Stable decompositions and rigidity for products of countable equivalence relations
aa r X i v : . [ m a t h . D S ] F e b STABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCTS OFCOUNTABLE EQUIVALENCE RELATIONS
PIETER SPAAS
Abstract.
We show that the “stabilization” of any countable ergodic p.m.p. equivalence relationwhich is not Schmidt, i.e. admits no central sequences in its full group, always gives rise to a stableequivalence relation with a unique stable decomposition, providing the first non-strongly ergodicsuch examples. In the proof, we moreover establish a new local characterization of the Schmidtproperty. We also prove some new structural results for product equivalence relations and orbitequivalence relations of diagonal product actions. Introduction and statement of the main results
Let (
X, µ ) be a standard probability space, and denote by Aut(
X, µ ) its automorphism group,i.e. the group of Borel automorphisms of X which preserve µ (and where we identify two suchautomorphisms if they agree µ -almost everywhere). Given a countable discrete group Γ, a proba-bility measure preserving (p.m.p.) action of Γ on ( X, µ ) is a homomorphism from Γ to Aut(
X, µ ).Every such action generates an orbit equivalence relation R := R (Γ y X ) ⊂ X × X , by letting( x, y ) ∈ R if and only if there exists g ∈ Γ such that y = gx . In this case R is a countable p.m.p.equivalence relation, i.e. every R -class is countable, and every Borel automorphism ψ of X forwhich ( ψ ( x ) , x ) ∈ R for almost every x ∈ X is measure-preserving. In fact, every countable p.m.p.equivalence relation arises this way (see [FM77]).Let R hyp denote the (unique up to isomorphism) hyperfinite ergodic p.m.p. equivalence relation.One of the main goals of this paper concerns the following general question: given countable ergodicp.m.p. equivalence relations R and S , when are their “stabilizations” R × R hyp and
S × R hyp isomorphic? To put this into context, recall from [JS85] that a countable ergodic p.m.p. equivalencerelation R is called stable if R admits a decomposition R ∼ = S × R hyp for some equivalence relation S . In recent years, there has been a lot of interest in the study of stable equivalence relations,see for instance [Ki12, TD14, Ki16, Ma17, IS18]. In [JS85], Jones and Schmidt also establish acharacterization of stability in terms of central sequences, providing a criterion to check when adecomposition as above exists. Below, we will be interested in the complementary question of whensuch a decomposition is unique.Following the terminology of [IS18], we say that a countable ergodic p.m.p. equivalence relation R admits a stable decomposition if R can be written as R = S × R hyp for some non-stable equivalencerelation S . We say R admits a unique stable decomposition if for any other stable decomposition R = T × R hyp , we necessarily have that S and T are stably isomorphic (see Section 2.1 for thenotion of stable isomorphism). Note that if S and T are stably isomorphic, we always have that S × R hyp ∼ = T × R hyp .Motivated by the breakthrough result of Popa in [Po06, Theorem 5.1], where he shows uniquenessof the McDuff decomposition of N ¯ ⊗ R when N is a non-Gamma II factor and R is the hyperfiniteII factor, the first unique stable decomposition result was proven in [IS18, Theorem G]. There itis shown that R × R hyp has a unique stable decomposition whenever R is strongly ergodic , i.e. hasno asymptotically invariant sequences in the measure space (see Section 2.1). In this paper, we willinstead study this problem in the absence of asymptotically central sequences in the full group [ R ]of R , i.e. when R is not Schmidt in the following sense. Definition 1.1 (cf. [Sc87, KTD18]) . Let R be a countable ergodic p.m.p. equivalence relationon a standard probability space ( X, µ ). We say R is Schmidt if there exists a non-trivial centralsequence in the full group [ R ], i.e. a sequence ( T n ) n ∈ [ R ] satisfying:(1) (central) ∀ S ∈ [ R ] : µ ( { x ∈ X | T n S ( x ) = ST n ( x ) } ) →
0, and(2) (non-trivial) lim inf n µ ( { x ∈ X | T n ( x ) = x } ) > R × R hyp when R is not Schmidt. More precisely, we have: Theorem A.
Suppose R is a countable ergodic non-Schmidt p.m.p. equivalence relation on astandard probability space, and let R hyp be the hyperfinite ergodic p.m.p. equivalence relation. If R × R hyp ∼ = R × R hyp for some countable ergodic p.m.p. equivalence relation R , then either(1) R is also not Schmidt, and R is stably isomorphic to R , or(2) R is stable, and thus R ∼ = R × R hyp ∼ = R × R hyp . It is known that whenever a free ergodic p.m.p. action of a countable group Γ is Schmidt in theabove sense, then Γ is necessarily inner amenable. Therefore, the above theorem applies for instancewhenever R is generated by a free ergodic p.m.p. action of any non-inner amenable group. Sinceevery non-inner amenable group without property (T), for instance any nontrivial free productgroup, admits tons of free ergodic non-strongly ergodic p.m.p. actions, Theorem A provides a largeclass of new examples of equivalence relations with a unique stable decomposition. We also notehere that the converse of the above observation is open, namely whether every inner amenablegroup admits a free ergodic p.m.p. action whose associated orbit equivalence relation is Schmidt. Remark 1.2.
Together with [IS18, Theorem G], Theorem A thus shows that equivalence rela-tions which either lack central sequences in the measure space (i.e. strongly ergodic ones) or lackcentral sequences in the full group (i.e. non-Schmidt ones), admit unique stable decompositions.In other words, some “control” over the central sequences allows for unique decomposition re-sults. Even though they are technically much more involved, both these results can therefore beseen as analogues for equivalence relations of the aforementioned unique McDuff decompositionresult [Po06, Theorem 5.1]. Also, they can be compared to further results about uniqueness ofMcDuff decompositions of von Neumann algebras as in [IS18, Theorems B, D].There are a few ingredients that go into the proof of Theorem A. Firstly, we prove a new localcharacterization of the Schmidt property in Lemma 4.1, which leads to a spectral gap criterionfor non-Schmidt equivalence relations. We will use this to characterize in Lemma 4.3 the centralsequences in the full group of
R × R hyp when R is not Schmidt. Secondly, we will use thischaracterization to “detect” from R×R hyp the absence of the Schmidt property for R (Lemma 4.5),and to establish certain intertwining results for equivalence relations of this type (Lemma 4.6). Acombination of these ingredients and Theorem B(2) below, will then lead to the desired conclusion.Theorem B will arise in the general framework of intertwining techniques for two subequivalencerelations S , T of a countable p.m.p. equivalence relation R , introduced in [Io11] as an analogue ofPopa’s intertwining-by-bimodules (see Section 2.4), and further investigated and used in [DHI16].We will further study and develop these techniques in Section 3. Intuitively, T intertwines into S if—upon restricting to appropriate subsets of X —there exist a subequivalence relation of bounded index(see Section 2.5) T ≤ T and an element θ ∈ [[ R ]] such that ( θ × θ )( T ) is a subequivalence relationof S . By analogy with Popa’s intertwining-by-bimodules in the von Neumann algebra setting, Ioanacharacterized this intertwining in [Io11, Lemma 1.7] using the function ϕ S : [[ R ]] → [0 ,
1] definedby ϕ S ( θ ) = µ ( { x ∈ dom( θ ) | ( θ ( x ) , x ) ∈ S} ) . Our next main Theorem shows that for product equivalence relations, the presence of intertwiningleads to certain rigidity phenomena. We refer to Propositions 3.6, 3.11 and Corollary 3.7 for theproofs and slightly more general statements, and to Definition 3.2 for the notation.
TABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCT EQUIVALENCE RELATIONS 3
Theorem B.
Let R , S , R , S be countable ergodic p.m.p. equivalence relations on standardprobability spaces such that R := R × S ∼ = R × S .(1) If, as subequivalence relations of R , we have R ≺ R R , then there exists an equivalencerelation T such that R is stably isomorphic to R × T .(2) If also R ≺ R R , then R is stably isomorphic to R , and S is stably isomorphic to S . The proofs of these results are inspired by the proof of the aforementioned [IS18, Theorem G],where it is shown that
R × R hyp admits a unique stable decomposition when R is strongly ergodic.In the language of this paper, the proof essentially shows that the strong ergodicity of R impliesthe intertwining condition in Theorem B(1).Next, we turn our attention to equivalence relations arising as orbit equivalence relations of diagonalproduct actions of countable discrete groups. On top of the product equivalence relations fromTheorem B, we will be able to apply the intertwining techniques in this setting as well. Morespecifically, we will study actions of the following type. Notation 1.3.
Let Γ , Γ , Σ , Σ be countable discrete groups, and ρ : Γ → Σ , ρ : Γ → Σ besurjective group homomorphisms. Denote Γ i := ker( ρ i ). Suppose Γ i y ( X i , µ i ) and Σ i y ( Y i , ν i )are free ergodic p.m.p. actions on standard probability spaces, and assume that also Γ i y ( X i , µ i )is ergodic for i = 1 ,
2. We then have the diagonal product actions Γ i y ( X i , µ i ) × ( Y i , ν i ) given by g · ( x, y ) = ( gx, ρ ( g ) y ), for all g ∈ Γ i , x ∈ X i , y ∈ Y i . For i = 1 ,
2, we will write R i := R (Γ i y X i ).Our last main result shows that, when taking such a product of a strongly ergodic action with anamenable action, we can “recover” parts of the strongly ergodic piece. Theorem C.
In the setting of Notation 1.3, assume the following extra conditions for i = 1 , (i) Γ i y ( X i , µ i ) is strongly ergodic,(ii) Σ i is amenable.If the actions Γ y ( X × Y , µ × ν ) and Γ y ( X × Y , µ × ν ) are orbit equivalent, then Γ y X and Γ y X are necessarily stably orbit equivalent. Remark 1.4.
