# AsymptoticC∗-Algebras from ℤk-Actions on Higher Rank Graphs

Authors: Inhyeop Yi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

For a dynamical system arising fromℤk-action on a higher rank graph with finite vertex set, we show that the semidirect product of the asymptotic equivalence relation groupoid is essentially principal if and only if the k-graph satisfies the aperiodic condition. Then we show that the corresponding asymptotic Ruelle algebra is simple if the graph is primitive with the aperiodic condition.


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## Body

## 1. Introduction

In [1], Kumjian and Pask constructed ℤk-action dynamical systems on higher rank graphs, which are higher dimensional analog of subshifts of finite type in symbolic dynamics. Naturally their dynamical systems exhibit many of the same dynamical properties in not only subshifts of finite type but also Smale spaces studied by  Putnam [2], Putnam and Spielberg [3], and Ruelle [4, 5]. Putnam associated several groupoid C*-algebras to Smale spaces using asymptotic, stable, and unstable equivalence relations generated by hyperbolic structures of Smale spaces [2]. Ruelle algebras from stable and unstable equivalence relations may be considered as higher dimensional analogs of Cuntz-Krieger algebras [2, 3, 6]. As in the case of irreducible Smale spaces, Kumjian and Pask built C*-algebras from the stable and unstable equivalence relations on their dynamical systems and the related Ruelle algebras [1]. Then they showed that stable Ruelle algebra Rs of a higher rank graph is strongly Morita equivalent to the graph C*-algebra and that Rs is simple, nuclear, and purely infinite if the graph satisfies the aperiodic condition. Although Kumjian and Pask showed that the groupoid algebra from asymptotic equivalence relation is a simple AF algebra if the graph is primitive, they left Ruelle algebra Ra from asymptotic equivalence relation behind, and it remains unclear whether or not Ra is simple. Here, we show that Ra is simple if the corresponding higher rank graph is primitive with aperiodic condition (Theorem 16). For this purpose, we first review the definitions and necessary properties of k-graphs and ℤk-action in Section 2. Then we follow Hou and Chen [7] to study structures of groupoids from asymptotic equivalence relation, and we obtain sufficient conditions for Ra to be simple in Section 3.


## 2. Dynamical Systems on Higher Rank Graphs

In this section, we briefly review the definitions and notations fork-graphs and ℤk-action on those established by Kumjian and Pask in [1].Definition 1 (see [1, 8]).
Ak-graph is a pair (Λ,d), where
(1)Λ=(Obj(Λ),Hom(Λ),r,s),
is a countable small category and d:Λ→ℕk is a morphism, called the degree map, satisfying the factorization property. For every λ∈Λ and m,n∈ℕk with d(λ)=m+n, there exist unique elements μ,ν∈Λ such that
(2)d(μ)=m,d(ν)=n,λ=μν.
Every λ∈Λ is called a path. For every nonzero n∈ℕk, by the factorization property we may define Λn=d-1(n) and identify Λ0 with Obj(Λ). Let r,s:Λ→Λ0 be the range and source maps. We abbreviate (Λ,d) to Λ when there is no confusion.Definition 2 (see [1, 8]).
Ak-graph Λ is called primitive if there is a nonzero p∈ℕk so that for every u,v∈Λ0 there is a λ∈Λp such that u=r(λ) and v=s(λ).Standing Assumption (see [1]). For every n∈ℕk, the restrictions of r and s on Λn are surjective and finite to one.


