Factors of type II_1 without non-trivial finite index subfactors

We call a subfactor trivial if it is isomorphic with the obvious inclusion of N into matrices over N. We prove the existence of type II_1 factors M without non-trivial finite index subfactors. Equivalently, every M-M-bimodule with finite coupling constant, both as a left and as a right M-module, is a multiple of L^2(M). Our results rely on the recent work of Ioana, Peterson and Popa, who proved the existence of type II_1 factors without outer automorphisms.


Introduction
We say that a subfactor N ⊂ M of finite index is trivial, if there exists n ∈ N such that N ⊂ M is isomorphic with 1 ⊗ N ⊂ M n (C) ⊗ N . We prove that there exist type II 1 factors all of whose finite index subfactors are trivial. Because of the constructions in the previous paragraph, the bifinite M -M -bimodules, should be considered as the generalized symmetries of the II 1 factor M . Our main statement then becomes that there exist type II 1 factors all of whose generalized symmetries are inner.
In general, computing the outer automorphism group Out(M ) of a II 1 factor M is very hard. Connes discovered in [3] that Out(M ) is countable whenever M is the group von Neumann algebra of an ICC property (T) group. Only very recently, Ioana, Peterson and Popa proved the existence of type II 1 factors M with Out(M ) trivial, see [5]. Their theorem is an existence result in the same way as is the main result in the current paper. We comment on that below. Explicit examples of II 1 factors with trivial outer automorphism group were constructed by Popa and the author in [14], using crossed products by generalized Bernoulli actions and relying on the techniques of Popa's breakthrough von Neumann strong rigidity results in [9,10]. Note that in [14], it is shown as well that any group of finite presentation can be explicitly realized as the outer automorphism group of a II 1 factor.
Also the fundamental group of a II 1 factor, introduced by Murray and von Neumann in [8], is very hard to compute, unless, of course, you deal with a McDuff factor and get R * + as its fundamental group. Connes proved in [3] that the fundamental group of the group von Neumann algebra of an ICC property (T) group is countable. The first example of a II 1 factor with trivial fundamental group was given by Popa in [11], as the group von Neumann algebra of SL(2, Z) ⋉ Z 2 . Many other such examples are given in [5,9,10,14]. In [9], Popa constructs type II 1 factors with an arbitrarily prescribed countable subgroup of R * + as a fundamental group. An alternative construction is given in [5].

Notations, terminology and preliminaries
Throughout, (M, τ ) denotes a von Neumann algebra M with a faithful normal tracial state τ . We denote, for all n ∈ N 0 and all (M, τ ), M n := M n (C) ⊗ M .
We use the convention N 0 = {1, 2, . . .}. If M is a II 1 factor and t > 0, we also introduce the usual notation M t = pM n p whenever p ∈ M n is a projection with non-normalized trace equal to t.
We make an extensive use of Popa's technique of intertwining subalgebras using bimodules. We explain a few notations and refer to the Appendix for more detailed statements. Let (M, τ ) be a von Neumann algebra with a fixed faithful normal tracial state τ . Let A, B ⊂ M be von Neumann subalgebras. We say that A embeds into B inside M and write A ≺ M B if L 2 (M ) contains a non-zero A-B-subbimodule H that is finitely generated as a right B-module. We write if for every non-zero projection p ∈ A ′ ∩ M , L 2 (pM ) contains a non-zero A-B-subbimodule that is finitely generated as a right B-module.
The normalizer of A ⊂ M consists of the unitaries u ∈ U(M ) satisfying uAu * = A and is denoted by N M (A). We say that A ⊂ M is regular if N M (A) ′′ = M .
If A ⊂ (M, τ ) is a von Neumann subalgebra, we say that a ∈ M quasi-normalizes A if there exist a 1 , . . . , a n , b 1 , . . . , b m ∈ M satisfying Aa ⊂ n i=1 a i A and aA ⊂ m j=1 Ab j . The set of elements quasinormalizing A is denoted by QN M (A) and is a unital * -subalgebra of M containing A. We call quasinormalizer of A inside M the von Neumann algebra QN M (A) ′′ generated by the elements quasi-normalizing A. If QN M (A) ′′ = M , we say that the inclusion A ⊂ M is quasi-regular.