This class of actions is also studied in [Io06a], where Ioana establishes several resultsthat are similar in spirit to Theorem C, see for instance [Io06a, Proposition 2.4, Proposition 3.3,and Theorem 4.1].We will end this introduction with several observations regarding the relation of orbit equivalenceon the space A (Γ , X, µ ) of p.m.p. actions of a fixed countable discrete group Γ on a standardprobability space ( X, µ ). We refer for instance to [Ke10] for a detailed study of this space. Buildingon earlier work by Ioana and Epstein ([Io06b, Ep07]), Epstein, Ioana, Kechris, and Tsankov showedin [IKT08] that for any non-amenable group Γ, free ergodic (in fact mixing) p.m.p. actions of Γ on(
X, µ ) are not classifiable by countable structures. Even though this settles the non-classifiabilityof actions of any non-amenable group up to orbit equivalence, it might be that certain naturalsubclasses of actions admit a classification. In particular, it is not clear from the co-inductionconstruction used to establish this result, whether the actions constructed in [IKT08] are in generalstrongly ergodic or not.Theorem C above can be used to show Theorem 1.5 below for a large class of groups, by associ-ating to any strongly ergodic action in a suitable “unclassifiable” family (see also Theorem 1.8),a non-strongly ergodic action via a diagonal product construction as in Notation 1.3. However,we observed that the more general statement of Theorem 1.5 below follows from a similar, albeiteasier, adaptation of the construction in [IKT08], without having to appeal to Theorem C above. Inparticular, the proofs of the remaining results in this introduction will largely be deduced from theexisting literature. Nevertheless, even though some of them may have been observed by experts,they seem not to have appeared in writing as of now, and so we collect them here.
P. SPAAS
Firstly, let Γ be any non-amenable group without property (T). Using the same proof as [IKT08,Theorem 3.12], one can establish the following result about the classification of non-strongly ergodic actions of Γ. Note that if Γ has property (T), this question would be void, since it is well-knownthat in this case all ergodic actions of Γ are strongly ergodic (see [Sc80]).
Theorem 1.5 ([IKT08]) . Let Γ be any non-amenable countable discrete group without property(T). Then free ergodic non-strongly ergodic p.m.p. actions of Γ up to orbit equivalence are notclassifiable by countable structures.Sketch of the proof. Fix any non-strongly ergodic action a of Γ, and consider the family of actions b ( π i ) constructed in [IKT08, Section 3]. It is then easy to check that the diagonal product actions b ( π i ) × a , which are obviously non-strongly ergodic, still satisfy the necessary rigidity propertiesin order to apply [IKT08, Lemma 3.13], and then the proof can be finished in exactly the sameway. (cid:3) Remark 1.6.
Using a co-induction construction similar to [IKT08], it was shown in [GL17] thatthe equivalence relations of (stable) orbit equivalence and (stable) von Neumann equivalence of freeergodic p.m.p. actions of a non-amenable group are not Borel. Using a similar diagonal productconstruction as above, one can show that this result also holds for free ergodic non-strongly ergodicactions of a non-property (T) group which contains a free group.In [ST09], it is shown that the isomorphism relation of II factors is not classifiable by countablestructures. The II factors constructed to establish this result arise from actions of the free group F . As observed in [Sa16], these actions are strongly ergodic, and thus, since F is not inneramenable, these II factors do not have property Gamma (cf. [Ch82]). Furthermore, using Popa’sunique McDuff decomposition result in the absence of property Gamma ([Po06, Theorem 5.1]), itis observed in [Sa16] that therefore McDuff II factors up to isomorphism are not classifiable bycountable structures either. Using Theorem 1.5, we can deduce that the same holds for II factorswith property Gamma that are not McDuff: Corollary 1.7.
The equivalence relation of isomorphism on the space of non-McDuff II factorswith property Gamma is not classifiable by countable structures.Proof. Consider for instance the free group F . By Theorem 1.5, non-strongly ergodic actions of F are not classifiable by countable structures. Now from [Si55] and the uniqueness of Cartansubalgebras [PV11, Theorem 1.2], it follows that the associated group measure space II factorsare not classifiable by countable structures either. Moreover, since the actions are non-stronglyergodic, these II factors have property Gamma. Finally, since F is not inner amenable, it followsfrom [Ch82] that they are not McDuff. (cid:3) Next, together with a recent result from Drimbe [Dr20, Proposition B], one can show that for manygroups, the actions constructed in [IKT08] can be made strongly ergodic.
Theorem 1.8 ([IKT08, Dr20]) . Let Γ be one of the following.(1) A group with property (T).(2) More generally, a group without the Haagerup property.(3) A group which contains a free subgroup which is not co-amenable in Γ (see, e.g., [Ey72]).Then strongly ergodic actions of Γ up to orbit equivalence are not classifiable by countable structures.Proof. If Γ has property (T), then every ergodic action is strongly ergodic, and more generally,if Γ does not have the Haagerup property, then every mixing action of Γ is strongly ergodic bya result of Jolissaint ([CCJJV, Chapter 2], see also [Ke10, Theorem 11.1]). In both cases, theresult thus follows from [IKT08, Theorem 5]. If Γ contains a non-co-amenable free group F n , thenfrom [Io06b, IKT08], we first of all get a family of actions of F n such that the family of associated TABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCT EQUIVALENCE RELATIONS 5 co-induced actions of Γ are not classifiable by countable structures. Using [Dr20, Proposition B]and the fact that F n is not co-amenable in Γ, it moreover follows that all these actions are stronglyergodic. (cid:3) Finally, together with [IS18, Theorem G] and Theorem A, we can deduce the following classificationresult for stable actions, where we say that an action is stable if its associated orbit equivalencerelation is stable.
Corollary 1.9.
Let Γ either be a group as in Theorem 1.8, or any non-inner amenable group, andlet Σ be an infinite amenable group. Then stable actions of Γ × Σ up to orbit equivalence are notclassifiable by countable structures.Proof. First, assume Γ is as in Theorem 1.8. Fix a free ergodic p.m.p. action a of Σ on somestandard probability space ( Y, ν ), and let ( b i ) i ∈ I be the family of pairwise non-orbit equivalentstrongly ergodic actions of Γ from Theorem 1.8. Using the rigidity result [BHI15, Lemma 7.4]instead of [IKT08, Lemma 3.13] in the proof, one can show that the actions b i are in fact pairwisenot stably orbit equivalent. We claim that then the b i × a are pairwise not orbit equivalent. Indeed,if b i × a would be orbit equivalent to b j × a for some i = j , it would follow from [IS18, Theorem G]that b i is stably orbit equivalent to b j , contradiction.Second, if Γ is not inner amenable, we can take any family of pairwise non-orbit equivalent freeergodic p.m.p. actions of Γ (for instance from [Ep07]) witnessing non-classifiability by countablestructures. Observing that these actions are necessarily not Schmidt, we can thus copy the firstpart of the proof using Theorem A instead of [IS18, Theorem G]. (cid:3) Organization of the paper.
Besides the Introduction, there are three other sections in thispaper. In Section 2 we collect some necessary preliminaries. In Section 3 we discuss intertwiningtechniques for equivalence relations, and we prove Theorems B and C. Finally in Section 4 wediscuss non-Schmidt equivalence relations and prove Theorem A.
Acknowledgments.
I am very grateful to both Andrew Marks and Adrian Ioana for severalstimulating discussions about results and topics in or related to this paper. I would also like tothank Adrian Ioana for several useful comments on an earlier draft of this paper.2.
Preliminaries
Equivalence relations.
Suppose (
X, µ ) is a standard probability space, and Γ y X is ap.m.p. action of a countable group Γ. Denote by R (Γ y X ) := { ( x, y ) ∈ X × X | ∃ g ∈ Γ : g · x = y } its orbit equivalence relation . By construction, this is a countable p.m.p. equivalence relation.Conversely, every countable p.m.p. equivalence relation is of this form, see [FM77].Given any countable p.m.p. equivalence relation R on ( X, µ ), we denote by [ x ] R the equivalenceclass of a point x ∈ X . We endow R ⊂ X × X with the (infinite) measure ¯ µ given by¯ µ ( A ) = Z X { y ∈ X | ( x, y ) ∈ A } dµ ( x ) , for every Borel set A ⊂ X × X , and where R ) of R is the set of all measure space automorphisms α of ( X, µ ) such that( α ( x ) , α ( y )) ∈ R for ¯ µ -almost every ( x, y ) ∈ R . The full group [ R ] of R is the subgroup of all α ∈ Aut( R ) such that ( x, α ( x )) ∈ R for µ -almost every x ∈ X . The full pseudogroup [[ R ]] of R consists of all isomorphisms α : ( A, µ A ) → ( B, µ B ) satisfying ( x, α ( x )) ∈ R for almost every x ∈ A ,where µ A and µ B denote the restrictions of µ to the measurable subsets A, B ⊂ X .We call R ergodic if every measurable set A ⊂ X such that µ ( α ( A )∆ A ) = 0 for all α ∈ [ R ],satisfies µ ( A ) ∈ { , } . We say R is strongly ergodic if for every asymptotically invariant sequence P. SPAAS of measurable sets A n ⊆ X , i.e. satisfying lim n →∞ µ ( α ( A n )∆ A n ) = 0 for all α ∈ [ R ], we havelim n →∞ µ ( A n )(1 − µ ( A n )) = 0. Similarly, the action Γ y X is called ergodic, respectively stronglyergodic, if these statements hold with α ∈ [ R ] replaced by g ∈ Γ. We note that a p.m.p. actionΓ y ( X, µ ) is (strongly) ergodic if and only if its orbit equivalence relation R (Γ y X ) is (strongly)ergodic.Assuming R is ergodic, we can define the t -amplification R t of R where t > N and consider a measurable set X t ⊂ X × N with ( µ × X t ) = t . Then R t is defined as the equivalence relation on X t given by (( x, m ) , ( y, n )) ∈ R t if and only if( x, y ) ∈ R . Since R is ergodic, this defines R t uniquely up to isomorphism.We call two countable ergodic p.m.p. equivalence relations R and S stably isomorphic if there exists t > R ∼ = S t . Similarly, we say two ergodic p.m.p. actions Γ y ( X, µ ) and Λ y ( Y, ν )are (stably) orbit equivalent , if their associated orbit equivalence relations are (stably) isomorphic.Finally, we note that for any t > R , S as above, we have isomorphisms R × S ∼ = R t × S /t .2.2. Tracial von Neumann algebras.
