The structure of crossed products by automorphisms of C(X,D)
aa r X i v : . [ m a t h . OA ] S e p THE STRUCTURE OF CROSSED PRODUCTS BYAUTOMORPHISMS OF C ( X, D ) DAWN ARCHEY, JULIAN BUCK, AND N. CHRISTOPHER PHILLIPS
Abstract.
We construct centrally large subalgebras in crossed products ofthe form C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) in which D is simple, X is compact metrizable, α induces a minimal homeomorphism h : X → X , and a mild technical as-sumption holds. We use this construction to prove structural properties ofthe crossed product, such as (tracial) Z -stability, stable rank one, real rankzero, and pure infiniteness, in a number of examples. Our examples are notaccessible via methods based on finite Rokhlin dimension, either because D isnot Z -stable or because X is infinite dimensional. Introduction
Significant progress has been made in recent years on the classification of crossedproduct C*-algebras arising from finite dimensional minimal dynamical systems.The long unpublished preprint [42] of Q. Lin and N. C. Phillips (see also the surveyarticles [40] and [41]) provides a thorough description of the transformation groupC*-algebras arising from minimal diffeomorphisms of finite dimensional smoothcompact manifolds in terms of a direct limit decomposition. In [38] and [62], itis shown that crossed products arising from minimal homeomorphisms of infinitecompact metrizable spaces with finite covering dimension are classified by their or-dered K-theory in the presence of sufficiently many projections (for instance, whenprojections separate traces). In [62] it is further proved that crossed products bysuch minimal homeomorphisms have finite nuclear dimension, and hence absorbthe Jiang-Su algebra Z tensorially (that is, are Z -stable). Finally, G. A. Elliottand Z. Niu ([17]) have shown that crossed products by minimal homeomorphismsof compact metric spaces with mean dimension zero (including all minimal homeo-morphisms of finite dimensional compact metric spaces) are Z -stable, from whichit follows that they are classifiable in the sense of the Elliott program by CorollaryD of Theorem A of [8].Not as much is known for crossed products of C*-algebras of the form C ( X, D ) fora noncommutative C*-algebra D . Hua ([25]) has shown that such crossed productshave tracial rank zero when X is the Cantor set, D has tracial rank zero, the actionon X is minimal, and some additional K-theoretic assumptions are made. In thispaper we consider the structure of crossed products of the form C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) ,in which X is a compact metric space, D is simple unital a C*-algebra, and α is Date : 25 September 2020.2010
Mathematics Subject Classification.
Primary 46L40, 46L55; Secondary 46L36.This material is based upon work supported by the US National Science Foundation un-der Grant DMS-1101742 and the Simons Foundation Collaboration Grant for Mathematicians an automorphism of C ( X, D ) which “lies over” a minimal homeomorphism (as de-scribed in Definition 1.2). In Section 1, we describe the types of actions on C ( X, D )which will be of interest here. In Section 2 we introduce the generalization of theorbit breaking subalgebras of [52] for actions on C ( X, D ), and show that these arelarge in various senses defined in [52]. Section 3 introduces a “ D -fibered” generaliza-tion of the recursive subhomogeneous algebras in [51], then demonstrates (followinga development analogous to that in [42]) that our orbit breaking subalgebras havesuch a recursive structure. In Section 4 we use the results of Sections 2 and 3 toobtain stronger structural properties for the orbit breaking subalgebras, and de-duce structural properties of the crossed product from those of the orbit breakingsubalgebra under appropriate additional assumptions. In Section 5 we establishminimality for products of certain Denjoy homeomorphisms that will be used toproduce examples. Section 6 gives a large collection of examples of crossed productsfor which we can use the theory developed here to deduce structural properties thatdo not seem accessible using previously known methods. Finally, in Section 7 wepose some open questions for further research.We recall Cuntz comparison and the Cuntz semigroup. For a much fuller dis-cussion, in a form useful for work with large subalgebras, we refer to Section 1of [52]. Notation 0.1. If A is a C*-algebra and a, b ∈ M ∞ ( A ) + , we write a - A b to meanthat a is Cuntz subequivalent to b over A , that is, there is a sequence ( v n ) ∞ n =1 in M ∞ ( A ) + such that lim n →∞ v n bv ∗ n = a . We write a ∼ A b to mean that a is Cuntzequivalent to b over A , that is, a - A b and b - A a .We specify A in the notation because Cuntz subequivalence with respect toproper subalgebras will play a key role.The Cuntz semigroup W ( A ) is then defined to be the set of Cuntz equivalenceclasses M ∞ ( A ) + / ∼ A , with addition given by direct sum and order coming fromCuntz subequivalence. Notation 0.2.
For any C*-algebra A , we denote the set of normalized 2-quasitraceson A by QT( A ). We use the word quasitrace to mean normalized 2-quasitrace.For a C*-algebra A , the topology on Aut( A ) is always pointwise convergencein the norm of A . That is, x α x is continuous if and only if x α x ( a ) iscontinuous for all a ∈ A . (This is the usual topology.) To be explicit, we point outthat α α − is continuous in this topology, as can be seen from the equation k α − ( a ) − β − ( a ) k = (cid:13)(cid:13) β − (cid:0) α ( α − ( a )) − β ( af − ( a )) (cid:1)(cid:13)(cid:13) for a ∈ A and α, β ∈ Aut( A ), and the fact that β − is isometric. Notation 0.3.
For a compact Hausdorff space X , we denote the covering dimensionof X (Definition 3.1.1 of [53]) by dim( X ). If h : X → X is a homeomorphism, itsmean dimension (Definition 2.6 of [43]) is denoted by mdim( h ).1. Preliminaries on actions on C ( X, D ) lying over actions on X In the section, we give a few basic facts about actions of groups on C*-algebrasof the form C ( X, D ) which “lie over” actions on X . We also introduce severaltechnical conditions which will be needed as hypotheses later, and give some casesin which they are automatically satisfied. ROSSED PRODUCTS OF C ( X, D ) 3
Notation 1.1.
Let G be a locally compact group and let X be a locally compactHausdorff space X on which G acts. We take the corresponding action α : G → Aut( C ( X )) to be be given by α g ( f )( x ) = f ( g − x ) for f ∈ C ( X ), g ∈ G , and x ∈ X .For a homeomorphism h : X → X , this means that the corresponding action of Z on C ( X ) is generated by the automorphism α ( f ) = f ◦ h − for f ∈ C ( X ). Definition 1.2.
Let X be a locally compact Hausdorff space, let G be a topologicalgroup, and let D be a C*-algebra. Let ( g, x ) gx be an action of G on X , and let α : G → Aut( C ( X, D )) be an action of G on C ( X, D ). We say that α lies over theaction ( g, x ) gx if there exists a function ( g, x ) α g,x from G × X to Aut( D )such that α g ( a )( x ) = α g,x ( a ( g − x )) for all g ∈ G , x ∈ X , and a ∈ C ( X, D ).We say that an automorphism α of C ( X, D ) lies over a homeomorphism h : X → X if the action generated by α lies over the action generated by h .In Definition 1.2, for g ∈ G and d ∈ D , the function x α g,x ( d ) must becontinuous. The following elementary lemma, which will be used without comment,shows that if G is discrete then this is the only continuity condition that is needed.If G is not discrete, there are additional continuity conditions. Lemma 1.3.
Let D be a C*-algebra, let X be a locally compact Hausdorff space,and let x α x be a continuous function from X to Aut( D ). The for every a ∈ C ( X, D ), the function b ( x ) = α x ( a ( x )) is also in C ( X, D ). Proof.
It is immediate that b vanishes at infinity. For continuity, let x ∈ X andlet ε >
0. Choose an open set U ⊂ X such that x ∈ U and for all x ∈ U we have k a ( x ) − a ( x ) k < ε k α x ( a ( x )) − α x ( a ( x )) k < ε . Then, using k α x k = 1 for all x ∈ X , one sees that x ∈ U implies k α x ( a ( x )) − α x ( a ( x )) k < ε . (cid:3) For any group G , there are also algebraic conditions relating the automorphisms α g,x , coming from the requirement that g α g be a group homomorphism. If G = Z , then we really need only the function x α ,x . For reference, we give therelevant statement as a lemma. Lemma 1.4.
Let X be a locally compact Hausdorff space, let h : X → X be ahomeomorphism, let D be a C*-algebra. Then there is a one to one correspondencebetween actions of Z on C ( X, D ) that lie over h and continuous functions from X to Aut( D ), given as follows.For any function x α x from X to Aut( D ) such that x α x ( d ) is continuousfor all d ∈ D , there is an automorphism α ∈ Aut( C ( X, D )) given by α ( a )( x ) = α x ( a ( h − ( x )) for all a ∈ C ( X, D ) and x ∈ X , and this automorphism lies over h .Conversely, if α ∈ Aut( C ( X, D )) lies over h , then there is a function x α x from X to Aut( D ) such that x α x ( d ) is continuous for all d ∈ D and such that α ( a )( x ) = α x ( a ( h − ( x )) for all a ∈ C ( X, D ) and x ∈ X . Proof.
Using Lemma 1.3, this is immediate. (cid:3)
There is a conflict in the notation in Lemma 1.4: if n ∈ Z then α n is one ofthe automorphisms in the action on C ( X, D ) (namely α n ), while if x ∈ X then α x ∈ Aut( D ). We use this notation anyway to avoid having more letters. Todistinguish the two uses, take α to be the action and write x α x when thefunction from X to Aut( D ) is intended. DAWN ARCHEY, JULIAN BUCK, AND N. CHRISTOPHER PHILLIPS If D is prime, then every action on C ( X, D ) lies over an action of G on X . Lemma 1.5.
Let X be a locally compact Hausdorff space, let D be a prime C*-algebra, let G be a topological group, and let α : G → Aut( C ( X, D )) be an actionof G on C ( X, D ). Then there exists an action of G on X such that α lies over thisaction. Proof.
The action of G on X is obtained from the identification X ∼ = Prim( C ( X, D )). (cid:3)
Proposition 1.6.
Let G be a discrete group, let X be a compact space, andsuppose G acts on X in such a way that the action is minimal and for every finiteset S ⊂ G \ { } , the set (cid:8) x ∈ X : gx = x for all g ∈ S (cid:9) is dense in X . Let D be a simple unital C*-algebra, and let α : G → Aut( C ( X, D ))be an action of G on C ( X, D ) which lies over the given action of G on X (in thesense of Definition 1.2). Then C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) is simple. Proof.
For any C*-algebra A , let b A be the space of unitary equivalence classes ofirreducible representations of A , with the hull-kernel topology. Since the primitiveideals of C ( X, D ) are exactly the kernels of the point evaluations, there is an obviousmap q : C ( X, D ) ∧ → X , and the open sets in C ( X, D ) ∧ are exactly the sets q − ( U )for open sets U ⊂ X . It is now immediate that for every finite set S ⊂ G \ { } , theset (cid:8) x ∈ C ( X, D ) ∧ : gx = x for all g ∈ S (cid:9) is dense in C ( X, D ) ∧ . That is, the action of G on C ( X, D ) ∧ is topologically free inthe sense of Definition 1 of [2].Let J ⊂ C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) be a nonzero ideal. Let π : C ∗ (cid:0) G, C ( X, D ) , α (cid:1) → C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) be the quotient map. Theorem 1 of [2] implies that π − ( J ) has nonzero intersectionwith the canonical copy of C ( X, D ) in C ∗ (cid:0) G, C ( X, D ) , α (cid:1) . Therefore J has nonzerointersection with the canonical copy of C ( X, D ) in C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) . Since D issimple, this intersection has the form C ( U, D ) for some nonempty open set U ⊂ X .Since the action of G on X is minimal and X is compact, there exist n ∈ Z > and g , g , . . . , g n ∈ Z > such that the sets g − U, g − U, . . . , g − n U cover X . Choose f , f , . . . , f n ∈ C ( X ) ⊂ C ( X, D ) ⊂ C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) such that supp( f k ) ⊂ g − k U for k = 1 , , . . . , n and P nk =1 f k = 1. For k = 1 , , . . . , n ,the functions α g k ( f k ) are in C ( U ) ⊂ C ( U, D ) ⊂ J , so1 = n X k =1 f k = n X k =1 u ∗ g k α g k ( f k ) u g k ∈ J. So J = C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) . (cid:3) Proposition 1.7.
Assume the hypotheses of Proposition 1.6, and in addition as-sume that G is amenable and D has a tracial state. Then C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) hasa tracial state and is stably finite. ROSSED PRODUCTS OF C ( X, D ) 5
Proof.
Since C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) is simple by Proposition 1.6, it suffices to showthat C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) has a tracial state. We know that the tracial state spaceT( C ( X, D )) is nonempty, since one can compose a tracial state on D with a pointevaluation C ( X, D ) → D . Since G is amenable, combining Theorem 2.2.1 and3.3.1 of [22] shows that C ( X, D ) has a G -invariant tracial state τ . Standardmethods show that the composition of τ with the conditional expectation from C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) to C ( X, D ) is a tracial state on C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) . (cid:3) The following condition is a technical hypothesis which we need for the proof ofthe main large subalgebra result (Theorem 2.9).
Definition 1.8.
Let D be a C*-algebra, and let S ⊂ Aut( D ). We say that S is pseudoperiodic if for every a ∈ D + \ { } there is b ∈ D + \ { } such that for every α ∈ S ∪ { id D } we have b - α ( a ).The interpretation of pseudoperiodicity is roughly as follows. Suppose S ⊂ Aut( D ) is pseudoperidic. Then there is no sequence ( α n ) n ∈ Z > in S for which thereis a nonzero element η ∈ W ( D ) such that the sequence ( α n ( η )) n ∈ Z > becomesarbitrarily small in W ( D ) in a heuristic sense.We give some conditions which imply pseudoperiodicity. Lemma 1.9.
Let D be a unital C*-algebra. Then the set of approximately innerautomorphisms of D is pseudoperiodic in the sense of Definition 1.8. Proof.
Let a ∈ D + \ { } . It suffices to prove that a - α ( a ) for every approximatelyinner automorphism α ∈ Aut( D ). To see this, let ε > u ∈ D such that k uα ( a ) u ∗ − a k < ε . (cid:3) Lemma 1.10.
Let D be a simple C*-algebra. Let S ⊂ Aut( D ) be a subset whichis compact in the topology of pointwise convergence in the norm on D . Then S ispseudoperiodic in the sense of Definition 1.8. Proof.
Let a ∈ D + \ { } . Without loss of generality k a k = 1. For β ∈ S ∪ { id D } set U β = (cid:26) α ∈ S ∪ { id D } : k α ( a ) − β ( a ) k < (cid:27) . By compactness, there are β , β , . . . , β n ∈ S ∪ { id D } such that U β , U β , . . . , U β n cover S ∪ { id D } . Since (cid:0) β ( a ) − (cid:1) + = 0 for all β ∈ Aut( D ), by Lemma 2.6 of [52]there is b ∈ D + \ { } such that b - (cid:0) β k ( a ) − (cid:1) + for j = 1 , , . . . , n . Let α ∈ S ∪ { id D } . Choose k ∈ { , , . . . , n } such that α ∈ U β k . Then k β k ( a ) − α ( a ) k < ,so b - (cid:0) β k ( a ) − (cid:1) + - α ( a ). (cid:3) The following result will not be used, since large subalgebras are not used in ourproofs when D is purely infinite. It is included as a further example of pseudope-riodicity. Lemma 1.11.
Let D be a purely infinite simple C*-algebra. Then Aut( D ) ispseudoperiodic in the sense of Definition 1.8. Proof.
Let a ∈ D + \ { } . Then a - b for all b ∈ D + \ { } . In particular, a - α ( a )for all α ∈ Aut( D ). (cid:3) Lemma 1.12.
Let D be a simple unital C*-algebra which has strict comparisonof positive elements. Then Aut( D ) is pseudoperiodic in the sense of Definition 1.8. DAWN ARCHEY, JULIAN BUCK, AND N. CHRISTOPHER PHILLIPS
Proof. If D is finite dimensional, the conclusion is immediate. Otherwise, let a ∈ D + \{ } . Without loss of generality k a k ≤
1. Then τ ( a ) ≤ d τ ( a ) for all τ ∈ QT( D ).Moreover, since QT( D ) is compact and D is simple, the number δ = inf τ ∈ QT( D ) τ ( a )satisfies δ >
0. Use Corollary 2.5 of [52] to find b ∈ D + \ { } such that d τ ( h b i ) < δ for all τ ∈ QT( D ). Then for every τ ∈ QT( D ), using τ ◦ α ∈ QT( D ) at the secondstep, we have d τ ( b ) < δ ≤ ( τ ◦ α )( a ) ≤ d τ ( α ( a )) . The strict comparison hypothesis therefore implies that b - A α ( a ). (cid:3) Definition 1.13.
Let A be a C*-algebra. We say that the order on projectionsover A is determined by quasitraces if whenever p, q ∈ M ∞ ( A ) are projections suchthat τ ( p ) < τ ( q ) for all τ ∈ QT( A ), then p - q . Lemma 1.14.
Let D be a simple unital C*-algebra with Property (SP) and suchthat the order on projections over A is determined by quasitraces. Then Aut( D ) ispseudoperiodic in the sense of Definition 1.8. Proof. If D is finite dimensional, the conclusion is immediate. Otherwise, let a ∈ D + \ { } . Choose a nonzero projection p ∈ aDa . Since QT( D ) is compact and D issimple, the number δ = inf τ ∈ QT( D ) τ ( p ) satisfies δ >
0. Choose n ∈ Z > such that n < δ . Use Lemma 2.3 of [52] to choose a unitary u ∈ A and a nonzero positiveelement b ∈ A such that the elements b, ubu − , u bu − , . . . , u n bu − n are pairwise orthogonal. Choose a nonzero projection q ∈ bDb . Then for every τ ∈ QT( D ), using τ ◦ α ∈ QT( D ) at the third step, we have τ ( q ) ≤ n + 1 < δ ≤ ( τ ◦ α )( p ) . The strict comparison hypothesis therefore implies that q is Murray-von Neumannequivalent to a subprojection of α ( p ). It follows that q - A α ( a ). (cid:3) The following definition is intended only for convenience in this paper. (Thecondition occurs several times as a hypothesis, and is awkward to state.)