If A ⊂ (M, τ ) is a von Neumann subalgebra, Jones' basic construction [7] is denoted by M, e A and defined as the von Neumann algebra acting on L 2 (M ) generated by A and the orthogonal projection e A of L 2 (M ) onto L 2 (A). Note that A commutes with e A and that e A xe A = E A (x)e A for all x ∈ M , where E A : M → A denotes the unique τ -preserving conditional expectation. Equivalently, M, e A equals the commutant of the right A-action on L 2 (M ).
If (A, τ ) is a von Neumann algebra with a fixed faithful normal tracial state τ and if H A is a right A-module, the commutant A ′ of the right A-action on H is equipped with a canonical normal faithful semifinite trace Tr that can be characterized as follows: One defines dim(H A ) := Tr (1) and one calls dim(H A ) the coupling constant or the relative dimension of the right A-module H A . As such, the definition of dim(H A ) depends on the choice of tracial state τ on A. Throughout this paper, either A will be a II 1 factor, in which case the coupling constant is canonically defined, or A will inherit a trace from a natural ambient II 1 factor.
For II 1 factors, the coupling constant is canonically defined and it is then a complete invariant of Hilbert A-modules. If A has a non-trivial center, a complete invariant of Hilbert A-modules can be given in terms of the center-valued trace. We shall only use the following corollary: if dim(H A ) < ∞ and ε > 0, there exists a central projection z ∈ Z(A), n ∈ N and a projection p ∈ A n such that τ (1−z) < ε and (Hz) for all a ∈ M . If now M K M is another M -M -bimodule, the Connes tensor product H ⊗ M K is defined as the separation and completion of the algebraic tensor product H ⊗ alg K for the scalar product The M -M -bimodule structure on H ⊗ alg K is given by When there is no risk for misunderstanding, the tensor product H ⊗ M K is sometimes simply denoted by HK. A subset F ⊂ FAlg(M ) is called a fusion subalgebra if F is closed under taking submodules, direct sums and tensor products. An important role is played in this paper by freeness between fusion subalgebras.
Definition 0.2. Let M be a II 1 factor. Two fusion subalgebras F 1 , F 2 ⊂ FAlg(M ) are said to be free if the following two conditions hold.
• Every tensor product of non-trivial irreducible bimodules, with factors alternatingly from F 1 and F 2 , is irreducible.
• Two tensor products of non-trivial irreducible bimodules, with factors alternatingly from F 1 and F 2 , are equivalent if and only if they are factor by factor equivalent.
Equivalently, F 1 and F 2 are free if every tensor product of non-trivial irreducible bimodules, with factors alternatingly from F 1 and F 2 , is disjoint from the trivial bimodule.
Whenever α ∈ Aut(M ), we defined in Notation 0.1 the bimodule H(α) ∈ FAlg(M ). So, if Γ M is an outer action, we can regard Γ as a fusion subalgebra of FAlg(M ). We prove some results on the almost normalizing bimodules for Γ N in Section 3. There, the terminology of bimodules almost normalizing Γ N , will become more clear as well. Right now, we already make the following observation.  The II 1 factors in the above theorem are of the form M = R ⋊ Γ, where R is the hyperfinite II 1 factor, Γ is the free product of two groups without non-trivial finite dimensional unitary representations and the outer action Γ N satisfies the following specific conditions. Theorem 1.2. Let Γ 0 , Γ 1 be infinite groups acting outerly on the II 1 factor N . Make the following assumptions.
• The groups Γ 0 , Γ 1 , Z are two by two not virtually isomorphic.
• The groups Γ 0 , Γ 1 are not virtually isomorphic to a non-trivial free product. • N ⊂ N ⋊ Γ 0 has the relative property (T).
