Strong pure infiniteness of crossed products
aa r X i v : . [ m a t h . OA ] M a r STRONG PURE INFINITENESS OF CROSSED PRODUCTS
E. KIRCHBERG AND A. SIERAKOWSKI
Abstract.
Consider an exact action of discrete group G on a separable C *-algebra A . It is shown that the reduced crossed product A ⋊ σ,λ G is strongly purely infinite– provided that the action of G on any quotient A/I by a G -invariant closed ideal I = A is element-wise properly outer and that the action of G on A is G -separating (cf. Definition 4.1). This is the first non-trivial sufficient criterion for strong pureinfiniteness of reduced crossed products of C *-algebras A that are not G -simple. Inthe case A = C ( X ) the notion of a G -separating action corresponds to the propertythat two compact sets C and C , that are contained in open subsets C j ⊆ U j ⊆ X ,can be mapped by elements of g j ∈ G onto disjoint sets σ g j ( C j ) ⊆ U j , but we do notrequire that σ g j ( U j ) ⊆ U j . A generalization of strong boundary actions [18] on com-pact spaces to non-unital and non-commutative C *-algebras A (cf. Definition 6.1) isalso introduced. It is stronger than the notion of G -separating actions by Proposi-tion 6.6, because G -separation does not imply G -simplicity and there are examplesof G -separating actions with reduced crossed products that are stably projection-lessand non-simple. Contents
1. Introduction 22. Preliminaries 43. Proper outerness and ideal structure 54. Strongly purely infinite crossed products 115. The case of commutative C *-algebras 146. Strong boundary actions versus G -separating actions 17Acknowledgments 27Appendix A. 27 Date : October 11, 2018.2010
Mathematics Subject Classification.
Primary: 46L35; Secondary: 19K99, 46L80, 46L55. eferences 291. Introduction
In this paper we pursue the study of C *-dynamical systems with applications inclassification via equivariant KK -theory. It was shown by the first named author thatfor any two separable nuclear strongly purely infinite C *-algebras, both with primitiveideal space isomorphic to a T -space X , the algebras are isomorphic if and only ifthey are KK X -equivalent. It is however far from understood when C *-algebra crossedproducts A ⋊ σ,λ G associated to C *-dynamical systems are strongly purely infinite interms of properties of the action σ , in particular in the non-simple case. Our mainfocus of this work is such characterisation for crossed products that are either simpleor more generally contain ideals coming from arbitrary G -invariant ideals of the algebra A on which the group G acts.We begin (in Section 2) by introducing crossed products and by proving the notationused throughout the paper.In Section 3 we look at results related to the ideal structure of crossed products. Itwas shown in [23] that residually properly outer (Definition 4.1) and exact (Definition3.5) actions σ : G → Aut( A ) on a separable C *-algebra A have the property that thelattice of (closed) ideals of the reduced crossed product A ⋊ σ,λ G is naturally isomorphicto the lattice of G -invariant ideals of A (by the map I A ∩ I ). We refine this resultby showing that for any exact and residually properly outer action σ of a discretegroup G on a separable or commutative C *-algebra A the set A + is a filling family(Definition A.3) for A ⋊ σ,λ G (which implies that I A ∩ I is injective, see RemarkA.5 for details).In Section 4 we introduce the notion of G -separating actions (Definition 4.1). Weshow in Theorem 4.3 that for any exact and residually properly outer action σ of adiscrete group G on a separable or commutative C *-algebra A and for any filling family F ⊆ A + , the crossed product A ⋊ σ,λ G is strongly purely infinite if and only if F hasthe diagonalization property (Definition A.9) in A ⋊ σ,λ G . Applying the work [15] weobtain (in Proposition 4.5) an equivalent characterisation of G -separating actions, fromwhich we can deduce that A + has the diagonalization property whenever the action on A is G -separating. By evoking [15] once again we prove our main result: heorem 1.1. Suppose that ( A, G, σ ) is a C*-dynamical system, where G is discreteand A is separable or commutative.If the action σ of G on A is exact (Def. 3.5), residually properly outer (Def. 3.1)and G -separating (Def. 4.1), then A ⋊ σ,λ G is strongly purely infinite. In Section 5 we look at actions on commutative C *-algebras. Here we characterisethe notion of G -separating action purely in terms of the underlying geometry. Morespecifically we consider actions α of a discrete group G on a locally compact Hausdorffspace X , and denote by σ the induced action on A := C ( X ). We show (in Lemma5.1) that the action of G on A is G -separating if and only if the following holds: Forevery open U , U ⊆ X and compact K , K ⊆ X with K ⊆ U , K ⊆ U , there exist g , g ∈ G such that α g ( K ) ⊆ U , α g ( K ) ⊆ U , α g ( K ) ∩ α g ( K ) = ∅ . This resultis what motives the choice of our terminology “ G -separating”. As a consequence ofthis characterisation we obtain in Corollary 5.2 a characterisation of when a crossedproduct C ( X ) ⋊ σ,λ G is a strongly purely infinite C *-algebra in terms of condition on α . In the final Section 6 we consider actions that produce simple and strongly purely infi-nite crossed products. We introduce (Definition 6.2 and 6.1) the notion of n -majorizing( n ≥
1) and n -covering actions ( n ≥ C *-algebras.These two notions aim to refine results on simple purely infinite crossed products in[18, 11] where the notion of strong boundary actions (Definition A.1) and n -fillings ac-tions (Definition A.2) was introduced. We prove in Remark 6.7 our notions are weaker:Any n -filling action on a unital C *-algebra A is n -covering, and for any action α ona compact spaces X with more than two points (on which strong boundary actionsare defined) the action α is a strong boundary action if and only if its adjoint action σ on C( X ) is 1-majorizing. Both our notions are G -simple. Therefore we call the -majorizing actions on not-necessarily unital or commutative C*-algebras also strongboundary actions . Despite our weaker assumptions we are able to prove: Theorem 1.2.
Suppose that the C*-dynamical system ( A, G, σ ) with discrete G is n -majorizing (Def. 6.1) for some n ≥ or n -covering (Def. 6.2) for some n ≥ , thelatter if A is unital. If the action σ is element-wise properly outer (Def. 3.1), and A is separable or commutative, then A ⋊ σ,λ G is simple strongly purely infinite. In Section 6 we also look at how the different properties relate to each other. InLemma 6.3 we show that each n -covering action (for n ≥
2) on a unital C *-algebra action is n -majorizings, and the latter properly (for n ≥
1) implies that the actionis ( n + 1)-covering. In Proposition 6.6 we prove that that any 1-majorizing action ona non-unital C *-algebra A is automatically G -separating. In Remark 6.7 we provethat any action on a unital commutative C *-algebra A is n -filling if and only if it is n -covering, and for n = 2 this is again equivalent to a strong boundary (i.e., 1-majorizing)action.We end with a number of remarks, including a proof of the fact that our notionsof G -separating, n -majorizing and n -covering actions can be expressed in terms ofprojections when A has real rank zero (see Remark 6.9).We hope that the study of crossed products – even those for actions of amenablediscrete groups on locally compact Polish spaces – can help to detect possible differencesbetween strong and weak pure infiniteness. This paper is a very first step in thisdirection, and gives a sufficient criterium by conditions on the action that impliesstrong pure infiniteness of reduced crossed products.2. Preliminaries
We let A + denote the set of positive elements in a C *-algebra A . We denote thepositive and the negative part of a selfadjoint element a ∈ A by a + := ( | a | + a ) / ∈ A + and a − := ( | a | − a ) / ∈ A + , where | a | := ( a ∗ a ) / . If a ∈ A + , then ( a − ε ) + , thepositive part of a − ε M ( A ), is again in A + . Here M ( A ) is the multiplier algebra of A . This notation will be used also for functions f : R → R , then e.g. ( f − ε ) + ( ξ ) =max( f ( ξ ) − ε ,
0) . A subset
F ⊆ A + is invariant under ε -cut-downs if for each a ∈ F and ε ∈ (0 , k a k ) we have ( a − ε ) + ∈ F . The minimal unitalisation of A is denoted ˜ A .Restriction of a map f to X is denoted f | X . We let C c (0 , ∞ ] + denote the set of allnon-negative continuous functions ϕ on [0 , ∞ ) with ϕ | [0 , η ] = 0 for some η ∈ (0 , ∞ ),such that lim t →∞ ϕ ( t ) exists. Remarks 2.1. (i) Suppose that a, b ∈ A + and ε > k a − b k < ε . Then thepositive part ( b − ε ) + ∈ A of ( b − ε · ∈ M ( A ) can be decomposed into d ∗ ad = ( b − ε ) + with some contraction d ∈ A ([14, lem. 2.2]).(ii) Let τ ∈ [0 , ∞ ) and 0 ≤ b ≤ a + τ · M ( A )), then for every ε > τ there is acontraction f ∈ A such that ( b − ε ) + = f ∗ a + f . (See [14, lem. 2.2] and [3, sec. 2.7].)We abbreviate C *-dynamical systems by ( A, G, σ ) with discrete groups G . We de-note by e the unit of G , and consider only closed and two-sided ideals of A . The reduced resp. the full) crossed product associated to ( A, G, σ ) is denoted by A ⋊ σ,λ G (resp. A ⋊ σ G ). The norm on A ⋊ σ,λ G will be sometimes written as k · k λ if it is necessary todistinguish it from other norms. Let I ( A ) denote the lattice of ideals in a C *-algebra A . The map η : A → A ⋊ σ G means the natural embedding into the full crossed prod-uct. Let π λ : A ⋊ σ G → A ⋊ σ,λ G be the natural epimorphism. We will sometimessuppress the canonical inclusion maps η : A → A ⋊ σ G and π λ ◦ η : A ⊆ A ⋊ σ,λ G .Let U denote the canonical unitary representation U : G → M ( A ⋊ σ G ). Notice herethat the linear span of η ( A ) U ( G ) is is a dense *-subalgebra of A ⋊ σ G . We denote by U λ : G → M ( A ⋊ σ,λ G ) the regular representation for some more precise explanations.The same happens with η λ := π λ ◦ η .The set C c ( G, A ) consists of the maps f : G → A with finite support F := G \ f − (0).There is a natural linear embedding of C c ( G, A ) into A ⋊ σ G by canonical identificationof f : G → A (of finite support) with an element of A ⋊ σ G : Let F ⊆ G be a finite subset,with f ( g ) = 0 for g F . Then f will be identified with the element P g ∈ F η ( a g ) U ( g )of A ⋊ σ G , where a g := f ( g ). In this way C c ( G, A ) becomes a *-subalgebra of A ⋊ σ G that contains A . The natural C *-morphism π λ : A ⋊ σ G → A ⋊ σ,λ G is faithful onC c ( G, A ), and we do not distinguish between π λ ( X g ∈ F η ( a g ) U ( g )) = X g ∈ F η λ ( a g ) U λ ( g )and P g ∈ F η ( a g ) U ( g ). In particular, η ( a ) ∈ A ⋊ σ G and η λ ( a ) ∈ A ⋊ σ,λ G will be denotedsimply by a ∈ A , and U λ ( g ) might be denoted U ( g ).We now recall the conditional expectation E : A ⋊ σ G → η ( A ) ∼ = A : The map E alg : C c ( G, A ) → A , P g ∈ F a g U ( g ) a e , extends by continuity to a faithful con-ditional expectation E λ : A ⋊ σ,λ G → A . In particular E λ is a completely positivecontraction, E λ ( A ⋊ σ,λ G ) = A , and E λ ( b ) = 0 imply b = 0 for b ∈ ( A ⋊ σ,λ G ) + .Since A is also contained in its full crossed product A ⋊ σ G , we can use the naturalepimorphism A ⋊ σ G → A ⋊ σ,λ G to define E by E := E λ ◦ π λ as a (not necessarilyfaithful) conditional expectation from A ⋊ σ G onto its C *-subalgebra A .3. Proper outerness and ideal structure
In this section we look at conditions on a C *-dynamical system ( A, G, σ ) ensuringthat the set A + is a filling family for A ⋊ σ,λ G in the sense of Definition A.3. This inparticular implies that the is a on-to-one correspondence between ideals of A ⋊ σ,λ G and G -invariant ideals of A , but (as wee shall see) also applies to the verification of when crossed product is strongly purely infinite. Proper outerness of the automorphisms σ t of A defining the action σ turns out also to be a crucial ingredient. We recall thedefinition below. Definition 3.1.