### 2.1.ℤk-Actions on k-Graphs (See [1, 2])

Suppose that(Δ,d) is a k-graph defined by
(3)Δ={(m,n)∣m,n∈ℤk,m≤n},
with the structure maps
(4)(l,m)·(m,n)=(l,n),r(m,n)=m,s(m,n)=n,d(m,n)=m-n.
Let Λ be a k-graph, and let the corresponding two-sided infinite path space be the set
(5)ΛΔ={x:Δ⟶Λ∣xisak-graphhomomorphism}.
Then, ΛΔ is a zero-dimensional space consisting of “two-sided” paths on Λ.A topology is endowed onΛΔ where its basis is given by
(6)Z(λ,n)={x∈ΛΔx∣(n,n+d(λ))=λ},
with n∈ℤk and λ∈Λ. It is not difficult to check that ΛΔ is compact (locally compact, resp.) if Λ0 is finite (infinite, resp.) so that ΛΔ is a metrizable space. A metric ρ on ΛΔ is defined as follows: for e=(1,…,1)∈ℤk and j∈ℕ, let θj∈Δ be the element (-je,je). Given x,y∈ΛΔ, set
(7)h(x,y)={1x(0)≠y(0)1+sup{j∣x(θj)=y(θj)}otherwise.
Then, for a fixed number r∈(0,1), a metric ρ is defined by the formula
(8)ρ(x,y)=rh(x,y)forx,y∈ΛΔ.
The metric topology is the same as the one generated by cylinder sets. And we remark that the cylinder set Z(λ,n) is always nonempty because of our standing assumption.Letσ be the action of ℤk on ΛΔ by the homeomorphism σp:ΛΔ→ΛΔ, p∈ℤk, defined by
(9)(σpx)(m,n)=x(m+p,n+p).Theorem 3 (see [1]).
Theℤk-action n↦σn on ΛΔ is expansive in the sense that there is an ϵ>0 such that, for x,y∈ΛΔ, ρ(σn(x),σn(y))<ϵ for every n∈ℤk implies x=y. Moreover, if Λ is primitive, then σ is topologically mixing.


### 2.2. Groupoids on(ΛΔ,σ,ρ)

We review Kumjian and Pask’s construction of the asymptotic groupoidGa and its corresponding C*-algebra C*(Ga) and Ruelle algebra Ra on (ΛΔ,σ,ρ), where ρ is the metric defined on ΛΔ as in (8).For a second countable locally compact groupoidG, we denote the source map and the range map of G by s and r, respectively, and G0 is the unit space of G. We refer to [9] for more detailed results on groupoids and groupoid C*-algebras.As in the case of subshifts of finite types, the asymptotic, stable, and unstable equivalence relations onΛΔ are defined as follows [1]: given x,y∈ΛΔ, we define
(10)x~ayiflim|j|→∞ρ(σje(x),σje(y))=0,x~syiflimj→∞ρ(σje(x),σje(y))=0,x~uyiflimj→-∞ρ(σje(x),σje(y))=0.
Then, we denote the asymptotic, stable, and unstable equivalence classes of x by Va(x), Vs(x), and Vu(x), respectively.Remark 4 (see [1, Remark 3.6]).
Forx,y∈ΛΔ, x~ay if and only if there is an m∈ℕk such that for every n≥m we have
(11)x(m,n)=y(m,n),x(-n,-m)=y(-n,-m).The asymptotic equivalence relation gives rise to a groupoid(12)Ga={(x,y)∈ΛΔ×ΛΔ:x~ay},
whose unit space is identified with ΛΔ, and structure maps are the natural ones. To give a topology on Ga, we construct conjugate homeomorphisms as in Ruelle [5]: for (x,y)∈Ga, let m∈ℕk be as in Remark 4 and define finite paths λ=x(-m,m) and ν=y(-m,m). We recall that the cylinder sets
(13)Z(λ,-m)={z∈ΛΔ:z(-m,m)=x(-m,m)},Z(ν,-m)={w∈ΛΔ:w(-m,m)=y(-m,m)},
are compact open subsets of ΛΔ. Since x and y satisfy the equalities in Remark 4, when we define φ:Z(λ,-m)→Z(ν,-m) by
(14)φ(x)=y,φ(z)(m,n)=z(m,n),φ(z)(-n,-m)=z(-n,-m),
for all z∈Z(λ,-m) and n≥m, φ is a bijective map by the factorization property and z~aφ(z) by the definition of φ and Remark 4. It is easy to check that φ is continuous so that it is a homeomorphism.A basis for the topology ofGa is given by the compact open sets
(15){(φ(z),z):z∈Z(λ,-m),m∈ℕk}.
Then, Ga is a second countable, locally compact, Hausdorff, r-discrete, and principal groupoid under this topology, and the counting measure is a Haar system [2, 7, 8]. The path space ΛΔ is identified with the unit space Ga0 by the embedding map x↦(x,x).To make the asymptotic Ruelle algebra, we need another groupoid: we imitate [2] to form the semidirect product Ga⋊ℤk given by
(16)Ga⋊ℤk={(u,n,v):u,v∈ΛΔ,n∈ℤk,(σn(u),v)∈Ga},
with the structure maps
(17)s(u,n,v)=u,r(u,n,v)=v,(u,n,v)(w,m,z)=(u,n+m,z)ifv=w,(u,n,v)-1=(v,-n,u).It is not difficult to check that the map defined by((x,y),n)↦(x,n,σn(y)) gives a groupoid isomorphism between Ga×ℤk and Ga⋊ℤk. We transfer the product topology on Ga×ℤk via this isomorphism so that Ga⋊ℤk is a locally compact, Hausdorff, and r-discrete groupoid, and the counting measure is a Haar system. The unit space is identified with ΛΔ via the map x↦(x,0,x).Definition 5 (see [1, 2]).
Suppose thatΛ is a k-graph and that Ga and Ga⋊ℤk are groupoids on Λ defined previously. The asymptoticC*-algebra of Λ is the reduced groupoid C*-algebra of Ga, denoted by A(Λ). And the asymptotic Ruelle algebra of Λ is the reduced groupoid C*-algebra of Ga⋊ℤk, denoted by Ra(Λ).


