C*-algebras from groupoids on self-similar groups

Volume 2013, Article ID 579042,7pages http://dx.doi.org/10.1155/2013/579042

Research Article

𝐶

∗

-Algebras from Groupoids on Self-Similar Groups

Inhyeop Yi

Department of Mathematics Education, Ewha Womans University, Seoul 120-750, Republic of Korea

Correspondence should be addressed to Inhyeop Yi; yih@ewha.ac.kr 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 that𝐶∗(𝑅_∞) associated to a postcritically finite hyperbolic rational function is an AT-algebra of real-rank zero with a unique trace state.

1. Introduction

Nekrashevych has developed a theory of dynamical systems and𝐶∗-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 to𝐶∗-algebras structures in [2] is based on Cuntz-Pimsner algebras generated by self-similar groups. However Smale spaces and their corresponding𝐶∗-algebras have rich dynamical structures, and it is conceivable that dynamical systems associated with self-similar groups may give another way to study 𝐶∗-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 𝐶∗-algebras from the stable equivalence relation on the limit solenoid(𝑆_𝐺, 𝜎) of a self-similar group (𝐺, 𝑋). Instead of using the groupoids𝐺_𝑠and𝐺_𝑠⋊ Z on the Smale space(𝑆_𝐺, 𝜎) as Putnam [3, 4] and Nekrashevych [2] did, we consider the essentially principal groupoids 𝑅_∞ and Γ(𝐽𝐺, 𝜎) of Anantharaman-Delaroche [8] and Deaconu [9] on

a presentation(𝐽_𝐺, 𝜎) of (𝑆_𝐺, 𝜎). While 𝐺_𝑠and𝐺_𝑠⋊ Z are not 𝑟-discrete groupoids, 𝑅_∞andΓ(𝐽_𝐺, 𝜎) are 𝑟-discrete. And 𝑅_∞ and Γ(𝐽_𝐺, 𝜎) are defined on (𝐽_𝐺, 𝜎) so that we do not need

to entail the inverse limit structure of(𝑆_𝐺, 𝜎). Thus 𝑅_∞and Γ(𝐽𝐺, 𝜎) are more manageable than 𝐺𝑠 and 𝐺𝑠 ⋊ Z for the

structures of their𝐶∗-algebras.

In this paper, we prove that, for a self-similar group (𝐺, 𝑋), its limit dynamical system (𝐽_𝐺, 𝜎) is topologically mixing so that(𝑆_𝐺, 𝜎) is an irreducible Smale space. And we show that𝑅_∞is equivalent to𝐺_𝑠andΓ(𝐽_𝐺, 𝑠) is equivalent to 𝐺𝑠⋊ Z in the sense of Muhly et al. [10]. Consequently, the

groupoid𝐶∗-algebras𝐶∗(𝑅_∞) and 𝐶∗(Γ(𝐽_𝐺, 𝜎)) are strongly Morita equivalent to the stable algebra𝑆 and the stable Ruelle algebra 𝑅_𝑠, respectively, of (𝑆_𝐺, 𝜎). Then we use 𝑅_∞ and Γ(𝐽_𝐺, 𝜎) to study structures of 𝐶∗-algebras from a self-similar group(𝐺, 𝑋). Finally we show that groupoid algebras of 𝑅_∞ from postcritically finite hyperbolic rational functions are 𝐴𝑇-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 𝑅_∞ is equivalent to 𝐺_𝑠 and Γ(𝐽_𝐺, 𝑠) is equivalent to 𝐺_𝑠 ⋊ Z. In Section 4, we give a proof that its groupoid algebra 𝐶∗(Γ(𝐽_𝐺, 𝜎)) is simple, purely infinite, separable, stable, and nuclear and satisfies the Universal Coefficient Theorem. For𝑅_∞, we show that𝐶∗(𝑅_∞) is simple and nuclear. And, when self-similar group is defined by a postcritically finite hyperbolic rational function and its Julia set, we show that𝐶∗(𝑅_∞) is an 𝐴𝑇-algebra.

2. Self-Similar Groups

We review the properties of self-similar groups. As for general references for the notions of self-similar groups, we refer to [1,2].

Suppose that𝑋 is a finite set. We denote by 𝑋𝑛 the set of words of length𝑛 in 𝑋 with 𝑋0 = {0}, and define 𝑋∗ = ∪∞

𝑛=0𝑋𝑛. A self-similar group(𝐺, 𝑋) consists of an 𝑋 and a

faithful action of a group𝐺 on 𝑋 such that, for all 𝑔 ∈ 𝐺 and𝑥 ∈ 𝑋, there exist unique 𝑦 ∈ 𝑋 and ℎ ∈ 𝐺 such that

𝑔 (𝑥𝑤) = 𝑦ℎ (𝑤) for every 𝑤 ∈ 𝑋∗. (1) The above equality is written formally as

𝑔 ⋅ 𝑥 = 𝑦 ⋅ ℎ. (2)

We observe that for any𝑔 ∈ 𝐺 and V ∈ 𝑋∗, there exists a unique elementℎ ∈ 𝐺 such that 𝑔(V𝑤) = 𝑔(V)ℎ(𝑤) for every 𝑤 ∈ 𝑋∗_{. The unique elementℎ is called the restriction of 𝑔 at}

V and is denoted by 𝑔|_V. For𝑢 = 𝑔(V) and ℎ = 𝑔|_V, we write

𝑔 ⋅ V = 𝑢 ⋅ ℎ. (3)

A self-similar group(𝐺, 𝑋) is called recurrent if, for all 𝑥, 𝑦 ∈ 𝑋, there is a 𝑔 ∈ 𝐺 such that 𝑔 ⋅ 𝑥 = 𝑦 ⋅ 1; that is, 𝑔(𝑥𝑤) = 𝑦𝑤 for every 𝑤 ∈ 𝑋∗. We say that (𝐺, 𝑋) is

contracting if there is a finite subset 𝑁 of 𝐺 satisfying the

following: for every𝑔 ∈ 𝐺, there is 𝑛 ≥ 0 such that 𝑔|_V∈ 𝑁 for everyV ∈ 𝑋∗of length|V| ≥ 𝑛. If the group is contracting, the smallest set𝑁 satisfying this condition is called the nucleus of the group.

