W*-rigidity paradigms for embeddings of II_1 factors
aa r X i v : . [ m a t h . OA ] M a r W ∗ -rigidity paradigms for embeddings of II factors by Sorin Popa and Stefaan Vaes Abstract
We undertake a systematic study of W ∗ -rigidity paradigms for the embeddability relation ֒ → between separable II factors and its stable, weaker version ֒ → s involving amplificationsof factors. We obtain concrete large families of non stably isomorphic II factors that aremutually embeddable ( many-to-one paradigm) and large families of II factors that aremutually non stably embeddable ( disjointness paradigm). We provide an augmentationfunctor G H G from the category of groups into icc groups, so that L ( H G ) ֒ → s L ( H G )iff G ֒ → G . We produce a large class of II factors for which we compute all stableself-embeddings, including II factors M without nontrivial endomorphisms M ֒ → M t andII factors with numerous prescribed outer automorphism groups. We construct concrete complete intervals of II factors, indexed over a large variety of partially ordered sets,including a strict chain of II factors ( M k ) k ∈ Z with the property that if N is any II factorwith M i ֒ → s N and N ֒ → s M j , then N ∼ = M tk for some i ≤ k ≤ j and t > Table of contents factors: proof of Theorem E . . . . . 287 Complete intervals of II factors: proof of Theorem G . . . . . . 318 II factors with prescribed semiring of self-embeddings . . . . . . 379 Realizing algebraic lattices as intermediate subgroup lattices . . 47 The embedding problem in von Neumann algebras asks whether factors arising from certaingeometric data can be embedded one into the other or not. It is a natural companion to theisomorphism/classification problem for factors. The two problems are particularly interestingwhen the factors involved are of type II . They are parallel, and often interconnected, withsimilar problems in geometric group theory, orbit equivalence ergodic theory and Lie groupdynamics.The isomorphism and embedding problems for II factors can by and large be divided into twotypes of phenomena, the many-to-one paradigm and the one-to-one (or W ∗ -rigidity ) paradigm.While opposite in nature and involving completely different techniques, both types of resultsare hard to establish and they often come as a surprise. For the isomorphism problem, a resultshowing that distinct geometric data G (like groups, groupoids, groups acting on spaces, etc.) Mathematics Department, UCLA, Los Angeles, CA 90095-1555 (United States), [email protected] in part by NSF Grant DMS-1955812 and the Takesaki Chair in Operator Algebras. KU Leuven, Department of Mathematics, Leuven (Belgium), [email protected] by FWO research project G090420N of the Research Foundation Flanders and by long term structuralfunding – Methusalem grant of the Flemish Government. factor L ( G ), is a many-to-one paradigm, while a W ∗ -rigidity paradigmamounts to identifying a class G of objects for which the functor G ∋ G L ( G ) ∈ II isone-to-one, modulo isomorphisms in G , respectively II .The typical many-to-one result for the embedding problem singles out a class of II factorsthat are mutually embeddable but not isomorphic. In turn, its W ∗ -rigidity counterpart has anarray of interesting questions, from the basic non-embeddability between specific II factors,to obtaining families of mutually non-embeddable factors ( disjointness ), calculating all self-embeddings of factors within a certain class, etc. Viewing the embeddability of II factors asa preorder relation in the class II , or alternatively as a weak subordination relation of theunderlying geometric objects, leads to a series of more nuanced problems.Despite the diversity of interesting questions of this type, and in sharp contrast with the recentspectacular progress on the isomorphism problem, the study of embedding phenomena hasbeen much less developed. We undertake in this paper a systematic study of the embeddingparadigms for II factors, using all tools presently available, especially deformation-rigiditytheory, and obtaining a multitude of completely new types of results.Thus, denoting by ֒ → the embedding relation between II factors and by ֒ → s its stable, weakerversion involving amplifications of factors, we exhibit one-parameter families of non stablyisomorphic separable II factors ( M t ) t ∈ R of each of the following types: examples where thefactors M t are mutually embeddable (a many-to-one result); examples where M t ֒ → s M r ifand only if t ≤ r (a strict chain); and examples that are mutually non-embeddable (an anti-chain). We actually produce such families of II factors ( M i ) i ∈ I indexed by different typesof partially ordered sets ( I, ≤ ). Moreover, we construct a large class of II factors for whichwe can compute all stable self-embeddings M ֒ → M t , including II factors M for which allembeddings M ֒ → M t are inner, II factors with a prescribed countable one-sided fundamentalgroup , and II factors with prescribed outer automorphism group from a wide range of Polishgroups. Also, we construct an augmentation functor assigning to any infinite group G an iccgroup H G such that L ( H G ) ֒ → s L ( H G ) if and only if G is isomorphic with a subgroup of G . Finally, we produce concrete complete intervals of II factors ( M i ) i ∈ I , indexed by a largevariety of partially ordered sets, where completeness means that any intermediate II factor N , with M i ֒ → s N and N ֒ → s M j , is stably isomorphic with M k for some i ≤ k ≤ j .Before stating these results in detail, and in order to put them in a proper perspective, letus briefly review the history of the embedding problem and the main results obtained in thisdirection over the years.Both the isomorphism and embedding problems have been initiated by Murray-von Neumannin [MvN43]. They obtained the first many-to-one results, by showing that all “locally finite”geometric objects G give rise to the hyperfinite II factor, R := ⊗ n ( M ( C ) , tr) n , which in turncan be embedded into any other II factor. Interestingly, they also construct two non stablyisomorphic, but mutually embeddable II factors in [MvN43, Appendix] and they comment on[MvN43, page 717]: “the possibility exists that any factor in the case II is isomorphic to asub-ring of any other such factor.” In other words, Murray and von Neumann did not excludethe possibility that no matter the geometric data they may come from, all separable II factorslook alike under the very weak equivalence relation given by mutual embeddability!It took more than two decades to prove that this is not the case. Thus, work in [Sch63, HT67]provided the right notion of amenability for II factors, showed that it is inherited by em-beddable factors and that L ( G ) is amenable as a II factor iff G is amenable (with its initialstructure), leading to the first non-embeddability result: if G is amenable and G is not, then L ( G ) ֒ → L ( G ), e.g. L ( F ) ֒ → L ( S ∞ ), where F is the free group on two generators and S ∞ is the group of finite permutations of N . Connes’ fundamental theorem [Con76] completedthe picture: all II factors L ( G ) arising from amenable geometric objects G , as well as all II R , are isomorphic to R . This reduced both the isomorphism and the embeddingproblems to II factors and initial data that are nonamenable.Two decades later, Connes and Jones obtained in [CJ83] a landmark non-embeddability resultfor nonamenable II factors, showing that if G is a group with Kazhdan’s property (T), e.g. G = SL(3 , Z ), and G is a group having the Haagerup property, e.g. G = F , then L ( G ) ֒ → L ( G ). A few years later, Cowling and Haagerup proved in [CH88] the striking, deep result thatif for each n ≥ G n = Sp(1 , n ) Z , then L ( G n ) ֒ → s L ( G m ) iff n ≤ m . Note that bothof these W ∗ -rigidity statements are about the non-embeddability of more rigid objects into lessrigid ones.During that same period of time, several embedding W ∗ -rigidity results modulo countableclasses were obtained in [Pop86], using a separability pigeon-hole principle trick, showing forinstance that, within the class T of property (T) II factors (as defined in [CJ83]), the equiv-alence relation given by mutual stable embedding with finite index (also called virtual isomor-phism ) is countable-to-one with respect to actual isomorphism. The separability trick togetherwith the abundance of simple property (T) groups in (see [Gro87]) was then used in [Oza02]to show that the embedding preorder relation does not have an upper bound in the class ofseparable II factors.Starting in 2001, deformation/rigidity and intertwining-by-bimodules techniques allowed a sys-tematic insight into W ∗ -rigidity phenomena. While particularly revealing for the isomorphismproblem, this led to several non-embeddability results as well. For instance, it is shown in[IPP05] that if n > m then the II factor arising from a free product of n property (T) in-finite groups cannot be embedded into one that comes from the free product of m property(T) groups. Also, it is shown in [OP07] that one cannot embed a nonamenable II factor withCartan subalgebra into a free group factor. On the other hand, a non-embeddability resultbased on amenability of boundary actions was obtained in [Oza03], showing that if N is anonamenable II factor that either has property Gamma of [MvN43], or is a tensor productof II factors, then it cannot be embedded into a free group factor. A number of results wereobtained about the non-embeddability of a tensor product of n factors of type II into a tensorproduct of a strictly smaller number of factors from a certain class, like free group factors, seee.g. [OP03].The main focus of deformation/rigidity theory has been on the isomorphism and classificationproblem for families of II factors. Nevertheless, several of these articles contained, bothimplicitly and explicitly, partial results on the structure of embeddings between specific groupII factors and group measure space II factors, most notably for Bernoulli crossed products,see e.g. [Pop03, Pop06, CH08, PV09, Ioa10, CS11, KV15].But overall, these recent tools have been very little used in a systematic study of the embed-ding problem. In particular, the two most prominent such problems, going back to [MvN43],remained open: finding large classes of mutually non stably embeddable II factors; findingnatural classes of mutually embeddable but non stably isomorphic II factors. While we weresolving these problems, stated as Theorem A and Corollary B below, a large number of ad-ditional questions naturally emerged and led to Theorems C – G, attesting to the strikingrichness of such W ∗ -rigidity phenomena.To state in details our main results, we need to fix some notations. As above, when N and M are II factors, we write N ֒ → M if there exists a unital normal ∗ -homomorphism from N to M . We write N ֒ → s M , to be read as N stably embeds into M , if N ֒ → M t for some t > N t ֒ → M for some t >
0. We write M ∼ = s N when M and N are stably isomorphic II factors, meaning that M ∼ = N t for some t >
0. Recall that a normal ∗ -homomorphismbetween II factors is necessarily a trace preserving embedding.3ur first result provides a family of II factors indexed by tracial amenable von Neumannalgebras and a precise characterization of when one can be (stably) embedded in the other. Theorem A.
Let
Γ = F n be the free group with ≤ n ≤ + ∞ generators. For every amenabletracial von Neumann algebra ( A , τ ) with A = C , we define the II factor M ( A , τ ) as theleft-right Bernoulli crossed product M ( A , τ ) = ( A , τ ) ⊗ Γ ⋊ (Γ × Γ) . (1.1) We have M ( A , τ ) ֒ → s M ( A , τ ) iff M ( A , τ ) ֒ → M ( A , τ ) iff there exists a trace preservingunital embedding ( A , τ ) ֒ → ( A , τ ) . A few particular choices of A lead to the following interesting examples, providing concretefamilies of II factors that are mutually incomparable for ֒ → s , as well as chains of II factors forthe relation ֒ → s and families of II factors that are mutually embeddable, yet nonisomorphicand even non virtually isomorphic, meaning that it is impossible to embed one as a finite indexsubfactor of an amplification of the other (see Remark 3.4). Corollary B.
We use the notation (1.1). • For a ∈ (0 , / , write A a = C ⊕ C with τ a ( x ⊕ y ) = ax + (1 − a ) y . Then M a = M ( A a , τ a ) isan anti-chain for the preorder relations ֒ → s and ֒ → , i.e. M a ֒ → s M b iff M a ֒ → M b iff a = b . • For a ∈ (0 , , denote by η a the probability measure on [0 , a ] ∪ { } given by the Lebesguemeasure on [0 , a ] and the atom η a (1) = 1 − a . Writing ( B a , τ a ) = L ∞ ([0 , a ] ∪ { } , η a ) , theII factors P a = M ( B a , τ a ) form a strict chain, i.e. P a ֒ → s P b iff P a ֒ → P b iff a ≤ b . • Let R be the hyperfinite II factor. For a ∈ (0 , / , define the trace τ a on R ⊕ R by τ a ( x ⊕ y ) = aτ ( x ) + (1 − a ) τ ( y ) . Write Q a = M ( R ⊕ R, τ a ) . We have Q a ֒ → M ( R, τ ) ֒ → Q b for all a, b ∈ (0 , / , while Q a ∼ = Q b iff Q a ∼ = s Q b iff Q a , Q b are virtually isomorphic iff a = b . Following [Ioa10, Definition 10.4], one defines the one-sided fundamental group F s ( M ) of a II factor M as the set of t > M ֒ → M t . Note that F s ( M ) also is the set of possiblefinite values of the right dimension dim( H M ) of a Hilbert M -bimodule H . Also note that F s ( M ) ⊆ R ∗ + is closed under addition and multiplication. We always have N ⊆ F s ( M ) and thefollowing dichotomy holds: either F s ( M ) ∩ (0 ,
1) = ∅ , or F s ( M ) = R ∗ + (see [Ioa10, Section 10]).The only existing computations of F s ( M ) yield N or R ∗ + . Combining our results in Theorem Awith the methods of [IPP05, Corollary 6.5], we realize any countable semiring N ⊆ F ⊆ [1 , + ∞ )as one-sided fundamental group. Theorem C.
Fix one of the II factors M = M a given in the first point of Corollary B. Let F ⊂ [1 , + ∞ ) be a countable subset satisfying F + F ⊆ F , F F ⊆ F and N ⊆ F . Define the freeproduct II factor P = ∗ t ∈F M /t . Then, F s ( P ) = F . We then turn to a systematic study of stable self-embeddings θ : M → M d . We say that such aself-embedding is trivial if d ∈ N and if θ is unitarily conjugate to M ֒ → M d ( C ) ⊗ M : x ⊗ x .It was conjectured in [Ioa10, Section 10] that there exist II factors M such that every stableself-embedding M ֒ → s M is trivial. This conjecture was proven in [Dep13], with an extremelyinvolved construction and the paper remaining unpublished. Our methods provide the followingsimpler example. 4enote by G the group of all finite, even permutations of N and let G < G be the subgroup ofpermutations fixing 1 ∈ N . Put Γ = G ∗ G and Γ = G ∗ G . Denote by A ⊂ G the subset ofelements of order 2 and by B ⊂ G the subset of elements of order 3. View A ⊂ Γ as elementsin the first copy free product factor G and view B ⊂ Γ as elements in the second free productfactor G . For all ( a, b ) ∈ A × B , define the subgroup Λ a,b < Γ × Γ generated by ( a, a ) and( b, b ). Note that Λ a,b ∼ = Z / Z ∗ Z / Z . Theorem D.
For all ( a, b ) ∈ A × B , choose distinct probability measures µ a,b on { , } with µ a,b (0) ∈ (0 , / . Consider the generalized Bernoulli action Γ × Γ y ( X, µ ) = Y ( a,b ) ∈A×B (cid:0) { , } , µ a,b (cid:1) (Γ × Γ) / Λ a,b and define M = L ∞ ( X, µ ) ⋊ (Γ × Γ) . Then every embedding M ֒ → M t is trivial. One may view Theorem A as a functorial construction that takes a base algebra ( A , τ ) asinput and produces a II factor such that embeddability holds if and only if the initial dataare embeddable. We develop this approach much further and provide the following concreteaugmentation functor assigning to any infinite group Γ an icc group H Γ , through a general-ized wreath product construction, such that L ( H Γ ) ֒ → s L ( H Λ ) iff L ( H Γ ) ֒ → L ( H Λ ) iff Γ isisomorphic with a subgroup of Λ. Theorem E.
Let Γ be an infinite group (not necessarily countable). Denote by G Γ ∼ = F | Γ | the free group with free generators a and ( a g ) g ∈ Γ . Define the surjective group homomorphism π Γ : G Γ → Z ∗ Γ by π Γ ( a ) = 1 ∈ Z and π Γ ( a g ) = g for all g ∈ Γ . Define the subgroup N Γ < G Γ × G Γ consisting of the elements ( g, g ) with π Γ ( g ) ∈ Γ . Consider the generalizedwreath product group H Γ = (cid:0) Z / Z (cid:1) (( G Γ × G Γ ) /N Γ ) ⋊ ( G Γ × G Γ ) . If Γ and Λ are arbitrary infinite groups, we have L ( H Γ ) ֒ → s L ( H Λ ) iff L ( H Γ ) ֒ → L ( H Λ ) iff Γ is isomorphic with a subgroup of Λ .The II factors L ( H Γ ) and L ( H Λ ) are virtually isomorphic iff they are stably isomorphic iffthey are isomorphic iff Γ ∼ = Λ . Both ֒ → and ֒ → s define preorder relations on the class of II factors. We similarly have apreorder relation ֒ → on the class of groups defined by Γ ֒ → Λ iff Γ is isomorphic to a subgroupof Λ. Combining Theorem E with the wildness of the relation ֒ → on countable groups, we mayconcretely realize numerous partial orders and total orders ( I, ≤ ) inside ( II , ֒ → ) and ( II , ֒ → s ),see Corollary F.More precisely, given a partially ordered set ( I, ≤ ), we want to construct a family of II factors( M i ) i ∈ I such that M i ֒ → M j (resp. M i ֒ → s M j ) if and only if i ≤ j . It is natural to ask whichpartial orders can be realized in this way within II factors having separable predual.We provide the following concrete interesting examples. The meaning of “concrete” will becomeclear in the proof of Corollary F, which is entirely constructive. In the formulation of thecorollary, we say that a subset I ⊆ I of a partially ordered set ( I, ≤ ) is sup-dense if everyelement of I is the supremum of a subset of I . We say that ( I, ≤ ) is separable if it admits acountable sup-dense subset. More generally, we say that ( I, ≤ ) has density character at most κ if κ is an infinite cardinal number and ( I, ≤ ) admits a sup-dense subset of cardinality atmost κ . We similarly say that a II factor M has density character at most κ if M admits a k · k -dense subset of cardinality at most κ . 5 orollary F. Let ( I, ≤ ) be a partially ordered set with density character at most κ . There existsa concrete family of II factors ( M i ) i ∈ I with density character at most κ such that M i ֒ → M j iff M i ֒ → s M j iff i ≤ j .There are concrete chains of II factors with separable predual, both in ( II , ֒ → ) and ( II , ֒ → s ) ,of order type ( R , ≤ ) and of order type ( ω , ≤ ) , where ω is the first uncountable ordinal. In specific cases, we may even determine complete intervals in ( II , ֒ → s ), meaning that we notonly provide families of II factors ( M i ) i ∈ I indexed by a partially ordered set ( I, ≤ ) with theproperty M i ֒ → s M j iff i ≤ j , but also having the property that no other II factors can sitbetween these M i : whenever N is a II factor and M i ֒ → s N and N ֒ → s M j , there exists a k ∈ I such that i ≤ k ≤ j and N ∼ = s M k .It is quite challenging to provide such complete intervals of II factors, since we should at thesame time control all possible embeddings M i ֒ → M tj and determine all intermediate subfactors M i ֒ → N ֒ → M tj . An appropriate variant of the construction in Theorem D allows to do allthis at once.We can then provide a concrete construction of such complete intervals of II factors indexedby several kinds of lattices (i.e. partially ordered sets in which every pair of elements has aninfimum and a supremum).Our result covers all “discrete” lattices like ( Z , ≤ ), arbitrary finite lattices or, more generally,any lattice ( I, ≤ ) with the property that { i ∈ I | a ≤ i ≤ b } is finite for all a, b ∈ I . Our resultactually covers many complete lattices, i.e. partially ordered sets in which every subset has aninfimum and a supremum, like ( Z ∪ {−∞ , + ∞} , ≤ ). In its most general form, our theoremrealizes every separable algebraic lattice as a complete interval of II factors w.r.t. ֒ → s .Separable algebraic lattices are abundant. Any partially ordered set can be canonically com-pleted into the algebraic lattice ( I, ≤ ), defined as the set of all downward closed subsets of I ,ordered by inclusion. If I is countable, ( I, ≤ ) is separable. Whenever Λ is a group and Λ Λis a subgroup, the lattice of intermediate subgroups ordered by inclusion is an algebraic lattice.It is separable if Λ is a countable group. By [Tum86], every algebraic lattice arises in thisway. For our proof of Theorem G, we will actually reprove this characterization with an extracontrol on the intermediate subgroups, like not having nontrivial finite dimensional unitaryrepresentations (see Theorem 9.1). We do this by adapting the proof of [Rep04].
Theorem G.
