Almost commuting matrices and stability for product groups

We prove that any product of two non-abelian free groups, $\Gamma=\mathbb F_m\times\mathbb F_k$, for $m,k\geq 2$, is not Hilbert-Schmidt stable. This means that there exist asymptotic representations $\pi_n:\Gamma\rightarrow \text{U}({d_n})$ with respect to the normalized Hilbert-Schmidt norm which are not close to actual representations. As a consequence, we prove the existence of contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$, with respect to the normalized Hilbert-Schmidt norm, but $A,B$ are not close to any matrices $A',B'$ such that $A'$ commutes with $B'$ and $B'^*$. This settles in the negative a natural version of a question concerning almost commuting matrices posed by Rosenthal in 1969.


Introduction and statement of main results
A famous question, which can be traced back to the foundations of quantum mechanics [vN29], is whether two matrices A, B, which almost commute with respect to a given norm, must be close to two commuting matrices A ′ , B ′ .It was first explicitly posed by Rosenthal [Ro69] for the normalized Hilbert-Schmidt norm and by Halmos [Ha76] for the operator norm.Almost commuting matrices have since been studied extensively and found applications to several areas of mathematics, including operator algebras and group theory, quantum physics and computer science (see, e.g, the introductions of [LS13,ES19]).The most interesting case of this question is when the matrices are contractions, and "almost" and "close" are taken independent of their sizes.The answer depends both on the types of matrices considered and the norms chosen.Historically, research has focused on the operator norm.In this situation, the answer is positive for self-adjoint matrices by a remarkable result of Lin [Li97] (see also [FR96,Ha08,KS14]), but negative for unitary [Vo83] and general matrices [Ch88] (see [Da85,EL89] for related results).More recently, several works [HL08,Gl10,FK10,FS11,Sa14,HS16,HS17] studied the question for the normalized Hilbert-Schmidt norm and obtained affirmative answers for pairs of self-adjoint, unitary and normal matrices.In fact, the answer is positive if at least one of the matrices is normal, see Remark 1.1(1).However, these results leave wide open the general situation when neither matrix is normal.
We make progress on this problem by proving that, in contrast to the case of normal matrices, a version of Rosenthal's question [Ro69] has a negative answer for non-normal matrices.The version that we consider is natural from the perspective of (self-adjoint) operator algebras.Indeed, it requires that A almost commutes not only with B but also with its adjoint, B * .Theorem A. There exist sequences of matrices A n , B n ∈ M dn (C), for some d n ∈ N, such that (a) A n , B n ≤ 1, for every n ∈ N, (b) The author was supported in part by NSF FRG Grant #1854074.
1 For A = (a i,j ) n i,j=1 ∈ M n (C), we denote by A , A 2 = 1 n n i,j=1 |a i,j | 2 1 2 and τ (A) = 1 n n i=1 a i,i the operator norm, normalized Hilbert-Schmidt norm and normalized trace of A.
Remark 1.1.We continue with two remarks on the statement of Theorem A.
(1) The conclusion of Theorem A fails if one of the matrices is normal.Moreover, the following holds: let A n , B n ∈ M dn (C) be contractions such that A n B n − B n A n 2 → 0 and B n is normal, for every n ∈ N. Then there are (2) Theorem A complements a result of von Neumann [vN42,Theorem 9.7] which implies the existence of contractions A n ∈ M kn (C), for some k n → ∞, such that any contractions B n ∈ M kn (C) which verify condition (b), must satisfy that B n − τ (B n )1 2 → 0. In particular, A n , B n are close to the commuting matrices A n , τ (B n )1.Thus, the pair A n , B n does not satisfy the conclusion of Theorem A, for any choice of contractions B n ∈ M kn (C).Moreover, the arguments used in [vN42] are probabilistic, which suggests that randomly chosen contractions A n ∈ M kn (C) should have the above property.This rules out examples as in Theorem A, where one of the matrices is chosen randomly.Nevertheless, we use matrices satisfying (a strengthening of) the property of [vN42] as building blocks in our construction of A n , B n as in Theorem A (see the comments at the end of the introduction).
Theorem A is a consequence of a non-stability result for the product group F 2 × F 2 , see Theorem B.
To motivate the latter result, we note that whether almost commuting matrices are near commuting ones is a prototypical stability problem.In general, following [Hy41,Ul60], stability refers to a situation when elements which "almost" satisfy an equation must be "close" to elements satisfying the equation exactly.In recent years, there has been a considerable amount of interest in the study of group stability (see [Th17,Io19]).For a countable group Γ, one can define stability with respect to any class C of metric groups endowed with bi-invariant metrics.This requires that any asymptotic homomorphism from Γ to a group in C is close to an actual homomorphism [AP14, AP17, CGLT17, Th17].Specializing to the class C of unitary groups endowed with the normalized Hilbert-Schmidt norms leads to the following notion of stability introduced in [HS17, BL18]: Definition 1.2.A sequence of maps ϕ n : Γ → U(d n ), for some d n ∈ N, is called an asymptotic homomorphism if it satisfies lim n→∞ ϕ n (gh) − ϕ n (g)ϕ n (h) 2 = 0, for every g, h ∈ Γ.The group Γ is called Hilbert-Schmidt stable (or HS-stable) if for any asymptotic homomorphism ϕ n : Γ → U(d n ), we can find homomorphisms ρ n : Γ → U(d n ) such that lim n→∞ ϕ n (g) − ρ n (g) 2 = 0, for every g ∈ Γ.