The M -M -bimodule H rep (θ) is defined as follows. The Hilbert space is given by C n ⊗ L 2 (M ) and Organization of the proof of Theorems 1.1 and 1.2. A given bifinite M -M -bimodule is of the form H(ψ), where ψ : N ⋊ Γ → (N ⋊ Γ) t is a finite index inclusion. Sections 2 and 3 will imply that we may assume that ψ(N ) ⊂ N t and that the latter is a finite index inclusion. This allows to prove Theorems 1.1 and 1.2 in Section 4. Theorem 1.1 follows once we have proven the existence of groups Γ 0 , Γ 1 without nontrivial finite-dimensional unitary representations, and actions of these groups on the hyperfinite II 1 factor R satisfying all conditions in Theorem 1.2. In order to prove this existence, we have to establish in Section 5 the following result: if F 1 and F 2 are countable fusion subalgebras of FAlg(R), where R is the hyperfinite II 1 factor, then the set α ∈ Aut(R) such that αF 1 α −1 and F 2 are free, is a G δ -dense subset of Aut(R). This last result generalizes A.3.2 in [5] 2 Results of Ioana, Peterson and Popa and some consequences Throughout this section, we fix infinite groups Γ 0 and Γ 1 . We set Γ = Γ 0 * Γ 1 and take an outer action Γ N of Γ on the II 1 factor N . We set We record from [5] the following result. The first statement follows from [5], Theorem 4.3 and the second one from [5], Theorem 1.2.1. [5]). The following results hold.
Since we assume that Q 0 has finite index in M , we arrive at a contradiction.
The following result is a first step towards the main theorem of the paper. Theorem 2.3. Let Γ 0 and Γ 1 be infinite groups, Γ = Γ 0 * Γ 1 their free product and Γ N an outer action on the II 1 factor N . Set M = N ⋊ Γ and suppose that N ⊂ M has the relative property (T).
Realize M t = pM n p. Since π(M ) ⊂ M t has finite index, we can take a projection p 1 ∈ π(M ) m , a finite index inclusion ψ : M t → p 1 π(M ) m p 1 and a non-zero partial isometry v ∈ p(M n,m (C) ⊗ M )p 1 satisfying xv = vψ(x) for all x ∈ M t . Write π(M ) s := p 1 π(M ) m p 1 . Cutting down if necessary, we may assume that E π(M) s (v * v) has support p 1 .
Then, ψ(N t ) ⊂ π(M ) s has the relative property (T). The quasi-normalizer of ψ(N t ) inside π(M ) s contains ψ(M t ) and hence, is of finite index. By Corollary 2.2, we get that ψ(N t ) ≺ π(M) s π(N ) s . So, we find a projection p 2 ∈ π(N ) k , a unital * -homomorphism θ : ψ(N t ) → p 2 π(N ) k p 2 and a non-zero partial isometry Since E π(M) s (v * v) has support p 1 and since w has coefficients in π(M ), it follows that vw = 0. Moreover,

Bifinite bimodules between crossed products and almost normalizing bimodules
The aim of this section is twofold. First of all, Propositions 3.1 and 3.3 describe the structure of irreducible bifinite (P ⋊ Λ)-(N ⋊ Γ)-bimodule containing a bifinite P -N -subbimodule.
The condition of containing a bifinite P -N -subbimodule is of course a very strong one. Typically, an application of the deformation/rigidity techniques explained in Section 2, yields the existence of a P -N -subbimodule of finite N -dimension and the existence of another P -N -subbimodule of finite P -dimension. In Proposition 3.5, we show that in good cases this suffices to get the existence of a bifinite P -N -subbimodule.
Then, there exists a projection p ∈ N n and an irreducible finite index inclusion ψ : • ψ(N 0 ) ⊂ pN n p and this inclusion has finite index; Remark 3.2. The method of the proof below also yields the following result, clarifying the notion of a bifinite bimodule almost normalizing Γ N . It follows that given such an almost normalizing bifinite N - See 0.1 for the notation H(σ g ).
Since the elements of A are M 0 -modular, we write A as acting on the right on H.
Take an irreducible bifinite N 0 -N -subbimodule K ⊂ H. Define H as the closed linear span of M 0 KA. We denote by z the orthogonal projection onto H and observe that z ∈ Z(A). Whenever v ∈ U(A), Kv ∼ = K as N 0 -N -bimodules. So, the regularity of N 0 ⊂ M 0 ensures that H is a direct sum of N 0 -N -bimodules isomorphic with one of the uK for u ∈ N M0 (N 0 ).
Since Z(A) is a finite-dimensional abelian algebra normalized by the unitaries u g , g ∈ Γ, we can define the finite index subgroup Γ 0 < Γ consisting of g ∈ Γ such that z and u g commute. Hence, for g ∈ Γ 0 , we have Ku g ⊂ H, implying that there exists u ∈ N M0 (N 0 ) satisfying Ku g ∼ = uK as N 0 -N -bimodules. Next define the subset I ⊂ Γ as It is easily checked that I is globally normalized by the elements of Γ 0 . Moreover, if g ∈ I, we have that H(σ g ) is contained in K ⊗ N0 K, implying that I is finite. The ICC property of Γ yields that I = {e}.