Definition 1.15.
Let X be a locally compact Hausdorff space, let G be a topolog-ical group, and let D be a C*-algebra. Let α : G → Aut( C ( X, D )) be an action of G on C ( X, D ) which lies over an action of G on X , and let ( g, x ) α g,x ∈ Aut( D )be as in Definition 1.2. We say that α is pseudoperiodically generated if (cid:8) α g,x : g ∈ G and x ∈ X (cid:9) is pseudoperiodic in the sense of Definition 1.8. Lemma 1.16.
Let X be a compact metric space, let h : X → X be a homeo-morphism, let D be a simple unital C*-algebra, and let α ∈ Aut( C ( X, D )) lieover h . As in Lemma 1.4, let ( α x ) x ∈ X be the family in Aut( D ) such that α ( a )( x ) = α x ( a ( h − ( x )) for all a ∈ C ( X, D ) and x ∈ X . Suppose that the subgroup H of Aut( D ) generated by { α x : x ∈ X } is pseudoperiodic. Then the action of Z generated by α is pseudoperiodically generated. Proof.
One checks that if ( n, x ) α n,x ∈ Aut( D ) is determined (following thenotation of Definition 1.2) by α n ( a )( x ) = α n,x ( a ( h − n ( x )), then α n,x ∈ H for all n ∈ Z and x ∈ X . (cid:3) ROSSED PRODUCTS OF C ( X, D ) 7 The orbit breaking subalgebra for a nonempty set meeting eachorbit at most once
Let h : X → X be a homeomorphism of a compact Hausdorff space X , and let D be a simple unital C*-algebra. For Y ⊂ X closed, following Putnam [55], inDefinition 7.3 of [52] we defined the Y -orbit breaking subalgebra C ∗ ( Z , X, h ) Y ⊂ C ∗ ( Z , X, h ). Here, for an automorphism α ∈ Aut( C ( X, D )) which lies over h we define C ∗ ( Z , C ( X, D ) , α ) Y . We prove that if X is infinite, h is minimal, Y intersects each orbit at most once, and an additional technical condition is satisfied(namely, that the action of Z generated by α is pseudoperiodically generated), then C ∗ ( Z , C ( X, D ) , α ) Y is a large subalgebra of C ∗ ( Z , C ( X, D ) , α ) of crossed producttype, in the sense of Definition 4.9 of [52]. This is a generalization of Theorem 7.10of [52]. Notation 2.1.
Let G be a discrete group, let A be a C*-algebra, and let α : G → Aut( A ) be an action of G on A . We identify A with a subalgebra of C ∗ r ( G, A, α )in the standard way. We let u g ∈ M ( C ∗ r ( G, A, α )) be the standard unitary corre-sponding to g ∈ G . When G = Z , we write just u for the unitary u correspondingto the generator 1 ∈ Z . We let A [ G ] denote the dense *-subalgebra of C ∗ r ( G, A, α )consisting of sums P g ∈ S a g u g with S ⊂ G finite and a g ∈ A for g ∈ S . We mayalways assume 1 ∈ S . We let E α : C ∗ r ( G, A, α ) → A denote the standard conditionalexpectation, defined on A [ G ] by E α (cid:16)P g ∈ S a g u g (cid:17) = a . When α is understood, wejust write E .When G acts on a compact Hausdorff space X , we use obvious analogs of thisnotation for C ∗ r ( G, X ), with the action as in Notation 1.1. In particular, if G = Z and the action of generated by a homeomorphism h : X → X , we have uf u ∗ = f ◦ h − . Notation 2.2.
For a locally compact Hausdorff space X and a C*-algebra D , weidentify C ( X, D ) = C ( X ) ⊗ D in the standard way. For an open subset U ⊂ X ,we use the abbreviation C ( U, D ) = (cid:8) a ∈ C ( X, D ) : a ( x ) = 0 for all x ∈ X \ U (cid:9) ⊂ C ( X, D ) . This subalgebra is of course canonically isomorphic to the usual algebra C ( U, D )when U is considered as a locally compact Hausdorff space in its own right.In particular, if Y ⊂ X is closed, then(2.1) C ( X \ Y, D ) = (cid:8) a ∈ C ( X, D ) : a ( x ) = 0 for all x ∈ Y (cid:9) . The following definition is the analog of Definition 7.3 of [52].
Definition 2.3.
Let X be a locally compact Hausdorff space, let h : X → X be a homeomorphism, let D be a C*-algebra, and let α ∈ Aut( C ( X, D )) be anautomorphism which lies over h . Let Y ⊂ X be a nonempty closed subset, and,following (2.1), define C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y = C ∗ (cid:0) C ( X, D ) , C ( X \ Y, D ) u (cid:1) ⊂ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) . We call it the Y -orbit breaking subalgebra of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) .We show that if Y intersects each orbit of h at most once, and the action of Z generated by α is pseudoperiodically generated, then C ∗ ( Z , C ( X, D ) , α ) Y is alarge subalgebra of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) of crossed product type. DAWN ARCHEY, JULIAN BUCK, AND N. CHRISTOPHER PHILLIPS
The following lemma is the analog of Proposition 7.5 of [52].
Lemma 2.4.
Let X be a compact Hausdorff space, let h : X → X be a home-omorphism, let D be a unital C*-algebra, and let α ∈ Aut( C ( X, D )) lie over h .Let u ∈ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) and E α : C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) → C ( X, D )be as in Notation 2.1. Let Y ⊂ X be a nonempty closed subset. For n ∈ Z , set Y n = S n − j =0 h j ( Y ) n > ∅ n = 0 S − nj =1 h − j ( Y ) n < . Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y (2.2) = (cid:8) a ∈ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) : E α ( au − n ) ∈ C ( X \ Y n , D ) for all n ∈ Z (cid:9) and(2.3) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y ∩ C ( X, D )[ Z ] = C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y . Proof.
By Lemma 1.4, there exists a function x α x from X to Aut( D ) such that x α x ( d ) is continuous for all d ∈ D and which satisfies α ( a )( x ) = α x ( a ( h − ( x ))for all a ∈ C ( X, D ) and x ∈ X .Most of the proof of Proposition 7.5 of [52] goes through with only the obviouschanges. In analogy with that proof, define B = (cid:8) a ∈ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) : E α ( au − n ) ∈ C ( X \ Y n , D ) for all n ∈ Z (cid:9) and B = B ∩ C ( X )[ Z ] . Then B is dense in B by the same reasoning as in [52] (using Ces`aro means andTheorem VIII.2.2 of [13]).The proof of Proposition 7.5 of [52] shows that when 0 ≤ m ≤ n and also when0 ≥ m ≥ n , we have Y m ⊂ Y n , that for n ∈ Z , we have(2.4) h − n ( Y n ) = Y − n , and that for m, n ∈ Z , we have Y m + n ⊂ Y m ∪ h m ( Y n ). It then follows, as in [52],that B is a *-algebra, and that C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y ⊂ B = B .We next claim that for all n ∈ Z and f ∈ C ( X \ Y n , D ), we have f u n ∈ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y . The changes to the proof of Proposition 7.5 of [52] at thispoint are more substantial.For n = 0 the claim is trivial. Let n > f ∈ C ( X \ Y n , D ). Define b = ( f ∗ f ) / (2 n ) . Let s ∈ C ( X \ Y n , D ) ′′ be the partial isometry in the polardecomposition of f , so that f = s ( f ∗ f ) / = sb n . It follows from Proposition 1.3of [10] that the element a = sb is in C ( X \ Y n , D ). Moreover, a ( f ∗ f ) − n = f .Define a ∈ C ( X, D ) by a ( x ) = α − h ( x ) (cid:0) b ( h ( x )) (cid:1) for x ∈ X , and for k = 1 , , . . . , n − a k +1 ∈ C ( X, D ) by a k +1 ( x ) = α − h ( x ) (cid:0) a k ( h ( x )) (cid:1) for x ∈ X . Thedefinition of Y n implies that a , a , . . . , a n − ∈ C ( X \ Y, D ), and we already have a ∈ C ( X \ Y n , D ) ⊂ C ( X \ Y, D ). Therefore the element a = ( a u )( a u ) · · · ( a n − u ) ROSSED PRODUCTS OF C ( X, D ) 9 is in C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y . For k = 1 , , . . . , n − x ∈ X , we have α ( a k +1 )( x ) = α x ( a k +1 ( h − ( x )) = α x (cid:0) α − x ( a k ( x )) (cid:1) = a k ( x ) , so α ( a k +1 ) = a k . Similarly, α ( a ) = b . Now a = a ( ua u − )( u a u − ) · · · (cid:0) u n − a n − u − ( n − (cid:1) u n = a α ( a ) α ( a ) · · · α n − ( a n − ) u n = ( sb ) b n − u n = f u n . So f u n ∈ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y . Finally, suppose n <
0, and let f ∈ C ( X \ Y n , D ).It follows from (2.4) that f ◦ h n ∈ C ( X \ Y − n , D ), whence also ( f ◦ h n ) ∗ ∈ C ( X \ Y − n , D ). Since − n >
0, we therefore get f u n = (cid:0) u − n f ∗ (cid:1) ∗ = (cid:0) ( f ◦ h n ) ∗ u − n (cid:1) ∗ ∈ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y . The claim is proved.It now follows that B ⊂ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y . Combining this result with B = B and C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y ⊂ B , we get C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y = B . (cid:3) The following lemma is the analog of Lemma 7.8 of [52].
Lemma 2.5.
Let G be a discrete group, let X be a compact space, and suppose G acts on X in such a way that for every finite set S ⊂ G \ { } , the set (cid:8) x ∈ X : gx = x for all g ∈ S (cid:9) is dense in X . Let D be a unital C*-algebra, and let α : G → Aut( C ( X, D )) be anaction of G on C ( X, D ) which lies over the given action of G on X (in the sense ofDefinition 1.2). Following Notation 2.1, let a ∈ C ( X, D )[ G ] ⊂ C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) and let ε >
0. Then there exists f ∈ C ( X ) ⊂ C ( X, D ) such that0 ≤ f ≤ , f a ∗ af ∈ C ( X, D ) , and k f a ∗ af k ≥ k E α ( a ∗ a ) k − ε. Proof.
We follow the proof of Lemma 7.8 of [52]. Set b = a ∗ a . There are a finite set T ⊂ G with 1 ∈ T , and elements b g ∈ C ( X, D ) for g ∈ T , such that b = P g ∈ T b g u g .If k b k ≤ ε take f = 0. Otherwise, define U = (cid:8) x ∈ X : k b ( x ) k > k E ( a ∗ a ) k − ε (cid:9) , which is a nonempty open subset of X . Choose V , W , f , and x as in the proof ofLemma 7.8 of [52]. Then, as there, f bf = f b f . Moreover, k f b f k ≥ f ( x ) k b ( x ) k f ( x ) = k b ( x ) k > k E α ( a ∗ a ) k − ε. This completes the proof. (cid:3)
The following lemma is the analog of Lemma 7.9 of [52].
Lemma 2.6.
Let G , X , the action of G on X , D , and α : G → Aut( C ( X, D )) beas in Lemma 2.5. Let B ⊂ C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) be a unital subalgebra such that,following Notation 2.1,(1) C ( X, D ) ⊂ B .(2) B ∩ C ( X, D )[ G ] is dense in B .Let a ∈ B + \ { } . Then there exists b ∈ C ( X, D ) + \ { } such that b - B a . Proof.
We follow the proof of Lemma 7.9 of [52], using our Lemma 2.5 in place ofLemma 7.8 of [52], except that c ∈ B ∩ C ( X, D )[ G ]. The element ( f c ∗ cf − ε ) + weobtain now satisfies ( f c ∗ cf − ε ) + ∈ C ( X, D ) + \ { } and ( f c ∗ cf − ε ) + - B a . (cid:3) In Lemma 2.6, we really want to have b ∈ C ( X ) + \ { } . When G = Z and underthe pseudoperiodicity hypothesis of Definition 1.15, this is possible. Lemma 2.7.
Let X be a compact metric space, let h : X → X be a minimalhomeomorphism, let D be a simple unital C*-algebra, and let α ∈ Aut( C ( X, D ))lie over h . Assume that the action generated by α is pseudoperiodically generated.Let Y ⊂ X be a compact set such that h n ( Y ) ∩ Y = ∅ for all n ∈ Z \ { } .Set B = C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y . Then for every a ∈ C ( X, D ) + \ { } there exists f ∈ C ( X ) + \ { } ⊂ B such that f - B a . Proof.
Use Kirchberg’s Slice Lemma (Lemma 4.1.9 of [58]) to find g ∈ C ( X ) + \{ } and d ∈ D + \ { } such that(2.5) g ⊗ d - C ( X,D ) a. As in Definition 1.2, let ( m, x ) α m,x be the function Z × X → Aut( D ) suchthat α m ( a )( x ) = α m,x ( a ( h − m ( x )) for m ∈ Z , a ∈ C ( X, D ), and x ∈ X . Since theaction generated by α is pseudoperiodically generated, there exists d ∈ D + \ { } such that(2.6) α m,x ( d ) - d for all x ∈ X and m ∈ Z . Without loss of generality k d k = 1. Set d = (cid:0) d − (cid:1) + .Use Corollary 1.14 of [52] to find k ∈ Z > and w , w , . . . , w k ∈ D such that(2.7) k X j =1 w j dw ∗ j = 1 . The set X \ Y is dense in X , so there is x ∈ X \ Y such that g ( x ) = 0. Choose g ∈ C ( X ) + such that g ( x ) = 1 and g | Y = 0. Then(2.8) g g ⊗ d - C ( X,D ) g ⊗ d . Set U = (cid:8) x ∈ X : ( g g )( x ) = 0 (cid:9) . Choose a nonempty open set U ⊂ X such that U ⊂ U . Set(2.9) ρ = inf x ∈ U ( g g )( x ) . Then ρ >
0. The set X \ [ n ∈ Z h n ( Y ) = \ n ∈ Z h n ( X \ Y )is dense by the Baire Category Theorem. So we can choose y ∈ U ∩ (cid:0) X \ S n ∈ Z h n ( Y ) (cid:1) .The forward orbits { h n ( y ) : n ≥ N } of y are all dense, so there exist n , n , . . . , n k ∈ Z ≥ with 0 = n < n < · · · < n k and h n j ( y ) ∈ U for j = 1 , , . . . , k .Choose an open set V containing y which is so small that the following hold:(1) V ⊂ X \ S n k m =0 h − m ( Y ).(2) h n j ( V ) ⊂ U for j = 1 , , . . . , k .(3) The sets V , h ( V ) , . . . , h n k ( V ) are disjoint.(4) For y ∈ V and m = 0 , , . . . , n k , we have (cid:13)(cid:13) α m,h m ( y ) ( d ) − α m,h m ( y ) ( d ) (cid:13)(cid:13) < . Choose f , f ∈ C ( X ) + such that:(5) k f k , k f k ≤ ROSSED PRODUCTS OF C ( X, D ) 11 (6) supp( f ) ⊂ V .(7) f f = f .(8) f ( y ) = 1.For m = 0 , , . . . , n k −
1, define c m ∈ C ( X, D ) by c m ( x ) = f ( h − m ( x )) α m,x ( d ) for x ∈ X . Thus(2.10) c m = α m ( f ⊗ d ) . Also set(2.11) β m = α m,h m ( y ) . We claim that for m = 0 , , . . . , n k we have c m - C ( X,D ) ( f ◦ h − m ) ⊗ β m ( d ) . To prove the claim, let ε >
0. Set L = h m (supp( f )). Define b ∈ C ( L, D ) by b ( x ) = α m,x ( d ) for x ∈ L . Since L ⊂ h m ( V ), condition (4) implies that for x ∈ L we have k α m,x ( d ) − β m ( d ) k < . In C ( L, D ) we therefore get k b − ⊗ β m ( d ) k ≤ < . So (cid:0) b − (cid:1) + - C ( L,D ) ⊗ β m ( d ) . Therefore there exists v ∈ C ( L, D ) such that (cid:13)(cid:13) v ∗ (1 ⊗ β m ( d )) v − (cid:0) b − (cid:1) + (cid:13)(cid:13) < ε . Define v, a ∈ C ( X, D ) by v ( x ) = ( f ( h − m ( x )) v ( x ) x ∈ L x L and a = v ∗ (cid:2) ( f ◦ h − m ) ⊗ β m ( d ) (cid:3) v. We show that k a − c m k < ε .For x ∈ X \ L , we have a ( x ) = c m ( x ) = 0. For x ∈ L , using f f = f at thesecond step and using d = (cid:0) d − (cid:1) + and the definition of b at the third step, weget k a ( x ) − c m ( x ) k = (cid:13)(cid:13) f ( h − m ( x )) f ( h − m ( x )) v ( x ) ∗ β m ( d ) v ( x ) − f ( h − m ( x )) α m,x ( d ) (cid:13)(cid:13) = (cid:13)(cid:13) f ( h − m ( x ))[ v ( x ) ∗ β m ( d ) v ( x ) − α m,x ( d )] (cid:13)(cid:13) = f ( h − m ( x )) (cid:13)(cid:13) v ( x ) ∗ β m ( d ) v ( x ) − (cid:0) b ( x ) − (cid:1) + (cid:13)(cid:13) ≤ f ( h − m ( x )) (cid:13)(cid:13) v ∗ (1 ⊗ β m ( d )) v − (cid:0) b − (cid:1) + (cid:13)(cid:13) < ε . Taking the supremum over x ∈ X gives k a − c m k ≤ ε < ε . Thus (cid:13)(cid:13) v ∗ [( f ◦ h − m ) ⊗ β m ( d )] v − c m (cid:13)(cid:13) < ε. Since ε > m = 0 , , . . . , n k , the functions f ◦ h − m are orthogonal since the sets h m ( V )are disjoint. The claim therefore implies that(2.12) k X j =1 c n j - C ( X,D ) k X j =1 ( f ◦ h − n j ) ⊗ β n j ( d ) . We now claim that(2.13) f ⊗ - B k X j =1 c n j . To prove this claim, for j = 1 , , . . . , k define v j = [( f ◦ h − n j ) ⊗ u n j (1 ⊗ w j ) ∗ ∈ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) . Combining (6) and (1), we see that f vanishes in particular on the sets h − ( Y ) , h − ( Y ) , . . . , h − n j ( Y ) , whence f ◦ h − n j vanishes on the sets Y, h ( Y ) , . . . , h n j − ( Y ) . So v j ∈ B by Lemma 2.4. Using (2.10) at the first step and f f = f at the laststep, we calculate: v ∗ j c n j v j = (1 ⊗ w j ) u − n j [( f ◦ h − n j ) ⊗ α n j ( f ⊗ d )[( f ◦ h − n j ) ⊗ u n j (1 ⊗ w j ) ∗ = (1 ⊗ w j )( f ⊗ f ⊗ d )( f ⊗ ⊗ w ∗ j ) = f ⊗ w j dw ∗ j . We apply (2.7) at the first step and use orthogonality of c , c , . . . , c n k and Lemma1.4(12) of [52] at the second step, to get f ⊗ k X j =1 f ⊗ w j dw ∗ j - B k X j =1 c n j . This proves the claim (2.13).We next claim that(2.14) k X j =1 ( f ◦ h − n j ) ⊗ β n j ( d ) - C ( X,D ) g g ⊗ d. To prove this, combine (2.6), (2.11), and Lemma 1.11 of [52] to get( f ◦ h − n j ) ⊗ β n j ( d ) - C ( X,D ) ( f ◦ h − n j ) ⊗ d for j = 1 , , . . . , k . Since the functions f ◦ h − n j are orthogonal, it follows that k X j =1 ( f ◦ h − n j ) ⊗ β n j ( d ) - C ( X,D ) k X j =1 ( f ◦ h − n j ) ⊗ d . Using orthogonality of the functions f ◦ h − n j again, together withsupp( f ◦ h − n j ) ⊂ h n j ( V ) ⊂ U and 0 ≤ f ◦ h − n j ≤ j = 1 , , . . . , k , we see that 0 ≤ P kj =1 f ◦ h − n j ≤ U . Since g g ≥ ρ on U by (2.9), we get the first step in the followingcomputation; the second step is clear: k X j =1 ( f ◦ h − n j ) ⊗ d ≤ ρ ( g g ⊗ d ) ∼ C ( X,D ) g g ⊗ d . The claim is proved.Now combine (2.5), (2.8), (2.14), (2.12), and (2.13) to get f ⊗ - B a . (cid:3) ROSSED PRODUCTS OF C ( X, D ) 13
Corollary 2.8.