The class of HS-stable groups includes the free groups F m , virtually abelian groups and one-relator groups with non-trivial center [HS17], certain graph product groups [At18], and is closed under free products.Moreover, the product of two HS-stable groups is HS-stable, provided that one of the groups is abelian [HS17, Theorem 1] or, more generally, amenable [IS19, Corollary D].
However, it remained a basic open problem whether HS-stability is closed under general direct products and, specifically, if F 2 × F 2 is HS-stable (see [Io19,Remark 1.4]).We settle this problem by proving that the product of two non-abelian free groups is not HS-stable.Moreover, we show: Theorem B. The group F k × F m is not flexibly HS-stable, for any integers k, m ≥ 2.
Before discussing the notion of flexible HS-stability and results related to Theorem B, let us outline how Theorem B implies Theorem A. Let ϕ n : F 2 × F 2 → U(d n ) be an asymptotic homomorphism which witnesses that F 2 × F 2 is not HS-stable and a 1 , a 2 ∈ F 2 be free generators.For 1 ≤ j ≤ 2 and n ∈ N, let h n,j , k n,j ∈ M dn (C) be self-adjoint matrices with spectrum contained in [− 1 2 , 1 2 ] such that ϕ n (a j , e) = exp(2πih n,j ) and ϕ n (e, a j ) = exp(2πik n,j ).We then prove that the matrices A n = h n,1 + ik n,1 and B n = h n,2 + ik n,2 satisfy the conclusion of Theorem A.
It was shown in [BL18] that all infinite residually finite property (T) groups Γ (e.g., SL n (Z), for n ≥ 3), are not HS-stable.The proof builds on the observation that any sequence of homomorphisms ρ n : Γ → U(d n ) with d n → ∞ can be perturbed slightly to obtain an asymptotic homomorphism ϕ n : Γ → U(d n − 1).To account for this method of constructing asymptotic homomorphisms, the following weakening of the notion of HS-stability was suggested in [BL18]: If a Connes-embeddable countable group Γ is flexibly HS-stable, then it must be residually finite.On the other hand, deciding if a residually finite group is flexibly HS-stable or not is a challenging problem.For instance, while the arithmetic groups SL n (Z), n ≥ 3, are not HS-stable by [BL18], it is open whether they are flexibly HS-stable.The first examples of residually finite groups which are not flexibly HS-stable were found only recently in [ISW20], where certain groups with the relative property (T), including Z 2 ⋊ SL 2 (Z), were shown to have this property.
Theorem B provides the only other known examples of non-flexibly HS-stable residually finite groups, and the first that do not have infinite subgroups with the relative property (T).Moreover, these are the first examples of residually finite non-HS-stable groups that neither satisfy property (T;FD) (see [BL18, Section 4.2]) nor have infinite subgroups with the relative property (T).
Remark 1.4.We now compare Theorem B with two related results concerning other notions of stability.A countable group Γ is called P-stable if it is stable with respect to the class of finite permutation groups endowed with the normalized Hamming distance (see [Io19] for a survey on P-stability).As shown in [Io19, Corollary B], P-stability is not closed under direct products.Given the similarity between the notions of HS-stability and P-stability [AP14], it should not be surprising that HS-stability is not closed under direct products.We note however that the methods of [Io19] cannot be adapted to prove Theorem B. The approach of [Io19], which exploits the discrete aspects of P-stability, allows to prove that the group F 2 × Z is not P-stable, despite being HS-stable by [HS17, Theorem 1].As we explain at the end of the introduction, to prove Theorem B we introduce an entirely new approach based on ideas from the theory of von Neumann algebras.
A countable group Γ is called W * -tracially stable if it is stable with respect to the class of unitary groups of tracial von Neumann algebras endowed with their 2-norms [HS17].Theorem B strengthens [IS19, Theorem E] which showed F k × F m is not W * -tracially stable, for any integers k, m ≥ 2. Indeed, being W * -tracial stable is stronger than being HS-stable, which corresponds to restricting to unitary groups of finite dimensional von Neumann algebras.
Next, we mention two reformulations of Theorem B in terms of operator algebras.Let (M n , τ n ), n ∈ N, be a sequence of tracial von Neumann algebras and ω be a free ultrafilter on N. The tracial ultraproduct von Neumann algebra ω M n is defined as the quotient ℓ First, by [IS19, Proposition C], if P, Q are commuting separable subalgebras of a tracial ultraproduct ω M n , and P is amenable, then there are commuting von Neumann subalgebras P n , Q n of M n , for all n ∈ N, such that P ⊂ ω P n and Q ⊂ ω Q n .In contrast, Theorem B implies that, without the amenability assumption, this lifting property fails in certain matricial ultraproducts: Corollary C.There exist a sequence (d n ) ⊂ N and commuting separable von Neumann subalgebras P, Q of ω M dn (C) such that the following holds: there are no commuting von Neumann subalgebras By [IS19, Theorem B], the conclusion of Corollary C holds for the ultrapower M ω of certain, fairly complicated, examples of II 1 factors M .Corollary C provides the first natural examples of tracial ultraproducts that satisfy its conclusion.We conjecture that this phenomenon holds for any ultraproduct II 1 factor ω M n .
