The class S as an ME invariant

0.15 MB·

by  Hiroki Sako

THE CLASS S AS AN ME INVARIANT
HIROKI SAKO
9 Abstract. We prove that being in Ozawa’s class S is a measure equivalence
0 invariant.
0
2

n
a
J 1. introduction

5
2 Measurable group theory is a new field and has been attracting many researchers
having various backgrounds. The discipline deals with how much information on

]
R countable groups is preserved through measure equivalence. The notion of measure
G equivalence was given by Gromov [Gr] as a variant of quasi-isometry. Two groups
. are said to be measure equivalent (ME) if there exists an ME coupling, instead of
h
t topological coupling.
a
In most cases, much information is lost through ME couplings. For example, any
m
two countable amenable groups are ME. This is a consequence of [OrWe], [CoFeWe].
[
Many people areinterested in finding small measure equivalence classes (higher rank
2
lattices [Fu1], mapping class groups with high complexity [Kid]) or in classifying
v
4 non-amenable groups up to measure equivalence.
7
We will prove that Ozawa’s class S defined in [Oz3] is an ME invariant class. The
3
3 class was defined by means of topological amenability [AD] on the largest boundary.
1. Ozawa and Popa proved classification results on group von Neumann algebras of the
0 class S ([Oz2], [OzPo1]).
9
0
Definition 1.1 ([Oz3]). A countable group G is said to be in S if the left-times-right
:
v translation action of G×G on βG∩Gc is amenable, where βG∩Gc is the Gelfand
i
X spectrum of the commutative C∗-algebra ℓ G/c G.
∞ 0
r
a The following is the main theorem of this paper.
Theorem 1.2. If G and Γ are ME and if Γ ∈ S, then G ∈ S.
The class S is an intermediate class between the set of exact groups and that of
amenable groups, which are also ME invariant classes. These three classes are also
characterized by topological amenability. A countable group is exact if and only if
there exists an amenable action on a compact space [Oz1]. A countable group is
amenable if and only if any continuous action on any compact space is amenable.
ByHjorth’stheorem[Hj],AcountablegroupGistreeableinthesense ofPemantle
and Peres [PemPer], if and only if G is ME to a free group (Z,F or F ). As
2 ∞
a corollary of Theorem 1.2, the class of treeable groups is an intermediate ME
invariant class between S and the set of amenable groups. We get the following fact
on group von Neumann algebras:
Key words and phrases. measurable group theory; solid von Neumann algebra.
1
2 HIROKISAKO
Corollary 1.3. If G is ME to a free group, then the group von Neumann algebra
L(G) is solid, namely, every diffuse subalgebra has the injective relative commutant.
Since the free groups are in the class S, if G is ME to a free group, then Γ ∈ S.
By [Oz2], L(G) is solid.
2. Measure Equivalence and Measure Embedding
The following is a generalization of Gromov’s measure equivalence (0.5.E in [Gr]).
Definition 2.1. Let G and Γ be countable groups. We say that the group G mea-
surably embeds into Γ, if there exist a standard measure space (Σ,ν), a measure
preserving action of G×Γ on Σ and measurable subsets X,Y ⊂ Σ with the follow-
ing properties:
Σ = γ(X) = g(Y), ν(X) < ∞.
γ∈Γ g∈G
G G
Then we use the notation G (cid:22) Γ. The infinite measure space Σ equipped with the
ME
G ×Γ-action is called a measure embedding of G into Γ. The measure embedding
Σ is said to be ergodic, if the G×Γ-action is ergodic.
If the subset Y also has finite measure, then Σ gives an ME coupling between G
and Γ and these groups are said to be measure equivalent (ME). As in the case of
an ME coupling, if there exists a measure embedding of G into Γ, there exists an
ergodic one by using ergodic decomposition. See Lemma 2.2 in Furman [Fu1] for
the proof.
Remark 2.2. (1) The relation (cid:22) is transitive; if H (cid:22) Λ and Λ (cid:22) Γ, then
ME ME ME
H (cid:22) Γ. The proof is same as that of transitivity of measure equivalence.
ME
(2) If countable groups G and Γ satisfy G (cid:22) Γ and if Γ is amenable, then G
ME
is also amenable. Exactness has the same property (see Remark 3.5).
