Uniformly recurrent subgroups and the ideal structure of reduced crossed products

0.23 MB·

by  Takuya Kawabe

UNIFORMLY RECURRENT SUBGROUPS AND THE IDEAL
STRUCTURE OF REDUCED CROSSED PRODUCTS
7
1
TAKUYAKAWABE
0
2

n
a
J
Abstract. Westudytheidealstructureofreducedcrossedproductoftopo-

2 logical dynamical systemsof acountable discretegroup. Moreconcretely, for
1 acompact HausdorffspaceX withanactionofacountable discretegroupΓ,
we consider the absence of a non-zero ideals in the reduced crossed product

] C(X)⋊r Γ which has a zero intersection with C(X). We characterize this
A
condition by a property for amenable subgroups of the stabilizer subgroups
O of X in terms of the Chabauty space of Γ. This generalizes Kennedy’s alge-
braic characterization of the simplicity for a reduced group C∗-algebra of a
.
h countable discretegroup.
t
a
m 1. Introduction
[ Throughout this paper, Γ denotes a countable discrete group. We say X is

a compact Γ-space if X is a compact Hausdorff space with a continuous Γ-action
1
v Γ×X →X, (t,x)7→tx. Westudytheidealstructureofthereducedcrossedprod-
3 uct C(X)⋊ Γ. The simplest situation is the following.
r
1
4 Definition 1.1. Let X be a compact Γ-space. We say C(X) separates the ideals
3 in C(X)⋊ Γ if for every ideal I in C(X)⋊ Γ, we have I =(I ∩C(X))⋊ Γ.
r r r
0
. Inotherwords,thereisone-to-onecorrespondencebetweentheidealsinC(X)⋊rΓ
1
and the Γ-invariant ideals in C(X) (see [15, Propostion 1.1]).
0
7 Definition 1.2. We saythatacompactΓ-spaceX satisfiesthe intersection prop-
1
erty if every non-zero ideal in C(X)⋊ Γ has a non-zero intersection with C(X).
: r
v
Then we have the following result.
i
X
Theorem 1.3 (Sierakowski, [15, Theorem 1.10]). Let X be a compact Γ-space.
r
a Then C(X) separates the ideals in C(X)⋊ Γ if and only if X satisfies the following
r
properties.
(i) The action of Γ on X is exact.
(ii) Every Γ-invariant closed set in X has the intersection property.
The purpose of this paper is to characterize the intersection property of Γ-
spaces in terms of dynamical systems. For an amenable group Γ, Kawamura and
TomiyamashowedthattheintersectionpropertiesofcompactΓ-spacesisequivalent
to topological freeness.
Theorem 1.4 (Kawamura–Tomiyama, [8, Theorem 4.1]). If Γ is amenable, the
following are equivalent.
(i) The space X has the intersection property.
(ii) For every t∈Γ\{e}, we have Fix(t)◦ =∅.
1
2 TAKUYAKAWABE
We say that Γ is C∗-simple if its reduced group C∗-algebra C∗Γ is simple. In
r
recent work [7], Kalantar and Kennedy established a dynamical characterization
of C∗-simplicity, and Breuillard, Kalantar, Kenndey and Ozawa proved that many
groupsare C∗-simple. In more recentwork [9], Kennedy showedan algebraicchar-
acterization of C∗-simplicity, as follows.
Theorem 1.5 (Kennedy, [9, Theorem 6.3]). A countable discrete group is C∗-
simpleifandonlyifitsatisfiesthefollowingcondition: Foreveryamenablesubgroup
Λ ≤ Γ, there exists a sequence (g ) such that for every subsequence (g ) of (g ),
n nk n
we have
g Λg−1 ={e}.
nk nk
k
\
Equivalently, the sequence (g Λg−1) converges to {e} in the Chabauty topology.
n n
The setSub(Γ) ofallsubgroupsof Γ admits a naturaltopology,calledChabauty
topology. WetreatSub(Γ)asacompactΓ-spacewiththistopologyandtheΓ-action
by conjugation (see Definition 5.1).
The first main result of this paper is the characterization of the intersection
property by a property for stabilizer subgroups, which is motivated by the above
results Theorem 1.4 and 1.5.
Theorem 1.6. Let X be a compact Γ-space. The following are equivalent.
(i) Every Γ-invariant closed set in X has the intersection property.
(ii) For every point x in X and every amenable subgroup Λ in Γ , there is a
x
net (g ) in Γ such that (g x) converges to x and (g Λg−1) converges to {e}
i i i i
in the Chabauty topology.