Standing Assumption. We assume that our self-similar group

(𝐺, 𝑋) is a contracting, recurrent, and regular group and that the group𝐺 is finitely generated.

Path Spaces. For a self-similar group(𝐺, 𝑋), the set 𝑋∗has a

natural structure of a rooted tree: the root is0, the vertices are words in𝑋∗, and the edges are of the formV to V𝑥, where V ∈ 𝑋∗ _{and𝑥 ∈ 𝑋. Then the boundary of the tree 𝑋}∗ _is

identified with the space𝑋𝜔of right-infinite paths of the form 𝑥₁𝑥₂⋅ ⋅ ⋅ , where 𝑥_𝑖∈ 𝑋. The product topology of discrete set 𝑋 is given on 𝑋𝜔_.

We say that a self-similar group(𝐺, 𝑋) is regular if, for every𝑔 ∈ 𝐺 and every 𝑤 ∈ 𝑋𝜔, either𝑔(𝑤) ̸= 𝑤 or there is a neighborhood of𝑤 such that every point in the neighborhood is fixed by𝑔.

We also consider the space 𝑋−𝜔 of left-infinite paths ⋅ ⋅ ⋅ 𝑥₋₂𝑥₋₁ over 𝑋 with the product topology. Two paths ⋅ ⋅ ⋅ 𝑥₋₂𝑥₋₁and⋅ ⋅ ⋅ 𝑦₋₂𝑦₋₁in𝑋−𝜔are said to be asymptotically

equivalent if there is a finite set𝐼 ⊂ 𝐺 and a sequence 𝑔_𝑛 ∈ 𝐼

such that

𝑔_𝑛(𝑥_−𝑛⋅ ⋅ ⋅ 𝑥₋₁) = 𝑦_−𝑛⋅ ⋅ ⋅ 𝑦₋₁, (4) for every 𝑛 ∈ N. The quotient of the space 𝑋−𝜔 by the asymptotic equivalence relation is called the limit space of (𝐺, 𝑋) and is denoted by 𝐽_𝐺. Since the asymptotic equivalence relation is invariant under the shift map ⋅ ⋅ ⋅ 𝑥₋₂𝑥₋₁ 󳨃→ ⋅ ⋅ ⋅ 𝑥₋₃𝑥₋₂, the shift map induces a continuous map𝜎 : 𝐽_𝐺 → 𝐽𝐺. We call the induced dynamical system(𝐽𝐺, 𝜎) the limit

dynamical system of(𝐺, 𝑋) (see [1,2] for details).

Remark 1. Recurrent and finitely generated conditions imply

that𝐽_𝐺is a compact, connected, locally connected, metrizable space of a finite dimension by Corollary2.8.5 and Theorem 3.6.4 of [1]. And regular condition implies that𝜎 is an |𝑋|-fold self-covering map by Proposition 6.1 of [2].

A cylinder set𝑍(𝑢) for each 𝑢 ∈ 𝑋∗= ∪_𝑛≥0𝑋𝑛is defined as follows:

𝑍 (𝑢) = {𝜉 ∈ 𝑋−𝜔: 𝜉 = ⋅ ⋅ ⋅ 𝑥_−𝑛−1𝑥_−𝑛⋅ ⋅ ⋅ 𝑥₋₁ such that𝑥_−𝑛⋅ ⋅ ⋅ 𝑥₋₁ = 𝑢} .

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Then the collection of all such cylinder sets forms a basis for the product topology on𝑋−𝜔. And we recall that a dynamical system(𝑌, 𝑓) is called topologically mixing if, for every pair of nonempty open sets𝐴, 𝐵 in 𝑌, there is an 𝑛 ∈ N such that 𝑓𝑘_{(𝐴) ∩ 𝐵 ̸= 0 for every 𝑘 ≥ 𝑛.}

Theorem 2. (𝐽_𝐺, 𝜎) is a topologically mixing system.

Proof. As𝑋−𝜔has the product topology and𝐽_𝐺has the

quo-tient topology induced from asymptotic equivalence relation, it is sufficient to show that, for arbitrary cylinder sets𝑍(𝑢) and 𝑍(V) of 𝑋−𝜔_{, there are infinite paths𝜉 = ⋅ ⋅ ⋅ 𝑥}

−2𝑥−1 ∈ 𝑍(𝑢) and

𝜂 = ⋅ ⋅ ⋅ 𝑦₋₂𝑦₋₁ ∈ 𝑍(V) such that 𝜉 is asymptotically equivalent to𝜂. Moreover we can assume that 𝑢, V ∈ 𝑋𝑛for some𝑛 ∈ N so that𝑢 = 𝑥_−𝑛⋅ ⋅ ⋅ 𝑥₋₁andV = 𝑦_−𝑛⋅ ⋅ ⋅ 𝑦₋₁.

We choose sufficiently large𝑚 and let 𝑎, 𝑏 ∈ 𝑋𝑚−𝑛so that 𝑎𝑢 and 𝑏V are elements of 𝑋𝑚_{. Then by recurrent condition}

and [1, Corollary2.8.5], for 𝑎𝑢 and 𝑏V in 𝑋𝑚, there is a𝑔 ∈ 𝐺 such that𝑔(𝑎𝑢) = 𝑔(𝑎)𝑔|_𝑎(𝑢) = 𝑏V. Since we chose large 𝑚, by contracting condition,𝑔|_𝑎is an element of the nucleus of (𝐺, 𝑋).