Suppose that (
A, G, σ ) is a C *-dynamical system and that G isdiscrete. The action σ will be called element-wise properly outer if, for each g ∈ G \{ e } , the automorphism σ g of A is properly outer in the sense of [6, def. 2.1], i.e., k σ g | I − Ad( U ) k = 2 for any σ g -invariant non-zero ideal I of A and any unitary U inthe multiplier algebra M ( I ) of I . See also [19, thm. 6.6(ii)].We call here an action σ residually properly outer if, for every G -invariant ideal J = A of A , the induced action [ σ ] J of G on A/J is element-wise properly outer. Remarks 3.2. (i) Notice that element-wise proper outerness passes to subgroups, i.e.,for each subgroup H ⊆ G the system ( A, H, σ | H ) is element-wise properly outer on A if ( A, G, σ ) is element-wise properly outer. But residual proper outerness does notnecessarily pass to subgroups. The system (
A, H, σ | H ) is not necessarily residuallyproperly outer, if ( A, G, σ ) is residual proper outer, because possibly there could bemore H -invariant ideals than G -invariant ideals of A .(ii) If A is non-commutative, then topological freeness of ( A, G, σ ) in sense of [1, def. 1]is – at least formally – stronger than the assumption of element-wise proper outernessof (
A, G, σ ) in Definition 3.1 (cf. [1, prop. 1]). We do not know examples where theyactually differ. Thus, for non-commutative A , “essential freeness” of the correspondingaction of G on b A in the sense of [23, def. 1.17] (inspired by [22, def. 4.8]) is – formally– stronger than our residual proper outerness of ( A, G, σ ).(iii) If G is countable and acting on C ( X ), one can show – using the Baire property of X – that elementwise proper outerness is the same as the requirement (for the action α of G on X with σ g ( f ) := f ◦ α g − ) that points with trivial fix-point subgroup (trivialisotropy) are dense in X , i.e., Definition [23, def. 1.17] holds. We can reformulate thisas: stability subgroups of non-empty open subsets are trivial. Remark 3.3.
We recall [19, lem. 7.1.] (cf. also [17, lem. 3.2]): If α , α , ..., α n are properly outer automorphisms of a separable C*-algebra A , thereis, for each a , a , a , ..., a n ∈ e A , with = a ≥ , and each ε > , an element x ∈ A + with k x k = 1 such that k xa x k > k a k − ε , k xa i α j ( x ) k < ε , ≤ i, j ≤ n . f A is commutative , i.e., A ∼ = C ( X ) for X = b A ⊆ A ∗ , then it is not necessaryto suppose that A is separable in the quoted lemma of D. Olesen and G. Pedersen(Compare also [7]): An automorphism σ ∈ Aut( A ) is properly outer, if and only if,for every open subset ∅ 6 = U ⊆ X there exists y ∈ U with b σ ( y ) = y . Thus, for everyfinite set S ⊆ Aut( A ) of properly outer automorphisms, every non-empty open subset U ⊆ X contains a non-empty open subset V ⊆ U with b σ ( V ) ∩ V = ∅ for all σ ∈ S . Ifone takes U := a − ( k a k − ε, ∞ ) and non-empty V ⊆ U as above, then each x ∈ C ( X )with k x k = 1 and support in V satisfies k xa x k > k a k − ε and xσ ( x ) = 0 for σ ∈ S .The following Lemma 3.4 is a suitable modification of proofs of [19, lem.7.1,thm.7.2].It has been proved in [1] under the stronger assumption that the action σ is topologicallyfree, and part (iii) has been shown in [12, thm. 4.1] even to be equivalent to thetopological freeness of the action if A is commutative and unital and G is amenable.Compare also Remark 5.3 for a “residual” version. Lemma 3.4.
Suppose that A is separable or commutative, and that the action of G on A is element-wise properly outer. (i) For every b ∈ ( A ⋊ σ G ) + with E ( b ) = 0 and ε > there exist x ∈ A + satisfyingthat k x k = 1 , k xbx − xE ( b ) x k < ε , k xE ( b ) x k > k E ( b ) k − ε . This holds also for b ∈ ( A ⋊ σ,λ G ) + and E λ in place of E . (ii) If h : A ⋊ σ G → L ( H ) is a *-representation such that h | A is faithful, then k h ( b ) k ≥ k E ( b ) k for all b ∈ ( A ⋊ σ G ) + .In particular, the kernel of h is contained in the kernel I λ of the naturalepimorphism π λ : A ⋊ σ G → A ⋊ σ,λ G . (iii) Every non-zero ideal of A ⋊ σ,λ G , has non-zero intersection with A . Proof . (i): Let b ∈ ( A ⋊ σ G ) + with E ( b ) = 0, and ε > a := E ( b ). Since C c ( G, A ) is dense in A ⋊ σ G , there exists a ′ = c + P mj =1 a j U ( g j ) ∈ C c ( G, A ) with g − i g j = e and g j = e for i = j ∈ { , . . . , m } ,and k a ′ − b k < ε/
6. Since E is a contraction, it follows that k b − a k < ε/ E ( a ) = a = E ( b ) for g := e and a := a + ( a ′ − c ) = P mj =0 a j U ( g j ). By [19, lem. 7.1]and Remark 3.3 there exists x ∈ A + with k x k = 1, k xE ( a ) x k > k E ( a ) k − ε/ m , and k xa j σ g j ( x ) k < ε/ m for g j = e , j = 1 , . . . , m . In particular, k xE ( b ) x k = k xE ( a ) x k > k E ( a ) k − ε = k E ( b ) k − ε . ince k xa j U ( g j ) x k = k xa j σ g j ( x ) k we get in A ⋊ σ G that k x ( a − E ( a )) x k ≤ X g j = e k xa j σ g j ( x ) k ≤ ε/ . Thus, in the full crossed product A ⋊ σ G we have k xbx − xE ( b ) x k ≤ k x ( b − a ) x k + k x ( a − E ( a )) x k + k x ( E ( a ) − E ( b )) x k < ε . The same arguments work for b ∈ ( A ⋊ σ,λ G ) + and E λ in place of E .(ii): The restriction of h to A ⊆ A ⋊ σ G is faithful, hence k h ( a ) k = k a k for all a ∈ A . Let b ∈ ( A ⋊ σ G ) + be given. If E ( b ) = 0 then k h ( b ) k ≥ k E ( b ) k . If E ( b ) = 0,then select x ∈ A + as in (i). It follows that k h ( xE ( b ) x ) k = k xE ( b ) x k ≥ k E ( b ) k − ε .On the other hand, k h ( b ) k ≥ k h ( x ) h ( b ) h ( x ) k = k h ( xbx ) k and ε > k xbx − xE ( b ) x k ≥k h ( xbx ) − h ( xE ( b ) x ) k . Thus k h ( b ) k + ε ≥ k h ( xE ( b ) x ) k , and k h ( b ) k + 2 ε ≥ k E ( b ) k forall ε > E = E λ ◦ π λ , we have b ∈ ( A ⋊ σ G ) + and E ( b ) = 0 implies that b is containedin the kernel of π λ : A ⋊ σ G → A ⋊ σ,λ G . In particular, if h : A ⋊ σ G → L ( H ) is any*-representation with k h ( b ) k ≥ k E ( b ) k for all b ∈ ( A ⋊ σ G ) + , then the kernel h − (0) of h is contained in the kernel of the natural epimorphism π λ : A ⋊ σ G → A ⋊ σ,λ G .(iii): Let I a closed ideal of A ⋊ σ,λ G with I ∩ A = { } , consider ( A ⋊ σ,λ G ) /I as a C *-subalgebra of some L ( H ), and let h : A ⋊ σ G → L ( H ) the corresponding representationwith kernel h − (0) = J := π − λ ( I ) ⊇ π − λ (0). Then h is faithful on A and, therefore,satisfies π − λ (0) ⊇ h − (0). It follows I = π λ ( h − (0)) = { } . (cid:3) Definition 3.5 ([23, def. 1.2]) . Suppose that (
A, G, σ ) is a C *-dynamical system withlocally compact G . The action σ of G on A is exact , if, for every G -invariant ideal J in A , the sequence 0 → J ⋊ σ | J,λ G → A ⋊ σ,λ G → A/J ⋊ [ σ ] J ,λ G → Remarks 3.6. (i) Recall that a locally compact group G is exact , if and only if, everyaction σ : G → Aut( A ) is exact. If G is discrete , then this is equivalent to the exactnessof the C *-algebra C ∗ λ ( G ), cf. [16]. This applies to all amenable groups G , e.g. G = Z .Under Definition 3.5 each minimal (= G -simple) action is exact. In particular, non-exact discrete groups can have exact (and faithful) actions.(ii) Let F denote the (small) Thompson group and ρ : F → Homeo( R ) the minimalaction of F (or only of its commutator subgroup F ′ ) on the real line R as describedby Haagerup and Picioroaga in [10, rem. 2.5.]. One can use ρ to construct a F -separating, non-minimal and exact action α of F (or F ′ ) on the disjoint union of two ines X := R ∪ ( i + R ) ⊆ C if one considers the restriction α ( g ) := β ( g ) | X to X of theaction g ∈ F → β ( g ) on C given by β ( g )( s + it ) := ρ ( g )( s ) + it ( ). It is at presentunknown whether the Thompson group F is exact or not, cf. [2, 8, 9].(iii) It is not known if Gromov’s examples of non-exact groups can have non-exactactions on commutative C *-algebras. It is likely that it has to do with still missingnon-trivial geometric conditions for G -actions on locally compact spaces X that areequivalent to the exactness of the adjoint action σ : G → Aut(C ( X )) given by σ g ( f ) := f ◦ α g − . Therefore we use the trivial and non-geometric definition and define α to beexact if its adjoint action σ on C ( X ) is exact. Remark 3.7.