If X is minimal, the simplicity of C(X)⋊ Γ is characterizedby purely algebraic
r
conditions for the stabilizer subgroups of X, as follows.
Theorem 1.7. Let X bea minimal compact Γ-space. The following areequivalent.
(i) The reduced crossed product C(X)⋊ Γ is simple.
r
(ii) For every point x in X and every amenable subgroup Λ in Γ , there is a
x
sequence (g ) in Γ such that (g Λg−1) converges to {e} in the Chabauty
i i i
topology.
(iii) There is a point x in X such that for every amenable subgroup Λ in Γ ,
x
there is a sequence (g ) in Γ such that (g Λg−1) converges to {e} in the
i i i
Chabauty topology.
To prove these results, the equivariant injective envelope C(X˜) of C(X) plays a
centralrole. TheΓ-spaceX˜ hassomepropertiesanalogoustothoseofthe Hamana
boundary (or universal Furstenberg boundary, [7, §3]).
The simplicity of reduced crossedproducts is alsocharacterizedin terms of uni-
formly recurrent subgroups (URS in short) as with the C∗-simplicity of countable
discrete groups [9]. The notion of URS’s is introduced by Glasner–Weiss [3] as a
topologicaldynamicalanalogueofthenotionofinvariantrandomsubgroups,which
is an ergodic theoritic concept. A URS of Γ is defined as a minimal component of
the Γ-space Sub(Γ). The set of all URS’s of Γ has a natural partial order (denoted
by 4), introduced by Le Boudec–Matte Bon [11, §2.4].
The secondmain resultofthis paper is a property foramenable URS’s fromthe
aspect of its order structure.
UNIFORMLYRECURRENTSUBGROUPSANDTHEIDEALSTRUCTURE 3
Theorem 1.8. Let X be a compact Γ-space. Suppose that S is a URS (X is not
X
necessarily minimal). Then S contains a unique URS A . Moreover, A is the
X˜ X X
largest amenable URS dominated by S . Namely, for every amenable URS U such
X
that U 4S , we have U 4A .
X X
The notationS denotes the closedΓ-invariantsubspaceofSub(Γ)arisingfrom
X
stabilizer subgroups of X, called the stability system of X (see [3, §1] or Definition
6.3). If X is minimal, the space S is a URS. On the other hand, every URS is a
X
stability system of a transitive Γ-space, but it is not known whether every URS is
a stability systemofa minimalΓ-space. Using the aboveresult, we provethat it is
true for amenable URS’s.
Inthispaper,wealsostudytheidealsinthegroupC∗-algebraofΓ. Inparticular,
weseetherelationshipbetweenamenableURS’sofΓandtheidealsofC∗Γ. Foran
r
amenable subgroup Λ of Γ, we have the continuous ∗-representation π of C∗Γ on
Λ r
the Hilbert space ℓ (Γ/Λ) extending the canonical action of Γ on the coset space
2
Γ/Λ. We show that for stabilizer subgroup Λ of the Hamana boundary, the ideal
ker(π ) is maximal.
Λ
InSection2werecallthenotionofstabilizersubgroupsandstudyitsrelationship
to the intersection property. In Section 3 we recall the Γ-injective envelope and
showsome properties fromthe viewpoint of operatoralgebraswhich are analogous
tothoseoftheHamanaboundary. InSection4weproveatechnicalresulttoprove
themainresultTheorem1.6andweproveitinSection5. InSection6weestablish
the characterization of simplicity of reduced products. In Section 7 we show a
property for the Γ-injective envelope from the viewpoint of topological dynamical
system to provethe main result Theorem1.8. Finally, in Section 8 and 9 we study
the ideals arising from amenable URS’s.
Acknowledgements. The author would like to thank his supervisor, Professor
Narutaka Ozawa for his support and many valuable comments.
2. Preliminaries
Definition 2.1. For a compact Γ-space X and a point x in X, we denote by Γ
x
the stabilizer subgroup, i.e. Γ = {t ∈ Γ : tx = x}. Let Γ◦ denote the subgroup
x x
consisting the elements in Γ which act as identity on a neighborhood of x. We say
that a compact Γ-space is topologically free if Γ◦ ={e}for every x∈X. Note that
x
a Γ-space X is topologically free if and only if Fix(t)◦ = ∅ for every t ∈ Γ\{e},
where Fix(t) denotes the fixed point set in X of the homeomorphism t.
