . Thus τ ( r n ) = τ ( z n w N − n q − ( N − n ) w ∗ N − n z ∗ n )= τ ( w N − n q − ( N − n ) w ∗ N − n )= τ ( v (( N − n ) /N ) q v (( N − n ) /N ) ∗ )= τ (( q + p N ) q )= τ ( q ) . Thus τ (1 − p ) = N X n =1 τ ( r n ) = N X n =1 τ ( q ) < N ǫ/ ( N + N + 1) < ǫ. This proves the case where F is of the form ( G ⊗ { M q ∞ } ) ∪ { u ⊗ M q ∞ } .For the general case, let ˜ F ⊂ A ⊗ M q ∞ be a finite subset. Using the identification A ⊗ M q ∞ ∼ = A ⊗ M q r ⊗ M q ∞ ∼ = A ⊗ M q ∞ ⊗ M q r , for r ∈ N , we may assume that the finite set is of the form( { A } ⊗ { M q ∞ } ⊗ B ) ∪ ( ˜ G ⊗ { M q ∞ } ⊗ { M qr } ) ∪ ( { u } ⊗ { M q ∞ } ⊗ { M qr } )where r ∈ N , B is a finite subset of M q r and G is a finite subset of C ( X ).We may further assume that 1 X = 1 A ∈ G and also that 1 M qr ∈ B . Then F = ( G ⊗ { M q ∞ } ) ∪ { u ⊗ M q ∞ } and ˜ F = F ⊗ B .Let ǫ >
0. By the above, there exists a projection p ∈ A { y } ⊗ M q ∞ satisfyingproperties (i) – (iii) of the lemma for the finite set F = G ⊗ { M q ∞ } ∪ { u ⊗ M q ∞ } ,with ǫ/ max( {k b k | b ∈ B} ,
1) in place of ǫ .Define ˜ p := p ⊗ M qr ∈ A { y } ⊗ M q ∞ ⊗ M q r . We now show that ˜ p satisfiesproperties (i) – (iii) of the lemma for ˜ F and ǫ .Let ˜ a ∈ ˜ F . Then ˜ a = a ⊗ b for some a ∈ F and some b ∈ B . We have k ˜ p ˜ a − ˜ a ˜ p k = k ( p ⊗ a ⊗ b ) − ( a ⊗ b )( p ⊗ k = k ( pa ) ⊗ b − ( ap ) ⊗ b k = k ( pa − ap ) ⊗ b k = k pa − ap kk b k < ǫ. By the special case above, for every a ∈ F , there is some x ∈ p ( A { y } ⊗ M q ∞ ) p such that k pap − x k < ǫ/ (max b ∈B k b k ). Thus x ⊗ ∈ ˜ p ( A { y } ⊗ M q ∞ ⊗ M q r )˜ p . Itis clear that ˜ p (1 ⊗ b )˜ p ∈ ˜ p ( A { y } ⊗ M q ∞ ⊗ M q r )˜ p for any b ∈ B , and so x ⊗ b ∈ ˜ p ( A { y } ⊗ M q ∞ ⊗ M q r )˜ p . It follows that k ˜ p ( a ⊗ b )˜ p − ˜ p ( x ⊗ b )˜ p k = k ˜ p ( a ⊗ ⊗ b )˜ p − ˜ p ( x ⊗ ⊗ b )˜ p k = k ˜ p ( a ⊗ p (1 ⊗ b ) − ˜ p ( x ⊗ p (1 ⊗ b ) k = k ( pap − pxp ) ⊗ kk b k < ǫ. This shows that (i) and (ii) hold.
INIMAL DYNAMICS AND Z -STABLE CLASSIFICATION 17 To prove (iii), simply observe that τ ∈ T ( A ⊗ M q ∞ ⊗ M q r ) is of the form τ ⊗ τ where τ ∈ T ( A ⊗ M q ∞ ) and τ ∈ T ( M q r ). Then τ (1 − ˜ p ) = τ (1 ⊗ − p ⊗
1) = τ ((1 − p ) ⊗
1) = τ (1 − p ) τ (1) < ǫ. (cid:3) For the following lemma, we do not need any assumptions on the properties ofthe class S of separable unital C ∗ -algebras. However, for Theorem 4.6, we cutdown a TA S C ∗ -subalgebra B ⊂ A ⊗ M q ∞ by a projection; there we must make theadditional requirement that the property of being a member of S passes to unitalhereditary subalgebras. Lemma 4.5.