## 2.1.ℤk-Actions on k-Graphs (See [1, 2])

Suppose that(Δ,d) is a k-graph defined by
(3)Δ={(m,n)∣m,n∈ℤk,m≤n},
with the structure maps
(4)(l,m)·(m,n)=(l,n),r(m,n)=m,s(m,n)=n,d(m,n)=m-n.
Let Λ be a k-graph, and let the corresponding two-sided infinite path space be the set
(5)ΛΔ={x:Δ⟶Λ∣xisak-graphhomomorphism}.
Then, ΛΔ is a zero-dimensional space consisting of “two-sided” paths on Λ.A topology is endowed onΛΔ where its basis is given by
(6)Z(λ,n)={x∈ΛΔx∣(n,n+d(λ))=λ},
with n∈ℤk and λ∈Λ. It is not difficult to check that ΛΔ is compact (locally compact, resp.) if Λ0 is finite (infinite, resp.) so that ΛΔ is a metrizable space. A metric ρ on ΛΔ is defined as follows: for e=(1,…,1)∈ℤk and j∈ℕ, let θj∈Δ be the element (-je,je). Given x,y∈ΛΔ, set
(7)h(x,y)={1x(0)≠y(0)1+sup{j∣x(θj)=y(θj)}otherwise.
Then, for a fixed number r∈(0,1), a metric ρ is defined by the formula
(8)ρ(x,y)=rh(x,y)forx,y∈ΛΔ.
The metric topology is the same as the one generated by cylinder sets. And we remark that the cylinder set Z(λ,n) is always nonempty because of our standing assumption.Letσ be the action of ℤk on ΛΔ by the homeomorphism σp:ΛΔ→ΛΔ, p∈ℤk, defined by
(9)(σpx)(m,n)=x(m+p,n+p).Theorem 3 (see [1]).
Theℤk-action n↦σn on ΛΔ is expansive in the sense that there is an ϵ>0 such that, for x,y∈ΛΔ, ρ(σn(x),σn(y))<ϵ for every n∈ℤk implies x=y. Moreover, if Λ is primitive, then σ is topologically mixing.