Let X be a compact Γ-space. There is a canonical conditional expectation E
X
from C(X)⋊ Γ to C(X) defined by
r
f t=e
E (fλ )=
X t
(0 t6=e
andextendedbylinearity. NotethatE isfaithful(see[2,Chapter4.1]). Forevery
X
x in X, we define a conditional expectation E from C(X)⋊ Γ to C∗(Γ ) by
x r r x
E (fλ )=f(x)E (λ )
x t Γx t
where E is the canonical conditional expectation from C∗Γ to C∗(Γ ) (given by
Γx r r x
E (λ )=λ if t∈Γ and E (λ )=0 if t∈Γ\Γ , see [2, Corollary 2.5.12]).
Γx t t x Γx t x
4 TAKUYAKAWABE
In this paper, we often use the following fact about unital completely positive
maps. See [2, Proposition 1.5.7] for proof.
Definition 2.2. Let A and B be unital C∗-algebras and φ be a unital completely
positive map. The multiplicative domain of φ is the subspace mult(φ) of A defined
by
mult(φ)={a∈A:φ(a∗a)=φ(a)∗φ(a) and φ(aa∗)=φ(a)φ(a)∗}.
Proposition 2.3. Let A and B be unital C∗-algebras and φ be a unital completely
positive map. Then, for every a ∈ mult(φ) and b ∈ A, one has φ(ab) = φ(a)φ(b)
and φ(ba) = φ(b)φ(a). In particular, mult(φ) is the largest C∗-subalgebra of A to
which the restriction of φ is multiplicative.
The following is a generalization of the result Theorem 1.4 and [13, Theorem
14].
Lemma 2.4. Let X be a compact Γ-space. Then we have the following.
(i) If the set {x ∈ X : Γ is C∗-simple} is dense in X, then X has the
x
intersection property. In particular, if X is topologically free, then X has
the intersection property.
(ii) If X has the intersection property and Γ◦ is amenable for every point x in
x
X, then X is topologically free.
Proof. We prove(i) by contradiction. Suppose thatthere is anon-zeroclosedideal
I in C(X)⋊ Γ such that I ∩C(X) = 0. Then E (I) is a non-zero since E is
r X X
faithful. Therefore ev ◦E (I) 6= 0 for some x in X such that Γ is C∗-simple
x X x
(otherwise, we have ev (E (I)) = 0 densely, this implies that E (I) = 0). It
x X X
follows that E (I)6= 0 since ev ◦E = τ ◦E , where τ is the canonical tracial
x x X λ x λ
state on C∗(Γ ) defined by τ (a) = haδ ,δ i for any a ∈ C∗Γ. We observe that
r x λ e e r
E (I)⊂C∗(Γ ) is anidealof C(X) since C∗(Γ ) is containedin the multiplicative
x r x r x
domainofE . WeshowthatE (I)isnotdenseinC∗(Γ ),whichyieldsthedesired
x x r x
contradiction with C∗-simplicity of C∗(Γ ). The ∗-homomorphism
r x
C(X)+I →(C(X)+I)/I ∼=C(X)/(C(X)∩I)=C(X)−e−v→x C
extends a state φ on C(X)⋊ Γ. We show that φ ◦E = φ . This implies that
x r x x x
ker φ ⊃ E (I) since φ (I) = 0 (hence E (I) is not dense). Let t be an element
x x x x
of Γ\Γ . There is a function f ∈ C(X) such that f(x) = 1 and f(tx) = 0. Since
x
C(X) is contained in the multiplicative domain of φ , we have
x
φ (λ ) = f(x)φ (λ )=φ (f)φ (λ )=φ (fλ )
x t x t x x t x t
= φ (λ (t−1f))=φ (λ )φ (t−1f)=φ (λ )f(tx)=0,
x t x t x x t
therefore φ = φ ◦E on C∗Γ. This implies that for every f ∈ C(X) and t ∈ λ,
x x x r
we obtain
φ ◦E (fλ )=φ (f(x)E (λ ))=f(x)φ (E (λ ))=φ (f)φ (λ )=φ (fλ ),
x x t x Γx t x Γx t x x t x t
thus we have φ ◦E =φ .
x x x
Next, we show(ii). Since Γ◦ is amenable for anyx, we define the representation
x
π of C(X)⋊ Γ on ℓ (Γ/Γ◦), which given by π (fλ )δ = f(tpx)δ for p ∈Γ/Γ◦.
