Tracially amenable actions and purely infinite crossed products

We introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C$^*$-algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group $F_2$, we show that outer, tracially amenable actions on simple, unital, $\mathcal{Z}$-stable C$^*$-algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.


Introduction
The theory of amenable actions on C * -algebras is an important tool for studying approximation properties of crossed products.Already initiated in [2], this topic has recently received a lot of attention after it gained new impetus in [10] (see also [9,1,5,33]).As recently established in [33], the notion of amenability is equivalent to the so-called quasicentral approximation property (QAP) from [9].The QAP is a versatile tool making powerful averaging techniques accessible, with evidence being the classification of amenable, outer actions on Kirchberg algebras [14] or the equivariant O 2 -absorption theorem [41].
The motivation for this paper originates from Elliott's classification program, a long-time endeavour in the theory of C * -algebras aiming to classify nuclear C *algebras by K-theoretic and tracial data.Elliott's program is now considered to be essentially completed, with unital, simple, separable, nuclear, Z-stable C * -algebras satisfying the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet [39] being classified up to isomorphism by their Elliott invariant (see [49] for an exhaustive bibliography on the matter).In the rest of this introduction, and although this terminology is not standard, algebras satisfying all these assumptions will be called classifiable.In view of this recent progress, an important further step is to identify prominent classes of C * -algebras that satisfy the assumptions of the classification theorem.This problem has attracted considerable attention in connection to crossed products, and most of the work in the literature has focused on actions of discrete (and usually amenable) groups on C * -algebras that are either abelian or simple.This work deals with the latter setting.
If α : G → Aut(A) is an action of a discrete group G on a simple, unital C *algebra A, then the crossed product A ⋊ r G is simple whenever α is outer (see [25]).If A is moreover nuclear, then A⋊ r G is nuclear if and only if α is amenable (see [2]).Whether A ⋊ r G satisfies the UCT is a very subtle question: the answer is always "yes" if A satisfies the UCT, and G is torsion-free with the Haagerup property (see [31,22]), but the problem is equivalent to the UCT question for torsion groups (see [6,Example 23.15.12 (d)] and [4]).Therefore, modulo the UCT, classifiability of A ⋊ r G for an amenable, outer action on a simple, unital, nuclear C * -algebra reduces to proving Z-stability for A ⋊ r G.
The problem of establishing Z-stability for A ⋊ r G when A is simple and nuclear has been largely investigated in the case where G is amenable and A is Z-stable.In this case, Z-stability of A⋊ r G is conjectured to always hold (see [44, Conjecture A]), and this has been verified in full generality if A is purely infinite ( [43]), and under various degrees of generality when A is stably finite and its trace space T (A) is a Bauer simplex with finite dimensional boundary ( [29,30,40,17,50]).Further progress has been recently obtained in [16], where Z-stability of A⋊ r G is obtained in a number of cases where the boundary of T (A) is compact but not necessarily finite dimensional.On the other hand, Z-stability may fail for actions of nonamenable groups; see [18,Theorem B].
In this paper, we study amenable actions of nonamenable groups on simple C *algebras, inspired by the results in [15] for commutative C * -algebras.Our original motivation was to show that the crossed product of an outer, amenable action of a nonamenable group on a simple, unital, nuclear, Z-stable C * -algebra is automatically purely infinite and simple.For groups containing the free group F 2 , this follows from Corollary 3.15.As it turns out, the assumptions on both the algebra and the action can be weakened significantly.For once, we do not need to assume A to be nuclear or even Z-stable (see Corollary 3.15 for the minimal set of assumptions).More important for this work is the fact that amenability of α is also stronger than necessary.This observation led us to identify and isolate the following notion: Definition A (Definition 2.2).An action α : G → Aut(A) of a countable, discrete group G on a separable, unital, tracial C * -algebra A is said to be tracially amenable if there is a sequence (ξ n ) n∈N of finitely supported functions for all a ∈ A and all g ∈ G.
In the above definition, we denote by α the diagonal action on ℓ 2 (G, A) with left translation on G and α on A, and we denote by • 2,u the so-called uniform trace norm on ℓ 2 (G, A); see the comments before Lemma 2.1.Thanks to the characterizations obtained in [33], the usual notion of amenability is obtained by replacing the uniform trace norm by the usual Hilbert C * -module-norm on ℓ 2 (G, A).Since • 2,u ≤ • , it follows that any amenable action is tracially amenable.Section 2 is devoted to obtaining several characterizations of tracial amenability, inspired by the work done in [33,Theorem 3.2,Theorem 4,4].We reproduce some of them below: Theorem B (Theorem 2.5).Let α : G → Aut(A) be an action of a countable, discrete, exact group G on a separable, unital, tracial C * -algebra A. The following are equivalent: (1) The action α is tracially amenable.
