arXiv:1211.2712v2 [math.OA] 23 Mar 2013 onie(o some/all (for coincide Here and and ovxst fqatmcreainmtie ftoidpnets independent two of matrices correlation quantum of sets convex Conjecture. with hog h etefrSmer n Deformation. and Symmetry for Centre the through oiiesm-ent prtr naHletspace Hilbert a on operators semi-definite positive SRLO’ RBE N SMTTCLYCOMMUTING ASYMPTOTICALLY AND PROBLEM TSIRELSON’S atal upre yJP 2503)adb h aihNational Danish the by and (23540233) JSPS by supported Partially OM(oiieoeao audmaue with measure) valued operator (positive POVM A e od n phrases. and words Key 2010 Date Q m ,j ,l k, j, i, 6Fbur 2012. February 16 : s Q ahmtc ujc Classification. Subject Mathematics oiieoeao audmaue.W locnie h ytm o systems Kirchberg Stronger the the consider formulate also and We matrices asympto unitary finite-dimensional hen measures. commuting of valued (and terms problem operator in Tsirelson’s positive conjectures) reinterpret Connes’s and interaction, and slight a ing Abstract. upt by outputs r lsdcne ust of subsets convex closed are s Q closure = c = r nie and indices are         nti oe ecnie unu orltoso iatt system bipartite of correlations quantum consider we note, this In h ,A ξ,         ,d m, h i k ,A ξ, B srlo’ rbe,KrhegsCnetr,Cne Embedding Connes Conjecture, Kirchberg’s Problem, Tsirelson’s j l ξ i k ≥ i B  NTR MATRICES UNITARY k,l i,j A ,( 2, j l ξ i k i :  AUAAOZAWA NARUTAKA osntma the mean not does 1. ,d m, k,l i,j H ( ( Introduction : A B ibr space, Hilbert a 11;4L6 46L07. 46L06, 81P15; ) M i k j l [ ( ( ) A ) (2 6= A dim B i m md j m =1 i k =1 i k j l [ B , ) 1 A ) ( , , i m j m R =1 H , i k =1 k l j l B , ) stewl-nw srlo problem, Tsirelson well-known the is 2)) ≥ o all for 0 = ] 1 = , 1 = , 0 < uhthat such ) k l j l o all for 0 = ] + 1 = 1 = d , . . . , H d , . . . , ∞ m uhthat such d , . . . , d , . . . , , ξ k ξ upt san is outputs t oe of power -th H ∈ OM on POVMs , OM on POVMs , H ∈ ,j i, OM on POVMs , OM on POVMs , Q ntvector unit a eerhFudto (DNRF) Foundation Research ,j i, ntvector unit a and s P ial commuting tically Q ⊂ Conjecture. and A asymptotically f sesof ystems ,l k, eKirchberg’s ce i .W rt the write We 1. = m c A ,l k, H H hte they Whether . tpe( -tuple i h sets The . , , H H hav- s        , , d        A POVMs . i ) i m =1 Q of c 2 NARUTAKA OZAWA and the matricial version of it is known to be equivalent to Kirchberg’s and Connes’s conjectures. We refer the reader to [Fr, J+, Oz2, Ts] for the literature and the proof of n n the equivalence. The matricial version of Tsirelson’s problem asks whether c = s for all n, where n and n are defined as follows: Q Q Qc Qs n a Hilbert space, V : ℓ2 an isometry H k m → H n k l (Ai )i=1, k =1,...,d, POVMs on , c =  V ∗Ai BjV k,l : l m H  Q  i,j (Bj)j=1, l =1,...,d, POVMs on ,   k l H    [Ai , Bj] = 0 for all i, j and k,l   and   n dim < + , V : ℓ2 an isometry kHm ∞ → H n k l (Ai )i=1, k =1,...,d, POVMs on , s = closure  V ∗Ai BjV k,l : l m H  . Q  i,j (Bj)j=1, l =1,...,d, POVMs on ,   k l H    [Ai , Bj] = 0 for all i, j and k,l   In this note, we consider “slightly interacting” systems. Suppose Alice and Bob 1/2 m 1/2 m conduct measurements by systems of operators (Ai )i=1 and (Bj )j=1 respectively. If Bob conducts a measurement immediately after Alice’s measurement of a state ξ, then 1/2 1/2 2 the probability of the output (i, j) is Bj Ai ξ —and vice versa. Therefore, when they conduct measurements of a state ξk at the samek time, the probability of the output 1/2 1/2 1/2 1/2 (i, j) is given by ξ, (Ai Bj)ξ , where A B =(A BA + B AB )/2. Thus, for ε> 0, we define theh quantum• i correlation• matrices of slightly interacting systems to be