We remind that the nucleus of(𝐺, 𝑋) is a finite set and equal to

𝑁 = ∪𝑔∈𝐺∩𝑛≥0{𝑔|V: V ∈ 𝑋∗, |V| ≥ 𝑛} . (6)

So an element of the nucleus is a restriction of another element of the nucleus. Hence𝑔|_𝑎 ∈ 𝑁 implies that there exist a letter𝑥_−𝑛−1 and a𝑔_−𝑛−1 ∈ 𝑁 such that 𝑔_−𝑛−1|_{𝑥−𝑛−1} = 𝑔|_𝑎. Then, for𝑔_−𝑛−1(𝑥_−𝑛−1) = 𝑦_−𝑛−1, we have

𝑔_−𝑛−1(𝑥_−𝑛−1𝑢) = 𝑦_−𝑛−1V. (7) So by induction there are a letter𝑥_−𝑚and a𝑔_−𝑚∈ 𝑁 for every 𝑚 ≥ 𝑛 such that

𝑔_−𝑚(𝑥_−𝑚⋅ ⋅ ⋅ 𝑥_−𝑛−1𝑢) = 𝑦_−𝑚⋅ ⋅ ⋅ 𝑦_−𝑛−1V. (8) Let𝜉 = ⋅ ⋅ ⋅ 𝑥₋₂𝑥₋₁ and let𝜂 = ⋅ ⋅ ⋅ 𝑦₋₂𝑦₋₁. Then it is trivial that 𝜉 ∈ 𝑍(𝑢) and 𝜂 ∈ 𝑍(V). And 𝜉 is asymptotically equivalent to𝜂. Therefore the limit dynamical system (𝐽_𝐺, 𝜎) is topologically mixing.

Let𝑋Zbe the space of bi-infinite paths⋅ ⋅ ⋅ 𝑥₋₁𝑥₀⋅ 𝑥₁𝑥₂⋅ ⋅ ⋅ over the alphabet 𝑋. The direct product topology of the discrete set𝑋 is given on 𝑋Z. We say that two paths⋅ ⋅ ⋅ 𝑥₋₁𝑥₀⋅ 𝑥1𝑥2⋅ ⋅ ⋅ and ⋅ ⋅ ⋅ 𝑦−1𝑦0⋅ ⋅ ⋅ 𝑦1𝑦2⋅ ⋅ ⋅ in 𝑋Z are asymptotically

equivalent if there is a finite set𝐼 ⊂ 𝐺 and a sequence 𝑔_𝑛 ∈ 𝐼

such that

𝑔_𝑛(𝑥_𝑛𝑥_𝑛+1⋅ ⋅ ⋅ ) = 𝑦_𝑛𝑦_𝑛+1. . . , (9) for every 𝑛 ∈ Z. The quotient of 𝑋Z by the asymptotic equivalence relation is called the limit solenoid of(𝐺, 𝑋) and is denoted by𝑆_𝐺. As in the case of𝐽_𝐺, the shift map on𝑋Zis transferred to an induced homeomorphism on𝑆_𝐺, which we will denote by𝜎.

Theorem 3 (see [1,2]). The limit solenoid𝑆_𝐺is homeomorphic

to the inverse limit space of(𝐽_𝐺, 𝜎)

𝐽_𝐺← 𝐽𝜎 _𝐺← ⋅ ⋅ ⋅ = {( 𝜉𝜎 ₀, 𝜉₁, 𝜉₂, . . .) ∈∏∞

𝑛≥0

𝐽_𝐺: 𝜎 ( 𝜉_𝑛+1)

= 𝜉_𝑛 for every𝑛 ≥ 0} ,

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and𝜎 : 𝑆_𝐺 → 𝑆_𝐺is the induced homeomorphism defined by

(𝜉0, 𝜉1, 𝜉2, . . .) 󳨃󳨀→ (𝜎 (𝜉0) , 𝜎 ( 𝜉1) , 𝜎 ( 𝜉2) , . . .)

= ( 𝜎 ( 𝜉₀) , 𝜉₀, 𝜉₁, . . .) . (11)

Moreover, the limit solenoid system(𝑆_𝐺, 𝜎) is a Smale space.

Then we have the following fromTheorem 2.

Corollary 4. (𝑆_𝐺, 𝜎) is topologically mixing.

We have a natural projection𝜋 : 𝑆_𝐺 → 𝐽_𝐺induced from the map

⋅ ⋅ ⋅ 𝑥₋₁𝑥₀⋅ 𝑥₁𝑥₂⋅ ⋅ ⋅ 󳨃󳨀→ ⋅ ⋅ ⋅ 𝑥₋₁𝑥₀, (12) and the relation that⋅ ⋅ ⋅ 𝑥_𝑛−1𝑥_𝑛 ∈ 𝑋−𝜔represents𝜉_𝑛 ∈ 𝐽_𝐺. Then it is easy to check𝜋 ∘ 𝜎 = 𝜎 ∘ 𝜋. The stable equivalence

relation on(𝑆_𝐺, 𝜎) is defined as follows [2, Proposition 6.8]:

Definition 5. One says that two elements𝛼 and 𝛽 in 𝑆_𝐺are

stably equivalent and write𝛼∼_𝑠𝛽 if there is a 𝑘 ∈ Z such that 𝜋𝜎𝑘(𝛼) = 𝜋𝜎𝑘(𝛽).