Second, Theorem B can be reformulated as a property of the full group C * -algebra C * (F 2 × F 2 ).This has been an important object of study since the work of Kirchberg [Ki93] showing that certain properties of C * (F 2 ×F 2 ) (being residually finite or having a faithful trace) are equivalent to Connes' embedding problem (see [Oz04,Pi20]).
Theorem B implies the existence of a * -homomorphism ϕ : Specifically, there is no * -homomorphism ϕ such that π • ϕ = ϕ, where π : ℓ ∞ (N, M dn ) → ω M dn is the quotient homomorphism.We do not know if ϕ admits a unital completely positive (ucp) lift ϕ.If no ucp lift exists, then it would follow that C * (F 2 × F 2 ) does not have the local lifting property (LLP) (see [Oz04, Corollary 3.12]).Whether C * (F 2 × F 2 ) has the LLP is an open problem which goes back to [Oz04] (see also [Oz13,Pi20]).
Comments on the proof of Theorem B. We end the introduction with a detailed outline of the proof of Theorem B. Let us first reduce it to a simpler statement.As we prove in Lemma 2.6, if Γ 1 , Γ 2 are HS-stable, then Γ 1 × Γ 2 is flexibly HS-stable if and only if it is HS-stable.Also, if (Γ 1 * Λ 1 ) × (Γ 2 * Λ 2 ) is HS-stable, for groups Γ 1 , Γ 2 , Λ 1 , Λ 2 , then Γ 1 × Γ 2 must be HS-stable.These facts imply that proving Theorem B is equivalent to showing that F 2 × F 2 is not HS-stable.To prove the latter statement, we will reason by contradiction assuming that F 2 × F 2 is HS-stable.
The proof of Theorem B is divided into two parts, which we discuss separately below.A main novelty of our approach is the use of ideas and techniques from the theory of (infinite dimensional) von Neumann algebras to prove a statement concerning finite unitary matrices.We combine small perturbations results for von Neumann algebras with finite dimensional analogues of two key ideas (the use of deformations and spectral gap arguments) from Popa's deformation/rigidity theory.
The first part of the proof, which occupies Sections 3-5, is devoted to proving the following: Proposition 1.5.Assume that F 2 × F 2 is HS-stable.Then for every ε > 0, there exists δ > 0 such that the following holds: for every k, m, n ∈ N and every To illustrate the strength of the conclusion of Proposition 1.5, we make the following remark: Remark 1.6.Let k, m ∈ N. Then F k × F m is HS-stable if and only if for every ε > 0, there exists δ > 0 such that the following holds: for every n ∈ N and every In view of this, Proposition 1.5 can be interpreted as follows: if F 2 × F 2 is HS-stable then F k × F m is HS-stable and, moreover, it satisfies an "averaged" version of HS-stability, uniformly over all k, m ∈ N.
We continue with some comments on the proof of Proposition 1.5 under the stronger assumption that F 3 × F 3 is HS-stable.The proof of Proposition 1.5 has three main ingredients.All subalgebras of matrix algebras considered below are taken to be von Neumann (i.e., self-adjoint) subalgebras.
The first is a small perturbation result for subalgebras of a tensor product of three matrix algebras Roughly speaking, we prove that any subalgebra Here, for subalgebras P, Q ⊂ M and ε > 0, we say that . A crucial aspect of Lemma 3.5 is that the constants involved are independent of k, n, m ∈ N. Its proof is based on ideas from [Ch79,Po01,Po03] and in particular uses the basic construction as in [Ch79].
The second ingredient in the proof of the Proposition 1.5 is the existence of pairs of unitaries satisfying the following "spectral gap" condition: for a universal constant κ > 0 and every n ∈ N, we can find 3).This is a consequence of a result of Hastings [Ha07,Pi12] on quantum expanders.
To finish the proof of Proposition 1.5 we combine the first two ingredients with a tensor product trick.
and Y 1 , Y 2 ∈ U(m) be pairs of unitaries with spectral gap.We define Then [Z i , T j ] 2 ≈ 0, for every 1 ≤ i, j ≤ 3. Since F 3 × F 3 is assumed HS-stable, there are unitaries ≈ 0 and T j − T ′ j 2 ≈ 0, for every 1 ≤ i, j ≤ 3. Let P the subalgebra of M generated by Z ′ 1 , Z ′ 2 , Z ′ 3 and let Q be its commutant.Since P almost commutes with T 1 , T 2 , using the spectral gap property of Y 1 , Y 2 via an argument inspired by [Po06a,Po06b] we deduce that At this point, the conclusion of Proposition 1.5 follows easily.
In the second part of the proof of Theorem B, presented in Section 6, we construct a counterexample to the conclusion of Proposition 1.5 and derive that F 2 × F 2 is not HS-stable.Our construction, which we describe in detail below, is inspired by Popa's malleable deformation for noncommutative Bernoulli actions, see [Po03,Va06], and its variant introduced in [Io06].
(1) We denote We view A n as a subalgebra of M n , where we embed C 2 ⊂ M 2 (C) as the diagonal matrices.
(2) For 1 on the i-th tensor position.(3) Let G n be a finite group of unitaries which generates A n ⊗ M n .