n dim < + , V : ℓ2 an isometry kHm ∞ → H n k l (Ai )i=1, k =1,...,d, POVMs on , ε = closure  V ∗(Ai Bj)V k,l : l m H  , Q • i,j (Bj)j=1, l =1,...,d, POVMs on ,  H    [Ak, Bl ] ε for all i, j and k,l  k i j k ≤   where [A, B] denotes the operator norm of the commutator [A, B]= AB BA. We note thatk n kis a closed convex subset of M (M (C) ). Recall that a POVM− (A )m Qε md n + i i=1 is said to be projective if all Ai’s are orthogonal projections. We also introduce the projective analogue of n: Qε n dim < + , V : ℓ2 an isometry Hk m ∞ → H n k l (Pi )i=1 projective POVMs on , ε = closure  V ∗(Pi Qj)V k,l : l m H  . P • i,j (Qj)j=1 projective POVMs on ,  H    [P k, Ql ] ε for all i, j and k,l  k i j k ≤  1  We simply write ε for ε . The following is the main result of this note. It probably suggests that Pis moreP natural than (cf. Introduction of [Fr]). Qc Qs TSIRELSON’S PROBLEM AND ASYMPTOTICALLY COMMUTING UNITARY MATRICES 3

Theorem. For every m, d, and n, one has n = n = n. In particular, Qc ε>0 Qε ε>0 Pε an affirmative answer to Tsirelson’s problem is equivalent to that ε s. T T ε>0 P ⊂ Q Hence, the matricial version of Tsirelson’s problem would have anT affirmative answer if the following assertion holds for some/all (m, d). Strong Kirchberg Conjecture (I). Let m, d 2 be such that (m, d) = (2, 2). For ≥ 6 k m every κ > 0, there is ε > 0 with the following property. If dim < + , and (Pi )i=1 and (Ql )m is a pair of d projective POVMs on such thatH [P k, Q∞l ] ε, then j j=1 H k i j k ≤ there are a finite-dimensional Hilbert space ˜ containing and projective POVMs ˜k m ˜l m ˜ ˜k ˜Hl H ˜k k (Pi )i=1 and (Qj)j=1 on such that [Pi , Qj] = 0 and Φ (Pi ) Pi κ and H ˜l l H k k k − k ≤ Φ (Qj) Qj κ, where Φ is the compression to . k H − k ≤ H H We will deal in Section 4 with a parallel and equivalent conjecture in the unitary setting.

2. Preliminary from C∗-algebra theory As it is observed in [Fr, J+, Ts], the study of quantum correlation matrices is essen- d d tially about the algebraic tensor product F F of the C∗-algebra m ⊗ m d m m Fm = ℓ ℓ , ∞ ∗···∗ ∞ m d the unital full free product of d-copies of ℓ . We note that Fm is -isomorphic to the ∞ d ∗ full group C∗-algebra C∗Γ of the group Γ = (Z/mZ)∗ . The condition m, d 2 m,d m,d ≥ and (m, d) = (2, 2) is equivalent to that Γm,d contains the free groups Fr. We denote by m 6 m k m (ei)i=1 the standard basis of minimal projections in ℓ , and by (ei )i=1 the k-th copy of d k ∞ k d d l it in the free product Fm. We also write ei for the elements ei 1 in Fm Fm and fj l d d ⊗ ⊗ for 1 ej. Thus, the maximal tensor product Fm max Fm is the universal C∗-algebra ⊗ k m l m⊗ generated by projective POVMs (ei )i=1 and (fj)j=1 under the commutation relations k l [e , f ] = 0. In passing, we note that C∗Γ C∗Γ is canonically -isomorphic to i j ⊗max ∗ C∗(Γ Γ) for any group Γ. By Stinespring’s dilation theorem (Theorem 1.5.3 in [BO]), one has× n k l d d M C M M C c = ϕ(ei fj) k,l : ϕ: Fm max Fm n( ) u.c.p. md( n( )+). Q { i,j ⊗ → } ⊂ See [Fr, J+] for the proof. Here u.c.p. stands for “unital completely positive.” We recall the notion of quasi-diagonality. We say a subset C of B( ) is quasi-diagonal if there is an increasing net (P ) of finite-rank orthogonal projectionsH on such that r H Pr 1 in the strong operator topology and [C,Pr] 0 for every C C. A C∗-algebra C isր said to be quasi-diagonal if there is a faithfulk k→-representation π∈of C on a Hilbert space such that π(C) is a quasi-diagonal subset.∗ A -representation π : C B( ) is H ∗ → H 4 NARUTAKA OZAWA said to be essential if π(C) does not contain non-zero compact operators. The following theorem of Voiculescu is the most fundamental result on quasi-diagonal C∗-algebras. See Section 7 of [BO] (Theorems 7.2.5 and 7.3.6) for the details. Theorem 1 (Voiculescu [Vo2]). The following statements hold.