In other words, when𝛼 and 𝛽 are represented by infinite paths (𝑥_𝑛)_𝑛∈Z and (𝑦_𝑛)_𝑛∈Z in 𝑋Z, 𝛼∼_𝑠𝛽 if and only if the corresponding left-infinite paths⋅ ⋅ ⋅ 𝑥_𝑘−1𝑥_𝑘and⋅ ⋅ ⋅ 𝑦_𝑘−1𝑦_𝑘in 𝑋−𝜔_{are asymptotically equivalent for some𝑘 ∈ Z.}

Groupoids on(𝐽_𝐺, 𝜎) and (𝑆_𝐺, 𝜎). Suppose that (𝐺, 𝑋) is a

self-similar group and(𝑆_𝐺, 𝜎) is its corresponding limit solenoid. We recall from [3] that the stable equivalence groupoid𝐺_𝑠on 𝑆_𝐺and its semidirect product byZ are defined to be

𝐺_𝑠= {( 𝛼, 𝛽) ∈ 𝑆_𝐺× 𝑆_𝐺: 𝛼∼_𝑠𝛽} , 𝐺_𝑠⋊ Z = {( 𝛼, 𝑛, 𝛽) ∈ 𝑆_𝐺× Z × 𝑆_𝐺: 𝑛 ∈ Z,

(𝜎𝑛_{(𝛼) , 𝛽) ∈ 𝐺}

𝑠} .

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Then𝐺_𝑠and𝐺_𝑠⋊ Z are groupoids with the natural structure maps. The unit spaces of𝐺_𝑠and𝐺_𝑠⋊ Z are identified with 𝑆_𝐺 via the maps𝛼 ∈ 𝑆_𝐺󳨃→ (𝛼, 𝛼) ∈ 𝐺_𝑠and𝛼 󳨃→ (𝛼, 0, 𝛼) ∈ 𝐺_𝑠⋊Z, respectively.

To give topologies on these groupoids, we consider subgroupoids of𝐺_𝑠. For each𝑛 ≥ 0, set

𝐺_𝑠,𝑛= {(𝛼, 𝛽) ∈ 𝑆_𝐺× 𝑆_𝐺: 𝜋𝜎𝑛_{(𝛼) = 𝜋𝜎}𝑛_{(𝛽)} . (14)}

Then𝐺_𝑠,𝑛is a subgroupoid of𝐺_𝑠. Note that if𝜇 and ] in 𝑆_𝐺 are stably equivalent with𝜋𝜎𝑙(𝜇) = 𝜋𝜎𝑙(]) for some negative integer𝑙, then

𝜋 ( 𝜇) = 𝜎−𝑙𝜋𝜎𝑙(𝜇) = 𝜎−𝑙𝜋𝜎𝑙(]) = 𝜋 (]) (15) implies that(𝜇, ]) ∈ 𝐺_𝑠,0. So we obtain the stable equivalence groupoid

𝐺_𝑠= ⋃

𝑛≥0

𝐺_𝑠,𝑛. ₍₁₆₎

Each𝐺_𝑠,𝑛is given the relative topology from𝑆_𝐺× 𝑆_𝐺, and𝐺_𝑠 is given the inductive limit topology. Under this topology, it is not difficult to check that𝐺_𝑠is a locally compact Hausdorff principal groupoid with the natural structure maps. For𝐺_𝑠⋊ Z, we transfer the product topology of 𝐺𝑠 × Z to 𝐺𝑠 ⋊ Z

via the map((𝛼, 𝛽), 𝑛) 󳨃→ (𝛼, 𝑛, 𝜎(𝛽)). Amenability and Haar systems on𝐺_𝑠and𝐺_𝑠× Z are explained in [2–4]. We denote the groupoid𝐶∗-algebra of𝐺_𝑠by𝑆 and that of 𝐺_𝑠⋊ Z by 𝑅_𝑠 and call it stable Ruelle algebra on(𝑆_𝐺, 𝜎).

For the limit dynamical system(𝐽_𝐺⋅ 𝜎) of a self-similar group (𝐺, 𝑋), we construct groupoids 𝑅_∞ and Γ(𝐽_𝐺, 𝜎) of Anantharaman-Delaroche [8] and Deaconu [9]. Let 𝑅_𝑛 = {(𝜉, 𝜂) ∈ 𝐽_𝐺× 𝐽_𝐺: 𝜎𝑛_{(𝜉) = 𝜎}𝑛_{(𝜂)} for 𝑛 ≥ 0 and define}

𝑅∞= ⋃

𝑛≥0𝑅𝑛,

Γ ( 𝐽𝐺, 𝜎) = { ( 𝜉, 𝑛, 𝜂) ∈ 𝐽𝐺× Z × 𝐽𝐺: ∃𝑘, 𝑙 ≥ 0,

𝑛 = 𝑘 − 𝑙, 𝜎𝑘(𝜉) = 𝜎𝑙(𝜂)}

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with the natural structure maps. The unit spaces of𝑅_∞and Γ(𝐽_𝐺, 𝜎) are identified with 𝐽_𝐺via𝜉 󳨃→ (𝜉, 𝜉) and 𝜉 󳨃→ (𝜉, 0, 𝜉). We give the relative topology from𝐽_𝐺× 𝐽_𝐺 on𝑅_𝑛 and the inductive limit topology on𝑅_∞. Then 𝑅_∞ is a second countable, locally compact, Hausdorff,𝑟-discrete groupoid with the Haar system given by the counting measures. A topology onΓ(𝐽_𝐺, 𝜎) is given by basis of the form