So, whenever x g = 0, Ku g ∼ = K ⊗ N H(σ g ) ∼ = K and hence g = e. It follows that our relative commutant lives inside p 1 N n p 1 and so, is trivial by the irreducibility of ψ 1 (N 0 ) ⊂ p 1 N n p 1 . The claim is proven.
In particular, we conclude that v * v = p 1 and that vv * is a minimal projection in qM m q ∩ θ(N 0 ) ′ . Also, v * θ(N 0 )v ⊂ p 1 N n p 1 and this is a finite index inclusion.
Set B = qM m q ∩θ(N 0 ) ′ . By irreducibility of θ(M 0 ) ⊂ qM m q, we know that Ad θ(N M0 (N 0 )) yields an ergodic action on B. Since B admits the minimal projection vv * , B is finite-dimensional. Denote by z the central support of vv * in B. Let (f ij ) be matrix units for zB with f 00 = vv * . Take a finite set of u k ∈ N M0 (N 0 ) such that k u k zu * k = q. Finally, take partial isometries v ki in N n (enlarging n if necessary) satisfying v ki v * ki = p 1 for all k, i and p = k,i v * ki v ki a projection in N n . Defining we are done. • ψ(P ) ⊂ pN n p and this is a finite index inclusion satisfying p(N ⋊ Γ) n p ∩ ψ(P ) ′ = pN n p ∩ ψ(P ) ′ ; • for some non-zero projection z ∈ Z(pN n p ∩ ψ(P ) ′ ), commuting with ψ(P ⋊ Λ 0 ), we have Remark 3.4. A close inspection of the proof below, implies that z can be chosen in such a way that the following holds. Define ψ 0 : P ⋊ Λ 0 → z(N ⋊ Γ 0 ) n z : ψ 0 (y) = ψ(y)z and consider the obvious inclusion bimodules Then, Proof of Proposition 3.3. By Proposition 3.1, we get H ∼ = H(ψ) where ψ : P ⋊ Λ → p(N ⋊ Γ) n p is a finite index inclusion satisfying p ∈ N n , ψ(P ) ⊂ pN n p a finite index inclusion and p(N ⋊ Γ) n p ∩ ψ(P ) ′ = pN n p ∩ ψ(P ) ′ .
The second condition in the next proposition is quite artificial. In the application in this paper, one might as well suppose that A ⊂ M is a quasi-regular inclusion, i.e. M = QN M (A) ′′ . Elsewhere, we plan another application of the proposition: there it is known that whenever H ⊂ L 2 (M, τ ) is an A-A-subbimodule with dim(H A ) < ∞, then actually H ⊂ L 2 (A).
It suffices to prove that pq = 0. Indeed, approximating p and q, we get p 0 with Tr A (p 0 ) < ∞ and q 0 with Tr B (q 0 ) < ∞, satisfying p 0 q 0 = 0. Taking a spectral projection of the positive operator q 0 p 0 q 0 , we arrive at an orthogonal projection r ∈ M, e A ∩ B ′ satisfying Tr A (r), Tr B (r) < ∞. Taking K = rL 2 (M, τ ), the lemma is proved.
Since B f ≺ M A, we may assume that v(1 ⊗ w) = 0. Note that ww * ∈ M ∩ B ′ , so that we may assume that v = v(1⊗ww * ). By construction, the right A-module generated by the (finitely many) coefficients of v(1⊗w), is also a left A-module. Our assumptions imply that the coefficients of v(1 ⊗ w) belong to QN M (A) ′′ . With p defined by (1), it is easily checked that H 0 := pL 2 (M, τ ) is a right QN M (A) ′′ -module. By construction, the coefficients of w belong to H 0 and hence, the coefficients of v * = w(v(1 ⊗ w)) * belong to H 0 as well. By construction, the coefficients of v * belong to qL 2 (M, τ ). So, we have shown that pq = 0. • pM n p ∩ ψ(N ) ′ = pN n p ∩ ψ(N ) ′ ,

Proof of the main theorems
• ψ(u g )z = x δ(g) u δ(g) for all g ∈ Λ, where Λ < Γ is a finite index subgroup, δ : Λ → Γ an injective homomorphism with finite index image, x h a unitary in zN n σ h (z) for all h ∈ δ(Λ) and z a central projection in pN n p ∩ ψ(N ) ′ commuting with ψ(N ⋊ Λ).