3. The class S as an ME Invariant
Theorem 3.1 is stronger than Theorem 1.2 and follows from Proposition 3.2.
Theorem 3.1. If G (cid:22) Γ and Γ ∈ S, then G ∈ S.
ME
Proposition 3.2. Suppose Γ ∈ S. Let β be a free measure preserving (m.p.) action
of Γ on a standard measure space (Y,µ) and let α be a free m.p. action of G on a
measurable subset X ⊂ Y with measure 1. If their orbits satisfy α(G)(x) ⊂ β(Γ)(x)
for a.e. x ∈ X, then G ∈ S.
In this proposition, Y can be an infinite standard measure space.
Proof of Theorem 3.1 from Proposition 3.2. Suppose that (Σ,ν) gives an ergodic
measure embedding of G into Γ. Choose G fundamental domain Y and Γ fun-
∼
damental domain X. We can replace Σ so that the G-action on Γ\Σ = X is free,
by taking a product of Σ and a G-probability space on which free m.p. G-action is
given. Then we can consider Σ comes from a stable orbit equivalence with constant
s = ν(Y)/ν(X) ∈ (0,∞]. This argument is covered by Lemma 3.2 in [Fu2] and
Remark 2.14 in [MoSh] if s < ∞. We do the same argument in the case of s = ∞.
THE CLASS S AS AN ME INVARIANT 3
We note that Γ ∈ S if and only if Γ×Z/nZ ∈ S. Replacing Γ with Γ×Z/nZ,
if necessary, we get R(G y X)s ∼= R(Γ y Y) with constant 1 ≤ s, where R(G y
X),R(Γ y Y) mean the orbit equivalence relations of free m.p. actions of G and Γ.
By Proposition 3.2, Γ ∈ S implies G ∈ S. (cid:3)
To prove Proposition 3.2, we use the notation given as follows.
We introduce a measure ν on R as the push forward under the map
β
Y ×Γ ∋ (y,γ) 7→ (α(γ)(y),y) ∈ R ,
β
where the measure on Y ×Γ is given by the product of µ and the counting measure.
The measure ν coincides with the measure defined in Feldman and Moore [FeMoo].
Theactionβ (resp.α)givesagroupactionofΓ(resp.G)onL∞(Y)(resp.L∞(X)).
We use the same notation β (resp. α) for this action. Let p ∈ L∞(Y) be the
characteristic function of X. The algebra L∞(Y) and the group Γ are represented
on L2(R ,ν) as
β
(Fξ)(x,y) = F(x)ξ(x,y), F ∈ L∞(Y),
(u ξ)(x,y) = ξ(β(γ−1)(x),y), γ ∈ Γ,ξ ∈ L2(R ),(x,y) ∈ R .
γ β β
Let B be the C∗-algebra generated by the images, which is the reduced crossed
product algebra B = L∞(Y) ⋊ Γ. Its weak closure is the group measure space
red
construction M = L∞(Y)⋊Γ [MvN]. We denote by tr the canonical faithful normal
semi-finite trace on M with normalization tr(p) = 1. The unitary involution J of
(M,tr) is written as (Jξ)(x,y) = ξ(y,x), (ξ ∈ L2(R ), (x,y) ∈ R ).
β β
The group G is represented on pL2(R ) = L2(R ∩(X ×Y)) by
β β
(v ξ)(x,y) = ξ(α(g−1)(x),y), g ∈ G,ξ ∈ pL2(R ),(x,y) ∈ R ∩(X ×Y).
g β β
We denote by C∗(G) the C∗-algebra generated by these operators. The algebra
λ
is isomorphic to the reduced group C∗-algebra of G. The Hilbert space L2(R ,ν)
α
can be identified with a closed subspace of pL2(R ). The algebra C∗(G) is also
β λ
represented on L2(R ) faithfully. We denote by P the orthogonal projection from
α
L2(R ) onto L2(R ). We note that the algebra pBp does not contain C∗(G) in
β α λ
general, although there exists an inclusion between their weak closures.