In this paper we will also work with tracial von Neumannalgebras ( M, τ ), i.e. M is a von Neumann algebra equipped with a faithful normal tracial state τ : M → C . We denote by k x k := p τ ( x ∗ x ) the of x and denote by L ( M ) the completionof M with respect to this norm. Unless stated otherwise, we will always assume L ( M ) to bea separable Hilbert space, in which case we also call M a separable von Neumann algebra. Wewill further denote by U ( M ) the unitary group of M , and by Aut( M ) the group of τ -preservingautomorphisms of M equipped with the Polish topology of pointwise k . k -convergence.Let P ⊂ M be a von Neumann subalgebra, which unless stated otherwise is assumed to be unital.We denote by E P : M → P the unique τ -preserving conditional expectation from M onto P . Wefurther denote by P ′ ∩ M := { x ∈ M | ∀ y ∈ P : xy = yx } the relative commutant of P in M , andby N M ( P ) := { u ∈ U ( M ) | uP u ∗ = P } the normalizer of P in M . We say that P is regular in M if N M ( P ) ′′ = M , i.e. N M ( P ) generates M as a von Neumann algebra. Most von Neumann algebras M we encounter will be II factors, i.e. the center Z ( M ) := M ′ ∩ M satisfies Z ( M ) = C Cartan subalgebras and the von Neumann algebra of an equivalence relation.
Given a countable p.m.p. equivalence relation R , we can associate a canonical von Neumannalgebra L ( R ) ⊆ B ( L ( R , ¯ µ )) to it (see [FM77]). This von Neumann algebra is generated by partialisometries u ϕ for ϕ ∈ [[ R ]], and contains a canonical copy of L ∞ ( X ) as a Cartan subalgebra, i.e. amaximal abelian regular von Neumann subalgebra. This von Neumann algebra L ( R ) is a II factorif and only if R is ergodic.Conversely, if ( M, τ ) is a separable tracial von Neumann algebra and A ⊂ M is an abelian vonNeumann subalgebra, we get a countable p.m.p. equivalence relation in the following way. Identify A = L ∞ ( X ), for some standard probability space ( X, µ ). Then for every u ∈ N M ( A ), we can findan automorphism α u of ( X, µ ) such that a ◦ α u = uau ∗ , for every a ∈ A . The equivalence relation R ( A ⊂ M ) of the inclusion A ⊂ M is then defined to be the smallest countable p.m.p. equivalencerelation on ( X, µ ) whose full group contains α u , for every u ∈ N M ( A ).Now, assume that M is a II factor and A ⊂ M is a Cartan subalgebra. Then R ( A ⊂ M ) is ergodic.Moreover, if ( A ⊂ M ) ∼ = ( L ∞ ( X ) ⊂ L ( R )) for some countable ergodic p.m.p. equivalence relation R , then R ( A ⊂ M ) ∼ = R , i.e. we can recover the equivalence relation R from the von Neumannalgebra inclusion ( L ∞ ( X ) ⊂ L ( R )). Furthermore, one can show that any Cartan inclusion ( A ⊂ M )arises as an inclusion ( L ∞ ( X ) ⊂ L w ( R )) of L ∞ ( X ) inside the “twisted” von Neumann algebra L w ( R ) for some 2-cocycle w ∈ H ( R , T ), see [FM77].For t >
0, we define the t -amplification of a II factor M , denoted by M t , as the isomorphismclass of p ( B ( ℓ ) ¯ ⊗ M ) p , where p ∈ B ( ℓ ) ¯ ⊗ M is a projection with (Tr ⊗ τ )( p ) = t , and Tr is theusual trace on B ( ℓ ). Similarly, the inclusion ( A t ⊂ M t ) is defined as the isomorphism class of the TABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCT EQUIVALENCE RELATIONS 7 inclusion ( p ( ℓ ∞ ⊗ A ) p ⊂ p ( B ( ℓ ) ¯ ⊗ M ) p ), where p ∈ B ( ℓ ) ¯ ⊗ A is a projection with (Tr ⊗ τ )( p ) = t ,and ℓ ∞ ⊂ B ( ℓ ) is the subalgebra of diagonal operators. With this notation, we then have that A t ⊂ M t is a Cartan subalgebra, and R ( A t ⊂ M t ) ∼ = R ( A ⊂ M ) t .2.4. Popa’s intertwining-by-bimodules.
In [Po03], Popa introduced a powerful theory for de-ducing unitary conjugacy of subalgebras of tracial von Neumann algebras, which we briefly recallhere.Let
P, Q be von Neumann subalgebras of a tracial von Neumann algebra (
M, τ ). We say that acorner of P embeds into Q inside M (or, P intertwines into Q inside M ), and write P ≺ M Q ifwe can find non-zero projections p ∈ P , q ∈ Q , a ∗ -homomorphism θ : pP p → qQq , and a non-zeropartial isometry v ∈ qM p satisfying θ ( x ) v = vx , for all x ∈ pP p . Moreover, if P p ′ ≺ M Q , forevery non-zero projection p ′ ∈ P ′ ∩ M , we write P ≺ sM Q , and say that P strongly intertwinesinto Q inside M . The main technical tool from Popa’s deformation/rigidity theory is the followingcharacterization of intertwining. Theorem 2.1 ([Po03]) . Let
P, Q be von Neumann subalgebras of a tracial von Neumann algebra ( M, τ ) , and let U ⊂ U ( P ) be a subgroup which generates P . Then the following conditions areequivalent:(1) P ≺ M Q .(2) There is no sequence u n ∈ U satisfying k E Q ( au n b ) k → , for all a, b ∈ M . Finite index subequivalence relations.
In Section 3, we will discuss Ioana’s intertwiningtechniques for equivalence relations, which were inspired by Popa’s aforementioned techniques forvon Neumann algebras. For this, we will have to deal with subequivalence relations of boundedindex, which we briefly recall here.Let R be a countable p.m.p. equivalence relation on a standard probability space ( X, µ ), andsuppose
T ≤ R is a subequivalence relation. Then we can decompose X = ⊔ N ∈ N ∪{∞} X N , wherefor every N ∈ N ∪ {∞} , X N := { x ∈ X | [ x ] R is the union of N T -classes } is the R -invariant set consisting of all points in X whose R -class contains exactly N T -classes.If µ ( X ∞ ) = 0, we say that the inclusion T ≤ R has (essentially) finite index . If there exists k ≥ µ ( X N ) = 0 for all N > k , we say that the inclusion
T ≤ R has bounded index .For future reference, we record the following Lemma, which follows almost immediately from[DHI16, Lemma 3.3].
Lemma 2.2.
Let R on ( X, µ ) = ( X × X , µ × µ ) be a product of two countable p.m.p. equivalencerelations, R = R × R . Suppose Y ⊂ X is a subset of positive measure, T ≤ ( R × id) | Y is asubequivalence relation, and θ ∈ [[ R ]] with dom( θ ) = Y satisfies ( θ × θ )( T ) ≤ ( R × id) | θ ( Y ) .If T ≤ ( R × id) | Y has bounded (respectively, essentially finite) index, then there is a sequence of T -invariant positive measure subsets Y n ⊂ Y with Y = ∪ ∞ n =1 Y n such that ( θ × θ )( T | Y n ) ≤ ( R × id) | θ ( Y n ) has bounded (respectively, essentially finite) index for each n ≥ .Proof. Since R on X and R on X are countable p.m.p. equivalence relations, by [FM77] wecan write R i = R (Γ i y X i ) for i = 1 ,
2. Hence R = R (Γ × Γ y X ) and R × id = R (Γ y X ).We can thus apply [DHI16, Lemma 3.3], and the result follows. (cid:3) P. SPAAS Intertwining equivalence relations
Let R be a countable p.m.p. equivalence relation on the standard probability space ( X, µ ). Considera subequivalence relation
S ≤ R on (
X, µ ) such that every R -class contains infinitely many S -classes. Following [IKT08] we define a function ϕ S : [[ R ]] → [0 ,
1] by ϕ S ( θ ) = µ ( { x ∈ dom( θ ) | ( θ ( x ) , x ) ∈ S} ) . Lemma 3.1 ([Io11, Lemma 1.7]) . Let E ⊆ X be a positive measure subset and T ≤ R| E asubequivalence relation. Then the following are equivalent.(1) There is no sequence { θ n } ∞ n =1 ⊆ [ T ] such that ϕ S ( ψθ n ψ ′ ) → for all ψ, ψ ′ ∈ [ R ] .(2) There exist a T -invariant subset E ′ ⊆ E of positive measure and a subequivalence relation T ≤ T such that for any positive measure subset E ⊆ E ′ , there is a positive measure subset Y ⊆ E and θ ∈ [[ R ]] , θ : Y → Z , such that(a) T | Y ≤ T | Y has bounded index, and(b) ( θ × θ )( T | Y ) ≤ S| Z . Definition 3.2.
Whenever the equivalent conditions from Lemma 3.1 hold, we will write
T ≺ R S ,and say that T intertwines into S inside R .The above Lemma is phrased entirely in the context of equivalence relations. However, as indicatedbefore, it was discovered in the context of Popa’s intertwining-by-bimodules techniques, from whichwe also borrowed the above notation and terminology. Moreover, there is the following directcorrespondence between the two frameworks. Lemma 3.3 ([Io11, Lemma 1.8]) . Let E ⊆ X be a positive measure subset and T ≤ R| E asubequivalence relation. Then the following are equivalent.(1) T ≺ R S , and(2) L ( T ) ≺ L ( R ) L ( S ) . Note that each of the von Neumann algebras L ( T ), L ( S ), and L ( R ) contains the canonical Cartansubalgebra L ∞ ( X ) (cut down by the projection 1 E where necessary). In the product setting, thisgives the following. Corollary 3.4.