Combination of Lemma 3.4(iii) and of the exactness of an action σ : G → Aut( A ) on a separable or commutative C *-algebra A shows that the lattice of (closed)ideals of the reduced crossed product A ⋊ σ,λ G is naturally isomorphic to the latticeof G -invariant ideals of A (by the map J A ∩ J ), if σ is exact and residual properlyouter . (See [23, Remark 2.23] for details.) Theorem 3.8.
Let ( A, G, σ ) a C*-dynamical system, with discrete G and separable orcommutative A . If the action σ of G on A is exact and residually properly outer, thenthe elements of A + build a filling family for A ⋊ σ,λ G in the sense of Definition A.3. Proof . We show that for every hereditary C *-subalgebra D of A ⋊ σ,λ G and every(closed) ideal I of A ⋊ σ,λ G with D I there exist f ∈ A + and z ∈ A ⋊ σ,λ G such that z ∗ z ∈ D and zz ∗ = f I .Suppose that D is a hereditary C *–subalgebra of A ⋊ σ,λ G and that I is an idealof A ⋊ σ,λ G with D I . Let J := I ∩ A , then J is a G -invariant ideal of A with J ⋊ σ | J,λ G ⊆ I and g ∈ G [ σ g ] J is an element-wise properly outer action on A/J byour assumptions on σ . We denote this action by α , i.e., α g ( a + J ) := σ g ( a ) + J .By Remark 3.7, the exactness and residual proper outerness of σ : G → Aut( A ) allowa natural identification ( A ⋊ σ,λ G ) /I = ( A/J ) ⋊ α,λ G .
Since D I implies D + I , there exists d ∈ D + , d / ∈ I . The epimorphism π I : A ⋊ σ,λ G → ( A ⋊ σ,λ G ) /I is equal to the quotient map π J from A ⋊ σ,λ G onto( A/J ) ⋊ α,λ G ∼ = ( A ⋊ σ,λ G ) /I (under natural identifications). We denote the conditional This action is not topologically free. xpectation E λ : ( A/J ) ⋊ α,λ G → A/J (temporarily) by E and define b := π I ( d ) , and ε := 14 k E ( b ) k > . Lemma 3.4(i) gives an element x ∈ ( A/J ) + such that k x k = 1 , k xbx − xE ( b ) x k < ε, k xE ( b ) x k > k E ( b ) k − ε = 34 k E ( b ) k . By Remark 2.1(i), there is a contraction y ∈ ( A/J ) ⋊ α,λ G such that y ∗ xbxy = ( xE ( b ) x − ε ) + ∈ ( A/J ) + . Note that y ∗ xbxy = 0 , because k ( xE ( b ) x − ε ) + k = k xE ( b ) x k − k E ( b ) k > k E ( b ) k = 2 ε . Since π I | A = π J | A and ( xE ( b ) x − ε ) + ∈ ( A/J ) + , there is exists c ∈ A + such that π J ( c ) = ( xE ( b ) x − ε ) + . Since π J (= π I ) is surjective, there exists a contraction w ∈ A ⋊ σ,λ G with π J ( w ) = xy . We obtain that c = w ∗ dw + v for some v ∈ I . The set C c ( G, J ) is dense in I , because I = J ⋊ σ,λ G and G is discrete.This allows us to see, that J IJ is dense in I . It follows that { e ∈ J + ; k e k < } is anapproximate unit of I . In particular, there exists e ∈ J + with k v − ev k < ε .Let 1 denote the unit of e A ⋊ σ,λ G , then A ⋊ σ,λ G is an ideal of e A ⋊ σ,λ G . With g := (1 − e ) ∈ e A + , k g k ≤ k gw ∗ dwg − gcg k = k gvg k ≤ k v − ev k < ε = 14 k E ( b ) k . Since gzg = z + eze − ( ze + ez ) and π I ( e ) = π J ( e ) = 0, we have π I ( gzg ) = π I ( z ) forall z ∈ A ⋊ σ,λ G .By Remark 2.1(i), there exists a contraction h ∈ A ⋊ σ,λ G such that h ∗ ( gw ∗ dwg ) h = ( gcg − ε ) + ∈ A + . With z := ( d / wgh ) ∗ we have that z ∗ z ∈ D and zz ∗ = ( gcg − ε ) + =: f ∈ A + . Finally,we see from π I ( gcg ) = π I ( c ) that k π I ( f ) k = k π I (( gcg ) − ε ) + ) k = ( k π I ( gcg ) k − ε ) + = ( k π I ( c ) k − ε ) + = ( k ( xE ( b ) x − ε ) + k − ε ) + = k xE ( b ) x k − k E ( b ) k > k E ( b ) k > . Hence, f I . (cid:3) . Strongly purely infinite crossed products
In this section we prove out main result Theorem 1.1. We start with the definitionof an G -separating action. Definition 4.1.
Suppose that (
A, G, σ ) is a C *-dynamical system with discrete group G . The action of G on A is G -separating if for every a, b ∈ A + , c ∈ A , ε >
0, thereexist elements s, t ∈ A and g, h ∈ G such that k s ∗ a s − σ g ( a ) k < ε , k t ∗ b t − σ h ( b ) k < ε and k s ∗ c t k < ε . (1) Remarks 4.2. (i) Notice that Definition A.6 and Remark A.7 immediately impliesthat every action σ : G → Aut( A ) is G -separating if A itself is strongly purely infinite :Take h = g = e ∈ G . If the contractions s, t ∈ A satisfy the defining inequalities (7) ofstrongly p.i. algebras A then they also satisfy the inequalities (1).(ii) G -separating actions on a locally compact space X are not necessarily minmal. Onecan show that above mentioned example of an exact and non-minimal action of the(small) Thompson group F on two parallel lines R ∪ ( i + R ) ⊆ C is also F -separating.(iii) The existence of a G -separating action on A imposes requirement on A itself,e.g. the cases a = b = c = p and a = b = c = 1 with ε = 1 / A can not contain minimal non-zero projections p ∈ A and that A must beproperly infinite in A if A is unital. Therefore, C *-algebras, that are commutative and unital, can not have any G -separating actions.(iv) Further variations of the concepts that we introduce here are possible, e.g. onecould start with conditions that are weaker than conditions for G -separating actions.Also one could require the existence of n ∈ N such that for a, b ∈ A + and ε > d , . . . , d n ∈ A and g , . . . , g n ∈ G of the inequality (3) in Definition 6.1 of n -majorizing actions whenever b is in the smallest G -invariant closed ideal that contains a . Or one could attempt to replace the filling family F := A + by smaller filling families F ⊆ A + and require more elaborate local matrix diagonalization formulas involvingalso G -translates, cf. Definition A.8.Combing Theorem 3.8 with Theorem A.13 we obtain the following result Theorem 4.3.
Let G be a discrete group acting by σ : G → Aut( A ) on a separableor commutative C*-algebra A . Suppose that the action is residually properly outer cf. Def. 3.1) and exact (cf. Def. 3.5). Let F ⊆ A + be a filling family for A . Then thefollowing are equivalent: (i) The crossed product A ⋊ σ,λ G is strongly purely infinite. (ii) The family F has the diagonalization property in A ⋊ σ,λ G . Proof . Let B := A ⋊ σ,λ G . The assumptions ensure that A + is a filling family for B by Theorem 3.8. Since F is filling for A , F is also filling for B by Lemma A.4.(i) ⇒ (ii): If B is s.p.i., then B + has the diagonalization property (see DefinitionA.9) in B , cf. [14, lem. 5.7]. This implies that our family F ⊆ A + ⊆ B + has thediagonalization property in B .(ii) ⇒ (i): Since our family F ⊆ A + is filling for B , and since F has the diagonalizationproperty in B , we get that B is s.p.i. by Theorem A.13. (cid:3) Remark 4.4.