Let ( I, ≤ ) be any separable algebraic lattice. There exists a family of separableII factors ( M i ) i ∈ I with the following properties: • we have M i ֒ → s M j iff i ≤ j iff M i ֒ → M j , • if N is any II factor such that M i ֒ → s N and N ֒ → s M j for some i, j ∈ I , there exists a k ∈ I with i ≤ k ≤ j and N ∼ = s M k , • if N is any II factor such that M i ֒ → N and N ֒ → M j for some i, j ∈ I , there exists a k ∈ I with i ≤ k ≤ j and N ∼ = M k . The level of concreteness of the family ( M i ) i ∈ I depends on the nature of ( I, ≤ ). When ( I, ≤ ) isthe completion ( I , ≤ ) of a countable partially ordered set, we give a separate proof of TheoremG that is entirely constructive and essentially “functorial” in ( I, ≤ ): see Proposition 7.3. Note An algebraic lattice is a complete lattice ( I, ≤ ) in which every element can be written as a supremum of compactelements. An element a ∈ I is called compact if whenever a ≤ sup J for some J ⊆ I , then a ≤ sup J for afinite subset J ⊆ J . A subset J ⊆ I is called downward closed if for all j ∈ J and all k ∈ I with k ≤ j , we have k ∈ J . Identifying i ∈ I with the subset { j ∈ I | j ≤ i } of I , we embed ( I, ≤ ) in its completion ( I, ≤ ). factors realizing complete intervals of the form( Z , ≤ ), or of the form ([0 , λ ] , ≤ ), where λ is any countable ordinal. For arbitrary separablealgebraic lattices, our proof depends on a much less explicit inductive procedure.We also provide the following example to illustrate how much weaker is the equivalence relationgiven by mutual embedding “ N ֒ → M and M ֒ → N ” from isomorphism N ∼ = M or stableisomorphism N ∼ = s M .Indeed, while the third point of Corollary B already provides an uncountable family of II factorsthat are mutually embeddable but pairwise nonisomorphic, things can become considerablyworse. Thus, denote by A the set of nonzero subgroups of Λ ∞ = Q ( N ) , the direct sum ofcountably many copies of the additive group Q . For every Λ ∈ A , using the notation ofTheorem E, we define P Λ = L ( H Λ × Λ ∞ ). By construction, P Λ ֒ → P Λ ′ for all Λ , Λ ′ ∈ A . On theother hand, P Λ ∼ = P Λ ′ iff P Λ ∼ = s P Λ ′ iff Λ × Λ ∞ ∼ = Λ ′ × Λ ∞ . By [DM07, Theorem 2.1], thisis a complete analytic equivalence relation on the standard Borel space A , i.e. an equivalencerelation that is as complicated as it can be in this context.Finally, variants of our method to prove Theorem D also lead to constructions of II factors M for which the semiring of stable self-embeddings M ֒ → s M can be explicitly determined.We consider such stable embeddings up to unitary conjugacy. Since we can take direct sumsand compositions of embedings, we obtain a semiring that we denote as Emb s ( M ). We provein Theorem 8.1 that for any countable structure G (like a countable group or a countablefield) with semigroup of self-embeddings S = Emb( G ), there exists a II factor M such thatEmb s ( M ) ∼ = N [ S ], the semiring of formal sums of elements in S . The construction of M isentirely explicit.This even provides new results on the possible outer automorphism groups Out( M ) of II factors. We prove in Corollary 8.2 that all of the following Polish groups can be realized asouter automorphism groups of full II factors: any closed subgroup of the Polish group ofall permutations of N , the unitary group U ( H ) of a separable Hilbert space and the unitarygroup U ( N ) of any von Neumann algebra with separable predual. As we discuss in Section 8,this encompasses all known realizations of outer automorphism groups and this is the firstsystematic result to realize a large class of non locally compact groups as Out( M ).We conclude this introduction with a few comments on our methods and proofs. We proveTheorem A as a special case of Theorem 3.2, which deals with a larger family of groups Γ anda possibly nontrivial action Γ y ( A , τ ) as initial data for the construction. To analyze allpossible embeddings θ : B ⋊ (Λ × Λ) → ( A ⋊ (Γ × Γ)) d between such generalized Bernoulliactions, we use several results from deformation/rigidity theory for Bernoulli crossed products,see [Pop03, Ioa10, IPV10, KV15]. This leads to a point where we know that after a unitaryconjugacy, θ maps L (Λ × Λ) into L (Γ × Γ) d such that the height of θ (Λ × Λ) inside L (Γ × Γ) d is positive.In [IPV10], it was proven that for isomorphisms θ : L (Λ) → L (Γ), with moreover d = 1,positive height implies that θ can be unitarily conjugated to an isomorphism that comes froman isomorphism between the groups Λ and Γ. In Section 2, we prove a generalization of thisresult for arbitrary embeddings θ : L (Λ) → L (Γ) d , also allowing for an arbitrary amplification.This unavoidably includes the appearance of finite index subgroups and finite-dimensionalunitary representations.Applying this result to our initial embedding of Bernoulli crossed products, the restriction of θ to L (Λ × Λ) must be group-like. From that point onward, it is easy to completely describeall possible embeddings, in terms of embeddings between the initial data Λ y ( B , τ ) andΓ y ( A , τ ).One may thus summarize the approach to Theorems A and 3.2 as follows: the context of7ernoulli crossed products is so rigid that any embedding must come from an embedding ofthe “base spaces”. And then Corollary B results from variants of the trivial observation thata base space with 3 points cannot be embedded into a base space with 2 points. Our resultson II factors with prescribed embeddings semiring, or prescribed outer automorphism group,arise in the same way, by constructing initial data with prescribed symmetries.The main focus and the main novelty of this paper are to consider arbitrary embeddings betweenarbitrary II factors N, M and their amplifications by any t >
0. This allows us to obtain newtypes of results even in the case N = M , where Ioana obtained in [Ioa10] numerous rigidityresults on self-embeddings M → pM p between Bernoulli crossed product factors of productgroups Γ × Γ, including cases where this forces p to be 1 and the embedding to be canonical,but where the cases t > factors that aremutually embeddable, but not (stably) isomorphic. Deformation/rigidity theory has provideda wealth of classification theorems up to stable isomorphisms. They can be exploited to givemany such families of mutually embeddable, but not (stably) isomorphic II factors. For in-stance, whenever the fundamental group F ( M ) is nontrivial, this fundamental group containsarbitrarily small t >
0. Taking direct sums of such M ∼ = M t , we find that M ֒ → M s forall s >
0. Taking representatives s > R ∗ + / F ( M ), the II factors M s aremutually embeddable, but not isomorphic. Of course, they are stably isomorphic by construc-tion. One can obtain mutually embeddable, but not stably isomorphic II factors by applying[Ioa10, IPV10] to Bernoulli actions of mutually embeddable, nonisomorphic groups. Also thefactor actions of a fixed Bernoulli action, as studied in [Pop04], lead to such examples since1-cohomology provides a method to distinguish these II factors up to stable isomorphism.As we already mentioned, our aim in this paper is to reveal unexpected W ∗ -rigidity paradigmsand illustrate them with classes of examples that have intrinsic interest. But there are famousembedding problems that remain wide open. For instance, a natural embedding version ofConnes’ rigidity conjecture asks whether for G n = PSL( n, Z ), n ≥
3, one has L ( G n ) ֒ → L ( G m )iff n ≤ m . Also, it is not known whether the class of nonamenable II factors admits a leastelement for the preorder relations ֒ → and ֒ → s . A well known conjecture predicts that L ( F ) issuch a least element, i.e. a nonamenable II factor that embeds into any other nonamenableII factor. This can be viewed as the W ∗ -version of the von Neumann-Day conjecture, thatany nonamenable group contains a copy of F , shown to be false in [Ols80]. But this W ∗ -version is believed to be true. Since the free group factors L ( F n ), 2 ≤ n ≤ ∞ , are mutuallyembeddable, they would all be least elements in ( II , ֒ → ) and define a single class N underthe mutual embedding equivalence relation. How large that class N would be, with respectto plain isomorphism (respectively stable isomorphism) of II factors N ∈ N , depends on theanswer to two other famous problems. On the one hand, it has been speculated (see e.g. [PS18])that if N is nonamenable and N ֒ → s L ( F ), then N must be an interpolated free group factor L ( F t ), for some 1 < t ≤ ∞ . Assuming this to be true as well, N would thus consist of allinterpolated free group factors and its further description would depend on the answer to thefree group factor problem. Recall in this respect that Voiculescu’s free probability theory hasbeen used in [Dyk92, Rad92] to show that L ( F n ), 2 ≤ n ≤ ∞ , are either all isomorphic, or allnonisomorphic. It was also shown that L ( F n ), 2 ≤ n < ∞ , are stably isomorphic and that theirnonisomorphism implies that L ( F ∞ ) is not stably isomorphic to any of them. Let Γ be a countable group and G U ( p ( M n ( C ) ⊗ L (Γ)) p ) a subgroup. As in [Ioa10, Section4] and [IPV10, Section 3], we say that G has zero height , denoted as h Γ ( G ) = 0, if there exists8 sequence v n ∈ G such thatsup g ∈ Γ k (id ⊗ τ )( v n (1 ⊗ u ∗ g )) k → n → + ∞ .More intuitively, h Γ ( G ) = 0 if and only if G contains a sequence whose Fourier coefficients tendto zero uniformly in g ∈ Γ.Recall that [IPV10, Theorem 3.1] is saying the following: if Γ is an icc group and π : L (Λ) → L (Γ) is a ∗ -isomorphism, then the following two statements are equivalent.1. We have h Γ ( π (Λ)) > w ∈ U ( L (Γ)) such that wπ ( T Λ) w ∗ = T Γ.As mentioned in [IPV10, Remark 3.2], it is unclear whether a similar result holds when π : L (Λ) → L (Γ) is an (irreducible) embedding rather than an isomorphism. It is even less clearwhat can be said if π : L (Λ) → L (Γ) t is an embedding into an amplification of L (Γ).When t ≤ π : L (Λ) → L (Γ) t is weakly mixing , it was proven in [KV15,Theorem 4.1] that the condition h Γ ( π (Λ)) > t = 1 and the embeddingbeing standard as above.In the main result of this section, we prove that when Γ is an icc group in Ozawa’s class ( S ) (see [Oza04] and [BO08, Definition 15.1.2]), one can handle arbitrary t > factors into arbitrary amplifications.Let Γ be a countable group and let G be any group. We start by describing the standard/trivialways to obtain group homomorphisms from G to the unitary group of an amplification of L (Γ), by combining group homomorphisms G →
Γ, finite-dimensional unitary representationsand finite index considerations. First, whenever δ : G →
Γ is a group homomorphism and γ : G → U ( n ) is a finite dimensional unitary representation, we may consider π γ,δ : G → U ( M n ( C ) ⊗ L (Γ)) : π γ,δ ( v ) = γ ( v ) ⊗ u δ ( v ) . Secondly, if G G is a finite index subgroup and π : G → U ( N ) is a group homomorphism tothe unitary group of a von Neumann algebra N , we write m = [ G : G ], choose representatives v , . . . , v m ∈ G for the cosets G / G and get a natural induction π : G → U ( M m ( C ) ⊗ N ) : π ( v ) ij = ( π ( w ) if vv j = v i w for some w ∈ G ,0 if vv j v i G . Definition 2.1.
Let Γ be an icc group and t >
0. Let G be a group. We say that a homomor-phism π : G → U ( L (Γ) t ) is standard if t ∈ N and if π is unitarily conjugate to a finite directsum of inductions to G of homomorphisms of the form π γ,δ : G → U ( M n ( C ) ⊗ L (Γ)) where G G is a finite index subgroup, δ : G → Γ is a group homomorphism and γ : G → U ( n ) isan n -dimensional unitary representation. Remark 2.2.
Definition 2.1 of a standard homomorphism of a group to U ( L (Γ) t ) may sounda bit cumbersome, but this is unavoidable since one may take inductions and direct sumsof arbitrary homomorphisms. Note that if π : G → U ( L (Γ) t ) is a standard homomorphism,then t = n ∈ N and there exists a finite index subgroup G G , a unitary representation γ : G → U ( C n ) and a group homomorphism δ : G → Γ such that the restriction π | G isunitarily conjugate to π γ,δ : G → U ( M n ( C ) ⊗ L (Γ)).9learly, if π : G → U ( L (Γ) t ) is a standard homomorphism, we have h Γ ( π ( G )) >
0. The mainresult of this section shows that the converse holds if Γ belongs to Ozawa’s class ( S ) and π ( G ) ′′ is not too small. For our applications in this paper, the important advantage of Theorem2.3 compared to [KV15, Theorem 4.1] is to make no assumptions at all on the nature of thehomomorphism π .Since we allow arbitrary amplifications, the formulation of the theorem has to take into accountdirect sums of homomorphisms. The theorem then says that an arbitrary homomorphism isthe direct sum of a standard homomorphism, a homomorphism whose image has zero heightand a homomorphism whose image is small (i.e. amenable). Theorem 2.3.
Let Γ be a countable group in Ozawa’s class ( S ). Assume that the centralizerof every element g ∈ Γ \ { e } is amenable.If G is a group, t > and π : G → U ( L (Γ) t ) is a group homomorphism, there exist projections p , p , p ∈ L (Γ) t ∩ π ( G ) ′ with p + p + p = 1 such that • The homomorphism
G ∋ v π ( v ) p is standard. • We have h Γ ( π ( G ) p ) = 0 . • We have that π ( G ) ′′ p is amenable. If Γ belongs to class ( S ), the centralizer of every infinite subgroup of Γ is amenable. So,Theorem 2.3 applies to all torsion free groups in class ( S ).We start by proving the following lemma, generalizing [KV15, Theorem 4.1] to arbitraryamplifications, with a very similar proof. For every element g in a group Γ, we denote by C Γ ( g ) = { h ∈ Γ | gh = hg } its centralizer. Lemma 2.4.
Let Γ be an icc group and t > . Write P = L (Γ) t . Let G U ( P ) be a subgroupsuch that the unitary representation (Ad v ) v ∈G on L ( P ) ⊖ C is weakly mixing. Assume thatfor all g ∈ Γ \ { e } , we have G ′′ L ( C Γ ( g )) .If h Γ ( G ) > , then t = 1 and there exists a unitary W ∈ L (Γ) such that W G W ∗ ⊆ T Γ .Proof. Take an integer n ≥ t , write M = M n ( C ), Q = L (Γ) and realize P = p ( M ⊗ Q ) p , where p is a projection with (Tr ⊗ τ )( p ) = t . We use in this proof multiple tensor products of M and Q . We use the tensor leg numbering notation, where e.g. ( a ⊗ b ⊗ c ⊗ d ) = b ⊗ a ⊗ d ⊗ c .When v ∈ M ⊗ Q , we denote by v ij ∈ Q the matrix coefficients. When a ∈ Q = L (Γ), we write( a ) g = τ ( au ∗ g ) for all g ∈ Γ.By the assumption that h Γ ( G ) >
0, there exists a δ > g ∈ Γ n X k,l =1 | ( v kl ) g | ≥ δ for all v ∈ G .Denote by ∆ : L (Γ) → L (Γ) ⊗ L (Γ) : ∆( u g ) = u g ⊗ u g the comultiplication. Note that for all v ∈ G ⊆ M ⊗ Q ,(Tr ⊗ Tr ⊗ τ ⊗ τ ⊗ τ ) (cid:0) ( v ⊗ (id ⊗ ∆)( v )) ((id ⊗ ∆)( v ∗ ) ⊗ v ∗ ) (cid:1) = n X i,j,k,l =1 ( τ ⊗ τ ⊗ τ ) (cid:0) ( v ij ⊗ ∆( v kl )) (∆( v ∗ ij ) ⊗ v ∗ kl ) (cid:1) = X g ∈ Γ n X i,j,k,l =1 | ( v ij ) g | | ( v kl ) g | ≥ (cid:16) sup g ∈ Γ n X k,l =1 | ( v kl ) g | (cid:17) ≥ δ > . X ∈ M ⊗ M ⊗ Q ⊗ Q ⊗ Q as the unique element of minimal k · k in the closed linearspan of (cid:8) ( v ⊗ (id ⊗ ∆)( v )) ((id ⊗ ∆)( v ∗ ) ⊗ v ∗ ) (cid:12)(cid:12) v ∈ G (cid:9) , the estimate above implies that(Tr ⊗ Tr ⊗ τ ⊗ τ ⊗ τ )( X ) ≥ δ , so that X = 0. Write q = (id ⊗ ∆)( p ). By construction and by the uniqueness of X ,( p ⊗ q ) X = X = X ( q ⊗ p ) and( v ⊗ (id ⊗ ∆)( v )) X = X ((id ⊗ ∆)( v ) ⊗ v ) for all v ∈ G .By our assumption that G ′′ L ( C Γ ( g )) if g ∈ Γ \ { e } and by [IPV10, Proposition 7.2(3)], itfollows that the unitary representation (Ad(id ⊗ ∆)( v )) v ∈G of G on the orthogonal complementof (id ⊗ ∆)( p ( M ⊗ Q ) p ) inside L ( q ( M ⊗ Q ⊗ Q ) q ) is weakly mixing. We also assumed thatthe unitary representation (Ad v ) v ∈G on L ( p ( M ⊗ Q ) p ) ⊖ C p is weakly mixing. Applying ∆,it follows that (Ad(id ⊗ ∆)( v )) v ∈G is weakly mixing on L ( q ( M ⊗ Q ⊗ Q ) q ) ⊖ C q . Taking thetensor product with the representation (Ad v ) v ∈G , it follows that the unitary representation(Ad( v ⊗ (id ⊗ ∆)( v ))) v ∈G on L ( p ( M ⊗ Q ) p ⊗ q ( M ⊗ Q ⊗ Q ) q ) ⊖ C ( p ⊗ q )is weakly mixing. Since the left support of X is invariant under this representation (afterreshuffling the tensor factors), we conclude that after multiplying X with a scalar, X is apartial isometry with left support ( p ⊗ q ) . Making the same reasoning on the right, theright support of X then equals ( q ⊗ p ) .Since the left and right supports match correctly, the element Y ∈ ( M ⊗ M ⊗ M ) ⊗ ( Q ⊗ Q ⊗ Q ⊗ Q ) given by Y = X X is a partial isometry with left support ( p ⊗ p ⊗ q ) and right support ( q ⊗ p ⊗ p ) and we have( v ⊗ v ⊗ (id ⊗ ∆)( v )) Y = Y ((id ⊗ ∆)( v ) ⊗ v ⊗ v ) for all v ∈ G .Consider the Hilbert space K = ( p ⊗ p ) (cid:0) ( C n ⊗ C n )( C n ) ∗ ⊗ L ( Q ⊗ Q ) (cid:1) q and the unitary representation ζ : G → U ( K ) : ζ ( v )( T ) = ( v ⊗ v ) T (id ⊗ ∆)( v ∗ ) . We can view Y as a nonzero invariant vector of the unitary representation ζ ⊗ ζ . Therefore, ζ is not a weakly mixing representation. We thus find an integer k ∈ N , an irreducible unitaryrepresentation ρ : G → U ( C k ) and a nonzero element Z ∈ ( p ⊗ p ) (cid:0) ( C n ⊗ C n )( C k ⊗ C n ) ∗ ⊗ L ( Q ⊗ Q ) (cid:1) (1 ⊗ q )satisfying ( v ⊗ v ) Z = Z ( ρ ( v ) ⊗ (id ⊗ ∆)( v )) for all v ∈ G . By weak mixing of both Ad v and Ad(id ⊗ ∆)( v ), we find that ZZ ∗ is a multiple of ( p ⊗ p ) in the matrix algebra over L ( Q ⊗ Q ), while Z ∗ Z is a multiple of 1 ⊗ q . We may thus assume that ZZ ∗ = ( p ⊗ p ) and Z ∗ Z = 1 ⊗ q . In particular, Z ( M k ( C ) ⊗ ⊗ ⊗ Z ∗ is a finite dimensional subspace of( M ⊗ M ) that is globally invariant under Ad( v ⊗ v ) . It follows that k = 1. It thenfollows that p ⊗ p and q have the same trace. Hence, t = 1 and we may assume that n = 1.We are now in precisely the same situation as in the last paragraph of the proof of [KV15,Theorem 4.1]. Invoking [IPV10, Theorem 3,3], we find a unitary W ∈ L (Γ) such that W G W ∗ ⊆ T Γ. 11 roof of Theorem 2.3.
Define p as the maximal projection in L (Γ) t ∩ π ( G ) ′ such that π ( G ) ′′ p is amenable. Of course, p could be zero. By construction, π ( G ) ′′ (1 − p ) has no amenabledirect summand. Let ( q i ) i ∈ I be a maximal orthogonal family of projections in L (Γ) t ∩ π ( G ) ′ with the properties that q i ≤ − p and h Γ ( π ( G ) q i ) = 0 for all i ∈ I . Define p = P i ∈ I q i .One checks that h Γ ( π ( G ) p ) = 0. Write p = 1 − p − p . By construction, π ( G ) ′′ p has noamenable direct summand and for every nonzero projection p ∈ π ( G ) ′ ∩ p L (Γ) t p , we havethat h Γ ( π ( G ) p ) > π : G → U ( L (Γ) t ) is a group homomorphismwith the properties that π ( G ) ′′ has no amenable direct summand and that h Γ ( π ( G ) p ) > p ∈ L (Γ) t ∩ π ( G ) ′ . We have to prove that π is standard.Define A ⊆ L (Γ) t as the subset of elements a ∈ L (Γ) t with the property that span { π ( v ) aπ ( v ) ∗ | v ∈ G} is finite dimensional. Note that A is a unital ∗ -subalgebra of L (Γ) t . Denote by A theweak closure of A . Let z ∈ Z ( A ) be the maximal central projection such that Az is diffuse.We prove that z = 0, so that A is discrete.By construction, π ( G ) is a subgroup of the normalizer of A inside L (Γ) t . Define α : G →
Aut( A ) : α ( v )( a ) = π ( v ) aπ ( v ) ∗ . By construction, the action α is compact, meaning that theclosure of α ( G ) in Aut( A ) is a compact group. Since π ( G ) ′′ has no amenable direct summand,a fortiori π ( G ) ′′ is diffuse. We can thus pick a sequence v n ∈ G such that π ( v n ) → α ( v n )) n is convergent in Aut( A ).We then find among the elements v − m v n a sequence w n ∈ G such that π ( w n ) → k π ( w n ) a − aπ ( w n ) k → a ∈ A . Since Γ belongs to Ozawa’s class ( S ), by [Oza10,Section 4], the II factor L (Γ) t is ω -solid, implying that A is amenable. It then follows from[OP07, Proposition 3.2] that the action α of G on A is weakly compact.Since π ( G ) normalizes A , we get that z commutes with π ( G ). So, Az ⊆ ( A ∪ π ( G )) ′′ z is a regularand weakly compact inclusion. Since Γ belongs to Ozawa’s class ( S ), if z = 0, it follows from[CS11, Theorem 4.1] that ( A ∪ π ( G )) ′′ z is amenable, contradicting the fact that π ( G ) ′′ has noamenable direct summand. So we have proven that z = 0 and that A is discrete.The center Z ( A ) is atomic and α restricts to an action of G on Z ( A ). We then find nonzeroprojections ( z n ) n ∈ I in Z ( A ) such that the following holds. • P n ∈ I z n = 1. • α ( v )( z n ) = z n for all v ∈ G and n ∈ I . • For every n ∈ I , the algebra Z ( A ) z n is finite dimensional and the restriction of α to Z ( A ) z n has trivial fixed point algebra C z n .It suffices to prove that for every n ∈ I , the homomorphism v π ( v ) z n is standard. To provethis, we multiply all data with z n and may thus assume that Z ( A ) is finite dimensional with Z ( A ) α = C z ∈ Z ( A ) be a minimal projection. Define the finite index subgroup G G by G = { v ∈ G | α ( v )( z ) = z } . We find that 1 = P v ∈G / G α ( v )( z ) and that π is induced from thehomomorphism π : G → L (Γ) t z : π ( v ) = π ( v ) z . Since G G has finite index, we still havethat h Γ ( π ( G )) > π ( G ) ′′ has no amenable direct summand.By construction, Az ∼ = M k ( C ) is a matrix algebra. We may thus realize L (Γ) t z as M k ( C ) ⊗ L (Γ) s in such a way that Az corresponds to M k ( C ) ⊗
1. For every v ∈ G , we have that α ( v ) restrictsto an automorphism of M k ( C ). We can thus choose unitaries γ ( v ) ∈ M k ( C ) and π ( v ) ∈ L (Γ) s such that π ( v ) = γ ( v ) ⊗ π ( v ). Since the unitaries γ ( v ) are uniquely determined up to a scalar,we find that G := T π ( G ) is a subgroup of U ( L (Γ) s ). It also follows that h Γ ( G ) > ′′ has no amenable direct summand. For every g ∈ Γ \ { e } , we have that C Γ ( g ) is amenableand thus, G ′′ L ( C Γ ( g )).Write P = L (Γ) s . We claim that the unitary representation (Ad v ) v ∈G on L ( P ) ⊖ C B ⊆ P ⊖ C v ) v ∈G . Define B = M k ( C ) ⊗ B . Then, B is globally invariant under(Ad π ( v )) v ∈G . Since G G has finite index, also B := span { α ( v )( b ) | v ∈ G , b ∈ B } is finite dimensional. By definition, B ⊆ A . Hence, B ⊆ Az . By construction, B isorthogonal to Az . This contradiction implies that the unitary representation (Ad v ) v ∈G on L ( P ) ⊖ C s = 1 and after a unitary conjugacy G ⊆ T Γ. We have thus realized L (Γ) t z as M k ( C ) ⊗ L (Γ) in such a way that π ( v ) = γ ( v ) ⊗ u δ ( g ) for all v ∈ G . This forces γ : G → U ( C k )to be a unitary representation and δ : G → Γ to be a group homomorphism. Since π is theinduction of π , the theorem is proven. Whenever ( A , τ ) is a tracial von Neumann algebra and I is a countable set, we denote by( A , τ ) I the tensor product von Neumann algebra N i ∈ I ( A , τ ). For every i ∈ I , we denote by π i : ( A , τ ) → ( A , τ ) I the embedding in the i ’th tensor factor.Let Γ be a countable group with infinite subgroup Γ Γ and let Γ y α ( A , τ ) be a tracepreserving action on an amenable tracial von Neumann algebra ( A , τ ). We build the II factor M (Γ , Γ , α ) = ( A , τ ) Γ ⋊ σ (Γ × Γ) where σ ( g,h ) ( π k ( a )) = π gkh − ( α g ( a )) (3.1)for all g ∈ Γ , h, k ∈ Γ and a ∈ A .We will require that Γ belongs to the following well studied class of groups with a rank onebehavior. Definition 3.1.
We say that a countable group Γ belongs to the family C if Γ is nonamenable,weakly amenable, in Ozawa’s class ( S ) and if every nontrivial element g ∈ Γ \ { e } has anamenable centralizer.Note that all free groups F n with 2 ≤ n ≤ + ∞ and all free products Γ ∗ Γ of amenablegroups with | Γ | ≥ | Γ | ≥ C . Since groups Γ in the family C have amenable centralizers, every nonamenable subgroup Γ ′ Γ is relatively icc, meaning that { ghg − | g ∈ Γ ′ } is infinite for every h ∈ Γ \ { e } . In particular, Γ is itself icc. Finally note thatif Γ ∈ C , then every nonamenable subgroup Γ ′ Γ still belongs to C .We now return to the construction in (3.1). In most cases, we consider Γ = Γ and denotethe II factor as M (Γ , α ). When Λ and Γ belong to the family C and when Γ y α ( A , τ )and Λ y β ( B , τ ) are arbitrary trace preserving actions with nontrivial kernel, we describe allpossible embeddings of M (Λ , β ) into amplifications of M (Γ , α ). In particular, we establish thefollowing result that provides numerous families of II factors that cannot be embedded oneinto the other.Note that Theorem A is a special case of Theorem 3.2 by considering the trivial actions Γ y ( A , τ ) and Λ y ( B , τ ). We explicitly state this special case as Corollary 3.3 below.13 heorem 3.2. Let Λ be a nonamenable icc group and let Γ be a group in the family C ofDefinition 3.1. Let Γ y α ( A , τ ) and Λ y β ( B , τ ) be trace preserving actions on amenabletracial von Neumann algebras such that A = C = B and such that Ker β is nontrivial. Thenthe following two statements are equivalent.1. There exists a t > such that M (Λ , β ) ֒ → M (Γ , α ) t .2. There exists an injective group homomorphism δ : Λ → Γ and a trace preserving unital ∗ -homomorphism ψ : ( B , τ ) → ( A , τ ) such that ψ ◦ β g = α δ ( g ) ◦ ψ for all g belonging to afinite index subgroup Λ Λ .Moreover, if these statements hold and if t > , we have M (Λ , β ) ֒ → M (Γ , α ) t if and only if t is an integer that can be written as a sum of integers of the form [Λ : Λ ] with Λ Λ as in 2. Applying Theorem 3.2 in the case t = 1, we get in particular that M (Λ , β ) ֒ → M (Γ , α ) if andonly if there exists an injective group homomorphism δ : Λ → Γ and a trace preserving unital ∗ -homomorphism ψ : ( B , τ ) → ( A , τ ) such that ψ ◦ β g = α δ ( g ) ◦ ψ for all g ∈ Λ.Theorem 3.2 applies in particular to the trivial action Γ y ( A , τ ). The resulting II factor isthe generalized Bernoulli crossed product of the left-right action of Γ × Γ on Γ and base algebra( A , τ ) : M (Γ , A , τ ) = ( A , τ ) Γ ⋊ (Γ × Γ) . Corollary 3.3.
