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LEVEL 2 OF 3 · The strong Kadison–Kastler conjecture
Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings
expertly designed by an internal OpenAI model · released 2026-10-05
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IntroductionLet \(M\) and \(N\) be von Neumann algebras acting on a complex Hilbert space \(H\), both with identity \(I_H\). Their one-sided near-inclusion gap is \[ \gamma(M,N)=\sup_{\substack{x\in M\\\|x\|\leq 1}} \inf_{y\in N}\|x-y\|. \tag{1}\] The approximating element \(y\) is not required to be contractive. A small value of \(\gamma(M,N)\) says that every contraction in \(M\) is close to some element of \(N\); it does not require these choices of approximants to preserve algebraic operations. If a unitary \(u\) satisfies \(uMu^*\subseteq N\), then \(\gamma(M,N)\leq 2\|u-I_H\|\). The converse question asks whether a small near inclusion can always be obtained in this way. The underlying perturbation program began with Kadison and Kastler (Kadison and Kastler 1972). They measured the distance between operator algebras by the Hausdorff distance between their unit balls in the operator norm, now called the Kadison–Kastler distance, and proposed that sufficiently close von Neumann algebras should be related by a unitary close to the identity. Christensen, Johnson, and Raeburn–Taylor established the amenable case in the 1970s (Christensen 1977; Johnson 1977; Raeburn and Taylor 1977). Christensen subsequently introduced near inclusions and proved a spatial embedding theorem for an amenable source algebra and an arbitrary target (Christensen 1980, Definition 2.1 and Theorem 4.3). In particular, sufficiently small near inclusions in this setting admit implementing unitaries whose distance from the identity is bounded linearly by the near-inclusion tolerance. We use the following formulation of the unrestricted one-sided question posed by Cameron, Christensen, Sinclair, Smith, White, and Wiggins in the paragraph preceding (Cameron et al. 2012, Theorem 2):
Here \(\delta\) must be independent of the algebras and their representations. We answer this question negatively. Theorem 1. There exist \(\varepsilon_0>0\), separable complex Hilbert spaces \(H_n\), and von Neumann algebras \(M_n,N_n\subseteq\mathcal B(H_n)\) with common identity \(I_{H_n}\) such that \[\gamma(M_n,N_n)\longrightarrow 0,\] but, for all sufficiently large \(n\), every unitary \(u\in\mathcal B(H_n)\) satisfying \(uM_nu^*\subseteq N_n\) obeys \[\|u-I_{H_n}\|\geq\varepsilon_0.\] In these examples a unitary embedding of \(M_n\) into \(N_n\) does exist; the obstruction concerns its distance from the identity. The theorem therefore separates normwise approximation from small spatial implementation even on separable Hilbert spaces. Its hypothesis is one-sided. The construction supplies no reverse near inclusion, and hence does not give a counterexample to the two-sided Kadison–Kastler conjecture. The positive results for nonamenable factors in (Cameron et al. 2012), which concern two-sided perturbations under additional structural hypotheses, address a different assertion. The proof transfers a classical finite-dimensional obstruction into