(2) There is a sequence (ξ n ) n∈N of finitely supported functions ξ n : G → A with ξ n ≤ 1 such that fin : G → Aut(A * * fin ) is von Neumann-amenable.Some of the above conditions look similar to analogous characterizations of amenability on C * -algebras, while some do not admit a counterpart in that setting.For example, the difference between Definition A and item (2) above is that we do not require approximate centrality in (2).Also, (4) and ( 5) do not have analogues in the setting of amenable actions, since amenability of G S(A) does not imply amenability of α.
We study the structure of crossed products in Section 3.For groups containing the free group F 2 , we show that tracially amenable actions give rise to purely infinite crossed products: Theorem C (Corollary 3.15).Let G be a countable, discrete group containing the free group F 2 , let A be a simple, separable, unital, stably finite, nuclear, Z-stable C * -algebra, and let α : G → Aut(A) be a tracially amenable, outer action.Then A ⋊ r G is a unital, simple, purely infinite C * -algebra.
As mentioned before, the requirements on A can be weakened, and we in particular do not need to assume A to be either nuclear or Z-stable (see the statement of Corollary 3.15 for the precise assumptions on A).The condition on the group can also be relaxed, and it suffices to assume that G has what we call weak paradoxical towers; see Definition 3.10.To obtain Theorem C, we show in Proposition 3.9 that it suffices to prove that any action as in the statement satisfies what Bosa, Perera, Wu and Zacharias call dynamical strict comparison (Definition 3.8).That this is the case is shown in Theorem 3.14, by exploiting the tension between tracial amenability of α and the existence of weak paradoxical towers in G.
Theorem C is new even if α is amenable.In this setting, A ⋊ r G is nuclear and therefore a Kirchberg algebra, so in particular O ∞ -stable (and, therefore, also Zstable).If, moreover, G is torsion-free and has the Haagerup property (for example, G = F n ; see [8, Definition 12.2.1]),and A satisfies the UCT, then A ⋊ r G also satisfies the UCT and thus is completely determined by its K-theory by [34]; see Corollary 3.13.Thus, the assumptions of outerness and amenability on α do in fact guarantee classifiability of the crossed product.There is, however, a drawback: Problem D. Are there any amenable actions of nonamenable groups on simple, unital, stably finite C * -algebras?
The above problem has recently attracted a fair amount of attention, and to the best of our knowledge it remains open.If one drops unitality of the algebra, an example has been constructed in [42].
The We do not know if the actions that we construct here are amenable.
Acknowledgements.We are indebted to Sam Evington for his help with the proof of Lemma 3.1, to Thierry Giordano, to Mikael Rørdam for suggesting Example 2.11, and to Gábor Szabó and Stuart White for helpful comments throughout the development of this project.We are very grateful to the referee for their thorough reading of the manuscript as well as for the numerous helpful suggestions and corrections.The first and the second author would like to thank the University of Münster for the hospitality during the visit in which part of this research was done.The third author would like to thank the Chalmers University of Technology and the University of Gothenburg for their hospitality during his stay there.

Tracially amenable actions
Amenable actions of discrete groups on C * -algebras were introduced in [2] and later extended to locally compact groups in [10].In this section, we define a tracial analogue of amenability for actions of discrete groups (Definition 2.2).The motivation to study this notion is two-fold.On the one hand, tracial amenability allows us to obtain strong structural results for the crossed product; see Subsection 3. On the other hand, and unlike for amenable actions, it is possible to construct many tracially amenable actions of nonamenable groups on unital, simple, stably finite C * -algebras, by means of classification results (see Subsection 2.1).
Fix a unital C * -algebra A and a discrete group G.For an action α : G → Aut(A), we denote by ℓ 2 (G, A) the completion of C c (G, A) with respect to the norm • induced by the A-valued inner product given by ξ, η = g∈G ξ(g) * η(g) for all ξ, η ∈ C c (G, A).We equip ℓ 2 (G, A) with the A-bimodule structure given by pointwise multiplication and the diagonal G-action α given by α g (ξ)(h) = α g (ξ(g −1 h)), for all ξ ∈ ℓ 2 (G, A), and all g, h ∈ G.
Applying τ and taking the square roots gives the first inequality, while the second one is immediate.
The following definition is inspired by the quasicentral approximation property (QAP) from [9, Definition 3.1], which was shown to be equivalent to amenability for actions on C * -algebras in [33,Theorem 3.2].In view of this equivalence, we directly present the tracial analogue of [9, Definition 3.1] under the name of tracial amenability.
Remark 2.3.Note that conditions (3) and ( 4) in Definition 2.2 can be equivalently replaced by the condition ξ, α g (ξ) − 1 2,u < ε for all g ∈ K.If G is countable and discrete, and A is separable, Definition 2.2 is furthermore equivalent to the existence of a sequence (ξ n ) n∈N in C c (G, A) such that the terms in items ( 2)-( 4) converge to zero along n → ∞, for every a ∈ A and g ∈ G.
