Research Article -Algebras from Groupoids on Self-Similar...
Transcript of Research Article -Algebras from Groupoids on Self-Similar...
-
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 579042, 7 pageshttp://dx.doi.org/10.1155/2013/579042
Research ArticleπΆ
β-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; [email protected]
Received 10 May 2013; Accepted 2 July 2013
Academic Editor: Salvador Hernandez
Copyright Β© 2013 Inhyeop Yi. This is an open access article distributed under the Creative Commons Attribution License, whichpermits 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 topologicallymixing and their stable algebra and stableRuelle algebra arestrongly Morita equivalent to groupoid algebras of Anantharaman-Delaroche and Deaconu. And we show that πΆβ(π
β) 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 systemsandπΆβ-algebras for self-similar groups in [1, 2].These groupsinclude groups acting on rooted trees and finite automata anditerated monodromy groups of self-covering on topologicalspaces. From self-similar groups, Nekrashevych constructedSmale spaces of Ruelle and Putnam with their correspondingstable and unstable algebras and those of Ruelle algebras forvarious equivalence relations on the Smale spaces [3β7].
Main approach to πΆβ-algebras structures in [2] is basedon Cuntz-Pimsner algebras generated by self-similar groups.However Smale spaces and their corresponding πΆβ-algebrashave rich dynamical structures, and it is conceivable thatdynamical systems associated with self-similar groups maygive another way to study πΆβ-algebras from self-similargroups. Our intention is to elucidate self-similar groups fromthe perspective of dynamical systems.
This paper is concerned with groupoids and theirgroupoid πΆβ-algebras from the stable equivalence relationon the limit solenoid (π
πΊ, π) of a self-similar group (πΊ,π).
Instead of using the groupoids πΊπ and πΊ
π β Z on the Smale
space (ππΊ, π) as Putnam [3, 4] and Nekrashevych [2] did,
we consider the essentially principal groupoids π β
andΞ(π½πΊ, π) of Anantharaman-Delaroche [8] andDeaconu [9] on
a presentation (π½πΊ, π) of (π
πΊ, π). While πΊ
π and πΊ
π βZ are not
π-discrete groupoids,π βand Ξ(π½
πΊ, π) are π-discrete. Andπ
β
and Ξ(π½πΊ, π) are defined on (π½
πΊ, π) so that we do not need
to entail the inverse limit structure of (ππΊ, π). Thus π
βand
Ξ(π½πΊ, π) are more manageable than πΊ
π and πΊ
π β Z for the
structures of their πΆβ-algebras.In this paper, we prove that, for a self-similar group
(πΊ,π), its limit dynamical system (π½πΊ, π) is topologically
mixing so that (ππΊ, π) is an irreducible Smale space. And we
show that π β
is equivalent to πΊπ and Ξ(π½
πΊ, π ) is equivalent to
πΊπ β Z in the sense of Muhly et al. [10]. Consequently, the
groupoid πΆβ-algebras πΆβ(π β
) and πΆβ(Ξ(π½πΊ, π)) are strongly
Morita equivalent to the stable algebra π and the stable Ruellealgebra π
π , respectively, of (π
πΊ, π). Then we use π
βand
Ξ(π½πΊ, π) to study structures ofπΆβ-algebras from a self-similar
group (πΊ,π). Finally we show that groupoid algebras of π β
from postcritically finite hyperbolic rational functions areπ΄π-algebras of real-rank zero.
The outline of the paper is as follows. In Section 2, wereview the notions of self-similar groups and their groupoidsand show that the induced limit dynamical system andthe limit solenoid of a self-similar group are topologicallymixing. In Section 3, we observe that π
βis equivalent
to πΊπ and Ξ(π½
πΊ, π ) is equivalent to πΊ
π β Z. In Section 4,
we give a proof that its groupoid algebra πΆβ(Ξ(π½πΊ, π)) is
simple, purely infinite, separable, stable, and nuclear andsatisfies the Universal CoefficientTheorem. For π
β, we show
that πΆβ(π β
) is simple and nuclear. And, when self-similargroup is defined by a postcritically finite hyperbolic rationalfunction and its Julia set, we show that πΆβ(π
β) is an π΄π-
algebra.
-
2 Abstract and Applied Analysis
2. Self-Similar Groups
We review the properties of self-similar groups. As for generalreferences for the notions of self-similar groups, we refer to[1, 2].