the near-inclusion problem. Voiculescu’s clock and shift matrices \(U_n,V_n\in\mathcal B(\mathbb C^n)\) satisfy \(\|U_nV_n-V_nU_n\|\to0\), but cannot be approximated in norm by commuting unitary pairs (Voiculescu 1983). Exel and Loring gave an elementary winding-number proof of this phenomenon (Exel and Loring 1989); we use its trace-logarithm form (Exel 1993, Lemma 3.1). We recall the short obstruction argument in Section 4. To place these matrices in operator algebras, we use the free group \(F\) on countably many generators. The starting representation is the real-parameter Pytlik–Szwarc representation (Pytlik and Szwarc 1986, sec. 2.3, Equation (5)), whose cyclic coefficient is Haagerup’s positive-definite function \(g\mapsto t^{|g|}\), \(0<t<1\), where \(|g|\) is reduced word length (Haagerup 1979, Lemma 1.2). Section 2 gives a self-contained tree realization at \(t=1/2\) and adds a circle parameter \(z\). The norms of its Fourier coefficients are bounded, uniformly over the group, by a sequence with summable first moment. Along successive distinct free generators, the values of the representations converge in the weak operator topology to \(tz\) times the projection onto the identity vector in \(\ell^2(F)\). Substituting \(U_n\) and \(V_n\) for the circle parameter therefore produces two unitary representations of \(F\) whose weak limits retain the original matrices. Two further steps connect this representation family to the theorem. In Section 3, a diagonal unitary built from the two representations conjugates a copy of the group von Neumann algebra of \(F\times F\). The target \(N_n\) is the commutant of an amplified right regular representation of this product group. Reordering the Fourier expansions gives an approximant in \(N_n\) and an error bounded by a constant times \(\|U_nV_n-V_nU_n\|\). Crucially, this estimate holds on the entire unit ball of the source algebra. In Section 4, a hypothetical sequence of spatial embeddings tending to the identity places the conjugated left regular representation in \(N_n\), where it commutes with the right regular representation. Their pointwise product is then a representation of \(F\times F\), so the von Neumann algebras generated by its two factors commute. Weak operator limits in these algebras recover operators close to copies of \(U_n\) and \(V_n\) supported on projections. Perturbing those projections, compressing to a common finite-dimensional corner, and taking polar parts would yield commuting unitary matrices close to \(U_n,V_n\), contradicting the winding-number obstruction. A circle family with uniform Fourier boundsWe first construct representations in which a scalar circle parameter can later be replaced by a unitary matrix. Two properties are needed: Fourier coefficients with a summable first moment, uniformly over the group, and a weak operator limit that retains the parameter. Let \(F\) be the free group on generators \(s_j\), \(j\geq 1\), with identity \(e\). Put \(K=\ell^2(F)\), write \((\delta_x)_{x\in F}\) for its standard orthonormal basis, and let \(p_e\) be the projection onto \(\mathbb C\delta_e\). We use the left regular representation \(l(g)\delta_x=\delta_{gx}\). Throughout the construction, \[t=\frac12,\qquad r=(1-t^2)^{1/2},\qquad \mathbb T=\{z\in\mathbb C:|z|=1\}.