Let R , S , R , S be countable ergodic p.m.p. equivalence relations, on standardprobability spaces X , Y , X , Y respectively, such that R := R ×S ∼ = R ×S . Then the followingare equivalent, where we identify R i := R i × id Y i ≤ R (1) R ≺ R R ,(2) L ( R ) ¯ ⊗ L ∞ ( Y ) ≺ L ( R ) L ( R ) ¯ ⊗ L ∞ ( Y ) ,(3) L ∞ ( Y ) ≺ L ( R ) L ∞ ( Y ) .Proof. The equivalence between (1) and (2) is immediate from Lemma 3.3. The equivalence between(2) and (3) follows from [Va08, Lemma 3.5]. (cid:3)
In the von Neumann algebra setting, one can generally deduce unitary conjugacy from intertwiningbetween different tensor factors of a given II factor, see for instance [OP03, Proposition 12].However, for equivalence relations, more work needs to be done. Indeed, from the above Corollarywe see that the intertwining R ≺ R R is not equivalent to L ( R ) ≺ L ( R ) L ( R ). Rather, one alsohas to take the Cartan subalgebras L ∞ ( Y i ) into account, which leads to a much different analysis.Note for instance that for the equivalence between (2) and (3) in the above Corollary, we couldapply the very useful result [Va08, Lemma 3.5] about the intertwining of relative commutants in thevon Neumann algebra setting. Unfortunately, there is no direct analogue of this for the intertwiningof equivalence relations. Nevertheless, we will be able to deduce some results of a similar flavor inthe product setting (see for instance Proposition 3.11 below). TABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCT EQUIVALENCE RELATIONS 9
For the remainder of this Section, we will turn to the study of intertwining results for productequivalence relations, and orbit equivalence relations arising from diagonal product actions. Firstly,we record the following easy Lemma, see for instance [IS18, Lemma 7.1].
Lemma 3.5.
Suppose M , N are II factors, and A ⊂ M , B ⊂ N are Cartan subalgebras. Then R ( A ⊂ M ) × R ( B ⊂ N ) ∼ = R ( A ¯ ⊗ B ⊂ M ¯ ⊗ N ) . Given an intertwining between factors of product equivalence relations, we can now show thefollowing.
Proposition 3.6.
Let R , S , R , S be countable ergodic p.m.p. equivalence relations on standardprobability spaces ( X , µ ) , ( Y , ν ) , ( X , µ ) , ( Y , ν ) respectively such that R := R ×S ∼ = R ×S .Assume that, as subequivalence relations of R , we have R ≺ R R .Then there exists an equivalence relation T such that R is stably isomorphic to R × T . Moreover,if S is hyperfinite, then either R is stably isomorphic to R , or R ∼ = R × S ∼ = R × S .Proof. By Lemma 3.1 we can find a subequivalence relation T ≤ R = R × id Y , positive measuresubsets W, Z ⊆ X := X × Y , and θ ∈ [[ R ]] , θ : W → Z , such that(a) T | W ≤ R | W has bounded index, say bounded by k , and(b) ( θ × θ )( T | W ) ≤ R | Z .Note that, as a subequivalence relation of R , R is a fibered equivalence relation over Y : R × id Y = Z ⊕ Y R dν ( y ) , where inside the integral we look at R as an equivalence relation on X . We note that this is infact the ergodic decomposition of R × id Y . Given the subequivalence relation T ≤ R × id Y , wecan thus write T = Z ⊕ Y T ,y dν ( y ) . Here for every y ∈ Y , T ,y is an equivalence relation on X . By (a) above, we note that moreover T ,y | W y has bounded index at most k inside R | W y for almost every y ∈ Y , where W y = { x ∈ X | ( x, y ) ∈ W } . Since R is ergodic, this implies in particular that for almost every y ∈ Y , theergodic decomposition of T ,y | W y is atomic with at most k atoms. By restricting to further subsetsif necessary, we can thus assume that for almost every y , T ,y is ergodic on W y . Taking yet furthersubsets if necessary, we can also assume that there is some c > y ∈ Y , wehave either µ ( W y ) = c or µ ( W y ) = 0.Fix any measurable set C ⊆ X with µ ( C ) = c . By ergodicity of R and a standard uniformizationargument, it follows that there is a measurable map ψ : Y → [ R ] such that ψ y ( W y ) = C whenever µ ( W y ) = 0. Writing D := { y ∈ Y | µ ( W y ) = 0 } we can thus assume that W = C × D ⊂ X × Y by composing with ψ = R ⊕ Y ψ y dν ( y ) ∈ [ R ].Passing to the corresponding von Neumann algebras, it follows from (b) above that u θ ( p W L ( T ) p W ) u ∗ θ ⊆ p Z ( L ( R ) ¯ ⊗ L ∞ ( Y )) p Z , where p W and p Z denote the projections in L ∞ ( X ) onto W and Z respectively. The foregoingreasoning implies that we can assume p W = p C ⊗ p D ∈ L ∞ ( X ) ¯ ⊗ L ∞ ( Y ) = L ∞ ( X ) and hence theprevious equation can be written as u θ (cid:18)Z ⊕ D p C L ( T ,y ) p C dν ( y ) (cid:19) u ∗ θ ⊆ p Z ( L ( R ) ¯ ⊗ L ∞ ( Y )) p Z . Moreover, since θ ( W ) = Z , we have u θ p W u ∗ θ = p Z and hence moving the u θ ’s to the right, we get Z ⊕ D p C L ( T ,y ) p C dν ( y ) ⊆ ( p C ⊗ p D ) u ∗ θ ( L ( R ) ¯ ⊗ L ∞ ( Y )) u θ ( p C ⊗ p D ) . For notational convenience, we will write A = L ∞ ( X ), A = L ∞ ( X ), B = L ∞ ( Y ), A = L ∞ ( X ),and B = L ∞ ( Y ). Also, we let P := u ∗ θ L ( R ) u θ , A := u ∗ θ A u θ , and B := u ∗ θ B u θ . With thisnotation, the last inclusion reads(3.1) Z ⊕ D p C L ( T ,y ) p C dν ( y ) ⊆ ( p C ⊗ p D )( P ¯ ⊗ B )( p C ⊗ p D ) . Since u θ normalizes A , we also have(3.2) A = A ¯ ⊗ B = A ¯ ⊗ B = A ¯ ⊗ B . Now, using the fact that commutants go through direct integrals (see for instance [KR97, Theo-rem 14.1.24]), we get by taking relative commutants in (3.1) that B ( p C ⊗ p D ) ⊆ (cid:18)Z ⊕ D p C L ( T ,y ) p C dν ( y ) (cid:19) ′ ∩ ( p C ⊗ p D ) L ( R )( p C ⊗ p D )= Z ⊕ D ( p C L ( T ,y ) p C ) ′ ∩ ( p C L ( R ) p C ) dν ( y ) . For the last equality, we also used the fact that R ⊕ D p C L ( T ,y ) p C dν ( y ) contains the maximal abeliansubalgebra L ∞ ( X )( p C ⊗ p D ), and thus its relative commutant is contained in L ∞ ( X )( p C ⊗ p D ) ⊂ p C L ( R ) p C ¯ ⊗ L ∞ ( Y ) p D = R ⊕ D ( p C L ( R ) p C ) dν ( y ).Since T ,y | C is ergodic, it follows that p C L ( T ,y ) p C is a factor. Moreover, this factor contains A p C which is a Cartan subalgebra, and thus maximal abelian, in the II factor p C L ( R ) p C . In particular,it follows that ( p C L ( T ,y ) p C ) ′ ∩ ( p C L ( R ) p C ) = C p C . We conclude that(3.3) B ( p C ⊗ p D ) ⊆ Z ⊕ D C p C dν ( y ) = p C ¯ ⊗ B p D . From this point, one can apply almost verbatim the end of the proof of [IS18, Theorem G]. For thereader’s convenience, and later reference, we reproduce the argument here. Thanks to (3.3), thereexists a von Neumann subalgebra B ⊂ B such that B ( p C ⊗ p D ) = p C ⊗ B p D . Taking relativecommutants in this equality, we get p C L ( R ) p C ¯ ⊗ [( B p D ) ′ ∩ p D L ( S ) p D ] = ( p C ⊗ p D )( P ¯ ⊗ B )( p C ⊗ p D ) . In particular, since P is a factor, we see that the center of the above algebra equals B ( p C ⊗ p D ) = p C ⊗ B p D . Identifying this center with L ∞ ( Y ) for some probability space ( Y, ν ) and disintegratingin the above equality we get Z ⊕ Y p C L ( R ) p C ¯ ⊗ N y dν ( y ) = Z ⊕ Y p y P p y dν ( y ) , where we decomposed N := ( B p D ) ′ ∩ p D L ( S ) p D = R ⊕ Y N y dν ( y ), and p C ⊗ p D = R ⊕ Y p y dν ( y ) ∈ A ¯ ⊗ B ⊂ P ¯ ⊗ B . It follows from for instance [Ta01, Theorem IV.8.23] that the above identificationsplits, i.e. for almost every y ∈ Y we necessarily have p C L ( R ) p C ¯ ⊗ N y = p y P p y . Moreover, we have B p D ⊂ B p D ⊂ ( B p D ) ′ ∩ p D L ( S ) p D , so we can also decompose B p D = R ⊕ Y B ,y dν ( y ) ⊂ R ⊕ Y N y dν ( y ) = N , where B ,y ⊂ N y is a unital inclusion for all y . Now B ⊂ L ( S )is a Cartan subalgebra, and hence so is B p D ⊂ p D L ( S ) p D . Since B p D ⊂ N ⊂ p D L ( S ) p D , itfollows from [Dy63] that also B p D ⊂ N is a Cartan subalgebra. From [Sp17, Lemma 2.2] we then TABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCT EQUIVALENCE RELATIONS 11 deduce that B ,y ⊂ N y is a Cartan subalgebra for almost every y ∈ Y . Furthermore, it follows from(3.2) that Z ⊕ Y A p C ¯ ⊗ B ,y dν ( y ) = Z ⊕ Y A p y dν ( y ) . This identification again splits by [Ta01, Theorem IV.8.23], i.e. for almost every y ∈ Y we have A p C ¯ ⊗ B ,y = A p y .From the above discussion we now deduce the following identification of inclusions of Cartan sub-algebras, for almost every y ∈ Y :(3.4) ( A p C ¯ ⊗ B ,y ⊂ p C L ( R ) p C ¯ ⊗ N y ) = ( A p y ⊂ p y P p y ) . Writing T y := R ( B ,y ⊂ N y ), t := τ ( p C ), and s := τ ( p y ), we thus get for almost every y ∈ Y :(3.5) R s ∼ = R ( A s ⊂ L ( R ) s ) ∼ = R ( A s ⊂ P s ) ∼ = R ( A t ¯ ⊗ B ,y ⊂ L ( R ) t ¯ ⊗ N y ) ∼ = R t × T y , where the last isomorphism follows from Lemma 3.5. Taking any y such that the above equationshold and setting T := T y , we get that R is stably isomorphic to R × T , as desired.For the moreover part, assume S is hyperfinite. Then L ( S ) is the hyperfinite II factor. Beinga subalgebra of L ( S ), N is thus amenable by [Co75]. Hence, N y is an amenable tracial factor foralmost every y , and is therefore either isomorphic to M n ( C ) for some n ∈ N , or to the hyperfiniteII factor. Choose again y such that (3.5) holds. Then in the first case we get that T y is finiteand hence R is stably isomorphic to R . In the second case, T y arises as the equivalence relationof a Cartan inclusion B ⊂ R , and as such is a hyperfinite ergodic p.m.p. equivalence relationby [CFW81]. In this case we thus have R ∼ = R × R hyp where R hyp ∼ = S is a hyperfinite ergodicp.m.p. equivalence relation. This finishes the proof of the Proposition. (cid:3) Corollary 3.7.