Suppose that π is a finite set. We denote by ππ the setof words of length π in π with π0 = {0}, and define πβ =βͺβ
π=0ππ. A self-similar group (πΊ,π) consists of an π and a
faithful action of a group πΊ on π such that, for all π β πΊand π₯ β π, there exist unique π¦ β π and β β πΊ such that
π (π₯π€) = π¦β (π€) for every π€ β πβ. (1)
The above equality is written formally as
π β π₯ = π¦ β β. (2)
We observe that for any π β πΊ and V β πβ, there exists aunique element β β πΊ such that π(Vπ€) = π(V)β(π€) for everyπ€ β π
β. The unique element β is called the restriction of π atV and is denoted by π|V. For π’ = π(V) and β = π|V, we write
π β V = π’ β β. (3)
A self-similar group (πΊ,π) is called recurrent if, for allπ₯, π¦ β π, there is a π β πΊ such that π β π₯ = π¦ β 1; that is,π(π₯π€) = π¦π€ for every π€ β πβ. We say that (πΊ,π) iscontracting if there is a finite subset π of πΊ satisfying thefollowing: for every π β πΊ, there is π β₯ 0 such that π|V β π forevery V β πβ of length |V| β₯ π. If the group is contracting, thesmallest set π satisfying this condition is called the nucleusof the group.
Standing Assumption.We assume that our self-similar group(πΊ,π) is a contracting, recurrent, and regular group and thatthe group πΊ is finitely generated.
Path Spaces. For a self-similar group (πΊ,π), the set πβ has anatural structure of a rooted tree: the root is 0, the verticesare words inπβ, and the edges are of the form V to Vπ₯, whereV β πβ and π₯ β π. Then the boundary of the tree πβ isidentifiedwith the spaceππ of right-infinite paths of the formπ₯1π₯2β β β , where π₯
πβ π. The product topology of discrete set
π is given on ππ.
We say that a self-similar group (πΊ,π) is regular if, forevery π β πΊ and every π€ β ππ, either π(π€) ΜΈ=π€ or there is aneighborhoodofπ€ such that every point in the neighborhoodis fixed by π.
We also consider the space πβπ of left-infinite pathsβ β β π₯β2
π₯β1
over π with the product topology. Two pathsβ β β π₯β2
π₯β1
and β β β π¦β2
π¦β1
in πβπ are said to be asymptoticallyequivalent if there is a finite set πΌ β πΊ and a sequence π
πβ πΌ
such thatππ(π₯βπ
β β β π₯β1
) = π¦βπ
β β β π¦β1
, (4)
for every π β N. The quotient of the space πβπ by theasymptotic equivalence relation is called the limit space of(πΊ,π) and is denoted by π½
πΊ. Since the asymptotic equivalence
relation is invariant under the shift map β β β π₯β2
π₯β1
β
β β β π₯β3
π₯β2, the shift map induces a continuous map π : π½
πΊβ
π½πΊ. We call the induced dynamical system (π½
πΊ, π) the limit
dynamical system of (πΊ,π) (see [1, 2] for details).
Remark 1. Recurrent and finitely generated conditions implythat π½πΊis a compact, connected, locally connected,metrizable
space of a finite dimension by Corollary 2.8.5 and Theorem3.6.4 of [1]. And regular condition implies that π is an |π|-fold self-covering map by Proposition 6.1 of [2].
A cylinder set π(π’) for each π’ β πβ = βͺπβ₯0
ππ is defined
as follows:
π (π’) = {π β πβπ
: π = β β β π₯βπβ1
π₯βπ
β β β π₯β1
such that π₯βπ
β β β π₯β1
= π’} .
(5)
Then the collection of all such cylinder sets forms a basis forthe product topology onπβπ. And we recall that a dynamicalsystem (π, π) is called topologically mixing if, for every pairof nonempty open sets π΄, π΅ in π, there is an π β N such thatππ
(π΄) β© π΅ ΜΈ= 0 for every π β₯ π.
Theorem 2. (π½πΊ, π) is a topologically mixing system.
Proof. As πβπ has the product topology and π½πΊhas the quo-
tient topology induced from asymptotic equivalence relation,it is sufficient to show that, for arbitrary cylinder setsπ(π’) andπ(V) ofπβπ, there are infinite paths π = β β β π₯
β2π₯β1
β π(π’) andπ = β β β π¦
β2π¦β1
β π(V) such that π is asymptotically equivalentto π. Moreover we can assume that π’, V β ππ for some π β Nso that π’ = π₯
βπβ β β π₯β1
and V = π¦βπ
β β β π¦β1.
We choose sufficiently largeπ and let π, π β ππβπ so thatππ’ and πV are elements of ππ. Then by recurrent conditionand [1, Corollary 2.8.5], for ππ’ and πV inππ, there is a π β πΊsuch that π(ππ’) = π(π)π|
π(π’) = πV. Since we chose large π,
by contracting condition, π|πis an element of the nucleus of
(πΊ,π).We remind that the nucleus of (πΊ,π) is a finite set and
equal to
π = βͺπβπΊ
β©πβ₯0
{π|V : V β πβ
, |V| β₯ π} . (6)
So an element of the nucleus is a restriction of anotherelement of the nucleus. Henceπ|
πβ π implies that there exist
a letter π₯βπβ1
and a πβπβ1
β π such that πβπβ1
|π₯βπβ1
= π|π.