Let C B( ) be a faithful essential -representation of a quasi-diagonal C∗- • algebra⊂C. Then,H C is a quasi-diagonal∗ subset of B( ). Quasi-diagonality is a homotopy invariant. H • The following is based on Brown’s idea ([Br] and Proposition 7.4.5 in [BO]). d d Theorem 2. The C∗-algebras F F and C∗F C∗F are quasi-diagonal. m ⊗max m d ⊗max d d M C M C Proof. We consider Fm as a C∗-subalgebra of M = m( ) m( ). Since the M C m ∗···∗ ˜ d conditional expectation Φ from m( ) onto ℓ extends to a u.c.p. map Φ from M to Fm which restricts to Φ on each free product component∞ ([Bo]), the canonical embedding d ˜ d Fm ֒ M is indeed faithful and Φ is a conditional expectation from M onto Fm. It → d d follows that Fm max Fm M max M. We will prove that the latter is quasi-diagonal. Let θ : M ⊗ M ⊂B( )⊗ be a faithful -representation on a separable Hilbert ⊗max → H ∗ space . We omit writing θ for a while and denote by M′′ the von Neumann algebra generatedH by θ(M C1). We write e m for the matrix units in M (C) and ⊗ { i,j}i,j=1 m ek for the k-th copy of it in M. We note that the matrix units ek is unitarily { i,j} { i,j} equivalent to the first copy e inside M′′. This is a well-known fact, but we include { i,j} the proof for the reader’s convenience. Let z M′′ be the central projection such ∈ that zM′′ is finite and (1 z)M′′ is properly infinite (Theorem V.1.19 in [Ta]). Then, − k the projections ze1,1 and ze1,1 are equivalent since they have the same center valued k trace z/n (Corollary V.2.8 in [Ta]). The projections (1 z)e1,1 and (1 z)e1,1 are also equivalent, since they are properly infinite and have full− central support− 1 z (Theorem − V.1.39 in [Ta]). Therefore, for each k, there is a partial isometry wk M′′ such that k k ∈ wk∗wk = e1,1 and wkwk∗ = e1,1. Now, Uk = i ei,1wk∗e1,i is a unitary element in M′′ such that U ek U = e for all i, j and k. Since M is a von Neumann algebra, there is a k i,j k∗ i,j P ′′ norm-continuous path Uk(t) of unitary elements connecting Uk(0) = 1 to Uk(1) = Uk. k k It follows that the -homomorphisms π : M M′′, e U (t)e U (t)∗, give rise ∗ t 7→ i,j 7→ k i,j k to a homotopy from π : M ֒ M′′ to π : M M (C) M′′. Likewise, there is a 0 → 1 → m ⊂ homotopy ρ : M θ(C1 M)′′ between the embedding ρ of M as the second tensor t → ⊗ 0 component and ρ1 which ranges in Mm(C). Thus, πt ρt : M max M B( ) is a homotopy between the embedding θ and π ρ . Therefore,× M ⊗ M is→ embeddableH 1 × 1 ⊗max into a C∗-algebra which is homotopic to M (C) M (C). Now quasi-diagonality of m ⊗ m M max M follows from Theorem 1. The case for C∗Fd is similar (Proposition 7.4.5 in [BO]).⊗  TSIRELSON’S PROBLEM AND ASYMPTOTICALLY COMMUTING UNITARY MATRICES 5