Denote by K the bifinite N -N -bimodule defined by the inclusion N → zN n z : x → ψ(x)z. We prove that K is a multiple the trivial N -N -bimodule, which will almost end the proof of the theorem.
Our claim is proven and we find a non-zero partial isometry v ∈ p(M n,1 (C) ⊗ N ) satisfying Then, v * v = 1 and (2) remains true replacing v by qψ(u g )vu * g whenever g ∈ Γ and q ∈ pN n p ∩ ψ(N ) ′ . It follows that we can find a unitary w ∈ p(M n,k (C) ⊗ N ) satisfying w * ψ(x)w = 1 ⊗ x for all x ∈ N . It is now an exercise to check that w * ψ(u g )w = θ(g) ⊗ u g for some representation θ : Γ → U(k).
Finally, we prove the existence of groups and actions satisfying all the requirements in Theorem 1.2 and moreover such that the groups do not admit finite-dimensional unitary representations.
Proof of Theorem 1.1. We have to prove that there exist infinite groups Γ 0 , Γ 1 together with outer actions on the hyperfinite II 1 factor R such that all conditions of Theorem 1.2 are satisfied and such that all finite dimensional unitary representations of Γ 0 and Γ 1 are trivial.
In particular, Γ 0 and Γ 1 do not have non-trivial finite index subgroups. Both SL(3, Z) and A ∞ are freely indecomposable. Then, the Kurosh theorem implies that Γ 0 is freely indecomposable as well.
We next claim that there exists an outer action of Γ 0 on the hyperfinite II 1 factor R such that R ⊂ R ⋊ Γ 0 has the relative property (T). First take an outer action of SL(3, Z) on R such that R ⊂ R ⋊ SL(3, Z) has the relative property (T). A way of doing so, goes as follows. Consider the semi-direct product SL(3, Z)⋉(Z 3 ×Z 3 ), where A · (x, y) = (Ax, (A −1 ) t y) for all A ∈ SL(3, Z) and x, y ∈ Z 3 . It is clear that Z 3 × Z 3 is a subgroup with the relative property (T). Take an SL(3, Z)-invariant non-degenerate 2-cocycle ω on Z 3 × Z 3 . We then get the required action of SL(3, Z) on R = L ω (Z 3 × Z 3 ). Next, take any outer action of A ∞ on R. By Connes' uniqueness theorem for outer actions of finite cyclic groups on R (see [4]), we may assume that the actions of Z/3Z ⊂ A ∞ and Z/3Z ⊂ SL(3, Z) coincide. Hence, we get an action of Γ 0 on R. Further modifying the action of A ∞ by applying Proposition 5.2, we have shown that there exists an outer action of Γ 0 on R that extends the SL(3, Z) action. Then, R ⊂ R ⋊ Γ 0 still has the relative property (T).
Finally, take any outer action of Γ 1 on the hyperfinite II 1 factor R. Denote by F the fusion subalgebra of FAlg(R) generated by the bifinite R-R-bimodules almost normalizing Γ 0 R. By Lemma 4.1 below, F is countable. It follows from Theorem 5.1 below that there exists an automorphism α ∈ Aut(R) such that F and αΓ 1 α −1 are free in the sense of Definition 0.2. Replacing Γ 1 by αΓ 1 α −1 , all conditions of Theorem 1.2 are fulfilled and moreover, Γ only has trivial finite dimensional unitary representations. So, we are done.
Assume for convenience that 1 ∈ F and consider the ψ i as non-unital homomorphisms M → M n . By the pigeon hole principle, we can find i = j such that ψ i (x) − ψ j (x) 2 < ε q i 2 for all x ∈ F . Consider the M -M -bimodule p i L 2 (M n )p j with left action given by ψ i and right action by ψ j . The vector ξ = p i −1 2 p i p j satisfies the above conditions and we conclude that p i L 2 (M n )p j contains a non-zero N -central vector. It follows that there exist irreducible N -N -subbimodules K i ⊂ H i and K j ⊂ H j with K i ∼ = K j as N -Nbimodules. To conclude to proof, it suffices to observe that for every i, H i as an N -N -bimodule is a direct sum of irreducible N -N -bimodules isomorphic with H(σ g )K j H(σ h ), g, h ∈ Γ.

