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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 579042, 7 pages
-Algebras from Groupoids on Self-Similar Groups
Department of Mathematics Education, Ewha Womans University, Seoul 120-750, Republic of Korea
Received 10 May 2013; Accepted 2 July 2013
Academic Editor: Salvador Hernandez
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.
We show that the Smale spaces from self-similar groups are topologically mixing and their stable algebra and stable Ruelle algebra are strongly Morita equivalent to groupoid algebras of Anantharaman-Delaroche and Deaconu. And we show that associated to a postcritically finite hyperbolic rational function is an AT-algebra of real-rank zero with a unique trace state.
Nekrashevych has developed a theory of dynamical systems and -algebras for self-similar groups in [1, 2]. These groups include groups acting on rooted trees and finite automata and iterated monodromy groups of self-covering on topological spaces. From self-similar groups, Nekrashevych constructed Smale spaces of Ruelle and Putnam with their corresponding stable and unstable algebras and those of Ruelle algebras for various equivalence relations on the Smale spaces [3–7].
Main approach to -algebras structures in  is based on Cuntz-Pimsner algebras generated by self-similar groups. However Smale spaces and their corresponding -algebras have rich dynamical structures, and it is conceivable that dynamical systems associated with self-similar groups may give another way to study -algebras from self-similar groups. Our intention is to elucidate self-similar groups from the perspective of dynamical systems.
This paper is concerned with groupoids and their groupoid -algebras from the stable equivalence relation on the limit solenoid of a self-similar group . Instead of using the groupoids and on the Smale space as Putnam [3, 4] and Nekrashevych  did, we consider the essentially principal groupoids and of Anantharaman-Delaroche  and Deaconu  on a presentation of . While and are not -discrete groupoids, and are -discrete. And and are defined on so that we do not need to entail the inverse limit structure of . Thus and are more manageable than and for the structures of their -algebras.
In this paper, we prove that, for a self-similar group , its limit dynamical system is topologically mixing so that is an irreducible Smale space. And we show that is equivalent to and is equivalent to in the sense of Muhly et al. . Consequently, the groupoid -algebras and are strongly Morita equivalent to the stable algebra and the stable Ruelle algebra , respectively, of . Then we use and to study structures of -algebras from a self-similar group . Finally we show that groupoid algebras of from postcritically finite hyperbolic rational functions are -algebras of real-rank zero.
The outline of the paper is as follows. In Section 2, we review the notions of self-similar groups and their groupoids and show that the induced limit dynamical system and the limit solenoid of a self-similar group are topologically mixing. In Section 3, we observe that is equivalent to and is equivalent to . In Section 4, we give a proof that its groupoid algebra is simple, purely infinite, separable, stable, and nuclear and satisfies the Universal Coefficient Theorem. For , we show that is simple and nuclear. And, when self-similar group is defined by a postcritically finite hyperbolic rational function and its Julia set, we show that is an -algebra.
2. Self-Similar Groups
Suppose that is a finite set. We denote by the set of words of length in with , and define . A self-similar group consists of an and a faithful action of a group on such that, for all and , there exist unique and such that The above equality is written formally as
We observe that for any and , there exists a unique element such that for every . The unique element is called the restriction of at and is denoted by . For and , we write
A self-similar group is called recurrent if, for all , there is a such that ; that is, for every . We say that is contracting if there is a finite subset of satisfying the following: for every , there is such that for every of length . If the group is contracting, the smallest set satisfying this condition is called the nucleus of the group.
Standing Assumption. We assume that our self-similar group is a contracting, recurrent, and regular group and that the group is finitely generated.
Path Spaces. For a self-similar group , the set has a natural structure of a rooted tree: the root is , the vertices are words in , and the edges are of the form to , where and . Then the boundary of the tree is identified with the space of right-infinite paths of the form , where . The product topology of discrete set is given on .
We say that a self-similar group is regular if, for every and every , either or there is a neighborhood of such that every point in the neighborhood is fixed by .
We also consider the space of left-infinite paths over with the product topology. Two paths and in are said to be asymptotically equivalent if there is a finite set and a sequence such that for every . The quotient of the space by the asymptotic equivalence relation is called the limit space of and is denoted by . Since the asymptotic equivalence relation is invariant under the shift map , the shift map induces a continuous map . We call the induced dynamical system the limit dynamical system of (see [1, 2] for details).
Remark 1. Recurrent and finitely generated conditions imply that is a compact, connected, locally connected, metrizable space of a finite dimension by Corollary 2.8.5 and Theorem 3.6.4 of . And regular condition implies that is an -fold self-covering map by Proposition 6.1 of .
A cylinder set for each is defined as follows: Then the collection of all such cylinder sets forms a basis for the product topology on . And we recall that a dynamical system is called topologically mixing if, for every pair of nonempty open sets in , there is an such that for every .
Theorem 2. is a topologically mixing system.
Proof. As has the product topology and has the quotient topology induced from asymptotic equivalence relation, it is sufficient to show that, for arbitrary cylinder sets and of , there are infinite paths and such that is asymptotically equivalent to . Moreover we can assume that for some so that and .