3. Proof of Theorem n n k,l n We start the proof of the inclusion ε>0 ε c . Take m, d, n and [Xi,j ] ε>0 ε N Q ⊂ Q k m ∈ l Qm arbitrary. Then, for every r , there are a pair of d POVMs (Ai (r))i=1 and (Bj(r))j=1 ∈ B T M C k l T 1 on r and a u.c.p. map ϕr : ( r) n( ) such that [Ai (r), Bj(r)] r− and H k l k,l 1H → k k ≤ ϕ (A (r) B (r)) X r− . We consider the C∗-algebras k r i • j − i,j k ≤ ∞ B B M = ( r)= (C(r))r∞=1 : C(r) ( r), sup C(r) < + , H { ∈ H r k k ∞} r=1 Y ∞ B B K = ( r)= (C(r))r∞=1 : C(r) ( r), lim C(r) =0 , H { ∈ H r k k } r=1 M k k and Q = M/K, with the quotient map π : M Q. Then Ai = π((Ai (r))r∞=1) and l l → Bj = π((Bj(r))r∞=1) are commuting POVMs in Q. Fix an ultra-limit Lim and consider the u.c.p. mapϕ ˜: M M (C) defined byϕ ˜((C(r))∞ ) = Lim ϕ (C(r)) M (C). It → n r=1 r r ∈ n factors through Q and one obtains a u.c.p. map ϕ: Q Mn(C) such thatϕ ˜ = ϕ π. k,l → k,l ◦ It follows that ϕ(AkBl )= ϕ(Ak Bl )= X , and hence [X ] n. i j i • j i,j i,j ∈ Qc For the inclusion n n, take m, d, n and [Xk,l] n arbitrary. Then, there Qc ⊂ ε>0 Pε i,j ∈ Qc d d M C k l k,l is a u.c.p. map ϕ: Fm max Fm n( ) such that ϕ(ei fj) = Xi,j . By Stinespring’s ⊗ T → d d dilation theorem, there are a -representation of Fm max Fm on a separable Hilbert ∗n ⊗ d d space and an isometry V : ℓ2 such that ϕ(C) = V ∗CV for C Fm max Fm. By inflatingH the -representation,→ we H may assume it is faithful and essential.∈ ⊗ Since d d ∗ F F is quasi-diagonal (Theorem 2), there is an increasing sequence (P )∞ m ⊗max m r r=1 of finite-rank orthogonal projections on such that Pr 1 in the strong operator topology and [C,P ] 0 for C Fd H Fd . Thus, P րekP and P f lP are close to k r k→ ∈ m ⊗max m r i r r j r projections (as r ) and one can find projective POVMs (Ek(r))m and (F l(r))m →∞ i i=1 j j=1 on P such that P ekP Ek(r) 0 and P f lP F l(r) 0. We note that rH k r i r − i k → k r j r − j k → P V V 0. It follows that [Ek(r), F l(r)] 0 and k r − k→ k i j k→ k l k l k l k,l lim V ∗(Ei (r) Fj (r))V = lim V ∗Ei (r)Fj (r)V = V ∗ei fjV = Xi,j . r r →∞ • →∞ This implies [Xk,l] n.  i,j ∈ ε>0 Pε 4. AsymptoticallyT commuting unitary matrices

Kirchberg’s conjecture ([Ki]) asserts that C∗F C∗F = C∗F C∗F for d ⊗min d d ⊗max d some/all d 2. By Choi’s theorem (Theorem 7.4.1 in [BO]), C∗F is residually finite ≥ d dimensional (RFD) and so is C∗Fd min C∗Fd. Since finite-dimensional representations factor through the minimal tensor product,⊗ Kirchberg’s conjecture is equivalent to the 6 NARUTAKA OZAWA assertion that C∗F C∗F is RFD. For the following, let u ,...u be the standard d ⊗max d 1 d unitary generators of C∗F . We also write u for the elements u 1 in C∗F C∗F and v d i i⊗ d⊗ d j for 1 uj. We denote by ( ) the set of unitary operators on . For α Md(Mn(C)), we consider⊗ U H H ∈