The definition of the QAP in [9, Definition 3.1] differs from Definition 2.2 in that all estimates are formulated in the C * -norm, rather than in the tracial seminorm.Since the C * -norm dominates • 2,u , it follows from [33, Theorem 3.2] that every amenable action is tracially amenable.
Our next goal is to obtain several characterizations of tracial amenability; see Theorem 2.5.For this, we need some preparation.Given an action α : G → Aut(A), a function θ : G → A is said to be of positive type with respect to α if for every finite set K ⊆ G, the matrix (α g (θ(g −1 h))) g,h∈K ∈ M |K| (A) is positive.We reproduce here two results from [2] in a way that will be more convenient for us later.The center of a C * -algebra A is denoted Z(A).
Theorem 2.4.[2, Theorem 3.3, Theorem 4.9] Let α : G → Aut(A) be an action of a discrete group G on a unital C * -algebra A. Suppose that for every finite subset K ⊆ G and every ε > 0, there exists a finitely supported positive type function θ : G → Z(A) (with respect to α) satisfying θ(e) ≤ 1 and θ(g) − 1 < ε for all g ∈ K. Then α is amenable.If A is commutative, the converse holds as well.
Given a unital C * -algebra A, its tracial state space T (A) is a convex compact topological space, and we denote by ∂ e T (A) the set of its extremal points.
For a fixed free ultrafilter ω ∈ βN \ N, we denote by A ω the quotient of ℓ ∞ (N, A) by the ideal of sequences (a n ) n∈N satisfying lim n→ω a n 2,u = 0.This is the tracial ultrapower of A. There exists a natural unital homomorphism ι : A → A ω , which maps each element in A to the class of the corresponding constant sequence.Note that the map ι is not necessarily injective, unless traces separate positive elements (which is always the case if A is simple and tracial).We denote by A ω ∩ ι(A) ′ the C * -algebra consisting of all elements in A ω that commute with (the images of) all constant sequences.Any action α : G → Aut(A) canonically induces an action on A ω and also on A ω ∩ ι(A) ′ , by acting coordinatewise.With a slight abuse of notation, we shall denote both actions by α ω .
The finite part A * * fin of the bidual A * * of a C * -algebra A can be identified with the weak closure of A with respect to the sum of all GNS-representations for all traces.Namely, Every action α on A canonically extends to an action α * * fin on A * * fin .An action α : such that ξ λ , ξ λ = 1, for all λ ∈ Λ, and the net ( ξ λ , α g (ξ λ ) ) λ∈Λ converges to 1 ∈ Z(M ) pointwise on G in the ultraweak topology (see [5,Theorem 1.1]).
In the following result we give several conditions that are equivalent to tracial amenability of an action.Some of the conditions below look similar to analogous characterizations of amenability obtained in (and, indeed, motivated by) [33, Theorem 3.2, Theorem 4,4].Others do not have a counterpart in that setting and reveal phenomena that are very special to tracial amenability.We discuss this in more detail after the proof of Theorem 2.5; see also Remark 2.8.
Theorem 2.5.Let α : G → Aut(A) be an action of a discrete group G on a unital, separable C * -algebra A with T (A) = ∅.Consider the following conditions: (1) The action α is tracially amenable.
(2) For ε > 0 and any finite set there is an equivariant, unital, completely positive map Φ : of positive type with respect to α ω , which satisfy θ n (e) ≤ 1 for all n ∈ N, and lim n→∞ θ n (g) − 1 = 0, for all g ∈ G.
(5) The induced action . If moreover G is countable and exact then all of the above conditions are equivalent.
For the proof we need the following well-known lemmas.
Lemma 2.6.Let A and B be unital C * -algebras, let π : A → B be a surjective homomorphism, and let F be a finite set.Denote the induced map ℓ 2 (F, A) → ℓ 2 (F, B) again by π.Given η ∈ ℓ 2 (F, B), there exists ξ ∈ ℓ 2 (F, A) such that π(ξ) = η and ξ = η .Proof.Note that there is an isometric embedding ℓ Lemma 2.7.Let K be a compact convex set and denote by Aff c (K) ⊆ C(K) the set of all continuous affine functions K → R. Then the restriction of the weak topology of C(K) to Aff c (K) coincides with the topology of pointwise convergence.
Proof.This follows from the fact that for every f ∈ Aff c (K) and any Radon probability measure µ on K, we have where β(µ) ∈ K denotes the barycenter of µ (see [45,Lemma IV.6.3]).