Λ (𝑈, 𝑉, 𝑘 ⋅ 𝑙) = {(𝜉, 𝑘 − 𝑙, (𝜎𝑙|𝑉)−1∘ 𝜎𝑘(𝜉)) : 𝜉 ∈ 𝑈} , (18)

where𝑈 and 𝑉 are open sets in 𝐽_𝐺and𝑘, 𝑙 ≥ 0 such that 𝜎𝑘|_𝑈 and 𝜎𝑙|_𝑉are homeomorphisms with the same range. Then Γ(𝐽_𝐺, 𝜎) is a second countable, locally compact, Hausdorff, 𝑟-discrete groupoid, and the counting measure is a Haar system [9,11]. Amenability of𝑅_∞andΓ(𝐽_𝐺, 𝜎) is explained in Proposition 2.4 of [12]. We denote the groupoid𝐶∗-algebras of𝑅_∞andΓ(𝐽_𝐺, 𝜎) by 𝐶∗(𝑅_∞) and 𝐶∗(Γ(𝐽_𝐺, 𝜎)), respectively.

3. Groupoid Equivalence

We follow Kumjian and Pask [13, Section 5] to obtain equivalence of groupoids between𝐺_𝑠and 𝑅_∞and between 𝐺_𝑠⋊ Z and Γ(𝐽_𝐺, 𝜎), respectively, in the sense of Muhly et al. [10].

We repeat Kumjian and Pask’s observation [13]. Suppose that𝑌 is a locally compact Hausdorff space and that Γ is a locally compact Hausdorff groupoid. For a continuous open surjection𝜙 : 𝑌 → Γ0, we set a topological space

𝑍 = 𝑌 ∗ Γ = {(𝑦, 𝛾) : 𝑦 ∈ 𝑌, 𝛾 ∈ Γ, 𝜙 (𝑦) = 𝑠 (𝛾)} (19) with the relative topology in 𝑌 × Γ and a locally compact Hausdorff groupoid

Γ𝜙= {( 𝑦₁, 𝛾, 𝑦₂) : 𝑦₁, 𝑦₂∈ 𝑌, 𝛾 ∈ Γ,

𝜙 ( 𝑦₁) = 𝑠 (𝛾) , 𝑟 (𝛾) = 𝜙 (𝑦₂)} (20) with the relative topology.

Theorem 6 (see [13, Lemma 5.1]). Suppose that𝑌, Γ, 𝜙, 𝑍, and

Γ𝜙_{are as previous. Then𝑍 implements an equivalence between}

Γ and Γ𝜙_{in the sense of Muhly-Renault-Williams.}

Now we consider 𝜙 : 𝑆_𝐺 → 𝑅0_∞ defined by 𝛼 󳨃→ (𝜋(𝛼), 𝜋(𝛼)). Since 𝜙 is the composition of the projection map 𝜋 : 𝑆_𝐺 → 𝐽_𝐺and the identity map from𝐽_𝐺to𝑅0_∞,𝜙 is a continuous open surjection. Then we have

𝑅𝜙_∞= {( 𝛼, ( 𝜋 (𝛼) , 𝜋 (𝛽)) , 𝛽) : 𝛼, 𝛽 ∈ 𝑆𝐺,

(𝜋 (𝛼) , 𝜋 ( 𝛽)) ∈ 𝑅∞} .

(21) It is not difficult to check that𝑅𝜙_∞= ∪_𝑛≥0𝑅𝜙_𝑛, where

𝑅𝜙

𝑛 = {( 𝛼, ( 𝜋 (𝛼) , 𝜋 (𝛽)) , 𝛽) : 𝛼, 𝛽 ∈ 𝑆𝐺,

𝜎𝑛(𝜋 (𝛼)) = 𝜎𝑛(𝜋 (𝛽))} , (22) and that the relative topology on 𝑅𝜙_∞ is equivalent to the inductive limit topology.

Lemma 7. Suppose that (𝑆𝐺, 𝜎) is the limit solenoid system

induced from a self-similar group(𝐺, 𝑋) and that 𝐺_𝑠 is the

stable equivalence groupoid associated with(𝑆_𝐺, 𝜎). Then ̃𝜙 :

𝐺_𝑠 → 𝑅𝜙

∞ defined by (𝛼, 𝛽) 󳨃→ (𝛼, (𝜋(𝛼), 𝜋(𝛽)), 𝛽) is a

groupoid isomorphism.

Proof. Remember that 𝐺_𝑠 = ∪_𝑛≥0𝐺_𝑠,𝑛 and 𝑅𝛼_∞ = ∪_𝑛≥0𝑅𝛼_𝑛.

From the commutative relation𝜎𝜋 = 𝜋𝜎, we observe (𝛼, 𝛽) ∈ 𝐺𝑠,𝑛⇐⇒ 𝜋𝜎𝑛(𝛼) = 𝜋𝜎𝑛(𝛽) ⇐⇒ 𝜎𝑛𝜋 (𝛼)

= 𝜎𝑛𝜋 ( 𝛽) . (23)

Hence ̃𝜙|_{𝐺𝑠,𝑛}is a well-defined bijective map between𝐺_𝑠,𝑛and 𝑅𝜙

𝑛.

Since topologies on 𝐺_𝑠,𝑛 and 𝑅𝜙_𝑛 are relative topologies from 𝑆_𝐺 × 𝑆_𝐺, ̃𝜙|_{𝐺𝑠,𝑛} is a homeomorphism. Then ̃𝜙 is a homeomorphism as the inductive limit topologies are given on𝐺_𝑠 and 𝑅𝜙_∞. It is routine to check that ̃𝜙 is a groupoid morphism.

The groupoid equivalence between 𝑅_∞ and 𝐺_𝑠 follows

fromTheorem 6and Lemma 7. Strong Morita equivalence

is from [10, Proposition 2.8] as both groupoids have Haar systems.