Let Λ be a nonamenable icc group and let Γ be a group in the family C .Let ( A , τ ) , ( B , τ ) be nontrivial amenable tracial von Neumann algebras. Then the followingstatements are equivalent.1. There exists a t > and an embedding M (Λ , B , τ ) ֒ → M (Γ , A , τ ) t .2. There exists an embedding M (Λ , B , τ ) ֒ → M (Γ , A , τ ) .3. There exists an injective group homomorphism δ : Λ → Γ and a trace preserving unital ∗ -homomorphism ψ : ( B , τ ) → ( A , τ ) .Moreover, if these statements hold, we have M (Λ , B , τ ) ֒ → M (Γ , A , τ ) t if and only if t ∈ N . Remark 3.4.
We say that the II factors M and N are virtually isomorphic if M can beembedded as a finite index subfactor of N t for some t >
0. Note that M and N are virtuallyisomorphic if and only if there exists a Hilbert M - N -bimodule M H N with dim − N ( H ) < + ∞ and dim M − ( H ) < + ∞ . Virtual isomorphism is thus an equivalence relation on the class of II factors.Our proof of Theorem 3.2 and Corollary 3.3 also implies the following. In the setting of Theorem3.2 and given t >
0, we have that the following statements are equivalent. • M (Λ , β ) ∼ = M (Γ , α ) t . • t = 1 and there exists a group isomorphism δ : Λ → Γ and a trace preserving unital ∗ -isomorphism ψ : ( B , τ ) → ( A , τ ) such that ψ ◦ β g = α δ ( g ) ◦ ψ for all g ∈ Λ. • M (Λ , β ) and M (Γ , α ) are virtually isomorphic.In particular, the II factors M (Γ , α ) have trivial fundamental group. In the setting of Corollary3.3, we similarly have that the following statements are equivalent. • M (Λ , B , τ ) ∼ = M (Γ , A , τ ) t . • t = 1, Λ ∼ = Γ and there exists a trace preserving isomorphism ( B , τ ) ∼ = ( A , τ ).14 M (Λ , B , τ ) and M (Γ , A , τ ) are virtually isomorphic.In particular, the third family of II factors in Corollary B consists of II factors that aremutually embeddable, yet not virtually isomorphic.Before proving Theorem 3.2, we need a few lemmas. For later use (for instance, in the proofof Theorem D below), we state and prove the first lemma without assuming that Ker β isnontrivial and by allowing the left acting group Λ to be a proper subgroup of Λ. Lemma 3.5.
Let Λ Λ and Γ Γ be nonamenable icc groups. Assume that Γ ∈ C . Let Γ y α ( A , τ ) and Λ y β ( B , τ ) be trace preserving actions on amenable tracial von Neumannalgebras such that A = C = B . Write ( A, τ ) = ( A , τ ) Γ and ( B, τ ) = ( B , τ ) Λ . Let t > and let θ : M (Λ , Λ , β ) → M (Γ , Γ , α ) t be a normal unital ∗ -homomorphism.Then t ∈ N and there exist finite index subgroups Λ Λ and Λ Λ such that the restrictionof θ to B ⋊ (Λ × Λ ) is unitarily conjugate to a finite direct sum of embeddings θ i of the form θ i : B ⋊ (Λ × Λ ) → M n i ( C ) ⊗ ( A ⋊ (Γ × Γ)) : ( θ i ( B ) ⊆ M n i ( C ) ⊗ A ,θ i ( u ( g,h ) ) = γ i ( g, h ) ⊗ u π i ( g,h ) , with γ i : Λ × Λ → U ( C n i ) a unitary representation and π i : Λ × Λ → Γ × Γ an injective grouphomomorphism that is either of the form π i ( g, h ) = ( η i ( g ) , δ i ( h )) or π i ( g, h ) = ( δ i ( h ) , η i ( g )) .Moreover, Λ Λ and Λ Λ can be chosen such that (crudely) [Λ : Λ ] [Λ : Λ ] ≤ exp( t ) .Proof. Write Q = θ ( L (Λ × { e } )) and Q = θ ( L ( { e } × Λ)). With some abuse of notation, wewrite Q ⊗ Q = θ ( L (Λ × Λ)). Similarly write P = L (Γ × { e } ) and P = L ( { e } × Γ), as wellas P ⊗ P = L (Γ × Γ). We also write M = M (Γ , Γ , α ).By [KV15, Lemma 5.3], after a unitary conjugacy, we may assume that Q ⊗ Q ⊆ ( P ⊗ P ) t .By [OP03, Theorem 7], there exists a nonempty countable set I = I ⊔ I , minimal projections( p i ) i ∈ I in ( Q ⊗ Q ) ′ ∩ ( P ⊗ P ) t and r i , s i > P i ∈ I p i = 1 and such thatFor every i ∈ I , after unitary conjugacy, p i Q ⊆ P r i ⊗ p i Q ⊆ ⊗ P s i .For every i ∈ I , after unitary conjugacy, p i Q ⊆ ⊗ P s i and p i Q ⊆ P r i ⊗
1. (3.2)Since both Λ and Λ are nonamenable icc groups, we have that Q and Q are nonamenablefactors. Since P i ∈ I p i = 1, it then follows from (3.2) that for every nonzero projection p ∈ ( Q ⊗ Q ) ′ ∩ M t , the subalgebra p ( Q ⊗ Q ) is not amenable relative to A ⋊ (Γ × { e } ) andnonamenable relative to A ⋊ ( { e } × Γ). Since the normalizer of θ ( B ) contains Q ⊗ Q and usingthe standard notation of full intertwining-by-bimodules, it follows from [PV12, Theorem 1.4]that θ ( B ) ≺ f A ⋊ (Γ × { e } ) and θ ( B ) ≺ f A ⋊ ( { e } × Γ). So, θ ( B ) ≺ f A .Fix i ∈ I . Take a unitary conjugacy such that p i θ ( u ( g,h ) ) = v g ⊗ w h for all ( g, h ) ∈ Λ × Λ,where G = { v g | g ∈ Λ } is a subgroup of U ( P r i ) and G = { w h | h ∈ Λ } is a subgroupof U ( P s i ). By [KV15, Lemma 5.10] and using that θ ( B ) is diffuse and θ ( B ) ≺ f A , we findthat h Γ ( G ) > h Γ ( G ) >
0. By Theorem 2.3, the embeddings Λ ∋ g v g andΛ ∋ h w h are standard. In particular, r i , s i ∈ N .The same reasoning holds for i ∈ I . In particular, all the amplifications r i , s i are integers andthere thus are only finitely many of them. It also follows that t ∈ N .We have proven that the restriction of θ to L (Λ × Λ) is a finite direct sum of embeddingsarising as the tensor product of standard embeddings L (Λ ) ֒ → L (Γ ) r i and L (Λ) ֒ → L (Γ) s i ,or the other way around. By the definition of a standard embedding, for each i , we findfinite index subgroups Λ ,i Λ and Λ ,i Λ such that the embeddings are induced from15mbeddings given by group homomorphisms and unitary representations defined on Λ ,i andΛ ,i . By construction, X i [Λ : Λ ,i ] [Λ : Λ ,i ] ≤ t . Define Λ = T i Λ ,i and Λ = T i Λ ,i . We have [Λ : Λ ] [Λ : Λ ] ≤ exp( t ). We find unitaryrepresentations γ i : Λ × Λ → U ( C n i ) and group homomorphisms π i : Λ × Λ → Γ × Γ suchthat the restriction of θ to L (Λ × Λ ) is given by the direct sum of L (Λ × Λ ) → M n i ( C ) ⊗ L (Γ × Γ) : u ( g,h ) → γ i ( g, h ) ⊗ u π i ( g,h ) (3.3)with i ∈ { , . . . , n } . Moreover, each π i is either of the form π i ( g, h ) = ( η i ( g ) , δ i ( h )) or π i ( g, h ) =( δ i ( h ) , η i ( g )). Since each of the embeddings in (3.3) is an embedding of II factors, each π i hasfinite kernel. Since the groups Λ and Λ are icc, it follows that the homomorphisms π i areinjective.Whenever π i , π j : Λ × Λ → Γ × Γ are conjugate by an element in Γ × Γ, we can perform aunitary conjugacy and regroup the two direct summands. We may thus assume that for i = j ,the homomorphisms π i and π j are not conjugate.Denote by p i ∈ M t ( C ) ⊗ i = j , we have p i θ ( B ) p j = { } . Since π i and π j are not conjugate and since π i (Λ × Λ ) is relatively icc inside Γ × Γ, it follows that { π i ( g, h )( a, b ) π j ( g, h ) − | ( g, h ) ∈ Λ × Λ } is infinite for every ( a, b ) ∈ Γ × Γ. We can then take ( g n , h n ) ∈ Λ × Λ such that π i ( g n , h n )( a, b ) π j ( g n , h n ) − → ∞ for all ( a, b ) ∈ Γ × Γ.For every finite subset S ⊂ Γ × Γ, denote by P S : M → M : P S ( X ) = X ( g,h ) ∈ S E A ( Xu ∗ ( g,h ) ) u ( g,h ) the orthogonal projection of M onto span { Au ( g,h ) | ( g, h ) ∈ S } . Note thatlim n →∞ (cid:13)(cid:13) (id ⊗ P S )( p i θ ( u ( g n ,h n ) ) X θ ( u ( g n ,h n ) ) ∗ p j ) (cid:13)(cid:13) = 0 (3.4)for all X ∈ M t ( C ) ⊗ M and all finite subsets S ⊂ Γ × Γ.Fix b ∈ U ( B ) and write b n = σ ( g n ,h n ) ( b ). Fix ε >
0. Since θ ( B ) ≺ f A , we can take a finitesubset S ⊂ Γ × Γ such that (cid:13)(cid:13) p i θ ( b n ) p j − (id ⊗ P S )( p i θ ( b n ) p j ) (cid:13)(cid:13) < ε for all n .Note that p i θ ( b n ) p j = p i θ ( u ( g n ,h n ) ) θ ( b ) θ ( u ( g n ,h n ) ) ∗ p j . So by (3.4), we have that (cid:13)(cid:13) (id ⊗ P S )( p i θ ( b n ) p j ) (cid:13)(cid:13) → . It follows that lim sup n (cid:13)(cid:13) p i θ ( b n ) p j (cid:13)(cid:13) ≤ ε , for all ε >
0, so that lim n (cid:13)(cid:13) p i θ ( b n ) p j (cid:13)(cid:13) = 0. Since (cid:13)(cid:13) p i θ ( b ) p j (cid:13)(cid:13) = (cid:13)(cid:13) θ ( u ( g n ,h n ) ) p i θ ( b ) p j θ ( u ( g n ,h n ) ) ∗ (cid:13)(cid:13) = (cid:13)(cid:13) p i θ ( b n ) p j k , it follows that p i θ ( b ) p j = 0 for all b ∈ U ( B ). So our claim is proven.By this claim, we get that θ ( B ) ⊆ L ni =1 ( M n i ( C ) ⊗ M ). To conclude the proof of the lemma,we show that p i θ ( B ) ⊆ M n i ( C ) ⊗ A . Since π i (Λ × Λ ) is relatively icc in Γ × Γ, we cantake ( g n , h n ) ∈ Λ × Λ such that π i ( g n , h n )( a, b ) π i ( g n , h n ) − → ∞ for all ( a, b ) ∈ Γ × Γ with( a, b ) = ( e, e ). It follows thatlim n →∞ (cid:13)(cid:13) (id ⊗ P S )( p i θ ( u ( g n ,h n ) ) X θ ( u ( g n ,h n ) ) ∗ p i ) (cid:13)(cid:13) = 016or all X ∈ M t ( C ) ⊗ ( M ⊖ A ) and all finite subsets S ⊂ Γ × Γ. When S ⊂ Γ × Γ is a finitesubset with ( e, e ) ∈ S and when b ∈ U ( B ), we define a = (id ⊗ E A )( θ ( b )) and X = θ ( b ) − a .We again write b n = σ ( g n ,h n ) ( b ). It follows that(id ⊗ P S )( p i θ ( b n ) p i ) = p i θ ( u ( g n ,h n ) ) a θ ( u ( g n ,h n ) ) ∗ p i + (id ⊗ P S )( p i θ ( u ( g n ,h n ) ) X θ ( u ( g n ,h n ) ) ∗ p i ) . The first term at the right hand side lies in M n i ( C ) ⊗ A and the second term tends to zeroin k · k . As in the previous paragraph, the left hand side lies uniformly close in k · k to p i θ ( b n ) p i . We have thus proven thatlim n → + ∞ (cid:13)(cid:13) p i θ ( b n ) p i − p i θ ( u ( g n ,h n ) ) a θ ( u ( g n ,h n ) ) ∗ p i (cid:13)(cid:13) = 0 . Conjugating with θ ( u ( g n ,h n ) ), it follows that p i θ ( b ) π i = p i ap i ∈ M n i ( C ) ⊗ A . This concludesthe proof of the lemma. Lemma 3.6.
Make the same assumptions as in Lemma 3.5. Assume further that Λ Λ isrelatively icc and that Ker β is nontrivial. There then exist finite index subgroups Λ Λ and Λ Λ such that Λ Λ and such that the restriction of θ to B ⋊ (Λ × Λ ) is unitarilyconjugate to a finite direct sum of embeddings θ i : B ⋊ (Λ × Λ ) → M n i ( C ) ⊗ ( A ⋊ (Γ × Γ)) satisfying one of the following conditions labeled (3.5) and (3.6). θ i ( π g ( B )) ⊆ ⊗ π ρ i ( g ) ( A ) for all g ∈ Λ , and θ i ( u ( g,h ) ) = γ i ( g, h ) ⊗ u ( δ i ( g ) ,δ i ( h )) for all ( g, h ) ∈ Λ × Λ , (3.5) where γ i : Λ × Λ → U ( C n i ) is a unitary representation, δ i : Λ → Γ is a injective grouphomomorphism satisfying δ i (Λ ) ⊆ Γ and ρ i : Λ → Γ is an injective map satisfying ρ i ( e ) = e and ρ i ( gkh ) = δ i ( g ) ρ i ( k ) δ i ( h ) for all g ∈ Λ , k ∈ Λ , h ∈ Λ . θ i ( π g ( B )) ⊆ ⊗ π ρ i ( g ) − ( A ) for all g ∈ Λ , and θ i ( u ( g,h ) ) = γ i ( g, h ) ⊗ u ( δ i ( h ) ,δ i ( g )) for all ( g, h ) ∈ Λ × Λ , (3.6) where γ i : Λ × Λ → U ( C n i ) is a unitary representation, δ i : Λ → Γ is a injective group homo-morphism and ρ i : Λ → Γ is an injective map satisfying ρ i ( e ) = e and ρ i ( gkh ) = δ i ( g ) ρ i ( k ) δ i ( h ) for all g ∈ Λ , k ∈ Λ , h ∈ Λ .Proof. Since Ker β is a nontrivial normal subgroup of the icc group Λ , we get that Ker β is aninfinite normal subgroup of Λ . We use the notation and conclusion of Lemma 3.5. Since thefinite index subgroup Λ of Λ can be embedded into Γ and since Γ belongs to C , the group Λ has no infinite amenable normal subgroups. It follows that Ker β is nonamenable. ReplacingΛ by a smaller finite index subgroup, we may assume that Λ is normal in Λ. Then replacingΛ by Λ ∩ Λ , we may also assume that Λ Λ . Define Λ = Λ ∩ Ker β . Since Λ Ker β has finite index, also Λ is nonamenable.To prove the lemma, we consider each of the direct summands θ i in Lemma 3.5 separately andthus drop the index i . First assume that π ( g, h ) = ( η ( g ) , δ ( h )) for all ( g, h ) ∈ Λ × Λ . Weclaim that there exists a k ∈ Γ such that kδ ( h ) k − = η ( h ) for all h ∈ Λ . Assume the contrary.Since η (Λ ) is relatively icc in Γ, it follows that { η ( h ) kδ ( h ) − | h ∈ Λ } is infinite for every k ∈ Γ. We can then take h n ∈ Λ such that η ( h n ) kδ ( h n ) − → ∞ for all k ∈ Γ. Given the formof θ ( u ( g,h ) ), it follows that θ ( u ( h n ,h n ) ) Xθ ( u ( h n ,h n ) ) ∗ → X ∈ M n ( C ) ⊗ ( A ⊖ C u ( h n ,h n ) commutes with π e ( B ), it follows that θ ( π e ( B )) ⊆ M n ( C ) ⊗
1. Conjugating with u ( e,g ) , g ∈ Λ , it follows that θ ( B Λ ) ⊆ M n ( C ) ⊗
1, which is17bsurd because B Λ is diffuse. So the claim is proven and after a further unitary conjugacywith 1 ⊗ u ( e,k ) , we may assume that η ( h ) = δ ( h ) for all h ∈ Λ . If now g ∈ Λ , since Λ is anormal subgroup of Λ , we get that η ( g ) δ ( g ) − commutes with η ( h ) for all h ∈ Λ . Since η (Λ )is relatively icc in Γ, it follows that η ( g ) = δ ( g ) for all g ∈ Λ .The same argument of the previous paragraph now also implies that θ ( π e ( B )) ⊆ M n ( C ) ⊗ π e ( A ) and hence θ ( π g ( B )) ⊆ M n ( C ) ⊗ π δ ( g ) ( A ) for all g ∈ Λ .Now let g ∈ Λ be arbitrary, not necessarily belonging to Λ . We use that Λ ⊳ Λ is normal,so that g − Λ g = Λ . We have that π g ( B ) commutes with u ( h,g − hg ) for all h ∈ Λ . Weclaim that there exists a unique element ρ ( g ) ∈ Γ such that δ ( h ) ρ ( g ) δ ( g − h − g ) = ρ ( g ) for all h ∈ Λ . The existence of ρ ( g ) follows because otherwise, the same reasoning as above leads to θ ( π g ( B )) ⊆ M n ( C ) ⊗ ρ ( g ) follows from the relativeicc property of δ (Λ ) in Γ. By uniqueness of ρ ( g ) and because Λ is normal in Λ , it followsthat ρ ( hgk ) = δ ( h ) ρ ( g ) δ ( k ) for all h ∈ Λ , g ∈ Λ and k ∈ Λ . By construction, ρ : Λ → Γextends δ : Λ → Γ, we have ρ ( e ) = e and θ ( π g ( B )) ⊆ M n ( C ) ⊗ π ρ ( g ) ( A ).We claim that ρ : Λ → Γ is injective. For every g ∈ Λ, we have that δ ( h ) ρ ( g ) = ρ ( g ) δ ( g − hg )for all h ∈ Λ . If g, k ∈ Λ and ρ ( g ) = ρ ( k ), it follows that δ ( g − hg ) = δ ( k − hk ) for all h ∈ Λ .Since δ : Λ → Γ is injective, we conclude that gk − commutes with Λ . Since Λ Λ isrelatively icc, we get that g = k .For every k ∈ Λ, define the von Neumann algebra D k ⊆ M n ( C ) by D k = (cid:8) (id ⊗ ω )( θ ( π k ( b ))) (cid:12)(cid:12) ω ∈ A ∗ , b ∈ B (cid:9) ′′ . Note that D gkh − = γ ( g, h ) D k γ ( g, h ) ∗ for all k ∈ Λ and ( g, h ) ∈ Λ × Λ .The subalgebras θ ( π k ( B )), k ∈ Λ, commute among each other. Since θ ( π k ( B )) ⊆ M n ( C ) ⊗ π ρ ( k ) ( A ) and since the map ρ : Λ → Γ is injective, the subalgebras D k ⊆ M n ( C ), k ∈ Λ, alsocommute among each other. For a given k ∈ Λ, we have that D gk = γ ( g, e ) D k γ ( g, e ) ∗ ∼ = D k forall g ∈ Λ . Since M n ( C ) has no room for infinitely many commuting subalgebras that are allnonabelian, we conclude that all D k , k ∈ Λ, are abelian. So it follows that D = (cid:0)S k ∈ Λ D k (cid:1) ′′ isan abelian von Neumann subalgebra of M n ( C ). Since D gkh − = γ ( g, h ) D k γ ( g, h ) ∗ , the unitaries γ ( g, h ) normalize D . Since D is finite dimensional and abelian, the subgroups { g ∈ Λ | γ ( g, e ) ∈ D ′ ∩ M n ( C ) } Λ and { h ∈ Λ | γ ( e, h ) ∈ D ′ ∩ M n ( C ) } Λ have finite index. Replacing Λ and Λ by these finite index subgroups and reducing theembedding θ by minimal projections in D , the conclusion of the lemma holds.In the case where π ( g, h ) = ( δ ( h ) , η ( g )), we reason analogously.We now turn to the assumptions of Theorem 3.2. Before proving Theorem 3.2, we will actuallyprovide a complete description of all possible embeddings M (Λ , β ) ֒ → M (Γ , α ) t , which is ofcourse of independent interest.To formulate the result, we first describe the canonical irreducible embeddings. Let G Λ × Λbe a finite index subgroup, γ : G → U ( C n ) an irreducible unitary representation and δ : Λ → Γan injective group homomorphism. For every k ∈ Λ, let ψ k : ( B , τ ) → ( A , τ ) be a unital tracepreserving ∗ -homomorphism such that ψ gkh − = α δ ( g ) ◦ ψ k ◦ β − g for all ( g, h ) ∈ G and k ∈ Λ. (3.7)Define θ : B ⋊ G → M n ( C ) ⊗ ( A ⋊ (Γ × Γ)) as the unique normal ∗ -homomorphism satisfying θ ( π k ( b )) = 1 ⊗ π δ ( k ) ( ψ k ( b )) , θ ( u ( g,h ) ) = γ ( g, h ) ⊗ u ( δ ( g ) ,δ ( h )) g, h ) ∈ G , k ∈ Λ, b ∈ B . Then, θ is an irreducible embedding and we define θ : M (Λ , β ) ֒ → M (Γ , α ) mn with m = [Λ × Λ : G ] as the induction of θ . Then also θ isirreducible.Finally, denote by ζ the flip automorphism of M (Λ , β ) given by ζ ◦ π k = π k − ◦ β − k and ζ ( u ( g,h ) ) = u ( h,g ) for all g, h, k ∈ Λ. Proposition 3.7.