Let S be a class of separable unital C ∗ -algebras. Let A be a simpleunital C ∗ -algebra and q ∈ N \{ } . Suppose that for every finite subset F ⊂ A ⊗ M q ∞ ,every ǫ > , and every nonzero positive c ∈ A ⊗ M q ∞ , there exists a projection p ∈ A ⊗ M q ∞ and a simple unital C ∗ -subalgebra B ⊂ p ( A ⊗ M q ∞ ) p which is TA S ,satisfies B = p and (i) k pa − ap k < ǫ for all a ∈ F , (ii) dist( pap, B ) < ǫ for all a ∈ F , (iii) 1 A − p is Murray–von Neumann equivalent to a projection in c ( A ⊗ M q ∞ ) c .Then A ⊗ M q ∞ is TA S .Proof. Although a TA S C ∗ -algebra may not have property (SP), the C ∗ -algebra A ⊗ M q ∞ always will, since A ⊗ M q ∞ has strict comparison (cf. [28]) and con-tains nonzero projections which are arbitrarily small in trace. After noting this,the proof is essentially the same as that of Lemma 4.4 of [20], replacing the C ∗ -subalgebra of tracial rank zero with the TA S C ∗ -subalgebra B , and replacing thefinite dimensional C ∗ -subalgebra with a C ∗ -subalgebra from the class S . (cid:3) Theorem 4.6.
Let S be a class of separable unital C ∗ -algebras such that the prop-erty of being a member of S passes to unital hereditary C ∗ -subalgebras. Let X bean infinite compact metric space, α : X → X a minimal homeomorphism, let u bethe unitary implementing α in A := C ( X ) ⋊ α Z and q ∈ N \ { } . Suppose there isa y ∈ X such that A { y } ⊗ M q ∞ is TA S . Then A ⊗ M q ∞ is TA S .Proof. We show that A ⊗ M q ∞ satisfies the conditions of Lemma 4.5.Let ǫ > F a finite subset of A ⊗ M q ∞ and a positive nonzero element c in A ⊗ M q ∞ be given. Use Lemma 4.4 to find a projection p ∈ A { y } ⊗ M q ∞ with respectto F , c , and ǫ = min( ǫ, min τ ∈ T ( A ⊗ M q ∞ ) τ ( c )). Put B = p ( A { y } ⊗ M q ∞ ) p . It is aunital simple C ∗ -subalgebra of p ( A ⊗ M q ∞ ) p and is TA S by the assumptions madeon S and Lemma 2.3 of [8]. Conditions (i) and (ii) of Lemma 4.5 are satisfied by thechoice of p . Since τ (1 A − p ) < min σ ∈ T ( A ⊗ M q ∞ ) σ ( c ) < τ ( c ) for every tracial state τ ∈ T ( A ⊗ M q ∞ ), it follows from Theorem 5.2(a) of [28] that 1 A − p is Murray–vonNeumann equivalent to a projection in c ( A ⊗ M q ∞ ) c . Thus A ⊗ M q ∞ is TA S byLemma 4.5. (cid:3) Classification. Outlook.
Recall that for a separable simple unital stably finite nuclear C ∗ -algebra A , theElliott invariant of A is given by(( K ( A ) , K ( A ) + , [1 A ]) , K ( A ) , T ( A ) , r A : T ( A ) → S ( K ( A ))) consisting of the ordered K -groups, the Choquet simplex of tracial states T ( A ) and r A : T ( A ) → S ( K ( A )) , the canonical affine map to the state space of ( K ( A ) , K ( A ) + , [1 A ]) given by r A ( τ )([ p ]) = τ ( p ) [29]. Since we are interested in applying our results to Elliott’sclassification program, the most immediate application for Theorem 4.6 is when S is the set of finite dimensional C ∗ -algebras, where we are able to apply Lin’sclassification theorem for C ∗ -algebras of tracial rank zero. A simple unital C ∗ -algebra with tracial rank zero always has real rank zero ([15], Theorem 3.4). When A has real rank zero, the map r A is bijective, and as such the invariant becomesthe ordered K -theory [29]. Applying Theorem 4.6 to this special case, we have thefollowing classification result up to tensoring with the Jiang–Su algebra Z . Corollary 5.1.