Theorem 8. Suppose that (𝐺, 𝑋) is a self-similar group, that

𝑅_∞is the groupoid associated with(𝐽_𝐺, 𝜎), and that 𝐺_𝑠is the

stable equivalence groupoid associated with(𝑆_𝐺, 𝜎). Then 𝑅_∞

and𝐺_𝑠are equivalent in the sense of Muhly-Renault-Williams.

Therefore𝐶∗(𝑅_∞) is strongly Morita equivalent to the stable

algebra𝑆 on the limit solenoid system (𝑆_𝐺, 𝜎).

Analogous assertions hold forΓ(𝐽_𝐺, 𝜎) and 𝐺_𝑠⋊Z. For 𝜓 : 𝑆𝐺 → Γ(𝐽𝐺, 𝜎)0defined by𝛼 󳨃→ (𝜋(𝛼), 0, 𝜋(𝛼)), we observe

Γ(𝐽𝐺, 𝜎)𝜓= {( 𝛼, ( 𝜋 (𝛼) , 𝑛, 𝜋 ( 𝛽)) , 𝛽) : 𝛼, 𝛽 ∈ 𝑆𝐺,

(𝜋 (𝛼) , 𝑛, 𝜋 ( 𝛽)) ∈ Γ ( 𝐽𝐺, 𝜎)} .

(24)

Lemma 9. Suppose that 𝐺_𝑠is the stable equivalence groupoid

of(𝑆_𝐺, 𝜎) and that 𝐺_𝑠⋊ Z is the semidirect product groupoid. Then ̃𝜓 : 𝐺_𝑠 ⋊ Z → Γ(𝐽_𝐺, 𝜎)𝜓 defined by (𝛼, 𝑛, 𝛽) 󳨃→

(𝛼, (𝜋(𝛼), 𝑛, 𝜋(𝛽)), 𝛽) is a groupoid isomorphism.

Proof. Recall that(𝛼, 𝑛, 𝛽) ∈ 𝐺_𝑠⋊ Z ⇔ (𝜎𝑛(𝛼), 𝛽) ∈ 𝐺_𝑠. Then

𝐺_𝑠= ∪_𝑛≥0𝐺_𝑠,𝑛implies that(𝜎𝑛(𝛼), 𝛽) ∈ 𝐺_𝑠,𝑙for some𝑙 ≥ 0. So from the proof ofLemma 7, we obtain that

(𝜎𝑛(𝛼) , 𝛽) ∈ 𝐺_𝑠,𝑙⇐⇒ 𝜎𝑛+𝑙(𝜋 (𝛼))

= 𝜎𝑙( 𝜋 ( 𝛽)) ⇐⇒ ( 𝜋 (𝛼) , 𝑛, 𝜋 ( 𝛽)) ∈ Γ ( 𝐽𝐺, 𝜎) .

(25)

Thus̃𝜓 is a well-defined bijective map. As 𝐺_𝑠⋊Z has the prod-uct topology, we notice that ̃𝜓|_{𝐺𝑠⋊{0}}is the homeomorphism ̃𝜙 defined inLemma 7and that ̃𝜓|_{𝐺𝑠⋊{𝑛}}is homeomorphism onto

{ ( 𝛼, ( 𝜋 (𝛼) , 𝑛, 𝜋 ( 𝛽)) , 𝛽) : 𝛼, 𝛽 ∈ 𝑆𝐺,

𝜎𝑛+𝑙_{(𝜋 (𝛼)) = 𝜎}𝑙_{( 𝜋 ( 𝛽))} .} (26)

It is trivial that̃𝜓 is a groupoid morphism.

Theorem 10. Suppose that (𝐺, 𝑋) is a self-similar group. Then

Γ(𝐽_𝐺, 𝜎) and 𝐺_𝑠 ⋊ Z are equivalent in the sense of

Muhly-Renault-Williams. Therefore 𝐶∗(Γ(𝐽_𝐺, 𝜎)) is strongly Morita

equivalent to the stable Ruelle algebra𝑅_𝑠on(𝑆_𝐺, 𝜎).

Remark 11. In [11], 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(𝐺, 𝑋) is a self-similar group. We use its corre-sponding𝑅_∞andΓ(𝐽_𝐺, 𝜎) to study 𝐶∗-algebraic structures of stable algebra and stable Ruelle algebra from(𝐺, 𝑋).

Following Renault [15], we say that a topological groupoid Γ with an open range map is essentially principal if Γ is locally compact and, for every closed invariant subset𝐸 of its unit

spaceΓ0,{𝑢 ∈ 𝐸 : 𝑟−1(𝑢) ∩ 𝑠−1(𝑢) = {𝑢}} is dense in 𝐸. A subset𝐸 of Γ0is called invariant if𝑟 ∘ 𝑠−1(𝐸) = 𝐸. And Γ is called minimal if the only open invariant subsets ofΓ0are the empty set0 and Γ0itself. We refer [15] for details.

Proposition 12. The groupoid Γ(𝐽_𝐺, 𝜎) is essentially principal.

Proof. Let

𝐴 = {𝜉 ∈ 𝐽_𝐺: for 𝑘, 𝑙 ≥ 0, 𝜎𝑘(𝜉) = 𝜎𝑙(𝜉) implies 𝑘 = 𝑙} , 𝐵 = {𝑏 ∈ Γ(𝐽𝑓, 𝑓)0: 𝑟−1(𝑏) ∩ 𝑠−1(𝑏) = {𝑏}} .

(27) Then we observe𝜉 ∈ 𝐴 ⇔ (𝜉, 0, 𝜉) ∈ 𝐵. Hence 𝐴 is dense in𝑋 implying that 𝐵 is dense in Γ(𝐽_𝑓, 𝑓)0so thatΓ(𝐽_𝐺, 𝜎) is essentially principal.