\] Proposition 2. There are unitary representations \(\pi_z:F\to\mathcal B(K)\), indexed by \(z\in\mathbb T\), and operators \(A_k(g)\in\mathcal B(K)\), indexed by \(k\in\mathbb Z\) and \(g\in F\), such that \[\begin{align*} \pi_z(g)&=\sum_{k\in\mathbb Z}z^kA_k(g), &\sup_{g\in F}\|A_k(g)\|&\leq c_k, &c_k&=\mathbf 1_{\{k=0\}}+\frac{2t^{|k|}}{1-t}, \tag{2}\\ \pi_z(s_j)&\xrightarrow[j\to\infty]{\mathrm{WOT}}tzp_e. \tag{3}\end{align*}\] The series in (2) converges absolutely in operator norm, uniformly in \(g\) and \(z\), and \[ S_0:=\sum_{k\in\mathbb Z}c_k<\infty, \qquad S_1:=\sum_{k\in\mathbb Z}|k|c_k<\infty. \tag{4}\] Proof. The underlying representation is the real-parameter Pytlik–Szwarc representation (Pytlik and Szwarc 1986, sec. 2.3, Equation (5)); its cyclic coefficient is the positive-definite function \(g\mapsto t^{|g|}\) of Haagerup (Haagerup 1979, Lemma 1.2). We give the tree construction before introducing the circle parameter. Use the Cayley tree whose edges join \(x\) to \(xs_j^{\pm1}\), rooted at \(e\), and write \(p(x)\) for the parent of \(x\ne e\). Define \[ \xi_e=\delta_e,\qquad \xi_x=t\xi_{p(x)}+r\delta_x\quad(x\ne e). \tag{5}\] All these vectors have norm one. If \(v\) is the last common vertex of the paths from \(e\) to \(x\) and \(y\), the two tails beyond \(v\) are orthogonal to each other and to \(\xi_v\). Thus \[\langle\xi_x,\xi_y\rangle =t^{d(x,v)+d(y,v)}=t^{d(x,y)},\] where \(d\) is the tree distance. Their linear span contains every \(\delta_x\), by (5). Left multiplication preserves \(d\), so the rule \[\pi(g)\xi_x=\xi_{gx}\] extends to a unitary on \(K\). These unitaries form a representation. The difference from the regular representation is confined to a single finite path. To see this precisely, put \(T_g=\pi(g)l(g)^*\). For \(y\ne g\), (5) gives \[ T_g\delta_y= \frac{\xi_y-t\xi_{g p(g^{-1}y)}}{r}, \qquad T_g\delta_g=\xi_g. \tag{6}\] The vertex \(g p(g^{-1}y)\) is the first step from \(y\) toward \(g\). Let \(e=h_0,h_1,\ldots,h_d=g\) be the path from \(e\) to \(g\). For \(y\) outside this path, the first steps toward \(e\) and \(g\) coincide; hence \(T_g\delta_y=\delta_y\). On the path, \[ T_g\delta_{h_b}=r\xi_{h_b}-t\delta_{h_{b+1}}\quad(0\leq b<d), \qquad T_g\delta_{h_d}=\xi_{h_d}. \tag{7}\] In particular, the path subspace and its orthogonal complement reduce \(T_g\). Expanding the vectors along the path gives \[\xi_{h_b}=t^b\delta_{h_0} +r\sum_{a=1}^b t^{b-a}\delta_{h_a}.\] By (7), the subdiagonal entries are \(-t\); every other nonzero entry in row \(a\), column \(b\) is \(t^{|a-b|}\) times \(1\), \(r\), or \(r^2\). Thus the matrix entries on this subspace satisfy \[ |(T_g)_{a,b}|\leq t^{|a-b|}\qquad(0\leq a,b\leq d). \tag{8}\] This also includes \(g=e\), when \(T_e=I\). Let \(q:F\to\mathbb Z\) be the homomorphism with \(q(s_j)=1\) for every \(j\), and define \(D_z\delta_x=z^{-q(x)}\delta_x\). The desired representation is \[ \pi_z(g)=z^{q(g)}D_z\pi(g)D_z^* =(D_zT_gD_z^*)l(g). \tag{9}\] The first expression proves that it is a unitary representation. The second follows from \(D_zl(g)D_z^*=z^{-q(g)}l(g)\) and supplies its Fourier bounds. On the path subspace, the entry in row \(a\), column \(b\) acquires the Laurent factor \(z^{q(h_b)-q(h_a)}\). Each edge changes \(q\) by \(1\) or \(-1\), so an entry of exponent \(k\) can occur only when \(|a-b|\geq|k|\). For each row or column there are at most two entries at any given distance \(m\). By (8), the absolute row and column sums of the coefficient of \(z^k\) are therefore at most \[2\sum_{m\geq|k|}t^m=\frac{2t^{|k|}}{1-t}.