Let (
A, G, σ ) a C *-dynamical system.(i) For each a , a ∈ A + , x, d , d ∈ A , g , g , g ∈ G and s := d U ( g ), s := σ g − ( d ) U ( g − g g ), c := xU ( g ), b := a , and b := σ g ( a ) the followingequalities hold: k s ∗ j a j s j − a j k = k d ∗ j b j d j − σ g j ( b j ) k and k s ∗ cs k = k d ∗ xd k . (ii) With g = e in (i) the equalities reduce to: k s ∗ j a j s j − a j k = k d ∗ j a j d j − σ g j ( a j ) k and k s ∗ cs k = k d ∗ cd k . Proposition 4.5.
Suppose that ( A, G, σ ) is a C*-dynamical system with discrete G .The following properties (i)–(ii) are equivalent: (i) The action of G on A is G -separating in the sense of Definition 4.1. (ii) For every a , a ∈ A + , c ∈ A ⋊ σ,λ G and ε > there exist d , d ∈ A and g , g ∈ G such that the elements s j = d j U ( g j ) of C c ( G, A ) satisfy, for j = 1 , , k s ∗ j a j s j − a j k < ε , and k s ∗ cs k < ε . (2) Proof . (ii) ⇒ (i): If we take c ∈ A , a := a and a := b for a, b ∈ A + and ε >
0, then(ii) implies, using Remark 4.4, that there exist d , d ∈ A and g , g ∈ G such that k d ∗ j a j d j − σ g j ( a j ) k < ε and k d ∗ cd k < ε , so the inequalities (1) of Definition 4.1 aresatisfied with d , d , g , g in place s, t, g, h . i) ⇒ (ii): Define C := { dU ( g ) ; d ∈ A , g ∈ G } and S := C . Select any ε > C is equal to A ⋊ σ,λ G . If we can show that F := A + , C and S satisfy the assumptions (i)-(iii) of Lemma A.11 – with A ⋊ σ,λ G in place of A –, then it follows from Lemma A.11 that for every a , a ∈ A + , c ∈ A ⋊ σ,λ G and ε > d , d ∈ A and g , g ∈ G such that s j = d j U ( g j ) ∈ S fulfil (2), which inturn gives (ii).It is evident that our C and S satisfy properties (ii) and (iii) in Lemma A.11. Since A + is closed under ε -cut-downs, property (i) becomes automatic if each pair ( a , a ),with a , a ∈ A + , has the matrix diagonalization property of Definition A.8 with respectto S and C :If a , a ∈ A + , c = xU ( g ) ∈ C with x ∈ A , g ∈ G , and ε > b := a , b := σ g ( a ). (If we instead of ε are given positive ε , ε and τ , set ε := min( ε , ε , τ ).) Since the action σ is G -separating, we can find d , d ∈ A and g , g ∈ G with k d ∗ b d − σ g ( b ) k , k d ∗ b d − σ g ( b ) k and k d ∗ xd k all strictly below ε . Remark 4.4 provides elements s j ∈ C satisfying (2). Thus ( a , a ) has the matrixdiagonalization property with respect to S and C . (cid:3) Theorem 4.6.
Let ( A, G, σ ) a C*-dynamical system, with discrete G . Suppose that A + is a filling family for A ⋊ σ,λ G and that the action of G on A is G -separating. Then A ⋊ σ,λ G is strongly purely infinite. Proof . By Theorem A.13 it remains to show that A + has the diagonalization propertyin A ⋊ σ,λ G . Since A + is closed under ε -cut-downs Lemma A.10 applies, and thereforeit is enough to show that each pair ( a , a ) with a , a ∈ A + , has the matrix diagonal-ization property in A ⋊ σ,λ G . But this follows from the G -separation property of theaction σ by Proposition 4.5. (cid:3) . Proof of Theorem 1.1 : By Theorems 3.8 and 4.6 the assumptions imply that A ⋊ σ,λ G is strongly purely infinite. (cid:3) Remark 4.7.
Suppose that (
A, G, σ ) is a C *-dynamical system and that G is discrete.Then a family F ⊆ A + ⊆ A ⋊ σ,λ G which is invariant under ε -cut-downs has thediagonalization properly in A ⋊ σ,λ G , if and only if, for every a , a ∈ F , c ∈ C c ( G, A )and > s , s ∈ C c ( G, A ) such that, for j = 1 , k s ∗ j a j s j − a j k < ε and k s ∗ cs k < ε . his follows from Lemma A.10, Lemma A.11 and the fact that C c ( G, A ) is dense in A ⋊ σ,λ G . Remark 4.8.
Notice that for an exact locally compact group G the reduced group C *-algebra C ∗ λ ( G ) is an exact C *-algebra (cf. [16, p. 171]). By Theorem A.12, the minimal C *-tensor product A ⊗ min B of a s.p.i. C *-algebra A with an exact C *-algebra B isagain s.p.i. Hence, if G is an exact locally compact group, σ ( g ) := id A is the trivialaction on a s.p.i. C *-algebra A then A ⋊ σ,λ G ∼ = A ⊗ min C ∗ λ ( G ) is s.p.i.This shows that there is room for refinements of our sufficient conditions on theactions that imply strong pure infiniteness of the reduced crossed products: Here theaction σ is even not element-wise properly outer, but satisfies the G -separation propertyand is an exact action by C *-exactness of C ∗ λ ( G ).5. The case of commutative C *-algebras The case of G -actions on commutative C *-algebras allows some topological interpre-tation. The next lemma has inspired our choice of the name G -separating in Definition4.1. Notice that we do not require α g j ( U j ) ⊆ U j in its part (ii). Lemma 5.1.
Suppose that ( A, G, σ ) is a C*-dynamical system, that A ∼ = C ( X ) iscommutative, and that the action σ of G on C ( X ) is induced by the action α of G on X ∼ = b A (i.e., σ g ( f ) := f ◦ α − g for f ∈ A, g ∈ G ). Then the following properties areequivalent: (i) The action of G on A is G -separating, i.e., for every a, b ∈ A + , c ∈ A , ε > ,there exist elements d , d ∈ A and g , g ∈ G such that k d ∗ ad − σ g ( a ) k < ε , k d ∗ bd − σ g ( b ) k < ε and k d ∗ cd k < ε . (ii) For every open U , U ⊆ X and compact K , K ⊆ X with K ⊆ U , K ⊆ U ,there exist g , g ∈ G such that α g ( K ) ⊆ U , α g ( K ) ⊆ U , α g ( K ) ∩ α g ( K ) = ∅ . Proof . (ii) ⇒ (i): Let a, b ∈ A + , c ∈ A and ε >
0. We use assumption (ii) on U := a − ( ε/ , ∞ ) = { x ∈ X ; a ( x ) > ε/ } , U := { x ∈ X ; b ( x ) > ε/ } ,K := { x ∈ X ; a ( x ) ≥ ε/ } , K := { x ∈ X ; b ( x ) ≥ ε/ } , nd find g , g ∈ G such that α g ( K ) ⊆ U , α g ( K ) ⊆ U , α g ( K ) ∩ α g ( K ) = ∅ . Since a, b ∈ C ( X ) + , we have that U ⊆ a − [ ε/ , ∞ ) and U ⊆ b − [ ε/ , ∞ ) are compactsubsets of X .Since the compact sets α g ( K ) and α g ( K ) are disjoint, applications of Tietzeextension theorem gives elements e , e ∈ A + with k e j k ≤ / √ ε and a contraction f = f ∗ ∈ A such that e | U = a − / | U , e | U = b − / | U , f | α g ( K ) = − , f | α g ( K ) = 1 . Let f + , f − ∈ A + be the canonical decomposition f = f + − f − with f + f − = 0. Wedefine d := e ( σ g ( a ) − ε/ / f − and d := e ( σ g ( b ) − ε/ / f + . Then d ∗ cd = 0 because f + f − = 0.Since ( σ g ( a ) − ε/ + ( x ) = 0 implies a ( α g − ( x )) > ε/
2, we get α g − ( x ) ∈ K , and x ∈ α g ( K ) ⊆ U ⊆ U . It implies that f − ( x ) = 1 and e ( x ) = a − / ( x ). We obtainthat d ∗ ad = e a ( σ g ( a ) − ε/ + ( f − ) = ( σ g ( a ) − ε/ + . In a similar way we see that d ∗ bd = ( σ g ( b ) − ε/ + .(i) ⇒ (ii): Let U , U ⊆ X be open and K , K ⊆ X compact subsets such that K ⊆ U and K ⊆ U . We can assume that the intersection K ∩ K is non-empty.There exists an open set W with a compact closure W such that K ∪ K ⊆ W ⊆ W ⊆ U ∪ U . By the Tietze extension theorem, there are contractions a, b, c ∈ A + such that a | K = 1 , b | K = 1 , c | W = 1 , supp( a ) ⊆ U ∩ W, supp( b ) ⊆ U ∩ W, supp( c ) ⊆ U ∪ U . We apply assumption (i) to a , b , c and ε := 1 /
4, and get elements d , d ∈ A and g , g ∈ G such that k d ∗ ad − σ g ( a ) k < / , k d ∗ bd − σ g ( b ) k < / , k d ∗ cd k < / . If x ∈ W , then c ( x ) = 1 and | d ( x ) || d ( x ) | ≤ k d ∗ cd k < /
4. Thus, V := { x ∈ W ; | d ( x ) | ≥ / } , V := { x ∈ W ; | d ( x ) | ≥ / } re disjoint sets. If x ∈ α g ( K ), then α g − ( x ) ∈ K and σ g ( a )( x ) = a ( α g − ( x )) = 1. Itfollows | d ( x ) | a ( x ) ≥ / | d ( x ) | > / ≥ a ( x ) > x ∈ U ∩ W and x ∈ V . It shows α g ( K ) ⊆ U ∩ V . In a similar way we get α g ( K ) ⊆ U ∩ V . It implies α g ( K ) ⊆ U , α g ( K ) ⊆ U and that α g ( K ) ∩ α g ( K ) = ∅ . (cid:3) The following condition (i) in Corollary 5.2 is satisfied if the action α has the residualversion of the topological freeness in sense of [1, def. 1], see e.g. the essential freenessdefined in [23, def. 1.17] (inspired by [22, def. 4.8]) when G is countable. Corollary 5.2.