Then, for πβπβ1
(π₯βπβ1
) = π¦βπβ1
, we have
πβπβ1
(π₯βπβ1
π’) = π¦βπβ1
V. (7)
So by induction there are a letter π₯βπ
and a πβπ
β π for everyπ β₯ π such that
πβπ
(π₯βπ
β β β π₯βπβ1
π’) = π¦βπ
β β β π¦βπβ1
V. (8)
Let π = β β β π₯β2
π₯β1
and let π = β β β π¦β2
π¦β1. Then it is trivial
that π β π(π’) and π β π(V). And π is asymptoticallyequivalent to π. Therefore the limit dynamical system (π½
πΊ, π)
is topologically mixing.
LetπZ be the space of bi-infinite paths β β β π₯β1
π₯0β π₯1π₯2β β β
over the alphabet π. The direct product topology of thediscrete setπ is given onπZ. We say that two paths β β β π₯
β1π₯0β
π₯1π₯2β β β and β β β π¦
β1π¦0β β β π¦1π¦2β β β in πZ are asymptotically
-
Abstract and Applied Analysis 3
equivalent if there is a finite set πΌ β πΊ and a sequence ππβ πΌ
such that
ππ(π₯ππ₯π+1
β β β ) = π¦ππ¦π+1
. . . , (9)
for every π β Z. The quotient of πZ by the asymptoticequivalence relation is called the limit solenoid of (πΊ,π) andis denoted by π
πΊ. As in the case of π½
πΊ, the shift map on πZ is
transferred to an induced homeomorphism on ππΊ, which we
will denote by π.
Theorem 3 (see [1, 2]). The limit solenoid ππΊis homeomorphic
to the inverse limit space of (π½πΊ, π)
π½πΊ
π
β π½πΊ
π
β β β β = {(π0, π1, π2, . . .) β
β
β
πβ₯0
π½πΊ
: π (ππ+1
)
= ππfor every π β₯ 0} ,
(10)
and π : ππΊ
β ππΊis the induced homeomorphism defined by
(π0, π1, π2, . . .) β (π (π
0) , π (π
1) , π (π
2) , . . .)
= (π (π0) , π0, π1, . . .) .
(11)
Moreover, the limit solenoid system (ππΊ, π) is a Smale space.
Then we have the following fromTheorem 2.
Corollary 4. (ππΊ, π) is topologically mixing.
We have a natural projection π : ππΊ
β π½πΊinduced from
the map
β β β π₯β1
π₯0β π₯1π₯2β β β β β β β π₯
β1π₯0, (12)
and the relation that β β β π₯πβ1
π₯π
β πβπ represents π
πβ π½πΊ.
Then it is easy to check π β π = π β π. The stable equivalencerelation on (π
πΊ, π) is defined as follows [2, Proposition 6.8]:
Definition 5. One says that two elements πΌ and π½ in ππΊare
stably equivalent and write πΌβΌπ π½ if there is a π β Z such that
πππ
(πΌ) = πππ
(π½).
In other words, when πΌ and π½ are represented by infinitepaths (π₯
π)πβZ and (π¦π)πβZ in π
Z, πΌβΌπ π½ if and only if the
corresponding left-infinite paths β β β π₯πβ1
π₯πand β β β π¦
πβ1π¦πin
πβπ are asymptotically equivalent for some π β Z.
Groupoids on (π½πΊ, π) and (π
πΊ, π). Suppose that (πΊ,π) is a self-
similar group and (ππΊ, π) is its corresponding limit solenoid.
We recall from [3] that the stable equivalence groupoidπΊπ on
ππΊand its semidirect product by Z are defined to be
πΊπ = {(πΌ, π½) β π
πΊΓ ππΊ
: πΌβΌπ π½} ,
πΊπ β Z = {(πΌ, π, π½) β π
πΊΓ Z Γ π
πΊ: π β Z,
(ππ
(πΌ) , π½) β πΊπ } .