Let R , S , R , S be countable ergodic p.m.p. equivalence relations on standardprobability spaces ( X , µ ) , ( Y , ν ) , ( X , µ ) , ( Y , ν ) respectively such that R := R ×S ∼ = R ×S .Assume that, as subequivalence relations of R , we have R ≺ R R and R ≺ R R .Then R is stably isomorphic to R .Proof. Note that under the given assumptions, Proposition 3.6 in particular implies that R is stablyisomorphic to a subequivalence relation of R , and vice versa. Using ergodicity, a straightforwardapplication of the measurable Schr¨oder-Bernstein theorem then implies the result. (cid:3) Remark 3.8.
On the von Neumann algebra level, using Corollary 3.4, this gives the following resultin the setting of Corollary 3.7: If L ( R ) ¯ ⊗ L ∞ ( Y ) ≺ L ( R ) L ( R ) ¯ ⊗ L ∞ ( Y ) and L ( R ) ¯ ⊗ L ∞ ( Y ) ≺ L ( R ) L ( R ) ¯ ⊗ L ∞ ( Y ), then R is stably isomorphic to R , and hence also L ( R ) is stably isomorphic to L ( R ).For the next Lemma, we will need the following stronger version of intertwining, which is closelytied together with strong intertwining of the corresponding von Neumann algebras. As above, let R be a countable p.m.p. equivalence relation on ( X, µ ). Suppose
S ≤ R is a subequivalence relationsuch that every R -class contains infinitely many S -classes. Let E ⊆ X be a positive measure subset,and T ≤ R| E a subequivalence relation. Definition 3.9.
If for every T -invariant subset E ⊆ X of positive measure, there is no sequence( θ n ) n in [ T | E ] such that ϕ S ( ψθ n ψ ′ ) → ψ, ψ ′ ∈ [ R ], then we say that T strongly intertwinesinto S , and write T ≺ s R S .Note that this condition is in particular implied by strong intertwining of the associated von Neu-mann algebras, L ( T ) ≺ sL ( R ) L ( S ). We now record a result which appeared in [DHI16] phrased interms of measure equivalence for groups, but whose proof applies verbatim to get the followingresult. Lemma 3.10 ([DHI16, Theorem 3.1]) . Let R , S , R , S be countable ergodic p.m.p. equivalencerelations on standard probability spaces ( X , µ ) , ( Y , ν ) , ( X , µ ) , ( Y , ν ) respectively such that R := R × S ∼ = R × S . Assume that, as subequivalence relations of R , we have R ≺ R R and R ≺ s R R .Then there are positive measure subsets W, Z ⊂ X := X × Y , a subequivalence relation S ≤ R ,and φ ∈ [[ R ]] , φ : Z → W , such that(1) S | Z ≤ R | Z has bounded index, and(2) ( φ × φ )( S | Z ) ≤ R | W has bounded index.In particular, upon restricting to appropriate subsets, R and R admit isomorphic subequivalencerelations of bounded index.Proof. The proof of [DHI16, Theorem 3.1] applies verbatim, upon using Lemma 2.2 from the pre-liminaries instead of [DHI16, Lemma 3.3] where it appears in the proof. (cid:3)
We will use this result to establish the following Proposition for product equivalence relations,which, albeit not equivalent, can be seen as an analogue of the intertwining of relative commutantsin the von Neumann algebra setting ([Va08, Lemma 3.5], compare also with Corollary 3.7 above).
Proposition 3.11.
Let R , S , R , S be countable ergodic p.m.p. equivalence relations on stan-dard probability spaces ( X , µ ) , ( Y , ν ) , ( X , µ ) , ( Y , ν ) respectively such that R := R × S ∼ = R × S . Assume that, as subequivalence relations of R , we have R ≺ R R and R ≺ R R .Then S is stably isomorphic to S .Proof. Firstly, we observe that we in fact have that R ≺ s R R . Indeed, by Corollary 3.4, R ≺ R R is equivalent to the fact that L ( R ) ¯ ⊗ L ∞ ( Y ) ≺ L ( R ) L ( R ) ¯ ⊗ L ∞ ( Y ). Since L ( R ) ¯ ⊗ L ∞ ( Y ) is regu-lar inside L ( R ), this implies that L ( R ) ¯ ⊗ L ∞ ( Y ) ≺ sL ( R ) L ( R ) ¯ ⊗ L ∞ ( Y ) by [DHI16, Lemma 2.4(3)],and therefore also R ≺ s R R .By Lemma 3.10, we can thus find positive measure subsets W, Z ⊂ X , a subequivalence relation S ≤ R , and φ ∈ [[ R ]], φ : Z → W , such that S | Z ≤ R | Z has bounded index k , and R :=( φ × φ )( S | Z ) ≤ R | W has bounded index l . As in the beginning of the proof of Proposition 3.6,we can write R = Z ⊕ Y R ,y dν ( y ) , and S = Z ⊕ Y S ,y dν ( y ) , such that for almost every y the ergodic decomposition of R ,y | W y is atomic with at most k atoms,and the ergodic decomposition of S ,y | Z y is atomic with at most l atoms. By taking further subsetsif necessary, we can thus assume that R ,y | W y and S ,y | Z y are ergodic. For u = u φ , p = proj W , q = proj Z , we then get(3.6) pL ( R ) p = u qL ( S ) q u ∗ , where we can write pL ( R ) p = Z ⊕ Y p y L ( R ,y ) p y dν ( y ) , and qL ( S ) q = Z ⊕ Y q y L ( S ,y ) q y dν ( y ) . TABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCT EQUIVALENCE RELATIONS 13
Moreover, by the above we have that p y L ( R ,y ) p y and q y L ( S ,y ) q y are factors for almost every y .Therefore the center of pL ( R ) p equals L ∞ ( Y ) p and the center of qL ( S ) q equals L ∞ ( Y ) q . From(3.6), it then follows that L ∞ ( Y ) p = u L ∞ ( Y ) q u ∗ . In particular, through conjugating by u , we get R ( L ∞ ( Y ) p ⊂ pM p ) ∼ = R ( L ∞ ( Y ) q ⊂ qM q ) . Since R ( L ∞ ( Y ) p ⊂ pM p ) ∼ = (id X ×S ) | W and R ( L ∞ ( Y ) q ⊂ qM q ) ∼ = (id X ×S ) | Z , we get on thevon Neumann algebra level a trace preserving automorphism Z ⊕ X p x L ( S ) p x dµ ( x ) ∼ = Z ⊕ X q x L ( S ) q x dµ ( x ) , in such a way that also the canonical Cartan subalgebras are identified, and where we decomposed p = R ⊕ X p x dµ ( x ) and q = R ⊕ X q x dµ ( x ). Using [Ta01, Theorem IV.8.23] (see also [Sp17, Theo-rem 1.13]), we can find null sets N ⊆ X , N ⊆ X , a Borel isomorphism Φ : X \ N → X \ N ,with Φ( µ ) equivalent to µ , such that the above isomorphism decomposes into tracial isomorphisms p x L ( S ) p x ∼ = q Φ − ( x ) L ( S ) q Φ − ( x ) . Moreover, by the above reasoning, these isomorphisms can bechosen in such a way that also the Cartan subalgebras L ∞ ( Y ) p x and L ∞ ( Y ) q Φ − ( x ) are identified.Taking some x ∈ X \ N , we thus get S t ∼ = R ( L ∞ ( Y ) p x ⊂ p x L ( S ) p x ) ∼ = R ( L ∞ ( Y ) q Φ − ( x ) ⊂ q Φ − ( x ) L ( S ) q Φ − ( x ) ) ∼ = S s , where we wrote t = τ ( p x ), and s = τ ( q Φ − ( x ) ). In other words, S is stably isomorphic to S ,finishing the proof of the Proposition. (cid:3) For the remainder of this section, we consider diagonal product actions. Using the establishedresults, this will in particular lead to a proof of Theorem C. For convenience we briefly recall theframework established in the Introduction.