Let G be a discrete group, α : G → Homeo( X ) an action of G on alocally compact Hausdorff space X . Suppose that (i) for every closed G -invariant subset Y of X and every e = g ∈ G the set { y ∈ Y : α g ( y ) = y } has empty interior, (ii) the action σ : G → Aut(C ( X )) , given by σ g ( f ) := f ◦ ( α g ) − , is exact on theC*-algebra C ( X ) , and (iii) the action α is G -separating, i.e., by Lemma 5.1, for every U , U ⊆ X openand K , K ⊆ X compact such that K ⊆ U , K ⊆ U , there exist g , g ∈ G such that α g ( K ) ⊆ U , α g ( K ) ⊆ U , α g ( K ) ∩ α g ( K ) = ∅ . Then C ( X ) ⋊ σ,λ G is a strongly purely infinite C*-algebra. Proof . Let A := C ( X ). It is easy to see that property (i) implies that the action onany quotient A/I by a G -invariant closed ideal I = A is element-wise properly outer.Now Theorem 1.1 applies to A ⋊ σ,λ G . (cid:3) The following remark shows that in case of commutative A and discrete amenable G several of the previously considered properties are equivalent. Remark 5.3. If A is commutative and G is a discrete amenable group that acts on A by σ , then the following properties are equivalent:(i) A separates the ideals of A ⋊ σ G , i.e., I A ∩ I is an injective map from I ( A ⋊ σ G ) into I ( A ) (see [23, def. 1.9]).(ii) The action σ : G → Aut( A ) is residually properly outer (Definition 3.1).(iii) The family F := A + is filling for A ⋊ σ G (Definition A.3). Proof . (i) ⇒ (ii): By [12, thm. 4.1] (in case of unital A , and [1, thm. 2] for the general– non-unital – case) the separation property implies that the adjoint action of G on the elfand spectrum of A is topologically free, which is equivalent to element-wise properouterness by [1, prop. 1].This applies also to the quotients ( A/J ) ⋊ [ σ ] J ,λ G , because the property (i) passes toquotients by amenability of G . See also [23, thm. 1.13].(ii) ⇒ (iii): Since amenable G are exact, the residual proper outerness of the actionimplies that F := A + is filling for A ⋊ σ,λ G by Theorem 3.8.(iii) ⇒ (i): By Remark A.5, the subalgebra A separates the ideals of B := A ⋊ σ,λ G if F := A + is filling for B . (cid:3) Asking G to be amenable can be weakened to exactness of σ and A ⋊ σ,λ G ∼ = A ⋊ σ G .One might also expect nuclearity of A ⋊ σ,λ G would suffice in place of amenability of G (this is know at least in the unital case).6. Strong boundary actions versus G -separating actions In this section we prove our Theorem 1.2. We state with the definition of n -majorizing and n -covering actions. Definition 6.1.
Let n ∈ N and A a non-zero C *-algebra, that is not isomorphic toa subalgebra of M n +1 ( C ) if A is unital . An action σ : G → Aut( A ) will be called an n -majorizing action of G on A , if, for every non-zero a ∈ A + , every non-invertible b ∈ A + and every ε >
0, there exist d , . . . , d n ∈ A and g , . . . , g n ∈ G such that k n X j =1 d ∗ j σ g j ( a ) d j − b k < ε . (3) Definition 6.2.
Let n ∈ N , n ≥ ). Suppose that A is a unital C *-algebra, that isnot isomorphic to a *-subalgebra of M n ( C ). An action σ of G on A is an n -coveringaction if, for every non-zero a ∈ A + , and every ε >
0, there exist d , . . . , d n ∈ A and g , g , . . . , g n ∈ G and such that k n X j =1 d ∗ j σ g j ( a ) d j − k < ε . (4)The following lemma denies the existence of non-zero “socles” in C *-algebras A thatadmit n -majorizing or n -covering actions considered in Definitions 6.1 and 6.2. If n = 1 then property (4) holds if and only if A is a unital simple purely infinite C *-algebra. emma 6.3. Let ( A, G, σ ) C*-dynamical system.Consider the following properties ( α ) or ( β ) of ( A, G, σ ) depending on n ∈ N : ( α ) There is n ≥ such that, for each non-zero a ∈ A + , non-invertible b ∈ A + and ε > , there exist d , . . . , d n ∈ A and g , . . . , g n ∈ G that satisfy the inequality (3) in Definition 6.1. ( β ) A is unital, and there is n ≥ such that, for each non-zero a ∈ A + and ε > ,there exist d , . . . , d n ∈ A and g , . . . , g n ∈ G that satisfy the inequality (4) inDefinition 6.2.If A is unital and ( A, G, σ ) satisfies ( α ) then it satisfies ( β ) with n replaced by n +1 , andif ( A, G, σ ) satisfies ( β ) then it satisfies ( α ) – with same n ∈ N . If ( A, G, σ ) satisfies( α ) or ( β ), then the algebra A is G -simple, i.e., A and { } are the only G -invariantclosed ideals of A .If A contains a projection p = 0 with pAp = C · p , then A is a C*-subalgebra of M n +1 (respectively of M n ) if ( A, G, σ ) has property ( α ) (respectively has property ( β )).The shift action σ of the cyclic group Z n +1 on A := C( Z n +1 ) satisfies ( α ) for n ∈ N and is element-wise properly outer. Proof . If A is unital and σ has property ( α ), then let b := 1 − ( k a k − a ) , and find d , . . . , d n and g , . . . , g n that satisfy the inequality (3). If we let g n +1 := e and d n +1 := k a k − / a , then a and g , . . . , g n , g n +1 satisfy (4). It shows that actions on unital A that satisfy property ( α ) are also actions that satisfy ( β ) with n + 1 in place of n . If( A, G, σ ) satisfies ( β ) and non-zero elements a, b ∈ A + are given with k b k = 1, then d b / , . . . , d n b / and g , . . . , g n is a solution of the inequality (3) in Definition 6.1 if d , . . . , d n and g , . . . , g n satisfy the inequality (4) of Definition 6.2.The properties ( α ) and ( β ) imply that { } and A are the only G -invariant closedideals of A : If A is non-unital in case ( α ), then the approximation, as expressed by theinequalities (3), shows that for each non-zero a ∈ A + the smallest closed G -invariantideal of A containing a contains each b ∈ A + . If A is unital and the actions hasproperty ( β ) then each G -invariant closed ideal of A contains 1. If A is unital and theC*-dynamical system ( A, G, σ ) satisfy property ( α ) then it satisfies property ( β ) with n replaced by n + 1. Thus, again, A and { } are the only closed G -invariant ideals. rom now on, we suppose that there exists a projection 0 = p ∈ A + with pAp = C p .We call those projections “minimal”, even if minimal non-zero projections of a C *-algebra A do not have the property pAp = C p in general, e.g. the unit of the Jiang-Sualgebra Z is a minimal projection. We show that this assumption of the existence ofsuch p ∈ A , together with the assumption that σ satisfies ( α ), implies that A is unital.Thus A satisfies ( β ) with n + 1 in place of n . Then we derive that property ( β ) andthe existence of such p ∈ A imply that A is a C *-subalgebra of M n .It is obvious that the ideal socle( A ) generated by those “minimal” projections is(universally) invariant under all automorphisms of A , i.e., α (socle( A )) = socle( A ) forall α ∈ Aut( A ). This happens also for the closure J of socle( A ). Thus, J must be G -invariant. It follows that J = A using socle( A ) = ∅ .It is not difficult to see, that J is isomorphic to the c -direct sum of a family ofalgebras K ( H τ ) of compact operators on suitable Hilbert spaces H τ , and that p is arank-one projection on some H τ . Let H denote the Hilbert space sum of the Hilbertspaces H τ . Then A becomes isomorphic to a non-degenerate C *-subalgebra of the alge-bra of compact operators K ( H ) on H , in a way that each minimal non-zero projection p ∈ A becomes a rank one projection on H . This happens also for all σ g ( p ). Recallthat every projection in A ⊆ K ( H ) has finite rank in L ( H ). Since A is a C *-subalgebraof K ( H ), A is in particular an AF-algebra, and – therefore – contains an approximateunit ( q τ ) consisting of an upward directed net of projections of finite rank in H .We show that A must be unital if ( A, G, σ ) satisfies ( α ) in addition: Suppose that A is not unital, then none of the projections ( q τ ) are invertible in A . Therefore, we cantake b := q τ , a := p and ε = 1 / q τ has linear rank ≤ n .This implies that the approximate unit ( q τ ) must be constant q τ = e for all τ ≥ τ with suitable τ . Then e ∈ A is necessarily the unit element of A , in contradiction toour assumption that A is not unital.It follows that A must be unital, and – as above observed – the action σ satisfiesproperty ( β ) with n replaced by n + 1.If A is unital and ( A, G, σ ) satisfies property ( β ), then we take a := p and ε := 1 / A in its representation is ≤ n . Thus A isa C *-subalgebra of M n in case ( β ).The crossed product C( Z n +1 ) ⋊ Z n +1 is naturally isomorphic to M n +1 . Hence, byRemark 5.3, the action of Z n +1 on C( Z n +1 ) is element-wise properly outer. If a ∈ + := C( Z n +1 ) + is non-zero and b ∈ A + is not invertible, then there are non-zerominimal projections p, q ∈ A + and δ > a ≥ δp and b ≤ k b k · (1 − q ). Select g , . . . , g n ∈ Z n +1 with P nj =1 σ g j ( p ) = 1 − q . It implies that δ − P nj =1 σ g j ( a ) ≥ − q .Thus, there exists T ∈ (1 − q ) A + (1 − q ) with T ( P nj =1 σ g j ( a )) T = 1 − q . Then a , b , d j := T b / , j = 1 , . . . , n and g , . . . , g n satisfy the inequality (3) for each ε > (cid:3) Remark 6.4.