(13)
Then πΊπ and πΊ
π βZ are groupoids with the natural structure
maps.The unit spaces of πΊπ and πΊ
π βZ are identified with π
πΊ
via themapsπΌ β ππΊ
β (πΌ, πΌ) β πΊπ andπΌ β (πΌ, 0, πΌ) β πΊ
π βZ,
respectively.To give topologies on these groupoids, we consider
subgroupoids of πΊπ . For each π β₯ 0, set
πΊπ ,π
= {(πΌ, π½) β ππΊΓ ππΊ
: πππ
(πΌ) = πππ
(π½)} . (14)
Then πΊπ ,π
is a subgroupoid of πΊπ . Note that if π and ] in π
πΊ
are stably equivalent with πππ(π) = πππ(]) for some negativeinteger π, then
π (π) = πβπ
πππ
(π) = πβπ
πππ
(]) = π (]) (15)
implies that (π, ]) β πΊπ ,0. So we obtain the stable equivalence
groupoid
πΊπ = β
πβ₯0
πΊπ ,π
. (16)
Each πΊπ ,π
is given the relative topology from ππΊΓ ππΊ, and πΊ
π
is given the inductive limit topology. Under this topology, itis not difficult to check thatπΊ
π is a locally compact Hausdorff
principal groupoid with the natural structure maps. For πΊπ β
Z, we transfer the product topology of πΊπ Γ Z to πΊ
π β Z
via the map ((πΌ, π½), π) β (πΌ, π, π(π½)). Amenability and Haarsystems on πΊ
π and πΊ
π Γ Z are explained in [2β4]. We denote
the groupoid πΆβ-algebra of πΊπ by π and that of πΊ
π β Z by π
π
and call it stable Ruelle algebra on (ππΊ, π).
For the limit dynamical system (π½πΊ
β π) of a self-similargroup (πΊ,π), we construct groupoids π
βand Ξ(π½
πΊ, π) of
Anantharaman-Delaroche [8] and Deaconu [9]. Let π π
=
{(π, π) β π½πΊΓ π½πΊ
: ππ
(π) = ππ
(π)} for π β₯ 0 and define
π β
= β
πβ₯0
π π,
Ξ (π½πΊ, π) = { (π, π, π) β π½
πΊΓ Z Γ π½
πΊ: βπ, π β₯ 0,
π = π β π, ππ
(π) = ππ
(π)}
(17)
with the natural structure maps. The unit spaces of π β
andΞ(π½πΊ, π) are identified with π½
πΊvia π β (π, π) and π β (π, 0, π).
We give the relative topology from π½πΊ
Γ π½πΊon π πand
the inductive limit topology on π β. Then π
βis a second
countable, locally compact, Hausdorff, π-discrete groupoidwith the Haar system given by the counting measures. Atopology on Ξ(π½
πΊ, π) is given by basis of the form
Ξ (π,π, π β π) = {(π, π β π, (ππ
|π)
β1
β ππ
(π)) : π β π} , (18)
whereπ and π are open sets in π½πΊand π, π β₯ 0 such that ππ|
π
and ππ|πare homeomorphisms with the same range. Then
Ξ(π½πΊ, π) is a second countable, locally compact, Hausdorff,
π-discrete groupoid, and the counting measure is a Haarsystem [9, 11]. Amenability of π
βand Ξ(π½
πΊ, π) is explained in
Proposition 2.4 of [12]. We denote the groupoid πΆβ-algebrasof π βand Ξ(π½
πΊ, π) by πΆβ(π
β) and πΆβ(Ξ(π½
πΊ, π)), respectively.
-
4 Abstract and Applied Analysis
3. Groupoid Equivalence
We follow Kumjian and Pask [13, Section 5] to obtainequivalence of groupoids between πΊ
π and π
βand between
πΊπ β Z and Ξ(π½
πΊ, π), respectively, in the sense of Muhly et al.
[10].We repeat Kumjian and Paskβs observation [13]. Suppose
that π is a locally compact Hausdorff space and that Ξ is alocally compact Hausdorff groupoid. For a continuous opensurjection π : π β Ξ0, we set a topological space
π = π β Ξ = {(π¦, πΎ) : π¦ β π, πΎ β Ξ, π (π¦) = π (πΎ)} (19)
with the relative topology in π Γ Ξ and a locally compactHausdorff groupoid
Ξπ
= {(π¦1, πΎ, π¦2) : π¦1, π¦2β π, πΎ β Ξ,
π (π¦1) = π (πΎ) , π (πΎ) = π (π¦
2)}
(20)
with the relative topology.
Theorem6 (see [13, Lemma 5.1]). Suppose thatπ, Ξ, π,π, andΞπ are as previous.Thenπ implements an equivalence between
Ξ and Ξπ in the sense of Muhly-Renault-Williams.
Now we consider π : ππΊ
β π 0
βdefined by πΌ β
(π(πΌ), π(πΌ)). Since π is the composition of the projectionmapπ : π
πΊβ π½πΊand the identity map from π½
πΊto π 0β, π is a
continuous open surjection. Then we have
π π
β= {(πΌ, (π (πΌ) , π (π½)) , π½) : πΌ, π½ β π
πΊ,
(π (πΌ) , π (π½)) β π β
} .
(21)
It is not difficult to check that π πβ
= βͺπβ₯0
π π
π, where
π π
π= {(πΌ, (π (πΌ) , π (π½)) , π½) : πΌ, π½ β π
πΊ,
ππ
(π (πΌ)) = ππ
(π (π½))} ,
(22)
and that the relative topology on π πβ
is equivalent to theinductive limit topology.