Under the same assumptions as in Theorem 3.2, each embedding M (Λ , β ) ֒ → M (Γ , α ) t is a finite direct sum of irreducible embeddings and each irreducible embedding isunitarily conjugate to either θ or θ ◦ ζ , with θ and ζ being as above.Proof. Let θ : M (Λ , β ) → M (Γ , α ) t be a unital normal ∗ -homomorphism. We apply Lemma3.6. Since Λ = Λ, we may assume that Λ = Λ and that Λ is a finite index normalsubgroup of Λ. We claim that the injective maps ρ : Λ → Γ appearing in (3.5) and (3.6)automatically are group homomorphisms. Since Λ is a normal subgroup of Λ, we have that δ ( g ) ρ ( k ) = ρ ( k ) δ ( k − gk ) for all g ∈ Λ and k ∈ Λ. Applying this equality twice, we get that δ ( g ) ρ ( k ) ρ ( h ) = ρ ( k ) ρ ( h ) δ (( kh ) − gkh ))for all g ∈ Λ and k, h ∈ Λ. We also have that δ ( g ) ρ ( kh ) = ρ ( kh ) δ (( kh ) − gkh )). It followsthat ρ ( k ) ρ ( h ) ρ ( kh ) − commutes with δ (Λ ). Since δ (Λ ) Γ is relatively icc, we find that ρ ( k ) ρ ( h ) = ρ ( kh ) for all k, h ∈ Λ. So, ρ : Λ → Γ is an injective group homomorphism.We can thus reformulate the conclusion of Lemma 3.6 in the following way. We have t ∈ N and we find a finite index normal subgroup Λ ⊳ Λ, an n ∈ N and for all i ∈ { , . . . , n } , aninjective group homomorphism δ i : Λ → Γ, a unitary representation γ i : Λ × Λ → U ( C n i ) andan embedding θ i : B ⋊ (Λ × Λ ) → M n i ( C ) ⊗ ( A ⋊ (Γ × Γ))satisfying θ i ( π k ( B )) ⊆ ⊗ π δ i ( k ) ( A ) and θ i ( u ( g,h ) ) = γ i ( g, h ) ⊗ u ( δ i ( g ) ,δ i ( h )) for all g, h ∈ Λ , k ∈ Λ, b ∈ B , such that after a unitary conjugacy, the restriction of θ to B ⋊ (Λ × Λ ) is given by B ⋊ (Λ × Λ ) → M d ( C ) ⊗ ( A ⋊ (Γ × Γ)) : θ | B ⋊ (Λ × Λ ) = m M i =1 θ i ⊕ n M i = m +1 ( θ i ◦ ζ ) , with t = d = P ni =1 n i . After a further unitary conjugacy, we may assume that for all 1 ≤ i, j ≤ m and for all m + 1 ≤ i, j ≤ n , either δ i = δ j or δ i and δ j are not conjugate. Denote by p i ∈ M d ( C ) ⊗ i ’th direct summand.Let 1 ≤ i < j ≤ m such that δ i and δ j are not conjugate. Take ( a, b ) ∈ Λ × Λ. The element X = p i θ ( u ( a,b ) ) p j (1 ⊗ u ∗ ( δ j ( a ) ,δ j ( b )) )satisfies (cid:0) γ i ( g, h ) ⊗ u ( δ i ( g ) ,δ i ( h )) (cid:1) X = X (cid:0) γ j ( a − ga, b − hb ) ⊗ u ( δ j ( g ) ,δ j ( h )) (cid:1) for all ( g, h ) ∈ Λ × Λ . Since δ i and δ j are not conjugate and since δ i (Λ ) Γ is relativelyicc, there exists a sequence g n ∈ Λ such that δ i ( g n ) kδ j ( g n ) − → ∞ for all k ∈ Γ. It followsthat X = 0. Denoting the entire crossed product as M = B ⋊ (Λ × Λ), we conclude that p i θ ( M ) p j = { } . A similar reasoning holds for m + 1 ≤ i < j ≤ n with δ i , δ j not conjugate,and also for all 1 ≤ i ≤ m and m + 1 ≤ j ≤ n .19his means that the original embedding can be decomposed as a direct sum of embeddings andfor each of these direct summands, in the description above, we have that all δ i are equal andthat either m = n , or m = 0. For the rest of the proof, we may consider each of these directsummands separately. Composing the embedding with the flip ζ if necessary, we may thusassume that all δ i are equal and that m = n . Write δ = δ i . This means that the restriction of θ to B ⋊ (Λ × Λ ) satisfies θ ( π k ( B )) ⊆ D n ( C ) ⊗ π δ ( k ) ( A ) and θ ( u ( g,h ) ) = γ ( g, h ) ⊗ u ( δ ( g ) ,δ ( h )) for all k ∈ Λ, g, h ∈ Λ , where D n ( C ) ⊂ M n ( C ) is the subalgebra of diagonal matrices and γ : Λ × Λ → U ( C n ) is a unitary representation.The same argument as above then forces that for all ( a, b ) ∈ Λ × Λ, the unitary θ ( u ( a,b ) ) is ofthe form γ ( a, b ) ⊗ u ( δ ( a ) ,δ ( b )) for some γ ( a, b ) ∈ U ( C n ).Define D ⊆ D n ( C ) as the smallest von Neumann subalgebra satisfying θ ( π k ( B )) ⊆ D ⊗ π δ ( k ) ( A ) for all k ∈ Λ. It follows that γ ( a, b ) Dγ ( a, b ) ∗ = D for all ( a, b ) ∈ Λ × Λ. Whenever p ∈ D is a projection that commutes with γ (Λ × Λ), we have that p ⊗ θ ( M ). Soafter a further decomposition into direct summands, we may assume that γ (Λ × Λ) ′ ∩ D = C p ∈ D and defining the finite index subgroup G Λ × Λ by G = { ( a, b ) ∈ Λ × Λ | γ ( a, b ) p γ ( a, b ) ∗ = p } , this means that θ is induced from the embedding θ : B ⋊ G → p M n ( C ) p ⊗ ( A ⋊ (Γ × Γ)) : θ ( x ) = θ ( x )( p ⊗ . Since p is a minimal projection in D , we find that θ ( π k ( b )) = p ⊗ π δ ( k ) ( ψ k ( b )), where ψ k :( B , τ ) → ( A , τ ) is a trace preserving unital ∗ -homomorphism. So, θ is precisely of the formas described before the proposition. Write γ : G → U ( p M n ( C ) p ) : γ ( a, b ) = γ ( a, b ) p . Toconclude the proof of the proposition, it suffices to decompose γ as a direct sum of irreduciblerepresentations.We are finally ready to prove Theorem 3.2. Proof of Theorem 3.2.
Assume that the first statement holds and let θ : M (Λ , β ) ֒ → M (Γ , α ) t be an embedding. By Proposition 3.7, θ is a finite direct sum of irreducible embeddingsand each irreducible embedding has a concrete description given before Proposition 3.7. Inparticular, we find for each of these irreducible embeddings a finite index subgroup G Λ × Λ,an injective group homomorphism δ : Λ → Γ and a family ( ψ k ) k ∈ Λ of trace preserving unital ∗ -homomorphisms ψ k : ( B , τ ) → ( A , τ ) such that (3.7) holds. Note that t equals a sum ofmultiples of [Λ × Λ : G ].Denote by ∆ : Λ → Λ × Λ the diagonal embedding. Defining the finite index subgroup Λ Λsuch that ∆(Λ ) = G ∩ ∆(Λ), it follows from (3.7) that α δ ( g ) ◦ ψ e = ψ e ◦ β g for all g ∈ Λ . So,the second statement of the theorem holds.In view of the “moreover” statement in the theorem, consider the left action of ∆(Λ) on thecoset space (Λ × Λ) /G . Let ( g j , h j ) G be representatives for the orbits. Define the finite indexsubgroups Λ j Λ such that ∆(Λ j ) is the stabilizer of ( g j , h j ) G . Note that we could take( g , h ) = ( e, e ) and then Λ equals the subgroup Λ in the previous paragraph. By definition,Λ j = (cid:8) k ∈ Λ (cid:12)(cid:12) ( g − j kg j , h − j kh j ) ∈ G (cid:9) . It thus follows from (3.7) that α δ ( g − j kg j ) ◦ ψ g − j h j = ψ g − j h j ◦ β g − j kg j for all k ∈ Λ j .20efining ψ j = α δ ( g j ) ◦ ψ g − j h j ◦ β g − j , this means that α δ ( k ) ◦ ψ j = ψ j ◦ β k for all k ∈ Λ j . Since[Λ × Λ : G ] = (cid:12)(cid:12) (Λ × Λ) /G (cid:12)(cid:12) = X j [Λ : Λ j ] , we have indeed written t as a sum of indices [Λ : Λ ], where Λ Λ is a finite index subgroupas in the second statement of the theorem.Conversely, whenever the second statement of the theorem holds, we have a canonical embed-ding ( B , τ ) Λ ⋊ (Λ × Λ) ֒ → ( A , τ ) Γ ⋊ (Γ × Γ) , which can be induced to an embedding M (Λ , β ) ֒ → M (Γ , α ) n with n = [Λ : Λ ].As mentioned above, we are formulating Lemmas 3.5 and 3.6 in a more general context, withpossibly proper subgroups Λ Λ and Γ Γ. This will be crucial to prove Theorem D, wherethe canonical flip automorphism ζ has to be avoided. To make the picture complete, we alsodescribe when such asymmetric crossed products embed one into the other. Compared to theformulation of Theorem 3.2, the formulation of Theorem 3.8 may sound a bit cumbersome, butthis is unavoidable. In Remark 3.9, we discuss the finite index subtleties that may arise. Theorem 3.8.
Let Λ Λ and Γ Γ be nonamenable groups such that Λ Λ is relativelyicc and Γ ∈ C . Let Γ y α ( A , τ ) and Λ y β ( B , τ ) be trace preserving actions on amenabletracial von Neumann algebras such that A = C = B and such that Ker β is nontrivial. Thenthe following statements are equivalent.1. There exists a t > and an embedding M (Λ , Λ , β ) ֒ → M (Γ , Γ , α ) t .2. There exist finite index subgroups Λ Λ and Λ Λ , an injective group homomorphism δ : Λ → Γ , an injective map ρ : Λ → Γ and a unital trace preserving ∗ -homomorphism ψ : ( B , τ ) → ( A , τ ) such that Λ ⊆ Λ , δ (Λ ) ⊆ Γ , ρ ( gkh ) = δ ( g ) ρ ( k ) δ ( h ) and ψ ◦ β g = α δ ( g ) ◦ ψ (3.8) for all g ∈ Λ , k ∈ Λ , h ∈ Λ .Moreover, if an embedding as in 1 exists, then t ∈ N .Proof. ⇒
2. By Lemma 3.6, we find that t ∈ N and we find finite index subgroups Λ Λ and Λ Λ with Λ ⊆ Λ , and an embedding θ of B ⋊ (Λ × Λ ) into M n ( C ) ⊗ ( A ⋊ (Γ × Γ))that is either of the form (3.5) or (3.6). In both cases, we define ψ : ( B , τ ) → ( A , τ ) suchthat θ ( π e ( b )) = 1 ⊗ π e ( ψ ( b )) for all b ∈ B and check that 2 holds.2 ⇒
1. Assume that we are given the data of (3.8). There then is a unique normal unital ∗ -homomorphism θ : B ⋊ (Λ × Λ ) → A ⋊ (Γ × Γ) satisfying θ ( π k ( b )) = π ρ ( k ) ( ψ ( b )) and θ ( u ( g,h ) ) = u ( δ ( g ) ,δ ( h )) for all k ∈ Λ , g ∈ Λ , h ∈ Λ and b ∈ B .This embedding can now be induced to an embedding of M (Λ , Λ , β ) into M (Γ , Γ , α ) n with n = [Λ : Λ ] [Λ : Λ ]. Remark 3.9.
In light of Theorem 3.2, it is tempting to try to formulate (3.8) better, interms of an injective group homomorphism δ : Λ → Γ. This is however impossible, as thefollowing example illustrates. Take Λ = F , freely generated by elements a and b . Then putΛ = Λ ∗ Z / Z , freely generated by Λ and an element c of order 2. Let ( A , τ ) be any nontrivialtracial amenable von Neumann algebra, e.g. A = C with τ (1 ,
0) = 1 / M (Λ , Λ , A ) = ( A , τ ) Λ ⋊ (Λ × Λ) embeds intoan amplification of M ( F , A ) = ( A , τ ) F ⋊ ( F × F ), although there is no injective grouphomomorphism Λ ֒ → F because F is torsion free. The corresponding map η : Λ → F canbe constructed as follows. Let F be freely generated by elements u, v . Define the index 2subgroup Λ Λ by Λ = h Λ , c Λ c i . Define the injective group homomorphism δ : Λ → F by δ ( a ) = u , δ ( b ) = vuv − , δ ( cac ) = v uv − , δ ( cbc ) = v uv − . Then define the injective map η : Λ → F by η ( g ) = δ ( g ) and η ( cg ) = v − δ ( g ) for all g ∈ Λ .One checks that η ( gkh ) = δ ( g ) η ( k ) δ ( h ) for all g ∈ Λ , k ∈ Λ and h ∈ Λ . Remark 3.10.
The W ∗ -rigidity paradigms for embeddings of II factors introduced in thisarticle have natural analogues in measurable group theory, for countable probability measurepreserving (pmp) equivalence relations.Already as such, most of the results in this paper have an immediate orbit equivalence counter-part. Given countable pmp equivalence relations R on ( X, µ ) and S on ( Y, ν ), we write R ֒ → S if R admits an extension that is isomorphic with a subequivalence relation of S (cf. [Pop05,Definition 1.4.2], [Fur06, Definition 1.6] and [AP15, Definition 2.7]). In terms of the associatedCartan inclusions A = L ∞ ( X ) ⊆ L ( R ) = M and B = L ∞ ( Y ) ⊆ L ( S ) = N , this amountsto the existence of an embedding θ : M → N satisfying θ ( A ) ⊆ B and θ ( N M ( A )) ⊆ N N ( B ).When S is ergodic, we similarly define the stable embedding relation R ֒ → s S .The generic construction (3.1) has the following analogue on the level of equivalence relations.Given Γ Γ, a countable pmp equivalence relation R on ( X , µ ) and a measure preservingaction α : Γ → Aut( R ), we define the countable pmp equivalence relation R on ( X, µ ) =( X , µ ) Γ by ( x, y ) ∈ R iff there exists g ∈ Γ and h ∈ Γ such that ( x gkh , α g ( y k )) ∈ R forall k ∈ Γ. Then, L ( R ) ∼ = M (Γ , Γ , α ), where Γ y α L ( R ) is the canonical trace preservingaction.We can thus apply Theorem 3.2 and provide in this way orbit equivalence variants of the mainresults stated in the introduction. Proof of Theorem C.
We use the following properties of the II factor M that appears in theinitial data. First, by Theorem 3.2, if M ֒ → M t for some t >
0, then t ∈ N . Second, M containscommuting nonamenable subfactors Q ⊂ M and Q ⊂ M given by the two subgroups Γ × { e } and { e } × Γ, and the subfactor Q ⊂ Q ⊗ Q given by the diagonal embedding of Γ into Γ × Γ,such that M is generated by M = (cid:0) Q ∪ ( Q ′ ∩ M ) ∪ ( Q ′ ∩ M ) (cid:1) ′′ . For any countably infinite subset
H ⊂ (0 , + ∞ ), we write P ( H ) = ∗ r ∈H M r . So by definition, P = P ( F − ). By [DR99, Theorem 1.5], we have P ( H ) t ∼ = P ( t H ) for every t >
0. By construction, P ( H ) ֒ → P ( H ) if H ⊆ H . Whenever t ∈ F , we have t F ⊆ F andthus F − ⊆ t F − . We conclude that P ֒ → P t so that F ⊆ F s ( P ).Conversely, assume that t ∈ F s ( P ). Since 1 ∈ F , we have that M ֒ → P and thus, M ֒ → P t ∼ = P ( t F − ). Fix such an embedding ψ : M → P ( t F − ). Combining [HU15, Theorem 4.3] with the22ontrol of relative commutants provided by [IPP05, Theorem 1.1], we find a countable family( p i ) i ∈ I of nonzero projections p i ∈ P ( t F − ) ∩ ψ ( Q ) ′ with P i p i = 1 and distinct s i ∈ F suchthat for every i , the subfactor ψ ( Q ) p i can be unitarily conjugated into the corner q i M ts − i q i of the canonical copy of M ts − i in the free product P ( t F − ). We have τ ( p i ) = τ ( q i ) and wedenote r i = τ ( q i ). Since P i p i = 1, we have that P i r i = 1.If a ∈ Q ′ ∩ M and i = j , we must have that p i ψ ( a ) p j = 0, since otherwise we can createa nonzero conjugacy between isomorphic copies of Q in different factors of the free product P ( t F − ). We conclude that p i commutes with ψ ( Q ). Using again [IPP05, Theorem 1.1], itfollows that ψ ( Q ⊗ Q ) p i can be unitarily conjugated into the corner q i M ts − i q i . Then reasoningsimilarly with ψ ( Q ), we find that p i ∈ ψ ( M ) ′ ∩ P t and that ψ ( M ) p i can be unitarily conjugatedinto the corner q i M ts − i q i . This means that M ֒ → M tr i s − i for every i . So, tr i s − i = n i ∈ N . Itfollows that tr i = n i s i ≥ i . Since P i r i = 1, we conclude that I is a finite set andthat t = P i n i s i belongs to F . Before proving Theorem D, we build a more abstract context and prove a more general resultthat we will also use in the proof of Theorem G.
Notation 5.1.
When Λ , Λ are subgroups of a group Γ, we write Λ ≺ Γ Λ if a finite indexsubgroup of Λ can be conjugated into Λ , i.e. if there exists a g ∈ Γ such that Λ ∩ g Λ g − has finite index in Λ .Fix a group Γ in the class C of Definition 3.1. Let Λ n < Γ < Γ Γ be subgroups and makethe following assumptions.
Assumptions 5.2.
1. We have that Γ is a proper subgroup of Γ and that Γ is a normalsubgroup of Γ. We have that all Λ n are nonamenable.2. The groups Γ and Γ have no nontrivial finite dimensional unitary representations.3. If n = m and g ∈ Γ, the group g Λ n g − ∩ Λ m is amenable.4. If g ∈ Γ \ Λ n , the group g Λ n g − ∩ Λ n is amenable.5. If i ∈ { , } and δ : Γ i → Γ is an injective group homomorphism such that δ (Λ n ) ≺ Γ Λ n forall n , there exists a g ∈ Γ such that δ ( k ) = gkg − for all k ∈ Γ i . Example 5.3.
To prove Theorem D, we will use Γ = G ∗ G , where G = A ∞ is the group offinite even permutations of N . We take Γ = Γ and Γ = G ∗ G , where G is the subgroup offinite even permutations of N that fix 1 ∈ N . We denote by A ⊂ G the set of all elements oforder 2 and we denote by B ⊂ G the set of all elements of order 3. For all ( a, b ) ∈ A × B , definethe subgroup Λ a,b = h a i ∗ h b i of Γ = G ∗ G . In Lemma 5.8, we prove that (an enumerationof) the groups Λ a,b satisfy the assumptions in 5.2.To prove Theorem G, we will use a similar family of groups Λ i < Γ < Γ Γ that we introducein Section 7.In all these cases, the assumptions 3, 4 and 5 in 5.2 follow from an analysis of reduced wordsin free product groups (see Lemma 5.6) combined with the Kurosh theorem.Given the assumptions in 5.2, we choose probability measures µ n on the two point set { , } such that µ n (0) takes distinct values in (0 , / → Γ × Γ be the diagonal23mbedding and consider the generalized Bernoulli actionΓ × Γ y ( X, µ ) = Y n ∈ N (cid:0) { , } , µ n (cid:1) (Γ × Γ) / ∆(Λ n ) . We denote by M = L ∞ ( X, µ ) ⋊ (Γ × Γ) the crossed product and consider the subfactor M = L ∞ ( X, µ ) ⋊ (Γ × Γ ). The following is the main technical result. Lemma 5.4.
Let Γ be a countable group in the class C of Definition 3.1. Let Λ n < Γ < Γ Γ be subgroups satisfying the assumptions in 5.2. Define M and M as above.If d > and θ : M → M d is an embedding, then d ∈ N and θ is unitarily conjugate to thetrivial embedding M → M n ( C ) ⊗ M : a ⊗ a .Proof. Before starting the actual proof of the lemma, we deduce a few other group theoreticproperties from the assumptions in 5.2. For convenience, we continue the numbering of 5.2.6. If δ : Γ → Γ is an injective group homomorphism and δ (Λ n ) ≺ Γ Λ n for all n , then δ isconjugate to the identity. Indeed, by assumption 5, we find g ∈ Γ such that δ ( k ) = gkg − for all k ∈ Γ . We have to prove that g ∈ Γ . Fix an n ∈ N . Take h ∈ Γ such that a finiteindex subgroup of hδ (Λ n ) h − is contained in Λ n . So, hg conjugates a finite index subgroupof Λ n into Λ n . By assumption 4, we have that hg ∈ Λ n < Γ . Thus g ∈ Γ .7. There is no injective group homomorphism δ : Γ → Γ such that δ (Λ n ) ≺ Γ Λ n for all n .Indeed, by the previous point, the restriction of δ to Γ would be an inner automorphism ofΓ and, in particular, surjective. Since Γ < Γ is a proper subgroup, this is in contradictionwith the injectivity of δ : Γ → Γ .For each n ∈ N and ( g, h ) ∈ Γ × Γ, we denote by A n ( g, h ) ∼ = C the algebra L ∞ ( { , } , µ n ) ⊂ L ∞ ( X, µ ) viewed in coordinate ( g, h )∆(Λ n ) ∈ (Γ × Γ) / ∆(Λ n ). Defining ( A , τ ) as the vonNeumann algebra generated by A n ( g, g ) for all n ∈ N and g ∈ Γ , we have the natural actionΓ y α ( A , τ ) implemented by the diagonal embedding ∆ : Γ → Γ × Γ . By definition,and using the notation (3.1), we have that M = M (Γ , Γ , α ). Writing M = M (Γ , Γ , α ), wecanonically have M ⊆ M .Denote A = L ∞ ( X, µ ) and define B ⊆ A as the von Neumann subalgebra generated by A n ( g, h ), n ∈ N , ( g, h ) ∈ Γ × Γ . Note that M = B ⋊ (Γ × Γ ), while M = A ⋊ (Γ × Γ ).Denote by θ the restriction of θ to M , so that θ : M → M d is an embedding that fits into thesetting of Lemma 3.5. We prove that d ∈ N and that after a unitary conjugacy θ ( u ( g,h ) ) = u ( g,h ) for all ( g, h ) ∈ Γ × Γ . Because Γ and Γ have no nontrivial finite dimensional unitaryrepresentations, it follows from Lemma 3.5 that d ∈ N and that, after a unitary conjugacy, θ is a direct sum of finitely many embeddings θ i : M → M d i ( C ) ⊗ M satisfying θ i ( B ) ⊂ M d i ( C ) ⊗ A and θ i ( u ( g,h ) ) = 1 ⊗ u π i ( g,h ) for all ( g, h ) ∈ Γ × Γ ,where π i : Γ × Γ → Γ × Γ is an injective group homomorphism that is either of the form π i ( g, h ) = ( η i ( g ) , δ i ( h )) or of the form π i ( g, h ) = ( δ i ( h ) , η i ( g )).We analyze each θ i separately and thus drop the index i . Fix n ∈ N . We prove that π (∆(Λ n )) ≺ Γ × Γ ∆(Λ n ). We first prove that π (∆(Λ n )) ≺ Γ × Γ ∆(Λ m ) for some m ∈ N .Indeed, if this is not the case, since A n ( e, e ) ⊂ B commutes with L (∆(Λ n )), it would followthat θ ( A n ( e, e )) ⊆ M d ( C ) ⊗ θ ( A n ( g, h )) ⊆ M d ( C ) ⊗ g, h ) ∈ Γ × Γ , whichis absurd.Fix m ∈ N such that π (∆(Λ n )) ≺ Γ × Γ ∆(Λ m ). We now use assumptions 3 and 4, and the factthat every nontrivial element of Γ has an amenable centralizer. Since A n ( e, e ) commutes with24 (∆(Λ n )), it follows that there exists ( g, h ) ∈ Γ × Γ such that θ ( A n ( e, e )) ⊆ M d ( C ) ⊗ A m ( g, h ).Denote by π m, ( g,h ) : C → A m ( g, h ) the canonical isomorphism. Define the von Neumannsubalgebra D ⊆ M d ( C ) ⊗ C such that θ ( A n ( e, e )) = (id ⊗ π m, ( g,h ) )( D ). Let k ∈ Γ \ { e } be anyelement. We have θ ( A n ( k, e )) = (id ⊗ π m,π ( k,e )( g,h ) )( D ). Note that θ ( A n ( k, e )) commutes with θ ( A n ( e, e )). Also note that given the special form of π , we must have π ( k, e )( g, h )∆(Λ m ) =( g, h )∆(Λ m ). The commutation thus forces D ⊆ D ⊗ C , where D ⊆ M d ( C ) is an abelianvon Neumann subalgebra.Defining C n as the von Neumann algebra generated by A n ( g, h ), ( g, h ) ∈ Γ × Γ and N n = C n ⋊ (Γ × Γ ), we have proven that θ ( N n ) ⊆ D ⊗ M . Let q ∈ D be any minimal projection.Since N n and M are II factors, the map N n → q ⊗ M : a θ ( a )( q ⊗
1) is (normalized) tracepreserving. Therefore, a θ ( a )( q ⊗
1) induces a trace preserving embedding of A n ( e, e ) into A m ( g, h ). When n = m , we have that µ n (0) and µ m (0) are distinct elements of (0 , /
2) andsuch an embedding does not exist. If n = m , because µ n (0) < /
2, such an embedding mustbe the canonical “identity” map. We have thus proven that n = m and that θ ( π n, ( e,e ) ( a ))( q ⊗
1) = q ⊗ π n, ( g,h ) ( a )for all a ∈ C and all minimal projections q ∈ D . Writing 1 ∈ D as a sum of minimalprojections, we conclude that θ ( π n, ( e,e ) ( a )) = 1 ⊗ π n, ( g,h ) ( a ) for all a ∈ C .We now return to the full setting where we had decomposed θ as a direct sum of embeddings θ i . We have in particular shown that π i (∆(Λ n )) ≺ Γ × Γ ∆(Λ n )for all i and all n . If π i ( g, h ) = ( δ i ( h ) , η i ( g )), we get that δ i : Γ → Γ is an injective grouphomomorphism satisfying δ i (Λ n ) ≺ Γ Λ n for all n . By property 7, this is impossible. Thus, π i ( g, h ) = ( η i ( g ) , δ i ( h )). We find that η i : Γ → Γ is an injective group homomorphismsatisfying η i (Λ n ) ≺ Γ Λ n for all n , while δ i : Γ → Γ is an injective group homomorphismsatisfying δ i (Λ n ) ≺ Γ Λ n for all n . Using property 6 and assumption 5, we get after a unitaryconjugacy that π i ( g, h ) = ( g, h ) for all ( g, h ) ∈ Γ × Γ .We finally return to the initial embedding θ : M → M d . We have proven that d ∈ N andthat, after a unitary conjugacy, θ ( u ( g,h ) ) = 1 ⊗ u ( g,h ) for all ( g, h ) ∈ Γ × Γ . Fix n ∈ N and( a, b ) ∈ Γ × Γ. Since Γ ⊳ Γ is normal, we have that ( a, b )∆(Λ n )( a, b ) − is a nonamenablesubgroup of Γ × Γ . This subgroup acts trivially on A n ( a, b ). Analyzing commutants as above,it follows that θ ( A n ( a, b )) ⊆ M d ( C ) ⊗ A n ( a, b ). Making the same reasoning as above, we getthat θ ( π n, ( a,b ) ( x )) = 1 ⊗ π n, ( a,b ) ( x )for all x ∈ C . So we have shown that θ ( x ) = 1 ⊗ x for all x ∈ M .We start by proving the following well known result. Lemma 5.5.