Let e be the projection from L2(R ) onto the set of L2-functions supported on
∆ β
the diagonal subset of R . This is the Jones projection for L∞(Y) ⊂ M. For γ ∈ Γ
β
and a finite subset Γ ⊂ Γ, we define the projections e(γ),e(Γ ) by
0 0
e(γ) = Ju Je Ju∗J, e(Γ ) = e(γ).
γ ∆ γ 0
γX∈Γ0
For g ∈ G and a finite subset G ⊂ G, we define the projections f(g),f(G ) by
0 0
f(g) = v e v∗ = v pe v∗, f(G ) = f(g).
g ∆ g g ∆ g 0
gX∈G0
Let K ⊂ B(L2(R )) be the hereditary subalgebra of B(L2(R )) with approximate
β β
units {e(Γ ) | Γ ⊂ Γ finite}, that is,
0 0
k·k
K = e(Γ )B(L2(R ))e(Γ ) .
0 β 0
[Γ0
It is easy to see that B and JBJ are in the multiplier of K, so is D = C∗(B,JBJ).
4 HIROKISAKO
The algebra B satisfies the following continuity property, which is similar to
Proposition 4.2 in [Oz3].
Proposition 3.3. The ∗-homomorphism
Ψ: B ⊗C JBJ → (D +K)/K
given by Ψ(b⊗c) = bc+K is continuous with respect to the minimal tensor norm.
e
For the rest of this paper, ⊗ stands for the minimal tensor product between two
e
C∗-algebras.
Proof. Consider the representation of ℓ Γ on L2(R ) given by
∞ β
(m ξ)(γy,y) = φ(γ)ξ(γy,y), φ ∈ ℓ Γ,ξ ∈ L2(R ),γ ∈ Γ,y ∈ Y.
φ ∞ β
Let D be the C∗-algebra generated by D and m(ℓ Γ). The algebra D is also in
∞
the multiplier of K. Since preimage m−1(K ∩ image(m)) is c Γ, m also gives an
0
embeedding of ℓ Γ/c Γ into (D+K)/K. The embedding m and the reperesentation
∞ 0
of L∞Y give a ∗-homomorphism
Ψ: E = ℓ Γ/c Γe⊗L∞Y ⊗JL∞YJ −→ (D +K)/K.
∞ 0
Here, we used the fact that abelian C∗-algebras are nuclear ([Tak]). Consider the
e e
action of Γ×Γ on E given by
A(γ ,γ )((φ+c Γ)⊗f ⊗Jf J) = (l r φ+c Γ)⊗β(γ )(f )⊗Jβ(γ )(f )J,
1 2 0 1 2 γ λ 0 1 1 2 2
where l and r stand for the left and the right translations. Since ℓ Γ/c Γ is
· · ∞ 0
in the center of E, by [AD], the full crossed product coincides with the reduced
crossed product F = E⋊ (Γ×Γ) and this is nuclear. The unitary representations
red
u and JuJ give a ∗-homomorphism Ψ: F → (D + K)/K. By restricting Ψ on
· ·
(L∞Y ⊗JL∞YJ)⋊ (Γ×Γ), we get a map satisfying Proposition 3.3. (cid:3)
red
e e e
If the ∗-homomorphism Ψ: B ⊗C JBJ given by Ψ(b⊗c) = bc is continuous with
respect to the minimal tensor norm, then the group Γ is amenable. The above
proposition can be regarded as a weakened amenability property for the Γ-action.
We make use of the following characterization of S. Proposition 15.2.3 and a
variant of Lemma 15.1.4 in [BrOz] imply the following.
Proposition 3.4. A countable group G is in S if and only if G is exact and there
exists a contractive c.p. map Φ: C∗(G)⊗C∗(G) → B(ℓ G), satisfying
λ ρ 2
Φ(b⊗c)−bc ∈ K(ℓ G), b ∈ C∗(G),c ∈ C∗(G),
2 λ ρ
where C∗(G) and C∗(G) are the C∗-algebras generated by the left and right regular
λ ρ
representations, respectively.
Remark 3.5. By using the notion of weak exactness introduced in Kirchberg [Kir],
we get that the exactness on Γ implies that on G. Indeed, the algebra M is weakly
exact by the exactness of Γ. The subalgebra L(G) ⊂ pMp is also weakly exact.