Lemma 7. Suppose that (ππΊ, π) is the limit solenoid system
induced from a self-similar group (πΊ,π) and that πΊπ is the
stable equivalence groupoid associated with (ππΊ, π). Then Μπ :
πΊπ
β π π
βdefined by (πΌ, π½) β (πΌ, (π(πΌ), π(π½)), π½) is a
groupoid isomorphism.
Proof. Remember that πΊπ
= βͺπβ₯0
πΊπ ,π
and π πΌβ
= βͺπβ₯0
π πΌ
π.
From the commutative relation ππ = ππ, we observe(πΌ, π½) β πΊ
π ,πββ ππ
π
(πΌ) = πππ
(π½) ββ ππ
π (πΌ)
= ππ
π (π½) .
(23)
Hence Μπ|πΊπ ,π
is a well-defined bijective map between πΊπ ,π
andπ π
π.Since topologies on πΊ
π ,πand π π
πare relative topologies
from ππΊ
Γ ππΊ, Μπ|πΊπ ,π
is a homeomorphism. Then Μπ is ahomeomorphism as the inductive limit topologies are givenon πΊπ and π π
β. It is routine to check that Μπ is a groupoid
morphism.
The groupoid equivalence between π β
and πΊπ follows
from Theorem 6 and Lemma 7. Strong Morita equivalenceis from [10, Proposition 2.8] as both groupoids have Haarsystems.
Theorem 8. Suppose that (πΊ,π) is a self-similar group, thatπ β
is the groupoid associated with (π½πΊ, π), and that πΊ
π is the
stable equivalence groupoid associated with (ππΊ, π). Then π
β
and πΊπ are equivalent in the sense of Muhly-Renault-Williams.
Therefore πΆβ(π β
) is strongly Morita equivalent to the stablealgebra π on the limit solenoid system (π
πΊ, π).
Analogous assertions hold for Ξ(π½πΊ, π) andπΊ
π βZ. Forπ :
ππΊ
β Ξ(π½πΊ, π)0 defined by πΌ β (π(πΌ), 0, π(πΌ)), we observe
Ξ(π½πΊ, π)
π
= {(πΌ, (π (πΌ) , π, π (π½)) , π½) : πΌ, π½ β ππΊ,
(π (πΌ) , π, π (π½)) β Ξ (π½πΊ, π)} .
(24)
Lemma 9. Suppose that πΊπ is the stable equivalence groupoid
of (ππΊ, π) and that πΊ
π β Z is the semidirect product groupoid.
Then οΏ½ΜοΏ½ : πΊπ β Z β Ξ(π½
πΊ, π)π defined by (πΌ, π, π½) β
(πΌ, (π(πΌ), π, π(π½)), π½) is a groupoid isomorphism.
Proof. Recall that (πΌ, π, π½) β πΊπ βZ β (π
π
(πΌ), π½) β πΊπ . Then
πΊπ = βͺπβ₯0
πΊπ ,π
implies that (ππ(πΌ), π½) β πΊπ ,πfor some π β₯ 0. So
from the proof of Lemma 7, we obtain that
(ππ
(πΌ) , π½) β πΊπ ,π
ββ ππ+π
(π (πΌ))
= ππ
(π (π½)) ββ (π (πΌ) , π, π (π½)) β Ξ (π½πΊ, π) .
(25)
Thus οΏ½ΜοΏ½ is a well-defined bijectivemap. AsπΊπ βZ has the prod-
uct topology, we notice that οΏ½ΜοΏ½|πΊπ
β{0}is the homeomorphism
Μπ defined in Lemma 7 and that οΏ½ΜοΏ½|
πΊπ
β{π}is homeomorphism
onto
{ (πΌ, (π (πΌ) , π, π (π½)) , π½) : πΌ, π½ β ππΊ,
ππ+π
(π (πΌ)) = ππ
(π (π½))} .
(26)
It is trivial that οΏ½ΜοΏ½ is a groupoid morphism.
Theorem 10. Suppose that (πΊ,π) is a self-similar group. ThenΞ(π½πΊ, π) and πΊ
π β Z are equivalent in the sense of Muhly-
Renault-Williams. Therefore πΆβ(Ξ(π½πΊ, π)) is strongly Morita
equivalent to the stable Ruelle algebra π π on (ππΊ, π).
Remark 11. In [11], Chen and Hou showed similar resultunder an extra condition that a Smale space is the inverselimit of an expanding surjection on a compact metric space.