Proof of Theorem 2.5.We prove the following set of implications where, in the implications labeled by dashed arrows, we additionally assume that G is countable and exact: (5 (3) (1) ⇒ (3).Fix a state ϕ ∈ S(C).Since G is countable and A is separable, we can find a sequence We define a map Φ = (Φ n for n ∈ N and c ∈ C. (Note that sup n∈N Φ n ≤ 1 by (a.1) above, so that the resulting map Φ does indeed take values in ℓ ∞ (N, A).) Denote by Φ : C → A ω the composition of Φ with the canonical quotient map ℓ ∞ (N, A) → A ω .We claim that Φ is an equivariant, unital, completely positive map, and that its image is contained in A ω ∩ ι(A) ′ .The fact that Φ is unital follows from the equality Φ n (1) − 1 2,u = ξ n , ξ n − 1 2,u together with (a.3).As each Φ n is completely positive, the same is true for Φ and thus also for Φ.To see that Φ takes values in the operator given by pointwise multiplication with the function g → ϕ(γ g −1 (d)), and note that T d ≤ d .Using the Cauchy-Schwarz inequality (see part (2) of Lemma 2.1) and ξ n ≤ 1 at the last step, we get with the latter term converging to zero as n → ∞, by (a.2).To see that Φ is equivariant, fix g ∈ G and c ∈ C. Using again the Cauchy-Schwarz inequality and ξ n = α g (ξ n ) ≤ 1 at the last step, we obtain with the latter term converging to zero as n → ∞, by (a.4).This proves (3).
(3) ⇒ (4).Recall that since G is countable and exact, there is an amenable action of G on a compact, metrizable space X (see [8,Theorem 5.1.7]).By Theorem 2.4, there exists a sequence of finitely supported positive type functions θ n : G → C(X) with respect to α satisfying θ n (e) ≤ 1 and lim n→∞ θ n (g) − 1 = 0 for all g ∈ G.
If Ψ : C(X) → A ω ∩ ι(A) ′ is an equivariant, unital, completely positive map, then Ψ • θ n : G → A ω ∩ ι(A) ′ , for n ∈ N, gives a sequence of positive type functions with respect to α ω with the desired properties.
(4) ⇒ (5).We prove amenability of the action α ω : G → Aut(A ω ∩ ι(A) ′ ) by showing that it satisfies the QAP, which can be stated as our Definition 2.2 once every occurrence of the tracial norm is replaced with the C * -norm on A ω (see [9, Definition 3.1]), and then resorting to [33,Theorem 3.2].Fix a finite subset K ⊆ G and ε > 0. By assumption, there is a finitely supported positive type function θ : G → A ω ∩ ι(A) ′ satisfying θ(e) ≤ 1 and θ(g) − 1 < ε for all g ∈ K.By [2, Proposition 2.5], there exists η ∈ ℓ 2 (G, A ω ∩ ι(A) ′ ) with θ(g) = η, α ω g (η) for all g ∈ G.By slightly perturbing η (and therefore θ), we can assume that η has finite support.This almost gives the QAP for α ω , except for the condition of almost centrality, which follows from a standard reindexing argument.To show this, fix a finite set A) that represents η and satisfies η n ≤ 1 and supp(η n ) ⊆ supp(η) for every n ∈ N. Since we have θ(g) − 1 < ε for all g ∈ K, we can find (b 2,u = 0 for all g ∈ K, and for all n ∈ N and g ∈ K. Now choose a dense sequence (a n ) n∈N in A, and for each x ∈ F find a representative (x n ) n∈N ∈ ℓ ∞ (N, A).Using that the (η n ) n∈N have uniformly finite support (contained in supp(η)) and that each (η n (g)) n∈N , for g ∈ supp(η), represents the element η(g) ∈ A ω ∩ ι(A) ′ , we can find inductively an increasing sequence (k(n)) n∈N of natural numbers satisfying (b.1) n , for all n ∈ N and j = 1, . . ., n; n , for all n ∈ N and g ∈ K. Then the sequence ξ := (η k(n) ) n∈N gives rise to a contraction in ℓ 2 (G, A ω ∩ ι(A) ′ ) of finite support satisfying xξ = ξx for all x ∈ F .Moreover, ≤ ε, for all g ∈ K.This proves (5).