Let S G ∗ K be a subgroup of a free product group. If the word length functionis bounded on S , then S is conjugate to a subgroup of G or K .Proof. We may assume that S = { e } . We distinguish two cases. First assume that S containsan element s ∈ S that is conjugate to a cyclically reduced word of length at least two, i.e. s = wgw − . Then the word length of s n = wg n w − tends to infinity, because | g n | = n | g | .If we are not in the first case, every element of S is conjugate to an element of G or K .Conjugating S , we may assume that S contains a nontrivial element of G or K . By symmetry,we may assume that S ∩ G = { e } and take s ∈ S ∩ G , s = e . If S G , the lemma is25roven. Otherwise, we find r ∈ S \ G . Write r = wgw − with g ∈ G ∪ K , g = e , and withthe concatenation wgw − being a reduced word. If the first letter of w belongs to G , write w = w w with w ∈ S being this first letter. Otherwise, take w = e and w = w . Considerthe element sr ∈ S . Then, sr = w (( w − sw ) w gw − ) w − and ( w − sw ) w gw − is a cyclicallyreduced word. We are thus back in the first case and the lemma is proven. Lemma 5.6.
Let
Γ = G ∗ K be a free product group. Assume that a ∈ G ∪ K is an elementof order in either G or K . Assume that b ∈ G ∪ K is an element of order in either G or K . Let w ∈ G ∗ K be such that a and wb w − are free and denote by Λ G ∗ K the subgroupgenerated by these two elements.1. Let a ∈ G be of order , b ∈ K of order and write Λ = h a i ∗ h b i .If z ∈ Γ \ Λ , then z Λ z − ∩ Λ is finite.If there exists a z ∈ Γ such that z Λ z − ∩ Λ is infinite, then a ∈ G , b ∈ K and w can bewritten as w = u − w v where u ∈ G , v ∈ K and w ∈ Λ such that ua u − = a , vb v − = b ± and such that w ∈ h a i ∗ h b i is either equal to e or starting with the letter b ± and endingwith the letter a .2. Let a ∈ G be of order and b ∈ G of order . Let k ∈ K \ { e } and write Λ = h a i ∗ k h b i k − .If z ∈ Γ \ Λ , then z Λ z − ∩ Λ is finite.If there exists a z ∈ Γ such that z Λ z − ∩ Λ is infinite, then a , b ∈ G and w can be writtenas w = u − w kv where u, v ∈ G and w ∈ Λ such that ua u − = a , vb v − = b ± and suchthat w ∈ h a i ∗ h kbk − i is either equal to e or starting with the letter kb ± k − and endingwith the letter a .Proof. Throughout the proof of this lemma, we use the following property: if two cyclicallyreduced words in G ∗ K are conjugate as elements of G ∗ K , then they are cyclic permutationsof each other. It follows in particular that in both settings 1 and 2, zGz − ∩ Λ and zKz − ∩ Λare finite groups for all z ∈ Γ.For the following reason, we have in both settings 1 and 2 that z ∈ Λ whenever z Λ z − ∩ Λ isinfinite. Indeed, if z Λ z − ∩ Λ is infinite, by Lemma 5.5, we can take an arbitrarily long word x ∈ Λ such that zxz − ∈ Λ. When the word length of x is large enough, the middle lettersof x remain untouched when reducing zxz − . Since this reduction must belong to Λ, we alignthese middle letters with the canonical reduced words defining elements of Λ and conclude that z ∈ Λ.To prove the second part of statements 1 and 2, write L = G or L = K so that a ∈ L . Denoteby L ′ the other group. Similarly denote R = G or R = K so that b ∈ R and R ′ is the othergroup. Assume that an infinite subgroup of Λ can be conjugated into Λ. The first paragraphof the proof now implies that Λ G and Λ K . So if L = R , we must have w L .Uniquely write w = u − w v with u ∈ L , v ∈ R and with w either equal to e (which can onlyoccur if L = R ), or w being a reduced word that starts with a letter in L ′ \ { e } and endswith a letter from R ′ \ { e } . Put a = ua u − and b = vb v − . So, an infinite subgroup of h a i ∗ h w b w − i can be conjugated into Λ. By the second paragraph of the proof, we can takean element of the form x = a w b ± w − a · · · w b ± w − (5.1)having arbitrarily large length such that x can be conjugated into Λ.In setting 1, with Λ = h a i ∗ h b i and a ∈ G , b ∈ K , it follows that all letters appearing in x mustbe either a or b ± . Since a , a have order 2, while b , b have order 3, we must have a = a , b = b ± and w ∈ Λ. In particular, L = G and R = K . So all conclusions of 1 indeed hold.26n setting 2, with Λ = h a i ∗ h kbk − i and a, b ∈ G , k ∈ K \ { e } , we make the following twoobservations: any cyclically reduced word in a and kb ± k − only contains the letters a , k ± , b ± and has the property that between any two occurrences of the letter a , there sits a wordthat belongs to Λ. We know that the word x in (5.1) can be cyclically permuted to such acyclically reduced word in a and kb ± k − . So, only the letters a , k ± , b ± appear in the word x . Since a has order 2, while b has order 3, we must have that a ∈ { a, k, k − } . Similarly, b ∈ { b, b − , k, k − } .First assume that a = a . By the previous paragraph, between any two occurrences of a in x , there sits a word that belongs to Λ. It follows that w b w − ∈ Λ. The only elements in Λof order 3 are conjugate (inside Λ) to kb ± k − . It follows that b = b ± and w = w k with w ∈ Λ. So all conclusions of 2 hold in this case.Next assume that a = a , which should lead to a contradiction. Then a ∈ { k, k − } . Inparticular, k has order 2. Since b ∈ { b, b − , k, k − } and b has order 3, it follows that b = b ± .So, k has order 2 and a = k . In any cyclically reduced word in a and kb ± k , the letter k appears twice as much as the letters b ± , and every occurrence of b ± is preceded and followedby k . Comparing with the description of x in (5.1), we must have that k appears in w andthat thus, w ends with the letter k . Write w = w k . Since a = k , we have that L = K , sothat w must start with a letter in G \ { e } . It follows that w = e and that w starts and endswith one of the letters a, b, b − . We have written the word x in (5.1) as x = ( k w kb ± k w − ) n for some n ∈ N .We know that this word can be cyclically permuted so as to become a cyclically reduced wordin a and kb ± k . Reading x from left to right, the only possibilities are w ∈ b ± k Λ and w ∈ Λ.In the first case, x is a reduced word in kb ± k and a followed by the “remainder” kb ∓ . Inthe second case kxk is a reduced word in a and kb ± k followed by the “remainder” k . In bothcases, we reached the required contradiction. Lemma 5.7.
Let G be the group of finite even permutations of N . Denote by G the subgroupof finite even permutations of N that fix ∈ N . Denote by A ⊂ G the elements of order anddenote by B ⊂ G the elements of order .Let δ be a group homomorphism from either G or G to G . If δ ( a ) = a for every a ∈ A ,then δ is the identity homomorphism. If δ ( b ) = b ± for every b ∈ B , then δ is the identityhomomorphism.Proof. Denote by δ the restriction of δ to G . Since the elements of order 2 generate G , if δ ( a ) = a for all a ∈ A , we have that δ ( σ ) = σ for all σ ∈ G .Next assume that δ ( b ) = b ± for all b ∈ B . Denote by B ⊂ B the subset of permutations ofthe form ( ijk ) with 2 ≤ i < j < k . We prove that δ ( b ) = b for all b ∈ B . Put b = (234).By symmetry, it suffices to prove that δ ( b ) = b . Assume that this is not the case. Then, δ ( b ) = b − . Write b = (456). We have b b b − = (256), while b − b b = (356). It is thusimpossible to get δ ( b ) = b ± . So δ ( b ) = b for all b ∈ B . Since B generates the group G ,we again have that δ ( σ ) = σ for all σ ∈ G .Every homomorphism δ : G → G whose restriction to G is the identity, must itself be theidentity. This concludes the proof of the lemma. Lemma 5.8.
When Λ a,b < Γ < Γ = Γ are defined as in Example 5.3, then the assumptionsin 5.2 all hold. roof. Note that 1 is immediate. For 2, it suffices to note that A ∞ has no nontrivial finitedimensional unitary representations. This is well known and can be seen as follows: since A ∞ contains infinitely many commuting copies of A , any finite dimensional representation π hasthe property that π ( A ) is abelian and hence trivial, which forces π to be trivial on A ∞ .Conditions 3 and 4 follow from the first statement of Lemma 5.6. To prove 5, we denote by G r the second copy of G in the free product Γ = G ∗ G r . Let δ : G ∗ G r → G ∗ G r be an injectivegroup homomorphism such that δ (Λ a,b ) ≺ Γ Λ a,b for all ( a, b ) ∈ A × B . After replacing δ by(Ad y ) ◦ δ for some y ∈ Γ, by the Kurosh theorem, we can take
L, R ∈ {
G, G r } , injective grouphomomorphisms δ : G → L and δ r : G r → R , and an element w ∈ Γ such that δ ( g ) = δ ( g )for all g ∈ G and δ ( h ) = wδ r ( h ) w − for all h ∈ G r . Denote by R ′ ∈ { G, G r } the “other”group, so that { R, R ′ } = { G, G r } . We may assume that either w = e or that w ends with aletter in R ′ \ { e } .Given ( a, b ) ∈ A × B , we view a ∈ G and b ∈ G r . Since a and b are free, also δ ( a ) and wδ r ( b ) w − are free. Since δ (Λ a,b ) ≺ Γ Λ a,b , we get that a finite index subgroup of h δ ( a ) i ∗ w h δ r ( b ) i w − can be conjugated into Λ a,b .By the first statement in Lemma 5.6, we first conclude that L = G , R = G r . In particular,either w = e or w ends with a letter in G \ { e } . In the case where w = e , Lemma 5.6 says that δ ( a ) = a and δ r ( b ) = b ± for all ( a, b ) ∈ A × B . By Lemma 5.7, both δ and δ r are the identitymap.In the case where w = e , we write w = u − w where either w = e or w is a word starting with b ± and ending with a . The latter option can never hold for all ( a, b ) ∈ A × B . In the formercase, after replacing δ by (Ad u ) ◦ δ , Lemma 5.6 again says that δ ( a ) = a and δ ( b ) = b ± forall ( a, b ) ∈ A × B . Again, δ is the identity homomorphism.So also condition 5 in 5.2 holds and the lemma is proven. Proof of Theorem D.
The theorem immediately follows from Lemmas 5.8 and 5.4. factors: proof of Theorem E Note that the augmentation functor Γ H Γ is functorial in the following precise sense. To anyinjective group homomorphism Γ → Λ corresponds a canonical injective group homomorphism H Γ → H Λ and thus, a canonical embedding of II factors L ( H Γ ) ֒ → L ( H Λ ). Theorem E, whichwe now prove, says that if L ( H Γ ) ֒ → s L ( H Λ ), then Γ ֒ → Λ, but does not say that any embeddingof L ( H Γ ) into L ( H Λ ) is of such a canonical form.The proof of Theorem E follows rather directly from Lemmas 3.5 and 3.6. We however haveto do a few tweaks in the case where the groups Γ and Λ are uncountable. Proof of Theorem E.
For every infinite group Γ, we denote by B Γ = (cid:0) Z / Z (cid:1) (( G Γ × G Γ ) /N Γ ) theabelian normal subgroup of H Γ such that H Γ = B Γ ⋊ ( G Γ × G Γ ). We also write A Γ = L ( B Γ ).Note that whenever Γ Γ is a subgroup, we canonically have H Γ H Γ . Through thisinclusion, A Γ A Γ .Also note that whenever Γ is a countable infinite group, the II factor L ( H Γ ) is of the formstudied in Section 3. Defining D Γ = (cid:0) Z / Z (cid:1) (( Z ∗ Γ) / Γ) and G Γ y α L ( D Γ ) through π Γ : G Γ → Z ∗ Γ, (6.1)we canonically have L ( H Γ ) = M ( G Γ , α ). 28hroughout the proof, we say that a group homomorphism π : G × G → G × G is of productform if π is either of the form π ( g, h ) = ( η ( g ) , δ ( h )) or the form π ( g, h ) = ( δ ( h ) , η ( g )) for grouphomomorphisms δ, η : G → G .Let Γ and Λ be arbitrary infinite groups. Take d > θ : L ( H Γ ) → L ( H Λ ) d . We have to prove that Γ is isomorphic with a subgroup of Λ. Fix a countable infinitesubgroup Γ Γ. Since H Γ is countable, we can take a countable subgroup Λ Λ such that θ ( L ( H Γ )) ⊆ L ( H Λ ) d . We denote by θ the restriction of θ to L ( H Γ ).As explained above, the embedding θ : L ( H Γ )) → L ( H Λ ) d fits into the context of Section 3.By Lemma 3.5, d ∈ N and we can unitarily conjugate the original embedding and replace Γ by a finite index subgroup such that θ becomes the direct sum of embeddings θ i : L ( H Γ )) → M d i ( C ) ⊗ L ( H Λ ) satisfying θ i ( A Γ ) ⊂ M d i ( C ) ⊗ A Λ and θ i ( u ( g,h ) ) ∈ U ( C d i ) ⊗ u π i ( g,h ) for all g, h ∈ G Γ , (6.2)where the π i : G Γ × G Γ → G Λ × G Λ are injective group homomorphisms of product form. Aftera further unitary conjugacy and regrouping direct summands, we may assume that for i = j ,the group homomorphisms π i and π j are non conjugate, as homomorphisms from G Γ × G Γ to G Λ × G Λ . Denote by p i ∈ M d ( C ) the projections corresponding to the i ’th direct summand.Whenever Γ Γ and L G Γ are subgroups and p ∈ M d ( C ) is a nonzero projection, wesay that θ is standard on (Γ , L , p ) if p can be written as the sum of nonzero projections q j ∈ M d ( C ) such that for all j , the projection q j ⊗ θ ( A Γ ⋊ ( L × L )) and( q j ⊗ θ ( A Γ ) ⊂ q j M d ( C ) q j ⊗ A Λ and ( q j ⊗ θ ( u ( g,h ) ) ∈ U ( p C d ) ⊗ u ζ j ( g,h ) for all g, h ∈ L and injective group homomorphisms ζ j : L × L → G Λ × G Λ of product form.By construction, θ is standard on (Γ , G Γ , θ is standard on (Γ , L,
1) for some finite index normal subgroup L G Γ .Let Γ Γ be an arbitrary countable subgroup with Γ Γ . We claim that there exists afinite index subgroup L G Γ with [ G Γ : L ] ≤ exp( d ) such that θ is standard on (Γ , L , p i )for every i .Applying as above Lemma 3.5 to the restriction of θ to L ( H Γ ), we find integers n j ∈ N with P j n j = d and we find a unitary V ∈ M d ( C ) ⊗ L ( H Λ ) such that the restriction of (Ad V ) ◦ θ to L ( H Γ ) satisfies V θ ( A Γ ) V ∗ ⊂ M j (cid:0) M n j ( C ) ⊗ A Λ (cid:1) and V θ ( u ( g,h ) ) V ∗ ∈ M j (cid:0) U ( C n j ) ⊗ u ζ j ( g,h ) (cid:1) (6.3)for all g, h in a finite index subgroup L G Γ , where ζ j : L × L → H Λ × H Λ are injectivegroup homomorphisms of product form and [ G Γ : L ] ≤ exp( d ). Denote by q j ∈ M d ( C ) theprojection corresponding to the j ’th direct summand in the above decomposition.Define L = G Γ ∩ L . Since G Γ G Γ , we have that L G Γ has finite index.Whenever ( q j ⊗ V ( p i ⊗
1) is nonzero, comparing (6.2) and (6.3), there must exist x, y ∈ G Λ such that { ζ j ( g, h ) ( x, y ) π i ( g, h ) − | g, h ∈ L } is a finite set. By the relative icc property of π i ( L × L ) in G Λ × G Λ , this set must be thesingleton { ( x, y ) } . Since moreover the action of π i ( L × L ) on A Λ is weakly mixing, we getthat ( q j ⊗ V ( p i ⊗ ∈ q j M d ( C ) p i ⊗ u ( x,y ) . In particular, the restrictions of π i and ζ j to L × L are conjugate. 29ix an index j . We claim that there is precisely one i such that ( q j ⊗ V ( p i ⊗
1) is nonzero. If i = i ′ and if both ( q j ⊗ V ( p i ⊗
1) and ( q j ⊗ V ( p i ′ ⊗
1) are nonzero, we find that π i and π i ′ are conjugate on the finite index subgroup L × L of G Γ × G Γ . By the relative icc property, π i and π i ′ are conjugate on G Γ × G Γ , contradicting the choices made above. So, the claim isproven and we can partition the indices j into subsets J i such that V ∗ ( q j ⊗ V = p i,j ⊗ j ∈ J i , with P j ∈ J i p i,j = p i . Also,( q j ⊗ V ( p i ⊗
1) = V i,j ⊗ u ( x i,j ,y i,j ) , where V i,j is a partial isometry with left support q j and right support p i,j whenever j ∈ J i . Itnow follows from (6.3) that θ is standard on (Γ , L , p i ) for all i , as claimed above.Since this holds for all countable subgroups Γ Γ with Γ Γ and since the index of L in G Γ stays bounded by exp( d ), it follows that θ is standard on (Γ , L,
1) for a finite indexsubgroup L G Γ , that we may next assume to be normal.Since θ is standard on (Γ , L, θ : L ( H Γ ) → L ( H Λ ) d . Next, also theconclusion of Lemma 3.6 holds, because the proof of Lemma 3.6 was entirely self containedand only exploited the control of relative commutants using the relative icc property, whichworks equally well in an uncountable setting. The same holds for the first paragraphs of theproof of Proposition 3.7. Using the notation of (6.1), there thus exists a trace preservingembedding ψ : L ( D Γ ) → L ( D Λ ) and an injective group homomorphism δ : G Γ → G Λ such that α δ ( g ) ◦ ψ = ψ ◦ α g for all g ∈ L .For every x ∈ Z ∗ Γ, denote by E x Γ ∼ = L ( Z / Z ) the natural subalgebra of L ( D Γ ). We similarlydefine E y Λ ⊂ L ( D Λ ) for all y ∈ Z ∗ Λ.If K G Λ and K N Λ , the action of K on L ( D Λ ) is ergodic. Since the elements of E e Λ arefixed by α g for al g ∈ N Γ , it follows that δ ( L ∩ N Γ ) ≺ N Λ . Conjugating δ and replacing L by a smaller finite index normal subgroup of G Γ , we may assume that δ ( L ∩ N Γ ) N Λ . Let y ∈ G Λ \ N Λ . Note that Λ Z ∗ Λ is malnormal, so that N Λ ∩ yN Λ y − = Ker π Λ . Similarly, N Γ ∩ xN Γ x − = Ker π Γ for all x ∈ G Γ \ N Γ .We claim that δ ( L ∩ N Γ ) N Λ ∩ yN Λ y − = Ker π Λ . Otherwise, because Ker π Λ is a normalsubgroup of G Λ , we find a finite index subgroup K L ∩ N Γ with δ ( K ) Ker π Λ . Theequivariance of ψ then implies that α g = id for all g ∈ K , which is absurd. By the claim, thesubalgebra of δ ( L ∩ N Γ )-invariant elements in D Λ equals E e Λ . So, ψ ( E e Γ ) ⊆ E e Λ . Since bothare two-dimensional and ψ is a trace preserving embedding, we have ψ ( E e Γ ) = E e Λ Fix a ∈ Z ∗ Γ \ Γ. Choosing x ∈ G Γ with π Γ ( x ) = a and making a similar reasoning for thefixed points under L ∩ xN Γ x − , we find b ∈ Z ∗ Λ such that ψ ( E a Γ ) = E b Λ . Since ψ is injective,we must have that b ∈ Z ∗ Λ \ Λ. Take y ∈ G Λ with π Λ ( y ) = b .Since δ ( L ∩ N Γ ) N Λ , we find that π Λ ( δ ( L ∩ N Γ )) Γ is a finite index normal subgroup of π Λ ( δ ( N Γ )). Since Γ Z ∗ Λ is malnormal, it follows that π Λ ( δ ( N Γ )) Λ. Thus, δ ( N Γ ) N Λ .Since L ∩ xN Γ x − fixes E a Γ pointwise, we have that δ ( L ∩ xN Γ x − ) fixes E b Λ pointwise. There-fore, δ ( L ∩ xN Γ x − ) yN Λ y − . Reasoning as above, it follows that δ ( xN Γ x − ) yN Λ y − .Since δ is injective, it follows that δ (Ker π Γ ) = δ ( N Γ ∩ xN Γ x − ) = δ ( N Γ ) ∩ δ ( xN Γ x − ) N Λ ∩ yN Λ y − = Ker π Λ . Conversely, L ∩ δ − ( N Λ ) must fix E e Γ pointwise, so that L ∩ δ − ( N Λ ) N Γ . Since L ∩ δ − ( N Λ )is a normal finite index subgroup of δ − ( N Λ ), malnormality again implies that δ − ( N Λ ) N Γ .Similarly, δ − ( yN Λ y − ) xN Γ x − . Taking intersections, we conclude that δ − (Ker π Λ ) Ker π Γ . Altogether, we have proven that g ∈ G Γ satisfies δ ( g ) ∈ Ker π Λ if and only if g ∈ Ker π Γ .30here is thus a unique, well defined, injective group homomorphism δ : Z ∗ Γ → Z ∗ Λ such that π Λ ◦ δ = δ ◦ π Γ . Since δ ( N Γ ) N Λ , we have δ (Γ) Λ. So, Γ is isomorphic with a subgroup ofΛ.Finally note that if the embedding θ : L ( H Γ ) → L ( H Λ ) d at the start of the proof has an imagethat is of finite index in L ( H Λ ) d , then δ must be surjective and it follows that Γ ∼ = Λ. Thisends the proof of the theorem. Proof of Corollary F.
Let ( I, ≤ ) be a partially ordered set with density character κ . Take a sup-dense subset I ⊆ I of cardinality at most κ . Then the map ϕ : I → I : ϕ ( i ) = { j ∈ I | j ≤ i } is injective and has the property that ϕ ( i ) ⊆ ϕ ( j ) if and only if i ≤ j . It thus suffices to realizethe partially ordered set (2 I , ⊆ ) in the class of II factors with density character at most κ and relation ֒ → s or ֒ → .By Theorem E, it suffices to realize (2 I , ⊆ ) in the class of infinite groups with cardinality atmost κ and with relation “is isomorphic with a subgroup of” that we denote as ֒ → . Formallyadd to I two extra elements a, b and put I = I ⊔ { a, b } . Assume that (Λ i ) i ∈ I is a family ofnontrivial groups of cardinality at most κ having the following properties: Λ i = Z , Λ i is freelyindecomposable and if i = j , then Λ i Λ j . Whenever J ⊆ I , define Γ J as the free productof the groups Λ i , i ∈ I ∪ { a, b } . For every J ⊆ I , the group Γ J is infinite (since it containsΛ a ∗ Λ b ), of cardinality at most κ , and by the Kurosh theorem, Γ J ֒ → Γ J ′ if and only if J ⊆ J ′ .When I is countable, it is very easy to give an explicit family of such mutually non embeddablegroups (Λ n ) n ∈ N . It suffices to enumerate the prime numbers ( p n ) n ∈ N and take Λ n = Z /p n Z .When κ is any cardinal number, the argument is obviously less explicit. By [FK77], we canchoose a family ( K i ) i ∈ I of infinite fields of cardinality at most κ with the property that for i = j , K i is not isomorphic to a subfield of K j . Then the groups Λ i = K i ⋊ K ∗ i are freelyindecomposable (even amenable) and for i = j , we have that Λ i ֒ → Λ j .Corollary B already provides the concrete chain of separable II factors ( M t ) t ∈ R of order type( R , ≤ ), both in ( II , ֒ → ) or ( II , ֒ → s ).We concretely realize as follows a chain of separable II factors of order type ω , the firstuncountable ordinal. Fix a prime number p . For every countable ordinal λ < ω , recall from[Ric71, Theorem 10] the standard construction of a countable abelian p -group B λ of length λ as the abelian group with generators a ( i , . . . , i n ) indexed by strictly increasing sequences0 ≤ i < · · · i n < λ of any finite length n ∈ N subject to the relations p a ( i ) = 0 and p a ( i , i , . . . , i n ) = a ( i , . . . , i n )for all 0 ≤ i < λ and n ≥
2, 0 ≤ i < · · · i n < λ . For all µ, λ < ω , we have that B λ ֒ → B µ iff λ ≤ µ . By Theorem E, the II factors M λ = M ( Z × Λ λ ) form a chain of separable II factors( M λ ) λ<ω of order type ω , both in ( II , ֒ → ) and ( II , ֒ → s ). factors: proof of Theorem G We prove Theorem G as a consequence of a general construction of II factors M ⊆ M wherewe have a complete control over all II factors N such that M ֒ → s N and N ֒ → s M .The construction is a variant of the construction in Theorem D. So we again denote by G thegroup of finite even permutations of N and let G G be the subgroup of permutations thatfix 1 ∈ N . Denote by A ⊂ G the elements of order 2 and denote by B ⊂ G the elements oforder 3. 31e then introduce an extra variable in the construction: let K be any countable group suchthat Γ = G ∗ K belongs to the family C of Definition 3.1. In our concrete applications, K willbe a free product of amenable groups, so that indeed Γ ∈ C .Define the normal subgroup Γ ⊳ Γ as the kernel of the natural homomorphism π : Γ → K .Put Γ = Γ ∩ ( G ∗ K ). For all a ∈ A , b ∈ B and k ∈ K \ { e } , consider the subgroup Λ a,b,k = h a i∗ k h b i k − of Γ . Choose distinct probability measures µ a,b,k on { , } with µ a,b,k (0) ∈ (0 , / → Γ × Γ the diagonal embedding. Consider the generalized Bernoulli actionΓ × Γ y ( X, µ ) = Y a ∈A ,b ∈B ,k ∈ K \{ e } ( { , } , µ a,b,k ) (Γ × Γ) / ∆(Λ a,b,k ) . (7.1)Write A = L ∞ ( X, µ ) and define the II factors M = A ⋊ (Γ × Γ) and M = A ⋊ (Γ × Γ ). Todescribe all II factors N such that M ֒ → s N and N ֒ → s M , we need the following notation. Notation 7.1.
