Proof. Since \(A'\) is weak-star closed, its preannihilator in the trace class computes the quotient norm \(\operatorname{dist}(h,A')\). Let \(\varphi\) be a trace-class functional of norm at most one vanishing on \(A'\). A singular-value decomposition gives vectors \(\xi,\eta\in\ell^2\otimes H\), each of norm at most one, with \[\varphi(x)=\langle (I\otimes x)\xi,\eta\rangle\quad(x\in\mathcal B(H)).\] This uses only countably many vectors even when \(H\) is nonseparable. Let \(p\) project onto \(\overline{(I\otimes A')\xi}\). This subspace reduces \(I\otimes A'\), so every matrix entry of \(p\) belongs to \(A\). Moreover \(p\xi=\xi\) and \(p\eta=0\). Therefore \[\varphi(h)=\langle [I\otimes h,p]\xi,\eta\rangle.\] If \(q_n\) projects onto the first \(n\) coordinates of \(\ell^2\), the compression of this commutator is \([I_n\otimes h,q_npq_n]\), of norm at most \(\lVert\operatorname{ad}_h|_A\rVert_{\mathrm{cb}}\) because \(q_npq_n\in M_n(A)_1\). Testing on finitely supported vectors and then using density proves the same bound for the full commutator. Taking the supremum over \(\varphi\) proves (3).

For \(h\in A'_1\), approximate each contraction in \(M_n(B)\) by one in \(M_n(A)\), with arbitrary slack above \(\gamma\). It follows that \(\lVert\operatorname{ad}_h|_B\rVert_{\mathrm{cb}}\le2\gamma\). The first assertion gives distance at most \(2\gamma\) to \(B'\), and normalization of approximants gives distance at most \(4\gamma\) to \(B'_1\). Reverse the algebras to finish. ◻

Proof. If \(e=0\), take \(u=I\). Otherwise let \[\pi(x)=\begin{pmatrix}\rho(x)&0\\0&\theta(x)\end{pmatrix}, \qquad S=\begin{pmatrix}0&I\\I&0\end{pmatrix}.\] The von Neumann image \(\pi(P)\) has the quotient matrix norms of \(P\). The commutator of \(S\) with \(\pi\) has cb norm at most \(e\), hence the same is true on \(\pi(P)\) by quotient lifts with arbitrarily small norm slack. By 3 choose \(D\in\pi(P)'\) with \(\lVert D-S\rVert<2e\). Its lower off-diagonal block \(T\) satisfies \[T\rho(x)=\theta(x)T,\qquad \lVert T-I\rVert<2e<1.\] Thus \(T\) is invertible and \(T^*T\) commutes with \(\rho(P)\). Its polar unitary \(u=T|T|^{-1}\) intertwines the representations. The singular values of \(T\) lie between \(1-r\) and \(1+r\), where \(r=\lVert T-I\rVert\), so \[\lVert u-I\rVert\le\lVert u-T\rVert+r=\lVert I-|T|\rVert+r\le2r<4e.\] ◻

Proof. Complete isometry of \(\rho\) gives \(\lVert F_n(X)\rVert\ge(1-e)\lVert X\rVert\) at every matrix size. Thus \(F\) is injective, its range \(V\) is norm closed, and \(\lVert F^{-1}:V\to P\rVert_{\mathrm{cb}}\le(1-e)^{-1}\). For \(y\in Q_1\), match \(y\) with \(\rho(x)\) for \(x\in P_1\), allowing arbitrary slack, to obtain \(\operatorname{dist}(y,V)\le g+e<1\). If \(V\) were proper, the quotient map \(Q\to Q/V\) would have norm one, contradicting this bound on \(Q_1\). Hence \(F\) is onto.