(2) ⇒ (6).Fix a finite set K ⊆ G and ε > 0. Choose an element ξ ∈ C c (G, A) satisfying ξ ≤ 1 and ξ, α g (ξ) − 1 2,u < ε, for g ∈ K.We define a positive type function θ : G → C(T (A)) by for all g ∈ G and all τ ∈ T (A).Using part (1) of Lemma 2.1 at the second step, we get fin .In order to verify that θ is equivariant, it is thus sufficient to check that θ g := (α * * fin ) g •θ•α * * g −1 also verifies equality (2.3) for every g ∈ G.To prove this, notice first that, given τ ∈ T (A) and g ∈ G, by uniqueness of the boundary measure µ τ •αg we have that (2.4) This proves equivariance of θ.Composing θ with the unital, equivariant inclusion of C ∂ e T (A) into B(∂ e T (A)), one obtains a unital, equivariant homomorphism C ∂ e T (A) → Z(A * * fin ).Since the action on ∂ e T (A) is assumed to be amenable, we conclude that the action on A * * fin is von Neumann-amenable (see, for example, [10, Lemma 3.21]).( 8) ⇒ (1).Fix finite sets F ⊆ A and K ⊆ G and ε > 0. We may assume without loss of generality that F consists of contractions.Throughout the proof, we will identify A with its natural (not necessarily isomorphic) copy inside A * * fin .This is possible thanks to Lemma 2.6 and the fact that • 2,u coincides on A and its copy inside A * * fin .We claim that for every δ > 0 and a finite subset T ⊆ T (A), there is η ∈ C c (G, A) satisfying η ≤ 1 and for all τ ∈ T .Set M = A * * fin and γ = α * * fin .Since γ is von Neumann amenable, there exists a net (ξ λ ) λ∈Λ of finitely supported contractions in ℓ 2 (G, Z(M )) such that ξ λ , ξ λ → 1, and ξ λ − γ g (ξ λ ), ξ λ − γ g (ξ λ ) → 0 pointwise on G in the ultraweak topology.Since each trace τ ∈ T extends to a normal state on M , there exists λ ∈ Λ such that, with ξ := ξ λ , we have for all τ ∈ T .By an application of Kaplansky's density theorem for Hilbert modules (see [10, Lemma 4.5]), there exists , for all τ ∈ T .Using the triangle inequality, one gets for all τ ∈ T .Since ξ takes values in the center of M , then η can be chosen so that a∈F aη − ηa 2 2,τ < δ 3 , for all τ ∈ T .This proves the claim.Given η ∈ C c (G, A), with η ≤ 1, set Q(η) := η, η and Note that the map D(η) : T (A) → [0, ∞) given by D(η)(τ ) = τ (D(η)), for all τ ∈ T (A), is continuous and affine.By the claim above, 0 is in the closure of the set with respect to the topology of pointwise convergence.Since the elements of Z are continuous, affine functions, it follows from Lemma 2.7 that 0 is also in the weak closure of Z. Thus, by the Hahn-Banach Theorem, there are η 1 , . . ., η n ∈ C c (G, A) with η j ≤ 1 for all j = 1, . . ., n, and λ 1 , . . ., λ n ≥ 0 with for all τ ∈ T (A).Assuming without loss of generality that G is infinite, and replacing each η j with an appropriate right translate, we can assume that the supports of α g (η i ) and α h (η j ) are disjoint whenever i = j and g, h ∈ K ∪ {1}.Set Using that η i , η j ≤ δ i,j for i, j = 1, . . ., n, one checks that ξ ≤ 1, since and that D(ξ) = n j=1 λ j D(η j ).Hence This proves (1).
As mentioned before, some of the conditions in Theorem 2.5 resemble similar characterizations of amenability for actions on C * -algebras.For example, the equivalence between (1) and ( 5) is the tracial version of the fact that amenability of an action on A is equivalent to amenability of the induced action on the norm-central sequence algebra A ω ∩ A ′ ; see [33,Theorem 4.4], while the equivalence between ( 1) and ( 8) is the tracial version of the fact that amenability of an action on A is equivalent to von Neumann amenability of the induced action on A * * ; see [33,Theorem 3.2] and [1,Lemma 6.5].On the other hand, ( 2), ( 6) and ( 7) do not have analogous statements for amenable actions.
The authors would like to thank Siegfried Echterhoff and Rufus Willett for pointing out to them the following observation.
Remark 2.8.The analogue of ( 1) ⇔ (6) in Theorem 2.5, where one replaces tracial amenability with amenability, and T (A) with the state space S(A), is not true.While amenability of an action α : G → Aut(A) always implies amenability of the induced action on the state space by [33,Proposition 3.5], the converse fails.An easy example can be constructed from any amenable action G X of a nonamenable group G on a compact space X by considering the associated inner action β on C(X) ⋊ G.As an inner action, β is not amenable as it induces the trivial action on Z((C(X) ⋊ G) * * ).However, β induces an amenable action on S(C(X) ⋊ G).Indeed, this follows from Theorem 2.4 and the equivariance of the unital, completely positive map ι : C(X) → C(S(C(X) ⋊ G)) given by ι(f )(ϕ) := ϕ(f ), for all f ∈ C(X) and ϕ ∈ S(C(X) ⋊ G).For the same reason, there is no analogue of (1) ⇔ (2) for amenable actions.
The key tool that allows to obtain these equivalences in the tracial setting is [32,Theorem 3].
Given an amenable action α : G → Aut(A) of a discrete group on a unital C *algebra, it is not hard to show that there exists an invariant state on A if and only if G is amenable.The next lemma, which will be needed in Section 3, shows that the tracial counterpart of this statement also holds.Lemma 2.9.Let α : G → Aut(A) be a tracially amenable action of a discrete group on a unital C * -algebra with T (A) = ∅.Then G is amenable if and only if A has an invariant trace.