Given a countable group L , we denote by H ( L, T ) the space of scalar 2-cocycles ω : L × L → T (up to coboundaries) that arise as the so-called obstruction 2-cocycle of a finite dimensional projective unitary representation π : L → U ( C n ) : π ( g ) π ( h ) = ω ( g, h ) π ( gh ).We always implicitly assume that ω is normalized, meaning that ω ( g, e ) = ω ( e, g ) = 1 and π ( e ) = 1. Theorem 7.2.
Let Γ , Γ and K be groups as above. Consider the action Γ × Γ y ( X, µ ) definedby (7.1) and define M ⊂ M as above. Whenever L K is a subgroup, define Γ L = π − ( L ) .When ω ∈ H ( L, T ) , define e ω ∈ H (Γ × Γ L ) by e ω = 1 × ( ω ◦ π ) . Put M ( L, ω ) = A ⋊ e ω (Γ × Γ L ) .1. A II factor N satisfies M ֒ → s N and N ֒ → s M if and only if N is stably isomorphic with M ( L, ω ) for a subgroup L K and an ω ∈ H ( L, T ) .2. Let L i K and ω i ∈ H ( L i , T ) for i = 1 , . We have M ( L , ω ) ֒ → s M ( L , ω ) if and onlyif there exists a g ∈ K such that L ∩ gL g − L has finite index.3. Let t > , L i K and ω i ∈ H ( L i , T ) for i = 1 , . We have M ( L , ω ) ∼ = M ( L , ω ) t ifand only if t = 1 and there exists a g ∈ K such that g − L g = L and ω = ω ◦ Ad g in H ( L , T ) (i.e. equal up to a coboundary).4. Let L i K and ω i ∈ H ( L i , T ) for i = 1 , . The II factors M ( L , ω ) and M ( L , ω ) are virtually isomorphic if and only if there exists a g ∈ K such that g − L g ∩ L has finiteindex in both g − L g and L .Proof. Note that Γ naturally is the semidirect product Γ = Γ ⋊ K . In this way, we obtain acanonical outer action α of K on M such that M = M ⋊ K . Under this identification, wehave that M ( L, ω ) = M ⋊ ω L whenever L K and ω ∈ H ( L, T ).Also note that by Lemma 7.4, all hypotheses of Lemma 5.4 are satisfied, so that all embeddingsof M into M t are trivial.1. This is now a consequence of Lemma 7.5 below.2. Assume that t > M ( L , ω ) ֒ → M ( L , ω ) t . Write N i = M ( L i , ω i ). We encode theembedding N ֒ → N t as a bimodule N H N with dim − N ( H ) = t < ∞ . Take a d -dimensionalprojective representation ζ : L → U ( C d ) with associated 2-cocycle ω . Denote by ρ : N → M d ( C ) ⊗ M the corresponding canonical embedding given by ρ ( a ) = 1 ⊗ a for all a ∈ M and ρ ( u g ) = ζ ( g ) ⊗ u g for all g ∈ L . Define N K M as the bimodule corresponding to ρ .Since dim − M ( K ) = d , the N - M -bimodule H ⊗ N K has finite right dimension td . Restrictingthis bimodule to an M - M -bimodule, Lemma 5.4 says that it must be a direct sum of the32rivial inclusion bimodule M L ( M ) M . We conclude that the M - N -bimodule H ⊗ N K ⊗ M K is isomorphic with a multiple of M K N .Note that K ⊗ M K = M d ( C ) ⊗ L ( M ) so that ρ provides an embedding of the trivial N - N -bimodule into K⊗ M K = M d ( C ) ⊗ L ( M ). We conclude that M H N is contained in a multiple of M K N . Restricting both to M - M -bimodules, it follows that M H M is contained in a multipleof M K M .For every g ∈ K , define M L gM as the irreducible bimodule encoding the automorphism α g ∈ Aut( M ), i.e. L g = L ( M ) and a · ξ · b = aξα g ( b ). Then, M K M is a direct sum ofbimodules of the form M L gM . We have thus shown that M H M is isomorphic with a directsum of bimodules of the form M L gM . So we uniquely find a subset I ⊂ K and a decomposition H = L g ∈ I H g of H into M - M -subbimodules such that M H gM is isomorphic with a directsum of d g ∈ N ∪ { + ∞} copies of L g .We now consider the left action by u s , s ∈ L . Since u s implements the automorphism α s on M , we must have u s · H g = H sg for every g ∈ I . Similarly, H g · u r = H gr for all r ∈ L . Since d g = dim − M ( H g ) = dim M − ( H g ), it follows that d sgr = d g for all g ∈ I , s ∈ L and r ∈ L . Inparticular, I = L IL . It also follows that t = dim − N ( H ) = X gL ∈ I/L d g . In particular, t ∈ N , d g < + ∞ for all g ∈ I and | I/L | < ∞ . Fix any g ∈ I . Since I/L is afinite set on which L acts, the stabilizer of gL has finite index in L . This stabilizer equals L ∩ gL g − .To prove the converse statement, write for brevity M ( L ) instead of M ( L,
1) with respect tothe trivial 2-cocycle 1 ∈ H ( L, T ). When L = L ∩ gL g − has finite index in L , note that M ( L ) ֒ → s M ( L ) and, via Ad u ∗ g , also M ( L ) ֒ → M ( L ). We also have M ( L , ω ) ֒ → s M ( L )and M ( L ) ֒ → s M ( L , ω ). Thus, M ( L , ω ) ֒ → s M ( L , ω ).3. Assume that t > M ( L , ω ) ∼ = M ( L , ω ) t . Again write N i = M ( L i , ω i ) and encodethe isomorphism N ∼ = N t as a bimodule N H N with dim − N ( H ) = t and dim N − ( H ) = 1 /t .Make the decomposition H = L g ∈ I H g as in the proof of 2. Since t = dim − N ( H ) = X gL ∈ I/L d g and 1 /t = dim N − ( H ) = X L g ∈ L \ I d g , it follows that t = 1, d g = 1 for all g ∈ I and both I/L and L \ I are singletons. So, we find g ∈ I such that I = g L = L g . Thus, g − L g = L .Since d g = 1 for all g ∈ I , we can identify H = L ( M ) ⊗ ℓ ( I ) such that the right action by u r , r ∈ L , is given by( a ⊗ δ g g ) · u r = ω ( g, r ) ( a ⊗ δ g gr ) for all a ∈ M , g, r ∈ L .Define γ : L → T such that the left action by u s , s ∈ L , satisfies u s · ( a ⊗ δ g ) = γ ( s ) ( α s ( a ) ⊗ δ sg ) for all a ∈ M , s ∈ L .Since u s u s ′ = ω ( s, s ′ ) u ss ′ for all s, s ′ ∈ L , we get that u s · ( a ⊗ δ s ′ g ) = γ ( s ′ ) ω ( s, s ′ ) γ ( ss ′ ) ( α s ( a ) ⊗ δ ss ′ g ) for all a ∈ M , s ∈ L .Since the left and right actions commute, we also get that u s · ( a ⊗ δ s ′ g ) = γ ( s ) ω ( g − sg , g − s ′ g ) ( α s ( a ) ⊗ δ ss ′ g ) for all a ∈ M , s ∈ L .33omparing both, it follows that ω is cohomologous to ω ◦ Ad g − .The converse statement is trivial.4. Assume that N = M ( L , ω ) and N = M ( L , ω ) are virtually isomorphic, through abimodule N H N with dim − N ( H ) < + ∞ and dim N − ( H ) < + ∞ . Making the decomposition H = L g ∈ I H g as in the proof of 2, we get that L \ I and I/L are finite sets. Taking any g ∈ I ,we find that g − L g ∩ L has finite index in both g − L g and L .Conversely, first note that M ( L i , ω i ) and M ( L i ,
1) are virtually isomorphic. If g − L g ∩ L has finite index in both g − L g and L , it follows immediately that M ( L ,
1) and M ( L ,
1) arevirtually isomorphic.We can now prove Theorem G as stated in the introduction. After that, we formulate aseemingly weaker result in Proposition 7.3, but with a much more concrete construction of thefamily of II factors. Proof of Theorem G.
Applying Theorem 9.1 to the countable group Λ = A ∞ of all finite evenpermutations of N , we find a group K with subgroup L K such that the following propertieshold.1. The intermediate subgroup lattice { L | L L K } is isomorphic with ( I, ≤ ).2. Every intermediate subgroup L L K is freely generated by isomorphic copies of A ∞ .3. For every g ∈ K , we have that g ∈ L ∨ gLg − .Since K is a free product of amenable groups, Theorem 7.2 applies. By property 2, the inter-mediate subgroups L L K have no nontrivial finite dimensional unitary representations.In particular, these groups do not have proper finite index subgroups. In Theorem 7.2, theembedding relations ֒ → and ֒ → s then amount to the relation gL g − L for some g ∈ K . Ifthis relation holds, property 3 and the fact that L L and L L imply that g ∈ L ∨ gLg − L ∨ gL g − L ∨ L = L , so that L L . This concludes the proof. Proposition 7.3.
Let ( I, ≤ ) be any countable partially ordered set and denote by ( I, ≤ ) itscompletion given by all downward closed subsets of ( I, ≤ ) . There is a concrete construction ofII factors ( M i ) i ∈ I with separable predual satisfying the conclusions of Theorem G.Proof. Choose any countably infinite locally finite field K . Define the K -vector space V =( K ) ( I ) as the direct sum of copies of K indexed by I . Note that V is countable. For every a ∈ I , we denote by V a ⊆ V the natural direct summand in position a ∈ I and we denote by V ′ a the direct sum of all V b with b = a , so that V = V a ⊕ V ′ a . Whenever v ∈ K and a ∈ I , wehave the natural vector v a ∈ V a .For every a ∈ I and A ∈ SL ( K ), define π a ( A ) ∈ GL( V ) by π a ( A )( v a ) = ( A ( v )) a for all v ∈ K and π a ( A )( w ) = w for all w ∈ V ′ a . Define the group T SL ( K ) by T = (cid:8)(cid:0) A X B (cid:1) (cid:12)(cid:12) A, B ∈ SL ( K ) , X ∈ M ( K ) (cid:9) . Whenever a, b ∈ I and a = b , denote by π a,b : T → GL( V ) the natural representation acting incoordinates a and b , given by π a,b (cid:0) A X B (cid:1) = S where S ( u ) = u if u ∈ V ′ a ∩ V ′ b , S ( v a + w b ) = ( A ( v ) + X ( w )) a + ( B ( w )) b . X = 0, then S = π a ( A ) π b ( B ). We define the countable subgroup K GL( V )as the subgroup generated by all π a,b ( T ) with a, b ∈ I and a < b . Note that π a (SL ( K )) K for all a ∈ I . We view V as a countable abelian group and K y V acting by automorphisms.Put K = V ⋊ K .For every subset J ⊆ I , denote by V ( J ) ⊆ V the direct sum of all V a with a ∈ J . We claimthat the intermediate subgroups K L K are precisely given by V ( J ) ⋊ K where J ⊆ I is a downward closed set. To prove this claim, fix such an intermediate subgroup L . Then, L = V ⋊ K where V V is an additive subgroup that is globally invariant under the actionof K .Note that if v ∈ K is any nonzero vector, then the additive subgroup of K generated by { A ( v ) − v | A ∈ SL ( K ) } equals K . Indeed, let w ∈ K be an arbitrary vector and take one ofthe two standard basis vectors e ∈ K such that w + e = 0. Then choose A, B ∈ SL ( K ) suchthat A ( e ) = w + e and B ( v ) = e . We get that( AB ( v ) − v ) − ( B ( v ) − v ) = AB ( v ) − B ( v ) = ( w + e ) − e = w . Denote by p a : V → K the natural projection map, so that v = P a ∈ I ( p a ( v )) a for all v ∈ V . If v ∈ V and a ∈ I , we have V ∋ π a ( A )( v ) − v = ( A ( p a ( v )) − p a ( v )) a for all A ∈ SL ( K ). The observation in the previous paragraph thus implies that V a ⊂ V whenever p a ( V ) = { } . Defining J ⊆ I as the set of all a ∈ I such that p a ( V ) = { } , we getthat V = V ( J ).When a ≤ b , we have π a,b ( T )( V b ) = V a . It follows that J is downward closed. So the claim isproven.Note that K is a locally finite group. In particular, K is amenable. Since SL ( K ) has nonontrivial finite dimensional unitary representations, the same holds for the group T . Then,every finite dimensional unitary representation of K has to be the identity on each copy π a,b ( T )of T , and hence on K . It then follows that also the intermediate subgroups V ⋊ K have nonontrivial finite dimensional unitary representations.For every downward closed set J ⊆ I , write L ( J ) = V ( J ) ⋊ K . We finally prove that gL ( J ) g − L ( J ) for some g ∈ K if and only if J ⊆ J . One implication being trivial,assume that gL ( J ) g − L ( J ). Since V ( J ) = gV ( J ) g − gL ( J ) g − L ( J ), we get that V ( J ) V ( J ) and thus, J ⊆ J .We can apply Theorem 7.2 and the proof is complete. Lemma 7.4.
The groups Λ a,b,k < Γ < Γ < Γ introduced in Theorem 7.2 satisfy all theassumptions in 5.2.Proof. Condition 1 is obvious. We can view Γ as the free product of the subgroups kGk − with k ∈ K . Similarly, Γ is the free product of the subgroups kG k − with k ∈ K . Therefore,Γ and Γ have no nontrivial finite dimensional unitary representations, confirming 2.The second statement of Lemma 5.6 implies that conditions 3 and 4 hold.To prove 5, let δ : Γ → Γ be an injective group homomorphism such that δ (Λ a,b,k ) ≺ Γ Λ a,b,k for all a ∈ A , b ∈ B , k ∈ K \ { e } . After composing δ with an inner automorphism of Γ andusing the Kurosh theorem, we find for all k ∈ K , T k ∈ { G, K } , injective group homomorphisms δ k : G → T k and elements w k ∈ Γ such that δ ( g ) = δ e ( g ) for all g ∈ G and δ ( kgk − ) = w k δ k ( g ) w − k for all k ∈ K \ { e } and g ∈ G .35enoting by T ′ k ∈ { G, K } the “other” group so that { T k , T ′ k } = { G, K } , we may furthermoreassume that for every k ∈ K \ { e } , either w k = e , or w k ends with a letter from T ′ k \ { e } .So, for every a ∈ A , b ∈ B and k ∈ K \ { e } , we get that δ (Λ a,b,k ) equals the free product h δ e ( a ) i ∗ w k h δ k ( b ) i w − k . Since δ (Λ a,b,k ) ≺ Γ Λ a,b,k , the second statement of Lemma 5.6 impliesthat T e = T k = G and that, because w k ends with a letter from K \ { e } , we can write w k asthe reduced word w k = u − k v k k , with u k ∈ G and v k ∈ Λ a,b,k . Given k , we must thus have v k ∈ Λ a,b,k for all a ∈ A , b ∈ B . It follows that v k = e . By Lemma 5.6, we also have that u k δ e ( a ) u − k = a and δ k ( b ) = b ± .For a fixed k ∈ K \ { e } , we thus get that δ k ( b ) = b ± for all b ∈ B . By Lemma 5.7, δ k isthe identity homomorphism from G to G . It also follows that, δ e ( g ) = u − k gu k for all g ∈ G and k ∈ K \ { e } . This forces all u k to be equal to a single u ∈ G . We have now proven that(Ad u ) ◦ δ is the identity homomorphism from Γ to Γ.When δ : Γ → Γ is an injective group homomorphism such that δ (Λ a,b,k ) ≺ Γ Λ a,b,k for all a ∈ A , b ∈ B , k ∈ K \ { e } , it follows from the previous paragraph that after a conjugacy, δ is the identity on Γ . In particular, δ ( kG k − ) = kG k − for all k ∈ K . This forces δ ( kGk − ) kGk − . A homomorphism G → G that is the identity on G must be the identityon G . We conclude that δ is the identity on kGk − for all k ∈ K . So, δ is the identity and alsocondition 5 holds. Lemma 7.5.
Let K be a countable group, P a II factor and K y P an outer action. Let N be a II factor. Then the following statements are equivalent.1. There exists a d ∈ N such that N is stably isomorphic with an intermediate subfactor of ⊗ P ⊆ M d ( C ) ⊗ ( P ⋊ K ) .2. There exists a subgroup L K and ω ∈ H ( L, T ) (see Notation 7.1) such that N is stablyisomorphic with P ⋊ ω L .Proof. ⇒
1. If L K is a subgroup and ω ∈ H ( L, T ), choose a finite dimensional projectiverepresentation π : L → U ( C d ) such that π ( g ) π ( h ) = ω ( g, h ) π ( gh ) for all g, h ∈ L . Define theembedding θ : P ⋊ ω L → M d ( C ) ⊗ ( P ⋊ K ) : θ ( au g ) = π ( g ) ⊗ au g for all a ∈ P , g ∈ L .Then, θ ( P ⋊ ω L ) is an intermediate subfactor for 1 ⊗ P ⊆ M d ( C ) ⊗ ( P ⋊ K ).1 ⇒
2. Assume that 1 ⊗ P ⊆ M ⊆ M d ( C ) ⊗ ( P ⋊ K ) is an intermediate subfactor. We provethat M is stably isomorphic with P ⋊ ω L for some subgroup L K and ω ∈ H ( L, T ).Write N = M d ( C ) ⊗ ( P ⋊ K ) and D = M d ( C ) ⊗ P . We can view N as the crossed product N = D ⋊ K . Denote by E D : N → D the unique trace preserving conditional expectation. Forevery g ∈ K , define φ g : N → Du g : φ g ( x ) = E D ( xu ∗ g ) u g . We claim that for every g ∈ K , there exists a vector subspace V g ⊂ M d ( C ) such that φ g ( M ) ⊆ V g ⊗ P u g ⊆ M . (7.2)To prove this claim, define V g = span (cid:8) (id ⊗ τ ) (cid:0) E D ( xu ∗ g )(1 ⊗ y ∗ ) (cid:1) (cid:12)(cid:12) x ∈ M, y ∈ P (cid:9) . By definition, φ g ( M ) ⊆ V g ⊗ P u g . We have to prove that V g ⊗ P u g ⊆ M .36ix x ∈ M , y ∈ P and write a = (id ⊗ τ ) (cid:0) E D ( xu ∗ g )(1 ⊗ y ∗ ) (cid:1) . Then, a = (id ⊗ τ ) (cid:0) E D ( x (1 ⊗ α g − ( y ∗ )) u ∗ g ) (cid:1) = (id ⊗ τ )( x (1 ⊗ α g − ( y ∗ )) u ∗ g ) . Since P ⊆ P ⋊ K is irreducible, it follows that a ⊗ u g belongs to the k · k -closure of the convexhull of the elements (1 ⊗ b ) x (1 ⊗ α g − ( y ∗ b ∗ )) where b ∈ U ( P ).Since 1 ⊗ P ⊆ M , all these elements belong to M and we conclude that V g ⊗ u g ⊆ M . Thenalso V g ⊗ P u g ⊆ M . So (7.2) is proven.Since φ g ( x ) = x for all x ∈ M d ( C ) ⊗ P u g , it follows from (7.2) that φ g ( M ) = V g ⊗ P u g = M ∩ ( M d ( C ) ⊗ P u g ) . (7.3)Write B = V e . By (7.3), B ⊆ M d ( C ) is a unital ∗ -subalgebra. It also follows from (7.3) that V g V ∗ g ⊆ B , V ∗ g V g ⊆ B and BV g B = V g .Let p ∈ B be a minimal projection. Since p ⊗ ∈ M , we can replace M by the stably isomorphic( p ⊗ M ( p ⊗
1) and we may thus assume that B = C
1. Define L = { g ∈ K | V g = { }} . Itfollows that for every g ∈ L , V g = C π ( g ) where π ( g ) ∈ U ( C d ). Since V g V h ⊆ V gh , it followsthat π is a projective representation. Denote by ω ∈ H ( L, ω ) the associated 2-cocycle. Wehave proven that M is generated by the elements π ( g ) ⊗ P u g , g ∈ L . This precisely means that M ∼ = P ⋊ ω L . factors with prescribed semiring of self-embeddings Given a II factor M , we consider all possible stable self-embeddings M ֒ → s M and identify twoembeddings when they are unitarily conjugate. Since we can take direct sums and compositionsof embeddings, we obtain the semiring Emb s ( M ) that we call the embeddings semiring of M .Note that Emb s ( M ) can be identified with the space of isomorphism classes of M - M -bimodules M H M having finite right dimension.The invariant Emb s ( M ) is extremely rich, encoding at the same time the outer automorphismgroup of M , the fundamental group of M and the fusion ring of finite index M - M -bimodules. Inthis section, we deduce from Theorems D and 7.2 several concrete computations of Emb s ( M ).As a consequence, we can prove the following result. Recall that a semigroup F is called leftcancellative if gh = gk implies h = k . Theorem 8.1.
Let S be one of the following semigroups.1. A countable, unital, left cancellative semigroup.2. The semigroup Emb( G ) of self-embeddings of a countable structure (like a countable group,or a countable field).3. The semigroup I ( N ) = { v ∈ N | v ∗ v = 1 } of isometries in a von Neumann algebra N withseparable predual.There then exists a II factor M with separable predual such that Emb s ( M ) ∼ = N [ S ] , the semiringof formal sums of elements in S . Point 1 of Theorem 8.1 was already stated in [Dep13]. There is however a gap in the proof of[Dep13, Lemma 6.2]. In Remark 8.10, we provide a more detailed discussion on the relationbetween our results and [Dep13]. 37or point 2, recall that a countable structure G consists of a countable set I together with acountable family of subsets R k ⊆ I n k , n k ∈ N . Given such a countable structure, its semigroupof self-embeddings, denoted Emb( G ), is defined as the set of all injective maps ϕ : I → I withthe property that for every k , we have( ϕ ( i ) , . . . , ϕ ( i n k )) ∈ R k if and only if ( i , . . . , i n k ) ∈ R k . Note that this semigroup can be smaller than the semigroup of monomorphisms, where onlythe implication ⇐ would be required.For a countable group Γ, we get the semigroup of injective group homomorphisms Γ → Γ.For a countable field K , we get the semigroup of injective field homomorphisms K → K .For a countable graph ( V, E ), we get the semigroup of graph embeddings, i.e. injective maps ϕ : V → V such that x, y are joined by an edge if and only if ϕ ( x ) , ϕ ( y ) are joined by an edge.Of course, Theorem 8.1 provides new classes of Polish groups that can be realized as the outerautomorphism group Out( M ) of a full II factor M and actually is the first systematic and ex-plicit realization result for non locally compact outer automorphism groups. For completeness,we explicitly state this as a corollary.Our method provides in particular a new method to realize countable groups, as well as compactgroups, as the outer automorphism group of a II factor with separable predual. This was firstproven in [IPP05] for compact abelian groups, in [PV06, Theorem 7.12] for finitely presentedcountable groups, in [Vae07, Theorem 2.13] for arbitrary countable groups and in [FV07] forarbitrary compact groups. The construction in the proof of Theorem 8.1 is substantially simplerand deals with all these groups at once. Corollary 8.2.