Realizing freeness between fusion subalgebras of FAlg(R)
In this section, we prove the following crucial result: whenever F 1 , F 2 are countable fusion subalgebras of FAlg(R), where R denotes the hyperfinite II 1 factor, there exists an automorphism α ∈ Aut(R) such that F α 1 := H(α −1 )F 1 H(α) and F 2 are free. In the terminology of [1], this implies that any two hyperfinite finite index subfactors admit a hyperfinite realization of their free composition (see page 94 in [1]).
Theorem 5.1. Let R be the hyperfinite II 1 factor. Let F 1 , F 2 be countable fusion algebras of bifinite R-Rbimodules. Then, {α ∈ Aut(R) | F α 1 and F 2 are free} is a G δ dense subset of Aut(R). In what follows, we make use of the following special property for a bifinite bimodules R H R over the hyperfinite II 1 factor R. Fix a free ultrafilter ω on N and consider the ultrapower algebra R ω . We claim that there exists n ∈ N and an R-R-bimodular isometric embedding V : H → L 2 (R ω ) ⊕n into the n-fold direct sum of Moreover, R H R does not contain the trivial bimodule if and only if (id ⊗E R ′ ∩R ω )(V ξ) = 0 for all ξ ∈ H.

Recall that if
We are now ready to prove Theorem 5.1 and the proof will be based on the technical Proposition 5.3 below.
3. Taking the intersection of W (ξ 0 , . . . , ξ 2k ; 1 m ) where m runs through N 0 and the ξ i run through a countable · 2 -dense subset of H i , we precisely obtain W .
By the Baire category theorem, these statements together show that W is a G δ dense subset of Aut(R).
To prove the first statement, observe that W (ξ 0 , . . . , ξ 2k ; ε) is the union of all where n runs through N 0 , where λ 1 , . . . , λ n runs through all n-tuples of positive real numbers with sum 1 and where u 1 , . . . , u n runs through all n-tuples of unitaries in R. All these sets are easily seen to be open.
To prove the second statement, set V i ξ i = y i = (y i (1), . . . , y i (n)) t ∈ M n,1 (C) ⊗ R ω . Then, extending an automorphism of R to an automorphism of R ω in the canonical way, we have Fix β ∈ Aut(R). We show that β is in the closure of W (ξ 0 , . . . , ξ 2k ; ε). Write R as the infinite tensor product of 2 by 2 matrices, yielding R = M 2 s (C) ⊗ R s . It is sufficient to prove that, for every s ∈ N 0 , there exists a unitary u ∈ R s such that (Ad u)β ∈ W (ξ 0 , . . . , ξ 2k ; ε). The existence of u follows combining (3), Proposition 5.3 and the following observations.
• If H i is disjoint from the trivial bimodule and β ∈ Aut(R) is arbitrary, H β i does not admit non-zero R-central vectors either and hence, does not even admit non-zero R s -central vectors. So, E R ′ s ∩R ω (β(y i (j))) = 0 for all j = 1, . . . , n, all s and all β ∈ Aut(R).
• By construction, the elements β(y i (j)) ∈ R ω quasi-normalize R for all β ∈ Aut(R). Hence, they quasi-normalize R s for all s.
• We apply Proposition 5.3 to the subfactor R s of the von Neumann algebra generated by R, the y 2i (j) and β(y 2i+1 (j)).
We have the following variant of Theorem 5.1, that we use in the proof of Theorem 1.1. Proof. One can almost entirely copy the proof of Theorem 5.1, using the following observation. Let α ∈ Aut(R) be such that σ k α is outer for every k ∈ K. Denote by R K the fixed point algebra of K. We claim that the R-R-bimodule H(α) has no non-zero R K -central vectors. If it would, the irreducibility of R K ⊂ R implies that there exists a unitary v ∈ R such that vα(x)v * = x for all x ∈ R K . By Jones' uniqueness theorem for outer actions of finite groups (see [6]), we may assume that the action of K is dual and conclude that (Ad v)α = σ k for some k ∈ K. This contradicts our assumption and proves that H(α) has no non-zero R K -central vectors. Writing R K as an infinite tensor product of 2 by 2 matrices, we get R K = M 2 k (C) ⊗ R k . If A ∈ R ω is a unitary implementing α, it follows as in the proof of 5.1 that E R ′ k ∩R ω (A) = 0. This is again the starting point to apply Proposition 5.3.