Proof.If G is amenable, then it is well-known that the affine action G T (A) has a fixed point.Conversely, assume that τ ∈ T (A) is G-invariant.Then the Dirac measure on T (A) associated to τ is also G-invariant.Since G T (A) is amenable by ( 1)⇒( 6) of Theorem 2.5 (the proof of this implication applies verbatim also if A is not separable), it follows from [15, Lemma 2.2] that G is amenable.

2.1.
Examples.In this subsection, we use our characterizations of tracial amenability from Theorem 2.5 (particularly ( 7)), together with results from the classification programme for simple, nuclear C * -algebras, to construct a wide class of examples of tracially amenable actions on unital, simple, stably finite C * -algebras.Since we lift automorphisms from the Elliott invariant to the C * -algebra in question, we work with free groups in order to guarantee that we get a group action.Recall that the Elliott invariant of a unital C * -algebra A is given by see [20,Definition 2.4] for its definition.
In the above construction there is a significant amount of freedom in the choice of the Elliott invariant of the C * -algebra A. The following variant of the construction above was suggested to the authors by Mikael Rørdam, and it has the advantage of readily producing actions on unital, simple AF-algebras.
Example 2.11.Let θ : F n X be an amenable action on the Cantor space X.By [36, Proposition 1.4.5],there is a unital AF-algebra A with where Q d denotes the rational numbers with the discrete topology, and C(X, Q d ) ++ denotes the strictly positive continuous functions X → Q d .Since (K 0 (A), K 0 (A) + ) is simple in the sense of [36,Definition 1.5.1], it follows from [36, Corollary 1.5.4] that A is simple.Use Elliott's classification of AF-algebras (in the form stated in [36, Theorem 1.3.3(ii)]) to lift the automorphisms of (K 0 (A), K 0 (A) + , [1]) induced by the homeomorphisms θ g1 , . . ., θ gn to automorphisms α g1 , . . ., α gn of A. We again obtain an action α : F n → Aut(A).Since there is a natural continuous affine homeomorphism T (A) ∼ = Prob(X) by [36, Proposition 1.5.5], it follows that the action that α induces on T (A) is conjugate to θ. Hence α is tracially amenable by Theorem 2.5.

Purely infinite crossed products
In this section, we investigate the internal structure of reduced crossed products by tracially amenable actions.For a large class of nonamenable groups, we show that outer, tracially amenable actions on simple, unital, Z-stable C * -algebras produce simple, purely infinite reduced crossed products.Our proof proceeds in two steps.The first one is Proposition 3.9, where we show in great generality that dynamical strict comparison is sufficient to conclude pure infiniteness of the reduced crossed product.The second step is Theorem 3.14, the main result of this work, where we show that tracial amenability implies dynamical strict comparison for a vast class of actions of nonamenable groups on simple C * -algebras.
3.1.From dynamical comparison to pure infiniteness of crossed products.We begin by recalling some preliminaries about Cuntz comparison, and refer the reader to [3,46,19] for modern introductions to the subject.
Let A be a C * -algebra.For positive elements a, b ∈ A + , we say that a is Cuntz subequivalent to b, written a b, if there exists a sequence (r n ) n∈N in A such that lim n→∞ r n br * n − a = 0.For a unital C * -algebra A, we denote by QT (A) the set of (everywhere defined) normalized (2-)quasitraces on A ([46, Definition 6.7]).We will be mainly interested in C * -algebras where QT (A) = T (A), which is the case, for instance, whenever A is exact ( [21]).
Given τ ∈ QT (A), there exists a unique extension of τ to a lower semicontinuous (unbounded) quasitrace on A ⊗ K.With a slight abuse of notation, we denote such extension again by τ .For a ∈ (A ⊗ K) + and τ ∈ QT (A), set The following lemma, which was pointed out to us by Sam Evington, will be needed in the proof of Theorem 3.14.For ε > 0, we denote by Lemma 3.1.Let A be a unital C * -algebra, and let γ, ε > 0.
(1) For every f ∈ C([0, 1]) there exists δ > 0 such that for all contractions a, b ∈ A + and all τ ∈ T (A) satisfying a − b 2,τ < δ, we have (2) There is δ > 0 such that for all contractions a, b ∈ A + and all τ ∈ T (A) satisfying a − b 2,τ < δ, we have Proof.(1).Suppose that there is n ∈ N such that f (t) = t n for all t ∈ [0, 1].Then Thus, in this case we may take δ = γ n .The case where f is a polynomial can be deduced from this, and the general case follows using a density argument.