Transport multiplication from \(Q\) to \(P\): \[m(x,y)=F^{-1}(F(x)F(y)).\] It is associative and preserves the involution. For rectangular \(X,Y\), insert \(\rho(XY)=\rho(X)\rho(Y)\) to see that \[\lVert F(XY)-F(X)F(Y)\rVert\le3e\lVert X\rVert\lVert Y\rVert.\] Here the first difference is bounded by \(e\lVert XY\rVert\) and the product difference by \(2e\lVert X\rVert\lVert Y\rVert\), since \(F\) and \(\rho\) are complete contractions. Equation (4) therefore gives \[q:=\lVert m-m_P\rVert_{\mathrm{cb}}\le\frac{3e}{1-e}<\frac1{11}.\] The cited theorem gives an involution-preserving \(L\) with \(\lVert L-\operatorname{id}\rVert_{\mathrm{cb}}\le\lambda:=30e/(1-e)<1\). Then \(\theta=FL^{-1}\) is a bijective \(*\)-homomorphism, and \[\lVert\theta-\rho\rVert_{\mathrm{cb}} \le e+\lVert L^{-1}-\operatorname{id}\rVert_{\mathrm{cb}} \le e+\frac{\lambda}{1-\lambda} = e+\frac{30e}{1-31e}\le32e.\] Bijective multiplicativity makes \(\theta\) unital. A \(*\)-isomorphism between von Neumann algebras is an order isomorphism on their selfadjoint parts, so it preserves bounded increasing suprema and is normal. Apply 4, since \(32e<1/4\). When \(e=0\), the preceding range argument gives \(Q=\rho(P)\) directly. ◻

Proof. Compress a contraction of \(M_n(A\mathbin{\overline\otimes}\mathcal B(K))\) by increasing finite-rank projections on \(K\). In a corner of rank \(r\), choose an opposite contraction in \(M_{nr}(B)\) within \(\gamma\) plus arbitrarily small slack, and extend it by zero on the complement. The compressions of the original contraction converge strongly to it. A bounded WOT cluster of the opposite contractions is in \(M_n(B\mathbin{\overline\otimes}\mathcal B(K))_1\) and is within \(\gamma\), by WOT closedness of the norm bound, choosing the approximation slack to tend to zero along the compressions. Reverse the argument and take the supremum over \(n\).

Now start with a contraction in \(M_n(A\mathbin{\overline\otimes}D)\). The first assertion supplies a nearby contraction in \(M_n(B\mathbin{\overline\otimes}\mathcal B(K))\). Average it by conjugation over the unitary group of each finite-dimensional stage of \(D'\). The input commutes with these groups, so the bound is preserved. A WOT cluster of the averages commutes with \(D'\) and still lies in \(M_n(B\mathbin{\overline\otimes}\mathcal B(K))\). The von Neumann tensor commutation theorem (Blackadar 2006, Theorem III.4.5.8 and Corollary III.4.5.9, pp. 319–320) places it in \(M_n(B\mathbin{\overline\otimes}D)\). Reverse and amplify as before. ◻

Proof. The coefficient formula defines a bounded sesquilinear form and hence an operator. Positivity at every matrix size and unitality follow by applying the positive scalar mean to matrix quadratic forms; the domain has the pointwise operations on bounded operator families. Fixed left and right factors pass through the coefficient formula. Testing the linear equations defining a weak-operator closed subspace proves range preservation. Testing vector coefficients at each rectangular size gives the norm bound. For the action average, left invariance makes the range invariant, while averaging fixes each invariant operator. An invariant algebra is preserved by the preceding range property. Predual duality gives the same argument for the general action on \(P\), including preservation of matrix norm bounds. ◻

Proof. There is nothing to prove if \(\delta=0\). Otherwise we use the triangular homomorphism associated with a derivation, as in (Haagerup 1983, 216): \[\Phi(a)= \begin{pmatrix}a&\delta^{-1}[h,a]\\0&a\end{pmatrix} \quad(a\in A).\] This is a unital homomorphism with \(\lVert\Phi\rVert\le2\). For \(\xi\in H\), the closed subspace \[K_\xi=\overline{\{\Phi(a)(0,\xi):a\in A\}} \subseteq H\oplus H\] is invariant for \(\Phi(A)\), and the restricted homomorphism is cyclic. Haagerup’s cyclic similarity theorem gives \(\lVert\Phi|_{K_\xi}\rVert_{\mathrm{cb}}\le\lVert\Phi|_{K_\xi}\rVert^3\le8\); see (Haagerup 1983, Theorem 1.1 and its proof, pp. 217, 225). Apply its column amplification to \((0,\xi)\in K_\xi\), and retain the upper component of each output. This gives \[\left(\sum_i\lVert[h,a_i]\xi\rVert^2\right)^{1/2} \le8\delta\,\lVert(a_i)_i\rVert_{\mathrm{col}}\lVert\xi\rVert.\] Taking the supremum over unit vectors proves the claim. ◻