The following is the crucial result to obtain Theorem 5.1. Most of the proof is taken almost literally from Lemmas 1.2, 1.3 and 1.4 in [13]. We repeat the argument for the convenience of the reader, since slight modifications are needed: in [13], the relative commutant N ′ ∩ M is assumed to be finite-dimensional, while we assume that N is a factor and the inclusion N ⊂ M quasi-regular. This forces us to prove the extra lemma 5.5 below. For every ε > 0 and every K ∈ N 0 , there exists a unitary u ∈ N such that  For every ε > 0 and every K ∈ N 0 , there exists a partial isometry v ∈ f N f satisfying vv * = v * v, τ (vv * ) ≥ τ (f )/4 and Here, and in what follows, we use the convention that v 0 := vv * and v −k := (v * ) k for k ∈ N 0 , whenever v is a partial isometry satisfying vv * = v * v.
Proof. We may assume that A ≤ 1 for all A ∈ V. Since z 2 2 ≤ z z 1 , we prove the following: for every ε > 0 and every K ∈ N 0 , there exists a partial isometry v ∈ f N f such that vv * = v * v, τ (vv * ) ≥ τ (f )/4 and Fix ε > 0 and K ∈ N 0 . Let ε 0 > 0 and define ε n = 2 n+1 ε n−1 , up to ε K . Take ε 0 small enough such that ε K < ε. Define I as the set of partial isometries v ∈ f N f satisfying vv * = v * v and Order I by inclusion of partial isometries. By Zorn's lemma, take a maximal element v ∈ I and set p = vv * . It might be that v = 0. If τ (p) ≥ τ (f )/4, we are done. Otherwise τ (p) < τ (f )/4 and we set p 1 := f − p. Note that τ (p)/τ (p 1 ) ≤ 1/3. Write M 1 := p 1 M p 1 , with normalized tracial state τ 1 and corresponding norms · 1,M1 and · 2,M1 . Applying Theorem A.1.4 in [12] to the inclusion p 1 N p 1 ⊂ p 1 M p 1 , take a non-zero projection q ∈ p 1 N p 1 , such that for all x = A 1 v k1 · · · v ks−1 A s and all 1 ≤ s ≤ K, 1 ≤ |k i | ≤ K and A 1 , . . . , A s ∈ V. We shall prove that a unitary w ∈ qN q can be chosen in such a way that v + w ∈ I. This then contradicts the maximality of v.
• There are n terms with w appearing at one place. Each term has its · 1 -norm bounded by εnτ (q) 4n . Altogether, their · 1 -norm is bounded by ε n τ (q)/4.
• There is 1 term with w appearing in position 1 and position n and with v's in the other positions. This term contains the subexpression qA 1 v k2 · · · v kn−1 A n−1 q .
• There are less than 2 n terms where w appears on at least two positions that are not exactly the positions 1, n. In every such term, we have the subexpression (4), the · 1 -norm of this subexpression is bounded by ε n−1 τ (q)/2. It follows that the sum of all the terms of this type has · 1 -norm bounded by 2 n−1 ε n−1 τ (q) ≤ ε n τ (q)/4.
Step 1. Let a ∈ M with a ≤ 1. The sequence E N ′ ∩M (aw n ) 2 converges to 0, whenever w n is a bounded sequence in N that converges weakly to zero. Indeed, writing E N ′ ∩M = E N ′ ∩M • E N ∨(N ′ ∩M) , we may assume that a ∈ N ∨ (N ′ ∩ M ). So, we may assume that a = xy with x ∈ N ′ ∩ M and y ∈ N . Because N is a factor, E N ′ ∩M (z) = τ (z)1 for all z ∈ N . Hence, E N ′ ∩M (xyw n ) = τ (yw n )x and this last sequence converges to 0 in · 2 .
Step 2. Let ξ ∈ L 2 (M ). The sequence E N ′ ∩M (ξw n ) 2 converges to 0, whenever w n is a bounded sequence in N that converges weakly to zero. This follows immediately from Step 1.