∗ ∗ α min = αi,j uivj Mn(C) C F minC F k k k ⊗ k ⊗ d⊗ d i,j X = sup α U V : k N, U ,V (ℓk) s.t. [U ,V ]=0 {k i,j ⊗ i jk ∈ i j ∈ U 2 i j } i,j X and

∗ ∗ α max = αi,j uivj Mn(C) C F maxC F k k k ⊗ k ⊗ d⊗ d i,j X = sup α U V : U ,V (ℓ ) s.t. [U ,V ]=0 . {k i,j ⊗ i jk i j ∈ U 2 i j } i,j X In the above expressions, one may assume U1 = 1 and V1 = 1 by replacing Ui and Vj with U1∗Ui and VjV1∗. It follows that α min = α max for d = 2. By Pisier’s linearization trick, Kirchberg’s conjecture is equivalentk k to thek k assertion that α = α holds k kmin k kmax for every d 3 (or just d = 3) and every α Md(Mn(C)). See Section 12 of [Pi], Chapter 13≥ in [BO], and [Oz1] for the proof of∈ this fact and more information. The proof of the following lemma is omitted because it is almost the same as that of the main theorem. Lemma 3. For every α M (M (C)), one has ∈ d n N k α max = inf sup αi,j UiVj : k , Ui,Vj (ℓ2) s.t. [Ui,Vj] ε . k k ε>0 {k ⊗ k ∈ ∈ U k k ≤ } i,j X We observe the following fact. Suppose dim < and U, V ( ) are such that [U, V ] ε. It is well-known that the pair (U,H V )∞ need not be∈ close U H to a commuting kpair ofk unitary ≤ matrices ([Vo1]), but after a dilation it is. Indeed, this follows from amenability of Z2. Let m = 1/√ε and F = 0,...,m 2 Z2. We define an isometry Z2 ⌊ 1⌋/2 { } ⊂ p q W : ℓ2 by Wξ = F − x F δx ϕ(x)ξ, where ϕ((p, q)) = U V ( ) H→ ⊗ H | | ∈ ⊗ ∈2 U H for (p, q) F . Then, for the commuting unitary operators u and v, acting on ℓ2Z by shifting∈ indices in Z2 by ( 1, 0) andP (0, 1) respectively, one has ⊗ H − − 1 W ∗uW U = ϕ(x)∗ϕ(x + (1, 0)) U k − k k F − k x F (( 1,0)+F ) | | ∈ ∩ X− mε +1/(m + 1) < 2√ε. ≤ TSIRELSON’S PROBLEM AND ASYMPTOTICALLY COMMUTING UNITARY MATRICES 7