Notation 1.3.
Let Γ , Γ , Σ , Σ be countable discrete groups, and ρ : Γ → Σ , ρ : Γ → Σ besurjective group homomorphisms. Denote Γ i := ker( ρ i ). Suppose Γ i y ( X i , µ i ) and Σ i y ( Y i , ν i )are free ergodic p.m.p. actions on standard probability spaces, and assume that also Γ i y ( X i , µ i )is ergodic for i = 1 ,
2. We then have the diagonal product actions Γ i y ( X i , µ i ) × ( Y i , ν i ) given by g · ( x, y ) = ( gx, ρ ( g ) y ), for all g ∈ Γ i , x ∈ X i , y ∈ Y i . For i = 1 ,
2, we will write R i := R (Γ i y X i ). Proposition 3.12.
In the setting of Notation 1.3, suppose the actions Γ y ( X × Y , µ × ν ) and Γ y ( X × Y , µ × ν ) generate the same orbit equivalence relation R (i.e. the actions areorbit equivalent). Assume that, as subequivalence relations of R , we have R × id Y ≺ R R × id Y .Then R is stably isomorphic to a subequivalence relation of R . Remark 3.13.
We note that by Lemma 3.3, the condition R × id Y ≺ R R × id Y is equivalent to L ( R ) ¯ ⊗ L ∞ ( Y ) ≺ L ( R ) L ( R ) ¯ ⊗ L ∞ ( Y ), where L ( R i ) = L ∞ ( X i ) ⋊ Γ i . Here we used the fact that,by construction, Γ i acts trivially on Y i . Taking relative commutants, we see that this is moreoverequivalent to L ∞ ( Y ) ≺ L ( R ) L ∞ ( Y ). We note that if this condition would instead be imposed onthe X i ’s, one can exploit the fact that Γ i acts freely on X i , and hence L ∞ ( X i ) is maximal abelianin L ∞ ( X i ) ⋊ Γ i (see for instance [Dr20, Io06a] for some results in this direction for such diagonalactions). However, L ∞ ( Y i ) is not maximal abelian in L ∞ ( Y i ) ⋊ Γ i , and so in the proof below wetake a different approach, similar to the product case considered above. Proof of Proposition 3.12.
The proof follows much the same lines as the proof of Proposition 3.6,so we only indicate the main differences. Firstly, since Γ acts trivially on Y , we note that as asubequivalence relation of R , we can indeed write R × id Y as a fibered equivalence relation over Y : R × id Y = Z ⊕ Y R dν ( y ) . A similar statement holds for R . In particular, writing M = L ( R ) = ( L ∞ ( X ) ¯ ⊗ L ∞ ( Y )) ⋊ Γ =( L ∞ ( X ) ¯ ⊗ L ∞ ( Y )) ⋊ Γ , with the Cartan subalgebras identified, we have on the von Neumannalgebra level that L ∞ ( Y ) ′ ∩ M = ( L ∞ ( X ) ⋊ Γ ) ¯ ⊗ L ∞ ( Y ) = L ( R ) ¯ ⊗ L ∞ ( Y ) , and L ∞ ( Y ) ′ ∩ M = ( L ∞ ( X ) ⋊ Γ ) ¯ ⊗ L ∞ ( Y ) = L ( R ) ¯ ⊗ L ∞ ( Y ) . Given the aforementioned facts, we can now apply verbatim the first half of the proof of Propo-sition 3.6 to find that, up to conjugating by a unitary, there exists a projection p = p C ⊗ p D ∈ L ∞ ( X ) ¯ ⊗ L ∞ ( Y ) such that (cf. (3.3)) L ∞ ( Y )( p C ⊗ p D ) ⊂ p C ⊗ L ∞ ( Y ) p D , and thus a von Neumann subalgebra B ⊂ L ∞ ( Y ) such that L ∞ ( Y )( p C ⊗ p D ) = p C ⊗ B p D . Taking relative commutants yields(3.7) ( p C ⊗ B p D ) ′ ∩ pM p = p ( L ( R ) ¯ ⊗ L ∞ ( Y )) p. Now since B ⊆ L ∞ ( Y ), we see that(3.8) ( L ∞ ( Y ) p ) ′ ∩ pL ( R ) p = p C L ( R ) p C ¯ ⊗ L ∞ ( Y ) p D ⊆ ( p C ⊗ B p D ) ′ ∩ pM p. Identifying the center of the algebras in (3.7) with L ∞ ( Y ) = L ∞ ( Y )( p C ⊗ p D ) = p C ⊗ B p D forsome probability space ( Y, ν ), and disintegrating over this center, we get Z ⊕ Y N y dν ( y ) = Z ⊕ Y p y L ( R ) p y dν ( y ) , where we wrote N := ( p C ⊗ B p D ) ′ ∩ pM p = R ⊕ Y N y dν ( y ) and p C ⊗ p D = R ⊕ Y p y dν ( y ). From (3.8),it then follows that p C L ( R ) p C ⊆ N y for almost every y ∈ Y . Moreover, the canonical Cartansubalgebras are still identified (cf. the proof of Proposition 3.6). Using once more [Ta01, Theo-rem IV.8.23], we thus conclude that R is stably isomorphic to a subequivalence relation of R . (cid:3) Proposition 3.14.
Assume the setting of Notation 1.3. Furthermore assume that Σ is amenable, Γ y ( X , µ ) is strongly ergodic, and the actions Γ y ( X × Y , µ × ν ) and Γ y ( X × Y , µ × ν ) generate the same orbit equivalence relation R . Then R × id Y ≺ R R × id Y .Proof. Write M := L ( R ) and X := X × Y = X × Y . As observed in Remark 3.13, it sufficesto show that L ∞ ( Y ) ≺ M L ∞ ( Y ). To show this, we first observe that since Σ is amenable andΣ y ( Y , ν ) is free, ergodic, and p.m.p., the associated orbit equivalence relation is a hyperfiniteergodic p.m.p. equivalence relation. Thus, we can identify ( Y , ν ) = ( Y , ν ) N , where Y = { , } and ν = ( δ + δ ), in such a way that ( y k ) k and ( z k ) k belong to the same orbit if and only if y k = z k for all but finitely many k ∈ N . We can then identify L ∞ ( Y ) = ¯ ⊗ k ∈ N L ∞ ( Y , ν ) k , and weput B n = ¯ ⊗ k ≥ n L ∞ ( Y , ν ) k , for n ∈ N . Fixing an ultrafilter ω on N , it is then not hard to showthat Y ω B n ⊂ M ′ ∩ L ∞ ( X ) ω , see for instance the first half of the proof of [IS18, Lemma 6.1]. Moreover, by strong ergodicity ofthe action of Γ on X , we also have M ′ ∩ L ∞ ( X ) ω ⊂ L ∞ ( Y ) ω . Combining both inclusions, we thus get Q ω B n ⊂ L ∞ ( Y ) ω . Applying [IS18, Lemma 2.3] then givesa sequence ε n → B n ⊂ ε n L ∞ ( Y ). In particular B n ≺ M L ∞ ( Y ) for large enough n .Since B n has finite index in L ∞ ( Y ), this also gives L ∞ ( Y ) ≺ M L ∞ ( Y ) and thus finishes the proofof the Proposition. (cid:3) TABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCT EQUIVALENCE RELATIONS 15
Proof of Theorem C.
From Proposition 3.14 applied twice, we get that R × id Y ≺ R R × id Y and R × id Y ≺ R R × id Y . As in the proof of Corollary 3.7, Proposition 3.12 then implies the desiredconclusion. (cid:3) (Non-)Schmidt equivalence relations In this section we consider (non-)Schmidt equivalence relations and prove Theorem A. First of all,identifying elements of [[ R ]] with partial isometries in L ( R ), and denoting by k . k the 2-norm on L ( R ) corresponding to the canonical trace, we can give a “local” characterization of the Schmidtproperty, similar to the local characterization of stability in [Ma17, Theorem 2.1]. Lemma 4.1.