Let X be a compact metric space, let h : X → X be a minimalhomeomorphism, let D be a simple unital C*-algebra, and let α ∈ Aut( C ( X, D ))lie over h . Assume that the action generated by α is pseudoperiodically generated.Let Y ⊂ X be a compact set (possibly empty) such that h n ( Y ) ∩ Y = ∅ for all n ∈ Z \ { } . Set B = C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y . Let a ∈ B + \ { } . Then there exists f ∈ C ( X ) + \ { } ⊂ B such that f - B a . Proof.
Lemma 2.4 implies that B satisfies the hypotheses of Lemma 2.6 with G = Z .Therefore, by Lemma 2.6, there exists b ∈ C ( X, D ) + \ { } such that b - B a . NowLemma 2.7 provides f ∈ C ( X ) + \ { } such that f - B a . (cid:3) We can now prove the analog of Theorem 7.10 of [52].
Theorem 2.9.
Let X be a compact metric space, let h : X → X be a minimalhomeomorphism, let D be a simple unital C*-algebra which has a tracial state,and let α ∈ Aut( C ( X, D )) lie over h . Assume that the action generated by α ispseudoperiodically generated. Let Y ⊂ X be a compact subset such that h n ( Y ) ∩ Y = ∅ for all n ∈ Z \ { } . Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a large subalgebra of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) of crossed product type in the sense of Definition 4.9 of [52]. Proof.
We verify the hypotheses of Proposition 4.11 of [52]. We follow Notation 2.1.Set A = C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) , B = C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y ,C = C ( X, D ) , and G = { u } . The algebra A is simple by Proposition 1.6 and finite by Proposition 1.7. Inparticular, condition (1) in Proposition 4.11 of [52] holds.We next verify condition (2) in Proposition 4.11 of [52]. All parts are obviousexcept (2d). So let a ∈ A + \ { } and b ∈ B + \ { } . Apply Corollary 2.8 twice, thefirst time with Y = ∅ and a as given and the second time with Y as given and with b in place of a . We get a , b ∈ C ( X ) + \ { } such that a - A a and b - B b .We can now argue as in the corresponding part of the proof of Theorem 7.10of [52] to find f ∈ C ( X ) such that f - C ( X ) a - A a and f - B b - B b, completing the proof of condition (2d). Alternatively, a , b ∈ C ∗ ( Z , X, h ) Y ⊂ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y , and C ∗ ( Z , X, h ) Y is a large subalgebra of C ∗ ( Z , X, h ) ofcrossed product type by Theorem 7.10 of [52], so the existence of f follows fromcondition (2d) in Proposition 4.11 of [52].We next verify condition (3) in Proposition 4.11 of [52]. Let m ∈ Z > , let a , a , . . . , a m ∈ A , let ε >
0, and let b ∈ B + \ { } . We follow the correspondingpart of the proof of Theorem 7.10 of [52]. Choose c , c , . . . , c m ∈ C ( X, D )[ Z ]such that k c j − a j k < ε for j = 1 , , . . . , m . (This estimate is condition (3b).)Choose N ∈ Z > such that for j = 1 , , . . . , m there are c j,l ∈ C ( X, D ) for l = − N, − N + 1 , . . . , N − , N with c j = N X l = − N c j,l u l . Apply Corollary 2.8 to B , to find f ∈ C ( X ) + \ { } such that f - B b . Set U = { x ∈ X : f ( x ) = 0 } , and choose nonempty disjoint open sets U l ⊂ U for l = − N, − N + 1 , . . . , N − , N . For each such l , use Lemma 7.7 of [52] to choose f l , r l ∈ C ( X ) + such that r l ( x ) = 1 for all x ∈ h l ( Y ), such that 0 ≤ r l ≤
1, suchthat supp( f l ) ⊂ U l , and such that r l - C ∗ ( Z ,X,h ) Y f l . Then also r l - B f l .Choose an open set W containing Y such that h − N ( W ) , h − N +1 ( W ) , . . . , h N − ( W ) , h N ( W )are disjoint, and choose r ∈ C ( X ) such that 0 ≤ r ≤ r ( x ) = 1 for all x ∈ Y , andsupp( r ) ⊂ W . Set g = r · N Y l = − N r l ◦ h l . Set g l = g ◦ h − l for l = − N, − N + 1 , . . . , N − , N . Then 0 ≤ g l ≤ r l ≤
1. Set g = P Nl = − N g l . The supports of the functions g l are disjoint, so 0 ≤ g ≤
1. This iscondition (3a) in Proposition 4.11 of [52]. The proof of Theorem 7.10 of [52] showsthat g - C ∗ ( Z ,X,h ) Y f - B b . Since C ∗ ( Z , X, h ) Y ⊂ B , it follows that g - B b . Thisis condition (3d) in Proposition 4.11 of [52].It remains to verify condition (3c) in Proposition 4.11 of [52]. This is done thesame way as the corresponding part of the proof of Theorem 7.10 of [52], exceptusing Lemma 2.4 in place of Proposition 7.5 of [52]. (cid:3) The following corollary is the analog of Corollary 7.11 of [52].
Corollary 2.10.
Let X be a compact metric space, let h : X → X be a minimalhomeomorphism, let D be a simple unital C*-algebra which has a tracial state,and let α ∈ Aut( C ( X, D )) lie over h . Assume that the action generated by α ispseudoperiodically generated. Let Y ⊂ X be a compact subset such that h n ( Y ) ∩ Y = ∅ for all n ∈ Z \ { } . Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a stably large subalgebraof C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) in the sense of Definition 5.1 of [52]. Proof.
Since C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is stably finite (by Proposition 1.7), we can com-bine Theorem 2.9, Proposition 4.10 of [52], and Corollary 5.8 of [52]. (cid:3) Corollary 2.11.
Let X be a compact metric space, let h : X → X be a minimalhomeomorphism, let D be a simple unital C*-algebra which has a tracial state,and let α ∈ Aut( C ( X, D )) lie over h . Assume that the action generated by α ispseudoperiodically generated. Let Y ⊂ X be a compact subset such that h n ( Y ) ∩ Y = ∅ for all n ∈ Z \{ } . Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a centrally large subalgebraof C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) in the sense of Definition 3.2 of [4]. Proof.
Since C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is stably finite (by Proposition 1.7), we can com-bine Theorem 2.9 and Theorem 4.6 of [4]. (cid:3) We conclude by giving some conditions on D and α which guarantee the hy-potheses of Corollary 2.11. These are more natural to consider than the awkwardpseudoperiodicity hypothesis. Corollary 2.12.
Let X be a compact metric space, let h : X → X be a minimalhomeomorphism, let Y ⊂ X be a compact subset such that h n ( Y ) ∩ Y = ∅ for all n ∈ Z \ { } , let D be a simple unital C*-algebra which has a tracial state, and let α ∈ Aut( C ( X, D )) lie over h . Let x α x be the corresponding map from X toAut( D ), as in Lemma 1.4. Assume one of the following conditions holds.(1) All elements of { α x : x ∈ X } ⊂ Aut( D ) are approximately inner.(2) D has strict comparison of positive elements. ROSSED PRODUCTS OF C ( X, D ) 15 (3) D has property (SP) and the order on projections over D is determined byquasitraces.(4) The set { α x : x ∈ X } is contained in a subgroup of Aut( D ) which is compactin the topology of pointwise convergence in the norm on D .Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a centrally large subalgebra of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) . Proof.
We claim that, under any of the conditions (1), (2), (3), or (4), the set { α x : x ∈ X } is contained in a pseudoperiodic subgroup of Aut( D ). For (1), thisfollows from Lemma 1.9; for (2), from Lemma 1.12; for (3), from Lemma 1.14; andfor (4), from Lemma 1.10. Now apply Lemma 1.16 to see that the hypotheses ofCorollary 2.11 are satisfied. (cid:3) The case in which D is purely infinite and simple is much easier. We give adirect proof, not depending on large subalgebras, that C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) is purelyinfinite simple for any discrete group G . It is still true, and will be proved below(Proposition 2.16), that, under the other hypotheses of Theorem 2.9 the subalgebra C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a large subalgebra of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) of crossed producttype. This fact seems potentially useful, but does not help with the analysis of anyof the examples in Section 6. Theorem 2.13.
Let G be a discrete group, let X be a compact space, and suppose G acts on X in such a way that the action is minimal and for every finite set S ⊂ G \ { } , the set (cid:8) x ∈ X : gx = x for all g ∈ S (cid:9) is dense in X . Let D be a purely infinite simple unital C*-algebra, and let α : G → Aut( C ( X, D )) be an action of G on C ( X, D ) which lies over the given action of G on X (in the sense of Definition 1.2). Then C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) is purely infinitesimple.We saw in Proposition 1.6 that C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) is simple, but we won’t usethat in this proof. Proof of Theorem 2.13.
For convenience of notation, set A = C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) .We also freely identify C ( X, D ) with C ( X ) ⊗ D . We prove the result by showingthat 1 A - a for all a ∈ A + \ { } . By Lemma 2.6 (taking B there to be A ), we canassume that a ∈ C ( X, D ) + \ { } .Use Kirchberg’s Slice Lemma (Lemma 4.1.9 of [58]) to find f ∈ C ( X ) + \ { } and b ∈ D + \ { } such that(2.15) f ⊗ b - C ( X,D ) a. Without loss of generality k f k = 1. Since D is purely infinite and simple, we have1 D - D b , and so it follows that(2.16) f ⊗ D - D f ⊗ b. Set U = (cid:8) x ∈ X : f ( x ) > (cid:9) . By minimality of the action, the sets gU for g ∈ G cover X . So there are n ∈ Z > and g , g , . . . , g n ∈ G such that the sets g U, g U, . . . , g n U cover X . The function x P nk =1 f ( g − k x ) is strictly positiveon X . Using this fact at the first step, pure infiniteness of D at the second last step, and (2.16) and (2.15) at the last step, we get1 A - C ( X ) n X k =1 α g k ( f ⊗ D ) - C ( X ) diag (cid:0) α g ( f ⊗ D ) , α g ( f ⊗ D ) , . . . , α g n ( f ⊗ D ) (cid:1) = diag (cid:0) u g ( f ⊗ D ) u ∗ g , u g ( f ⊗ D ) u ∗ g , . . . , u g n ( f ⊗ D ) u ∗ g n (cid:1) ∼ A diag (cid:0) f ⊗ D , f ⊗ D , . . . , f ⊗ D (cid:1) = f ⊗ M n ( D ) - C ( X,D ) f ⊗ D - A a. This completes the proof. (cid:3)
As promised, we now give a result on large subalgebras when G = Z . We usetwo lemmas, the first of which has a similar proof to that of Theorem 2.13. Lemma 2.14.
Let X be a compact metric space, let h : X → X be a minimalhomeomorphism, let D be a purely infinite simple unital C*-algebra, and let α ∈ Aut( C ( X, D )) lie over h . Let Y ⊂ X be a compact subset such that h n ( Y ) ∩ Y = ∅ for all n ∈ Z \ { } . Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is purely infinite and simple. Proof.
For convenience of notation, set B = C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y . We also freelyidentify C ( X, D ) with C ( X ) ⊗ D . We prove the result by showing that 1 A - a for all a ∈ A + \ { } . By Lemma 2.6 and Lemma 2.4, we can assume that a ∈ C ( X, D ) + \ { } . Using Kirchberg’s Slice Lemma and pure infiniteness of D as inthe proof of Theorem 2.13, there is f ∈ C ( X ) + \ { } such that f ⊗ D - C ( X,D ) a .We now claim that for every x ∈ X there is l x ∈ C ( X ) + such that l x ⊗ D - B f ⊗ D and l x ( x ) > . Given this, the proof is finished as in the last computationin the proof of Theorem 2.13.To prove the claim, set U = (cid:8) x ∈ X : f ( x ) > (cid:9) . If x ∈ U , take l x = f .Suppose next that x U and x S ∞ m =0 h m ( Y ). By minimality of h , thereis n ∈ Z > such that x ∈ h n ( U ). Choose r ∈ C ( X ) + such that r ( y ) = 0 forall y ∈ S n − m =0 h m ( Y ) and r ( x ) = 1. Then, letting u ∈ C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) be thestandard unitary (as in Notation 0.2), we have ( r ⊗ D ) u n ∈ B by Lemma 2.4.Taking l x = r · ( f ◦ h − n ), we have l x ( x ) = f ( h − n ( x )) > . Also, ( ru n )( f ⊗ D )( ru n ) ∗ = l x ⊗ D , so l x ⊗ D - B f ⊗ D .Finally, suppose x U and x ∈ S ∞ m =0 h m ( Y ). Then x S ∞ m =1 h − m ( Y ). Byminimality of h , there is n ∈ Z > such that x ∈ h − n ( U ). Choose r ∈ C ( X ) + suchthat r ( y ) = 0 for all y ∈ S nm =1 h − m ( Y ) and r ( x ) = 1. Then ( r ⊗ D ) u − n ∈ B by Lemma 2.4. Taking l x = r · ( f ◦ h n ), we have l x ( x ) = f ( h n ( x )) > . Also,( ru − n )( f ⊗ D )( ru − n ) ∗ = l x ⊗ D , so l x ⊗ D - B f ⊗ D . This completes theproof. (cid:3) Lemma 2.15.
Let X be a compact Hausdorff space, let G be a discrete group,and let D be a C*-algebra. Let ( g, x ) gx be an action of G on X , and let α : G → Aut( C ( X, D )) be an action of G on C ( X, D ) which lies over ( g, x ) gx in the sense of Definition 1.2. Then for every b ∈ C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) and every ε >
0, there is a finite set T ⊂ G and a nonempty open set V ⊂ X such that,whenever x ∈ V and f ∈ C ( X ) satisfy 0 ≤ f ≤ f ( gx ) = 1 for all g ∈ T , then k f b k > k b k − ε .If D is not unital, then the product f b is realized via the inclusion of C ( X ) in M (cid:0) C ∗ (cid:0) G, C ( X, D ) , α (cid:1)(cid:1) . ROSSED PRODUCTS OF C ( X, D ) 17
Proof of Lemma 2.15.