2 Similarly, one has W ∗vW V < 2√ε. Since C∗Z is abelian (and RFD), one can find k − k a finite dimensional Hilbert space ˜ containing and commuting unitary matrices U˜ and V˜ on ˜ such that Φ (U˜)H U < 2√εHand Φ (V˜ ) V < 2√ε, where H k H − k k H − k Φ : B( ˜) B( ) is the compression. We note that Φ (U˜) U and Φ (V˜ ) V for H H → H H ≈ H ≈ any unitary elements imply Φ (U˜V˜ ) UV (see, e.g., Theorem 18 in [Oz2]). Keeping these facts in mind, we formulateH the≈ Strong Kirchberg Conjecture (II). Strong Kirchberg Conjecture (II). Let d 2. For every κ> 0, there is ε> 0 with the following property. If dim < + and U≥,...,U ,V ...,V ( ) are such that H ∞ 1 d 1 d ∈ U H [U ,V ] ε, then there are a finite-dimensional Hilbert space ˜ containing and k i j k ≤ H H U˜i, V˜j ( ˜) such that [U˜i, V˜j] = 0 and Φ (U˜i) Ui κ and Φ (V˜j) Vj κ. ∈ U H k k k H − k ≤ k H − k ≤ We note that the analogous statement for U1, U2,V is true, by the proof of the following theorem plus the fact that C∗(F2 Z) is RFD and has the LLP (local lifting property). See Chapter 13 in [BO] for the definition× of the LLP and relevant results. Theorem 4. The following conjectures are equivalent. (1) The Strong Kirchberg Conjecture (I) holds for some/all (m, d). (2) The Strong Kirchberg Conjecture (II) holds for some/all d. (3) Kirchberg’s conjecture holds and C∗(F F ) has the LLP for some/all d 2. d × d ≥ (4) The algebraic tensor product C∗F C∗F B(ℓ ) has unique C∗-norm. d ⊗ d ⊗ 2 We note that it is not known whether C∗(Fd Fd) has the LLP, but it is independent of d 2 and equivalent to that the LLP is closed× under the maximal tensor product. Also ≥ it is equivalent to the LLP for C∗(Γm,d Γm,d). This problem seems to be independent of Kirchberg’s conjecture. We will only× prove the equivalence (2) (3), because the proof of (1) (3) is very similar and (3) (4) is an immediate⇔ consequence of the tensor product⇔ characterization of the LLP⇔ (see [Ki] and Chapter 13 in [BO]). Lemma 5. The following conjectures are equivalent: (1) For every κ> 0, there is ε> 0 with the following property. If dim < + and U ,...,U ,V ...,V ( ) are such that [U ,V ] ε, then thereH are∞ a (not 1 d 1 d ∈ U H k i j k ≤ necessarily finite-dimensional) Hilbert space ˜ containing and U˜ , V˜ ( ˜) H H i j ∈ U H such that [U˜i, V˜j] =0 and Φ (U˜i) Ui κ and Φ (V˜j) Vj κ. k k k H − k ≤ k H − k ≤ (2) The C∗-algebra C∗(F F ) has the LLP. d × d Proof. (1) (2) : To prove the LLP of C∗-algebra C∗(F F ), it suffices to show ⇒ d × d that the surjective -homomorphism π from C∗(F ) = C∗(w ,...,w ,w′ ,...,w′ ) onto ∗ 2d 1 d 1 d C∗(F F ), w u and w′ v , is locally liftable. By the Effros–Haagerup theorem d × d i 7→ i j 7→ j (Theorem C.4 in [BO]), this follows once it is shown that the canonical surjection

Θ: B(ℓ ) C∗(F )/B(ℓ ) ker π B(ℓ ) C∗(F F ) 2 ⊗min 2d 2 ⊗min → 2 ⊗min d × d 8 NARUTAKA OZAWA is isometric. Let u =1= v and E = span u , v : 0 i, j d be the operator 0 0 { i j ≤ ≤ } subspace of C∗(Fd Fd). By Pisier’s linearization trick, it is enough to check that Θ is (completely) isometric× on B(ℓ ) E. For this, take α M (B(ℓ )) arbitrary and let 2 ⊗ ∈ d+1 2 ∗ λ = αi,j uivj B(ℓ2) minC (F2 )/B(ℓ2) minker π. k ⊗ k ⊗ d ⊗ F Let (en)n∞=1 be a quasi-centralX approximate unit for ker π in C∗( 2d), and let wi(n) = 1/2 1/2 (1 e ) w (1 e ) + e and w′ (n) likewise (although the proof will equally work − n i − n n j for wj′ (n)= wj′ ). Then, one has