\] Outside the path the operator is the identity, which contributes only to the coefficient at \(k=0\). The row and column sum bound for operator norm, followed by multiplication by \(l(g)\), proves (2). For each fixed \(g\) this is a finite Laurent expansion; the common bounds \(c_k\) give the asserted uniform absolute convergence and (4). Finally, (7) and (9) give \[\pi_z(s_j)\delta_e=tz\delta_e+r\delta_{s_j}.\] For any fixed \(x\ne e\), all sufficiently large \(j\) satisfy \(s_jx\notin\{e,s_j\}\), and then \(\pi_z(s_j)\delta_x=\delta_{s_jx}\). These basis-vector calculations, together with the uniform norm bound \(\|\pi_z(s_j)\|=1\), prove (3). ◻ From a unitary pair to a near inclusionWe now use Proposition 2 to encode two unitaries \(U,V\) on a nonzero finite-dimensional Hilbert space \(D\). The resulting von Neumann algebras will have one-sided gap bounded by a constant times \(\|UV-VU\|\). Matrix substitution and the algebrasOn \(L=K\otimes K\otimes D\), define \[ a(g)=\sum_k A_k(g)\otimes I_K\otimes U^k, \qquad b(g)=\sum_k I_K\otimes A_k(g)\otimes V^k \quad(g\in F). \tag{10}\] The series converge absolutely in norm. A spectral decomposition \(U=\sum_z zP_z\) rewrites the first expression as \(\sum_z\pi_z(g)\otimes I_K\otimes P_z\), so \(a\) is a unitary representation. Diagonalizing \(V\) gives the same conclusion for \(b\). There is no requirement that their ranges commute. The weak limits from Proposition 2 become \[ a(s_j)\xrightarrow{\mathrm{WOT}}t p_e\otimes I_K\otimes U, \qquad b(s_j)\xrightarrow{\mathrm{WOT}}t I_K\otimes p_e\otimes V. \tag{11}\] Only finite spectral sums are involved, so these limits follow directly from (3). Put \(G=F\times F\), \(S=\ell^2(G)\), and \(H=S\otimes L\). Our conventions for the commuting left and right regular representations on \(S\) are \[\lambda(g)\delta_x=\delta_{gx},\qquad \rho(g)\delta_x=\delta_{xg^{-1}}.\] For \(x=(x_1,x_2)\in G\), put \(f(x)=a(x_1)b(x_2)\) and define the diagonal unitary \(W\) by \[W(\delta_x\otimes v)=\delta_x\otimes f(x)v.\] Although \(a\) and \(b\) are representations, \(f\) need not be one. Define \[ M=W\bigl(\lambda(G)''\otimes I_L\bigr)W^*, \qquad N=\bigl(\rho(G)\otimes I_L\bigr)'. \tag{12}\] Primes denote commutants. Here \(\lambda(G)''\otimes I_L\) means the algebra of all \(X\otimes I_L\) with \(X\in\lambda(G)''\); it is weak operator closed, as is seen by taking matrix blocks relative to an orthonormal basis of \(L\). Thus \(M,N\) are von Neumann algebras with common identity \(I_H\). Notice also that \(W^*MW\subset N\), so these examples always admit a spatial embedding. The exactly commuting case explains the choice of \(W\). If \(U\) and \(V\) commute, then so do the ranges of \(a\) and \(b\), and \(f\) is a representation of \(G\). In that case \[W(\lambda(g)\otimes I_L)W^*=\lambda(g)\otimes f(g)\in N \quad(g\in G),\] so \(M\subset N\). When \(U\) and \(V\) only nearly commute, we must control the error on the entire unit ball of \(M\); estimates for the group elements alone would not suffice. A norm estimate on the whole algebraProposition 3. For the algebras in (12), \[ \gamma(M,N)\leq C\|UV-VU\|, \qquad C:=\sum_{k,m,p,h\in\mathbb Z} c_kc_mc_pc_h|m-p|\,|h|<\infty. \tag{13}\] Moreover, for every \(g\in G\), \[ \bigl\|W(\lambda(g)\otimes I_L)W^* -\lambda(g)\otimes f(g)\bigr\| \leq C\|UV-VU\|. \tag{14}\] The constant is independent of \(D,U,V\). Proof. Write \(\nu=\|UV-VU\|\). Fix \(X\in\lambda(G)''\) and denote its scalar matrix entries by \(X_{x,y}\), for \(x,y\in G\). Since \(X\) commutes with \(\rho(G)\), these entries are invariant under common right translation: \[ X_{xq,yq}=X_{x,y}\quad(x,y,q\in G). \tag{15}\] The operator \(Z=W(X\otimes I_L)W^*\) has \(\mathcal B(L)\)-valued blocks \[Z_{x,y}=X_{x,y}a(x_1)b(x_2)b(y_2)^*a(y_1)^*.\] We seek an approximant \(Y\) with the reordered blocks \[ Y_{x,y}=X_{x,y}a(x_1)a(y_1)^*b(x_2)b(y_2)^* =X_{x,y}f(xy^{-1}). \tag{16}\] The last equality uses the representation laws for \(a\) and \(b\) separately. Since \((xq)(yq)^{-1}=xy^{-1}\), these blocks are right-translation invariant by (15). Thus, once boundedness is established, \(Y\) commutes with \(\rho(G)\otimes I_L\) and belongs to \(N\). The Fourier bounds establish boundedness and control the reordering in operator norm. On \(S\otimes K\otimes K\), let \(D_{k,m}\) be the diagonal operator whose block at \(x\in G\) is \(A_k(x_1)\otimes A_m(x_2)\). Then \(\|D_{k,m}\|\leq c_kc_m\). Set \[B_{k,m,p,h}(X)= D_{k,m}(X\otimes I_K\otimes I_K)D_{h,p}^*.\] The \(A_k\) coefficients of \(a\) and \(b\) act on separate copies of \(K\); only their matrix factors \(U^k,V^m\) act on the same space. In the decomposition \(H=(S\otimes K\otimes K)\otimes D\), expansion of the four factors therefore gives \[\begin{align*} Z&=\sum_{k,m,p,h} B_{k,m,p,h}(X) \otimes U^kV^{m-p}U^{-h}, \tag{17}\\ Y&=\sum_{k,m,p,h} B_{k,m,p,h}(X) \otimes U^{k-h}V^{m-p}. \tag{18}\end{align*}\] Both sums converge absolutely because \[\|B_{k,m,p,h}(X)\| \leq c_kc_mc_pc_h\|X\|, \qquad \sum_{k,m,p,h}c_kc_mc_pc_h=S_0^4<\infty.\] Their blocks are precisely the expressions above, which also justifies the definition of \(Y\). For integers \(a,b\), telescoping gives \[ \|U^aV^b-V^bU^a\|\leq |a|\,|b|\nu. \tag{19}\] For positive powers, expand a commutator one factor at a time; negative powers follow from \([U^{-1},V]=-U^{-1}[U,V]U^{-1}\) and its analogue for \(V^{-1}\), where \([A,B]=AB-BA\). Consequently the two matrix factors in (17) and (18) differ by at most \(|m-p|\,|h|\nu\). Hence \[ \|Z-Y\|\leq C\nu\|X\|, \qquad C\leq 2S_0^2S_1^2<\infty. \tag{20}\] Since \(Y\in N\) and \(\|Z\|=\|X\|\), this proves (13); no bound of \(1\) on \(\|Y\|\) is needed. If \(X=\lambda(g)\), its nonzero entries have \(x=gy\). The representation laws for \(a\) and \(b\) turn (16) into \(Y=\lambda(g)\otimes a(g_1)b(g_2)=\lambda(g)\otimes f(g)\). Thus (14) is the same estimate for this choice of \(X\). ◻ Recovering the matrix obstructionThe preceding construction turns a small commutator into a small one-sided gap. We now show that a spatial embedding close to