(2).Let g ε ∈ C([0, 1]) be the function defined as g ε (t) := min{ε −1 t, 1}, for all t ∈ [0, 1].Using part (1), find δ > 0 such that whenever a, b ∈ A + are contractions satisfying a − b 2,τ < δ, then g ε (a) − g ε (b) 2,τ < γ.By part (1) of Lemma 2.1, we have Denote by µ the measure on the spectrum of a induced by the trace τ | C * (1,a) , so that Using that g ε = 1 on (ε, 1] at the second step, we get as desired. For a compact convex set K, we denote by LAff(K) + (respectively, LAff(K) ++ ) the set of lower semicontinuous positive (respectively, strictly positive) affine functions on K. Given a simple, unital C * -algebra A and a ∈ (A ⊗ K) + \ {0}, it is always true that d τ (a) > 0, for every τ ∈ QT (A).As rk(a) is a lower semicontinuous map on a compact set, it attains its minimum.It follows that rk(a) ∈ LAff(QT (A)) ++ , for every a ∈ (A ⊗ K) + \ {0}.We say that all ranks are realized in A if the rank map rk: One of the assumptions of Theorem 3.14 is that all ranks are realized for the coefficient C * -algebra.This is in fact a very mild assumption, which is automatic in many cases of interest.This is for example the case for nonelementary, simple, separable, unital C * -algebras of stable rank one by [47,Theorem 8.11].Moreover, it is an open question of Nate Brown whether all ranks are realized in every nonelementary, unital, simple, separable, stably finite C * -algebra (see [47,Question 1.1]).
We will need the notion of strict comparison.Although this is most commonly defined using tracial states, we choose to do it using normalized quasitraces since this version is more compatible with the Cuntz semigroup outside of the exact setting.Remark 3.4.When A is a simple, unital C * -algebra, and QT (A) = ∅, then strict comparison is equivalent to the fact that 1 a for all nonzero a ∈ (A ⊗ K) + .This, in turn, is equivalent to pure infiniteness of A (see [24,Definition 4.1] and [12]).
The following definition originates from an ongoing project by Bosa, Perera, Wu and Zacharias, and it provides a noncommutative generalization of the notion of dynamical subequivalence from [23,Definition 3.2].Definition 3.5.Let α : G → Aut(A) be an action of a countable, discrete group G on a C * -algebra A, and let a, b ∈ A + .
(1) Write a 0 b if for any ε > 0 there are δ > 0, n ∈ N, g 1 , . . ., g n ∈ G, and positive elements x 1 , . . ., x n ∈ M ∞ (A), such that The need for (2) stems for the fact that, unlike for , it is not clear whether the relation 0 is transitive.The notion of dynamical strict comparison below is due to Bosa, Perera, Wu and Zacharias, as part of their ongoing work on almost elementariness extending the ideas and techniques developed in [23] to the noncommutative setting.In the definition above, we deliberately allow for the possibility that there are no invariant quasitraces, in which case dynamical strict comparison is equivalent to the statement that a G b for all nonzero a, b ∈ (A ⊗ K) + .
Note that when α is trivial, dynamical strict comparison for α is equivalent to strict comparison for A. The following is a generalization of [28,Theorem 1.1] to the noncommutative setting (see also [15,Theorem 2.8]).Recall that an action α : G → Aut(A) is said to be outer if α g is not inner for all g ∈ G \ {1}.Proposition 3.9.Let α : G → Aut(A) be an outer action of a countable, discrete group G on a unital, simple C * -algebra A. Assume that α has dynamical strict comparison and no invariant quasitraces.Then A⋊ r G is simple and purely infinite.
Proof.Simplicity follows from [25,Theorem 3.1].Let a ∈ A ⋊ r G be a nonzero positive element.We will show that 1 a in A ⋊ r G, which implies that A ⋊ r G is purely infinite by Remark 3.4.By [38,Lemma 3.2] (see also [16,Lemma 4.2]) there exists a nonzero element b ∈ A + such that b a.In the absence of invariant quasitraces, dynamical strict comparison gives 1 G b.By Remark 3.7, we have that 1 b a in A ⋊ r G, as desired.
3.2.Groups with weak paradoxical towers.Next, we introduce the notion of weak paradoxical towers, based on ideas from [15].This is a technical assumption which allows us to link tracial amenability to strict comparison.The class of groups having weak paradoxical towers is quite large, and it for example includes all groups containing the free group F 2 ; see Corollary 3.13.Definition 3.10.Let n ∈ N. A group G is said to have weak n-paradoxical towers if for every m ∈ N, there exist a finite subset D ⊆ G with |D| ≥ m, subsets K 1 , . . .K n ⊆ G, and group elements g 1 , . . ., g n ∈ G such that (1) For each j = 1, . . ., n, the sets {dK j } d∈D are pairwise disjoint; (2) n j=1 g j K j = G.A group is said to have weak paradoxical towers if it has weak n-paradoxical towers for some n ∈ N.
We make the following observation: Lemma 3.11.Let G be a group with weak paradoxical towers.Then G is not amenable.