(4) We define U t ∈ U(C 2 ⊗ C 2 ) by U t = P + e it (1 − P ), where P : C 2 ⊗ C 2 → C 2 ⊗ C 2 be the orthogonal projection onto the one dimensional space spanned by e 1 ⊗ e 2 − e 2 ⊗ e 1 .(5) We identify M n ⊗M n = ⊗ n k=1 (M 2 (C)⊗M 2 (C)) and let θ t,n be the automorphism of M n ⊗M n given by θ t,n (⊗ n k=1 x k ) = ⊗ n k=1 U t x k U * t .(6) Finally, consider the following two sets of unitaries in Then U n and V t,n almost commute: . This is because U n commutes with G n and θ t,n (U ) − U 2 ≤ 2t, for every U ∈ U n .Using this, we show that if t > 0 is small enough, then the sets U n , V t,n contradict the conclusion of Proposition 1.5 for large n ∈ N.
To informally outline our argument, suppose that F 2 × F 2 is HS-stable.Then Proposition 1.5 provides commuting subalgebras P n , Q n of M n ⊗ M n so that P n almost contains U n and Q n almost contains V t,n .Thus, P n almost commutes with V t,n and hence with the generating groups G n , . By passing to commutants, we derive that P n is almost contained in both A n ⊗ 1 and θ t,n (A n ⊗ 1).By perturbing P n slightly, we can in fact assume that Assume for a moment that n = ∞ in the above construction.Then (θ t,∞ ) t∈R recovers the malleable deformation of the Bernoulli action on the hyperfinite II [Io06].While this result cannot be used in our finite dimensional setting, we use the intuition behind its proof and a dimension argument to derive a contradiction.
2 of length at most l, for some l independent on n.This forces the dimension of P n to be at most polynomial in n.But P n also almost contains U n and so all tensors of length 1.This forces the dimension of P n to be at least exponential in n, giving a contradiction as n → ∞.
Remark 1.8.Let Γ = F k × F m , for integers k, m ≥ 2. Note that U n and V t,n almost commute in the operator norm: [U, V ] ≤ 4t, for U ∈ U n , V ∈ V t,n .Using this fact, a close inspection of the proof of Theorem B shows that we prove the following stronger statement: the asymptotic homomorphism ϕ n : Γ → U(d n ) which witnesses that Γ is not flexibly HS-stable is an asymptotic homomorphism in the operator norm, i.e., lim n→∞ ϕ n (gh) − ϕ n (g)ϕ n (h) = 0, for all g, h ∈ Γ.Therefore, Γ fails a hybrid notion of stability which weakens both the notion of matricial stability studied in [ESS18,Da20] and (flexible) HS-stability.
1.1.Acknowledgements.I am grateful to Rémi Boutonnet for stimulating discussions and to Sorin Popa for helpful comments.

Preliminaries
While the main results of this paper concern matrix algebras, the proofs are based on ideas and techniques from the theory of von Neumann algebras.Moreover, our proofs often extend with no additional effort from matrix algebras to general tracial von Neumann algebras.As such, it will be convenient to work in the latter framework.In this section, we recall several basic notions and constructions concerning von Neumann algebras (see [AP] and [Ta79] for more information).
A tracial von Neumann algebra is a pair (M, τ ) consisting of a von Neumann algebra M and a trace τ , i.e., a faithful normal state τ : M → C which satisfies τ (xy) = τ (yx), for all x, y ∈ M .We endow M with the 1-and 2-norms given by Then xy 2 ≤ x y 2 and xy 2 ≤ x 2 y , for all x, y ∈ M .We denote by L 2 (M ) the Hilbert space obtained as the closure of M with respect to • 2 , and consider the standard representation M ⊂ B(L 2 (M )) given by the left multiplication action of M on L 2 (M ).For further reference, we recall the Powers-Størmer inequality (see [BO08, Proposition 6.2.4] and [AP,Theorem 7 2 , for every projections p, q ∈ M .The latter inequality holds because p − q 2 2 = τ (p) + τ (q) − 2τ (pq) and τ (pq) ≤ min{τ (p), τ (q)}.We also note that (2.1) and (2.2) more generally hold when τ : M → C is a semifinite trace.
The matrix algebra M n (C) = B(C n ), for n ∈ N, with its normalized trace τ : M n (C) → C given by is a tracial von Neumann algebra.The associated 2-norm is the normalized Hilbert-Schmidt norm Since M n (C) has trivial center, it is a tracial factor.Any tracial factor is either finite dimensional and isomorphic to M n (C), for some n ∈ N, or infinite dimensional and called a II 1 factor.
Moreover, any finite dimensional von Neumann algebra M is isomorphic to a direct sum of matrix algebras and therefore it is tracial.
where M z i is a finite dimensional factor and thus a matrix algebra, for 1 ≤ i ≤ k.We claim that there is a finite subgroup G ⊂ U (M ) which generates M .If M = M n (C), we can take G to be group of unitaries of the form A subalgebra of a matrix algebra M n (C) is a von Neumann subalgebra if and only if it is self-adjoint.Nevertheless, for consistency, we will call self-adjoint subalgebras of M n (C) von Neumann algebras.

The basic construction. Let (M, τ ) be a tracial von Neumann algebra together with a von
Jones' basic construction M, e Q of the inclusion Q ⊂ M is defined as the von Neumann subalgebra of B(L 2 (M )) generated by M and e Q .Let J : L 2 (M ) → L 2 (M ) be the involution given by J(x) = x * , for all x ∈ M .Then M, e Q is equal to both JQ ′ J, the commutant of the right multiplication action of Q on L 2 (M ), and the weak operator closure of the span of {xe Q y | x, y ∈ M }.