the identity would turn the original matrix pair into a nearby commuting pair. Choosing the clock and shift matrices will make this impossible. The winding obstructionThe matrices below are Voiculescu’s almost commuting unitary pair (Voiculescu 1983). We include the logarithmic winding argument, in the form developed in (Exel and Loring 1989; Exel 1993). Lemma 4. For \(n\geq3\), let \(U_n,V_n\) be the unitaries on \(\mathbb C^n\) given by \[ U_ne_j=\omega_n^je_j,\qquad V_ne_j=e_{j+1\bmod n},\qquad \omega_n=e^{2\pi i/n}\quad(0\leq j<n). \tag{21}\] Then \(\nu_n:=\|U_nV_n-V_nU_n\|=|\omega_n-1|\to0\). There is no infinite subsequence along which commuting unitaries \(\widehat U_n,\widehat V_n\) on \(\mathbb C^n\) satisfy \[\|\widehat U_n-U_n\|+\|\widehat V_n-V_n\|\longrightarrow0.\] Proof. The identity \(U_nV_n=\omega_nV_nU_n\) proves the first assertion. For any unitary matrices \(U,V\) whose commutator \(UVU^*V^*\) has no eigenvalue \(-1\), define \[\kappa(U,V)=\frac{1}{2\pi i}\mathop{\mathrm{Tr}}\log(UVU^*V^*),\] using the principal logarithm and the unnormalized trace. The determinant identity \[\exp\bigl(\mathop{\mathrm{Tr}}\log(UVU^*V^*)\bigr) =\det(UVU^*V^*)=1\] shows that \(\kappa(U,V)\) is an integer. It is continuous, and hence constant, along any path on which the logarithm remains defined. For the pair in (21), its value is \(1\), since \(U_nV_nU_n^*V_n^*=\omega_n I\) and \(0<2\pi/n<\pi\). For a commuting pair its value is \(0\). Suppose the asserted approximants existed along a subsequence. For all sufficiently large \(n\) on that subsequence, write \(U_n^*\widehat U_n=e^{iA_n}\) and \(V_n^*\widehat V_n=e^{iB_n}\), with \(A_n,B_n\) self-adjoint and spectra in \((-\pi,\pi)\). The paths \[U_n(s)=U_ne^{isA_n},\qquad V_n(s)=V_ne^{isB_n}\qquad(0\leq s\leq1)\] stay within the respective endpoint distances from \(U_n,V_n\). Indeed, for \(|\theta|<\pi\) and \(0\leq s\leq1\), \(|e^{is\theta}-1|\leq|e^{i\theta}-1|\). A four-factor telescoping estimate gives, uniformly in \(s\), \[\|U_n(s)V_n(s)U_n(s)^*V_n(s)^*-I\| \leq \nu_n+2\|\widehat U_n-U_n\| +2\|\widehat V_n-V_n\|\longrightarrow0.\] The principal logarithm is therefore defined along the whole path for all sufficiently large \(n\). Its integer changes from \(1\) to \(0\), a contradiction. ◻ A finite-dimensional cornerCompression of commuting operators to a prescribed subspace need not preserve commutation. The next lemma avoids this difficulty by replacing each support projection by a nearby projection in its corresponding algebra and then using their common range. Lemma 5. For each \(n\), let \(E_{1n},E_{2n}\) be commuting projections on a complex Hilbert space \(H_n\), with \(D_n=E_nH_n\) nonzero and finite-dimensional, where \(E_n=E_{1n}E_{2n}\). Let \(\widetilde R_{1n},\widetilde R_{2n}\) be unitaries commuting with both projections, and write \(R_{in}=\widetilde R_{in}|_{D_n}\). Suppose \(\mathcal A_{1n},\mathcal A_{2n}\subset\mathcal B(H_n)\) are commuting unital von Neumann algebras and there are \(T_{in}\in\mathcal A_{in}\) such that \[ \|T_{in}-E_{in}\widetilde R_{in}\|\longrightarrow0 \quad(i=1,2). \tag{22}\] Then, for all sufficiently large \(n\), there are commuting unitaries \(\widehat R_{1n},\widehat R_{2n}\) on \(D_n\) with \[\|\widehat R_{in}-R_{in}\|\longrightarrow0\quad(i=1,2).