Step 3, proof of the lemma. Define K as the closure of N bN in L 2 (M ). Since N ⊂ M is quasiregular, we may assume that dim(K N ) < ∞. We then find ξ ∈ M 1,n (C) ⊗ K and a, possibly non-unital, * -homomorphism ψ : N → M n (C)⊗N , such that xξ = ξψ(x) for all x ∈ N and such that K equals the closure of ξ(M n,1 (C) ⊗ N ). So, we may assume that b = ξd for some d ∈ M n,1 (C) ⊗ N . But then, aw n b = aξψ(w n )d.
Since ψ(w n )d is a bounded sequence in M n,1 (C) ⊗ N that converges weakly to zero, the lemma follows from Step 2.
We are now ready to prove Proposition 5.3, using an ultrapower argument.
Proof of Proposition 5.3. Let N ⊂ (M, τ ) be a quasi-regular inclusion. Suppose that N is a II 1 factor.

Claim 1.
Let ω be a free ultrafilter on N and f ∈ N ω a non-zero projection. If V ⊂ M ω is a countable set with E (N ′ ∩M) ω (f xf ) = 0 for all x ∈ V, there exists a non-zero partial isometry v ∈ f N ω f satisfying vv * = v * v and E (N ′ ∩M) ω (y) = 0 for every product y with factors alternatingly from V and {v k | k ∈ Z, k = 0}.

Claim 2.
Let ω be a free ultrafilter on N and V ⊂ M ω a countable set with E (N ′ ∩M) ω (x) = 0 for all x ∈ V.
There exists a unitary u ∈ N ω satisfying E (N ′ ∩M) ω (y) = 0 for every product y with factors alternatingly from V and {u k | k ∈ Z, k = 0}.
Proof of Claim 1. Write f = (f n ) where f n is a non-zero projection in N for every n. Write V = {x k | k ∈ N} and choose representatives x k = (x k,n ) n such that E N ′ ∩M (f n x k,n f n ) = 0 for all k, n. By Lemma 5.4, take partial isometries v n ∈ f n N f n such that v n v * n = v * n v n , τ (v n v * n ) ≥ τ (f n )/4 and E N ′ ∩M (y) 2 < 1/n whenever y is a product of at most 2n + 1 factors alternatingly from {x 0,n , . . . , x n,n } and {v k n | 1 ≤ |k| ≤ n}. Then, v := (v n ) does the job.
Proof of Claim 2. Define I as the set of partial isometries v ∈ N ω satisfying vv * = v * v and E (N ′ ∩M) ω (y) = 0 whenever y is a product with factors alternatingly from V and {v k | k ∈ Z, k = 0}. By Zorn's lemma, I admits a maximal element v. If v is a unitary, we are done. Otherwise, vv * = p < 1 and we set f = 1 − p. Define W as the (countable) set of products y with factors alternatingly from V and {v k | k ∈ Z, k = 0} and such that the product y starts and ends with a factor from V. Observe that E (N ′ ∩M) ω (f yf ) = 0 for all y ∈ W. Indeed, Using Claim 1, take a non-zero partial isometry w ∈ f N ω f satisfying ww * = w * w and E (N ′ ∩M) ω (y) = 0 for every product y with factors alternatingly from W and {w k | k ∈ Z, k = 0}. Then, v + w ∈ I, contradicting the maximality of v.
Proof of the Proposition. Consider V ⊂ M ⊂ M ω with E N ′ ∩M (x) = 0 for all x ∈ V. Claim 2 yields a unitary u ∈ N ω such that E (N ′ ∩M) ω (y) = 0 for every product y with factors alternatingly from V and {u k | k ∈ Z, k = 0}. Writing u = (u n ) with u n unitary for all n, some u n for n big enough will do the job since the elements of V are represented by constant sequences in M ω .

Appendix. Intertwining bimodules and quasi-normalizers
We briefly recall Popa's technique of intertwining subalgebras of a II 1 factor using bimodules, introduced in [9,11] (see also Appendix C in [15]). • There exists a projection p ∈ B n , a normal * -homomorphism ψ : A → pB n p and a non-zero partial isometry v ∈ M 1,n (C) ⊗ M satisfying xv = vψ(x) for all x ∈ A.
• There does not exist a generalized sequence (u i ) i∈I of unitaries in A satisfying E B (au i b) 2 → 0 for all a, b ∈ M .
We write A f ≺ M B, if one of the following equivalent conditions is satisfied.