4. Groupoid Algebras
Suppose that (πΊ,π) is a self-similar group. We use its corre-sponding π
βand Ξ(π½
πΊ, π) to studyπΆβ-algebraic structures of
stable algebra and stable Ruelle algebra from (πΊ,π).Following Renault [15], we say that a topological groupoid
Ξ with an open range map is essentially principal if Ξ is locallycompact and, for every closed invariant subset πΈ of its unit
-
Abstract and Applied Analysis 5
space Ξ0, {π’ β πΈ : πβ1(π’) β© π β1(π’) = {π’}} is dense in πΈ. Asubset πΈ of Ξ0 is called invariant if π β π β1(πΈ) = πΈ. And Ξ iscalledminimal if the only open invariant subsets of Ξ0 are theempty set 0 and Ξ0 itself. We refer [15] for details.
Proposition 12. The groupoid Ξ(π½πΊ, π) is essentially principal.
Proof. Let
π΄ = {π β π½πΊ
: for π, π β₯ 0, ππ (π) = ππ (π) implies π = π} ,
π΅ = {π β Ξ(π½π, π)
0
: πβ1
(π) β© π β1
(π) = {π}} .
(27)
Then we observe π β π΄ β (π, 0, π) β π΅. Hence π΄ is densein π implying that π΅ is dense in Ξ(π½
π, π)0 so that Ξ(π½
πΊ, π) is
essentially principal.To show thatπ΄ is dense in π½
πΊ, we assumeπ΄ is not dense in
π½πΊ. Then we can find an open set π β π½
πΊsuch that π β© π΄ = 0
as π½πΊis a compact Hausdorff space. Since
π½πΊβ π΄ =
β
β
π=1
β
β
π=0
πβπ
(Perπ) , (28)
where Perπ= {π β π½
πΊ: ππ
(π) = π}, we have
π = π β© (π½πβ π΄) =
β
β
π=1
β
β
π=0
π β© πβπ
(Perπ) . (29)
Then by Baire category theorem, there exist some integers π β₯1 and π β₯ 0 such that π β© πβπ(Per
π) has nonempty interior.
But Perπ
= {π β π½πΊ
: ππ
(π) = π} is a finite set because πis a finite set, and πβπ(Per
π) is a finite set as π is an |π|-fold
coveringmap, a contradiction.Thereforeπ΄ is dense in π½πΊ, and
Ξ(π½πΊ, π) is an essentially principal groupoid.
There are excellent criteria for groupoid algebras fromdynamical systems to be simple and purely infinite developedby Renault [12].
Lemma 13 (see [12]). For a topological space π and a localhomeomorphism π : π β π, let Ξ(π, π) be the groupoid ofAnantharaman-Delaroche and Deaconu. Suppose that Ξ(π, π)is an essentially principal groupoid andπΆβ(π, π) is its groupoidalgebra.
(1) Assume that for every nonempty open set π β π andevery π₯ β π, there exist π, π β N such that ππ(π₯) βππ
(π). Then πΆβ(π, π) is simple.(2) Assume that for every nonempty open setπ β π, there
exist an open setπ β π andπ, π β N such that ππ(π)is strictly contained in ππ(π). Then πΆβ(π, π) is purelyinfinite.
As Ξ(π½πΊ, π) is an essentially principal groupoid, we have
an alternative proof for Theorem 6.5 of [2].
Theorem 14. The algebra πΆβ(Ξ(π½πΊ, π)) is simple, purely infi-
nite, separable, stable, and nuclear and satisfies the UniversalCoefficient Theorem of Rosenberg-Schochet.
Proof. Suppose that π is an open set in π½πΊ. Then the inverse
image of π in πβπ, say π, is open, and there is a cylinderset π(π’) defined by some π’ β ππ such that π(π’) β π. Bydefinition of cylinder sets, we haveππ(π(π’)) = πβπ β ππ(π),which implies that ππ(π) = π½
πΊon the quotient space.Thus for
every π β π½πΊ, π β ππ(π) and πΆβ(Ξ(π½
πΊ, π)) is simple.
For an open setπ of π½πΊ, letπ be an open subset ofπ such
that the inverse image ofπ inπβπ is equal to the cylinder setπ(V), where V β ππ for some π β₯ 2. Then we obtain ππ(π) =π½πΊas in the previous, and ππ(π) is a proper subset of ππ(π)
for every 1 β€ π βͺ π. Hence πΆβ(Ξ(π½πΊ, π)) is purely infinite.
Since Ξ(π½πΊ, π) is locally compact and second countable,
πΆβ
(Ξ(π½πΊ, π)) is π-unital, nonunital, and separable. So Zhangβs
dichotomy [16, Theorem 1.2] implies that πΆβ(Ξ(π½πΊ, π)) is sta-
ble. By Proposition 2.4 of [12], nuclear is an easy consequencefrom amenability of Ξ(π½
πΊ, π). Because Ξ(π½
πΊ, π) is a locally
compact amenable groupoid with Haar system, πΆβ(Ξ(π½πΊ, π))
satisfied the Universal Coefficient Theorem by Lemma 3.5and Proposition 10.7 of [17].