Let ( g, x ) α g,x be as in Definition 1.2.Fix a faithful representation ρ of D on a Hilbert space H . Then for every x ∈ X there is a representation ρ ◦ ev x : C ( X, D ) → L ( H ). Let ( v x , π x ) be thecorresponding regular covariant representation of (cid:0) G, C ( X, D ) , α (cid:1) on l ( G, H ), andlet σ x : C ∗ (cid:0) G, C ( X, D ) , α (cid:1) → l ( G, H ) be its integrated form. We identify l ( G, H )with l ( G ) ⊗ H , and we write δ g ∈ l ( G ) for the standard basis vector correspondingto g ∈ G . For later use, we recall that if S , S ⊂ G are finite sets, d ∈ C ( X, D )[ G ]is given as d = P g ∈ S d g u g with d g ∈ C ( X, D ) for g ∈ S , and ξ ∈ l ( G ) ⊗ H hasthe form ξ = P h ∈ S δ h ⊗ ξ h with ξ h ∈ H for h ∈ S , then(2.17) σ x ( d ) ξ = X g ∈ S X h ∈ S δ gh ⊗ ρ (cid:0) α h − g − ,x ( d g ( ghx )) (cid:1) ξ h . The representation L x ∈ X ρ ◦ ev x is a faithful representation of C ( X, D ), so L x ∈ X σ x is a faithful representation of C ∗ r (cid:0) G, C ( X, D ) , α (cid:1) . Therefore there exists x ∈ X such that k σ x ( b ) k > k b k− ε . Choose c ∈ C ( X, D )[ G ] such that k b − c k < ε .In particular, k σ x ( c ) k > k b k − ε . Choose ξ ∈ l ( G, H ) with finite support suchthat k ξ k = 1 and k σ x ( c ) ξ k > k b k − ε . Write c = P g ∈ S c g u g and ξ = P h ∈ S δ h ⊗ ξ h with S , S ⊂ G finite, c g ∈ C ( X, D ) for g ∈ S , and ξ h ∈ H for h ∈ S . UseLemma 1.3 to choose an open set V ⊂ X such that x ∈ V and such that for all x ∈ V , g ∈ S , and h ∈ S , we have(2.18) (cid:13)(cid:13) α h − g − ,x ( c g ( ghx )) − α h − g − ,x ( c g ( ghx )) (cid:13)(cid:13) < ε S )card( S ) + 1 . Set T = (cid:8) gh : g ∈ S and h ∈ S (cid:9) . Now let x ∈ V , and suppose f ∈ C ( X ) satisfies 0 ≤ f ≤ f ( gx ) = 1 for all g ∈ T . The condition on f and the formula (2.17) imply that σ x ( f c ) ξ = σ x ( c ) ξ .Applying (2.17) again, and using (2.18) at the second step, we get(2.19) k σ x ( c ) ξ − σ x ( c ) ξ k ≤ X g ∈ S X h ∈ S (cid:13)(cid:13) α h − g − ,x ( c g ( ghx )) − α h − g − ,x ( c g ( ghx )) (cid:13)(cid:13) < ε . Therefore, using k f k ≤ k b − c k < ε and the second andfifth steps, as well as (2.19) at the fourth step, k f b k ≥ k σ x ( f b ) k > k σ x ( f c ) ξ k − ε k σ x ( c ) ξ k − ε > k σ x ( c ) ξ k − ε > k σ x ( b ) ξ k − ε > k σ x ( b ) k − ε > k σ x ( b ) k − ε > k b k − ε, as desired. (cid:3) Proposition 2.16.
Let X be a compact metric space, let h : X → X be a mini-mal homeomorphism, let Y ⊂ X be a compact subset such that h n ( Y ) ∩ Y = ∅ for all n ∈ Z \ { } , let D be a purely infinite simple unital C*-algebra, and let α ∈ Aut( C ( X, D )) lie over h . Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a large subalgebra of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) of crossed product type. Proof.
We verify directly the conditions of Definition 4.9 of [52]. We follow Nota-tion 2.1. Set A = C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) , B = C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y , C = C ( X, D ) , and G = { u } . All parts of condition (1) in Definition 4.9 of [52] are obvious.We verify condition (2) there. Let m ∈ Z > , let a , a , . . . , a m ∈ A , let ε > b ∈ A + satisfy k b k = 1, and let b ∈ B + \ { c , c , . . . , c m ∈ C ( X, D )[ Z ] such that k c j − a j k < ε for j = 1 , , . . . , m . (Thisestimate is condition (2b).) Choose N ∈ Z > such that for j = 1 , , . . . , m thereare c j,l ∈ C ( X, D ) for l = − N, − N + 1 , . . . , N − , N with c j = N X l = − N c j,l u l . Apply Lemma 2.15, getting a finite set T ⊂ Z and a nonempty open set U ⊂ X such that, whenever x ∈ U and f ∈ C ( X ) satisfy 0 ≤ f ≤ f ( h k ( x )) = 1 forall k ∈ T , then k f b / k > − ε . Define S = (cid:8) n − k : k ∈ T and n ∈ [ − N, N ] ∩ Z (cid:9) , which is a finite subset of Z . Then S n ∈ S h n ( Y ) is nowhere dense in X , so thereexists x ∈ W such that x S n ∈ S h n ( Y ). This choice implies that { h k ( x ) : k ∈ T } ∩ N [ n = − N h n ( Y ) = ∅ , so there is g ∈ C ( X ) such that 0 ≤ g ≤ g ( y ) = 1 for all y ∈ S Nn = − N h n ( Y ), and g ( h k ( x )) = 0 for all k ∈ T .Condition (1a) in Definition 4.9 of [52] holds by construction. The proof that(1 − g ) c j ∈ B for j = 1 , , . . . , m (condition (2c) in Definition 4.9 of [52]) is thesame as at the end of the proof of Theorem 2.9: follow the corresponding part of theproof of Theorem 7.10 of [52], except using Lemma 2.4 in place of Proposition 7.5of [52]. Lemma 2.14 implies 1 A - A b and 1 A - B b , so the requirements g - A b and g - B b (condition (2d) in Definition 4.9 of [52]) follow immediately. Forcondition (2e) in Definition 4.9 of [52], we estimate, using 1 − g ( h k ( x )) = 1 for all k ∈ T at the second step: (cid:13)(cid:13) (1 − g ) b (1 − g ) (cid:13)(cid:13) = k (1 − g ) b / k > (cid:0) − ε (cid:1) > ε. This completes the proof. (cid:3) Recursive structure for orbit breaking subalgebras
In this section, under appropriate conditions we will show that the orbit breakingsubalgebras of Definition 2.3 have a tractable recursive structure, as subalgebrasof certain homogeneous algebras tensored with D . We start by introducing theformalism for a generalization of the recursive subhomogeneous algebras introducedin [51] that were crucial for the analysis in [42] and [38]. Definition 3.1.
Let A , B , and C be C*-algebras, and let ϕ : A → C and ψ : B → C be homomorphisms. Then the associated pullback C*-algebra A ⊕ C,ϕ,ψ B is definedby A ⊕ C,ϕ,ψ B = (cid:8) ( a, b ) ∈ A ⊕ B : ϕ ( a ) = ψ ( b ) (cid:9) . ROSSED PRODUCTS OF C ( X, D ) 19
We frequently write A ⊕ C B when the maps ϕ and ψ are understood. If C = 0we just get A ⊕ B . Definition 3.2.
Let D be a simple unital C*-algebra. The class of recursive sub-homogeneous algebras over D is the smallest class R of C*-algebras that is closedunder isomorphism and such that:(1) If X is a compact Hausdorff space and n ≥
1, then C ( X, M n ( D )) ∈ R .(2) If B ∈ R , X is compact Hausdorff, n ≥ X (0) ⊂ X is closed (possiblyempty), ϕ : B → C ( X (0) , M n ( D )) is any unital homomorphism (the zerohomomorphism if X (0) is empty), and ρ : C ( X, M n ( D )) → C ( X (0) , M n ( D ))is the restriction homomorphism, then the pullback B ⊕ C ( X (0) , M n ( D )) , ϕ, ρ ) C ( X, M n ( D ))= (cid:8) ( b, f ) ∈ B ⊕ C ( X, M n ( D )) : ϕ ( b ) = f | ( X (0) (cid:9) is in R .Taking D = C in this definition gives the usual definition for the class of recursivesubhomogeneous algebras. (See [51].) This definition makes sense for any unitalC*-algebra D , but it is not clear whether it is appropriate in this generality. Definition 3.3.
We adopt the following standard notation for recursive subhomo-geneous algebras over D . The definition implies that if R is any recursive subho-mogeneous algebra over D , then R may be written in the form R ∼ = h · · · hh C ⊕ C (0)1 ,ϕ ,ρ C i ⊕ C (0)2 ,ϕ ,ρ i · · · i ⊕ C (0) l ,ϕ l ,ρ l C l , with C k = C ( X k , M n ( k ) ( D )) for compact Hausdorff spaces X k and positive inte-gers n ( k ), and with C (0) k = C (cid:0) X (0) k , M n ( k ) ( D ) (cid:1) for compact subsets X (0) k ⊂ X k (possibly empty), where the maps ρ k : C k → C (0) k are always the restriction maps.An expression of this type for R will be referred to as a decomposition of R , andthe notation that appears here will be referred to as the standard notation for adecomposition . We associate the following objects to this decomposition.(1) Its length l .(2) The k -th stage algebra R ( k ) = h · · · hh C ⊕ C (0)1 C i ⊕ C (0)2 C i · · · i ⊕ C (0) k C k . (3) Its base spaces X , . . . , X l and total space X = ` lk =0 X k .(4) Its matrix sizes n (0) , . . . , n ( l ) and matrix size function m : X → Z ≥ de-fined by m ( x ) = n ( k ) when x ∈ X k . (This is called the matrix size of R at x .)(5) Its minimum matrix size min k n ( k ) and maximum matrix size max k n ( k ).(6) Its topological dimension dim( X ) = max k dim( X k ) and topological dimen-sion function d : X → Z ≥ , defined by d ( x ) = dim( X k ) for x ∈ X k . (Thisis called the topological dimension of R at x .)(7) Its standard representation σ = σ R : R → L lk =0 C ( X k , M n ( k ) ( D )), definedby forgetting the restriction to a subalgebra in each of the fibered productsin the decomposition.(8) The associated evaluation maps ev x : R → M n ( k ) ( D ), defined to be therestriction of the usual evaluation map on L lk =0 C ( X k , M n ( k ) ( D )) to R , when R is identified with a subalgebra of this algebra through the standardrepresentation σ R .A decomposition of an algebra R as a recursive subhomogeneous algebra over D may be highly nonunique, which in the case D = C is clear from examples in [51].Moreover, the matrix sizes are not uniquely determined even if all the other datahas already been chosen, which is easily seen through the example of the 2 ∞ UHFalgebra.
Notation 3.4.
Throughout, we let E denote a separable, unital C*-algebra that isstrongly selfabsorbing in the sense of [63]: there exists an isomorphism E ∼ = E ⊗ E that is unitarily equivalent to a a ⊗ E .Examples include the Cuntz algebras O and O ∞ , and the Jiang-Su algebra Z ,which is a simple, separable, unital, infinite dimensional, strongly selfabsorbing,nuclear C*-algebra having the same Elliott invariant as the complex numbers C .(See [26] for its construction). Definition 3.5.
Adopt Notation 3.4. A separable C*-algebra D is called E -stable if there is an isomorphism D ⊗ E ∼ = D .It is clear that if D is E -stable, then so are M n ( D ) and C ( X, D ). With appropri-ate assumptions on the algebra D , we can give some results about the E -stabilityof recursive subhomogeneous algebras over C . Lemma 3.6.
Adopt Notation 3.4. Let A , B , and C be separable unital C*-algebras(allowing C = 0), and let ϕ : A → C and ψ : B → C be unital homomorphisms.Let P = A ⊕ C,ϕ,ψ B be the associated pullback (Definition 3.1). If ψ is surjectiveand both A and B are E -stable, then P is E -stable. Proof.
Define π : P → A by π ( a, b ) = a . There is an isomorphism ι : Ker( ψ ) → Ker( π ) given by ι ( b ) = (0 , b ) for b ∈ Ker( ψ ). Moreover, surjectivity of ψ is easilyseen to imply surjectivity of π . We obtain an exact sequence0 −→ Ker( ψ ) ι −→ P π −→ A −→ . Now Ker( ϕ ) is E -stable by Corollary 3.1 of [63]. Since B is also E -stable, Theorem4.3 of [63] implies that P is E -stable. (cid:3) Proposition 3.7.
Adopt Notation 3.4. Let D be a simple separable E -stable C*-algebra, and let R be a recursive subhomogeneous algebra over D . Then R is E -stable. Proof.
We proceed by induction on the length of a decomposition for R as a recur-sive subhomogeneous algebra over D . The base case is when R = C ( X, M n ( D )),and this is E -stable whenever D is E -stable. For the inductive step, we may as-sume that there are an E -stable unital C*-algebra R ′ , n ∈ Z > , a compact Haus-dorff space X , a closed subset X (0) ⊂ X , and a unital homomorphism ϕ : R ′ → C (cid:0) X (0) , M n ( D ) (cid:1) such that R = (cid:8) ( a, f ) ∈ R ′ ⊕ C ( X, M n ( D )) : ϕ ( a ) = f | X (0) (cid:9) Since f f | X (0) is surjective and both R ′ and C ( X, M n ( D )) are E -stable, it followsfrom Lemma 3.6 that R is E -stable, which completes the induction. (cid:3) ROSSED PRODUCTS OF C ( X, D ) 21
Proposition 3.8.
Let X be a compact Hausdorff space, let h : X → X be ahomeomorphism, let D be a unital C*-algebra, and let α ∈ Aut( C ( X, D )) lie over h .Let Y ⊂ X be closed, and let Y ⊃ Y ⊃ · · · be closed subsets of X such that T ∞ n =1 Y n = Y . Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y = [ ∞ n =1 A Y n ∼ = lim −→ A Y n . Proof.
The proof is easy, and is omitted. (cid:3)
The results in the remainder of this section are mostly generalizations of onesin Section 1 of [42]. We follow a slightly more modern development, adapted fromSection 11.3 of [20], since [42] has not been published. (The proofs in [20] aremostly taken from [42], with some changes in notational conventions.) Most proofsgo through with changes only to the notation, such as replacing the action of aminimal homeomorphism h with the action α on C ( X, D ). The biggest technicaldifferences are in the proof of Lemma 2.4. We begin with the well-known Rokhlintower construction.
Notation 3.9.
Let X be a compact Hausdorff space and let h : X → X be aminimal homeomorphism. Let Y ⊂ X be a closed set with int( Y ) = ∅ . For y ∈ Y , define r ( y ) = inf (cid:0) { m ≥ h m ( y ) ∈ Y } (cid:1) . Then sup y ∈ Y r ( y ) < ∞ (seeLemma 3.10(1)), so there are finitely many distinct values n (0) < n (1) < · · · < n ( l )in the range of r . For k = 0 , , . . . l , set Y k = { y ∈ Y : r ( y ) = n ( k ) } and Y ◦ k = int (cid:0)(cid:8) y ∈ Y : r ( y ) = n ( k ) (cid:9)(cid:1) . We warn that, while Y ◦ k ⊂ int( Y k ), the inclusion may be proper. Lemma 3.10.
Let Y ⊂ X be a closed set with int( Y ) = ∅ , and adopt Notation3.9. Then:(1) sup y ∈ Y r ( y ) < ∞ .(2) The sets h j ( Y ◦ k ) are pairwise disjoint for 0 ≤ k ≤ l and 1 ≤ j ≤ n ( k ).(3) S lk =0 Y k = Y .(4) S lk =0 S n ( k ) − j =0 h j ( Y k ) = X . Proof.
This is Lemma 11.3.5 of [20]. (cid:3)
We first consider the specific situation of the structure of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y when X is the Cantor set. This is needed later. Lemma 3.11.
Adopt Notation 3.9, let X be the Cantor set, and let Y ⊂ X be anonempty compact open subset. Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y ∼ = l M k =0 C ( Y k , M n ( k ) ( D )) . Proof.
This is a straightforward adaption of the proof of Lemma 11.2.20 of [20]. (cid:3)
If we set B k = C ( Y k , M n ( k ) ), then B k is an AF algebra for each k (since Y k is totally disconnected), and hence C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a direct sum of whatmight be called “AF D -algebras”. Proposition 3.12.
Let X be a compact Hausdorff space, let h : X → X be aminimal homeomorphism, let D be a unital C*-algebra, let α ∈ Aut( C ( X, D )) lieover h , let Y ⊂ X be a closed set with int( Y ) = ∅ , and adopt Notation 3.9. Follow-ing the notation of Lemma 2.4, there is N ∈ Z > such that C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y has the Banach space direct sum decomposition C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y = N M n = − N C ( X \ Y n , D ) u n . Proof.
The proof is the same as that of Corollary 11.3.7 of [20]. (cid:3)
Proposition 3.13.
Let X be a compact Hausdorff space, let h : X → X be aminimal homeomorphism, let D be a unital C*-algebra, let α ∈ Aut( C ( X, D )) lieover h , let Y ⊂ X be a closed set with int( Y ) = ∅ , and adopt Notation 3.9. Definethe unitary s k ∈ C ( Y k , M n ( k ) ( D )) by s k = · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . For k = 0 , , . . . , l there is a unique linear map γ k : C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y → C ( Y k , M n ( k ) ( D ))such that for m = 0 , , . . . , n ( k ) − f ∈ C ( X \ Z m , D ) we have:(1) γ k ( f u m ) = diag (cid:0) f | Y k , α − ( f ) | Y k , . . . , α − [ n ( k ) − ( f ) | Y k (cid:1) · s mk .(2) γ k ( u − m f ) = s − mk · diag (cid:0) f | Y k , α − ( f ) | Y k , . . . , α − [ n ( k ) − ( f ) | Y k (cid:1) .Moreover, the map γ : C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y → l M k =0 C ( Y k , M n ( k ) ( D ))given by γ ( a ) = ( γ ( a ) , γ ( a ) , . . . , γ l ( a )) is a ∗ -homomorphism. Proof.