## 2.2. Groupoids on(ΛΔ,σ,ρ)

We review Kumjian and Pask’s construction of the asymptotic groupoidGa and its corresponding C*-algebra C*(Ga) and Ruelle algebra Ra on (ΛΔ,σ,ρ), where ρ is the metric defined on ΛΔ as in (8).For a second countable locally compact groupoidG, we denote the source map and the range map of G by s and r, respectively, and G0 is the unit space of G. We refer to [9] for more detailed results on groupoids and groupoid C*-algebras.As in the case of subshifts of finite types, the asymptotic, stable, and unstable equivalence relations onΛΔ are defined as follows [1]: given x,y∈ΛΔ, we define
(10)x~ayiflim|j|→∞ρ(σje(x),σje(y))=0,x~syiflimj→∞ρ(σje(x),σje(y))=0,x~uyiflimj→-∞ρ(σje(x),σje(y))=0.
Then, we denote the asymptotic, stable, and unstable equivalence classes of x by Va(x), Vs(x), and Vu(x), respectively.Remark 4 (see [1, Remark 3.6]).
Forx,y∈ΛΔ, x~ay if and only if there is an m∈ℕk such that for every n≥m we have
(11)x(m,n)=y(m,n),x(-n,-m)=y(-n,-m).The asymptotic equivalence relation gives rise to a groupoid(12)Ga={(x,y)∈ΛΔ×ΛΔ:x~ay},
whose unit space is identified with ΛΔ, and structure maps are the natural ones. To give a topology on Ga, we construct conjugate homeomorphisms as in Ruelle [5]: for (x,y)∈Ga, let m∈ℕk be as in Remark 4 and define finite paths λ=x(-m,m) and ν=y(-m,m). We recall that the cylinder sets
(13)Z(λ,-m)={z∈ΛΔ:z(-m,m)=x(-m,m)},Z(ν,-m)={w∈ΛΔ:w(-m,m)=y(-m,m)},
are compact open subsets of ΛΔ. Since x and y satisfy the equalities in Remark 4, when we define φ:Z(λ,-m)→Z(ν,-m) by
(14)φ(x)=y,φ(z)(m,n)=z(m,n),φ(z)(-n,-m)=z(-n,-m),
for all z∈Z(λ,-m) and n≥m, φ is a bijective map by the factorization property and z~aφ(z) by the definition of φ and Remark 4. It is easy to check that φ is continuous so that it is a homeomorphism.A basis for the topology ofGa is given by the compact open sets
(15){(φ(z),z):z∈Z(λ,-m),m∈ℕk}.
Then, Ga is a second countable, locally compact, Hausdorff, r-discrete, and principal groupoid under this topology, and the counting measure is a Haar system [2, 7, 8]. The path space ΛΔ is identified with the unit space Ga0 by the embedding map x↦(x,x).To make the asymptotic Ruelle algebra, we need another groupoid: we imitate [2] to form the semidirect product Ga⋊ℤk given by
(16)Ga⋊ℤk={(u,n,v):u,v∈ΛΔ,n∈ℤk,(σn(u),v)∈Ga},
with the structure maps
(17)s(u,n,v)=u,r(u,n,v)=v,(u,n,v)(w,m,z)=(u,n+m,z)ifv=w,(u,n,v)-1=(v,-n,u).It is not difficult to check that the map defined by((x,y),n)↦(x,n,σn(y)) gives a groupoid isomorphism between Ga×ℤk and Ga⋊ℤk. We transfer the product topology on Ga×ℤk via this isomorphism so that Ga⋊ℤk is a locally compact, Hausdorff, and r-discrete groupoid, and the counting measure is a Haar system. The unit space is identified with ΛΔ via the map x↦(x,0,x).Definition 5 (see [1, 2]).
Suppose thatΛ is a k-graph and that Ga and Ga⋊ℤk are groupoids on Λ defined previously. The asymptoticC*-algebra of Λ is the reduced groupoid C*-algebra of Ga, denoted by A(Λ). And the asymptotic Ruelle algebra of Λ is the reduced groupoid C*-algebra of Ga⋊ℤk, denoted by Ra(Λ).