\] Proof. We first replace the intended projections by nearby projections in the commuting algebras. Set \(\delta_{in}=\|T_{in}-E_{in}\widetilde R_{in}\|\). Since \((E_{in}\widetilde R_{in})^*(E_{in}\widetilde R_{in})=E_{in}\), \[d_{in}:=\|T_{in}^*T_{in}-E_{in}\| \leq \delta_{in}(2+\delta_{in})\longrightarrow0.\] For sufficiently large \(n\), take the spectral projection \[Q_{in}=\mathbf1_{(1/2,\infty)}(T_{in}^*T_{in})\in\mathcal A_{in}.\] The spectrum of \(T_{in}^*T_{in}\) lies in \([0,d_{in}]\cup[1-d_{in},1+d_{in}]\). This follows from the resolvent bound for a self-adjoint operator within \(d_{in}\) of a projection. Functional calculus therefore gives \[\|Q_{in}-T_{in}^*T_{in}\|\leq d_{in}, \qquad \|Q_{in}-E_{in}\|\leq2d_{in}\longrightarrow0.\] Since the two algebras commute, \(P_n=Q_{1n}Q_{2n}\) is a projection, and \[\|P_n-E_n\| \leq\|Q_{1n}-E_{1n}\|+\|Q_{2n}-E_{2n}\|\longrightarrow0.\] We identify this perturbed corner with \(D_n\) by a small unitary. The operator \[S_n=P_nE_n+(I-P_n)(I-E_n) =I+(P_n-E_n)(2E_n-I)\] tends to the identity in norm and satisfies \(S_nE_n=P_nS_n\). It is invertible for large \(n\). Its absolute value commutes with \(E_n\), so its unitary polar part \(J_n=S_n(S_n^*S_n)^{-1/2}\) satisfies \[ J_nE_nJ_n^*=P_n,\qquad \|J_n-I\|\longrightarrow0. \tag{23}\] In particular, \(P_nH_n\) has exactly the dimension of \(D_n\). The compressions \(P_nT_{1n}P_n\) and \(P_nT_{2n}P_n\) commute, including across adjoints. This needs a check because \(Q_{in}\) need not commute with \(T_{in}\). Suppress \(n\) and put \(A=Q_1T_1Q_1\in\mathcal A_1\) and \(B=Q_2T_2Q_2\in\mathcal A_2\). The two compressions are \(AQ_2\) and \(Q_1B\). Since \(AQ_1=Q_1A=A\), \(BQ_2=Q_2B=B\), and every element of \(\mathcal A_1\) commutes with every element of \(\mathcal A_2\), both orders of their product equal \(AB\). Replacing \(A\) or \(B\) by its adjoint proves the other commutations. Transfer these compressions to \(D_n\) using (23): \[C_{in}=\left.E_nJ_n^*T_{in}J_nE_n\right|_{D_n}.\] The pair still commutes across adjoints. Moreover, \[\|C_{in}-R_{in}\| \leq\delta_{in}+2\|J_n-I\|\longrightarrow0,\] by comparing \(T_{in}\) with \(E_{in}\widetilde R_{in}\) before conjugation. Thus \(C_{in}\) is invertible for large \(n\) and \(\|C_{in}^*C_{in}-I_{D_n}\|\to0\). Its unitary polar part \[\widehat R_{in}=C_{in}(C_{in}^*C_{in})^{-1/2}\] remains close to \(R_{in}\). The unital \(C^*\)-algebras generated by \(C_{1n}\) and \(C_{2n}\) commute; continuous functional calculus puts each polar part in its respective algebra. Hence the two polar unitaries commute, as required. ◻ Proof of the main theoremApply the construction of Section 3 to \(D=\mathbb C^n\) and the clock–shift pair of Lemma 4. Use subscripts \(n\) for the resulting objects \(L_n,H_n,a_n,b_n,f_n,W_n, M_n,N_n\). All these Hilbert spaces are separable. By Proposition 3, \[ \gamma(M_n,N_n)\leq C\nu_n\longrightarrow0. \tag{24}\] Proof of Theorem 1. We show that no infinite subsequence admits unitaries \(u_n\) with \[ u_nM_nu_n^*\subset N_n,\qquad\|u_n-I\|\longrightarrow0. \tag{25}\] Suppose otherwise and work on that subsequence. Define, for \(g\in G\), \[Z_n(g)=W_n(\lambda(g)\otimes I_{L_n})W_n^*,\qquad \Sigma_n(g)=u_nZ_n(g)u_n^*(\rho(g)\otimes I_{L_n}).