Let B be a non-zero simple C *-algebra. In preperation for the proof ofTheorem 1.2 we display here a number of properties equivalent to strong pure infinite-ness of B .(i) B is strongly purely infinite.(ii) Each non-zero element of B + is properly infinite in sense of [13].(iii, n ) There exists n ∈ N such that, for each non-zero elements a, b ∈ B + and ε > n elements d , . . . , d n ∈ B with k d ∗ ad + · · · + d ∗ n ad n − b k < ε , (5)and B is not isomorphic to M k for any k ≤ n .(iv) B is locally purely infinite in sense of [3, def. 1.3], i.e., each non-zero hereditary C *-subalgebra of B contains a non-zero stable C *-subalgebra.(v) B is purely infinite in the sense of J. Cuntz [5, p. 186], i.e., each non-zerohereditary C *-subalgebra contains an infinite projection. Proof . Property (ii) implies (i) by [3, thm. 5.8] and (i) implies (ii) by [14, prop. 5.4].Property (iii, n = 1) is equivalent to (ii) by [13, thm. 4.16]. The properties (iii, n = 1),(iv) and (v) are equivalent by [3, prop. 3.1].(iii) ⇒ (ii): Suppose that B is elementary, i.e., that B ∼ = K ( H ) for some Hilbert space H . By (iii), applied on some rank-one projection a := p ∈ B + , b ∈ B + , and ε = 1 /
2, itfollows that each element of b ∈ B + has rank ≤ n . Thus, H has dimension k ≤ n , and B ∼ = M k . But the latter case was excluded in (iii, n ). Therefore B is non-elementaryand hence has the global Glimm halving property in sense of [3, def. 2.6]. This is easyto see for non-elementary simple B (or use Glimm halving [20, lem. 6.7.1]). Since B is simple property (iii, n ) ensures that B satisfies property (i) of [3, def. 1.2] of pi( n ).Therefore, [3, prop. 4.14] says that B is pi(1). Since B is simple and B = C , there areno non-zero characters on B . In particular pi(1) ensures that B is purely infinite inthe sense of [13, def. 4.1]. By [13, thm. 4.1] we obtain property (ii). (cid:3) roof of Theorem 1.2 : By Lemma 6.3, A is G -simple. Thus the action σ isautomatically exact by Definition 3.5. Since σ is element-wise properly outer by as-sumption, it is now also residually properly outer, and Theorem 3.8 applies. It saysthat F := A + is a filling family in A ⋊ σ,λ G . In particular, A separates the ideals ofthe reduced crossed product. It shows that B := A ⋊ σ,λ G is simple.If b ∈ B + with k b k = 1, then there exists non-zero z ∈ B such that z ∗ z ≤ b and zz ∗ ∈ A , because A + is filling for B : One way to see this is to use property (i) ofDefinition A.3 on a ′ := (3 b − + , b ′ := (3 b − + − a ′ and c ′ := 3 b − a ′ − b ′ to get elements z j , d ∈ B . This imply that z ∗ j z j ≤ e = P i z ∗ i z i = c ′ ec ′ ≤ k c ′ kk e k c ′ ≤ k c ′ kk e k b , where e = 0 because d ∗ ed = a ′ = 0.Take δ ∈ (0 , k z k / = z ( z ∗ z − δ ) + z ∗ = ϕ ( zz ∗ ) ∈ A + for some suitable ϕ ∈ C c (0 , ∞ ] + .We consider three cases: (i) the action is n -majorizing and A is non-unital, (ii) theaction is n -majorizing and A is unital and (iii) the action is n -covering and A is unital.Since G is discrete, A is a non-degenerate C *-subalgebra of B . In particular, A + contains an approximate unit ( e ν ) of positive contractions in A + for B , which we willuse for case (i). In case (ii)-(iii) let e ν := 1, where 1 is the unit of A . Define m := n + 1for case (ii) and m := n for case (i) and (iii). By each of the Definitions 6.1 and6.2 (and using Lemma 6.3 to get property ( β ) in case (ii) with n replaced by m ),for each ε > e ν ∈ A + there are d , . . . , d m ∈ A and g , . . . , g m ∈ G such that k e ν − P j f ∗ j ( z ( z ∗ z − δ ) + z ∗ ) f j k < ε for f j := U ( g − j ) d j in B .By Remark 2.1(ii), there exists a contraction d ∈ B with d ∗ bd = ( z ∗ z − δ ) + . Thenthe elements y j := d z ∗ f j ∈ B satisfy k e ν − P mj =1 y ∗ j by j k < ε . Since a / e ν a / convergesto a ∈ B + , we get that B has the following property: For any two non-zero elements b, a ∈ B + and ε > there exists m elements d , . . . , d m ∈ B such that k a − P mj =1 d ∗ j bd j k < ε . A simple C *-algebra B withthis property is strongly purely infinite by Remark 6.4 if B is not isomorphic to M k forsome k ≤ m . The case that B is isomorphic to a C *-subalgebra of M m has been ex-cluded by the definitions: For case (i) we know A is non-unital, hence B is non-unital.For case (ii)-(iii) we know A is not isomorphic to a C *-subalgebra of M m , hence B isnot isomorphic to a C *-subalgebra of M m . (cid:3) Lemma 6.5.
The following are equivalent for C*-algebras B . (i) B does not contain a non-zero projection p ∈ B + with pBp = C · p . ii) For every non-zero hereditary C*-subalgebra D of B , each maximal commutativeC*-subalgebra C of D has perfect Gelfand spectrum b C . i.e., b C does not containan isolated point. (iii) For every a , a ∈ B + \{ } and c ∈ B there are b , b ∈ B + \{ } with b cb = 0 ,and b j ≤ a j ( j = 1 , ). Proof . (iii) ⇒ (i): Let p ∗ = p = p ∈ B \{ } , put a k := c := p for k = 1 ,
2. Then thereare non-zero b , b ∈ ( pBp ) + with b b = 0. Thus pBp = C · p .(i) ⇒ (ii): Let D = { } a hereditary C *-subalgebra of B , and C a maximal commuta-tive C *-subalgebra of D . Suppose that b C is not perfect. Then b C contains an isolatedpoint χ . The point χ corresponds to a projection 0 = p ∈ C + ⊆ D + with pBp = C · p ,see [21, lem. 7.14].(ii) ⇒ (iii): It is easy to see that a commutative C *-algebra C with a perfect spectrum b C contains non-zero contractions e , e ∈ C + with e e = 0, because the locally compactHausdorff space b C must in particular contain two different points ( = ∞ ).Given c ∈ B and non-zero a , a ∈ B + , let d j := ( a j − k a j k / + and x := d / cd / .Notice that 0 = d j ∈ B + , and that 0 = d / j y j d / j ≤ a j for all non-zero contractions0 ≤ y j ∈ d j Bd j . If x = 0 then take b j := d j . If x = 0, consider the hereditary C *-subalgebra D := x ∗ Bx = x ∗ xBx ∗ x that is generated by x ∗ x , and is contained in d ∗ Bd . Let C be a maximal commutative C *-subalgebra of D with x ∗ x ∈ C . Then C contains non-zero contractions e , e ∈ D + with e e = 0 = e ( x ∗ x ) / e .It is well-known (and easy to see) that the polar decomposition x = v ( x ∗ x ) / in B ∗∗ has the property that vDv ∗ = xBx ∗ ⊆ d ∗ Bd . Thus f := ve v ∗ ∈ xBx ∗ and has theproperty f xe = ve ( x ∗ x ) / e = 0. It follows that b := d / f d / and b := d / e d / satisfy b cb = d / f xe d / = 0 and 0 = b j ≤ a j . (cid:3) Proposition 6.6.
Let ( A, G, σ ) a C*-dynamic system with non-zero non-unital A . Ifthe action σ of G on A is an -majorizing action in the sense of Definition 6.1, then A is G -simple and σ is G -separating for A . Proof . The algebra A is G -simple and A does not contain a projection p = 0 with pAp = C · p by Lemma 6.3. To show that σ is G -separating let a , a ∈ A + \{ } , c ∈ A and ε >
0. By Lemma 6.5, there exist b , b ∈ A + \{ } with b cb = 0 and b j ≤ a j ( j = 1 , e j ∈ A , h j ∈ G for = 1 , k e ∗ j σ h j (cid:0) b j a j b j (cid:1) e j − a j k < ε . With g j := h − j and d j := b j σ g j ( e j )we get k d ∗ j a j d j − σ g j ( a j ) k < ε and d ∗ cd = 0, i.e., σ is G -separating. (cid:3) Remark 6.7.