Thus G is exact by Ozawa’s theorem [Oz4].
We have only to show the existence of Φ in Proposition 3.4. However, lack of
inclusion “C∗(G) ⊂ B” requires some technical elaboration.
λ
Lemma 3.6. There exists a sequence {q } of projections in L∞(X) satisfying:
n n=1,2,···
THE CLASS S AS AN ME INVARIANT 5
(1) The sequence {q } is increasing and strongly converges to p;
n
(2) For any finite subset G ⊂ G and n, there exists a finite subset Γ ⊂ Γ
0 0
satisfying q f(G ) ≤ e(Γ );
n 0 0
(3) For any finite subset Γ ⊂ Γ and n, there exists a finite subset G ⊂ G
0 0
satisfying q e(Γ )P ≤ f(G ).
n 0 0
We note that projections P,p,e(γ),f(g),(γ ∈ Γ,g ∈ G) are in the commutative
von Neumann algebra L∞(R ) ⊂ B(L2(R )). Every projection in L∞(R ) which
β β β
is less than f(g) (resp. pe(γ)) is of the form qf(g) (resp. qe(γ)) for some q ∈
L∞(X). We also note that qf(g) ≤ e(Γ ) if and only if there exists a partition
0
{q ∈ L∞(X) | γ ∈ Γ } of q such that qv = q u .
γ 0 g γ∈Γ0 γ γ
Proof. We fix an index on G; {g ,g ,...} = GP. For any g , the projection e(Γ )f(g )
1 2 k 0 k
can be written as Q(g ,Γ )f(g ) by some projection Q(g ,Γ ) ∈ L∞(X). The net of
k 0 k k 0
projections {e(Γ )f(g ) | Γ ⊂ Γ finite} strongly converges tof(g ). For any natural
0 k 0 k
number n,thereexistsafinitesubsetΓ ⊂ Γsuchthattr(Q(g ,Γ )) ≥ 1−2−(n+k).
k,n k k,n
Then the projections Q = ∞ Q(g ,Γ ) satisfy tr(Q ) ≥ 1−2−n and
n k=1 k k,n n
Q f(g ) ≤ Q(g ,Γ )f(g ) = e(Γ )f(g ) ≤ e(Γ ).
n k Vk k,n k k,n k k,n
Let {q } bethe increasing sequence of projections { n Q }. Then we have tr(q ) ≥
n l=1 l n
1−2−n and
W
n
q f(g ) ≤ e Γ , g ∈ G.
n k k,l k
!
l=1
[
It turned out that the sequence {q } satisfies (1) and (2).
n
By a similar technique, we get a sequence {p } with (1) and (3), since
n
strlimf(G )e(γ) = e(γ)P, γ ∈ Γ.
0
G0
Taking products {p q }, we get a sequence which satisfies (1), (2) and (3) at the
n n
(cid:3)
same time.
The Hilbert space ℓ G embeds into L2(R ) by the isometry
2 α
ℓ G ∋ δ 7→ v ξ = Jv∗Jξ ∈ L2(R ),
2 g g ∆ g ∆ α
where the L2-function ξ is the characteristic function of the diagonal subset of R .
∆ α
We regard ℓ G as a subspace of L2(R ) by this map. The subspace ℓ G is invariant
2 α 2
under the action of C∗G and JC∗GJ.
λ λ
Lemma 3.7. For a projection q ∈ L∞(X), the following inequality on operator
norm holds true: k(1−qJqJ)| k ≤ (2−2tr(q))1/2.
ℓ2G
Proof. It suffices to show kη −qJqJηk2 ≤ (2−2tr(q))kηk2 for any vector η ∈ ℓ G.
2
Since qJqJP = PqJqJ and η = Pη, we get
kη−qJqJηk2 = kf(g)(η−qJqJη)k2 = kf(g)η−qJqJf(g)ηk2,
g∈G g∈G
X X
kηk2 = kf(g)ηk2,
g∈G
X
The claim reduces to the inequality kf(g)η−qJqJf(g)ηk2 ≤ (2−2tr(q))kf(g)ηk2.
6 HIROKISAKO
Wenotethatη takes aconstant valueη(g)ontheset {(α(g)(x),x) ∈ R |x ∈ X}.