Using elementary arguments, one can show that all nonabelian free groups have weak 2-paradoxical towers; see [15,Proposition 3.2] for a stronger result.More generally, groups that have paradoxical towers in the sense of [15, Definition 3.1] have weak paradoxical towers in our sense, and in particular all the groups considered in [15,Section 4] have weak paradoxical towers.The class of groups we consider here is really larger than the one considered in [15]: for example F 2 × Z does not have paradoxical towers by [15,Remark 3.7], but it has weak paradoxical towers by Corollary 3.13 below.The key difference is that weak paradoxical towers can be induced from subgroups: For an element x ∈ A and ℓ ∈ N, we will denote by x ⊕ℓ ∈ M ℓ (A) the ℓ-fold direct sum of x with itself.We will prove that 1 G a in two steps.Claim 3.14.1.We have 1 0 (b − ε) ⊕3n + .For j = 1, . . ., n and h ∈ D ∪{g 1 , . . ., g n }, we denote by 1 hKj ∈ L A (ℓ 2 (G, A)) the projection onto the elements supported on hK j ⊆ G. Using the Cauchy-Schwarz inequality (see part (2) of Lemma 2.1) for the third step, we obtain Combining (3.5) and (3.6), we get for all h ∈ D ∪ {g 1 , . . ., g n } and j = 1, . . ., n. Fix τ ∈ T (A).For j ∈ {1, . . ., n}, denote by µ j the measure on the spectrum of α gj (c j ) corresponding to τ | C * (1,αg j (cj)) .We have Furthermore, we have We conclude that there is j ∈ {1, . . ., n} for which  The following is the desired result for crossed products.Corollary 3.15.Let G be a countable, discrete group with weak paradoxical towers, let A be a unital, simple, separable C * -algebra with QT (A) = T (A) = ∅, with strict comparison and for which all ranks are realized, and let α : G → Aut(A) be a tracially amenable, outer action.Then A ⋊ r G is simple and purely infinite.
Proof.The statement follows from Theorem 3.14 and Proposition 3.9.
Let A be a simple, separable, exact, unital, stably finite C * -algebra.If A has strict comparison and stable rank one, then all ranks are realized on it by [47,Theorem 8.11], and so A satisfies the assumptions of Corollary 3.15.In particular, this is the case if A is Z-stable, by [37, Theorem 4.5 and Theorem 6.7].Instead of stable rank one, one can equivalently require A to be almost divisible.Indeed, simple, separable, unital, stably finite C * -algebras with strict comparison and almost divisibility have stable rank one by work in progress of Geffen and Winter (see also [27], where the assumption of almost divisibility is replaced by tracial approximate oscillation zero).
We close the paper with an application of our results to classifiable C * -algebras.In the following corollary, we require amenability of the action, as tracial amenability is not sufficient to grant nuclearity of the crossed product.Corollary 3.16.Let G be a countable, discrete group with weak paradoxical towers, and let A be a simple, separable, unital, nuclear, stably finite, Z-stable C * -algebra.Suppose that α : G → Aut(A) is an amenable, outer action1 .Then A ⋊ r G is a unital Kirchberg algebra.If A satisfies the UCT and G is torsion-free and satisfies the Haagerup property (for example, G = F n ; see [8, Definition 12.2.1]),then A⋊ r G also satisfies the UCT, and is therefore completely determined by its K-theory.
Proof.By Corollary 3.15 and the comments after it, A ⋊ r G is simple, separable, unital and purely infinite.By [2, Theorem 4.5], it is also nuclear.The fact that it satisfies the UCT follows from [22] and [31,Corollary 9.4] (see [39,Corollary 7.2] for the case G = F n ).

Definition 3 . 3 .
A unital, simple C * -algebra A is said to have strict comparison if we have a b whenever a, b ∈ (A ⊗ K) + \ {0} satisfy rk(a) < rk(b).

Remark 3 . 6 .
Assume that there are n ∈ N, elements g 1 , . . ., g n ∈ G, and positive elements x 1 , . . ., x n ∈ M ∞ (A) such that a ⊕ n j=1 α gj (x j ) and ⊕ n j=1 x j b.Then a 0 b.This follows from double application of [35, Proposition 2.4].Remark 3.7.One can check that a G b entails a b in A ⋊ r G, by putting together [35, Proposition 2.4], [46, Proposition 2.18], and the fact that the action on A is unitarily implemented inside A ⋊ r G.

Definition 3 . 8 .
An action α : G → Aut(A) of a countable, discrete group G on a unital C * -algebra A is said to have dynamical strict comparison if we have a G b for all a, b ∈ (A ⊗ K) + \ {0} satisfying d τ (a) < d τ (b) for all invariant quasitraces τ ∈ QT (A).
fact that Problem D remains open highlights another advantage of focusing on tracially amenable actions throughout: namely, we can construct several examples of actions satisfying the assumptions of Theorem C: Example E (Example 2.10; Example 2.11).For every n ≥ 2, there exist outer, tracially amenable actions of the free group F n on stably finite, classifiable C *algebras, including actions on simple, unital AF-algebras.