The computations in this proof are analogous to those in the proof of Propo-sition 11.3.9 of [20], using Lemma 2.4 in place of Proposition 11.3.6 of [20]. (cid:3)
Lemma 3.14.
Let X be a compact Hausdorff space, let h : X → X be a minimalhomeomorphism, let D be a unital C*-algebra, let α ∈ Aut( C ( X, D )) lie over h ,and let Y ⊂ X be a closed set with int( Y ) = ∅ . Adopt Notation 3.9. Identify C ( Y k , M n ( k ) ( D )) with M n ( k ) ( C ( Y k , D )) in the obvious way. Define maps E ( m ) k : C ( Y k , M n ( k ) ( D )) → C ( Y k , M n ( k ) ( D ))by E ( m ) k ( b ) m + j,j = b m + j,j for 1 ≤ j ≤ n ( k ) − m (if m ≥
0) and for − m +1 ≤ j ≤ n ( k )(if m ≤ E ( m ) k ( b ) i,j = 0 for all other pairs ( i, j ). (Thus, E ( m ) k is the projection ROSSED PRODUCTS OF C ( X, D ) 23 map on the m -th subdiagonal.) Write G m = l M k =0 E ( m ) k ( C ( Y k , M n ( k ) ( D )) . Then:(1) There is a Banach space direct sum decomposition l M k =0 C ( Y k , M n ( k ) ( D )) = n ( l ) M m = − n ( l ) G m . (2) For k = 0 , , . . . , l , m ≥ f ∈ C ( X \ Z m , D ), and x ∈ Y k , the expression γ k ( f u m )( x ) is given by the following matrix, in which the first nonzeroentry is in row m + 1: γ k ( f u m )( x ) = · · · · · · · · · · · · α − m ( f )( x ) 0 ...0 α − [ m +1] ( f )( x ) ...... . . . ...0 · · · · · · α − [ n ( k ) − ( f )( x ) 0 · · · . (3) For m ≥ f ∈ C ( X \ Z m , D ), we have γ k ( f u m ) ∈ E ( m ) k ( C ( Y k , M n ( k ) ( D ))) , γ ( f u m ) ∈ G m ,γ k ( u − m f ) ∈ E ( − m ) k ( C ( Y k , M n ( k ) ( D ))) , and γ ( u − m f ) ∈ G − m . (4) The homomorphism γ is compatible with the direct sum decomposition ofProposition 3.12 on its domain and the direct sum decomposition of part(1) on its codomain. Proof.
The direct sum decomposition is essentially immediate from the definition ofthe maps E ( m ) k , while the other statements follow from Proposition 3.12 and somestraightforward matrix calculations as in the proof of Lemma 11.3.15 of [20]. (cid:3) Corollary 3.15.
Let X be a compact Hausdorff space, let h : X → X be a mini-mal homeomorphism, let D be a unital C*-algebra, and let α ∈ Aut( C ( X, D )) lieover h . Let Y ⊂ X be closed with int( Y ) = ∅ . Then the homomorphism γ ofProposition 3.12 is injective. Proof.
The proof is the same as the proof of Lemma 11.3.17 of [20]. (cid:3)
Lemma 3.16.
Let X be a compact Hausdorff space, let h : X → X be a minimalhomeomorphism, let D be a unital C*-algebra, and let α ∈ Aut( C ( X, D )) lie over h .Let Y ⊂ X be closed with int( Y ) = ∅ . Let b = ( b , b , . . . , b l ) ∈ l M k =0 C ( Y k , M n ( k ) ( D )) . Then b ∈ γ (cid:0) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y (cid:1) if and only if whenever • r > • k, t , . . . , t r ∈ { , . . . , l } , • n ( t ) + n ( t ) + · · · + n ( t r ) = n ( k ), • x ∈ ( Y k \ Y ◦ k ) ∩ Y t ∩ h − n ( t ) ( Y t ) ∩ · · · ∩ h − [ n ( t )+ ··· + n ( t r − )] ( Y t r ),then b k ( x ) is given by the block diagonal matrix b k ( x ) = b t ( x ) α − n ( t ) ( b t )( x ) . . . α − [ n ( t )+ ··· + n ( t r − )] ( b t r )( x ) . Proof.
The proof is analogous (with appropriate changes to notation and exponents)to the proof of Lemma 11.3.18 of [20]. (cid:3)
We are now in position to give a decomposition of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y as arecursive subhomogeneous algebra over D . Theorem 3.17.
Let X be a compact Hausdorff space, let h : X → X be a minimalhomeomorphism, let D be a unital C*-algebra, and let α ∈ Aut( C ( X, D )) lie over h .Let Y ⊂ X be closed with int( Y ) = ∅ . Adopt Notation 3.9 and the notation ofProposition 3.13. Then the homomorphism γ : C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y → l M k =0 C ( Y k , M n ( k ) ( D ))of Proposition 3.13 induces an isomorphism of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y with the re-cursive subhomogeneous algebra over D defined, in the notation of Definition 3.3,as follows:(1) l and n (0) , n (1) , . . . , n ( l ) are as in Notation 3.9.(2) X k = Y k for 0 ≤ k ≤ l .(3) X (0) k = Y k ∩ S k − j =0 Y j .(4) For x ∈ X (0) k and b = ( b , b , . . . , b k − ) in the image in L k − j =0 C ( Y j , M n ( j ) ( D ))of the k − C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) ( k − Y , whenever x ∈ ( Y k \ Y ◦ k ) ∩ Y t ∩ h − n ( t ) ( Y t ) ∩ · · · ∩ h − [ n ( t )+ ··· + n ( t r − ] ( Y t r )with n ( t ) + n ( t ) + · · · + n ( t r ) = n ( k ), then ϕ k ( b ( x )) is given by the blockdiagonal matrix ϕ k ( b ( x )) = b t ( x ) α − n ( t ) ( b t )( x ) . . . α − [ n ( t )+ ··· + n ( t r − )] ( b t r )( x ) . (5) ρ k is the restriction map.The topological dimension of this decomposition is dim( X ), and the standard repre-sentation of σ (cid:0) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y (cid:1) is the inclusion map in L lk =0 C ( Y k , M n ( k ) ( D )). Proof.
The proof is analogous to the proof of Theorem 11.3.19 of [20], while theproof of the statement regarding the topological dimension is the same as for theproof given in [20] for the part of Theorem 11.3.14 that is not included in Theorem11.3.19. (cid:3)
ROSSED PRODUCTS OF C ( X, D ) 25 Structural properties of the orbit breaking subalgebra and thecrossed product
In this section, we give some general theorems on the structure of C*-algebrasof the form C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) , which we derive from results on the structure oforbit breaking subalgebras. We give many explicit examples in Section 6. Convention 4.1.
In this section, as in Definition 2.3 but with additional restric-tions, X will be an infinite compact metric space, h : X → X will be a minimalhomeomorphism, D will be a simple unital C*-algebra, and α ∈ Aut( C ( X, D )) willbe an automorphism which lies over h .For a few results, simplicity is not needed. Proposition 4.2.
Adopt Convention 4.1. For any nonempty closed set Y ⊂ X , C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a direct limit of recursive subhomogeneous algebras over D of the form C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y n for closed subsets Y n ⊂ X with int( Y n ) = ∅ . Proof.
Given Y ⊂ X closed, choose a sequence ( Y n ) ∞ n =1 of closed subsets of X with int( Y n ) = ∅ and Y n +1 ⊂ Y n for n ∈ Z > , and with T ∞ n =1 Y n = Y . That C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a direct limit of separable recursive subhomogeneous alge-bras over D follows by applying Theorem 3.17 and Proposition 3.8. (cid:3) Proposition 4.3.
Adopt Convention 4.1. Let E be a separable unital C*-algebrathat is strongly selfabsorbing in the sense of [63]. Assume that D is separableand E -stable. Let Y ⊂ X be closed and nonempty. Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is E -stable. Proof.
If int( Y ) = ∅ , this follows immediately from Theorem 3.17 and Proposition3.7. The case of a general nonempty closed set now follows from Proposition 4.2above and Corollary 3.4 of [63]. (cid:3) In particular, we get Z -stability. Corollary 4.4.
Adopt Convention 4.1. Let E be a separable unital C*-algebrathat is strongly selfabsorbing in the sense of [63]. Assume that D is separableand E -stable. Let Y ⊂ X be closed and nonempty. Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is Z -stable. Proof.
Strongly selfabsorbing C*-algebras are Z -stable by Theorem 3.1 of [64], and C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is E -stable by Proposition 4.3, so the conclusion is immediate. (cid:3) We now turn to Z -stability of the crossed product C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) . When D is nuclear, it can be obtained from the results of [3]. However, if nuclearity is not as-sumed, then the main conclusion of that paper only implies that C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is tracially Z -stable (in the sense of [23]). Theorem 4.5.
Adopt Convention 4.1. Assume that D is a simple separable unital Z -stable C*-algebra which has a quasitrace. Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is tracially Z -stable. If, in addition, D is nuclear, then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is Z -stable. Proof.
Theorem 3.3 of [23] implies that D has strict comparison of positive elements,and so C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a centrally large subalgebra of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) by Corollary 2.12(2). Also, C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is Z -stable by Corollary 4.4, so Theorem 2.2 of [3] implies that C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is tracially Z -stable. If D isnuclear, then it is Z -stable by Theorem 4.1 of [23], and C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) isnuclear. So Z -stability of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) follows from Theorem 2.3 of [3]. (cid:3) Corollary 4.6.
Adopt Convention 4.1. Let D be a simple separable unital nuclear Z -stable C*-algebra which has a quasitrace. Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) has nucleardimension at most 1. Proof.
This follows immediately from Theorem 4.5 above and Theorem B of [8]. (cid:3)
Corollary 4.4 and its consequences require no assumption about the under-lying dynamical system other than minimality. In particular, if D is Z -stablethen C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is Z -stable when X is infinite dimensional, and evenwhen mdim( X, h ) (as in Notation 0.3) is strictly positive. If moreover D is nu-clear then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is Z -stable. Thus, crossed products of the form C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) are Z -stable whenever h is minimal and D is simple, separa-ble, unital, nuclear, and Z -stable, regardless of anything else about the underlyingdynamics.If D is not assumed to be Z -stable, then we must make assumptions about thedynamical system ( X, h ) besides just minimality to expect C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y tohave tractable structure. The hypotheses in Proposition 4.7, Theorem 4.8, Propo-sition 4.13, and Theorem 4.14 (which also include conditions on D ) are surelymuch stronger than needed. They have the advantage that the proofs can easily byobtained from results already in the literature. Proposition 4.7.
Adopt Convention 4.1, and assume that X is the Cantor set.Let Y ⊂ X be closed and nonempty.(1) If tsr( D ) = 1, then tsr (cid:0) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y (cid:1) = 1.(2) If RR( D ) = 0, then RR (cid:0) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y (cid:1) = 0. Proof.
First suppose that int( Y ) = ∅ . Then, from Lemma 3.11 and the remarkimmediately after it, we see there are are AF algebras B , B , . . . , B k such that C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y ∼ = l M k =0 B k ⊗ D. The result follows immediately.Since stable rank one and real rank zero are preserved by direct limits, the generalcase now follows from Proposition 4.2. (cid:3)
Theorem 4.8.
Adopt Convention 4.1, and assume that X is the Cantor set. Fur-ther assume that one of the following conditions holds:(1) All elements of { α x : x ∈ X } ⊂ Aut( D ) are approximately inner.(2) D has strict comparison of positive elements.(3) D has property (SP) and the order on projections over D is determined byquasitraces.(4) The set (cid:8) α nx : x ∈ X and n ∈ Z (cid:9) is contained in a subset of Aut( D ) whichis compact in the topology of pointwise convergence in the norm on D .If tsr( D ) = 1, then tsr (cid:0) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1)(cid:1) = 1, and if also RR( D ) = 0, thenRR (cid:0) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1)(cid:1) = 0. ROSSED PRODUCTS OF C ( X, D ) 27
The action in this theorem has the Rokhlin property. However, it seems to beunknown whether the crossed product of even a simple unital C*-algebra with stablerank one by a Rokhlin automorphism still has stable rank one. See Problem 7.1and the discussion afterwards.We also emphasize that the conclusion for real rank zero is only valid if stablerank one is also assumed. The reason is the stable rank one hypothesis in Theo-rem 6.4 of [4]. That theorem likely holds without stable rank one, but this has notyet been proved. For now, this is no great loss, since, other than purely infinitesimple C*-algebras (which are covered by our Theorem 2.13), there are no knownexamples of simple unital C*-algebras which have real rank zero but not stable rankone.
Proof of Theorem 4.8.
Choose any one point subset Y ⊂ X . Since tsr( D ) = 1,Proposition 4.7(1) implies that the subalgebra C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y (see Defini-tion 2.3) has stable rank one. If also RR( D ) = 0, then Proposition 4.7(2) im-plies tsr (cid:0) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y (cid:1) = 1. Now C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y is a centrally largesubalgebra of C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) by the appropriate part of Corollary 2.12. Sotsr (cid:0) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1)(cid:1) = 1 by Theorem 6.3 of [4], and if RR( D ) = 0, then Theo-rem 6.4 of [4] gives RR (cid:0) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1)(cid:1) = 0. (cid:3) With more restrictive assumptions on D , enough is known to get results whendim( X ) = 1. Proposition 4.9.
Let A , B , and C be C*-algebras, and let ϕ : A → C and ψ : B → C be homomorphisms. Let D = A ⊕ C,ϕ,ψ B (Definition 3.1). Assume that ψ issurjective, and that A and B have stable rank one. Then D has stable rank one. Proof.
Set J = Ker( ψ ). We have a commutative diagram with exact rows (mapsdefined below): 0 −−−−→ J λ −−−−→ D π −−−−→ A −−−−→ y id B y σ y γ −−−−→ J ι −−−−→ B κ −−−−→ B/J −−−−→ . Here ι is the inclusion of J in B and κ is the quotient map, π : D → A and σ : D → B are the restrictions of the coordinate projections ( a, b ) → a and ( a, b ) → b , and λ ( b ) = ( b,
0) for b ∈ J . For a ∈ A , define γ ( a ) by first choosing b ∈ B such that ψ ( b ) = ϕ ( a ), and then setting γ ( a ) = κ ( b ). It is easy to check that γ is well defined,and that the diagram commutes. All parts of exactness are immediate except thatsurjectivity of π follows from surjectivity of ψ .The algebra A has stable rank one by hypothesis, and J has stable rank one byTheorem 4.4 of [56]. Next, ind : K ( B/J ) → K ( J ) is the zero map by Corollary 2of [46], so K ( ι ) is injective by the long exact sequence in K-theory for the bottomrow. Since K ( ι ) = K ( σ ) ◦ K ( λ ), it follows that K ( λ ) is injective. Thereforeind : K ( A ) → K ( J ) is the zero map by the long exact sequence in K-theory forthe top row. Now D has stable rank one by Corollary 2 of [46]. (cid:3) We state for convenient reference the result on the stable rank of direct limits.
Lemma 4.10.
Let ( A λ ) λ ∈ Λ be a direct system of C*-algebras. Then tsr (cid:0) lim −→ λ A λ (cid:1) ≤ lim inf λ tsr( A λ ). We do not assume that the maps of the direct system are injective.
Proof of Lemma 4.10.
When Λ = Z > , this is Theorem 5.1 of [56]. The proof forgeneral direct limits is the same. (cid:3) Proposition 4.11.
Let D be a simple unital C*-algebra with stable rank one andreal rank zero, and suppose that K ( D ) = 0. Let X be a compact Hausdorff spacewith dim( X ) ≤
1. Then C ( X, D ) has stable rank one.
Proof. If X is a point, this is trivial. If X = [0 , X is a one dimensional finite complex, the result follows from these factsand Proposition 4.9 by induction on the number of cells.Now suppose X is a compact metric space. By Corollary 5.2.6 of [60], there isan inverse system ( X n ) n ∈ Z ≥ of one dimensional finite complexes such that X ∼ =lim ←− n X n . Then C ( X, D ) ∼ = lim −→ n C ( X n , D ). Since C ( X n , D ) has stable rank one forall n ∈ Z ≥ , it follows from Lemma 4.10 that C ( X, D ) has stable rank one.Finally, let X be a general compact Hausdorff space with dim( X ) ≤
1. ByTheorem 1 of [44] (in Section 3 of that paper), there is an inverse system ( X λ ) λ ∈ Λ ofone dimensional compact metric spaces such that X ∼ = lim ←− λ X λ . It now follows that C ( X, D ) has stable rank one by the same reasoning as in the previous paragraph. (cid:3)
Proposition 4.12.
Let D be a simple unital C*-algebra. Let R be a recursivesubhomogeneous C*-algebra over D (Definition 3.2) with topological dimensionat most 1 (Definition 3.3(6)). If D has stable rank one and real rank zero, and K ( D ) = 0, then R has stable rank one. Proof.
The proof is the same as that of Proposition 3.7, except using Proposition 4.9in place of Lemma 3.6, and using Proposition 4.11. (cid:3)
Proposition 4.13.
Adopt Convention 4.1, and assume that dim( X ) ≤
1, that D has stable rank one and real rank zero, and that K ( D ) = 0. Let Y ⊂ X be closedand nonempty. Then tsr (cid:0) C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) Y (cid:1) = 1. Proof.
First suppose that int( Y ) = ∅ . Then the recursive subhomogeneous algebraover D in Theorem 3.17 has base spaces Y k which, following Notation 3.9, are closedsubsets of X . By Proposition 3.1.3 of [53], they therefore have covering dimensionat most 1. The result now follows from Proposition 4.12.Stable rank one is preserved by direct limits (Lemma 4.10), so the general casenow follows from Proposition 4.2. (cid:3) The obvious examples of minimal homeomorphisms of one dimensional spacesare irrational rotations on S , but there are others. See, for example, [21], and thework on minimal homeomorphisms of the product of the Cantor set and the circlein [36] and [37]. Theorem 4.14.