∗ αi,j wi(n)wj′ (n) B(ℓ2) minC (F2d) λ, k ⊗ k ⊗ ≥ 2 lim [wi(n),wXj′ (n)] = lim (1 en) [wi,wj′ ] = π([wi,wj′ ]) =0, n k k n k − k k k F and limn [wi∗(n),wj′ (n)] = 0. Since C∗( 2d) is RFD, one can find a finite-dimensional -representationk σ suchk that ∗ n 1 ∗ αi,j σn(wi(n)wj′ (n)) B(ℓ2) minσn(C (F2d)) λ . k ⊗ k ⊗ ≥ − n For every contractiveX matrices x and y, we consider the unitary matrices defined by x √1 xx∗ y √1 yy∗ − ∗ − ∗ √1 x∗x x y √1 yy − − ∗ ∗ − Ux = x √1 xx∗ and Vy = √1 y y y . − ∗ − ∗ − ∗ " √1 x∗x x # " √1 y y y # − − − − We observe that the (1, 1)-entry of U V is xy, and if [x, y] 0 and [x∗,y] 0, then x y k k ≈ k k ≈ [U ,V ] 0. Thus, applying the assumption (1) to U and V ′ , one may k x y k ≈ σn(wi(n)) σn(wj (n)) find unitary operators U˜i(n), V˜j(n) and the compression Φn such that [U˜i(n), V˜j(n)] = 0, Φ (U˜ (n)) U 0, and Φ (V˜ (n)) V ′ 0. It follows that k n i − σn(wi(n))k→ k n j − σn(wj (n))k→ ∗ ˜ ˜ αi,j uivj B(ℓ2) minC (Fd Fd) lim sup αi,j Ui(n)Vj(n) k ⊗ k ⊗ × ≥ n k ⊗ k X →∞ X lim sup αi,j Φn(U˜i(n)V˜j(n)) ≥ n k ⊗ k →∞ X ′ = lim sup αi,j Uσn(wi(n))Vσn(wj (n)) n k ⊗ k →∞ X lim sup αi,j σn(wi(n)wj′ (n)) ≥ n k ⊗ k →∞ λ. X ≥ This proves that Θ is isometric on B(ℓ2) E, and the assertion (2) follows. (2) (1) : Suppose that the assertion⊗ (1) does not hold for some κ> 0. Thus, there are unitary⇒ operators U (n) and V (n) on with [U (n),V (n)] 0 which witness i j Hn k i j k → a violation of the conclusion of (1). We consider the C∗-algebras M = B( ) and Hn Q TSIRELSON’S PROBLEM AND ASYMPTOTICALLY COMMUTING UNITARY MATRICES 9

Q = B( )/ B( ), with the quotient map π : M Q. Then, U = π((U (n))∞ ) Hn Hn → i i n=1 and Vj = π((Vj(n))∞ ) are commuting systems of unitary elements in Q, and the map Q L n=1 u U , v V extends to a -homomorphism on C∗(F F ). By the assumption (2), i 7→ i j 7→ j ∗ d × d one may find a u.c.p. map ϕ: C∗(F F ) M such that π(ϕ(u )) = U and π(ϕ(v )) = d × d → i i j Vj. We expand ϕ as (ϕn)n∞=1 and see Ui(n) ϕn(ui) 0 and Vj(n) ϕn(vj) 0. Take N such that U (N) ϕ (u )

The analogue of Lemma 5 also holds in the projective setting, and it can be proven using the following dilation lemma. Lemma 6. Let m N be fixed and (A (n))m and (B (n))m be sequences of POVMs on ∈ i i=1 j j=1 n such that limn [Ai(n), Bj(n)] =0. Then, there are sequences of projective POVMs H(P (n))m and (Q k(n))m on ℓm+1k ℓm+1 such that lim [P (n), Q (n)] =0 and i i=1 j j=1 2 ⊗ 2 ⊗ Hn n k i j k Φn(Pi(n)) = Ai(n), Φn(Qj(n)) = Bj(n), and Φn(Pi(n)Qj(n)) = Ai(n)Bj(n). Here Φn denotes the compression to Cδ Cδ = . 1 ⊗ 1 ⊗ Hn ∼ Hn 1/2 1/2 Proof. Let X(n) = [A1(n) Am(n) ] M1,m(B( n)), and consider the unitary element ··· ∈ H X(n) 0 U(n)= Mm+1(B( n)). 1 X(n) X(n) X(n)∗ ∈ H  − ∗ −  We denote by Ei(n) the orthogonalp projection in Mm+1(B( n)) onto the i-th coordinate, H and define Pi′(n) = U(n)Ei(n)U(n)∗ for i = 1,...,m 1 and Pm′ (n) = U(n)(Em(n)+ m − m+1 Em+1(n))U(n)∗. Then, (Pi′(n))i=1 is a projective POVM on ℓ2 n whose (1, 1)- m ⊗ H m entry is (Ai(n))i=1. Similarly, one obtains a projective POVM (Qj′ (n))j=1. Define σ : B(ℓm+1 ) B(ℓm+1 ℓm+1 ) by C D C 1 D if p = 1, and p,3 2 ⊗ Hn → 2 ⊗ 2 ⊗ Hn ⊗ 7→ ⊗ ⊗ C D 1 C D if p = 2; and let P (n)= σ (P ′(n)) and Q (n)= σ (Q′ (n)). Since ⊗ 7→ ⊗ ⊗ i 1,3 i j 2,3 j lim [A (n), B (n)] = 0, the entries of P ′(n) asymptotically commute with those of n k i j k i Qj′ (n). It follows that limn [Pi(n), Qj(n)] = 0. They also satisfy the other conditions. k k 