Corollary 15. πΆβ(Ξ(π½πΊ, π)) isβ-isomorphic to the stable Ruelle
algebra π π .
Proof. Because πΆβ(Ξ(π½πΊ, π)) and π
π are stable, this is trivial
fromTheorem 10.
For πΆβ(π β
), we use the fact that π β
= βͺπ πis a principal
groupoid representing an AP equivalence relation [18].
Proposition 16. The groupoid π β
is minimal, and itsgroupoid algebra πΆβ(π
β) is simple.
Proof. In the proof ofTheorem 14, we observed that for everycylinder set π(π’) of πβπ, there is an π > 0 such thatππ
(π(π’)) = πβπ. Since the inverse image of a nonempty
open setπ in π½πΊcontains a cylinder setπ(π’), this observation
induces that ππ(π) = π½πΊon the quotient space. Then π
βis a
minimal groupoid by [19, Proposition 2.1]. And simplicity ofπΆβ
(π β
) follows from [15, Proposition II.4.6] as π β
is an π-discrete principal groupoid.
Proposition 17. πΆβ(π β
) is the inductive limit ofπΆβ(π π). And
each πΆβ(π π) is strongly Morita equivalent to πΆ(π 0
π/π π) =
πΆ(π½πΊ/π π).
Proof. Note that π β
= βͺπβ₯0
π πis the groupoid representing
an AP equivalence relation on stationary sequence π½πΊ
π
β
π½πΊ
π
β β β β . Thus it is easy to check that Corollary 2.2 of [18]implies the inductive limit structure.
Clearly π π
= {(π’, V) β π½πΊ
Γ π½πΊ
: ππ
(π’) = ππ
(V)} is thegroupoid representing an equivalence relation on π½
πΊdefined
by π’βΌπV if and only if ππ(π’) = ππ(V). And (π Γ π)(π
π) =
(πβπ
Γ πβπ
)(Ξ), where Ξ = {(π’, π’) β π½πΊ
Γ π½πΊ} implies that
(π Γ π)(π π) is a closed subset of π½
πΊΓ π½πΊ. Thus we have
strong Morita equivalence of πΆβ(π π) and πΆ(π½
πΊ/π π) by [20,
Proposition 2.2].
-
6 Abstract and Applied Analysis
Corollary 18. πΆβ(π β
) is a nuclear algebra.
Proof. Since πΆ(π½πΊ/π π) is nuclear, πΆβ(π
π) is also nuclear by
[21, Theorem 15]. And it is a well-known fact that the classof nuclear πΆβ-algebras is closed under inductive limit. SoπΆβ
(π β
) is nuclear.
Postcritically Finite Rational Maps. Suppose that π : C β Cis a postcritically finite hyperbolic rational function of degreemore than one, that is, a rational function of degree morethan one such that the orbit of every critical point of πeventually belongs to a cycle containing a critical point.Thenπ is expanding on a neighborhood of its Julia set π½
π, the
group IMG(π) is contracting, recurrent, regular, and finitelygenerated, and the limit dynamical system π : π½IMG(π) βπ½IMG(π) is topologically conjugate with the action of π on itsJulia set π½
π(see [2, Sections 2 and 6] for details).
We borrowed the following theorem from Theorem 3.16and Remark 4.23 of [22].
Theorem 19. Let π : C β C be a postcritically finitehyperbolic rational function of degree more than one andlet π β
be the groupoid on its limit dynamical system as inSection 2. Then πΆβ(π
β) is an π΄π-algebra of real-rank zero
with a unique trace state.
Proof. To show that πΆβ(π β
) is an π΄π-algebra, we use thework of Gong [23, Corollary 6.7]. By Propositions 16 and17, πΆβ(π
β) is a simple algebra which is an inductive limit
of an π΄π» system with uniformly bounded dimensions oflocal spectra. And Nekrashevych showed that πΎ-groups ofπΆβ
(π β
) for postcritically finite hyperbolic rational functionsare torsion free in [2,Theorem 6.6]. HenceπΆβ(π
β) is anπ΄π-
algebra.As π : π½
πβ π½πis an expanding local homeomorphism
(see [2, Section 6.4]) and exact by Proposition 16 and [19,Proposition 2.1], πΆβ(π
β) has a unique trace state by Remark
3.6 of [19]. Simplicity and uniformly bounded dimensionconditions imply that πΆβ(π
β) is approximately divisible in
the sense of Blackadar et al. [24] as shown by Elliot et al.[14]. Therefore πΆβ(π
β) has real-rank zero by Theorem 1.4 of
[24].
Corollary 20. πΆβ(π β
) associated with postcritically finitehyperbolic rational functions of degreemore than one belongs tothe class ofπΆβ-algebras covered by Elliot classification program.
Acknowledgment
The author would like to express gratitude to the referees fortheir kind suggestions.