## 3. Main Results

In this section, we first show that for a primitivek-graph, the groupoids Ga and Ga⋊ℤk are minimal. Then, we show that if a k-graph Λ is such that Λ0 is finite, then the aperiodic condition is equivalent to essential principality of Ga⋊ℤk. So the asymptotic C*-algebra A(Λ) is simple if Λ is primitive, and the asymptotic Ruelle algebra Ra(Λ) is simple if in addition Λ satisfies the aperiodic condition.Lemma 6.
Suppose thatΛ is a primitive k-graph. For every x∈ΛΔ, Va(x)={y∈ΛΔ:x~ay} is dense in ΛΔ.Proof.
We need to show that, for any base elementZ(λ,n), there is a y∈Va(x)∩Z(λ,n). Since Λ is primitive, there is an N∈ℕk such that, for every p≥N and every u,v∈Λ0, there is a ν∈Λp with s(ν)=v and r(ν)=u.
First, we selectm∈ℕk satisfying N≤m-n-d(λ) and N≤m+n. Then, the primitive condition implies that there are α∈Λm+n and β∈Λm-n-d(λ) such that
(18)r(α)=x(-m,-m),s(α)=r(λ),s(λ)=r(β),s(β)=x(m,m).
So, by the factorization property (cf. [8, Remarks 2.2]), there is a unique y∈ΛΔ such that, for every l≥m,
(19)y(-l,-m)=x(-l,-m),y(-m,m)=αλβ,y(m,l)=x(m,l).
It is trivial that y∈Z(λ,n) from the construction, and y∈Va(x) by Remark 4. Therefore, the asymptotic equivalence class of any element in ΛΔ is dense in ΛΔ.Recall that a topological groupoidG with an open range map is called minimal if the only open invariant subsets of G0 are the empty set ∅ and G0 itself. A subset E of G0 is called invariant if r∘s-1(E)=E. We refer to [9] for details.Proposition 7.
Suppose thatΛ is a primitive k-graph. Then, Ga and Ga⋊ℤk are minimal groupoids.Proof.
We recall that the unit spaces ofGa and Ga⋊ℤk are identified with ΛΔ via the maps x↦(x,x) and x↦(x,0,x), respectively. When E is an invariant subset of Ga0, we observe
(20)r∘s-1(E)={y∈ΛΔ:y~axforsomex∈E}=⋃x∈EVa(x)=E.
We assume that there is a nontrivial open invariant subsetE of Ga0. Then, for z∈ΛΔ∖E, we have Va(z)∩E=∅ because the collection of asymptotic equivalence classes is a partition of ΛΔ. But this is a contradiction to Lemma 6 as E is a nontrivial open set. So the only open invariant subsets of Ga0 are the empty set and Ga0, and Ga is a minimal groupoid.
ForGa⋊ℤk and an open invariant subset H of its unit space, we note that
(21)r∘s-1(H)=⋃x∈H,n∈ℤkVa(σn(x))=⋃x∈HVa(x)=H.
Then, the result follows as in the case of Ga.For an irreducible Smale space(X,φ), Putnam and Spielberg [3] showed that groupoids defined by stable and unstable equivalence relations are essentially principal, and Hou and Chen [7] showed that Ga⋊ℤ is essentially principal. Recall that a Smale space is said to be irreducible if it is nonwandering and has a dense orbit. The basic ideas behind their proofs are the facts that Va(x) is at most countable for every x and the set of periodic points of period ℓ is finite for every nonzero ℓ∈ℤ (see [3, Section 5], and [4, 7.16] for details). We will follow [7, Section 2], and [3, Section 5], to show that Ga⋊ℤk is an essential principal.Because of the factorization property and thatΛm is a countable set for every m∈ℕk, it is not difficult to check that Va(x) is at most countable in k-graphs.Lemma 8.
LetΛ be a k-graph and x∈ΛΔ. Then, Va(x) is a countable set.Proof.
Lety∈ΛΔ be asymptotically equivalent to x. Then, there is an m∈ℕk such that x(m,n)=y(m,n) and x(-n,-m)=y(-n,-m) for every n≥m by Remark 4, and the factorization property implies that the set of finite paths
(22){x(-n,-m)y(-m,m)x(m,n):n≥m},