\] The first factor in the expression for \(\Sigma_n(g)\) lies in \(N_n\), by (25). Thus the two representations being multiplied have commuting ranges, and \(\Sigma_n\) is a unitary representation of \(G\). Proposition 3 gives the uniform estimate \[ \sup_{g\in G} \bigl\|\Sigma_n(g)-\lambda(g)\rho(g)\otimes f_n(g)\bigr\| \leq\beta_n, \qquad \beta_n=C\nu_n+2\|u_n-I\|\longrightarrow0. \tag{26}\] Let \(\mathcal A_{1n}\) and \(\mathcal A_{2n}\) be the von Neumann algebras generated by the restrictions of \(\Sigma_n\) to \(F\times\{e\}\) and \(\{e\}\times F\), respectively. These algebras commute. We will apply Lemma 5 by taking weak limits of the generators in each factor. Under \(S=K\otimes K\), write \[H_n=K\otimes K\otimes K\otimes K\otimes\mathbb C^n.\] Let \(E_{1n}\) use \(p_e\) in the first and third slots, and let \(E_{2n}\) use \(p_e\) in the second and fourth slots, with identities in all other slots. These projections commute, and \(E_n=E_{1n}E_{2n}\) has range naturally identified with \(\mathbb C^n\). Let \(\widetilde R_{1n}\) and \(\widetilde R_{2n}\) act as \(U_n,V_n\), respectively, on the final slot and as the identity elsewhere. Set \(g_{1j}=(s_j,e)\) and \(g_{2j}=(e,s_j)\). On the \(i\)-th factor of \(S\), the operator \(\lambda(g_{ij})\rho(g_{ij})\) sends \(\delta_x\) to \(\delta_{s_jxs_j^{-1}}\). For each fixed nonidentity word \(x\), this conjugate leaves every finite subset of \(F\) as \(j\to\infty\): for large \(j\), the letter \(s_j\) occurs neither in \(x\) nor in the finitely many words under consideration. Hence the conjugation operators converge weakly to \(p_e\) in that slot. Combining this observation with (11) gives, for each fixed \(n\) and \(i\), \[ \lambda(g_{ij})\rho(g_{ij})\otimes f_n(g_{ij}) \xrightarrow[j\to\infty]{\mathrm{WOT}}tE_{in}\widetilde R_{in}. \tag{27}\] Here tensor products of bounded weakly convergent sequences on separate slots converge weakly, as follows first on elementary tensors and then by density. The finite spectral decompositions used in (11) justify the matrix-valued factor. For each \(i,n\), choose a weak operator cluster point \(B_{in}\) of \(\Sigma_n(g_{ij})\) as \(j\to\infty\). Such a point exists by weak operator compactness of the closed unit ball of \(\mathcal B(H_n)\), and belongs to \(\mathcal A_{in}\). Norm balls are weak operator closed, so (26) and (27) imply \[\|B_{in}-tE_{in}\widetilde R_{in}\|\leq\beta_n.\] Thus \(T_{in}=t^{-1}B_{in}\in\mathcal A_{in}\) satisfies (22). No product of weak limits is used: commutation comes from membership in the two commuting algebras. Lemma 5 now produces commuting unitary matrices on \(\mathbb C^n\) approaching \(U_n,V_n\) in norm. This contradicts Lemma 4, proving that (25) cannot hold on any infinite subsequence. It follows that some \(\epsilon_0>0\) bounds \(\|u-I_{H_n}\|\) from below for every implementing unitary and all sufficiently large \(n\). Otherwise one could choose increasing \(n_j\) and implementing unitaries with distance less than \(1/j\), contrary to what we have just proved. Together with (24), this is Theorem 1. ◻
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