The basic construction admits a normal semifinite trace Tr : It also admits a normal semifinite center-valued tracial weight Φ : [AP,Section 9.4]).Here, Z(Q) + denotes the set of positive operators affiliated with Z(Q).If S and T are the left multiplication operators by x and y * , for x, y ∈ M , then ST * = xe Q y and T * S = E Q (yx).Thus, we conclude that We note that if M is finite dimensional, then M, e Q is finite dimensional and Tr and Φ are finite, i.e., Tr(1) < ∞ and Φ(1) ∈ Z(Q).We next record two well-known properties of Φ.
Lemma 2.1.Given two projections p, q ∈ M, e Q , the following hold: (1) p is equivalent to a subprojection of q if and only if Φ(p) ≤ Φ(q).
Proof.For (1), see [AP, Proposition 9.1.8.].For (2), it is easy to see that any maximal projection r ≤ p which is equivalent to a subprojection of q has the desired property.

Almost containment.
Let us recall the notion of ε-containment studied in [MvN43, Mc70,Ch79].Let P ⊂ pM p, Q ⊂ qM q be von Neumann subalgebras of a tracial von Neumann algebra (M, τ ), for projections p, q ∈ M .For ε ≥ 0, we write P ⊂ ε Q and say that We also define the distance between P and Q by letting Convention.To specify the trace τ , we sometimes write In the rest of this subsection, we prove several useful lemmas.We start with two well-known results.
Proof.Consider the basic construction M, e Q with its canonical semifinite trace Tr : Then an easy calculation shows that Since G is a group, f commutes with G and thus with P .Therefore, if x ∈ (P ) 1 , then using that x commutes with f and (2.5) we get that This proves the conclusion.
Our next goal is to establish the following useful elementary lemma.
Lemma 2.5.Let (M, τ ) be a tracial factor and P ⊂ M a von Neumann subalgebra.Let ε ∈ (0, 1 8 ] and assume that q ∈ M is a projection such that τ (1 − q) = ε.Then there is a von Neumann subalgebra Proof.We claim that P or , there is a minimal projection r ∈ P such that τ (r) > ε 1 2 .Let z ∈ P be the smallest central projection such that r ≤ z.Then we can find d ∈ N such that τ (z) = dτ (r), P z ∼ = M d (C) and there is a * -isomorphism θ : rM r → z(P ′ ∩ M )z such that τ (θ(x)) = dτ (x), for every x ∈ rM r.Since M is a factor and τ (1 − q) = ε, rM r contains a projection of trace ε.Thus, P ′ ∩ M contains a projection of trace dε.Since dε ≥ ε and dε = (dε Let p be a projection in and M is a factor, there is projection q 0 ∈ qM q such that τ . By Lemma 2.2(1) we can find a unitary u ∈ M such that u(1 − p)u * = q 0 and u − 1 2 ≤ 6ε Conversely, let y ∈ (Q) 1 and write y = uxu * + α(q − q 0 ), for some x ∈ (P ) 1 and α ∈ C with |α| ≤ 1.Then Hence, Q ⊂ 13ε 1 4 P , and the conclusion follows.We end this section by illustrating the usefulness of Lemma 2.5 in proving the following: Lemma 2.6.Let Γ 1 and Γ 2 be HS-stable countable groups.Then Γ 1 × Γ 2 is HS-stable if and only if it is flexibly HS-stable.
Proof.Let Γ = Γ 1 × Γ 2 .To prove the lemma, we only have to argue that if Γ is flexibly HS-stable, then it is HS-stable.To this end, assume that Γ is flexibly HS-stable.

Perturbation results
In this section we study the almost containment relation for tracial von Neumann algebras.A crucial feature of the results is that they do not depend on the dimensions of the algebras involved.

3.1.
A "small perturbation" lemma.Our main result is the following small perturbation lemma.If P and Q are subalgebras of a tracial von Neumann algebra such that P is almost contained in Q, we show that P must be close to a subalgebra of M 2 (C)⊗Q.For a tracial von Neumann algebra (M, τ ), we equip M 2 (C) ⊗ M with the trace τ given by τ ( 2 i,j=1 e i,j ⊗ x i,j ) = 1 2 τ (x 1,1 ) + τ (x 2,2 ) .Lemma 3.1.Let (M, τ ) be a tracial von Neumann algebra and let P, Q ⊂ M be von Neumann subalgebras.Assume that P ⊂ ε Q, for some ε ∈ (0, 1 200 ).Then there exists a * -homomorphism Lemma 3.1 implies that there is a non-trivial * -homomorphism from P to M 2 (C) ⊗ Q.This generalizes [Ch79, Theorem 4.7] where the same conclusion was proved assuming that Q is a factor.As shown in [Po01, Theorem A.2] under certain conditions (e.g., if P, Q ⊂ M are irreducible subfactors and P ⊂ M is regular) P ⊂ ε Q implies the existence of u ∈ U (M ) such that uP u * ⊂ Q.However, such a strong conclusion does not hold in general even for irreducible subfactors P, Q ⊂ M (see [PSS03, Proposition 5.5]).