Proof. The functionals \(a\mapsto\tau(ba)\), \(b\in P\), are norm dense in the predual \(P_*\). Indeed, their annihilator in \((P_*)^*=P\) is zero, by faithfulness of the trace, so Hahn–Banach gives the density. For a normal vector functional \(\varphi\), approximate \(\varphi\) in norm by one of these functionals. If \(\lVert x_j\rVert\le K_0\), then \[|\varphi(x_j^*x_j)| \le K_0^2\lVert\varphi-\tau(b\,\cdot)\rVert +\lVert b\rVert\lVert x_j\rVert_2^2.\] This proves (i). For (ii), the defining identity of the conditional expectation gives \[\lim_{j\to\omega}\tau(bx_j) =\tau_\omega(bx)=\tau(bE_P(x))\qquad(b\in P).\] The same predual-density argument, using boundedness, extends the identity to every normal functional. ◻

Proof. We estimate the centered words first, then restore their scalar parts. The decomposition below uses the creation, annihilation, and diagonal actions in the free-product Hilbert space, as in the framework developed by Voiculescu (Voiculescu 1985) and the reduced-free-product inequalities of Ricard and Xu (Ricard and Xu 2006, sec. 1 and Fact 2.6). The corresponding bound for words of fixed length in free groups is Haagerup’s inequality (Haagerup 1979, Lemma 1.4). Here the prescribed label multiplicities require the additional tuple-matrix and square-summation estimates proved below.

Proof. Fix \(r\) and a label \(a\). There are \(\binom{d-1}{r-1}\) distinct-label multisets containing \(a\), and their contribution is \[49(r-1)!d^{-(r-1)}\binom{d-1}{r-1}\leq49.\] For \(r=1\) this is the entire sum. Suppose \(r\geq2\). A repeated-label multiset has support of cardinality \(t\leq r-1\). If it contains \(a\), its other \(t-1\) support labels can be chosen in \(\binom{d-1}{t-1}\) ways. Its positive multiplicities, with total \(r\), can be chosen in \(\binom{r-1}{t-1}\) ways. Consequently the number of such multisets is at most \[\sum_{t=1}^{r-1}\binom{d-1}{t-1}\binom{r-1}{t-1} \leq A_r d^{r-2}, \qquad A_r=\sum_{u=0}^{r-2}\frac1{u!}\binom{r-1}{u}.\] Their contribution to (14) is therefore at most \(C_r^2 A_r/d\). Choose \(d_0(N)\) to be an integer at least all the \(d_r\) for \(1\leq r\leq N\) and all the numbers \(C_r^2A_r/15\) for \(2\leq r\leq N\). The latter condition is empty when \(N=1\). The sum is then at most \(49+15=64\), uniformly in \(a\) and simultaneously for the finitely many lengths \(r\leq N\). ◻

Proof. We construct the free unitary in two stages, then control its representatives.

Proof. Put \(v=sU\). It suffices first to show \[\tau(v^k)=0\quad(k\neq0),\qquad \tau(v^{k_1}x_1\cdots v^{k_r}x_r)=0\] for \(r\geq1\), nonzero integers \(k_i\), and \(x_i\in P\cap\ker\tau\). These cyclic mixed words suffice: if an alternating word starts and ends in the same algebra, cyclically bring those two letters together, split their product into its scalar and centered parts, and induct on word length. Regard an expanded trace word as a cyclic sequence of letters \(s\) and \(s^*\) separated by elements of \(P\). The expansion of \((sU)^k\) uses \((sU)^k\) itself for \(k>0\) and \((U^*s^*)^{-k}\) for \(k<0\). At a transition from an \(s\)-letter to an \(s^*\)-letter the separator is \(Ux_iU^*\); at a transition from \(s^*\) to \(s\) it is \(x_i\). Thus every separator between opposite signs is centered, including the cyclic junction.

Split every other separator into its scalar and centered parts. In each resulting term, removal of a scalar separator merges only powers of \(s\) with the same sign. Consequently every surviving power has nonzero exponent, and every surviving separator is a centered element of \(P\). If any separator remains, freeness of \(s\) from \(P\) gives trace zero. If no separator remains, all signs were the same and the term is a scalar multiple of a nonzero power of \(s\), again with trace zero. The argument for \(\tau(v^k)\) is identical, with all signs the same from the outset. This proves the displayed identities.