Let R be an ergodic p.m.p. equivalence relation on a standard probability space ( X, µ ) .Then the following are equivalent:(1) R is Schmidt.(2) For every finite set F ⊂ [[ R ]] and every ε > , there exists v ∈ [[ R ]] such that ∀ u ∈ F : k vu − uv k < ε (cid:13)(cid:13) v − id | dom( v ) (cid:13)(cid:13) . Proof. (1) ⇒ (2) is immediate. Indeed, by definition, we can find δ > T n ) n in [ R ] such that µ ( { x ∈ X | T n ( x ) = x } ) ≥ δ for all n . Moreover, we observe that for any v ∈ [[ R ]], we have (cid:13)(cid:13) v − id | dom( v ) (cid:13)(cid:13) = 2 µ ( { x ∈ dom( v ) | v ( x ) = x } ). Therefore, given any finite set F ⊂ [[ R ]], we can take n large enough so that ∀ u ∈ F : k T n u − uT n k < ε √ δ ≤ ε k T n − id k , and thus (2) holds.Before proving (2) ⇒ (1), we first show that if R satisfies (2), then for every subset Y ⊂ X ofpositive measure, R| Y satisfies (2) as well. The argument is, mutatis mutandis, essentially the sameas the first part of the proof of (iii) ⇒ (ii) in [Ma17, Theorem 2.1], but we include it for the reader’sconvenience.Let Y ⊂ X be a subset of positive measure and set p = 1 Y . Suppose that R| Y does not satisfy (2).Then we can find a finite set F ⊂ [[ R| Y ]] and a constant κ > v ∈ [[ R| Y ]], wehave(4.1) (cid:13)(cid:13) v − id dom( v ) (cid:13)(cid:13) ≤ κ X u ∈ F k vu − uv k . Since R is assumed ergodic, we can find a finite set K ⊂ [[ R ]] such that P w ∈ K w ∗ w = p ⊥ and ww ∗ ≤ p for all w ∈ K . Now take any v ∈ [[ R ]]. Then we can write (cid:13)(cid:13) v − id | dom( v ) (cid:13)(cid:13) = (cid:13)(cid:13) p ( v − id | dom( v ) ) (cid:13)(cid:13) + X w ∈ K (cid:13)(cid:13) w ( v − id | dom( v ) ) (cid:13)(cid:13) . At this point, we also observe that we can write id | dom( v ) = v ∗ v . Thus, for any w ∈ K , we have (cid:13)(cid:13) w ( v − id | dom( v ) ) (cid:13)(cid:13) = k wv − wv ∗ v k ≤ k wv − vw k + k vw − v ∗ vw k + k v ∗ vw − v ∗ wv k + k v ∗ wv − wv ∗ v k ) ≤ k wv − vw k + k vp − v ∗ vp k + k vw − wv k + k v ∗ w − wv ∗ k ) . Combining the above, we thus have (cid:13)(cid:13) v − id | dom( v ) (cid:13)(cid:13) ≤ | K | ( k p ( v − v ∗ v ) k + k ( v − v ∗ v ) p k )+ 8 X w ∈ K k vw − wv k + 4 X w ∈ K ∗ k vw − wv k . Furthermore, we see that v ∗ vp = pv ∗ v = pv ∗ vp = id dom( v ) ∩ Y = id dom( pvp ) , from which a directcalculation gives k p ( v − v ∗ v ) k + k ( v − v ∗ v ) p k = k p ( v − v ∗ v ) − ( v − v ∗ v ) p k + 2 k p ( v − v ∗ v ) p k = k pv − vp k + 2 (cid:13)(cid:13) pvp − id dom( pvp ) (cid:13)(cid:13) . Applying (4.1) to pvp ∈ [[ R| Y ]], we get (cid:13)(cid:13) pvp − id dom( pvp ) (cid:13)(cid:13) ≤ κ X u ∈ F k ( pvp ) u − u ( pvp ) k ≤ κ X u ∈ F k vu − uv k . Combining everything, we thus have (cid:13)(cid:13) v − id | dom( v ) (cid:13)(cid:13) ≤ | K | k pv − vp k + 8 | K | κ X u ∈ F k vu − uv k + 8 X w ∈ K k vw − wv k + 4 X w ∈ K ∗ k vw − wv k . This shows R does not satisfy (2), contradiction. Hence for every subset Y ⊂ X of positive measure, R| Y satisfies (2) as well.We now prove (2) ⇒ (1). Take a k . k -dense sequence ( S n ) n in [ R ]. Using (2), we can find for every n ∈ N an element v ∈ [[ R ]] such that ∀ i ≤ n : k vS i − S i v k < n (cid:13)(cid:13) v − id | dom( v ) (cid:13)(cid:13) . Let v n be such an element for which µ ( { x ∈ X | v n ( x ) = x } ) is maximal among such elements.Noting that v ∗ n v n is the identity map on A n := dom( v n ) and v n v ∗ n is the identity map on B n :=im( v n ), we see that ( A n ) n and ( B n ) n are asymptotically invariant sequences in X , and therefore soare ( A n ∩ B n ) n and ( A n \ ( A n ∩ B n )) n . Case 1. lim n →∞ µ ( A n \ ( A n ∩ B n )) = 0.Passing to a subsequence if necessary, we can in this case assume that there exists δ > n , µ ( A n \ ( A n ∩ B n )) ≥ δ . Restricting v n to A n \ ( A n ∩ B n ), we then find a sequencewitnessing condition (iii) in [Ma17, Theorem 2.1]. Therefore, R is stable and hence Schmidt. Case 2. lim n →∞ µ ( A n \ ( A n ∩ B n )) = 0.We claim that in this case, µ ( { x ∈ dom( v n ) | v n ( x ) = x } ) → µ (dom( v n )) → v n in any way on X \ dom( v n ) to an element T n ∈ [ R ], this thenshows that R is Schmidt.We first of all show that for v n chosen as above, lim n →∞ µ ( A n ) = 1. Indeed, if not, then taking asubsequence if necessary, there exists δ > µ ( A n ) ≤ − δ for all n . By the assumption ofCase 2, we thus get µ ( X \ ( A n ∪ B n )) ≥ δ/ n . Fix such n . Write Y := X \ ( A n ∪ B n ),and apply (2) to R| Y . This gives w n ∈ [[ R| Y ]] such that ∀ i ≤ n : k pS i pw n − w n pS i p k < n (cid:13)(cid:13) w n − id | dom( w n ) (cid:13)(cid:13) , where p = 1 Y denotes the projection onto Y . Extending v n to equal w n on dom( w n ) ⊂ Y thencontradicts the maximality of v n .Similarly, if lim n →∞ µ ( { x ∈ dom( v n ) | v n ( x ) = x } ) = 1, we consider the set Y n := { x ∈ dom( v n ) | v n ( x ) = x } . After passing to a subsequence if necessary, we can then assume µ ( Y n ) ≥ δ for all n and some δ >
0. Applying (2) to Y n , produces w n ∈ [[ R| Y n ]] such that ∀ i ≤ n : k pS i pw n − w n pS i p k < n (cid:13)(cid:13) w n − id | dom( w n ) (cid:13)(cid:13) , TABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCT EQUIVALENCE RELATIONS 17
Define the element ˜ v n ∈ [[ R ]] to equal v n on X \ Y n (which is obviously v n -invariant) and to equal w n on Y n . This element then once again contradicts the maximality of v n , finishing the proof ofCase 2, and therefore the Lemma. (cid:3) Remark 4.2.
Note that the proof in particular shows that if R is Schmidt, then there exists acentral sequence T n ∈ [ R ] such that µ ( { x ∈ X | T n ( x ) = x } ) = 1 for all n . This fact was firstobserved in [KTD18, Lemma 5.5].Next, we will use Lemma 4.1 to provide a characterization of central sequences in the full groupof a product equivalence relation, one of whose factors is not Schmidt. Before stating and provingthis result, we establish some notation which will be useful in the proof. We also refer to [Ma17,Section 3] where a similar notation is introduced and exploited.Let R be a p.m.p. equivalence relation on a standard probability space ( X, µ ), and identify L ( R , ¯ µ )with the canonical L -space of the von Neumann algebra L ( R ). For any x ∈ L ( R ), we will write ˆ x for the corresponding vector in L ( R , ¯ µ ). Most important for us is that for v ∈ [[ R ]], ˆ v correspondsto the indicator function of { ( x, v ( x )) | x ∈ dom( v ) } .Now if S is another p.m.p. equivalence relation on some standard probability space ( Y, ν ), then forany v ∈ [[ R × S ]], there exists a unique function v R ∈ L ( S , [[ R ]]) such thatˆ v ( x, x ′ , y, y ′ ) = \ v R ( y, y ′ )( x, x ′ )for almost every ( x, x ′ , y, y ′ ) ∈ R × S . In other words, v R satisfies v R ( y, y ′ )( x ) = x ′ ⇔ \ v R ( y, y ′ )( x, x ′ ) = 1 ⇔ ˆ v ( x, x ′ , y, y ′ ) = 1 ⇔ v ( x, y ) = ( x ′ , y ′ )for almost every ( x, x ′ , y, y ′ ) ∈ R × S . Lemma 4.3.
Let R and S be countable ergodic p.m.p. equivalence relations on standard probabilityspaces ( X, µ ) and ( Y, ν ) respectively, and assume R is not Schmidt. If ( v n ) n is a central sequencein [[ R × S ]] , then for almost every x ∈ X there exists a central sequence ( v n,x ) n ∈ [[ S ]] such that ( µ × ν )( { ( x, y ) ∈ X × Y | v n ( x, y ) = ( x, v n,x ( y )) } ) → as n → ∞ . Moreover, if v n ∈ [ R × S ] for every n , then v n,x ∈ [ S ] for every n and almost every x ∈ X .Proof. Let ( v n ) n ∈ [[ R × S ]] be a central sequence. Write v n ( x, y ) = ( v (1) n ( x, y ) , v (2) n ( x, y )), andconsider v n, R ∈ L ( S , [[ R ]]). The Lemma will follow if we prove that( µ × ν )( { ( x, y ) ∈ X × Y | v (1) n ( x, y ) = x } ) → . Since R is not Schmidt, it follows from Lemma 4.1 that there exists a finite set F ⊂ [[ R ]] and κ > ∀ v ∈ [[ R ]] : (cid:13)(cid:13) v − id | dom( v ) (cid:13)(cid:13) ≤ κ X u ∈ F k vu − uv k . Moreover, we recall the observation that (cid:13)(cid:13) v − id | dom( v ) (cid:13)(cid:13) = 2 µ ( { x ∈ dom( v ) | v ( x ) = x } ). We nowcompute: ( µ × ν )( { ( x, y ) ∈ X × Y | v (1) n ( x, y ) = x } ) == ( µ × ν )( { ( x, y ) ∈ X × Y | v n, R ( y, v (2) n ( x, y ))( x ) = x } )= ( µ × ¯ ν )( { ( x, y, y ′ ) ∈ X × S | v n, R ( y, y ′ )( x ) = x } )= Z S µ ( { x ∈ X | v n, R ( y, y ′ )( x ) = x } ) d ¯ ν ( y, y ′ ) ≤ κ/ X u ∈ F Z S (cid:13)(cid:13) v n, R ( y, y ′ ) u − uv n, R ( y, y ′ ) (cid:13)(cid:13) d ¯ ν ( y, y ′ )= κ/ X u ∈ F k v n ( u ⊗ − ( u ⊗ v n k . Since ( v n ) n is by assumption a central sequence in [[ R × S ]], the latter converges to zero, finishingthe proof of the main part of the Lemma. The moreover part follows immediately. (cid:3)
Before continuing, we record the following easy fact.
Lemma 4.4.