determines a unique element in ΛΔ. It is easy to check that the uniquely determined element is y as x(-n,-m)y(-m,m)x(m,n)=y(-n,n) by the factorization property. So the map defined by y↦y(-m,m) is an injection from Va(x) to ∪m∈ℕkΛ2m. Since Λm is a countable set, ∪m∈ℕkΛ2m is a countable set. Hence, Va(x) is countable.In irreducible Smale spaces, the set of periodic points of periodℓ∈ℤ, say Perℓ, is finite for every nonzero integer ℓ as covered in [7, Lemma 2.1] and [3, Lemma 5.2]. But Perl in (ΛΔ,σ) may not be a finite set for some nonzero l∈ℤk. For example, the 2-graph discussed in [8, Definition 1.9 and Example 6.1] has only one vertex and is irreducible, but every path has period (1,-1)∈ℤ2. So we need to add some extra conditions on our k-graphs to obtain that Perl is finite for every nonzero l∈ℤk.Definition 9 (see [8]).
LetΛ be a k-graph with its two-sided path space ΛΔ. A path x∈ΛΔ is called aperiodic if σlx(m,n)≠x(m,n) for every l∈ℤk∖{0} and every (m,n)∈Δ. We say that Λ satisfies the aperiodic condition if, for every v∈Λ0, there is an aperiodic path x∈ΛΔ such that x(0,0)=v.Observe thatΛ satisfies the aperiodic condition if and only if the set of aperiodic paths is dense in ΛΔ [8, Proposition 4.5]. So, for every l∈ℤk, the ℤ-action on ΛΔ given by j↦σjl is not the identity map.The following sublemma is definitely well known to experts, but we could not find any reference.Sublemma 10. Suppose that Λ is a k-graph with a finite Λ0. If Λ satisfies the aperiodic condition, then every isolated point in ΛΔ is aperiodic, and the derived set of ΛΔ is invariant under σl for every l∈ℤk.Proof.
Letx∈ΛΔ be an isolated point. Then, there is an aperiodic point y such that x(0,0)=y(0,0), that is, y∈Z(x(0,0),0). So we have x=y because x is an isolated point.
Letz∈ΛΔ be a limit point of ΛΔ and {zj} a sequence in ΛΔ such that z=limzj. Then, continuity of σl implies that σl(z)=limσl(zj) so that σl(z) is also a limit point of ΛΔ.Since our interest is periodic points which are included in the derived set ofΛΔ and the fact that the derived set is also compact, when there are isolated points in ΛΔ, we can consider ℤk-action of σ on the derived set instead of on ΛΔ. Thus, without loss of generality, we can assume that ΛΔ has no isolated point.We recall thatσ:ΛΔ→ΛΔ is expansive with the expansive constant ϵ>0 (Theorem 3).Lemma 11.
Suppose thatΛ is a k-graph with a finite Λ0. If Λ satisfies the aperiodic condition, then the set of periodic points of period l is finite for every l∈ℤk∖{0}.Proof.
For every nonzerol∈ℤk, we consider an induced ℤ-action ϕ on ΛΔ given by j↦ϕj=σjl. Then, ϕ is also an expansive homeomorphism on ΛΔ with the same expansive constant as that of σ, and Perl={x∈ΛΔ:σl(x)=x} is equal to Fix(ϕ)={x∈ΛΔ:ϕ(x)=x}.
Letx and y be two different points in Fix(ϕ). Then, we have ρ(x,y)≥ϵ because ρ(x,y)=ρ(ϕj(x),ϕj(y))<ϵ for every j∈ℤ implies that x=y by expansiveness. So the distance between any two fixed points of ϕ is at least ϵ, and Fix(ϕ) is a finite set as ΛΔ is a compact set without isolated points.Lemma 12.
LetΛ be a k-graph with a finite Λ0. If x∈ΛΔ is such that σl(x)∈Va(x) for some l∈ℤk∖{0}, then (1)
the sequence{σjl(x):j∈ℕ} converges to an element, say x1, in Perl as j→∞ and x∈Vs(x1),(2)
the sequence{σ-jl(x):j∈ℕ} converges to an element, say x2, in Perl as j→∞ and x∈Vu(x2).Proof.
Convergence of{σjl(x):j∈ℕ} to x1∈Perl is equivalent to (ΛΔ,σ) being an irreducible Smale space, so we refer to [3, 7] for details.
To show thatx∈Vs(x1), we need to find an m∈ℤk such that x(m,n)=x1(m,n) for every n≥m. Let m∈ℕk be the number given by σl(x)~ax so that x(m,n)=σl(x)(m,n) and x(-n,-m)=σl(-n,-m) for every n≥m.