Adopt Convention 4.1, and assume that dim( X ) ≤
1, that D hasstable rank one and real rank zero, and that K ( D ) = 0. Further assume that oneof the following conditions holds:(1) All elements of { α x : x ∈ X } ⊂ Aut( D ) are approximately inner.(2) The order on projections over D is determined by quasitraces.(3) The set (cid:8) α nx : x ∈ X and n ∈ Z (cid:9) is contained in a subset of Aut( D ) whichis compact in the topology of pointwise convergence in the norm on D . ROSSED PRODUCTS OF C ( X, D ) 29
Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) has stable rank one. Proof.
Combine Theorem 6.3 of [4], Corollary 2.12, and Proposition 4.12. (cid:3) Minimality of products of Denjoy homeomorphisms of the Cantorset
For use in examples in the next section, we prove minimality of products ofDenjoy homeomorphisms with rationally independent rotation numbers. We havenot found this result in the literature. The method we use here is not the traditionalapproach to this kind of problem, but we hope it will be useful elsewhere.
Notation 5.1.
For θ ∈ R we let r θ : S → S denote the rotation by 2 πθ , that is, r θ ( ζ ) = exp(2 πiθ ) ζ for ζ ∈ S . Proposition 5.2.
Let I be a set, and let ( θ i ) i ∈ I be a family of elements of R .Suppose that 1 and the numbers θ i for i ∈ I are linearly independent over Q . Thenthe product r : ( S ) I → ( S ) I of the rotations r θ i is minimal. Proof. If I is finite, this is Proposition 1.4.1 of [28]. The general case follows fromthe finite case by taking the inverse limit of the products over finite subsets of I ,by the discussion after Proposition II.4 of [18]. The proof is is written for actionsof the semigroup Z ≥ and for countable inverse limits, but the proof for actions of Z and arbitrary inverse limits is the same. (cid:3) Definition 5.3.
Let X and Y be topological spaces, and let g : Y → X . Let y ∈ Y .We say that g is strictly open at y if (cid:8) g − ( U ) : U ⊂ X is open and g ( y ) ∈ U (cid:9) is a neighborhood base for y in Y .Some of the theory below works without assuming g is continuous, so we do notrequire that neighborhoods be open.The next lemma gives an alternate interpretation of strict openness. Lemma 5.4.
In the situation in Definition 5.3, assume that g is continuous andsurjective, Y is compact, and X and Y are Hausdorff. Then g is strictly open at y if and only if the following conditions hold:(1) g − ( { g ( y ) } ) = { y } .(2) For every open set V ⊂ Y with y ∈ V , the set g ( V ) is a neighborhood of g ( y ).The lemma is false without compactness of Y . Take X = [0 ,
1] and Y = { (0 , } ∪ (cid:0) (0 , × [0 , (cid:1) ⊂ [0 , , let g be projection to the first coordinate, and take y = (0 , g is not strictly open at y . Proof of Lemma 5.4.
Assume that g is strictly open at y . To prove (1), let z ∈ Y \ { y } , choose V ⊂ Y open with y ∈ V and z V , and choose U ⊂ X open suchthat g ( y ) ∈ U and g − ( U ) ⊂ V . Then z g − ( U ), so g ( z ) = g ( y ).To prove (2), let V ⊂ Y be open with y ∈ V . Choose U ⊂ X open such that g ( y ) ∈ U and g − ( U ) ⊂ V . Surjectivity of g implies U ⊂ g ( g − ( U )) ⊂ g ( V ), so g ( V ) is a neighborhood of g ( y ). For the converse, assume (2), and suppose that g is not strictly open at y .Let N be the set of open subsets of X which contain g ( y ), ordered by reverseinclusion. Since g is continuous, failure of strict openness means we can choose V ⊂ Y open with y ∈ V and such that for every U ∈ N , we have g − ( U ) V .So there is z U ∈ Y such that z U V but g ( z U ) ∈ U . The net ( z U ) U ∈N satisfies g ( z U ) → g ( y ). Choose a subnet ( t λ ) λ ∈ Λ such that t = lim λ t λ exists. Then t V .Also g ( t λ ) → g ( t ) by continuity and g ( t λ ) → g ( y ) by construction, so g ( t ) = g ( y ).We have contradicted (1). (cid:3) Lemma 5.5.
Let X and Y be compact Hausdorff spaces, and let h : X → X and k : Y → Y be homeomorphisms. Let g : Y → X be continuous and surjective, andsatisfy g ◦ k = h ◦ g . Suppose that h is minimal and there is a dense subset S ⊂ Y such that g is strictly open at every point of S . Then k is minimal. Proof.
It is enough to show that for every y ∈ Y and every nonempty open set V ⊂ Y , there is n ∈ Z such that k n ( y ) ∈ V . Choose z ∈ S ∩ V . Since g is strictlyopen at z , there is an open set U ⊂ X such that g ( z ) ∈ U and g − ( U ) ⊂ V . Since h is minimal, there is n ∈ Z such that h n ( g ( y )) ∈ U . Then g ( k n ( y )) ∈ U , so k n ( y ) ∈ g − ( U ) ⊂ V . (cid:3) It is well known that Lemma 5.5 fails without strict openness, even if one assumesthat g − ( x ) is finite for all x ∈ X and has only one point for a dense subset of points x ∈ X . The following example was suggested by B. Weiss. Example 5.6.
Fix θ ∈ R \ Q . Take X = S and take h = r θ (rotation by 2 πθ ;Notation 5.1). Define Y ⊂ S × [0 ,
1] to be Y = (cid:0) S × { } (cid:1) ∪ (cid:26)(cid:18) e πinθ , | n | + 1 (cid:19) : n ∈ Z (cid:27) , and define k : Y → Y to be k ( ζ,
0) = ( r θ ( ζ ) ,
0) for ζ ∈ S and k (cid:18) e πinθ , | n | + 1 (cid:19) = (cid:18) e πi ( n +1) θ , | n + 1 | + 1 (cid:19) for n ∈ Z . Let g : Y → X be projection to the first coordinate. Then g ◦ k = h ◦ g , g is surjective, h is minimal, k is not minimal, g − ( x ) has at most two points forevery x ∈ X , and g − ( x ) has one point for all but countably many x ∈ X .The following two easy lemmas will be used in the construction of examples. Lemma 5.7.
Let I be a nonempty set, and for i ∈ I let g i : X i → Y i and y i ∈ Y i be as in Definition 5.3, with g i strictly open at y i . Set X = Y i ∈ I X i and Y = Y i ∈ I Y i , let g : Y → X be given by g ( z ) = ( g i ( z i )) i ∈ I for z = ( z i ) i ∈ I ∈ Y , and set y = ( y i ) i ∈ I .Then g is strictly open at y . Proof.
It is obvious that g − ( { g ( y ) } ) = { y } . Now let V ⊂ Y be open with y ∈ V .We need an open set U ⊂ X such that g ( y ) ∈ U , g − ( U ) is a neighborhood of y ,and g − ( U ) ⊂ V . Without loss of generality there are open subsets V i ⊂ Y i anda finite set F ⊂ I such that V = Q i ∈ I V i , V i = Y i for all i ∈ I \ F , and y i ∈ V i for all i ∈ I . For i ∈ F choose an open set U i ⊂ X i with g i ( y i ) ∈ U i and suchthat g − i ( U i ) ⊂ V i . Set U i = X i for i ∈ I \ F . Since g − i ( U i ) is a neighborhood ROSSED PRODUCTS OF C ( X, D ) 31 of y i for all i ∈ I and U i = X i for i ∈ I \ F , the set g − ( U ) = Q i ∈ I g − i ( U i ) is aneighborhood of y . So the set U = Q i ∈ I U i is the required set. (cid:3) Lemma 5.8.
In the situation in Definition 5.3, assume that g is strictly open at y .Let Y ⊂ Y be a subspace such that y ∈ Y . Then g | Y is strictly open at y . Proof.
The result is immediate from the definition of the subspace topology. (cid:3)
Lemma 5.9.
Let g : S → S be continuous and surjective. Suppose that thereare disjoint closed arcs I , I , . . . ⊂ S such that:(1) For n ∈ Z > , the function g is constant on I n .(2) S ∞ n =1 I n is dense in S .(3) With R = S \ S ∞ n =1 I n , the restriction g | R is injective.Then g is strictly open at every point of R . Proof.
For distinct λ, ζ ∈ S , we denote the open, closed, and half open arcs from λ to ζ using interval notation: ( λ, ζ ), [ λ, ζ ], [ λ, ζ ), and ( λ, ζ ]. Further write I n =[ β n , γ n ] with β n , γ n ∈ S , and set B = { β n : n ∈ Z > } and C = { γ n : n ∈ Z > } .The set g ( S \ R ) is countable and g is surjective, so S \ g ( R ) is countable.First, we claim that if E, F ⊂ S are disjoint, then int( g ( E )) ∩ int( g ( F )) = ∅ . Ifthe claim is false, then g ( E ) ∩ g ( F ) is uncountable. Now g ( E ) \ g ( E ∩ R ) ⊂ g ( S \ R )which is countable, so g ( E ) \ g ( E ∩ R ) is countable. Similarly g ( F ) \ g ( F ∩ R ) iscountable. So g ( E ∩ R ) ∩ g ( F ∩ R ) is uncountable. This contradicts E ∩ F = ∅ and injectivity of g | R .Second, we claim that if J ⊂ S is any nonempty open arc such that there is no n ∈ Z > with J ⊂ I n , then R ∩ J = ∅ . Suppose the claim is false. Write J = ( λ, ζ )with λ, ζ ∈ S . Define S = ( ∅ λ R { λ } λ ∈ R and T = ( ∅ ζ R { ζ } ζ ∈ R. Then J is the disjoint union of the closed sets S, T, J ∩ I , J ∩ I , . . . . Suppose there is n ∈ Z > such that J ∩ I m = ∅ for all m = n . Since J I n ,there is a nonempty open arc L ⊂ J such that L ∩ I n = ∅ , whence L ∩ I m = ∅ for all m ∈ Z > . This contradicts (2). Thus, at least two of the sets J ∩ I n arenonempty. But, according to Theorem 6 in Part III of Section 47 (in Chapter 5)of [32], a connected compact Hausdorff space cannot be the disjoint union of closedsubsets E , E , . . . with at least two of them nonempty. This contradiction provesthe claim.Third, we claim that, under the same hypotheses, R ∩ J is infinite. Again write J = ( λ, ζ ) with λ, ζ ∈ S . Apply the previous claim to choose ρ ∈ R ∩ J and set J = ( ρ , ζ ). Then ρ S ∞ n =1 I n so there is no n ∈ Z > with J ⊂ I n . Apply theprevious claim again, choosing ρ ∈ R ∩ J , and showing that J = ( ρ , ζ ) satisfies J I n for all n ∈ Z > . Proceed inductively.Fourth, we claim that if λ ∈ R ∪ C and ζ ∈ R ∪ B are distinct, then g (cid:0) ( λ, ζ ) (cid:1) is an open arc. For the proof, set J = ( λ, ζ ). We can assume that g ( J ) = S .Since g ( J ) is connected, there are µ, ν ∈ S such that g ( J ) is one of ( µ, ν ), [ µ, ν ](in this case we allow µ = ν ), [ µ, ν ), or ( µ, ν ]. Suppose µ ∈ g ( J ); we will obtain a contradiction. Choose ρ ∈ J such that g ( ρ ) = µ . There are two cases: ρ ∈ R and ρ ∈ S ∞ n =1 I n .In the first case, let L and L be the nonempty arcs L = ( λ, ρ ) and L = ( ρ, ζ ).For j = 0 ,
1, since ρ ∈ L j , the arc L j satisfies the hypotheses of the second claim.So R ∩ L j is infinite by the third claim. Since g | R is injective, g ( L j ) is infinite.Clearly µ ∈ g ( L j ) ⊂ [ µ, ν ]. Also, g ( L j ) is an arc by connectedness. So there is ω j ∈ ( µ, ν ) such that ( µ, ω j ) ⊂ g ( L j ). This implies int( g ( L )) ∩ int( g ( L )) = ∅ ,contradicting the first claim.In the second case, let n ∈ Z > be the integer such that ρ ∈ I n . Both λ ∈ [ β n , γ n ) and ζ ∈ ( β n , γ n ] are impossible, but [ β n , γ n ] ∩ ( λ, ζ ) = ∅ , so [ β n , γ n ] ⊂ ( λ, ζ ). Therefore the arcs L = ( λ, β n ) and L = ( γ n , ζ ) are nonempty and do notintersect I n . If there is m ∈ Z > such that L ⊂ I m , then clearly m = n , but then β n ∈ I m ∩ I n , contradicting I m ∩ I n = ∅ . So no such m exists, whence L satisfiesthe hypotheses of the second claim. Similarly L satisfies the hypotheses of thesecond claim. Since g is constant on I n , we have g ( β n ) = g ( γ n ) = g ( ρ ) = µ . Nowwe get a contradiction in the same way as in the first case.So µ g ( J ). Similarly ν g ( J ). The claim is proved.Fifth, we claim that if L ⊂ S is an open arc, then g − ( L ) is an open arc.To prove the claim, since g − ( L ) is open, there are an index set S and disjointnonempty open arcs J s for s ∈ S such that g − ( L ) = S s ∈ S J s , and there are λ s , ζ s ∈ S such that J s = ( λ s , ζ s ).Let s ∈ S . Suppose λ s ∈ I n for some n ∈ Z > . Since g is constant on I n , either I n ∩ g − ( L ) = ∅ or I n ⊂ g − ( L ). In the second case, I n ∪ ( λ s , ζ s ) is a connectedsubset of g − ( L ). Since ( λ s , ζ s ) is a maximal connected subset of g − ( L ), we have I n ⊂ ( λ s , ζ s ). This contradicts λ s ∈ I n . So the first case must apply, and therefore λ s = γ n . Combining this with the possibility λ s S ∞ n =1 I n , we conclude that λ s ∈ R ∪ C . Similarly, ζ s ∈ R ∪ B . By the previous claim, g (cid:0) ( λ s , ζ s ) (cid:1) is open.The first claim now implies that the sets g (cid:0) ( λ s , ζ s ) (cid:1) are all disjoint. Since L isconnected, it follows that card( S ) = 1, which implies the claim.To prove the lemma, let η ∈ R and let V ⊂ S be open with η ∈ V . We need anopen set U ⊂ S such that g ( η ) ∈ U and g − ( U ) ⊂ V . Choose λ , ζ ∈ S such that η ∈ ( λ , ζ ) ⊂ V . Use the second claim to choose λ ∈ ( λ , η ) ∩ R and ζ ∈ ( η, ζ ) ∩ R .Then U = g (cid:0) ( λ , ζ ) (cid:1) is open by the fourth claim, and by the fifth claim there are λ , ζ ∈ S such that g − ( U ) = ( λ , ζ ). If λ ∈ g − ( U ), then ( λ , λ ) is a nonemptyopen arc contained in g − ( U ) with λ ∈ R . The third claim and injectivity of g | R imply that g (cid:0) ( λ , λ ) (cid:1) is not a point. Since g (cid:0) ( λ , λ ) (cid:1) is connected, it hasnonempty interior. It is contained in U = g (cid:0) ( λ , ζ ) (cid:1) , contradicting the first claim.So λ ( λ , ζ ). Similarly ζ ( λ , ζ ). Since η ∈ ( λ , ζ ), η ∈ ( λ , ζ ), and ( λ , ζ )is connected, it now follows that ( λ , ζ ) ⊂ ( λ , ζ ). (In fact, we have equality, since g (cid:0) ( λ , ζ ) (cid:1) = U .) Thus g − ( U ) = ( λ , ζ ) ⊂ ( λ , ζ ) ⊂ ( λ , ζ ) ⊂ V. This completes the proof. (cid:3)
We have not seen a term in the literature for the following class of homeomor-phisms.
ROSSED PRODUCTS OF C ( X, D ) 33
Definition 5.10. A restricted Denjoy homeomorphism is a homeomorphism of theCantor set which is conjugate to the restriction and corestriction of a Denjoy home-omorphism of S , in the sense of Definition 3.3 of [54], to the unique minimal setof Proposition 3.4 of [54]. The rotation number of a restricted Denjoy homeomor-phism is the rotation number, as at the beginning of Section 3 of [54], of the Denjoyhomeomorphism of S of which it is the restriction; this number is well defined byRemark 3 at the end of Section 3 of [54]. Lemma 5.11.
Let X be the Cantor set, and let k : X → X be a restricted Denjoyhomeomorphism with rotation number θ , as in Definition 5.10. Then there are acontinuous surjective map g : X → S and a dense subset R ⊂ X such that g isstrictly open at x for every x ∈ R and, following Notation 5.1, g ◦ h = r θ ◦ g . Proof.
We may assume that k : S → S is a Denjoy homeomorphism as in Defi-nition 3.3 of [54], that X ⊂ S is the unique minimal set for k , and that k is therestriction and corestriction of k to X . Corollary 3.2 of [54] gives h : S → S suchthat h ◦ k = r θ ◦ h . Set g = h | X . Since X ⊂ S is compact and invariant and r θ is minimal, g must be surjective. The discussion after Proposition 3.4 of [54]implies that h satisfies the hypotheses of Lemma 5.9, with X = S \ S ∞ n =1 int( I n ).Moreover, the set R of Lemma 5.9 is nonempty by Theorem 6 in Part III of Sec-tion 47 (in Chapter 5) of [32], and k -invariant, hence dense in X . By Lemma 5.9and Lemma 5.8, g is strictly open at every point of R . (cid:3) Proposition 5.12.