Let ( X, µ ) be a standard probability space and f ∈ Aut(
X, µ ) . If f = id X , i.e. µ ( { x ∈ X | f ( x ) = x } ) > , then there exists a Borel subset Y ⊆ X of positive measure such that f ( Y ) ∩ Y = ∅ . The next Lemma tells us that we can recognize the absence of the Schmidt property in the stabi-lization of an equivalence relation.
Lemma 4.5.
Let R , R be countable ergodic p.m.p. equivalence relations on ( X , µ ) and ( X , µ ) respectively, and R , , R , be hyperfinite ergodic p.m.p. equivalence relations on ( Y , ν ) and ( Y , ν ) respectively, such that R × R , ∼ = R × R , . Assume that R is not Schmidt. Theneither(1) R is also not Schmidt, or(2) R is stable, and thus R ∼ = R × R , ∼ = R × R , .Proof. For notational convenience, we will identify ( X × Y , µ × ν ) and ( X × Y , µ × ν ) throughthe given isomorphism, and write µ = µ × ν = µ × ν . Assume by contradiction that R isSchmidt and not stable. Then, since R is Schmidt, we can find a central sequence ( T n ) n in [ R ]and δ > n , µ ( { x ∈ X | T n ( x ) = x } ) ≥ δ. Furthermore, since R is not stable, we must have by [JS85, Theorem 3.4] that for every asymp-totically invariant sequence ( A n ) n ⊆ X for R ,lim n →∞ µ ( T n A n ∆ A n ) → . By considering T n := T n × id Y ∈ [ R × R , ], we then get from the above that also(4.2) µ ( { ( x, y ) ∈ X × Y | T n ( x, y ) = ( x, y ) } ) = ( µ × ν )( { ( x, y ) ∈ X × Y | T n ( x ) = x } ) ≥ δ, and for any asymptotically invariant sequence ( A n ) n ⊆ X × Y ,(4.3) lim n →∞ µ ( T n A n ∆ A n ) → . For the latter, note that if ( A n ) n is an asymptotically invariant sequence in X × Y , it is inparticular asymptotically invariant under all maps of the form S × id Y ∈ [ R × R , ]. Therefore,writing A n,y := { x ∈ X | ( x, y ) ∈ A n } , the sets ( A n,y ) n have to be asymptotically invariant for R for almost every y ∈ Y . Thus, necessarily lim n →∞ µ ( T n A n,y ∆ A n,y ) → y ∈ Y .Integrating over Y , (4.3) then follows from the dominated convergence theorem. TABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCT EQUIVALENCE RELATIONS 19
Through the given isomorphism, we now view T n ∈ [ R ×R , ]. Since R is not Schmidt, Lemma 4.3provides central sequences T n,x ∈ [ R , ] for almost every x ∈ X such that(4.4) µ ( { ( x, y ) ∈ X × Y | T n ( x, y ) = ( x, T n,x ( y )) } ) → . Combining (4.4) and (4.2), we get that(4.5) ν ( { y ∈ Y | T n,x ( y ) = y } ) ≥ δ/ n and all x in some positive measure subset X ⊂ X .Since R , is ergodic, hyperfinite, and p.m.p., we can realize it as the orbit equivalence relationarising from the action of ⊕ N Z / Z on Y = Q N Z / Z by componentwise translation. Fix x ∈ X .Passing to a subsequence if necessary, and using the fact that ( T n,x ) n is a central sequence for thehyperfinite equivalence relation R , , we claim that we can find V n,x ∈ [ R , ] of the form V n,x = id Q i ≤ n Z / Z × ˜ V n,x such that(4.6) ν ( { y ∈ Y | T n,x ( y ) = V n,x ( y ) } ) → . Indeed, for a fixed m ≥
2, we can consider the (finitely many) elements { S k } of the full group [ R , ]corresponding to bijections { , } m → { , } m which only change the first m coordinates of a givenelement in Q N Z / Z . Note that these elements permute the clopen subsets { C a | a ∈ { , } m } where C a := { y = ( y i ) i ∈ Y | y i = a i for all i ≤ m } . Since ( T n ) n is a central sequence, we can take n large enough so that(4.7) ν ( { y ∈ Y | T n S k ( y ) = S k T n ( y ) for all k } ) > − m . Take y ∈ Y such that T n ( y ) i = y i for some i ≤ m , and let a = b ∈ { , } m be such that y ∈ C a and T n ( y ) ∈ C b . Choose any element S k as above that leaves C a invariant and maps C b onto C c for some c = a, b . Then T n ( S k ( y )) = T n ( y ) ∈ C b , but S k ( T n ( y )) ∈ C c , and therefore T n S k ( y ) = S k T n ( y ) . Combining this with (4.7), we conclude that the set Z := { y ∈ Y | T n ( y ) i = y i for some i ≤ m } has measure ν ( Z ) ≤ m . Now let V m,x equal T n,x on Y \ Z and define V m,x in any way on Z such that it leaves the first m coordinates of any element fixed. Then the V m,x will satisfy the above requirements.Combining (4.6) with (4.5), we now get ˜ δ > ν ( { y ∈ Y | V n,x ( y ) = y } ) ≥ ˜ δ, for large enough n . Applying Lemma 4.4 and a maximality argument, we can thus find subsets A n,x ⊂ Y such that ν ( A n,x ) ≥ ˜ δ/
2, and(4.8) V n,x ( A n,x ) ∩ A n,x = ∅ . By the construction of V n,x , we can moreover choose A n,x to be of the form A n,x = Q i ≤ n Z / Z × ˜ A n,x .Since T n,x depends on x in a measurable way, we can make sure that the same holds for V n,x and A n,x . Let A n := [ x ∈ X { x } × A n,x ⊂ X × Y . (One can also view this as a direct integral.) By the above, we have µ ( A n ) ≥ µ ( X )˜ δ/ > n . Moreover, by construction ( A n ) n is an asymptotically invariant sequence for R × R , ∼ = R × R , , and by (4.8) we have lim inf n →∞ µ ( V n A n ∆ A n ) > , where we defined V n ( x, y ) := ( x, V n,x ( y )) for ( x, y ) ∈ X × Y . Combining this with (4.6) and (4.4),we thus get lim inf n →∞ µ ( T n A n ∆ A n ) > . However, this is a contradiction with (4.3), and thus finishes the proof of the Lemma. (cid:3)
The last Lemma we will need in order to complete the proof of Theorem A deduces certain inter-twining phenomena in the presence of a non-Schmidt equivalence relation.
Lemma 4.6.
Let R , R be countable ergodic p.m.p. equivalence relations on ( X , µ ) and ( X , µ ) respectively, and suppose R , , R , are hyperfinite ergodic p.m.p. equivalence relations on ( Y , ν ) and ( Y , ν ) respectively. Assume that R is not Schmidt, and that R := R × R , ∼ = R × R , .Then R , ≺ R R , .Proof. For notational convenience, we will identify ( X × Y , µ × ν ) and ( X × Y , µ × ν ) throughthe given isomorphism, and denote it by ( Z, µ ). Assume that R , R R , . Claim.
We can find a sequence ( θ n ) ∞ n =1 ∈ [id X ×R , ] which is central for R , and such that ϕ id X ×R , ( θ n ) → Proof of the Claim.
Since R , is ergodic, hyperfinite, and p.m.p., we can again realize it as the orbitequivalence relation arising from the action of ⊕ N Z / Z on Q N Z / Z by componentwise translation.For a fixed n ∈ N , we denote by R ( n )0 , the equivalence relation R ( ⊕ m>n Z / Z y Q N Z / Z ). Byconstruction, R ( n )0 , ≤ R , has finite index. As subequivalence relations of R , since R , R R , ,we thus also get that R ( n )0 , R R , . Applying Lemma 3.1, we then get a sequence { θ ( n ) k } ∞ k =1 ⊂ [id X ×R ( n )0 , ] such that ϕ id X ×R , ( θ ( n ) k ) → k → ∞ . Moreover, by the construction of R ( n )0 , , the θ ( n ) k can be chosen to be of the form θ ( n ) k = Z ⊕ X (cid:16) id Q i ≤ n Z / Z × ˜ θ ( n ) k,x (cid:17) dµ ( x ) , where ˜ θ ( n ) k,x ∈ [ R ( ⊕ m>n Z / Z y Q N Z / Z )] for almost every x ∈ X . Choosing k n ≥ n such that ϕ id X ×R , ( θ ( n ) k n ) ≤ n and putting θ n := θ ( n ) k n we now have a sequence ( θ n ) n in [id X ×R , ] such that(a) ϕ id X ×R , ( θ n ) → n → ∞ , and(b) θ n is of the form θ n = R ⊕ X (cid:16) id Q i ≤ n Z / Z × ˜ θ n,x (cid:17) dµ ( x ) for some ˜ θ n,x ∈ [ R ( ⊕ m>n Z / Z y Q N Z / Z )].Since it follows from (b) that ( θ n ) n forms a central sequence in [ R × R , ] = [ R ], this finishes theproof of Claim. (End Claim (cid:3) )Taking ( θ n ) n as in the Claim, applying Lemma 4.3 together with the fact that R is not Schmidtgives central sequences ( θ n,x ) n in [[ R , ]] for almost every x ∈ X such that µ ( { ( x, y ) ∈ X × Y | θ n ( x, y ) = ( x, θ n,x ( y )) } ) → . In particular, denoting by θ (1) n ( x, y ) the first coordinate of θ n ( x, y ) ∈ X × Y , we get that(4.9) µ ( { ( x, y ) ∈ X × Y | θ (1) n ( x, y ) = x } ) → . However, the fact that ϕ id X ×R , ( θ n ) → µ ( { ( x, y ) ∈ X × Y | θ (1) n ( x, y ) = x } )= µ ( { ( x, y ) ∈ X × Y | θ n ( x, y )(id X ×R , )( x, y ) } )= ϕ id X ×R , ( θ n ) → . TABLE DECOMPOSITIONS AND RIGIDITY FOR PRODUCT EQUIVALENCE RELATIONS 21
This contradicts (4.9), and we conclude that indeed R , ≺ R R , . (cid:3) Proof of Theorem A.
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