References
[1] V. Nekrashevych, Self-Similar Groups, vol. 117 of MathematicalSurveys and Monographs, American Mathematical Society,Providence, RI, USA, 2005.
[2] V. Nekrashevych, βπΆβ-algebras and self-similar groups,β JournalfuΜr die Reine und Angewandte Mathematik, vol. 630, pp. 59β123,2009.
[3] I. F. Putnam, βπΆβ-algebras from Smale spaces,β CanadianJournal of Mathematics, vol. 48, no. 1, pp. 175β195, 1996.
[4] I. Putnam, Hyperbolic Systems and Generalized Cuntz-KriegerAlgebras, Lecture notes from Summer School in OperatorAlgebras, Odense, Denmark, 1996.
[5] I. F. Putnam and J. Spielberg, βThe structure of πΆβ-algebrasassociated with hyperbolic dynamical systems,β Journal of Func-tional Analysis, vol. 163, no. 2, pp. 279β299, 1999.
[6] D. Ruelle, Thermodynamic Formalism, vol. 5, Addison-Wesley,1978.
[7] D. Ruelle, βNoncommutative algebras for hyperbolic diffeomor-phisms,β Inventiones Mathematicae, vol. 93, no. 1, pp. 1β13, 1988.
[8] C. Anantharaman-Delaroche, βPurely infinite πΆβ-algebrasarising from dynamical systems,β Bulletin de la SocieΜteΜMatheΜmatique de France, vol. 125, no. 2, pp. 199β225, 1997.
[9] V. Deaconu, βGroupoids associated with endomorphisms,βTransactions of the AmericanMathematical Society, vol. 347, no.5, pp. 1779β1786, 1995.
[10] P. S. Muhly, J. N. Renault, and D. P. Williams, βEquivalence andisomorphism for groupoid πΆβ-algebras,β Journal of OperatorTheory, vol. 17, no. 1, pp. 3β22, 1987.
[11] X. Chen and C. Hou, βMorita equivalence of groupoid πΆβ-algebras arising from dynamical systems,β Studia Mathematica,vol. 149, no. 2, pp. 121β132, 2002.
[12] J. Renault, βCuntz-like algebras,β in Operator Theoretical Meth-ods, pp. 371β386, Bucharest, Romania, Theta Foundation, 2000.
[13] A. Kumjian and D. Pask, βActions of Zπ associated to higherrank graphs,β Ergodic Theory and Dynamical Systems, vol. 23,no. 4, pp. 1153β1172, 2003.
[14] G. A. Elliott, G. Gong, and L. Li, βApproximate divisibility ofsimple inductive limit πΆβ-algebras,β Contemporary Mathemat-ics, vol. 228, pp. 87β97, 1998.
[15] J. Renault, A Groupoid Approach to Cβ-Algebras, vol. 793 ofLecture Notes in Mathematics, Springer, Berlin, Germany, 1980.
[16] S. Zhang, βCertain πΆβ-algebras with real rank zero and theircorona and multiplier algebras. I,β Pacific Journal of Mathemat-ics, vol. 155, no. 1, pp. 169β197, 1992.
[17] J.-L. Tu, βLa conjecture de Baum-Connes pour les feuilletagesmoyennables,βK-Theory in theMathematical Sciences, vol. 17, no.3, pp. 215β264, 1999.
[18] J. Renault, βThe Radon-Nikodym problem for appoximatelyproper equivalence relations,β Ergodic Theory and DynamicalSystems, vol. 25, no. 5, pp. 1643β1672, 2005.
[19] A. Kumjian and J. Renault, βKMS states on πΆβ-algebras asso-ciated to expansive maps,β Proceedings of the American Mathe-matical Society, vol. 134, no. 7, pp. 2067β2078, 2006.
[20] P. S. Muhly and D. P.Williams, βContinuous trace groupoidπΆβ-algebras,βMathematica Scandinavica, vol. 66, no. 2, pp. 231β241,1990.
[21] A. An Huef, I. Raeburn, and D. P. Williams, βProperties pre-served under Morita equivalence of πΆβ-algebras,β Proceedingsof the American Mathematical Society, vol. 135, no. 5, pp. 1495β1503, 2007.
[22] K. Thomsen, βπΆβ-algebras of homoclinic and heteroclinicstructure in expansive dynamics,β Memoirs of the AmericanMathematical Society, vol. 206, no. 970, 2010.
-
Abstract and Applied Analysis 7
[23] G. Gong, βOn the classification of simple inductive limit πΆβ-algebrasβI.The reduction theorem,βDocumenta Mathematica,vol. 7, pp. 255β461, 2002.
[24] B. Blackadar, A. Kumjian, and M. RΓΈrdam, βApproximatelycentral matrix units and the structure of noncommutative tori,βK-Theory, vol. 6, no. 3, pp. 267β284, 1992.
-
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of