Let A denote the class of C ∗ -algebras with the following properties. (i) A ∈ A is of the form C ( X ) ⋊ α Z for some infinite compact metric space X and minimal homeomorphism α . (ii) The projections in A separate T ( A ) .Let A, B ∈ A and suppose there is a graded order isomorphism φ : K ∗ ( A ⊗ Z ) → K ∗ ( B ⊗ Z ) . Then there is a ∗ -isomorphism Φ : A ⊗ Z → B ⊗ Z inducing φ .Proof. Let A = C ( X ) ⋊ α Z and B = C ( Y ) ⋊ β Z be in A and suppose φ : K ∗ ( A ⊗Z ) → K ∗ ( B ⊗ Z ) is a graded order isomorphism. Let q be any prime number, let x ∈ X and y ∈ Y ; define the C ∗ -subalgebras A { x } := C ∗ ( C ( X ) , uC ( X \ { x } )) and B { y } := C ∗ ( C ( Y ) , vC ( Y \ { y } )), where u and v are the unitaries implementing α and β in C ( X ) ⋊ α Z and C ( Y ) ⋊ β Z , respectively. It follows from Section 3 of[21] and Corollary 2.2 of [23] that both A { x } ⊗ M q ∞ and B { y } ⊗ M q ∞ have locallyfinite decomposition rank. By Lemma 4.3 above and Theorem 1.2 (4) of [21], the K -groups and tracial state spaces of A { x } ⊗ M q ∞ and A ⊗ M q ∞ (respectively B { y } ⊗ M q ∞ and B ⊗ M q ∞ ) are identical. But then our assumptions on the class A imply that A { x } ⊗ M q ∞ and B { y } ⊗ M q ∞ have projections separating tracialstates. Since A { x } ⊗ M q ∞ and B { y } ⊗ M q ∞ are approximately divisible (cf. [29]),we deduce they have real rank zero by the results of [2], whence tracial rank zeroby Theorem 2.1 of [36]. Applying Theorem 4.6 (with S being the class of finitedimensional C ∗ -algebras), A ⊗ M q ∞ and B ⊗ M q ∞ have tracial rank zero. Sincethis is true for any q , we may employ Theorem 5.4 of [19] to conclude that thereis a ∗ -isomorphism Φ : A ⊗ Z → B ⊗ Z inducing φ . Note that [19] also requires A and B to satisfy the UCT, which automatically holds in our situation. (cid:3) Most notably, this is the missing link between existing classification results andthe work of Toms and the second named author in [33]. There it is shown thatif X has finite covering dimension, then the resulting crossed product is Z -stable([33], Theorem 4.4); in this case Corollary 5.1 becomes Theorem 0.1 of [33]; seealso Theorem A of [34]. In particular this solves the classification problem for C ∗ -algebras associated to uniquely ergodic minimal finite dimensional dynamicalsystems.A compelling set of examples are the C ∗ -algebras arising from minimal homeo-morphisms of odd spheres S n for n ≥ n = 3 and α a minimal diffeomorphism. It follows from Corollary 3of Section 5 of [4] that the crossed product of C ( S n ) by Z induced by a minimaldiffeomorphism has no nontrivial projections. The K -group of such a C ∗ -algebra INIMAL DYNAMICS AND Z -STABLE CLASSIFICATION 19 A is shown in Example 4.6 of [24] to be K ( A ) = Z with order given by ( m, n ) ≥ n > m, n ) = (0 , Z -stability. The main result of [33] (which in turn makesheavy use of [38]) then shows that Z -stability is automatic, so in the uniquely er-godic case Connes’ odd spheres are all isomorphic. The respective statement in thesmooth case was already derived in [38], using the inductive limit decomposition of[22].To use Corollary 5.1 to classify crossed products with more general tracial statespaces, at the current stage only Lin’s classification of C ∗ -algebras which are TAIafter tensoring with UHF algebras is available [18]. The algebras covered by thisare all rationally Riesz, i.e., their ordered K -groups become Riesz groups aftertensoring with the K -theory of a UHF algebra. However, for general transformationgroup C ∗ -algebras it is not clear when they are rationally Riesz, and one cannothope for TAI classification to be a sufficient tool in this case. The strategy wouldthen be to identify a suitable class S of unital C ∗ -algebras such that A { y } ⊗ M q ∞ can be verified to be TA S and such that TA S algebras can be classified in a similarmanner as TAF or TAI algebras. At least for odd spheres it is not hard to see thatthey are rationally Riesz, so it only remains to verify that they are indeed TAI upto tensoring with UHF algebras; the result would then be that they are entirelydetermined by their tracial state spaces, by virtue of [19] and our Corollary 5.1. Atleast in the case of finitely many ergodic measures we are confident that our resultswill lead to a complete solution; this will be pursued in a subsequent paper.We wish to point out that Corollary 5.1 does not require any condition on thedimension of the underlying space and that, without such a condition, classificationup to Z -stability is probably the best for which one can hope. The examples con-structed by Giol and Kerr in [9] suggest that counterexamples to the general case ofthe Elliott conjecture as exhibited by Toms in [31] can also occur as transformationgroup C ∗ -algebras. One would still expect classification up to Z -stability in thissetting, which would then also imply that at least the crossed products stabilizedby Z have finite topological dimension. Conversely, if the underlying space is fi-nite dimensional, then Z -stability is automatic by [33]. In this sense, our result isanalogous to [36], which in the real rank zero case also provided classification upto Z -stability under otherwise mild structural conditions. Acknowledgments
We would like to thank the referee for a number of helpful comments.
Supported by:
EPSRC First Grant EP/G014019/1.
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