Becauseσjl(x) converges to x1 as j→∞, for every open neighborhood U of x1, there is an N∈ℕ such that j≥N implies σjl(x)∈U. For
(23)U=Z(x1(m,n),m)={y∈ΛΔ:y(m,n)=x1(m,n)},
let N∈ℕ be such that σNl(x)∈Z(x1(m,n),m), that is,
(24)σNl(x)(m,n)=x(m+Nl,n+Nl)=x1(m,n).
Since x1 is a periodic point of period l, we have σ-Nl(x1)(m,n)=x1(m,n)=x(m,n) for every n≥m. Therefore, x is stably equivalent to x1.
The case ofx∈Vu(x2) is analogous.Lemma 13.
Suppose thatΛ is a k-graph and that Λ0 is a finite set. If Λ satisfies the aperiodic condition, then the set
(25){x∈ΛΔ:σl(x)∈Va(x)forsomel∈ℤk∖{0}},
is a countable set.Proof.
Letx∈ΛΔ be such that σl(x)∈Va(x) for some l∈ℤk∖{0}. First we notice that, by Lemma 12, we have x~sx1 and x~ux2 so that Vs(x)=Vs(x1) and Vu(x)=Vu(x2). Hence, Vs(x1)∩Vu(x2)=Va(x) is a countable set by Lemma 8.
Lemma12 implies that x∈Vs(x1)∩Vu(x2) so that the set
(26){x∈ΛΔ:σl(x)∈Va(x)forsomel∈ℤk∖{0}},
is a subset of
(27)⋃l∈ℤk⋃x1,x2∈PerlVs(x1)∩Vu(x2).
The union is a countable union by Lemma 11, and Vs(x1)∩Vu(x2) is a countable set by the previous paragraph and Lemma 8. This completes the proof.Recall that a topological groupoidG with an open range map is called essentially principal if G is locally compact and, for every closed invariant subset F of its unit space G0, {u∈F:r-1(u)∩s-1(u)={u}} is dense in F (see [9] for details).Theorem 14.
Suppose thatΛ is a k-graph and that Λ0 is a finite set. Then, Λ satisfies the aperiodic condition if and only if Ga⋊ℤk is an essentially principal groupoid.Proof.
LetB={u∈(Ga⋊ℤk)0:s-1(u)∩r-1(u)={u}}, collection of points of (Ga⋊ℤk)0 whose isotropy group is trivial. We observe that, for u=(x,0,x)∈(Ga⋊ℤk)0,u∉B if and only if there is an l∈ℤk∖{0} such that σl(x)~ax. And every element in B is aperiodic because a periodic point y∈ΛΔ with period l satisfies σl(x)=x, that is, σl(x)∈Va(x) and x∉B.
IfΛ satisfies the aperiodic condition, then Bc is a countable set by Lemma 13 so that B is a dense subset of (Ga⋊ℤk)0. Therefore Ga⋊ℤk is an essentially principal groupoid.
IfGa⋊ℤk is essentially principal, then B is dense in ΛΔ, and the set of aperiodic paths is dense in ΛΔ by the previous paragraph. Hence, for every v∈Λ0, the base element Z(v,0) must contain aperiodic paths. Thus, Λ satisfies the aperiodic condition.Before we go to the main theorem, let us check that our groupoidsGa and Ga⋊ℤk are amenable so that we do not need to use reduced groupoid algebras: let Gs and Gu be the locally compact principal groupoids defined by the stable and unstable equivalence relations, respectively, on ΛΔ. Kumjian and Pask showed that Gs and Gu are amenable in [1, Theorem 5.3]. Then, proof of the following lemma is exactly the same as [7, Lemma 2.5] and [1, Remark 5.8].Lemma 15.
Suppose thatΛ is a k-graph. Then, Ga and Ga⋊ℤk are amenable in the sense of Renault.Now we apply [9, Proposition 4.6], Proposition 7, and Theorem 14 to obtain simplicity of C*-algebras of Ga and Ga⋊ℤk. Remark that a primitive k-graph with our standing assumption automatically has a finite Λ0.Theorem 16.
Suppose thatΛ is a primitive k-graph. Then, the asymptotic C*-algebras A(Λ) generated by Ga is a simple nuclear C*-algebra. If in addition Λ satisfies the aperiodic condition, then the asymptotic Ruelle algebra Ra(Λ) generated by Ga⋊ℤk is also a simple nuclear C*-algebra.



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*Source: 752497-2013-06-19.xml*