Corollary 3.3.For any ε > 0, there is δ = δ 2 (ε) > 0 such that the following holds.Let (M, τ ) be a tracial von Neumann algebra and P, Q ⊂ M be finite dimensional von Neumann subalgebras such that P ⊂ δ Q. Assume that Q is abelian.Then there exists a von Neumann subalgebra R ⊂ Q such that d(P, R) ≤ ε.
Proof.Given ε > 0, we will prove that any δ > 0 such that δ < 1 200 and 200δ 1 8 < ε works.Assume that P ⊂ δ Q.We will first show that P has a large abelian direct summand.
Let z ∈ Z(P ) be the largest projection such that P z is abelian.Since P (1− z) has no abelian direct summand, we can find a projection p ∈ P (1 − z) with τ (p) ≥ τ (1−z) 3 and a unitary u ∈ P (1 − z) such that p and upu * are orthogonal.Thus, [u, p] 2,τ = √ 2 p 2,τ ≥ 2 3 1 − z 2,τ .On the other hand, By combining the last two inequalities, we derive that Let {p i } m i=1 and {q j } n j=1 be the minimal projections of P z and Q, so that P z = m i=1 Cp i and Q = n j=1 Cq j .Since P ⊂ δ Q, Lemma 3.1 provides a * -homomorphism θ : , for every x ∈ (P ) 1 .In particular, using (3.14) we get that Write θ(z) = n j=1 α j ⊗ q j , where α j ∈ M 2 (C) is a projection, for every 1 ≤ j ≤ n.Let S be the set of all j ∈ {1, • • • , n} such that α j has rank one.Define w = j∈S q j ∈ Q.If j / ∈ S, then α j is equal to 0 or 1 and thus In combination with (3.15), we derive that Since θ(z)(1 ⊗ w) = j∈S α j ⊗ q j and α j has rank one, for every j ∈ S, we get that there is a partition , for every x ∈ (P ) 1 .

Pairs of unitary matrices with spectral gap
The goal of this section is to prove the following two results giving pairs of unitary matrices with spectral gap properties.These provide the first step towards proving Theorem B. For n ∈ N, we denote by τ the normalized trace on M n (C), and by • 2 and • 1 the associated norms.
Proposition 4.1.There is a constant η > 0 such that the following holds.
Proposition 4.1 sufficces to prove that F 3 × F 3 , and thus F m × F n , for all m, n ≥ 3, is not HS-stable.However, to prove the failure of HS-stability for F 2 × F 2 , we will need the following result.
Proposition 4.2.There is a constant η > 0 such that the following holds.
4.1.Pairs of unitary matrices with spectral gap.The proofs of Propositions 4.1 and 4.2 rely on the following result.
Lemma 4.3.There exist a constant κ > 0, a sequence (k n ) of natural numbers with k n → ∞, and a pair of unitaries Moreover, we can take k n = n, for every n ∈ N.
This result is likely known to experts, but, for completeness, we indicate how it follows from the literature.We give two proofs of the main assertion based on property (T) and quantum expanders, respectively.The second proof will allow us to also derive the moreover assertion.
First proof of the main assertion of Lemma 4.3.The first proof combines an argument from the proof of [BCI15, Proposition 3.9(4)], which we recall below, with the fact that Γ := SL 3 (Z) is 2-generated.Indeed, by [Tr62], the following two matrices generate Γ: Since Γ has Kazhdan's property (T) (see, e.g., [BO08, Theorem 12.1.14]),we can find κ > 0 such that if ρ : Γ → U(H) is any unitary representation and P : H → H is the orthogonal projection onto the subspace of ρ(Γ)-invariant vectors, then (4.1) Since Γ is residually finite and has property (T), it has a sequence of finite dimensional irreducible representations π n : Γ → U(k n ), n ∈ N, with k n → ∞ (see the proof of [BCI15, Proposition 3.9(4)]).Alternatively, if p is a prime, then any nontrivial representation of SL 3 (Z/pZ) has dimension at least p−1 2 (see [Ta15, Exercise 3.0.9]).Thus, we can take π n to be any irreducible representation of Γ factoring through SL 3 (Z/p n Z), for any n ∈ N, where (p n ) is a sequence of primes with p n → ∞.
We next give a second proof of the main assertion of Lemma 4.3 showing that one can take k n = n.This relies on the notion on quantum expanders introduced in [BS07, Ha07] (see also [Pi12]).For a related application of quantum expanders, see the recent article [MS20].For k ≥ 2 and a k-tuple Endow M n (C) with the normalized Hilbert-Schmidt norm, note that the space M n (C) ⊖ C1 of matrices of trace zero is T u -invariant, and denote by T 0 u the restriction of T u to M n (C) ⊖ C1.Remark 4.4.We clearly have that T 0 u ≤ k.Moreover, equality holds if k = 2.To see this, Next, by the polar decomposition we can find a unitary z ∈ U(n) such that x = yz.Then we have Since x = 0, by combining the fact that δ ≥ 1 8κ 2 with the last inequality we conclude that Since κ ≥ 1 4 and ε ≤ 4, it follows that ε ≥ 1 10 5 κ 6 , which finishes the proof.
For completeness, let us recall the argument.Let {ξ i } i∈I be an orthonormal basis of B with respect to the scalar product given by its trace.Let x ∈ A ⊗ B and write This finishes the proof.