x r 2 x x t p tp x
Note that for t and s in Γ such that s−1t∈Γ◦, we have tx=sx, thus the notation
x
px is well-defined. Set π = π . The representation π is faithful by the
x∈X x
intersectionpropertysince ker(π)∩C(X)=0. This implies thatX is topologically
L
UNIFORMLYRECURRENTSUBGROUPSANDTHEIDEALSTRUCTURE 5
free. Otherwise,there is anelementt in Γ\{e}andanon-zerofunction f inC(X)
such that supp(f) is contained in Fix(t)◦, which implies that π(f(1−λ )) = 0 in
t
contradiction with faithfulness. (cid:3)
3. Equivariant injective envelopes
Definition 3.1. We say that an operator system (resp. unital C∗-algebra) V is a
Γ-operator system (resp. unital Γ-C∗-algebra) if it comes together with a complete
orderisomorphic(resp.unital ∗-isomorphic)Γ-actiononV. AΓ-equivariantunital
complete positive map between Γ-operator systems is called a Γ-morphism.
Definition 3.2. WesaythatΓ-operatorsystemV isΓ-injective ifV isaninjective
object in the category of all Γ-operator systems with Γ-morphisms. Namely, for
any Γ-operator systems W ⊂W and any Γ-morphism φ from W to V, there is a
0 0
Γ-morphism φ˜from W to V such that φ˜| =φ.
W0
For every compact Γ-space X, we denote by X˜ the Gelfand spectrum of the
Γ-injective envelope of C(X), i.e. C(X˜) satisfies the following properties (see [6]).
• The Γ-C∗-algebra C(X˜) is a Γ-injective operator system.
• The Γ-C∗-algebraC(X)is containedinC(X˜) asa unital Γ-C∗-subalgebra
and C(X)⊂C(X˜) is rigid, i.e. the identity map is the only Γ-morphisms
on C(X˜) which is the identity map on C(X).
If X is the one-point Γ-space, X˜ is called the Hamana boundary, denoted by ∂ Γ.
H
We prove some facts for X˜, a generalization of the properties for the Hamana
boundary ([13, Proposition 8 and Lemma 9]). Recall that a subgroup Λ ≤ Γ is
relatively amenable if there is a Λ-invariant state on ℓ Γ. Since there is a Λ-
∞
morphism from ℓ Λ to ℓ Γ, the notions of amenability and relative amenability
∞ ∞
coincidefordiscretegroups. WedenotebyqtheΓ-equivariantcontinuoussurjection
X˜ to X.
Proposition 3.3. Let X be a compact Γ-space. Then, one has the following.
(i) The space X˜ is a Stonean space.
(ii) For any closed Γ-invariant set Z in X˜, we have Z =X˜ if q(Z)=X.
(iii) The group Γ is amenable for every point y in X˜.
y
In particular, for any t∈Γ, the set Fix(t) is clopen, hence Γ =Γ◦ for any y ∈X˜.
y y
Proof. There is an including Γ-equivariant unital ∗-homomorphism from C(X) to
the Γ-injective C∗-algebraℓ (Γ,ℓ X), whichisdefined byf 7→(tf) . Itfollows
∞ ∞ t∈Γ
thatthereareΓ-morphismsφ: ℓ (Γ,ℓ X)→C(X˜)andψ: C(X˜)→ℓ (Γ,ℓ X),
∞ ∞ ∞ ∞
which extend the identity map on C(X). Since ℓ (Γ,ℓ X) is also an injective
∞ ∞
operator system, C(X˜) is an injective operator system, thus X˜ is Stonean. Then
Fix(t) is clopen by Frol´ık’s theorem.
Next we show the condition (ii). Suppose that there is a closed Γ-invariant set
Z ( X˜ suth that q(Z) = X, then the corresponding Γ-equivariant quotient map
π from C(X˜) to C(Z) is not faithful. Since q(Z) = X, there is a Γ-morphism φ
from C(Z) to C(X˜) such that φ◦π| =id by Γ-injectivity of C(X˜). This
C(X) C(X)
implies that φ◦π =id by rigidity, hence π is faithful, a contradiction.
C(X˜)
Next,weproveamenabilityofΓ . Thereisainclusionιfromℓ Γtoℓ (Γ,ℓ X)
y ∞ ∞ ∞
as a unital Γ-C∗-subalgebra. Since the map ev ◦φ◦ι is a Γ -invariant state on
x y
ℓ Γ, we obtain (relative) amenability of Γ . (cid:3)
∞ y
6 TAKUYAKAWABE
We obtain the following result the case X being trivial (see [7, Theorem 6.2]).
Theorem 3.4. Let X be a compact Γ-space. Then the following are equivalent.
(i) The space X has the intersection property.
(ii) The space X˜ has the intersection property.
(iii) The space X˜ is (topologically) free.
Proof. First,weprovethat(i)implies(ii). SupposeX hastheintersectionproperty.