Each of the following groups arises as the outer automorphism group of a fullII factor with separable predual.1. Permutation groups, i.e. closed subgroups of the group of all permutations of a countable setequipped with the topology of pointwise convergence.2. The unitary groups U ( N ) of von Neumann algebras with separable predual.3. Second countable compact groups.4. Second countable locally compact groups that are totally disconnected.5. Direct sums of the groups in 1, 2, 3, 4. To make the connection with Theorem 8.1, recall that the class of permutation groups in point 1coincides with the class of automorphism groups of countable structures.Before embarking on the proof of Theorem 8.1, we first note that we can easily describe theembeddings semiring for the II factors appearing in Theorem 7.2. Using the same notation,assume that L K is any subgroup without nontrivial finite dimensional unitary representa-tions. Consider the II factor N = L ∞ ( X, µ ) ⋊ (Γ × Γ L ) as in Theorem 7.2. The followingresult then follows immediately. Proposition 8.3.
Define F = { g ∈ K | gLg − L } and denote by F = L \F the (welldefined) quotient semigroup. There is a canonical isomorphism Emb s ( N ) ∼ = N [ F ] .Proof. In the proof of Theorem 7.2, we described all N - N -bimodules N H N with finite rightdimension as direct sums of embeddings induced by g ∈ F . Two such embeddings, inducedby g , g ∈ F , are unitarily conjugate if and only if Lg = Lg .38 emark 8.4. Already Proposition 8.3 can be used to realize several semirings N [ S ] as theembeddings semiring of a II factor. Indeed, assume that S is a countable unital semigroupthat can be embedded into a group T that is either amenable or a free product of amenablegroups. Let G be the group of finite even permutations of N . In the construction of Proposition8.3, we can then take K = G ∗ T with the subgroup L K defined as the free product of theconjugates hGh − , h ∈ S . Note that G ∗ K = G ∗ G ∗ T belongs to the family C of Definition3.1. Since G has no nontrivial finite dimensional unitary representations, also L does not havethem. So we can apply Proposition 8.3. Define F = { g ∈ K | gLg − ⊂ L } . Then, F = L S and L \F = S . We conclude that Emb s ( N ) ∼ = N [ S ].To prove Theorem 8.1, we need more flexibility and proceed in two steps. Assume that ( A , τ ) isan amenable tracial von Neumann algebra and let G Aut( A , τ ) be any closed subgroup of thePolish group of trace preserving automorphisms. We then define the semigroup End( A , τ, G )of all unital normal trace preserving ∗ -homomorphisms ψ : A → A that commute with G ,i.e. θ ◦ ψ = ψ ◦ θ for all θ ∈ G . We similarly denote by Aut( A , τ, G ) the subgroup of invertibleelements, i.e. the trace preserving ∗ -automorphisms ψ ∈ Aut( A , τ ) that commute with G .As a first step, we prove in Proposition 8.5 that an appropriate modification of the constructionin Theorem D allows to realize N [End( A , τ, G )] as the embeddings semiring of a full II factor M , with Aut( A , τ, G ) corresponding to its outer automorphism group Out( M ). Such a resultwas actually already proven in [Dep10, Theorem 8.5] for Out( M ), and in [Dep13, Theorem4.1] for Emb s ( M ). For completeness, we nevertheless include our construction here, since it issimpler and a direct consequence of the results in Section 3.As a second and new step, we provide in Propositions 8.8, 8.11 and 8.12 several results onwhich semigroups arise as End( A , τ, G ), leading to the proof of Theorem 8.1 and Corollary8.2. Proposition 8.5.
Let ( A , τ ) be an amenable tracial von Neumann algebra with separablepredual. Let G Aut( A , τ ) be a closed subgroup and write S = End( A , τ, G ) . Then, thesemiring N [ S ] arises as the embeddings semiring Emb s ( M ) of a full II factor M with separablepredual. We have Out( M ) ∼ = Aut( A , τ, G ) as Polish groups.Proof. Let Γ × Γ y ( X, µ ) be the group action defined in Theorem D. In that context,Γ = G ∗ G G ∗ G = Γ. Let c ∈ G and d ∈ G be elements of order 5. View c in thefirst free product factor and view d in de second free product factor. Define Λ G asΛ = h c i ∗ h d i ∼ = Z / Z ∗ Z / Z . Choose F ∞ ∼ = Λ Λ such that Λ Λ is malnormal: g Λ g − ∩ Λ = { e } for all g ∈ Λ \ Λ . Denote by ∆ : Γ → Γ × Γ the diagonal embedding.We prove the following properties of Λ Γ.1. For all g ∈ Γ and ( a, b ) ∈ A × B , we have that g Λ g − ∩ Λ a,b = { e } .2. For all g ∈ Γ \ Λ , we have that g Λ g − ∩ Λ = { e } .3. If δ : Γ → Γ is an injective group homomorphism and ( a, b ) ∈ A × B , we have that δ (Λ a,b ) Γ Λ .To prove 1, note that as in Lemma 5.6, we have that g Λ g − ∩ Λ a,b = { e } for all ( a, b ) ∈ A × B and g ∈ Γ. Similarly, g Λ g − ∩ Λ is finite for every g ∈ Γ \ Λ . Since Λ Λ and Λ istorsion free, we get that g Λ g − ∩ Λ = { e } for every g ∈ Γ \ Λ . Since Λ Λ is malnormal,statement 2 follows. To prove 3, let δ : Γ → Γ be an injective group homomorphism and( a, b ) ∈ A × B . By the Kurosh theorem, we can compose δ with an inner automorphism ofΓ such that δ ( G ∗ e ) is a subgroup of either G ∗ e or e ∗ G , and such that δ ( e ∗ G ) can beconjugated into either G ∗ e or e ∗ G . So, we may assume that δ ( a ) = a and δ ( b ) = wb w − a , b are elements of order 2 and 3 in either G ∗ e or e ∗ G , and w ∈ Γ. As in Lemma5.6, because of incompatible orders, no infinite subgroup of h a i ∗ w h b i w − can be conjugatedinto Λ = h c i ∗ h d i . A fortiori, statement 3 holds.Choose a group homomorphism α : Λ ∼ = F ∞ → G such that α (Λ ) G is dense and such thatKer α is infinite. We can view α as an action Λ y α ( A , τ ). We get the coinduced actionΓ × Γ y ( A , τ ) = ( A , τ ) (Γ × Γ) / ∆(Λ ) . Using the diagonal action, we define M = ( A ⊗ L ∞ ( X, µ )) ⋊ (Γ × Γ) . We prove that the embeddings semiring Emb s ( M ) is given by N [End( A , τ, G )]. To prove thisstatement, we have to repeat the proof of Lemma 5.4 in this slightly broader context. Write A = A ⊗ L ∞ ( X, µ ).Let d > θ : M → M d a normal unital ∗ -homomorphism. As in the proof of Lemma 5.4,the crossed product II factor M fits into the framework of Section 3. Thus, by Lemma 3.5,we find that d ∈ N and that θ is a direct sum of embeddings of a special form. It suffices toanalyze each of these direct summands separately and may thus assume that θ : M → M d ( C ) ⊗ M with θ ( A ) ⊆ M d ( C ) ⊗ A and θ ( u ( g,h ) ) = 1 ⊗ u π ( g,h ) , where π : Γ × Γ → Γ × Γ is an injective group homomorphism that is either of the form π ( g, h ) = ( η ( g ) , δ ( h )) or π ( g, h ) = ( δ ( h ) , η ( g )).Fix ( a, b ) ∈ A × B . By property 3 above, we have π (∆(Λ a,b )) Γ × Γ Λ . As in the proof ofLemma 5.4, we then find that π (∆(Λ a,b )) ≺ Γ × Γ Λ a,b for all ( a, b ) ∈ A × B . By Lemma 5.8,after a conjugacy, we may assume that π ( g, h ) = ( g, h ) for all ( g, h ) ∈ Γ × Γ. As in Lemma5.4, it then also follows that θ (1 ⊗ b ) = 1 ⊗ (1 ⊗ b ) for all b ∈ L ∞ ( X, µ ).Identify A with its copy in A sitting in the coordinate position ( e, e )∆(Λ ). Properties 1and 2 above imply that A ∩ L (∆(Ker α )) ′ = A ⊗
1. Therefore, θ ( a ⊗
1) = ψ ( a ) ⊗ ψ : A → M d ( C ) ⊗ A . Let g ∈ Γ \ { e } . Since θ ( A ⊗
1) commutes with θ ( u ( e,g ) ( A ⊗ u ∗ ( e,g ) ),we find that ψ ( A ) ⊆ D ⊗ A where D ⊂ M d ( C ) is an abelian von Neumann subalgebra. Itfollows that θ ( M ) ⊆ D ⊗ M . We can thus further decompose θ as a direct sum of embeddings θ : M → M such that θ ( a ⊗
1) = ψ ( a ) ⊗ a ∈ A , θ (1 ⊗ b ) = 1 ⊗ b for all b ∈ L ∞ ( X ), θ ( u ( g,h ) ) = u ( g,h ) for all ( g, h ) ∈ Γ × Γ, (8.1)where ψ : A → A is a trace preserving inclusion. We find that ψ ∈ End( A , τ, G ). Conversely,whenever ψ ∈ End( A , τ, G ), there is a unique embedding θ : M → M satisfying (8.1). Sincethe relative commutant of L (Γ × Γ) in M is trivial, distinct elements of End( A , τ, G ) give riseto embeddings that are not unitarily conjugate. We also have that the relative commutant of L (Γ × Γ) in the ultrapower M ω is trivial, so that M is full. This concludes the proof of theproposition.The following lemma is the key technical result to realize concrete semigroups as End( A , τ, G ).It was already stated as [Dep13, Step 1 of Lemma 6.2], but the proof there has a gap that werepair here. We refer to Remark 8.10 for a more detailed discussion. Lemma 8.6.
Let ( X , µ ) be a standard nonatomic probability space and put ( X, µ ) = ( X , µ ) N .For every measure preserving automorphism ∆ ∈ Aut( X , µ ) , consider the diagonal automor-phism β ∆ ∈ Aut(
X, µ ) : ( β ∆ ( x )) n = ∆( x n ) . et ψ : ( X, µ ) → ( X, µ ) be a nonsingular factor map satisfying ψ ◦ β ∆ = β ∆ ◦ ψ for all ∆ ∈ Aut( X , µ ) . Then ψ is measure preserving and there exists an injective map σ : N → N such that ( ψ ( x )) n = x σ ( n ) for all n ∈ N and a.e. x ∈ X .Proof. Choosing a mixing probability measure preserving action Λ y ( X , µ ), its diagonalproduct Λ y ( X, µ ) is ergodic and measure preserving. It follows that ψ ∗ ( µ ) is Λ-invariant andhence equal to µ . So, ψ is measure preserving.For notational convenience, we write A = L ∞ ( X , µ ) and view ( A, τ ) as the infinite tensorproduct ( A , τ ) N . We denote by π n : A → A the embedding in the n ’th tensor factor.Whenever p ∈ A is a projection, write A ( p ) = C p + C (1 − p ). We view ψ as a trace preservingunital ∗ -homomorphism ψ : A → A that commutes with all β ∆ . Claim 1.
For every projection p ∈ A , we have that ψ (cid:0) ( A ( p )) N (cid:1) ⊆ ( A ( p )) N .When p equals 0 or 1, there is nothing to prove. So assume that 0 < p <
1. Choose a tracepreserving action Λ y ( A , τ ) such that p is invariant and such that the actions Λ y A p andΛ y A (1 − p ) are mixing. This can be done by taking the disjoint union of two Bernoullishifts. Then ( A ( p )) N is precisely the fixed point algebra under the diagonal action Λ y A .Since ψ commutes with this diagonal action, claim 1 follows. Claim 2.
For every n, m ∈ N and every a ∈ A , we have that ψ ( π n ( a )) ∈ π m ( C C a ) A N \{ m } . Write A = A N \{ m } and view A as the tensor product A = π m ( A ) ⊗ A . Let ω be an arbitrarynormal state on A . Define the unital normal positive map θ : A → A : θ ( a ) = π − m ((id ⊗ ω ) ψ ( π n ( a ))) . It suffices to prove that θ ( a ) ∈ C C a for all a ∈ A .Choose an increasing sequence of finite dimensional unital ∗ -subalgebras B k ⊂ A such that S k B k is weakly dense in A . Fix k and denote by p , . . . , p N the minimal projections of B k .From claim 1, we know that θ ( p i ) ∈ C p i + C (1 − p i ) for all i . We can thus write θ ( p i ) = α i p i + β i (1 − p i ) with α i , β i ∈ [0 , η k ∈ ( B k ) ∗ + by η k ( p i ) = β i . Define b k ∈ B k by b k = P i ( α i − β i ) p i . Note that − ≤ b k ≤ θ ( a ) = b k a + η k ( a )1 for all a ∈ B k .Since θ is unital, we have in particular that η k (1)1 = 1 − b k . Therefore, η k (1) ≤ b k = (1 − η k (1)) 1. Since η k is positive, k η k k ≤
2. We conclude that θ ( a ) = η k ( a )1 + (1 − η k (1)) a for all a ∈ B k . (8.2)Let a ∈ A be an arbitrary element. By the Kaplansky density theorem, we can choose k n → ∞ and a n ∈ B k n such that k a n k ≤ k a k for all n and a n → a strongly. After passage to asubsequence, we may assume that the bounded sequences η k n ( a n ) and η k n (1) are convergent,say to α, β ∈ R . It follows from (8.2) that θ ( a n ) → α βa strongly. On the other hand, θ ( a n ) → θ ( a ) strongly, by normality of θ . So, claim 2 is proven.It follows from claim 2 that for all n ∈ N , F ⊂ N finite and a ∈ A ψ ( π n ( a )) ∈ ( C C a ) ⊗F A N \F . (8.3)41ix n ∈ N . We prove that there exists an m ∈ N such that ψ ( π n ( a )) = π m ( a ) for all a ∈ A .Fix a generating Haar unitary u ∈ ( A , τ ).For every finite subset F ⊂ N and every unitary v ∈ A , write π F ( v ) = Y m ∈F π m ( v ) . We use the convention that π ∅ ( v ) = 1. For every unitary v ∈ A with τ ( v ) = 0, we denote by H ( v ) ⊂ L ( A, τ ) the k · k -closed linear span of { π F ( v ) | F ⊂ N finite } . Note that the vectors π F ( v ), F ⊂ N finite, form an orthonormal basis of H ( v ). For every finite subset F ⊂ N , denoteby E F the unique trace preserving conditional expectation of A onto A F . By (8.3), we havethat E F ( ψ ( π n ( v ))) ∈ H ( v ) for every v ∈ U ( A ) with τ ( v ) = 0, and every finite F ⊂ N . Letting F → N , we have E F ( ψ ( π n ( v ))) → ψ ( π n ( v )) in k · k , and we conclude that ψ ( π n ( v )) ∈ H ( v )for every unitary v ∈ A with τ ( v ) = 0.Applying this to our fixed Haar unitary u ∈ A , we can uniquely write ψ ( π n ( u )) = X F⊂ N finite α F π F ( u ) , with convergence in k · k . We now use that ψ ( π n ( u )) = ψ ( π n ( u )) . When F 6 = G are differentfinite subsets of N , we have π F ( u ) π G ( u ) ⊥ H ( u ). Since ψ ( π n ( u )) ∈ H ( u ), it follows that ψ ( π n ( u )) = P H ( u ) (cid:0) ψ ( π n ( u )) (cid:1) = P H ( u ) (cid:0) ψ ( π n ( u )) ψ ( π n ( u )) (cid:1) = X F⊂ N finite α F π F ( u ) . Since ψ is trace preserving, both ψ ( π n ( u )) and ψ ( π n ( u )) have k · k equal to 1. We concludethat X F⊂ N finite | α F | = 1 = X F⊂ N finite | α F | . So there is precisely one finite subset
F ⊂ N with | α F | = 1 and α G = 0 for all G 6 = F .If F = ∅ , we have ψ ( π n ( u )) ∈ C ψ ( π n ( A )) ⊆ C
1, which is absurd because ψ ( π n ( A ))is a diffuse von Neumann algebra. If F has two or more elements, write a = u + u . Then, ψ ( π n ( a )) = α F π F ( u ) + α F π F ( u ). Take m ∈ F . Since F \ { m } is nonempty, it follows thatthe linear span of (id ⊗ ω ) ψ ( π n ( a )), ω ∈ ( A N \{ m } ) ∗ , contains both π m ( u ) and π m ( u ). Byclaim 2, this linear span should be contained in C C π m ( a ). We reached a contradiction andconclude that F = { m } . Write α = α F . It then follows from claim 2 that απ m ( u ) + α π m ( u )belongs to C C π m ( u + u ). This implies that α = 1. So, ψ ( π n ( u )) = π m ( u ). It follows that ψ ( π n ( a )) = π m ( a ) for all a ∈ A .Obviously, this m ∈ N is unique. We denote m = σ ( n ). Since ψ is an injective homomorphism, σ : N → N must be an injective map. We have proven that ψ ( π n ( a )) = π σ ( n ) ( a ) for all n ∈ N and a ∈ A . This concludes the proof of the lemma. Definition 8.7.
Let ( A , τ ) be an amenable tracial von Neumann algebra with separablepredual. Let End( A , τ ) be the semigroup of unital trace preserving ∗ -homomorphisms A → A . We say that a unital subsemigroup S ⊆
End( A , τ ) has the bicentralizer property if thefollowing two properties hold.1. The (automorphic) centralizer C ( S ) = { β ∈ Aut( A , τ ) | α ◦ β = β ◦ α for all α ∈ S} actsergodically on ( A , τ ).2. The bicentralizer equals S : if ψ ∈ End( A , τ ) and ψ ◦ β = β ◦ ψ for all β ∈ C ( S ), then ψ ∈ S . 42learly, a subsemigroup with the bicentralizer property must be closed in the usual Polishtopology on End( A , τ ). Also, if S ⊆
End( A , τ ) has the bicentralizer property, then automat-ically S ∩
Aut( A , τ ) equals the group S inv of invertible elements of S and with the notationof Definition 8.7, C ( C ( S inv )) = S inv . Finally note that if S ⊆
End( A , τ ) has the bicentralizerproperty and ψ : A → A is any unital, normal, injective ∗ -homomorphism that commuteswith C ( S ), the ergodicity of C ( S ) y ( A , τ ) implies that ψ is automatically trace preservingand thus, ψ ∈ S .When A = L ∞ ( X, µ ) is abelian, we identify End( A , τ ) with the semigroup Factor( X, µ ) ofmeasure preserving factor maps (
X, µ ) → ( X, µ ), even though this identification is an anti-isomorphism.It is easy to realize any unital, left cancellative semigroup S as the semigroup of self-embeddingsof a countable structure. Therefore, point 1 of Theorem 8.1 is a special case of point 2. Butin this special case, we can give the following more straightforward construction, which has anindependent interest (see Remark 8.9). Proposition 8.8.
Let S be a countable, unital, left cancellative semigroup. Let ( X , µ ) be astandard nonatomic probability space. Define ( X, µ ) = ( X , µ ) S . For every g ∈ S , define themeasure preserving factor map ψ g : ( X, µ ) → ( X, µ ) : ( ψ g ( x )) h = x gh for all g, h ∈ S and a.e. x ∈ X .Then, the semigroup { ψ g | g ∈ S} Factor(
X, µ ) has the bicentralizer property (in the senseof Definition 8.7). Note that we need the left cancellation property to ensure that ψ g is a well defined factor map. Remark 8.9.
It follows in particular from Proposition 8.8 that the bicentralizer of the Bernoulliaction Γ Aut(( X , µ ) Γ ) equals Γ whenever ( X , µ ) is a standard nonatomic probabilityspace and Γ is any countable group. In [Rud77], this bicentralizer property is proven for theBernoulli action of Γ = Z and an arbitrary, possibly atomic , standard base space ( X , µ ). Wedo not know whether this property holds for arbitrary countable groups Γ. Proof of Proposition 8.8.
Fix a measure preserving factor map ψ : ( X, µ ) → ( X, µ ) that com-mutes with C ( S ). For every measure preserving automorphism ∆ ∈ Aut( X , µ ), consider thediagonal automorphism β ∆ ∈ Aut(
X, µ ) : ( β ∆ ( x )) n = ∆( x n ) . Note that β ∆ ∈ C ( S ). Since we can choose a mixing ∆, it follows that C ( S ) acts ergodicallyon ( X, µ ). By Lemma 8.6, we can take an injective map σ : S → S such that ψ ( x ) h = x σ ( h ) for all h ∈ S and a.e. x ∈ X . Denote by e ∈ S the unit element. It remains to prove that σ ( h ) = σ ( e ) h for all h ∈ S , since it then follows that ψ = ψ g with g = σ ( e ).To prove this, we copy the proof of [Dep13, Step 2 in Lemma 6.2]. Fix a Borel set U ⊂ X with 0 < µ ( U ) <
1. Fix a measure preserving automorphism ∆ ∈ Aut( X , µ ) that is free,i.e. ∆( x ) = x for all x ∈ X , and that satisfies ∆( U ) = U . Fix k ∈ S . Define the measurepreserving automorphism β ∈ Aut(
X, µ ) by β : X → X : ( β ( x )) h = ( ∆( x h ) if x hk ∈ U , x h if x hk
6∈ U .We have β ∈ C ( S ). Therefore, ψ commutes with β . We have( β ( ψ ( x ))) h = ( ∆( x σ ( h ) ) if x σ ( hk ) ∈ U , x σ ( h ) if x σ ( hk )
6∈ U , and ( ψ ( β ( x ))) h = ( ∆( x σ ( h ) ) if x σ ( h ) k ∈ U , x σ ( h ) if x σ ( h ) k
6∈ U .43f for some h ∈ S , we have σ ( hk ) = σ ( h ) k , the set V = { x ∈ X | x σ ( hk ) ∈ U , x σ ( h ) k
6∈ U } has positive measure. By the freeness of ∆, we have β ( ψ ( x )) = ψ ( β ( x )) for all x ∈ V . So, weconclude that σ ( hk ) = σ ( h ) k for all h, k ∈ S . Writing g = σ ( e ), this means that σ ( k ) = gk forall k ∈ S . Remark 8.10.
We use the notation of Proposition 8.5. As mentioned above, in [Dep13,Theorem 4.1], it was already proven that N [End( A , τ , G )] arises as the embeddings semiringEmb s ( M ) of a II factor M , whenever G is an ergodic group of measure preserving transfor-mations of an abelian ( A , τ ). This is very similar to our result in Proposition 8.5, althoughwe do not require A to be abelian or the action G y A to be ergodic.Next, the statement of [Dep13, Lemma 6.2] is identical to the statement of our Proposition8.8. The proof of [Dep13, Lemma 6.2] consists of two steps. In step 1, it is implicitly used that( { , } , µ ) N is atomic when µ is a nontrivial probability measure on { , } , which is of coursefalse. We repair this step 1 here in Lemma 8.6. Step 2 of the proof of [Dep13, Lemma 6.2] iscorrect and is repeated here in our proof of Proposition 8.8.Next assume that G is a countable structure, given by a countable set I and a countable familyof subsets R k ⊆ I n k , k ≥
1. Let J be the disjoint union of I and the sets I n k . Let Emb( G ) acton J by defining g · ( i , . . . , i n k ) = ( g · i , . . . , g · i n k ) for all g ∈ Emb( G ) and ( i , . . . , i n k ) ∈ I n k .We identify in this way Emb( G ) with a closed subsemigroup of Inj( J ), the injective maps from J to J . Proposition 8.11.