The proof of Proposition 4.1 shows that (4.4) By Lemma 4.6, x 2 ≤ 10 5 κ 6 ( uxt − x 2 + vxt − x 2 ), for every t ∈ U (A) and x ∈ A. Then the argument from the proof of Proposition 4.1 implies that for every t ∈ U (A) and x ∈ A ⊗ B we have Then Z 1 , Z 2 satisfy conditions (2) and (3) from the conclusion and We will show that condition (1) is satisfied for η = 10 7 (1 + κ 6 ).To this end, fix , where x i,j ∈ A ⊗ B are such that x * i,j = x j,i , for every 1 ≤ i, j ≤ 3. Our goal is to show that (4.6) x Towards this goal, we denote ε = [Z 1 , x] 2 + [Z 2 , x] 2 and record the following elementary fact: where By using Fact 4.7, we also derive that (u Together with (4.7), this gives that (4.9) Since [Z 2 , x] 2 ≤ ε, we also get that x 2,3 w − x 1,2 2 ≤ ε and x 3,1 − x 1,2 2 ≤ ε.Together with (4.8), this gives that (4.10) Since x = x * , by using (4.7), (4.8), (4.9) and (4.10), we get that x 2 ] 2 2 , for every unitary u and by using (4.6) for x 1 and x 2 we get that This finishes the proof.

Proof of Proposition 1.5
This section is devoted to the proof of Proposition 1.5.We first prove Proposition 1.5 under the stronger assumption that F 3 × F 3 is HS-stable, since the proof is more transparent in this case and relies on the simpler Proposition 4.1 instead of Proposition 4.2.
The second paragraph of the proof implies that there are Denote by P ⊂ M the von Neumann subalgebra generated by Similarly, using that Q = P ′ ∩ M commutes with Z 1 , Z 2 and (5.3) we get that Since M is a finite dimensional factor, the bicommutant theorem gives that Q ′ ∩ M = P .By applying Lemma 2.3 we derive that A ⊗ 1 ⊗ 1 ⊂ 16ηε 0 P .Since 16ηε 0 < δ 3 ( ε 16 ), by combining the last two paragraphs and Lemma 3.5 we find a von Neumann subalgebra S ⊂ B such that (5.5) d(P, A ⊗ S ⊗ 1) ≤ ε 16 .
Denote T = S ′ ∩ B. By Lemma 2.3 we get that (5.6) Since Z 3 ∈ P , (5.5) gives that Finally, by Lemma 2.2 we can find Since S and T commute, the conclusion follows.
6. Proof of Theorem B 6.1.Construction.In this section we prove Theorem B by constructing a counterexample to the conclusion of Proposition 1.5.We start by recalling our construction presented in the introduction.
(1) We denote We view A n as a subalgebra of M n , where we embed C 2 ⊂ M 2 (C) as the diagonal matrices.
(2) For 1 orthogonal projection onto the one dimensional space spanned by e 1 ⊗ e 2 − e 2 ⊗ e 1 .(5) We identify We begin with the following elementary lemma.For t ∈ R, we let ρ t = 1+cos(t) 2 ∈ [0, 1].We endow M n with its unique trace τ and the scalar product given by x, y = τ (y * x), for every x, y ∈ M n .For 1 ≤ l ≤ n, we denote by e l : M n → M n the orthogonal projection onto the subspace of tensors of length at most l, i.e., the span of ⊗ n k=1 x k , with x k ∈ M 2 (C) and |{k | x k = 1}| ≤ l.Lemma 6.2.The following hold: (1) P (x ⊗ 1)P = τ (x)P , for every x ∈ M 2 (C).
To prove (4), for 0 ≤ i ≤ n, we denote by V i ⊂ M n the span of tensors of the form 2 , (6.1) implies part (4).Next, we show that the sets of unitaries U n and V t,n almost commute: 6.2.A consequence of HS-stability of F 2 × F 2 .To prove Theorem B, we show that if t > 0 is small enough, the almost commuting sets of unitaries U n and V t,n contradict the conclusion of Proposition 1.5 for large n ∈ N. To this end, we first use Proposition 1.5 to deduce the following: Corollary 6.4.Assume that F 2 × F 2 is HS-stable.Then for every ε > 0, there exists t > 0 such that the following holds: for every n ∈ N, we can find a von Neumann subalgebra 64 , where δ 2 : (0, +∞) → (0, +∞) is the function provided by Corollary 3.3.By Lemma 6.3 we have that , for every n ∈ N and t ∈ R. Since F 2 × F 2 is HS-stable, Proposition 1.5 implies that if t > 0 is small enough then the following holds: given any n ∈ N, we can find a von Neumann subalgebra Since M n ⊗ M n is a finite dimensional factor, the bicommutant theorem gives that (P ′ ) ′ = P .Since A n ⊂ M n is a maximal abelian subalgebra, we also have that (A n ⊗ M n ) ′ = A n ⊗ 1.By combining these facts with (6.4) and Lemma 2.3, we derive that P ⊂ 8 √ η A n ⊗ 1 and P ⊂ 8 √ η θ t,n (A n ⊗ 1).
Let {z j } m j=1 be an enumeration of the minimal projections of Z(C).Then C = m j=1 Cz j , where Cz j is a factor and thus isomorphic to a matrix algebra M n j (C), for some n j ∈ N.