α
By a direct computation, we get
kf(g)ηk2 = |(f(g)η)(y,x)|2dν = |η(α(g)(x),x)|2dµ = |η(g)|2.
ZRα Zx∈X
Let X ⊂ X be a measurable subset such that χ(X ) = p−q. The measure of subset
q q
X = {x ∈ X | x ∈ X or α(g)(x) ∈ X } satisfies ν(X ) ≤ 2−2tr(q). Then we get
0 q q 0
kf(g)η−qJqJf(g)ηk2 = |η(α(g)(x),x)|2dµ = µ(X )|η(g)|2.
0
Zx∈X0
(cid:3)
Our claim was confirmed.
We finish the proof of Proposition 3.2.
Proposition 3.2. We will show the existence of Φ in Proposition 3.4. We consider
that C∗(G) is a subalgebra of B(pL2(R )) and C∗(G) is JC∗(G)J. By P we denote
λ β ρ λ 0
the orthogonal projection from L2(R ) onto ℓ G.
β 2
Let {q } be the sequence satisfying Lemma 3.6. and let B be the C∗-algebra
n 0
generated by p and q C∗(G)q . The condition (2) in Lemma 3.6 means B ⊂ B.
n n λ n 0
We recall that B ⊗ JB J is separable and that F in the proof of Proposition
0 0
S
3.3 is nuclear. By Choi–Effros lifting theorem [ChEf], there exists a contractive
c.p. lifting Ψ: B ⊗JB J → D+K for Ψ| . We define contractive c.p. maps
0 0 B0⊗JB0J
Φ : C∗(G)⊗C∗(G) → B(ℓ G) by
n λ ρ 2
e
Φ (b⊗JcJ) = P Q Ψ(q bq ⊗Jq cq J)Q P ,
n 0 n n n n n n 0
where Q = q Jq J. By the condition (3) in Lemma 3.6, we get P Q KQ P ⊂
n n n 0 n n 0
K(ℓ G). The element Φ (b⊗JcJ) is in
2 n
P Q (bJcJ +K)Q P ⊂ P Q bJcJQ P +K(ℓ G).
0 n n 0 0 n n 0 2
Thesequence{P Q bJcJQ P +K(ℓ G)} ⊂ B(ℓ G)/K(ℓ G)convergestoP bJcJP +
0 n n 0 2 2 2 0 0
K(ℓ G), by the inequality
2
kP bJcJP −P Q bJcJQ P k
0 0 0 n n 0
≤ kP (1−Q )bJcJP k+kP Q bJcJ(1−Q )P k
0 n 0 0 n n 0
≤ 2k(1−Q )P kkbkkck
n 0
≤ 2(2−2tr(q ))1/2kbkkck.
n
Itfollowsthatthenatural∗-homomorphismfromtheminimaltensorproductΦ: C∗(G)⊗
λ
C∗(G) → B(ℓ G)/K(ℓ G) is given and Φ is a limit of liftable maps. By Theorem 6
ρ 2 2
e (cid:3)
of [Ar], there exists a contractive c.p. lifting Φ for Φ.
e
4. Final remarke
As a consequence of Proposition 3.2, we get the following indecomposability of an
equivalence relation given by a class S group.
Corollary 4.1. Let Γ be a countable group and let H ⊂ G be an inclusion of count-
able groups. Suppose that Γ ∈ S and that the centralizer Z (H) is non-amenable.
G
Let β be a free m.p. action of Γ on a standard measure space (Y,µ) and let α be a
THE CLASS S AS AN ME INVARIANT 7
free m.p. action of G on a measurable subset X ⊂ Y with measure 1. If the orbits
satisfy α(G)(x) ⊂ β(Γ)(x) for a.e. x ∈ X, then H is finite.
Proof. TheclassS hasthefollowingproperty: IfG ∈ S andZ (H)isnon-amenable,
G
then H is finite. (cid:3)
In particular, the group G is not a direct product group of an infinite group
and a non-amenable group. Word hyperbolic groups are typical examples of S
groups. Adams [Ad] showed a measurable indecomposability of non-amenable word
hyperbolic groups. The above corollary covers some part of Adams’ theorem.