We show that every quotient map π from C(X˜)⋊ Γ to a C∗-algebra A is faithful
r
if ker(π)∩C(X˜) = 0. Since ker(π)∩C(X) = 0, the quotient map π is faithful
on C(X)⋊ Γ by the intersection property for X. By Γ-injectivity of C(X˜), there
r
is a Γ-morphism φ from A to C(X˜) such that φ◦π|C(X)⋊rΓ = EX. This implies
that φ ◦ π| = id by rigidity of C(X) ⊂ C(X˜). Therefore, we obtain
C(X˜) C(X˜)
C(X˜)⊂mult(φ◦π). It follows that φ◦π =E , hence π is faithful.
X˜
Next,weprovethat(ii)implies(i). SupposeX˜ hastheintersectionproperty. Let
πbearepresentationofC(X)⋊ ΓonaHilbertspaceHsuchthatkerπ∩C(X)=0.
r
We prove that π is injective. By Arveson’s extension theorem, we extend π to a
unital completely positive map π˜ from C(X˜)⋊ Γ to B(H). We consider a C∗-
r
subalgebra of B(H) defined by
D =C∗(π˜(C(X˜)⋊ Γ))=closure(π˜(C(X˜))·π(C∗Γ)).
r r
We define Γ-actionon D as Ad π(·), then π˜ is Γ-equivariant. Since π is faithful on
C(X),thereisaΓ-morphismφfromC∗(π˜(C(X˜)))toC(X˜)suchthatφ◦π =id
C(X)
by Γ-injectivity of C(X˜), which implies that φ◦π˜| = id by rigidity. It
C(X˜) C(X˜)
followsthatC∗(π˜(C(X˜)))⊂mult(φ),henceφisa∗-homomorphism. Now,consider
a subset of D given by
L=closure(ker(φ)·π(C∗Γ)).
r
Since φ is Γ-equivariant, the set ker(φ) is Γ-invariant, therefore we have
ker(φ)·π(C(Γ))=π(C(Γ))·ker(φ).
ThisimpliesthatLisanidealofD. SinceL∩C∗(π˜(C(X˜)))=ker(φ),themapφex-
tendsthequotientmapφ˜fromDtoD/L. Itfollowsthatφ˜◦π˜ isa∗-homomorphism
which is faithful on C(X˜). Thus, we have ker(π)⊂ker(φ˜◦π˜)=0 since X˜ has the
intersection property.
Theequivalenceof(ii)and(iii)followsfromLemma2.4andProposition3.3. (cid:3)
4. Γ-morphisms to injective envelopes
Inthissection,weproveequivalenceoftheintersectionpropertyandthe“unique
traceproperty”forcrossedproducts. First,weshowalemmatoprovethetheorem.
Lemma 4.1. Let Y be compact Γ-space. If Y is topologically free, then the only
conditional expectation from C(Y)⋊ Γ to C(Y) is the canonical conditional expec-
r
tation E . Moreover, if Y is Stonean and Γ is amenable for every y in Y, then
Y y
the converse is also true.
Proof. Suppose that Y is topologically free and let Φ be a conditional expectation
from C(Y)⋊ Γ to C(Y). The space Y is topologically free if and only if {y ∈
r
Y : Γ = {e}} (denoted by Y ) is dense in Y since ∂Fix(t) has no interior
y 0 t∈Γ
S
UNIFORMLYRECURRENTSUBGROUPSANDTHEIDEALSTRUCTURE 7
by Baire category theorem. Fix an element t in Γ\{e}. For every y in Y , we
0
have ty 6= y. Then there is a non-zero function in C(X) such that f(y) = 1 and
fλ f =0. ItfollowsthatΦ(λ )(y)=Φ(fλ f)(y)=0,hence Φ(t)=0. This implies
t t t
that Φ=E .
Y
Next,weshowtheconverse. SupposethatY isStoneanspaceandΓ isamenable
y
for every y in Y. There is a conditional expectation Φ from C(Y)⋊ Γ to C(Y),
r
definedbyΦ(fλ )=f·χ . ContinuityofΦ followsfromthe equalityΦ(·)(y)=
t Fix(t)
τ ◦E , where τ is the unit character of Γ . Note that τ is continuous on C∗(Γ )
0 y 0 y 0 r y
sinceΓ isamenable. Itfollowsthatthereisanon-canonicalconditionalexpectation
y
if Y is not (topologically) free. (cid:3)
Theorem 4.2. Let X be a compact Γ-space. Then the following are equivalent.
(i) The space X has the intersection property.