If j ∈ S, then since Cz j is a isomorphic to the matrix algebra M n j (C), it admits an orthonormal basis B j whose every element is of the form u u 2 , for some u ∈ U (Cz j ).Then B = ∪ j∈S B j is an orthonormal basis for Cz = j∈S Cz j and (6.8) implies that e l (ξ) 2 2 ≥ 1 2 , for every ξ ∈ B. Recall that e l is the orthogonal projection onto the subspace W l ⊂ M n of tensors of length at most l and let O be an orthonormal basis for W l .Then we have that Combining (6.9), (6.10) and (6.11) implies that dim(Cz) ≤ 2(6n) 64 ε t 2 +1 , as desired.
Proof.Let p = 1 0 0 0 .For 1 ≤ i ≤ n, let , where p is placed on the i-th tensor position.Then X n,i = 2p i − 1 and so X n,i − E C (X n,i ) = 2(p i − E C (p i )), for every 1 ≤ i ≤ n.Thus, the hypothesis rewrites as (6.12) Let {q j } m j=1 be the minimal projections of C such that C = m j=1 Cq j .We claim that (6.13) p − E C (p) 2 2 = m j=1 τ (pq j )τ ((1 − p)q j ) τ (q j ) , for every projection p ∈ A n .
group Γ is called flexibly HS-stable if for any asymptotic homomorphism ϕ n : Γ → U(d n ), we can find homomorphisms ρ n : Γ → U(D n ), for some D n ≥ d n such that lim n→∞ Dn dn = 1 and lim n→∞ ϕ n (g) − p n ρ n (g)p n 2 = 0, for every g ∈ Γ, where p n : C Dn → C dn denotes the orthogonal projection for every n ∈ N.

2. 1 .
Von Neumann algebras.For a complex Hilbert space H, we denote by B(H) the algebra of bounded linear operators on H and by U(H) = {u ∈ B(H) | u * u = uu * = 1} the group of unitary operators on H.For x ∈ B(H), we denote by x its operator norm.A set of operators S ⊂ B(H) is called self-adjoint if x * ∈ S, for all x ∈ S. We denote by S ′ the commutant of S, i.e., the set of operators y ∈ B(H) such that xy = yx, for all x ∈ S. A self-adjoint subalgebra M ⊂ B(H) is a von Neumann algebra if it is closed in the weak operator topology.By von Neumann bicommutant's theorem, a unital self-adjoint subalgebra M ⊂ B(H) is a von Neumann algebra if and only if it is equal to its bicommutant, M = (M ′ ) ′ .From now on, we assume that all von Neumann algebras M are unital.We denote by Z(M ) = M ′ ∩ M the center of M , by (M ) 1 = {x ∈ M | x ≤ 1} the unit ball of M , by M + = {x ∈ M | x ≥ 0} the set of positive elements of M , and by U (M ) the group of unitary operators in M .We call M a factor if Z 1 − p belongs to P or P ′ ∩ M , (1 − p)P (1 − p) is a von Neumann algebra.Define Q Tr ≤ √ 2ε, for every u ∈ U (P ).Let C ⊂ M, e Q be the weak operator closure of the convex hull of the set {ue Q u * | u ∈ U (P )}.Then C is • 2,Tr -closed and admits an element h of minimal • 2,Tr -norm which satisfies 0 ≤ h ≤ 1, h ∈ P ′ ∩ M, e Q and h − e Q 2,Tr ≤ √ 2ε by (5.1) (see [Ch79, Section 3] or [AP17, Lemma 14.3.3]).
The following lemma is a simple application of the basic construction.Lemma 2.4.Let (M, τ ) be a tracial von Neumann algebra and let P, Q ⊂ M be von Neumann subalgebras.Assume that P is finite dimensional and G ⊂ U (P ) is a finite subgroup which generates P and satisfies that 1 such that lim n→∞ Kn kn = 1 and denoting by e n : C Kn → C kn the orthogonal projection, (2.6) lim n→∞ π n (g) − e n ρ n (g)e n 2,τ kn = 0, for every g ∈ Γ.Since lim n→∞ Kn kn = 1 and e n − 1 2,τ Kn = Kn−kn Kn , for every n ∈ N, we get that (2.7) lim n→∞ e n − 1 2,τ Kn = 0. Let P n ⊂ M Kn (C) be the von Neumann algebra generated by ρ n (Γ 1 ).By using (2.7) and applying Lemma 2.5 we can find a von Neumann subalgebra Q n ⊂ e n M Kn (C)e n ≡ M kn (C) such that (2.8) lim n→∞ d τ Kn (P n , Q n ) = 0. n→∞ d τ Kn From almost containment to containment.Let P and Q be von Neumann subalgebras of a tracial von Neumann algebra.If P is close to a subalgebra of Q, then P is almost contained in Q.In this subsection, we use Lemma 3.1 to prove that the converse holds provided that Q is a factor (see Corollary 3.2) or a finite dimensional abelian algebra (see Corollary 3.3).For any ε > 0, there is δ = δ 1 (ε) > 0 such that the following holds.Let (M, τ ) be a tracial von Neumann algebra and P, Q ⊂ M be von Neumann subalgebras such that P ⊂ δ Q. Assume that Q is a factor.Then there exists a von Neumann subalgebra R ⊂ Q such that d(P, R) ≤ ε.