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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 752497, 5 pages
Asymptotic -Algebras from -Actions on Higher Rank Graphs
Department of Mathematics Education, Ewha Womans University, Seoul 120-750, Republic of Korea
Received 17 March 2013; Accepted 5 June 2013
Academic Editor: Baodong Zheng
Copyright © 2013 Inhyeop Yi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For a dynamical system arising from -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 -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.
In , Kumjian and Pask constructed -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 , Putnam and Spielberg , and Ruelle [4, 5]. Putnam associated several groupoid -algebras to Smale spaces using asymptotic, stable, and unstable equivalence relations generated by hyperbolic structures of Smale spaces . 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 -algebras from the stable and unstable equivalence relations on their dynamical systems and the related Ruelle algebras . Then they showed that stable Ruelle algebra of a higher rank graph is strongly Morita equivalent to the graph -algebra and that 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 algebra if the graph is primitive, they left Ruelle algebra from asymptotic equivalence relation behind, and it remains unclear whether or not is simple. Here, we show that 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 -graphs and -action in Section 2. Then we follow Hou and Chen  to study structures of groupoids from asymptotic equivalence relation, and we obtain sufficient conditions for to be simple in Section 3.
2. Dynamical Systems on Higher Rank Graphs
In this section, we briefly review the definitions and notations for -graphs and -action on those established by Kumjian and Pask in .
Definition 1 (see [1, 8]). A -graph is a pair , where is a countable small category and is a morphism, called the degree map, satisfying the factorization property. For every and with , there exist unique elements such that Every is called a path. For every nonzero , by the factorization property we may define and identify with . Let be the range and source maps. We abbreviate to when there is no confusion.
Standing Assumption (see ). For every , the restrictions of and on are surjective and finite to one.
Suppose that is a -graph defined by with the structure maps Let be a -graph, and let the corresponding two-sided infinite path space be the set Then, is a zero-dimensional space consisting of “two-sided” paths on .
A topology is endowed on where its basis is given by with and . It is not difficult to check that is compact (locally compact, resp.) if is finite (infinite, resp.) so that is a metrizable space. A metric on is defined as follows: for and , let be the element . Given , set Then, for a fixed number , a metric is defined by the formula The metric topology is the same as the one generated by cylinder sets. And we remark that the cylinder set is always nonempty because of our standing assumption.
Let be the action of on by the homeomorphism , , defined by
Theorem 3 (see ). The -action on is expansive in the sense that there is an such that, for , for every implies . Moreover, if is primitive, then is topologically mixing.
2.2. Groupoids on
We review Kumjian and Pask’s construction of the asymptotic groupoid and its corresponding -algebra and Ruelle algebra on , where is the metric defined on as in (8).
For a second countable locally compact groupoid , we denote the source map and the range map of by and , respectively, and is the unit space of . We refer to  for more detailed results on groupoids and groupoid -algebras.
As in the case of subshifts of finite types, the asymptotic, stable, and unstable equivalence relations on are defined as follows : given , we define Then, we denote the asymptotic, stable, and unstable equivalence classes of by , , and , respectively.
Remark 4 (see [1, Remark 3.6]). For , if and only if there is an such that for every we have
The asymptotic equivalence relation gives rise to a groupoid whose unit space is identified with , and structure maps are the natural ones. To give a topology on , we construct conjugate homeomorphisms as in Ruelle : for , let be as in Remark 4 and define finite paths and . We recall that the cylinder sets are compact open subsets of . Since and satisfy the equalities in Remark 4, when we define by for all and , is a bijective map by the factorization property and by the definition of and Remark 4. It is easy to check that is continuous so that it is a homeomorphism.
A basis for the topology of is given by the compact open sets Then, is a second countable, locally compact, Hausdorff, -discrete, and principal groupoid under this topology, and the counting measure is a Haar system [2, 7, 8]. The path space is identified with the unit space by the embedding map .
To make the asymptotic Ruelle algebra, we need another groupoid: we imitate  to form the semidirect product given by with the structure maps
It is not difficult to check that the map defined by gives a groupoid isomorphism between and . We transfer the product topology on via this isomorphism so that is a locally compact, Hausdorff, and -discrete groupoid, and the counting measure is a Haar system. The unit space is identified with via the map .
Definition 5 (see [1, 2]). Suppose that is a -graph and that and are groupoids on defined previously. The asymptotic -algebra of is the reduced groupoid -algebra of , denoted by . And the asymptotic Ruelle algebra of is the reduced groupoid -algebra of , denoted by .
3. Main Results
In this section, we first show that for a primitive -graph, the groupoids and are minimal. Then, we show that if a -graph is such that is finite, then the aperiodic condition is equivalent to essential principality of . So the asymptotic -algebra is simple if is primitive, and the asymptotic Ruelle algebra is simple if in addition satisfies the aperiodic condition.
Lemma 6. Suppose that is a primitive -graph. For every , is dense in .
Proof. We need to show that, for any base element , there is a . Since is primitive, there is an such that, for every and every , there is a with and .
First, we select satisfying and . Then, the primitive condition implies that there are and such that So, by the factorization property (cf. [8, Remarks 2.2]), there is a unique such that, for every , It is trivial that from the construction, and by Remark 4. Therefore, the asymptotic equivalence class of any element in is dense in .
Recall that a topological groupoid with an open range map is called minimal if the only open invariant subsets of are the empty set and itself. A subset of is called invariant if . We refer to  for details.