(ii) The only Γ-morphism from C(X)⋊ Γ to C(X˜) which is the identity map
r
on C(X) is the canonical conditional expectation E .
X
Proof. Letφbe aΓ-morphismfromC(X)⋊ ΓtoC(X˜)suchthatφ| =id .
r C(X) C(X)
There is a Γ-morphism Φ from C(X˜)⋊ Γ onto C(X˜) extending φ. Then Φ is
r
a conditional expectation since C(X) ⊂ C(X˜) is rigid. Hence (ii) is equivalent
to the uniqueness of conditional expectations from C(X˜)⋊ Γ onto C(X˜), that is
r
equivalentto the (topological)freeness of X˜ by Proposition3.3 andLemma 4.1. It
follows that (i) and (ii) are equivalent by Theorem 3.4. (cid:3)
5. Stabilizer subgroups and the intersection property
In this section, we establish a characterization of the intersection property in
terms of stabilizer subgroups.
Definition 5.1. The Chabauty space of Γ is the set Sub(Γ) of all subgroups in Γ
with the relative topology of the product topology on {0,1}Γ.
Note that a sequence (Λ ) of subgroup in Γ converges to a subgroup Λ in the
i i
Chabauty topology if and only if it satisfies the following conditions.
(i) For every t∈Λ, one has t∈Λ eventually.
i
(ii) For every subsequence (Λ ) of (Λ ) , one has Λ ⊂Λ.
ik k i i k ik
Let X be a compact Γ-space. We set the compact Γ-space
T
S(X,Γ)={(x,Λ)∈X ×Sub(Γ):Λ≤Γ }
x
withtherelativetopologyoftheproducttopologyonX×Sub(Γ). Weconsiderthe
closed Γ-invariant subspace of S(X,Γ), defined by
S (X,Γ)={(x,Λ)∈X ×Sub(Γ):Λ≤Γ , Λ is amenable}.
a x
We denote by p the Γ-equivariant continuous surjection from S (X,Γ) to X de-
X a
fined by
p (x,Λ)=x
X
for (x,Λ)∈S (X,Γ), hence C(X)⊂C(S (X,Γ)) as a unital Γ-C∗-subalgebra.
a a
Theorem 5.2. Let X be a compact Γ-space. Then the following are equivalent.
(i) The space X has the intersection property.
(ii) For every closed Γ-invariant set Y in S (X,Γ) such that p (Y)=X, the
a X
space Y contains X ×{e}.
8 TAKUYAKAWABE
Proof. Suppose that X does not have the intersection property. We denote by q
the Γ-equivariant continuous surjection from X˜ to X. We define a Γ-equivariant
continuous map Φ from X˜ to S (X,Γ) by Φ(y) = (q(y),Γ ) for y ∈ X˜. We claim
a y
that Φ(X˜) 6⊃ X ×{e}, which means that (ii) is not true. Otherwise, the closed
Γ-invariantset Z :={y ∈X˜ :Γ ={e}}satisfies that q(Z)=X˜, therefore we have
y
Z = X by Proposition 3.3. Since X does not have the intersection property, the
space X˜ is not free by Theorem 3.4, a contradiction.
On the other hand, let Y be a closed Γ-invariant set in S (X,Γ) such that
a
p (Y)=X and Y 6⊃X ×{e}. There is a Γ-morphism θ from C(X)⋊ Γ to C(Y),
X r
defined by
f(x) t∈Λ
θ(fλ )(x,Λ)= .
t
(0 t6∈Λ
There is a Γ-morphism µ from C(Y) to C(X˜) which is the identity map on C(X)
since C(X˜) is Γ-injective. We show that µ◦θ 6= E . Suppose that µ◦θ = E .
X X
We claim that for every x∈X, one has p−1(x)∩Y ={x}×Y for a Y ⊂Sub(Γ).
X x x
Since Y 6⊃ X ×{e}, there is a point x in X such that Y 6∋ {e}. Let x˜ be a point
x
in X˜ such that q(x˜) = x. We observe that the support of ev ◦µ is contained
x˜
in {x}×Y . Indeed, let f be a function in C(Y) such that f| = 0. For
x {x}×Yx
everyopenneighborhoodU ofx, wetakea continuousfunctionh onX suchthat
U
0 ≤ h ≤ 1, the support of h contained in U and h (x) = 1. We denote by h˜
U U U U
the function h ◦q. For every ε > 0, there is a neighborhood U of x such that
U
|f(y)|<ε for every y ∈p−1(U)∩Y, hence we obtain
X
|ev ◦µ(f)|=|ev ◦µ(f)·h˜ (x˜)|=|ev ◦µ(fh˜ )|≤kfh˜ k≤ε.
x˜ x˜ U x˜ U U
Therefore ev ◦µ(f)=0, which implies that supp(ev ◦µ)⊂{x}×Y . We denote
x˜ x˜ x
by µ the Radon probability measure on Y which is the restriction of ev ◦µ on
x x x˜
Y . Then, for any t∈Γ, we obtain the following equation.
x
µ {Λ∈Y :t∈Λ} = µ (θ(λ )| )
x x x t Yx
= ev ◦µ◦θ(λ )
x˜ t
1 (t=e)
= .