The class C of [MoSh] also contains non-amenable word-hyperbolic groups. A
group G ∈ C satisfies an indecomposability property; G has no infinite normal
amenable subgroup. On the other hand, a non-amenable class S group can have an
infinite normal amenable subgroup (for example, Z2 ⋊SL(2,Z) ∈ S [Oz5]).
Acknowledgment . This paper was written during the author’s stay in UCLA. The
author is grateful to Professor Sorin Popa and Professor Narutaka Ozawa for their
encouragement and fruitful conversations. He is supported by JSPS Research Fel-
lowships for Young Scientists.
References
[Ad] S. Adams, Indecomposability of equivalence relations generated by word hyperbolic groups,
Topology 33 (1994), no. 4, 785–798.
[AD] C. Anantharaman-Delaroche, Syst`emes dynamiques non commutatifs et moyennabilit´e,
Math. Ann. 279 (1987), no. 2, 297–315.
[Ar] W. Arveson, Notes on extensions of C∗-algebras, Duke Math. J. 44 (1977), no. 2, 329–355.
[BrOz] N.P. Brown and N. Ozawa, C∗-algebras and finite-dimensional approximations, Graduate
Studies in Mathematics, 88. American Mathematical Society, Providence, RI, 2008.
[ChEf] M.D. Choi, E.D. Effros, The completely positive lifting problem for C∗-algebras, Ann. of
Math. (2) 104 (1976), no. 3, 585–609.
[CoFeWe] A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by
a single transformation, ErgodicTheory Dynamical Systems 1 (1981),no. 4, 431–450(1982).
[FeMoo] J.FeldmanandC.C.Moore,Ergodicequivalencerelations,cohomology, andvonNeumann
algebras. II, Trans. Amer. Math. Soc. 234 (1977), no. 2, 325–359.
[Fu1] A. Furman, Gromov’s measure equivalence and rigidity of higher rank lattices, Ann. of
Math. (2) 150 (1999), no. 3, 1059–1081.
[Fu2] A. Furman, Orbit equivalence rigidity, Ann. of Math. (2) 150 (1999), no. 3, 1059–1081.
[Gr] M. Gromov,Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2, 1–295,
London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993.
[Hj] G. Hjorth, A lemma for cost attained, Ann. Pure Appl. Logic 143 (2006), no. 1-3, 87–102.
[Kid] Y.Kida,Measureequivalence rigidity of themapping class group,toappearinAnn.ofMath.
[Kir] E. Kirchberg, Commutants of unitaries in UHF algebras and functorial properties of exact-
ness, J. Reine Angew. Math. 452 (1994), 39–77.
[MoSh] N. Monod and Y. Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. of
Math. (2) 164 (2006), no. 3, 825–878.
[MvN] F.J.MurrayandJ.vonNeumann,On rings of operators. IV,Ann.ofMath.(2) 44(1936),
116-229
[OrWe] D. S. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin
lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164.
[Oz1] N. Ozawa,Amenable actions and exactness for discrete groups, C.R. Acad.Sci.ParisS´er.I
Math. 330 (2000), no. 8, 691–695.
[Oz2] N. Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), no. 1, 111–117.
8 HIROKISAKO
[Oz3] N. Ozawa,A Kurosh-type theorem for type II1 factors, Int. Math. Res. Not. (2006), Art. ID
97560,21 pages, DOI 10.1155/IMRN/2006/97560.
[Oz4] N. Ozawa, Weakly exact von Neumann algebras, J. Math. Soc. Japan 59 (2007), no. 4,
985–991.
[Oz5] N. Ozawa, An example of a solid von Neumann algebra, preprint, arXiv:0804.0288.
[OzPo1] N. Ozawa and S. Popa, Some prime factorization results for type II1 factors, In-
vent. Math. 156 (2004), no. 2, 223–234.
[PemPer] R. Pemantle and Y. Peres, Nonamenable products are not treeable, Israel J. Math. 118
(2000), 147–155.
[Tak] M. Takesaki, On the cross-norm of the direct product of C∗-algebras, Tohoku Math. J. (2)
16 (1964), 111–122.
Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo,
153-8914, Japan.
E-mail address: [email protected]