Suppose that (
A, G, σ ) is a C *-dynamical system, A is unital and com-mutative, and G is discrete. Then the following properties (i)–(iv) of the action σ areequivalent:(i) The action is 2-covering in sense of Definition 6.2.(ii) The corresponding (adjoint) action b σ , on b A is a strong boundary action in thesense of Definition A.1.(iii) The action is 1-majorizing in sense of Definition 6.1.(iv) The action is 2-filling in sense of Definition A.2, and A is not isomorphic to asubalgebra of M ( C ).We do not know if, also for non-commutative and unital A , every 2-covering action isa 1-majorizing action, or a 2-filling action. Proof . We show more general implications, except for (i) ⇒ (ii). In particular we showthat an action σ on a unital abelian C *-algebra A is n -filling if and only if it is n -covering provided that the (linear) dimension of A is greater than n .(i) ⇒ (iv) (for A unital, commutative, any n ):Suppose that A ∼ = C( X ), and take any n ≥
2. Let α denote the action of G on X inducing σ , i.e., σ g ( f ) = f ◦ α − g for f ∈ A and g ∈ G . Since the action α is minimal byLemma 6.3, Remark [11, rem. 0.4] shows it suffices to prove that for each non-emptysubset U of X there exist g , . . . , g n ∈ G , such that α g ( U ) ∪ α g ( U ) ∪ · · · ∪ α g n ( U ) = X .Let such U be given. Select non-zero a ∈ A + with support contained in U . By (i) thereexist g , . . . , g n ∈ G and d , . . . , d n ∈ A such that P nj =1 d ∗ j σ g j ( a ) d j ≥ . In particularfor each x ∈ X , σ g j ( a )( x ) is non-zero for some j , so x ∈ α g i ( U ).(iv) ⇒ (i) (for A unital, commutative/non-commutative, any n ):Suppose that A is unital, and take any n ≥
2. Let 0 = a ∈ A + . Using (iv) there are g , . . . , g n ∈ G and δ > D := P nj =1 σ g j ( a ) ≥ δ , and A is not isomorphicto a C *-subalgebra of M n . Thus, D is invertible in A and P nj =1 d ∗ j σ g j ( a ) d j = 1 for d j := D − / in A .(iii) ⇒ (i) (for A unital, commutative/non-commutative, any n ):Each n -majorizing actions on unital A is an ( n + 1)-covering action by Lemma 6.3. i) ⇒ (ii) (for A unital, commutative, one n ):Suppose that A ∼ = C( X ) and let α denote the action of G on X inducing σ . Theequivalence of (i) and (iv), shows that for given 0 = a ∈ A + , there exists g ∈ G and δ > a + σ g ( a ) ≥ δ U ⊆ X open and non-empty. There is 0 = a ∈ C( X ) + with support a − (0 , ∞ ) ⊆ U . Choose h ∈ G and δ > a + σ h ( a ) ≥ δ
1. It implies that σ h ( a )( x ) > x ∈ X \ U . Thus, α g ( x ) ∈ U for all x ∈ X \ U and g := h − , i.e., there exists g ∈ G with α g ( X \ U ) ⊆ U .Given non-empty open subsets U and V of X . We let W := U ∩ V if U ∩ V = ∅ .Then g ∈ G with α g ( X \ W ) ⊆ W satisfies α g ( X \ U ) ⊆ V . If U ∩ V = ∅ then we find g, h ∈ G with α g ( X \ U ) ⊆ U ⊆ X \ V and α h ( X \ V ) ⊆ V . Then α hg ( X \ U ) ⊆ V .The space X contains more than two points because A is not isomorphic to a C *-subalgebra of M ( C ). Thus, ( X, G, α ) satisfies the conditions of Definition A.1 of astrong boundary action.(ii) ⇒ (iii) (for A unital/non-unital, commutative, one n ):We show (iii) using possibly less than (ii): Let X be a locally compact space that isnot necessarily compact and contains more than 2 points. Let α an action of G on X with the property that, for every compact subset K ⊆ X with K = X and eachnon-empty open subset U ⊆ X , there exists g ∈ G with α g ( K ) ⊆ U . Then the adjointaction σ of α on A := C ( X ) is an -majorizing action of G on A . Indeed: Let 0 = a ∈ A + , b ∈ A + non-invertible, and ε >
0. Put δ := ε/
3. Then,considered as functions on X , they have the property that U := a − ( k a k / , ∞ ) is non-empty and open and K := b − [ δ, ∞ ) is compact. Find h ∈ G with α h ( K ) ⊆ U . Then x ∈ K ⇔ b ( x ) ≥ δ implies that for g := h − we get σ g ( a )( x ) = a ( α h ( x )) > k a k / k d ∗ σ g ( a ) d − b k < ε with d ∈ A + given by d ( x ) := σ g ( a )( x ) − / ( b ( x ) − δ ) / for x ∈ K and d ( x ) := 0 for x ∈ X \ K . (cid:3) Remarks 6.8. (i) Let α be an action of a discrete group G on a locally compactHausdorff space X with more than two points, and σ the induced action on A := C ( X ).By Remark 6.7 the following properties are equivalent for X compact:(1) The action α is a strong boundary action (Definition A.1) in the sense of [18]:For each pair of non-empty open subsets U and V of X there exists g ∈ G with U ∩ α g ( V ) = X .
2) For any compact set K = X and any open set U = ∅ there exist t ∈ G suchthat α t ( K ) ⊆ U .(3) For every non-zero a ∈ A + , every non-invertible b ∈ A + and every ε >
0, thereexist d ∈ A and g ∈ G such that k d ∗ σ g ( a ) d − b k < ε .Clearly, this can not be the case if X is locally compact but is not compact. In general(i.e., when X is compact or non-compact) we know (1) ⇒ (2) ⇒ (3). Properties (2)-(3)are both candidates for a generalised notion of a strong boundary action, however only(3) applies when A is non-commutative.(ii) The notion of a strong boundary action (Definition A.1) is defined on compact Hausdorff spaces with more than two points. In view of Remark 6.8(i) and Remark 6.7,we call the 1-majorizing actions on not-necessarily unital or commutative C *-algebrasalso strong boundary actions.(iii) Suppose that a discrete group G acts by a topologically free action α on a compact Hausdorff space X , and that X contains more than two points. It was shown in [18,thm. 5] that the crossed product C( X ) ⋊ σ,λ G is purely infinite provided that theaction – in addition – is a strong boundary action. Since topological freeness implies σ is element-wise properly outer (by [1, prop. 1]) we conclude that, with the terminologyof Remark 6.8(ii), [18, thm. 5] is a special case of Theorem 1.2. (cid:3) (iv) Let α be an action on a non-compact locally compact Hausdorff space X with morethan two points and σ the induced action. It was shown in Proposition 6.6 that σ is G -separating if σ is a strong boundary (i.e., 1-majorizing) action. A simpler argumentapplies if we assume that for any compact set K = X and any non-empty open set U ⊆ X there exist g ∈ G such that α g ( K ) ⊆ U : Proof . Since any finite subset M of X is compact, it can be moved by suitable α g intoany non-empty open subset U of X . In particular X is perfect and each non-emptyopen set V ⊆ X contains at least two non-empty open disjoint subsets V and V . Let K ⊆ U and K ⊆ U with K j compact (hence K j = X ) and U j open. If U and U are disjoint, then we can take g = g = e in Lemma 5.1(ii). If V := U ∩ U = ∅ ,then consider the above disjoint non-empty open subsets V j ⊆ V . By assumption,there exist g , g ∈ G with α g j ( K j ) ⊆ V j ⊆ U j . Thus, the adjoint action σ of α is G -separating. (cid:3) (v) Suppose that a discrete group G acts by a topologically free action α on a non-compact locally compact Hausdorff space X , and that X contains more than two points. hen the crossed product C ( X ) ⋊ σ,λ G is purely infinite provided that the followingproperty holds: for any compact set K = X and any non-empty open set U ⊆ X thereexist t ∈ G such that α t ( K ) ⊆ U . This follows as a corollary of Theorem 1.1, also ofTheorem 1.2 or of Corollary 5.2. Proof . We must verify the following properties according to each of the listed results:(1.1) The action σ is exact, residually properly outer and G -separating.(1.2) The action σ is 1-majorizing, and element-wise properly outer.(5.2) The action σ is exact, G -separating and fulfills (*): For every closed G -invariantsubset Y of X and every g = e the set { y ∈ Y : α g ( y ) = y } has empty interior.By Remark 6.8(i) we know the action σ is a strong boundary (i.e., 1-majorizing) action.Hence A is G -simple, cf. Lemma 6.3. In particular it follows that the action α on X is minimal. This reduces property (*) to the definition of topological freeness, cf. [1,p.120]. The minimality of the action α implies that the corresponding adjoint action σ : G → Aut(C ( X )) is exact, and that it becomes residually properly outer if it iselement-wise properly outer. But σ is element-wise properly outer if and only if α is a topological free action (see [1, p.120] or [23, cor 2.22]). It remains to show σ is G -separating, but this is already contained in Remark 6.8(iv). (cid:3) (vii) It is an important point that a strong boundary action is often G -separating and(in fact always) minimal, but the notion of a G -separating action is not typically relatedto minimality. Consequently, working with G -separating actions allows us to considerideal-related classification of non-simple strongly purely infinite crossed products. Remark 6.9. If A has real rank zero, then one can restrict the conditions in Definitions4.1, 6.1 and 6.2 to projections p, q ∈ A in place of the elements a, b ∈ A + . Proof . Case of Definition 4.1: Let a , a ∈ A + , c ∈ A , ε > δ := ε/ (1 + k a k + k a k ) . By [4], D j := a j Aa j contains an approximate unit consisting of non-zero projec-tions. Thus, there are projections p j ∈ D j such that k a j − a / j p j a / j k < δ . Use [13,prop. 2.7(i)] and the comment following [13, prop. 2.6] to select z j ∈ D j satisfying z ∗ j a j z j = p j . Let c ′ := z ∗ cz .Suppose that there exists e j ∈ A , g j ∈ G such that k e ∗ j p j e j − σ g j ( p j ) k < δ and k e ∗ c ′ e k < δ . efine v j := σ g j ( a / j ) and d j := z j e j σ g j ( a / j ) . They satisfy d ∗ j a j d j = v ∗ j e ∗ j p j e j v j , v ∗ j σ g j ( p j ) v j = σ g j ( a / j p j a / j ). Thus, k d ∗ j a j d j − σ g j ( a j ) k < (1 + k a j k ) δ ≤ ε . Since d ∗ cd = v ∗ e ∗ c ′ e v , we get k d ∗ cd k < δ ( k a k · k a k ) / ≤ δ ( k a k + k a k ) ≤ ε .Case of Definitions 6.1 and 6.2: Let a , a ∈ A + and ε >
0, with a = 0 and a notinvertible in A (respectively a = 1). We can assume ε ≤
1. Define δ := ε/ (1 + k a k ) .Choose p j , z j ∈ D j := a j Aa j as above with k a / j p j a / j − a j k < δ and z ∗ j a j z j = p j . Then p = 0 and p is not invertible in A (i.e., p = 1) if a is not invertible, otherwise p = 1:If p is invertible then 1 ∈ a Aa , so a is invertible. Conversely, if a is invertible then k p − k < ε/
2, so p is invertible.If there are e , . . . , e n ∈ A and g , . . . , g n ∈ G with k p − P j e ∗ j σ g j ( p ) e j k < δ , then d j := σ g j ( z ) e j a / satisfies k a − P j d ∗ j σ g j ( a ) d j k < (1 + k a k ) δ ≤ ε . (cid:3) Acknowledgments
Parts of this work were conducted while the second named author was at the FieldsInstitute from 2009 to 2012. It is with great pleasure we forward our thanks to theFields Institute and in particular Professor George Elliott for all the support. Thisresearch was also supported by the Australian Research Council.