(0 (t6=e)
It contradicts that Y 6∋{e}. (cid:3)
x
Theorem 5.3. Let X be a compact Γ-space. The following are equivalent.
(i) Every Γ-invariant closed set in X has the intersection property.
(ii) For every point x in X and every amenable subgroup Λ in Γ , there is a
x
net (g ) in Γ such that (g x) converges to x and (g Λg−1) converges to {e}
i i i i
in the Chabauty topology.
Proof. Suppose that (ii) is true. Let Z be a Γ-invariant closed subset of X. Then,
for any (z,Λ) ∈ S(Z,Γ), we have Γ(z,Λ) ∋ (z,{e}). This implies that for every
Γ-invariant closed subset Y in S (Z,Γ), one has Z×{e}⊂Y, therefore Z has the
a
intersection property by Theorem 5.2.
Conversly,suppose that(i)istrue. Letxbe apointinX andΛ be anamenable
subgroup in Γ . Then, we have Γ(x,Λ) ⊃ p Γ(x,Λ) ×{e} by Theorem 5.2.
x X
(cid:16) (cid:17)
UNIFORMLYRECURRENTSUBGROUPSANDTHEIDEALSTRUCTURE 9
In particular, there is a net (g ) in such that (g x) converges to x and (g Λg−1)
i i i i
converges to {e}. (cid:3)
6. Minimal case
We consider the case where the compact Γ-space X is minimal, i.e. there are no
non-trivialclosedΓ-invariantsubspacesinX. Equivalently,therearenonon-trivial
Γ-invariantclosed ideals in C(X). For a minimal compactΓ-space X, the space X˜
is also minimal by Proposition 3.3 (ii).
We claim that for a minimal compact Γ-space X, the reduced crossed product
C(X)⋊ Γ is simple if and only if X has the intersection property. since for every
r
ideal I in C(X)⋊ Γ, C(X)∩I is Γ-invariant ideal.
r
Theorem 6.1. Let X bea minimal compact Γ-space. The following areequivalent.
(i) The reduced crossed product C(X)⋊ Γ is simple.
r
(ii) For every point x in X and every amenable subgroup Λ in Γ , there is a
x
sequence (g ) in Γ such that (g Λg−1) converges to {e} in the Chabauty
i i i
topology.
(iii) There is a point x in X such that for every amenable subgroup Λ in Γ ,
x
there is a sequence (g ) in Γ such that (g Λg−1) converges to {e} in the
i i i
Chabauty topology.
Proof. We show that (ii) and (iii) are equivalent to existence of a minimal Γ-
invariantsubspace Y in S (X,Γ) such that Y 6=X×{e}. This implies the desired
a
equivalence by Theorem 5.2. Note that for every Γ-invariant subspace Z of X, we
have p (Z)=X since X is minimal.
X
Suppose that there is a minimal Γ-invariant subspace Y in S (X,Γ) such that
a
X×{e}6=Y. Let (x,Λ) be an element in Y. We claim that Ad(Γ)Λ6∋{e}, which
means that (i) and (ii) are not true. Otherwise, there is a net (g ) in Γ such that
i
g Λg−1 → {e}. We may assume that g x → y for a point y ∈ X. Then we have
i i i
g (x,λ)→(y,{e}), this implies that Y ⊃X×{e}. By minimality of Y, we obtain
i
Y =X ×{e}, a contradiction.
Next,weshowtheconverse. Supposethatthereisanelemant(x,Λ)inS (X,Γ)
a
such that Ad(Γ)Λ 6∋ {e}. Then we have Γ(x,Λ)∩X ×{e} = ∅, hence there is a
minimalΓ-invariantsubspaceY inS (X,Γ)suchthatY 6=X×{e}(takeaminimal
a
component of Γ(x,Λ)). (cid:3)
We also characterize simplicity for reduced crossed products in terms uniformly
recurrent subgroups (Glasner–Weiss [3]).