Let I be a set, let ( θ i ) i ∈ I be a family of elements of R , andfor i ∈ I let k i : X i → X i be a restricted Denjoy homeomorphism with rotationnumber θ i , as in Definition 5.10. Suppose that 1 and the numbers θ i for i ∈ I are linearly independent over Q . Then the product k : Q i ∈ I X i → Q i ∈ I X i of thehomeomorphisms k i is minimal. Proof.
For i ∈ I , let g i : X i → S be as in Lemma 5.11. By Lemma 5.11, thereis a dense subset R i ⊂ X i such that g i is strictly open at every point of R i . Let g : Q i ∈ I X i → ( S ) I be the product of the maps g i . Lemma 5.7 implies that g is strictly open at every point of Q i ∈ I R i , this set is dense, and the product r : ( S ) I → ( S ) I of the rotations r θ i is minimal by Proposition 5.2. So k is minimalby Lemma 5.5. (cid:3) Examples
To take full advantage of Proposition 4.2, we need information on the structureof simple direct limits of recursive subhomogeneous algebras over the algebra D which appears there. For example, we need conditions for such direct limits tohave stable rank 1 (as originally done for D = C in [12]) and to have real rank zero(as originally done for D = C in [6]). We intend to study this problem in a futurepaper. Even without generalizations of those theorems, three cases are accessiblenow. They are h : X → X wild (for example, with strictly positive mean dimension,as in Notation 0.3) but with D being Z -stable; X is the Cantor set and D has stablerank one but is otherwise wild; and X = S , h is an irrational rotation, and D hasstable rank one, real rank zero, and trivial K -group, but is otherwise wild.We also give examples for Theorem 2.13, although only for G = Z .In the first type of example, if X is finite dimensional then the action has ahigher dimensional Rokhlin property, and results on crossed products by such ac-tions can be applied. As far as we know, however, there are no previously known general theorems which apply when X is infinite dimensional. In the second typeof example, the action has the Rokhlin property. Since the algebra is not simple,results of [48] can’t be applied, and, when D is not Z -stable and does not havefinite nuclear dimension, Theorems 4.1 and 5.8 of [24] can’t be applied. We knowof no previous general theorems which apply in this case. Actions in the third typeof example are further away from known results: the situation is like for the secondtype of example, but the Rokhlin property must be weakened to finite Rokhlindimension with commuting towers.Since we will use quasifree automorphisms of reduced C*-algebras of free groupsand the free shift on C ∗ r ( F ∞ ) in several examples, we establish notation for themseparately. Notation 6.1.
For n ∈ Z > ∪{∞} , we let F n denote the free group on n generators.We let u , u , . . . , u n ∈ C ∗ r ( F n ) (when n = ∞ , for u , u , . . . ∈ C ∗ r ( F ∞ )) be the“standard” unitaries in C ∗ r ( F n ), obtained as the images of the standard generatorsof F n . For ζ = ( ζ , ζ , . . . , ζ n ) ∈ ( S ) n (when n = ∞ , for ζ = ( ζ n ) n ∈ Z > ∈ ( S ) Z > ),we let ϕ ζ ∈ Aut( C ∗ r ( F n )) be the (quasifree) automorphism determined by ϕ ζ ( u k ) = ζ k u k for k = 1 , , . . . , n (when n = ∞ , for k = 1 , , . . . ). It is well known that ζ ϕ ζ is continuous. Notation 6.2.
Take the standard generators of the free group F ∞ to be indexedby Z , and for n ∈ Z let u n ∈ C ∗ r ( F ∞ ) be the unitary obtained as the image of thecorresponding generator of F ∞ . We denote by σ the free shift on C ∗ r ( F ∞ ), that is,the automorphism σ ∈ Aut( C ∗ r ( F ∞ )) determined by σ ( u n ) = u n +1 for n ∈ Z . Example 6.3.
Set X = ( S ) Z . Let h : X → X be the (forwards) shift, definedfor ζ = ( ζ n ) n ∈ Z ∈ X by h ( ζ ) = ( ζ n − ) n ∈ Z . Let D = N n ∈ Z > M be the 2 ∞ UHFalgebra. Choose a bijection σ : Z > → Z , and for ζ = ( ζ n ) n ∈ Z ∈ X define α (0) ζ = O n ∈ Z > Ad (cid:18)(cid:18) ζ σ ( n ) (cid:19)(cid:19) ∈ Aut( D ) . Then ζ α (0) ζ is continuous.We have mdim( h ) = 1 (see Notation 0.3) by Proposition 3.3 of [43], and wecan use X , h , D , and ζ α (0) ζ in Lemma 1.4. However, h is not minimal.We therefore proceed as follows. Identify [0 ,
1] with a closed arc in S , say via λ exp( πiλ ). Use this identification to identify [0 , Z with a closed subset of X = ( S ) Z . This identification is equivariant when both spaces are equipped withthe shift homeomorphisms. Let X ⊂ X and h = h | X : X → X be the minimalsubshift in [19], which can be taken to have mean dimension arbitrarily close to 1.For ζ ∈ X let α ζ = α (0) ζ . Let α : Z → Aut( C ( X, D )) be the corresponding action asin Lemma 1.4. Then α lies over the free minimal action of Z on X generated by h .The crossed product C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is simple by Proposition 1.6, and isnuclear, so it is Z -stable by Theorem 4.5.The action in Example 6.3 can also be described as follows. Realize D as thealgebra of the canonical anticommutation relations on generators a k for k ∈ Z > ,following Section 5.1 of [7]. Then α (0) ζ is the gauge automorphism α (0) ζ ( a k ) = ζ σ ( k ) a k . In [7], see Section 5.1 and the proof of Lemma 5.2.The next example is a slightly different version, with larger mean dimension. ROSSED PRODUCTS OF C ( X, D ) 35
Example 6.4.
Let X , h , D , and ζ α (0) ζ be as in Example 6.3. We will,however, use the homeomorphism h of X , which is the shift on ( S × S ) Z . Let( X, h ) be the minimal subspace of the shift on ([0 , ) Z constructed in Proposition3.5 of [43], which satisfies mdim( h ) > , in ( S ) to choose an equivariant homeomorphism g from ( X, h ) to an invariantclosed subset of ( X , h ). For x ∈ X let α x = α (0) g ( x ) . Let α : Z → Aut( C ( X, D )) bethe corresponding action as in Lemma 1.4. The crossed product C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is again simple by Proposition 1.6, so Z -stable by Theorem 4.5. Example 6.5.
Let X , h , D , and ζ α ζ be as in Example 6.3. Define E = D ⊗ C ∗ r ( F ∞ ), and, following Notation 6.1, for ζ ∈ X define β ζ = α ζ ⊗ ϕ ζ . ApplyLemma 1.4 to get an action β : Z → Aut( C ( X, E )) which lies over the free minimalaction of Z on X generated by h .We claim that C ∗ (cid:0) Z , C ( X, E ) , β (cid:1) is tracially Z -stable and has stable rank one.Tracial Z -stability follows from Theorem 4.5. For stable rank one, E is exact, sohas strict comparison of positive elements by Corollary 4.6 of [59]. Choose anyone point subset Y ⊂ X . Then C ∗ (cid:0) Z , C ( X, E ) , β (cid:1) Y (see Definition 2.3) is Z -stable by Corollary 4.4. This algebra is simple by Proposition 1.6, so by Theorem6.7 of [59] it has stable rank one. The algebra C ∗ (cid:0) Z , C ( X, E ) , β (cid:1) Y is centrallylarge in C ∗ (cid:0) Z , C ( X, E ) , β (cid:1) by Corollary 2.12(2), so Theorem 6.3 of [4] implies that C ∗ (cid:0) Z , C ( X, E ) , β (cid:1) has stable rank one.In Example 6.5, we don’t know whether C ∗ (cid:0) Z , C ( X, E ) , β (cid:1) is Z -stable. We alsodon’t know whether tracial Z -stability implies stable rank one, although this isexpected to be true.There is nothing special about the specific formulas for x α x in Example 6.3and Example 6.4, and x β x in Example 6.5. They were chosen merely to showthat interesting examples exist.Example 6.3, Example 6.4, and Example 6.5 were constructed so that the home-omorphism h does not have mean dimension zero. If X is finite dimensional, thenone can get all we do by using known results for crossed products by actions withfinite Rokhlin dimension with commuting towers. With finite dimensional X , oneeven gets Z -stability in examples like Example 6.3, Example 6.4, and Example 6.5,by Theorem 5.8 of [24]. However, we know of no results which apply to examplesof this type when X is infinite dimensional and h has mean dimension zero.The next most obvious choice for a minimal homeomorphism of an infinite dimen-sional space X seems to be as follows. Take X = ( S ) Z > , fix θ = ( θ n ) n ∈ Z > ∈ R Z > ,and define h : X → X by h (cid:0) ( ζ n ) n ∈ Z > (cid:1) = (cid:0) e πiθ n ζ n (cid:1) n ∈ Z > . If 1 , θ , θ , . . . are lin-early independent over Q , then h is minimal by Proposition 5.2. However, byconsidering the action in just one coordinate, one sees that, regardless of ζ α ζ ,the action has finite Rokhlin dimension with commuting towers. By Theorem 6.2of [24], irrational rotations have finite Rokhlin dimension with commuting towers.Remark 6.3 of [24], according to which the result extends to any homeomorphismwhich has an irrational rotation as a factor, applies equally well to any automor-phism of C ( S ) ⊗ B , for any unital C*-algebra B , which lies over an irrationalrotation on S in the sense of Definition 1.2. If we start with C (cid:0) ( S ) Z > , D (cid:1) , take B above to be C (cid:0) ( S ) Z > \{ } , D (cid:1) . Therefore Theorem 5.8 of [24] applies, and ourresults give nothing new. We now give some examples of the second type discussed in the introduction tothis section. For easy reference, we recall a result on the stable rank of reduced freeproducts. Many reduced free products have stable rank 1. The following result isfrom [15]. (It is not affected by the correction [16].) Reduced free products of unitalC*-algebras in [15] are implicitly taken to be amalgamated over C ; see Section 2.2of [15]. Proposition 6.6 (Corollary 3.9 of [15]) . Let G and H be discrete groups withcard( G ) ≥ H ) ≥
3. Then C ∗ r ( G ⋆ H ) has stable rank 1.
Example 6.7.
Let X be the Cantor set and let h be an arbitrary minimal homeo-morphism of X . Let σ ∈ Aut( C ∗ r ( F ∞ )) be as in Notation 6.2. Let α ∈ Aut (cid:0) C ( X ) ⊗ C ∗ r ( F ∞ ) (cid:1) be the tensor product of the automorphism f f ◦ h − of C ( X ) and σ .Then C ∗ (cid:0) Z , C ( X ) ⊗ C ∗ r ( F ∞ ) , α (cid:1) is simple by Proposition 1.6.We claim that C ∗ (cid:0) Z , C ( X ) ⊗ C ∗ r ( F ∞ ) , α (cid:1) has stable rank 1.To prove the claim, we apply Theorem 4.8 with D = C ∗ r ( F ∞ ) and α as given.Use Theorems 9.2.6 and 9.2.7 of [20] to see that D is simple and has a tracialstate. By Proposition 6.6, we have tsr( D ) = 1. By Proposition 6.3.2 of [57], D hasstrict comparison of positive elements. By construction, α lies over h . Now applyTheorem 4.8(2).Apparently no previously known results give anything about the crossed productin this example. In particular, as discussed after Problem 7.1, knowing that theaction has the Rokhlin property doesn’t seem to help.There are many other automorphisms of C ∗ r ( F ∞ ) which could be used in placeof σ . For example, one could take a quasifree automorphism, as in Notation 6.1.Here is a more interesting version. Example 6.8.
Fix any n ∈ { , , . . . , ∞} and take D = C ∗ r ( F n ). Adopt Nota-tion 6.1. Let h : X → X be a restricted Denjoy homeomorphism with rotationnumber θ ∈ R \ Q , as in Definition 5.10. Let ζ : X → S be the continuous surjec-tive map with ζ ( h ( x )) = e πiθ ζ ( x ) of Lemma 5.11 (gotten from Corollary 3.2 andProposition 3.4 of [54]). Following Notation 6.1 for the generators of D , for x ∈ X let α x ∈ Aut( D ) be determined by α x ( u k ) = ζ ( x ) u k for k = 1 , , . . . , n (or k ∈ Z > if n = ∞ ). Apply Lemma 1.4 to get an action α : Z → Aut (cid:0) C ( X, D ) (cid:1) , which wecan think of as a kind of noncommutative Furstenberg transformation. The algebra C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is simple by Proposition 1.6.We claim that C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) has stable rank one. To prove the claim, useTheorems 9.2.6 and 9.2.7 of [20] to see that D is simple and has a tracial state. ByProposition 6.6, we have tsr( D ) = 1. Now apply Theorem 4.8(4).Again, apparently no previously known results give anything about the crossedproduct. Since C ∗ r ( F n ) is not Z -stable, knowing that the action has the Rokhlinproperty doesn’t seem to help.We can generalize Example 6.8 as follows. Example 6.9.
Fix n ∈ { , , . . . , ∞} . Fix θ , θ , . . . , θ n ∈ R (or θ , θ , . . . ∈ R )such that 1 , θ , θ , . . . , θ n (or 1 , θ , θ , . . . ) are linearly independent over Q . For k =1 , , . . . , n , let h : X k → X k be a restricted Denjoy homeomorphism with rotationnumber θ k ∈ R \ Q , as in Definition 5.10. Take X = Q nk =1 X k , and let h : X → X act as h k on the k -th factor. Then h is minimal by Proposition 5.12. Define ζ k : X k → S analogously to the definition of ζ in Example 6.8, and take α x ( u k ) = ROSSED PRODUCTS OF C ( X, D ) 37 ζ k ( x k ) u k . Apply Lemma 1.4 to get an action α : Z → Aut (cid:0) C ( X, C ∗ r ( F n )) (cid:1) . Then C ∗ (cid:0) Z , C ( X, C ∗ r ( F n )) , α (cid:1) is simple and has stable rank one for the same reasons asin Example 6.8. Example 6.10.
Let X be the Cantor set and let h be an arbitrary minimal home-omorphism of X . Choose any decomposition X = X ∐ X of X as the disjointunion of two nonempty closed subsets. Set D = C ∗ r ( F ∞ ), with the generators of F ∞ indexed by Z . Fix any ζ = ( ζ n ) n ∈ Z ∈ ( S ) Z > . For x ∈ X take α x to be theautomorphism ϕ ζ of Notation 6.1, except using Z in place of Z > , and for x ∈ X take α x to be the automorphism σ of Notation 6.2. Apply Lemma 1.4 to get anaction α : Z → Aut( C ( X, D )). The algebra C ∗ ( Z , C ( X ) ⊗ D, α ) is simple by Propo-sition 1.6. The algebra D has strict comparison of positive elements by Proposition6.3.2 of [57], so C ∗ ( Z , C ( X ) ⊗ D, α ) has stable rank one by Theorem 4.8(2).Example 6.10 admits many variations. Here are several.
Example 6.11.
Let h : X → X be a restricted Denjoy homeomorphism, as inExample 6.8, and let ζ : X → S be as there. Choose any decomposition X = X ∐ X of X as the disjoint union of two nonempty closed subsets. FollowingNotation 6.1 for the generators of C ∗ r ( F ), for x ∈ X let α x ∈ Aut( C ∗ r ( F n ))be determined by α x ( u ) = ζ ( x ) u and α x ( u ) = u , and for x ∈ X let α x ∈ Aut( C ∗ r ( F n )) be determined by α x ( u ) = u and α x ( u ) = ζ ( x ) u . The algebra C ∗ (cid:0) Z , C ( X ) ⊗ C ∗ r ( F ) , α (cid:1) is simple by Proposition 1.6.We claim that this algebra has stable rank one. To prove the claim, use The-orems 9.2.6 and 9.2.7 of [20] to see that D is simple and has a tracial state. ByProposition 6.6, we have tsr( D ) = 1. Now apply Theorem 4.8(4). Example 6.12.
In Example 6.11, replace the definition of α x for x ∈ X with α x ( u ) = u and α x ( u ) = u . Proposition 1.6 and Theorem 4.8(4) still apply, so C ∗ (cid:0) Z , C ( X ) ⊗ C ∗ r ( F ) , α (cid:1) again is simple and has stable rank one.We now give examples in which we also get real rank zero. The following lemmawill be used for them. Lemma 6.13.
Let M be factor of type II , let α : G → Aut( M ) be an action ofa countable group G on M , and let P ⊂ M be a separable C*-subalgebra. Thenthere exists a simple separable unital C*-subalgebra D ⊂ M which contains P , isinvariant under σ , has a unique tracial state (the restriction to D of the uniquetracial state on M ), has real rank zero and stable rank one, and such that the orderon projections over D is determined by traces. Proof.
The proof is the same as that of Proposition 3.1 of [50], except for G -invariance. We construct by induction on n ∈ Z ≥ separable unital subalgebras A n , B n , C n , D n , E n , F n ⊂ M with P ⊂ A ⊂ B ⊂ C ⊂ D ⊂ E ⊂ F ⊂ · · · ⊂ A n ⊂ B n ⊂ C n ⊂ D n ⊂ E n ⊂ F n ⊂ · · · such that A n , B n , C n , D n , and E n have the properties in the proof of Propo-sition 3.1 of [50] ( A n is simple, etc.), and such that F n is G -invariant. In theinduction step, after constructing E n , we take F n to be the C*-subalgebra gener-ated by S g ∈ G α g ( E n ). Then S ∞ n =0 F n is G -invariant. The proof in [50] now givesthe desired conclusion, except that the conclusion is that K ( D ) → K ( M ) is anorder isomorphism onto its range instead of that the order on projections over D is determined by traces. But the conclusion we get implies the conclusion we wantif A has cancellation, and stable rank one implies cancellation by Proposition 6.4.1and Proposition 6.5.1 of [5]. (cid:3) Example 6.14.