To show that𝐴 is dense in 𝐽_𝐺, we assume𝐴 is not dense in 𝐽_𝐺. Then we can find an open set𝑈 ⊂ 𝐽_𝐺such that𝑈 ∩ 𝐴 = 0 as𝐽_𝐺is a compact Hausdorff space. Since

𝐽_𝐺− 𝐴 =⋃∞ 𝑛=1 ∞ ⋃ 𝑘=0 𝜎−𝑘(Per_𝑛) , (28)

where Per_𝑛 = {𝜉 ∈ 𝐽_𝐺: 𝜎𝑛(𝜉) = 𝜉}, we have 𝑈 = 𝑈 ∩ (𝐽𝑓− 𝐴) = ∞ ⋃ 𝑛=1 ∞ ⋃ 𝑘=0 𝑈 ∩ 𝜎−𝑘(Per𝑛) . (29)

Then by Baire category theorem, there exist some integers𝑛 ≥ 1 and 𝑘 ≥ 0 such that 𝑈 ∩ 𝜎−𝑘(Per𝑛) has nonempty interior.

But Per_𝑛 = {𝜉 ∈ 𝐽_𝐺 : 𝜎𝑛(𝜉) = 𝜉} is a finite set because 𝑋 is a finite set, and𝜎−𝑘(Per_𝑛) is a finite set as 𝜎 is an |𝑋|-fold covering map, a contradiction. Therefore𝐴 is dense in 𝐽_𝐺, and Γ(𝐽_𝐺, 𝜎) is an essentially principal groupoid.

There are excellent criteria for groupoid algebras from dynamical systems to be simple and purely infinite developed by Renault [12].

Lemma 13 (see [12]). For a topological space 𝑋 and a local

homeomorphism𝑇 : 𝑋 → 𝑋, let Γ(𝑋, 𝑇) be the groupoid of

Anantharaman-Delaroche and Deaconu. Suppose thatΓ(𝑋, 𝑇)

is an essentially principal groupoid and𝐶∗(𝑋, 𝑇) is its groupoid

algebra.

(1) Assume that for every nonempty open set𝑈 ⊂ 𝑋 and

every𝑥 ∈ 𝑋, there exist 𝑚, 𝑛 ∈ N such that 𝑇𝑛(𝑥) ∈

𝑇𝑚_{(𝑈). Then 𝐶}∗_{(𝑋, 𝑇) is simple.}

(2) Assume that for every nonempty open set𝑈 ⊂ 𝑋, there

exist an open set𝑉 ⊂ 𝑈 and 𝑚, 𝑛 ∈ N such that 𝑇𝑚(𝑉)

is strictly contained in𝑇𝑛(𝑉). Then 𝐶∗(𝑋, 𝑇) is purely

infinite.

AsΓ(𝐽_𝐺, 𝜎) is an essentially principal groupoid, we have an alternative proof for Theorem 6.5 of [2].

Theorem 14. The algebra 𝐶∗_(Γ(𝐽

𝐺, 𝜎)) is simple, purely infi-nite, separable, stable, and nuclear and satisfies the Universal Coefficient Theorem of Rosenberg-Schochet.

Proof. Suppose that𝑈 is an open set in 𝐽_𝐺. Then the inverse

image of𝑈 in 𝑋−𝜔, say𝑈󸀠, is open, and there is a cylinder set𝑍(𝑢) defined by some 𝑢 ∈ 𝑋𝑛 such that𝑍(𝑢) ⊂ 𝑈󸀠. By definition of cylinder sets, we have𝜎𝑛(𝑍(𝑢)) = 𝑋−𝜔⊆ 𝜎𝑛(𝑈󸀠), which implies that𝜎𝑛(𝑈) = 𝐽_𝐺on the quotient space. Thus for every𝜉 ∈ 𝐽_𝐺,𝜉 ∈ 𝜎𝑛(𝑈) and 𝐶∗(Γ(𝐽_𝐺, 𝜎)) is simple.

For an open set𝑈 of 𝐽_𝐺, let𝑉 be an open subset of 𝑈 such that the inverse image of𝑉 in 𝑋−𝜔is equal to the cylinder set 𝑍(V), where V ∈ 𝑋𝑛_{for some𝑛 ≥ 2. Then we obtain 𝜎}𝑛_{(𝑉) =}

𝐽_𝐺as in the previous, and𝜎𝑚(𝑉) is a proper subset of 𝜎𝑛(𝑉) for every1 ≤ 𝑚 ⪇ 𝑛. Hence 𝐶∗(Γ(𝐽_𝐺, 𝜎)) is purely infinite.

SinceΓ(𝐽_𝐺, 𝜎) is locally compact and second countable, 𝐶∗_(Γ(𝐽

𝐺, 𝜎)) is 𝜎-unital, nonunital, and separable. So Zhang’s

dichotomy [16, Theorem 1.2] implies that𝐶∗(Γ(𝐽_𝐺, 𝜎)) is sta-ble. By Proposition 2.4 of [12], nuclear is an easy consequence from amenability of Γ(𝐽_𝐺, 𝜎). Because Γ(𝐽_𝐺, 𝜎) is a locally compact amenable groupoid with Haar system,𝐶∗(Γ(𝐽_𝐺, 𝜎)) satisfied the Universal Coefficient Theorem by Lemma 3.5 and Proposition 10.7 of [17].

Corollary 15. 𝐶∗_(Γ(𝐽

𝐺, 𝜎)) is ∗-isomorphic to the stable Ruelle

algebra𝑅_𝑠.