We choose sufficiently large and let so that and are elements of . Then by recurrent condition and [1, Corollary 2.8.5], for and in , there is a such that . Since we chose large , by contracting condition, is an element of the nucleus of .
We remind that the nucleus of is a finite set and equal to So an element of the nucleus is a restriction of another element of the nucleus. Hence implies that there exist a letter and a such that . Then, for , we have So by induction there are a letter and a for every such that Let and let . Then it is trivial that and . And is asymptotically equivalent to . Therefore the limit dynamical system is topologically mixing.
Let be the space of bi-infinite paths over the alphabet . The direct product topology of the discrete set is given on . We say that two paths and in are asymptotically equivalent if there is a finite set and a sequence such that for every . The quotient of by the asymptotic equivalence relation is called the limit solenoid of and is denoted by . As in the case of , the shift map on is transferred to an induced homeomorphism on , which we will denote by .
Then we have the following from Theorem 2.
Corollary 4. is topologically mixing.
We have a natural projection induced from the map and the relation that represents . Then it is easy to check . The stable equivalence relation on is defined as follows [2, Proposition 6.8]:
Definition 5. One says that two elements and in are stably equivalent and write if there is a such that .
In other words, when and are represented by infinite paths and in , if and only if the corresponding left-infinite paths and in are asymptotically equivalent for some .
Groupoids on and Suppose that is a self-similar group and is its corresponding limit solenoid. We recall from  that the stable equivalence groupoid on and its semidirect product by are defined to be Then and are groupoids with the natural structure maps. The unit spaces of and are identified with via the maps and , respectively.
To give topologies on these groupoids, we consider subgroupoids of . For each , set Then is a subgroupoid of . Note that if and in are stably equivalent with for some negative integer , then implies that . So we obtain the stable equivalence groupoid Each is given the relative topology from , and is given the inductive limit topology. Under this topology, it is not difficult to check that is a locally compact Hausdorff principal groupoid with the natural structure maps. For , we transfer the product topology of to via the map . Amenability and Haar systems on and are explained in [2–4]. We denote the groupoid -algebra of by and that of by and call it stable Ruelle algebra on .
For the limit dynamical system of a self-similar group , we construct groupoids and of Anantharaman-Delaroche  and Deaconu . Let for and define with the natural structure maps. The unit spaces of and are identified with via and .
We give the relative topology from on and the inductive limit topology on . Then is a second countable, locally compact, Hausdorff, -discrete groupoid with the Haar system given by the counting measures. A topology on is given by basis of the form where and are open sets in and such that and are homeomorphisms with the same range. Then is a second countable, locally compact, Hausdorff, -discrete groupoid, and the counting measure is a Haar system [9, 11]. Amenability of and is explained in Proposition 2.4 of . We denote the groupoid -algebras of and by and , respectively.
3. Groupoid Equivalence
We repeat Kumjian and Pask's observation . Suppose that is a locally compact Hausdorff space and that is a locally compact Hausdorff groupoid. For a continuous open surjection , we set a topological space with the relative topology in and a locally compact Hausdorff groupoid with the relative topology.
Theorem 6 (see [13, Lemma 5.1]). Suppose that , , , , and are as previous. Then implements an equivalence between and in the sense of Muhly-Renault-Williams.
Now we consider defined by . Since is the composition of the projection map and the identity map from to , is a continuous open surjection. Then we have It is not difficult to check that , where and that the relative topology on is equivalent to the inductive limit topology.
Lemma 7. Suppose that is the limit solenoid system induced from a self-similar group and that is the stable equivalence groupoid associated with . Then defined by is a groupoid isomorphism.
Proof. Remember that and . From the commutative relation , we observe
Hence is a well-defined bijective map between and .
Since topologies on and are relative topologies from , is a homeomorphism. Then is a homeomorphism as the inductive limit topologies are given on and . It is routine to check that is a groupoid morphism.
Theorem 8. Suppose that is a self-similar group, that is the groupoid associated with , and that is the stable equivalence groupoid associated with . Then and are equivalent in the sense of Muhly-Renault-Williams. Therefore is strongly Morita equivalent to the stable algebra on the limit solenoid system .
Analogous assertions hold for and . For defined by , we observe
Lemma 9. Suppose that is the stable equivalence groupoid of and that is the semidirect product groupoid. Then defined by is a groupoid isomorphism.
Proof. Recall that . Then implies that for some . So from the proof of Lemma 7, we obtain that Thus is a well-defined bijective map. As has the product topology, we notice that is the homeomorphism defined in Lemma 7 and that is homeomorphism onto It is trivial that is a groupoid morphism.
Theorem 10. Suppose that is a self-similar group. Then and are equivalent in the sense of Muhly-Renault-Williams. Therefore is strongly Morita equivalent to the stable Ruelle algebra on .
Remark 11. In , Chen and Hou showed similar result under an extra condition that a Smale space is the inverse limit of an expanding surjection on a compact metric space.
4. Groupoid Algebras
Suppose that is a self-similar group. We use its corresponding and to study -algebraic structures of stable algebra and stable Ruelle algebra from .