Let X be the Cantor set and let h be an arbitrary minimal home-omorphism of X . Let σ ∈ Aut( C ∗ r ( F ∞ )) be as in Notation 6.2.We regard C ∗ r ( F ∞ ) as a subalgebra of the group von Neumann algebra W ∗ ( F ∞ )in the usual way, and we let σ ∈ Aut( W ∗ ( F ∞ )) be the von Neumann algebraautomorphism which shifts the generators of F ∞ in the same way that σ does.Thus σ | C ∗ r ( F ∞ ) = σ . Choose a σ -invariant subalgebra D ⊂ W ∗ ( F ∞ ) which contains C ∗ r ( F ∞ ) as in Lemma 6.13, with the properties there. Set γ = σ | D ∈ Aut( D ). Thendefine α ∈ Aut( C ( X ) ⊗ D ) in the same was as in Example 6.7, using γ in placeof σ . The algebra C ∗ (cid:0) Z , C ( X ) ⊗ D, α (cid:1) is simple by Proposition 1.6. It has stablerank 1 and real rank zero by Theorem 4.8(3).The algebra D is not nuclear because the Gelfand-Naimark-Segal representationfrom its tracial state gives a nonhyperfinite factor.It is perhaps interesting to point out that every quasitrace on the algebra D inExample 6.14 is a trace. This is a consequence of the following lemma. Lemma 6.15.
Let A be a unital C*-algebra with real rank zero. Suppose that τ is aquasitrace on A and that whenever p, q ∈ M ∞ ( A ) are projections with τ ( p ) < τ ( q ),then p - q . Then τ is the only quasitrace on A . Proof.
Suppose the conclusion is false, and let σ be some other quasitrace on A .By definition, for any quasitrace ρ on A and any a, b ∈ A sa , we have ρ ( a + ib ) = ρ ( a ) + iρ ( b ). Therefore there is a ∈ A sa such that σ ( a ) = τ ( a ). Since quasitracesare continuous, it follows from real rank zero that there is a ∈ A sa such that a has finite spectrum and σ ( a ) = τ ( a ). Since quasitraces are linear on commutativeC*-subalgebras, there is a projection p ∈ A such that σ ( p ) = τ ( p ).Suppose σ ( p ) < τ ( p ). Choose m, n ∈ Z ≥ such that mσ ( p ) < n < mτ ( p ). Defineprojections e, f ∈ M ∞ ( A ) by e = 1 M m ⊗ p and f = 1 M n ⊗ A . Then τ ( f ) < τ ( e ), so f - e . But σ ( f ) > σ ( e ), a contradiction. Similarly, σ ( p ) > τ ( p ) is also impossible.This contradiction shows that σ does not exist. (cid:3) Example 6.16.
Let n ∈ { , , . . . } . Following Notation 6.1 for the generatorsof C ∗ r ( F n ), for ρ in the symmetric group S n let ψ ρ ∈ Aut( C ∗ r ( F n )) be the auto-morphism determined by ψ ρ ( u k ) = u ρ − ( k ) for k = 1 , , . . . , n , and let ψ ρ be thecorresponding automorphism of the group von Neumann algebra W ∗ ( F n ).By Theorems 9.2.6 and 9.2.7 of [20], the algebra C ∗ r ( F n ) is simple and has aunique tracial state. Use Lemma 6.13 to find a simple separable unital C*-algebra D ⊂ W ∗ ( F n ) which contains C ∗ r ( F n ), is invariant under all the automorphisms ψ ρ for ρ ∈ S n , and has the other properties given in Lemma 6.13.Let X be the Cantor set, and let h : X → X be any minimal homeomorphism.Choose any decomposition X = ` ρ ∈ S n X ρ of X as the disjoint union of nonemptyclosed subsets. For x ∈ X ρ let α x = ψ ρ | D . Let α : Z → Aut( C ( X, D )) be thecorresponding action as in Lemma 1.4.The algebra C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is simple by Proposition 1.6. It has stable rankone and real rank zero by Theorem 4.8(4). The algebra D is not nuclear for thesame reason as in Example 6.14.The next example is of the third type discussed in the introduction to this section. ROSSED PRODUCTS OF C ( X, D ) 39
Example 6.17.
Let ( π , π , . . . ) be a sequence of unital finite dimensional repre-sentations of C ∗ ( F ) such that, for every n ∈ Z > , the representation L ∞ k = n π k isfaithful. For n ∈ Z ≥ , let d ( n ) be the dimension of π n , and define l ( n ) = d ( n ) + 4and r ( n ) = Q nk =1 l ( k ). Let u , u ∈ C ∗ ( F ) be the “standard” unitaries in C ∗ ( F ),obtained as the images of the standard generators of F , as in Notation 6.2 exceptthat we are now using the full C*-algebra instead of the reduced C*-algebra. Let γ , γ , γ ∈ Aut( C ∗ ( F )) be the automorphisms determined by γ ( u ) = u − and γ ( u ) = u ,γ ( u ) = u and γ ( u ) = u − , and γ ( u ) = u − and γ ( u ) = u − . For n ∈ Z > define D n = M r ( n ) ⊗ C ∗ ( F ), which we identify as M l ( n ) ⊗ M l ( n − ⊗ · · · ⊗ M l (1) ⊗ C ∗ ( F ) , and define γ n,n − : D n − → D n by, for a ∈ D n − = M r ( n − ⊗ C ∗ ( F ), γ n,n − ( a ) = diag (cid:0) a, (id M r ( n − ⊗ ϕ )( a ) , (id M r ( n − ⊗ ϕ )( a ) , (id M r ( n − ⊗ ϕ )( a ) , ( π n ⊗ id M r ( n − ( a ) ⊗ (cid:1) ∈ M l ( n ) ⊗ D n − = D n . Let D be the direct limit of the resulting direct system. It follows by methods of [11]that D is simple, separable, and has tracial rank zero. In particular, D has stablerank one and real rank zero by Theorem 3.4 of [34], and the order on projectionsover D is determined by traces by Theorem 6.8 of [33]. Moreover, using the knownresult for K ∗ ( C ∗ ( F n )) (see [9]), one gets K ( D ) = 0.For ζ ∈ S define inductively unitaries u n ( ζ ) ∈ D n as follows. Set u ( ζ ) = 1,and, given u n − ( ζ ) ∈ D n − , set u n ( ζ ) = (cid:0) diag (cid:0) ζ, , , . . . , (cid:1) ⊗ D n − (cid:1) γ n,n − ( u n − ( ζ )) ∈ M l ( n ) ⊗ D n − = D n . Then the actions ζ Ad( u n ( ζ )) ∈ Aut( D n ) of S on D n are compatible with thedirect system, and so yield a continuous map (in fact, an action) ζ α ζ : S → Aut( D ). Let h : S → S be an irrational rotation. Apply Lemma 1.4 to get anaction α : Z → Aut( C ( S , D )).The algebra C ∗ (cid:0) Z , C ( S , D ) , α (cid:1) is simple by Proposition 1.6 and has stable rankone by Theorem 4.14(2).Methods of [47] will probably show that the algebra D in Example 6.17 is not Z -stable (see Question 7.7), although it definitely is tracially Z -stable.Finally, we give purely infinite examples. Example 6.18.
Let (
X, h ) be the minimal subshift (of the shift on ( S ) Z ) withnonzero mean dimension used in Example 6.3. Let D = O ∞ , but with standardgenerators indexed by Z , say s k for k ∈ Z . Let α ζ be the gauge automorphism, givenby α ζ ( s k ) = ζ k s k . Let α : Z → Aut( C ( X, D )) be the corresponding action as inLemma 1.4. Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is purely infinite and simple by Theorem 2.13.Since C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is nuclear, it is necessarily O ∞ -stable by Theorem 3.15of [30].In Example 6.18, Theorems 4.1 and 5.8 of [24] don’t apply, since there is noreason to think that α has finite Rokhlin dimension with commuting towers. Example 6.19.
Let D be the reduced free product D = ( M ⊗ M ) ⋆ r C ([0 , C ([0 , M ⊗ M given by tensor product of the usual tracial state tr with the state ρ ( x ) =tr (cid:0) diag (cid:0) , (cid:1) x (cid:1) on M . It is shown in Example 5.8 of [1] that D is purely infiniteand simple but not Z -stable.Take X = ( S ) , and for ζ = ( ζ , ζ , ζ , ζ ) ∈ X take α ζ to be the free productautomorphism which is given byAd (cid:18)(cid:18) ζ ζ (cid:19)(cid:19) ⊗ Ad (cid:18)(cid:18) ζ ζ (cid:19)(cid:19) on M ⊗ M and is trivial on C ([0 , θ , θ , θ , θ ∈ R such that 1 , θ , θ , θ , θ are linearly independent over Q . Take h : X → X to be( ζ , ζ , ζ , ζ ) (cid:0) e πiθ ζ , e πiθ ζ , e πiθ ζ , e πiθ ζ (cid:1) . Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is purely infinite and simple by Theorem 2.13.The action in Example 6.19 has finite Rokhlin dimension with commuting towers.However, we don’t know any theorem on pure infiniteness for crossed productsby such actions when the original algebra is not O ∞ -stable. We address this inQuestion 7.8 below. Our result also does not imply that the crossed product is O ∞ -stable, or even Z -stable, although it seems plausible that it might be.In the next two examples, D is again not Z -stable. Also, X isn’t finite dimen-sional, and h doesn’t even have mean dimension zero. So a positive answer toQuestion 7.8 presumably would not help. Example 6.20.
Let D be as in Example 6.19, and let ( X, h ) be as in Example 6.3(and reused in Example 6.18). For ζ = ( ζ n ) n ∈ Z ∈ X take α ζ to be the free productautomorphism which is given byAd (cid:18)(cid:18) ζ ζ (cid:19)(cid:19) ⊗ Ad (cid:18)(cid:18) ζ ζ (cid:19)(cid:19) on M ⊗ M and is trivial on C ([0 , , , , C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is purely infinite and simple by Theo-rem 2.13, but may well not be Z -stable.With a small modification, we can give an example of this type in which theunderlying action of ( S ) Z is effective. Example 6.21.
Let B be the 2 ∞ UHF algebra, and let τ be its (unique) tracialstate. Let tr be the tracial state on M , and let ρ be the state on M given by ρ ( x ) = tr (cid:0) diag (cid:0) , (cid:1) x (cid:1) . Let D be the reduced free product D = ( B ⊗ M ) ⋆ r C ([0 , τ ⊗ ρ on B ⊗ M and the Lebesgue measure stateon C ([0 , A is purely infinite and simple but not Z -stable.To prove pure infiniteness, in Examples 3.9(iii) of [14] take A = B with thestate τ , take F = M with the state ρ , and take B = C ([0 , A is is purelyinfinite and simple.To prove that A is not Z -stable, let π be the Gelfand-Naimark-Segal representa-tion of B associated with τ , set N = τ ( B ) ′′ , and also write τ for the correspondingtracial state on N . In Proposition 5.6 of [1], take P = N ⊗ M with the state τ ⊗ ρ , ROSSED PRODUCTS OF C ( X, D ) 41 and take P = L ∞ ([0 , a = (cid:20)(cid:18) − (cid:19) ⊗ ⊗ ⊗ · · · (cid:21) ⊗ ∈ ∞ O n =1 M ! ⊗ M = B ⊗ M ⊂ P . Take G = { , a } ⊂ P , and take G ⊂ P to be the set of functions λ e πinλ for n ∈ Z . These choices satisfy the hypotheses there. Moreover, G ⊂ B ⊗ M and G ⊂ C ([0 , A = ( B ⊗ M ) ⋆ r C ([0 , P in Proposition 5.6 of [1] which contains a , b , and c , sois not Z -stable by Proposition 5.6 of [1]. The claim is proved.Let ( X, h ) be as in Example 6.3 (and reused in Example 6.18). Let σ : Z > → Z be a bijection (as in Example 6.3). For ζ = ( ζ n ) n ∈ Z ∈ X take α ζ to be the freeproduct automorphism which is the identity on C ([0 , O n ∈ Z > Ad (cid:18)(cid:18) ζ σ ( n ) (cid:19)(cid:19) ⊗ Ad (cid:18)(cid:18) ζ σ (1) ζ σ (2) (cid:19)(cid:19) ∈ Aut( B ⊗ M )on B ⊗ M . Then C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) is purely infinite and simple by Theorem 2.13,but may well not be Z -stable. 7. Open problems
In this section, we collect some open questions suggested by the examples andresults in this paper.
Problem 7.1.
Let A be a unital C*-algebra, and let α be an action of Z on A which has finite Rokhlin dimension with commuting towers. Suppose that A hasstable rank one and C ∗ ( Z , A, α ) is simple. Does it follow that C ∗ ( Z , A, α ) has stablerank one?This seems to be unknown even if A is simple (in which case simplicity of C ∗ ( Z , A, α ) is automatic) and α has the Rokhlin property. Without assuming sim-plicity of C ∗ ( Z , A, α ), the answer is definitely no. For aperiodic homeomorphismsof the Cantor set whose transformation group C*-algebras don’t have stable rankone, see Theorem 3.1 of [55] (it is easy to construct examples there which have morethan one minimal set) or Example 8.8 of [49]. Corollary 2.6 of [61] implies thatthe corresponding actions of Z have Rokhlin dimension with commuting towers atmost 1. In fact, though, at least in Example 8.8 of [49] we get the Rokhlin property. Lemma 7.2.
Let h : X → X be the aperiodic homeomorphism of the Cantor setin Example 8.8 of [49]. Then the induced automorphism of C ( X ) has the Rokhlinproperty. Proof.
Recall from [49] that X = Z ∪ {±∞} , h : X → X is n n + 1, X isthe Cantor set, h : X → X is minimal, X = X × X , and h = h × h .Let N ∈ Z > . The standard first return time construction provides n, r (1) , r (2) , . . . , r ( n ) ∈ Z > with N < r (1) < r (2) < · · · < r ( n ), and compact open subsets Z , Z , . . . , Z n ⊂ X ,such that X = n a k =1 r ( k ) − a j =0 h j ( Z k ) . Then the sets X × Z , X × Z , . . . , X × Z n form a system of Rokhlin towers for h , with heights r (1) , r (2) , . . . , r ( n ), all of whichexceed N . (cid:3) In fact, it seems to be known that any aperiodic homeomorphism of the Cantorset X induces an automorphism of C ( X ) with the Rokhlin property. We have notfound a reference, and we do not prove this here.The following question asks for a plausible generalization of Lemma 2.7. Question 7.3.
Let D be a simple unital C*-algebra. Let X be a compact metricspace, let G be a discrete group, and let ( g, x ) gx be a minimal and essentiallyfree action of G on X . Let α : G → Aut( C ( X, D )) be an action of G which liesover the action of G on X and which is pseudoperiodically generated. Set A = C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) .Does it follow that for every a ∈ C ( X, D ) + \{ } there exists f ∈ C ( X ) + \{ } ⊂ A such that f - A a ?We expect the techniques in the proof of Lemma 2.7 (taking Y to be the emptyset) can be used to answer this question in the affirmative. Question 7.4.
Is there a simple unital C*-algebra D such that Aut( D ) is notpseudoperiodic in the sense of Definition 1.8?Presumably such examples exist, but we don’t know of any. Indeed, we don’tsee any reason why there should not be α ∈ Aut( D ) such that { α n : n ∈ Z } is notpseudoperiodic. Question 7.5.
Consider the crossed product C ∗ (cid:0) Z , C ( X, E ) , β (cid:1) in Example 6.5.Is this algebra Z -stable? What about crossed products by similarly constructedactions? Question 7.6.
Consider the crossed product C ∗ (cid:0) Z , C ( X ) ⊗ C ∗ r ( F ∞ ) , α (cid:1) in Ex-ample 6.7. Is this algebra Z -stable? What about crossed products by similarlyconstructed actions? What about Example 6.8? Question 7.7.
Consider the crossed product C ∗ (cid:0) Z , C ( S , D ) , α (cid:1) in Example 6.17.Is this algebra Z -stable?The following question is motivated by Example 6.19. Question 7.8.
Let A be a nonsimple unital C*-algebra which is purely infinitein the sense of Definition 4.1 of [31]. Let α : Z → Aut( A ) be an action with finiteRokhlin dimension with commuting towers. Does it follow that C ∗ ( Z , A, α ) is purelyinfinite?If A is O ∞ -stable, then C ∗ ( Z , A, α ) is at least Z -stable by Theorem 5.8 of [24].If A is also exact, then so is C ∗ ( Z , A, α ) (by Proposition 7.1(v) of [29]), and C ∗ ( Z , A, α ) is traceless (as at the beginning of Section 5 of [59]) because A is,so C ∗ ( Z , A, α ) is purely infinite by Corollary 5.1 of [59]. (In this case, one obvi-ously wants O ∞ -stability. See Question 7.10 below.) But the question as statedseems to be open, even if one assumes that C ∗ ( Z , A, α ) is simple. If A itself issimple, then C ∗ ( Z , A, α ) is purely infinite even just assuming that α is pointwiseouter, by Corollary 4.4 of [27]. Provided one uses the reduced crossed product, thisremains true if Z is replaced by any discrete group. ROSSED PRODUCTS OF C ( X, D ) 43
Question 7.9.
Consider the crossed product C ∗ (cid:0) Z , C ( X, D ) , α (cid:1) in Example 6.20.Is this algebra Z -stable? Is it O ∞ -stable?We hope that Z -stability should come from the action in the “( S ) direction”.We suppose that if a purely infinite simple C*-algebra is Z -stable, then it is probably O ∞ -stable, but this seems to be open in general. Question 7.10.
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