Appendix
A.In this appendix we have included a few recent definitions and results that arefrequently cited throughout this paper. The results quoted from [15] are available aspreprint.
Definition A.1 ([18]) . Let α be an action of a discrete group G on a compact spaces X with at least three points. The action α is as strong boundary action if for every pair U and V of non-empty open subsets of X there exists t ∈ G such that α t ( X \ U ) ⊆ V . Definition A.2 ([11]) . An action σ of a discrete group G on a unital C *-algebra A is n -filling ( n ≥
2) if, for all b , . . . , b n ∈ A + , with k b j k = 1 for each j , and all ε > g , . . . , g n ∈ G such that P nj =1 σ g j ( b j ) ≥ − ε . Definition A.3 ([15]) . Let F be a subset of A + . The set F is a filling family for A ,if F satisfies the following equivalent conditions (i) and (ii). i) For every a, b, c ∈ A with 0 ≤ a ≤ b ≤ c ≤
1, with ab = a = 0 and bc = b ,there exists z , z , . . . , z n ∈ A and d ∈ A with z j ( z j ) ∗ ∈ F , such that ec = e and d ∗ ed = a for e := z ∗ z + . . . + z ∗ n z n .(ii) For every hereditary C *-subalgebra D of A and every primitive ideal I of A with D I there exist f ∈ F and z ∈ A with z ∗ z ∈ D and zz ∗ = f I . Lemma A.4 ([15]) . Suppose that A ⊆ B is a C*-subalgebra of B and F ⊆ A + is asubset of A + . If F is filling for A , and A + is filling for B , then F is a filling familyfor B . Remark A.5 ([15]) . Let A ⊆ B be C *-algebras and F ⊆ A + . If F := A + ⊆ B isfilling for B , then the map I ∈ I ( B ) I ∩ A ∈ I ( A ) is injective, i.e., A separates theclosed ideals of B . Definition A.6 ([15]) . A C *-algebra A is strongly purely infinite (for short: s.p.i. )if, for every a, b ∈ A + and ε >
0, there exist elements s, t ∈ A such that k s ∗ a s − a k < ε , k t ∗ b t − b k < ε and k s ∗ abt k < ε . (6) Remark A.7 ([15]) . A C *-algebra A is strongly purely infinite if and only if for every a, b ∈ A + , c ∈ A and ε >
0, there exist contractions s, t ∈ A such that k s ∗ as − a k < ε , k t ∗ bt − b k < ε and k s ∗ ct k < ε . (7) Definition A.8 ([15]) . Let
S ⊆ A be a multiplicative sub-semigroup of a C *-algebra A and C ⊆ A a subset of A . An n -tuple ( a , . . . , a n ) of positive elements in A has the matrix diagonalization property with respect to S and C , if for every [ a ij ] ∈ M n ( A ) + with a jj = a j and a ij ∈ C (for i = j ) and ε j > , τ > s , . . . , s n ∈ S with k s ∗ j a jj s j − a jj k < ε j , and k s ∗ i a ij s j k < τ for i = j . (8)If S = C = A then this is the matrix diagonalization property of ( a , . . . , a n ) as definedin [14, def. 5.5], and we say that ( a , . . . , a n ) has matrix diagonalization (in A ). Definition A.9 ([15]) . Let F be a subset of A + . The family F has the (matrix)diagonalization property (in A ) if each finite sequence a , . . . , a n ∈ F has the matrixdiagonalization property (in A ) of Definition A.8. Lemma A.10 ([15]) . Suppose that
F ⊆ A + is invariant under ε -cut-downs, i.e., thatfor each a ∈ F and ε ∈ (0 , k a k ) we have ( a − ε ) + ∈ F .Then the family F has the matrix diagonalization property, if and only if, each pairof elements in F has the matrix diagonalization property of Definition A.8. emma A.11 ([15]) . Let ε > and non-empty subsets F ⊆ A + , C ⊆ A be given,and let S ⊆ A be a (multiplicative) sub-semigroup of A that satisfies s ∗ C s ⊆ C for all s , s ∈ S . Suppose that the following properties hold: (i) For every ε > δ > , the pair (( a − δ ) + , ( a − δ ) + ) the matrix diagonalizationproperty with respect to S and C of Definition A.8. (ii) ϕ ( a ) cϕ ( a ) ∈ C for each c ∈ C and ϕ ∈ C c (0 , ∞ ] + . (iii) ϕ ( a ) s, ϕ ( a ) s ∈ S for each s ∈ S and ϕ ∈ C c (0 , ∞ ] + .Then, for every c ∈ span( C ) , a , a ∈ F , ε / ≥ ε > , and τ > , there exists s , s ∈ S that fulfil k s j k ≤ k a j k /ε and k s ∗ a s − a k < ε , k s ∗ a s − a k < ε and k s ∗ cs k < τ . (9) Theorem A.12 ([15]) . The minimal tenor product of a strongly purely infinite and anexact C*-algebra is strongly purely infinite.
Theorem A.13 ([15]) . Suppose that A + contains a filling family F that has the diag-onalization property (in A ). Then A is strongly purely infinite. References [1] R.J. Archbold and J.S. Spielberg,
Topologically free actions and ideals in discrete C*-dynamicalsystems , Proc. Edinburgh Math. Soc. (2) (1994), 119–124. 6, 7, 16, 17, 25, 26[2] G. Arzhantseva, V. Guba, and M. Sapir, Metrics on diagram groups and uniform embeddings inHilbert space , Comment. Math. Helv. (2006), 911–929. 9[3] E. Blanchard and E. Kirchberg, Non-simple purely infinite C*-algebras: the Hausdorff case , J.Funct. Anal. (2004), 461–513. 4, 20[4] L.G. Brown and G.K. Pedersen,
C*-algebras of real rank zero , J. Funct. Anal. (1991), 131–149.26[5] J. Cuntz, K-theory for certain C*-algebras , Ann. of Math. (1981), 181–197. 20[6] G.A. Elliott,
Some simple C*-algebras constructed as crossed products with discrete outer auto-morphism groups , Publ. Res. Inst. Math. Sci. (1980), 299–311. 6[7] R. Exel, M. Laca, and J. Quigg, Partial dynamical systems and C*-algebras generated by partialisometries , J. Operator Theory (2002), 169–186. 7[8] D.S. Farley, Proper isometric actions of Thompson’s groups on Hilbert space , Int. Math. Res.Notes (2003), 2409–2414. 9[9] E. Guentner and J. Kaminker, Exactness and uniform embeddability of discrete Groups , J. LondonMath. Soc.(2) (2004), 703–718. 9[10] U. Haagerup and G. Picioroaga, New presentations of Thompson’s groups and applications , J.Operator Theory (2011), 217–232. 8
11] P. Jolissaint and G. Robertson,
Simple purely infinite C*-algebras and n -filling actions , J. Funct.Anal. (2000), 197–213. 3, 23, 27[12] S. Kawamura and J. Tomiyama, Properties of topological dynamical systems and correspondingC*-algebras , Tokyo J. Math. (1990), 251–257. 7, 16[13] E. Kirchberg and M. Rørdam, Non-simple purely infinite C*-algebras , Amer. J. Math. (2000),637–666. 20, 26[14] ,
Infinite non-simple C*-algebras: absorbing the Cuntz algebras O ∞ , Adv. Math. (2002), no. 2, 195–264. 4, 12, 20, 28[15] E. Kirchberg and A. Sierakowski, Filling families and strong pure infiniteness , preprint 2014. 2,27, 28, 29[16] E. Kirchberg and S. Wassermann,
Exact groups and continuous bundles of C*-algebras ,Math. Ann. (1999), 169–203. 8, 14[17] A. Kishimoto,
Outer automorphisms and reduced crossed products of simple C*-algebras ,Comm. Math. Phys. (1981), 429–435. 6[18] M. Laca and J. Spielberg, Purely infinite C*-algebras from boundary actions of discrete groups ,J. Reine Angew. Math. (1996), 125–139. 1, 3, 24, 25, 27[19] D. Olesen and G.K. Pedersen,
Applications of the Connes spectrum to C*-dynamical systems. III ,J. Funct. Anal. (1982), 357–390. 6, 7[20] G.K. Pedersen, C*-algebras and their automorphism groups , LMS Monographs, vol. 14, AcademicPress Inc., London, 1979. 20[21] M. Rørdam, F. Larsen, and N. Laustsen,
An introduction to K-theory for C*-algebras , LondonMathematical Society Student Texts , Cambridge University Press, Cambridge, 2000. 22[22] J. Renault, The ideal structure of groupoid crossed product C*-algebras , With an appendix byGeorges Skandalis.