Following Renault , we say that a topological groupoid with an open range map is essentially principal if is locally compact and, for every closed invariant subset of its unit space , is dense in . A subset of is called invariant if . And is called minimal if the only open invariant subsets of are the empty set and itself. We refer  for details.
Proposition 12. The groupoid is essentially principal.
Then we observe . Hence is dense in implying that is dense in so that is essentially principal.
To show that is dense in , we assume is not dense in . Then we can find an open set such that as is a compact Hausdorff space. Since where , we have Then by Baire category theorem, there exist some integers and such that has nonempty interior. But is a finite set because is a finite set, and is a finite set as is an -fold covering map, a contradiction. Therefore is dense in , and is an essentially principal groupoid.
There are excellent criteria for groupoid algebras from dynamical systems to be simple and purely infinite developed by Renault .
Lemma 13 (see ). For a topological space and a local homeomorphism , let be the groupoid of Anantharaman-Delaroche and Deaconu. Suppose that is an essentially principal groupoid and is its groupoid algebra.(1)Assume that for every nonempty open set and every , there exist such that . Then is simple.(2)Assume that for every nonempty open set , there exist an open set and such that is strictly contained in . Then is purely infinite.
As is an essentially principal groupoid, we have an alternative proof for Theorem 6.5 of .
Theorem 14. The algebra is simple, purely infinite, separable, stable, and nuclear and satisfies the Universal Coefficient Theorem of Rosenberg-Schochet.
Proof. Suppose that is an open set in . Then the inverse image of in , say , is open, and there is a cylinder set defined by some such that . By definition of cylinder sets, we have , which implies that on the quotient space. Thus for every , and is simple.
For an open set of , let be an open subset of such that the inverse image of in is equal to the cylinder set , where for some . Then we obtain as in the previous, and is a proper subset of for every . Hence is purely infinite.
Since is locally compact and second countable, is -unital, nonunital, and separable. So Zhang's dichotomy [16, Theorem 1.2] implies that is stable. By Proposition 2.4 of , nuclear is an easy consequence from amenability of . Because is a locally compact amenable groupoid with Haar system, satisfied the Universal Coefficient Theorem by Lemma 3.5 and Proposition 10.7 of .
Corollary 15. is -isomorphic to the stable Ruelle algebra .
Proof. Because and are stable, this is trivial from Theorem 10.
For , we use the fact that is a principal groupoid representing an AP equivalence relation .
Proposition 16. The groupoid is minimal, and its groupoid algebra is simple.
Proof. In the proof of Theorem 14, we observed that for every cylinder set of , there is an such that . Since the inverse image of a nonempty open set in contains a cylinder set , this observation induces that on the quotient space. Then is a minimal groupoid by [19, Proposition 2.1]. And simplicity of follows from [15, Proposition II.4.6] as is an -discrete principal groupoid.
Proposition 17. is the inductive limit of . And each is strongly Morita equivalent to .
Proof. Note that is the groupoid representing an AP equivalence relation on stationary sequence . Thus it is easy to check that Corollary 2.2 of  implies the inductive limit structure.
Clearly is the groupoid representing an equivalence relation on defined by if and only if . And , where implies that is a closed subset of . Thus we have strong Morita equivalence of and by [20, Proposition 2.2].
Corollary 18. is a nuclear algebra.
Proof. Since is nuclear, is also nuclear by [21, Theorem 15]. And it is a well-known fact that the class of nuclear -algebras is closed under inductive limit. So is nuclear.
Postcritically Finite Rational Maps. Suppose that is a postcritically finite hyperbolic rational function of degree more than one, that is, a rational function of degree more than one such that the orbit of every critical point of eventually belongs to a cycle containing a critical point. Then is expanding on a neighborhood of its Julia set , the group is contracting, recurrent, regular, and finitely generated, and the limit dynamical system is topologically conjugate with the action of on its Julia set (see [2, Sections 2 and 6] for details).
We borrowed the following theorem from Theorem 3.16 and Remark 4.23 of .
Theorem 19. Let be a postcritically finite hyperbolic rational function of degree more than one and let be the groupoid on its limit dynamical system as in Section 2. Then is an -algebra of real-rank zero with a unique trace state.
Proof. To show that is an -algebra, we use the work of Gong [23, Corollary 6.7]. By Propositions 16 and 17, is a simple algebra which is an inductive limit of an system with uniformly bounded dimensions of local spectra. And Nekrashevych showed that -groups of for postcritically finite hyperbolic rational functions are torsion free in [2, Theorem 6.6]. Hence is an -algebra.
As is an expanding local homeomorphism (see [2, Section 6.4]) and exact by Proposition 16 and [19, Proposition 2.1], has a unique trace state by Remark 3.6 of . Simplicity and uniformly bounded dimension conditions imply that is approximately divisible in the sense of Blackadar et al.  as shown by Elliot et al. . Therefore has real-rank zero by Theorem 1.4 of .
Corollary 20. associated with postcritically finite hyperbolic rational functions of degree more than one belongs to the class of -algebras covered by Elliot classification program.
The author would like to express gratitude to the referees for their kind suggestions.
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