Definition 6.2. A subset U of Sub(Γ) is called a uniformly recurrent subgroup
(URS) of Γ if U is a minimal closed subset of the Chabauty space Sub(Γ). A URS
U is amenable if any subgroup contained in U is amenable.
Definition 6.3. For a compact Γ-space X, we define the subspace S of Sub(Γ)
X
as the closure of the set {Γ :x∈X, Γ =Γ◦}. We call S by stability system of
x x x X
X. If X is minimal, the set S is a URS ([3, Proposition 1.4]).
X
For a normal subgroup N in Γ, the singleton {N} in Sub(Γ) is a URS. By [9,
Theorem 4.1], C∗-simplicity of Γ is equivalent to absence of non-trivial amenable
URS’s. There is a non-C∗-simple countable group which has no non-trivialnormal
10 TAKUYAKAWABE
amenable subgroup, e.g. given by Le Boudec [10]. This implies that there is a
countable group which admits a non-singleton amenable URS.
We define an partial order of the set of uniformly recurrent subgroups. Let U
and V be URS’s, we denote U 4 V if there exist H ∈ U and K ∈ V such that
H ≤ K. Note that U 4V if and only if for every H ∈U, there exists K ∈V such
that H ≤K (see [11, Propostiton 2.14]).
Corollary6.4. LetX beaminimalcompactΓ-space. Thefollowing areequivalent.
(i) The reduced crossed product C(X)⋊ Γ is simple.
r
(ii) For any non-trivial amenable URS U, one has U 64S .
X
Proof. IfC(X)⋊ Γisnotsimple,X˜ isnottopologicallyfreebyTheorem3.4. Hence
r
S is a non-trivial amenable URS and S 4S .
X˜ X˜ X
We showthe converse. SupposethatC(X)⋊ Γ issimple. LetU be anamenable
r
URS such that U 4 S . For Λ ∈ U, there is a point x ∈ X such that Λ ≤ Γ ,
X x
this implies that Ad(Γ)Λ ∋ {e} by Theorem 6.1. Therefore we have U = {e} by
minimality of U. (cid:3)
7. Strongly proximallity and amenable URS’s
Definition 7.1. Let X be a compact Γ-space. X is strongly proximal if for every
RadonprobabilitymeasureµonX,the weak∗-closureofΓµcontainsapointmass.
X is called a Γ-boundary if X is minimal and strongly proximal.
ItisknownthattheHamanaboundary∂ ΓisaΓ-boundary(Kalantar–Kennedy
H
[7]). In this section, we proof an analogousproperty for every compact Γ-space X.
We denote by X˜ the inverse image of a point z ∈ X under the Γ-equivariant
z
continuoussurjectionfromX˜ toX. ForanycompactHausdorffspaceY,wedenote
by M(Y) the set of all Radon probability measures on Y.
Theorem 7.2. Let X be a compact Γ-space and Z be a subset in X such that ΓZ
is dense in X. Then for every family {µ } such that µ ∈M(X˜ ), the space X˜
z z∈Z z z
is contained in the weak∗-closure of {tµ :t∈Γ, z ∈Z} in M(X˜).
z
Proof. We define Γ-morphism φ from C(X˜) to ℓ (Γ×Z) by
∞
φ(f)(t,z)=hf,tµ i, t∈Γ, z ∈Z.
z
Observe that for f ∈ C(X) one has φ(f) = (f(tz)) . Since ΓZ is dense in X,
t,z
φ| is the inclusion map from C(X) to ℓ (Γ×Z) as a unital Γ-C∗-subalgebra.
C(X) ∞
Hence by Γ-injectivity of C(X˜), there is a Γ-morphismψ from ℓ (Γ×Z)to C(X˜)
∞
which satisfies that (ψ ◦φ)| = id (then ψ ◦φ = id by rigidity). It
C(X) C(X) C(X˜)
implies that for any y ∈X˜, there is a state ω on ℓ (Γ×Z) such that ω◦φ=ev .
∞ y
Since thereis anet (ξ )inℓ (Γ×Z)suchthatξ →ω inweak∗-topology,itimplies
i 1 i
that
ξ ◦φ(f)= ξ (t,z)hf,tµ i→f(y)
i i z
t,z
X
for any f ∈ C(X˜). It means that ev ∈ conv{tµ }, therefore we have ev ∈ {tµ }
y z y z
by Milman’s converse since ev is an extreme point of M(X˜). (cid:3)
y
Next, we consider some applications to properties for amenable URS’s, inspired
from Le Boudec–Matte Bon [11, §2.4].