Proof. Because𝐶∗(Γ(𝐽_𝐺, 𝜎)) and 𝑅_𝑠 are stable, this is trivial

fromTheorem 10.

For𝐶∗(𝑅_∞), we use the fact that 𝑅_∞= ∪𝑅_𝑛is a principal groupoid representing an AP equivalence relation [18].

Proposition 16. The groupoid 𝑅∞ is minimal, and its

groupoid algebra𝐶∗(𝑅_∞) is simple.

Proof. In the proof ofTheorem 14, we observed that for every

cylinder set 𝑍(𝑢) of 𝑋−𝜔, there is an 𝑛 > 0 such that 𝜎𝑛(𝑍(𝑢)) = 𝑋−𝜔. Since the inverse image of a nonempty open set𝑈 in 𝐽_𝐺contains a cylinder set𝑍(𝑢), this observation induces that𝜎𝑛(𝑈) = 𝐽_𝐺on the quotient space. Then𝑅_∞is a minimal groupoid by [19, Proposition 2.1]. And simplicity of 𝐶∗_(𝑅

∞) follows from [15, Proposition II.4.6] as𝑅∞is an

𝑟-discrete principal groupoid.

Proposition 17. 𝐶∗_(𝑅

∞) is the inductive limit of 𝐶∗(𝑅𝑛). And each 𝐶∗(𝑅_𝑛) is strongly Morita equivalent to 𝐶(𝑅0_𝑛/𝑅_𝑛) =

𝐶(𝐽𝐺/𝑅𝑛).

Proof. Note that𝑅_∞ = ∪_𝑛≥0𝑅_𝑛is the groupoid representing

an AP equivalence relation on stationary sequence𝐽_𝐺 󳨀→𝜎 𝐽_𝐺 󳨀→ ⋅ ⋅ ⋅ . Thus it is easy to check that Corollary 2.2 of [𝜎 18] implies the inductive limit structure.

Clearly𝑅_𝑛 = {(𝑢, V) ∈ 𝐽_𝐺× 𝐽_𝐺 : 𝜎𝑛(𝑢) = 𝜎𝑛(V)} is the groupoid representing an equivalence relation on𝐽_𝐺defined by 𝑢∼_𝑛V if and only if 𝜎𝑛(𝑢) = 𝜎𝑛(V). And (𝑠 × 𝑟)(𝑅_𝑛) = (𝜎−𝑛_{× 𝜎}−𝑛_{)(Δ), where Δ = {(𝑢, 𝑢) ∈ 𝐽}

𝐺× 𝐽𝐺} implies that

(𝑠 × 𝑟)(𝑅_𝑛) is a closed subset of 𝐽_𝐺 × 𝐽_𝐺. Thus we have strong Morita equivalence of𝐶∗(𝑅_𝑛) and 𝐶(𝐽_𝐺/𝑅_𝑛) by [20, Proposition 2.2].

Corollary 18. 𝐶∗_(𝑅

∞) is a nuclear algebra.

Proof. Since𝐶(𝐽_𝐺/𝑅_𝑛) is nuclear, 𝐶∗(𝑅_𝑛) is also nuclear by

[21, Theorem 15]. And it is a well-known fact that the class of nuclear 𝐶∗-algebras is closed under inductive limit. So 𝐶∗_(𝑅

∞) is nuclear.

Postcritically Finite Rational Maps. Suppose that𝑓 : C → C

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 𝑓 eventually belongs to a cycle containing a critical point. Then 𝑓 is expanding on a neighborhood of its Julia set 𝐽𝑓, the

group IMG(𝑓) is contracting, recurrent, regular, and finitely generated, and the limit dynamical system𝜎 : 𝐽_IMG(𝑓) →

𝐽_IMG(𝑓)is topologically conjugate with the action of𝑓 on its

Julia set𝐽_𝑓(see [2, Sections 2 and 6] for details).

We borrowed the following theorem from Theorem 3.16 and Remark 4.23 of [22].

Theorem 19. Let 𝑓 : C → C be a postcritically finite hyperbolic rational function of degree more than one and

let 𝑅_∞ be the groupoid on its limit dynamical system as in

Section 2. Then 𝐶∗(𝑅_∞) is an 𝐴𝑇-algebra of real-rank zero

with a unique trace state.

Proof. To show that 𝐶∗(𝑅_∞) is an 𝐴𝑇-algebra, we use the

work of Gong [23, Corollary 6.7]. By Propositions16 and

17, 𝐶∗(𝑅_∞) is a simple algebra which is an inductive limit of an𝐴𝐻 system with uniformly bounded dimensions of local spectra. And Nekrashevych showed that𝐾-groups of 𝐶∗_(𝑅

∞) for postcritically finite hyperbolic rational functions

are torsion free in [2, Theorem 6.6]. Hence𝐶∗(𝑅_∞) is an 𝐴𝑇-algebra.

As𝑓 : 𝐽_𝑓 → 𝐽_𝑓is an expanding local homeomorphism (see [2, Section 6.4]) and exact by Proposition 16 and [19, Proposition 2.1],𝐶∗(𝑅_∞) has a unique trace state by Remark 3.6 of [19]. Simplicity and uniformly bounded dimension conditions imply that𝐶∗(𝑅_∞) is approximately divisible in the sense of Blackadar et al. [24] as shown by Elliot et al. [14]. Therefore𝐶∗(𝑅_∞) has real-rank zero by Theorem 1.4 of [24].

Corollary 20. 𝐶∗_(𝑅

∞) associated with postcritically finite hyperbolic rational functions of degree more than one belongs to

the class of𝐶∗-algebras covered by Elliot classification program.

Acknowledgment

The author would like to express gratitude to the referees for their kind suggestions.

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