We are now ready for the proof of Theorem 4.

Proof. (2) (3) : Assume the assertion (2). Then, Lemma 3 implies that α max = α for every⇒ α M (M (C)) and hence Kirchberg’s conjecture follows.k Lemmak 5 k kmin ∈ d+1 n implies that C∗(F F ) has the LLP. d × d 10 NARUTAKA OZAWA

(3) (2) : Assume the assertion (3). Then, by Lemma 5, one has the Strong ⇒ Kirchberg Conjecture (II) for a possibly infinite-dimensional ˜. Since Kirchberg’s con- F F F F H ˜ jecture is assumed and C∗( d d) ∼= C∗ d min C∗ d is RFD, one can reduce to a finite-dimensional Hilbert space,× up to a perturbation.⊗ See Theorem 1.7.8 in [BO].H  Final Remarks and Acknowledgment. The main theorem equally holds for three or more commuting systems. Although it is stated as “the Strong Kirchberg Conjecture,” the author thinks that both Kirchberg’s and the LLP conjectures for C∗(Fd Fd) would have negative answers. This research came out from the author’s lectures for× “Master- class on sofic groups and applications to operator algebras” (University of Copenhagen, 5–9 November 2012). The author gratefully acknowledges the kind hospitality provided by University of Copenhagen during his stay in Fall 2012. References [Bo] F. Boca; Free products of completely positive maps and spectral sets. J. Funct. Anal. 97 (1991), 251–263. [Br] N.P. Brown; On quasidiagonal C∗-algebras. Operator algebras and applications, 19–64, Adv. Stud. Pure Math., 38, Math. Soc. Japan, Tokyo, 2004. [BO] N.P. Brown and N. Ozawa; C∗-algebras and Finite-Dimensional Approximations. Graduate Stud- ies in Mathematics, 88. American Mathematical Society, Providence, RI, 2008. [Fr] T. Fritz; Tsirelson’s problem and Kirchberg’s conjecture. Rev. Math. Phys. 24 (2012), 1250012, 67 pp. [J+] M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V.B. Scholz, and R.F. Werner; Connes embedding problem and Tsirelson’s problem. J. Math. Phys. 52 (2011), 012102, 12 pp. [Ki] E. Kirchberg; On nonsemisplit extensions, tensor products and exactness of group C∗-algebras. Invent. Math. 112 (1993), 449–489. [Oz1] N. Ozawa; About the QWEP conjecture. Internat. J. Math. 15 (2004), 501–530. [Oz2] N. Ozawa; About the Connes embedding conjecture—Algebraic approaches. Jpn. J. Math. 8 (2013), 147–183. [Pi] G. Pisier; Grothendieck’s theorem, past and present. Bull. Amer. Math. Soc. (N.S.) 49 (2012), 237–323. [Ta] M. Takesaki; Theory of operator algebras. I. Encyclopaedia of Mathematical Sciences, 124. Op- erator Algebras and Non-commutative Geometry, 5. Springer-Verlag, Berlin, 2002. [Ts] B.S. Tsirelson; Some results and problems on quantum Bell-type inequalities. Fundamental ques- tions in quantum physics and relativity, 32–48, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1993. [Vo1] D. Voiculescu; Asymptotically commuting finite rank unitary operators without commuting ap- proximants. Acta Sci. Math. (Szeged) 45 (1983), 429–431. [Vo2] D.V. Voiculescu; A note on quasi-diagonal C∗-algebras and homotopy. Duke Math. J. 62 (1991), 267–271.

RIMS, Kyoto University, 606-8502 Kyoto, Japan E-mail address: [email protected]