Let G be a countable structure and write S = Emb( G ) . Define S ⊆
Inj( J ) asabove. Let ( X , µ ) be a standard nonatomic probability space. Define ( X, µ ) = ( X , µ ) J . Forevery g ∈ S , define the measure preserving factor map ψ g : ( X, µ ) → ( X, µ ) : ( ψ g ( x )) j = x g · j for all g ∈ S , j ∈ J and a.e. x ∈ X .Then, the semigroup { ψ g | g ∈ S} Factor(
X, µ ) has the bicentralizer property (in the senseof Definition 8.7).Proof. For every k ∈ N , write J k = I n k and write J = I . We then view J as the disjoint unionof the subsets J k , k ≥
0. For every ∆ ∈ Aut( X , µ ) and every k ≥
0, consider β ∆ ,k ∈ Aut(
X, µ )defined by (cid:0) β ∆ ,k ( x ) (cid:1) j = ( x j if j ∈ J \ J k ,∆( x j ) if j ∈ J k .By construction, β ∆ ,k ∈ C ( S ) for all ∆ ∈ Aut( X , µ ) and k ≥
0. Using a mixing ∆ andvarying k , it already follows that C ( S ) acts ergodically.Let ψ : ( X, µ ) → ( X, µ ) be a measure preserving factor map that commutes with C ( S ). Forevery k ≥
0, denote by π k : ( X, µ ) → ( X , µ ) J k the natural factor map. Then π k ◦ ψ isinvariant under β ∆ ,m for all ∆ and m = k . By ergodicity, it follows that π k ◦ ψ = ψ k ◦ π k where ψ k : ( X , µ ) J k → ( X , µ ) J k is a measure preserving factor map that commutes with all β ∆ ,k .It then follows from Lemma 8.6 that there exist injective maps σ k : J k → J k such that( ψ ( x )) j = x σ k ( j ) for all k ∈ N , j ∈ J k and a.e. x ∈ X . (8.4)We write σ = σ , which is an injective map from I to I . We will prove that σ k ( i , . . . , i n k ) =( σ ( i ) , . . . , σ ( i n k )) for all k ≥ i , . . . , i n k ) ∈ J k . To prove this statement, we may assumethat X = T and that µ is the Lebesgue measure on T . Fix k ≥ ≤ l ≤ n k . Forevery i ∈ J k = I n k , denote by i l ∈ I its l ’th coordinate and recall that I = J . We will provethat ( σ k ( i )) l = σ ( i l ) for all i ∈ J k . 44efine γ l,k ∈ Aut(
X, µ ) by( γ l,k ( x )) i = ( x i if i ∈ J \ J k , x i l x i if i ∈ J k , with l ’th coordinate i l ∈ I = J .By construction, γ l,k ∈ C ( S ). So, ψ commutes with γ l,k . Using (8.4), it follows that x σ ( i l ) x σ k ( i ) = x ( σ k ( i )) l x σ k ( i ) for a.e. x ∈ X .Hence, σ ( i l ) = ( σ k ( i )) l . This holds for all k ≥
1, 1 ≤ l ≤ n k and i ∈ J k . We have thus proventhat σ k ( i , . . . , i n k ) = ( σ ( i ) , . . . , σ ( i n k )).Recall that the countable structure G is defined by the sets R k ⊆ I n k = J k . For every ∆ ∈ Aut( X , µ ) and every k ≥
1, define η ∆ ,k ∈ Aut(
X, µ ) by( η ∆ ,k ( x )) i = ( x i if i ∈ J \ J k or i ∈ J k \ R k ,∆( x i ) if i ∈ R k .By construction, η ∆ ,k ∈ C ( S ). Expressing that ψ commutes with η ∆ ,k for all ∆ and k says that( σ ( i ) , . . . , σ ( i n k )) ∈ R k if and only if ( i , . . . , i n k ) ∈ R k . We have thus proven that σ ∈ Emb( G ),meaning that ψ = ψ g for some g ∈ S .For the formulation of the next result, we make use of Gaussian probability spaces. To every realHilbert space H R is associated a canonical standard probability space ( X, µ ) and a generatingfamily of random variables X → R : x
7→ h x, ξ i for every ξ ∈ H R with a Gaussian distributionwith mean zero and variance k ξ k , such that ξ
7→ h · , ξ i is real linear.We thus have the Gaussian Hilbert space H R ⊂ L R ( X, µ ). The construction is functorial, in thesense that to any real linear isometric v : H R → H R corresponds an essentially unique measurepreserving factor map ψ v : ( X, µ ) → ( X, µ ) such that h ψ v ( x ) , ξ i = h x, vξ i for all ξ ∈ H R anda.e. x ∈ X .The von Neumann bicommutant theorem can then be translated as follows to a bicentralizerproperty. Proposition 8.12.
Let N ⊆ B ( H ) be a von Neumann algebra acting on a separable Hilbertspace H . Assume that pH has infinite dimension for every nonzero projection p ∈ N . Define I ( N ) = { v ∈ N | v ∗ v = 1 } , the semigroup of isometries in N .View H as a real Hilbert space by forgetting multiplication by i ∈ C and using the scalar product Re h · , · i . Then the semigroup { ψ v | v ∈ I ( N ) } ⊆ Factor(
X, µ ) has the bicentralizer propertyin the sense of Definition 8.7. Of course, the only situation where pH is finite dimensional arises when N admits a type Ifactor B ( K ) as direct summand represented on a finite direct sum of K . We can always realize pH to be infinite dimensional by choosing a representation of N with infinite multiplicity. Proof.
Write S = { ψ v | v ∈ I ( N ) } . Denote P = N ′ ∩ B ( H ). By construction, ψ u ∈ C ( S ) forevery u ∈ U ( P ). By our assumption, the representation of U ( P ) on H has no nonzero finitedimensional invariant subspace, because they would be of the form pH for a nonzero projection p ∈ N . Thus, C ( S ) acts ergodically on ( X, µ ).Take ψ ∈ Factor(
X, µ ) such that ψ commutes with all ψ u , u ∈ U ( P ). We prove that ψ ∈ S .Define the real linear isometry θ : L R ( X, µ ) → L R ( X, µ ) : θ ( F ) = F ◦ ψ .
45e write H R instead of H whenever we want to stress that we view H as a real Hilbert space.Then, U ( H ) ⊂ O ( H R ) and this is a proper inclusion. For every u ∈ O ( H R ), we denote by θ u the orthogonal transformation of L R ( X, µ ) given by θ u ( F ) = F ◦ ψ u .The real Hilbert space L R ( X, µ ) has a canonical direct sum decomposition into R H k ⊂ L R ( X, µ ), k ≥
1, with H = H R and H k being the k -fold symmetric real tensor productof H R . For every u ∈ O ( H R ), the subspaces H k are invariant subspaces of the orthogonaltransformation θ u and the restriction of θ u to H k is given by the k -fold tensor product u ⊗· · ·⊗ u .So whenever u n ∈ O ( H R ) is a sequence that converges weakly to t < t <
1, it followsthat θ u n ( F ) → t k F weakly for every F ∈ H k and k ≥ P does not admit a matrix algebra as directsummand. Therefore, P contains a diffuse abelian von Neumann subalgebra. In particular,we can choose u n ∈ U ( P ) such that u n → t t = 1 /
2. By the discussion in theprevious paragraph, θ u n ( F ) → t k F weakly for every F ∈ H k and k ≥
1. Define the real linearoperator T on L R ( X, µ ) by T (1) = 1 and T ( F ) = t k F for every F ∈ H k and k ≥
1. Since θ u n → T weakly and since θ commutes with all θ u n , we find that θ commutes with T . Since H R = H is the spectral subspace of T with eigenvalue t , we conclude that θ ( H R ) ⊆ H R . Wethus find a real linear isometry v : H R → H R such that the restriction of θ to H R equals v .Since θ commutes with θ u for all u ∈ U ( P ), it follows that v commutes with all u ∈ U ( P ) ⊂O ( H R ). In particular, v commutes with the multiplication by i ∈ C , so that v : H → H isactually complex linear and v ∈ P ′ . By the bicommutant theorem, v ∈ I ( N ).By construction, h ψ ( x ) , ξ i = h ψ v ( x ) , ξ i for a.e. x ∈ X . So, ψ = ψ v ∈ S .We have now gathered enough material to prove Theorem 8.1 and Corollary 8.2. Proof of Theorem 8.1.
The theorem is an immediate consequence of Proposition 8.5 and Propo-sition 8.8 (for point 1), Proposition 8.11 (for point 2) and Proposition 8.12 (for point 3, bytaking an infinite multiplicity representation of N ). Proof of Corollary 8.2.
We first recall that the closed subgroups of the group of all permuta-tions of a countable set can be characterized in two ways. These are precisely the automorphismgroups of countable structures. And they are precisely the Polish groups for which the neu-tral element admits a neighborhood basis consisting of open subgroups. In particular, secondcountable, locally compact, totally disconnected groups belong to this class, so that point 4 isa special case of point 1.By Propositions 8.11 and 8.12 and the remarks in the previous paragraph, each of the groupsin points 1, 2 and 4 can be realized as closed subgroups G Aut(
X, µ ) with C ( G ) actingergodically and G = C ( C ( G )). If K is a second countable, compact group, using the lefttranslation action K y ( K, Haar), the same holds for the groups in point 3.We finally make the following easy observation: if G i Aut( A i , τ i ) are closed subgroups suchthat C ( G i ) acts ergodically and G i = C ( C ( G i )), then G × G Aut( A ⊗ A , τ ⊗ τ ) hasthe same property. Indeed, if ψ ∈ Aut( A ⊗ A , τ ⊗ τ ) commutes with C ( G ) × C ( G ), theergodicity of C ( G i ) implies that ψ ( A ⊗
1) = A ⊗ ψ (1 ⊗ A ) = 1 ⊗ A . Thus, ψ = ψ ⊗ ψ with ψ i ∈ C ( C ( G i )) = G i .Therefore, we can make direct products of the groups in points 1, 2, 3 and 4, so that the proofof the corollary is complete. 46 Realizing algebraic lattices as intermediate subgroup lattices
Let ( I, ≤ ) be a partially ordered set. We say that a subset J ⊆ I admits a supremum if theset { b ∈ I | a ≤ b for all a ∈ J } is nonempty and admits a least element, which is then unique.We similarly define the notion of infimum. We call ( I, ≤ ) a lattice of for all a, b ∈ I , the set { a, b } admits a supremum, denoted a ∨ b , and an infimum, denoted a ∧ b .A lattice ( I, ≤ ) is called complete if every subset admits an infimum and a supremum. Notethat ( I, ≤ ) then admits a least element, denoted as 0 ∈ I , and a greatest element. Let ( I, ≤ )be a complete lattice. An element a ∈ I is called compact if the following property holds:whenever J ⊆ I is a nonempty subset such that a ≤ sup J , there exists a finite subset J ⊆ J such that a ≤ sup J . An algebraic lattice is a complete lattice in which every element can bewritten as the supremum of a set of compact elements.Whenever K is a group and L K , the set of intermediate subgroups { L | L L K } partially ordered by inclusion, is a complete lattice. It is actually algebraic: the compactelements are precisely the intermediate subgroups that can be generated by L and a finitesubset F ⊆ K .Conversely, it was proven in [Tum86] that every algebraic lattice arises in this way as anintermediate subgroup lattice of L K . To prove Theorem G, we need a certain control over L , K and the intermediate subgroups. We want K to be a free product of amenable groupsand we want that the intermediate subgroups have no nontrivial finite dimensional unitaryrepresentations. We also want a control over conjugacy between intermediate subgroups. Allthis can be realized at once by adapting the proof of [Rep04]. We thus prove the followingresult. Theorem 9.1.
Let ( I, ≤ ) be an algebraic lattice and denote by I ⊆ I the set of compactelements. Let κ be the cardinal number max {| I | , | N |} . Let Λ be any nontrivial countablegroup. Define K = Λ ∗ κ as the free product of κ copies of Λ . There exists a subgroup L K with the following properties.1. The intermediate subgroup lattice { L | L L K } is isomorphic with ( I, ≤ ) .2. Every intermediate subgroup L L K is freely generated by conjugates of the copies of Λ in K .3. For every g ∈ K , we have that g ∈ L ∨ gLg − . We prove Theorem 9.1 by straightforwardly adapting the proof of [Rep04]. For completeness,we include the details here. In this paper, we only use Theorem 9.1 in the case where I iscountable. But again for completeness, we prove the general version here. The only placewhere this level of generality influences the proof is in Lemma 9.3, where we need a transfiniteinduction rather than a standard inductive argument.As in [Rep04], we use a dual point of view. Fix an algebraic lattice ( I, ≤ ) and denote by I ⊆ I the set of compact elements. Note that the least element 0 belongs to I and that a ∨ b ∈ I when a, b ∈ I . In technical terms, ( I , ≤ ) is a join semilattice with 0. A subset J ⊆ I is calledan ideal if for all a, b ∈ J , we have a ∨ b ∈ J and x ∈ J whenever x ∈ I and x ≤ a . The mapsending a ∈ I to the ideal { x ∈ I | x ≤ a } is an isomorphism between ( I, ≤ ) and the lattice ofideals in ( I , ≤ ).Let G be a group and let ( I , ≤ ) be a join semilattice with 0. As in [Rep04], a map δ : G → I is called a valuation if δ ( e ) = 0, δ ( g ) = δ ( g − ) and δ ( gh ) ≤ δ ( g ) ∨ δ ( h ) for all g, h ∈ G . Wedenote G δ = { g ∈ G | δ ( g ) = 0 } . More generally, for every a ∈ I , we define the subgroup G δ ( a ) = { g ∈ G | δ ( g ) ≤ a } . So, G δ = G δ (0). 47 emma 9.2. Let G be a group and let ( I , ≤ ) be a join semilattice with . Let δ : G → I bea valuation. Fix g ∈ G and write a = δ ( g ) . Let K , K be any nontrivial groups and define G = G ∗ K ∗ K .There exists a valuation δ : G → I with the following properties.1. For every h ∈ G , we have δ ( h ) = δ ( h ) .2. For every h ∈ G with δ ( h ) ≤ a , we have that h ∈ G δ ∨ g G δ g − .3. For every b ∈ I , the subgroup G δ ( b ) G is a freely generated by conjugates of G δ ( a ) , G δ ( b ) , K and K (which need not all occur).Proof. Note that for any group K , we can extend δ to the valuation δ ∗ G ∗ K , which isdefined as follows. Whenever x = u v u · · · v n u n is a reduced word in u i ∈ G , v i ∈ K , wedefine ( δ ∗ x ) = δ ( u ) ∨ δ ( u ) ∨ · · · ∨ δ ( u n ). It is easy to check that δ ∗ δ ∗ G δ ( b ) ∗ K .If g = e , we take δ = δ ∗ ∗
0. For the rest of the proof, assume that g = e . Pick nontrivialelements k i ∈ K i \ { e } . Write S = G ∗ K and define the valuation δ ′ = δ ∗ S . Define thesubgroup G S by G = k G δ ( a ) k − .We view G as the free product G = S ∗ K . Define h ∈ S by h = g k g − . Whenever b ∈ I satisfies a b , define the subgroup G ( b ) G by G ( b ) = S δ ′ ( b ) ∨ k G k − ∨ h K h − . We will construct the valuation δ : G → I such that G δ ( b ) = G ( b ) whenever a b and G δ ( b ) = S δ ′ ( b ) ∗ K if a ≤ b . We will define δ by considering words in three types of “letters”: L = { s ∈ S | s = e, δ ′ ( s ) = a } , L = k ( G \ { e } ) k − and L = h ( K \ { e } ) h − . We claim that if two alternating words v · · · v n and w · · · w m in the letters L , L , L definethe same element x ∈ G , then n = m and v i = w i for all i . By induction, it suffices to prove that v = w . View x as an element of the free product S ∗ K . The only reductions that happenin v · · · v n and w · · · w m arise when a letter s ∈ L is preceded and/or followed by a letterin L , leading to one of the following elements in S : s , sh , h − s or h − sh . Since δ ′ ( s ) = a and δ ′ ( h ) = a , the elements sh and h − s differ from e . Since s = e , also the elements s and h − sh differ from e . So no further reductions happen.Note here, as a side remark for later use, that this already means that if a b , then G ( b ) isthe free product of the three groups S δ ′ ( b ), k G k − and h K h − .If v = w , up to exchanging the v ’s and w ’s if needed, we must have v ∈ L , v ∈ L , w ∈ L , w ∈ L . The reduced word defining x ∈ S ∗ K then starts as( v h ) ( K \ { e } ) ( h − s ) · · · = w k ( G \ { e } ) k − · · · where depending on the subsequent v i , i ≥
3, we have h − s ∈ { h − , h − L , h − L h } . This setdoes not intersect G \ { e } , because h G and because all elements in h ( G \ { e } ) ∪ h ( G \{ e } ) h − have δ ′ -valuation equal to a , which is the case because h and G are free inside S .So, the claim is proven.Define the valuation δ ′′ : G → I by δ ′′ ( e ) = 0 and δ ′′ ( s z s · · · z n s n ) = a ∨ δ ′ ( s ) ∨ · · · ∨ δ ′ ( s n )when s z s · · · z n s n is a reduced word with s , s n ∈ S , s i ∈ S \ { e } for all 1 ≤ i ≤ n − z j ∈ K \ { e } for all j . It is easy to check that δ ′′ is indeed a valuation.48hen define the map δ : G → I as follows. If x ∈ G can be written as an alternating wordwith letters from L , L and L , and if s , · · · , s n are the letters from L that occur in thisword, put δ ( x ) = δ ′ ( s ) ∨ · · · ∨ δ ′ ( s n ) . If x cannot be written in such a way, then put δ ( x ) = δ ′′ ( x ). By the claim above, δ is welldefined.We prove that δ is indeed a valuation. Since δ ′′ ( v ) ≤ a for all x ∈ L ∪ L , we have for all x ∈ G \ { e } that a ∨ δ ( x ) = δ ′′ ( x ). In particular, δ ( x ) ≤ δ ′′ ( x ) for all x ∈ G .Fix x, x ′ ∈ G \ { e } . We have to prove that δ ( xx ′ ) ≤ δ ( x ) ∨ δ ( x ′ ). If at least one of the x, x ′ cannot be written as an alternating word with letters from L , L and L , we have that δ ( x ) ∨ δ ( x ′ ) ≥ a and thus, δ ( x ) ∨ δ ( x ′ ) = a ∨ δ ( x ) ∨ δ ( x ′ ) = δ ′′ ( x ) ∨ δ ′′ ( x ′ ) ≥ δ ′′ ( xx ′ ) ≥ δ ( xx ′ ) . If both x, x ′ can be written as an alternating word with letters from L , L and L , we denotethe letters from L as s , . . . , s n and s ′ , . . . , s ′ m , respectively. Put b = δ ′ ( s ) ∨ · · · ∨ δ ′ ( s n ) ∨ δ ′ ( s ′ ) ∨ · · · ∨ δ ′ ( s ′ m ) = δ ( x ) ∨ δ ( x ′ ) . If a b , we have that x, x ′ ∈ G ( b ) and thus, xx ′ ∈ G ( b ). This implies that δ ( xx ′ ) ≤ b = δ ( x ) ∨ δ ( x ′ ). If a ≤ b , we get that δ ( x ) ∨ δ ( x ′ ) = b = a ∨ b = a ∨ δ ( x ) ∨ δ ( x ′ ) = δ ′′ ( x ) ∨ δ ′′ ( x ′ ) ≥ δ ′′ ( xx ′ ) ≥ δ ( xx ′ ) . So, δ is a valuation.By construction, if a b , we have G δ ( b ) = G ( b ). If a ≤ b , we have G δ ( b ) = S δ ′ ( b ) ∗ K = G δ ( b ) ∗ K ∗ K .So, δ satisfies properties 1 and 3 of the lemma. If h ∈ G and δ ( h ) ≤ a , we have that h ∈ G δ ( a ). So, h ∈ k − G k G ∨ G δ because δ ( k ) = δ ′ ( k ) = 0. By construction, h K h − G δ , so that k ∈ h − G δ h . Since k ∈ G δ , we have h ∈ g G δ g − . We concludethat k ∈ G δ ∨ g G δ g − . But also by construction, k G k − G δ . Thus, G k − G δ k G δ ∨ g G δ g − . We proved above that h ∈ G ∨ G δ . We thus conclude that h ∈ G δ ∨ g G δ g − and the lemma is proven. Lemma 9.3.
Let G be a group and let ( I , ≤ ) be a join semilattice with . Let δ : G → I be avaluation. Let κ be the cardinal number max {| G | , | N |} . Let Λ be any nontrivial group. Define K = G ∗ Λ ∗ κ . There exists a valuation δ : K → I with the following properties.1. For every h ∈ G , we have δ ( h ) = δ ( h ) .2. If h, g ∈ K and δ ( h ) ≤ δ ( g ) , then h ∈ K δ ∨ gK δ g − .3. For every b ∈ I , the subgroup K δ ( b ) is freely generated by conjugates of copies of Λ K and subgroups of the form G δ ( c ) G , c ∈ I .Proof. Put K = G ∗ Λ ∗ κ . We first prove that there exists a valuation δ : K → I such that 1and 3 hold, and such that for all h, g ∈ G with δ ( h ) ≤ δ ( g ), we have h ∈ ( K ) δ ∨ g ( K ) δ g − .To prove this statement, let κ be the smallest ordinal with | κ | = | G | and let ( g µ ) µ<κ be a wellordering of G . We define the inductive system of extensions ( G µ , δ µ ) by transfinite induction.Put ( G , δ ) = ( G, δ ). For a successor ordinal µ + 1, we define ( G µ +1 , δ µ +1 ) by applying49emma 9.2 to ( G µ , δ µ ), the element g µ ∈ G G µ and the nontrivial groups K = K = Λ. Inparticular, G µ +1 = G µ ∗ Λ ∗ Λ and whenever h ∈ G with δ ( h ) ≤ δ ( g µ ), we have that h ∈ ( G µ +1 ) δ µ +1 ∨ g µ ( G µ +1 ) δ µ +1 g − µ . (9.1)When µ is a limit ordinal, we define ( G µ , δ µ ) as the direct limit of the inductive system( G λ , δ λ ) λ<µ .We put ( K , δ ) = ( G κ , δ κ ). If h, g ∈ G and δ ( h ) ≤ δ ( g ), we have that g = g µ for some µ < κ . By (9.1), we have that h ∈ ( K ) δ ∨ g ( K ) δ g − .We now inductively apply the previous construction, defining an inductive system of extensions( K n , δ n ) such that properties 1 and 3 hold, and such that for all h, g ∈ K n with δ ( h ) ≤ δ ( g ),we have that h ∈ ( K n +1 ) δ n +1 ∨ g ( K n +1 ) δ n +1 g − . Defining ( K, δ ) as the direct limit of ( K n , δ n ),the lemma is proven.We are now ready to prove Theorem 9.1. Proof of Theorem 9.1.
Define G = Λ ∗ I as the free product of copies of Λ indexed by I .Denote by Λ a G the copy of Λ in position a ∈ I . Every nontrivial element g ∈ G \ { e } can be uniquely written as a reduced word with letters from Λ a \ { e } , a ∈ F , where F ⊂ I isthe finite subset of those types of letters that occur in the reduced expression of g . We define δ ( g ) = sup F . It is easy to check that δ : G → I is a well defined valuation.By construction, δ ( h ) ≤ a if and only if h belongs to the free product of Λ b , b ≤ a . We alsonote that δ : G → I is surjective, because δ ( g ) = a whenever g ∈ Λ a \ { e } . We finallyclaim that for all a, b ∈ I , there exist g, h ∈ G with δ ( g ) = a , δ ( h ) = b and δ ( gh ) = a ∨ b .When a = b and a = 0, we pick g ∈ Λ a \ { e } and k ∈ Λ \ { e } . Defining h = kg , the requiredproperties hold. When a = b = 0, we take g = h = e . When a = b , we take g ∈ Λ a \ { e } and h ∈ Λ b \ { e } any nontrivial elements.We now apply Lemma 9.3 to ( G , δ ). Since G = Λ ∗ I and Λ is countable, we have that | G | = κ . We thus find an extension of ( G , δ ) to ( K, δ ) such that for all h, g ∈ K with δ ( h ) ≤ δ ( g ), we have h ∈ K δ ∨ gK δ g − , and such that the subgroups K δ ( b ) are freely generatedby conjugates of Λ. Define the subgroup L K by L = K δ .The original algebraic lattice ( I, ≤ ) is identified with the set of ideals in ( I , ≤ ), partiallyordered by inclusion. For every ideal J ⊆ I , define the intermediate subgroup L K ( J ) K by K ( J ) = { g ∈ K | δ ( g ) ∈ J } . Since δ : G → I is surjective, the extension δ : K → I is surjective. The map J K ( J ) isthus injective and K ( J ) K ( J ) if and only if J ⊆ J .We prove that every intermediate subgroup L S K is of the form S = K ( J ) for an ideal J ⊆ I . Fix L S K . Define J = δ ( S ). We have to prove that J is an ideal and that S = K ( J ).Let a ∈ J and let b ∈ I with b ≤ a . We have to prove that b ∈ J . Take g ∈ S with δ ( g ) = a . Since δ is surjective, take any h ∈ G with δ ( h ) = b . Since δ ( h ) ≤ δ ( g ), we have that h ∈ L ∨ gLg − . Since L S , we get that h ∈ S . So, b ∈ J . Next, take a, b ∈ J . We haveto prove that a ∨ b ∈ J . Take g, h ∈ S such that δ ( g ) = a and δ ( h ) = b . By the propertiesof δ : G → I , we can take g , h ∈ G such that δ ( g ) = a , δ ( h ) = b and δ ( g h ) = a ∨ b .Since δ ( g ) ≤ δ ( g ) and g ∈ S , it follows as in the previous point that g ∈ S . Similarly, h ∈ S .Since δ ( g h ) = a ∨ b , we get that a ∨ b ∈ J . So, J ⊆ I is an ideal.50y definition, S K ( J ). Take h ∈ K ( J ). We have to prove that h ∈ S . Since δ ( h ) ∈ J , wecan take g ∈ S with δ ( g ) = δ ( h ). In particular, δ ( h ) ≤ δ ( g ), so that h ∈ L ∨ gLg − . Because L S , we get that h ∈ S .Statements 1 and 2 of the theorem are now proven. Statement 3 follows because δ ( g ) ≤ δ ( g ). References [AP15] A.N. Aaserud and S. Popa, Approximate equivalence of group actions.
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