Proposition 7. Suppose that is a primitive -graph. Then, and are minimal groupoids.
Proof. We recall that the unit spaces of and are identified with via the maps and , respectively. When is an invariant subset of , we observe
We assume that there is a nontrivial open invariant subset of . Then, for , we have because the collection of asymptotic equivalence classes is a partition of . But this is a contradiction to Lemma 6 as is a nontrivial open set. So the only open invariant subsets of are the empty set and , and is a minimal groupoid.
For and an open invariant subset of its unit space, we note that Then, the result follows as in the case of .
For an irreducible Smale space , Putnam and Spielberg  showed that groupoids defined by stable and unstable equivalence relations are essentially principal, and Hou and Chen  showed that 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 is at most countable for every and the set of periodic points of period is finite for every nonzero (see [3, Section 5], and [4, 7.16] for details). We will follow [7, Section 2], and [3, Section 5], to show that is an essential principal.
Because of the factorization property and that is a countable set for every , it is not difficult to check that is at most countable in -graphs.
Lemma 8. Let be a -graph and . Then, is a countable set.
Proof. Let be asymptotically equivalent to . Then, there is an such that and for every by Remark 4, and the factorization property implies that the set of finite paths determines a unique element in . It is easy to check that the uniquely determined element is as by the factorization property. So the map defined by is an injection from to . Since is a countable set, is a countable set. Hence, is countable.
In irreducible Smale spaces, the set of periodic points of period , say , is finite for every nonzero integer as covered in [7, Lemma 2.1] and [3, Lemma 5.2]. But in may not be a finite set for some nonzero . For example, the -graph discussed in [8, Definition 1.9 and Example 6.1] has only one vertex and is irreducible, but every path has period . So we need to add some extra conditions on our -graphs to obtain that is finite for every nonzero .
Definition 9 (see ). Let be a -graph with its two-sided path space . A path is called aperiodic if for every and every . We say that satisfies the aperiodic condition if, for every , there is an aperiodic path such that .
Observe that satisfies the aperiodic condition if and only if the set of aperiodic paths is dense in [8, Proposition 4.5]. So, for every , the -action on given by 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 is a -graph with a finite . If satisfies the aperiodic condition, then every isolated point in is aperiodic, and the derived set of is invariant under for every .
Proof. Let be an isolated point. Then, there is an aperiodic point such that , that is, . So we have because is an isolated point.
Let be a limit point of and a sequence in such that . Then, continuity of implies that so that is also a limit point of .
Since our interest is periodic points which are included in the derived set of and the fact that the derived set is also compact, when there are isolated points in , we can consider -action of on the derived set instead of on . Thus, without loss of generality, we can assume that has no isolated point.
We recall that is expansive with the expansive constant (Theorem 3).
Lemma 11. Suppose that is a -graph with a finite . If satisfies the aperiodic condition, then the set of periodic points of period is finite for every .
Proof. For every nonzero , we consider an induced -action on given by . Then, is also an expansive homeomorphism on with the same expansive constant as that of , and is equal to .
Let and be two different points in . Then, we have because for every implies that by expansiveness. So the distance between any two fixed points of is at least , and is a finite set as is a compact set without isolated points.
Lemma 12. Let be a -graph with a finite . If is such that for some , then (1)the sequence converges to an element, say , in as and , (2)the sequence converges to an element, say , in as and .
Proof. Convergence of to is equivalent to being an irreducible Smale space, so we refer to [3, 7] for details.
To show that , we need to find an such that for every . Let be the number given by so that and for every .
Because converges to as , for every open neighborhood of , there is an such that implies . For let be such that , that is, Since is a periodic point of period , we have for every . Therefore, is stably equivalent to .
The case of is analogous.
Lemma 13. Suppose that is a -graph and that is a finite set. If satisfies the aperiodic condition, then the set is a countable set.
Proof. Let be such that for some . First we notice that, by Lemma 12, we have and so that and . Hence, is a countable set by Lemma 8.
Lemma 12 implies that so that the set is a subset of The union is a countable union by Lemma 11, and is a countable set by the previous paragraph and Lemma 8. This completes the proof.
Recall that a topological groupoid with an open range map is called essentially principal if is locally compact and, for every closed invariant subset of its unit space , is dense in (see  for details).
Theorem 14. Suppose that is a -graph and that is a finite set. Then, satisfies the aperiodic condition if and only if is an essentially principal groupoid.
Proof. Let , collection of points of whose isotropy group is trivial. We observe that, for if and only if there is an such that . And every element in is aperiodic because a periodic point with period satisfies , that is, and .
If satisfies the aperiodic condition, then is a countable set by Lemma 13 so that is a dense subset of . Therefore is an essentially principal groupoid.
If is essentially principal, then is dense in , and the set of aperiodic paths is dense in by the previous paragraph. Hence, for every , the base element must contain aperiodic paths. Thus, satisfies the aperiodic condition.
Before we go to the main theorem, let us check that our groupoids and are amenable so that we do not need to use reduced groupoid algebras: let and be the locally compact principal groupoids defined by the stable and unstable equivalence relations, respectively, on . Kumjian and Pask showed that and 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 that is a -graph. Then, and are amenable in the sense of Renault.
Theorem 16. Suppose that is a primitive -graph. Then, the asymptotic -algebras generated by is a simple nuclear -algebra. If in addition satisfies the aperiodic condition, then the asymptotic Ruelle algebra generated by is also a simple nuclear -algebra.
The author would like to express his deep gratitude to referees for their kind suggestions.
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