Centered Laurent polynomials in \(v\) are \(2\)-norm dense, with bounded approximants, in the centered part of \(W^*(v)\): use the Fejér sums of the corresponding bounded functions on the circle. The displayed identities therefore imply scalar freeness of \(W^*(v)\) and \(P\). Here bounded \(2\)-norm approximation can be passed through each trace word by telescoping and the inequality \(\lvert\tau(a y b)\rvert\leq\lVert a\rVert\lVert y\rVert_2\lVert b\rVert\). Finally apply the proved assertion to \(s^*\) and \(U^*\) and take adjoints to obtain the assertion for \(Us\). ◻

Proof. Choose \(t\) with \(d(M,N)<t<1/10\). If \(v\in N\) were a proper isometry, choose \(x\in M_1\) with \(\lVert x-v\rVert<t\). Then \(x^*x\ge(1-t)^2I\), so the polar part \(w\) of \(x\) is an isometry in \(M\). Finiteness makes \(w\) unitary. Since \(1-t\le |x|\le1\), \(\lVert w-v\rVert\le\lVert w-x\rVert+\lVert x-v\rVert<2t<1\). This is impossible: a proper isometry within distance less than one of a unitary would be invertible. Thus \(N\) is finite.

Suppose that \(q\in Z(N)\) is a projection. Choose \(a\in M_1\) with \(\lVert a-q\rVert<t\). For \(v\in\mathcal U(M)\) choose \(b\in N_1\) with \(\lVert v-b\rVert<t\). Since \(q\) commutes with \(b\), \[\lVert vav^*-q\rVert\le\lVert a-q\rVert+\lVert[v,q]\rVert<3t.\] The trace-Hilbert closed convex hull of the conjugates \(vav^*\) contains \(\tau(a)I\). Indeed, its unique vector of least norm is conjugacy invariant, is bounded by one by weak compactness of the operator ball, and is therefore scalar; its trace is \(\tau(a)\). Consequently \(\lVert q-\tau(a)I\rVert\le3t<1/2\). A nontrivial projection has distance \(1/2\) from the scalars, so \(q\) is zero or one. The spectral projections of every central selfadjoint element are therefore trivial, proving \(Z(N)=\mathbb CI\). ◻

Proof. We first make a single selfadjoint element nearly scalar under a matrix average. Let \(P\) be a \(\mathrm{II}_1\) factor, \(y=y^*\in P\), and \(\varepsilon>0\). Choose a spectral step approximation \(y_0=\sum_{i=1}^l\lambda_i p_i\), where \(\sum_i p_i=I\). For a sufficiently large integer \(b\), split \(\lfloor b\tau(p_i)\rfloor\) orthogonal projections of trace \(1/b\) from each \(p_i\). The remaining projection has trace \[\frac{b-\sum_i\lfloor b\tau(p_i)\rfloor}{b}<\frac lb,\] and also splits into projections of trace \(1/b\). The resulting \(b\) equivalent projections extend to matrix units for a unital \(F\cong M_b(\mathbb C)\) in \(P\). Let \(y_1\in F\) be diagonal, equal to \(\lambda_i\) on the pieces taken from \(p_i\) and zero on the remainder. Then \[\lVert y-y_1\rVert_2\le\lVert y-y_0\rVert_2+\lVert y_0\rVert\sqrt{l/b}.\] Haar averaging over \(\mathcal U(F)\) sends \(y_1\) to \(\tau(y_1)I\). Contractivity of this average in \(L^2\) gives \[ \lVert E_{F'\cap P}(y)-\tau(y)I\rVert_2 \le2\lVert y-y_1\rVert_2<\varepsilon, \tag{42}\] by first choosing \(y_0\) and then \(b\ge2\).

Choose an \(L^2\)-dense sequence \((x_j)\) in the selfadjoint unit ball of \(M\), and enumerate the requirements \((j,k)\in\mathbb N^2\). Start with \(Q_0=\mathbb CI\). Given a full matrix algebra \(Q_n\), its relative commutant \(P_n=Q_n'\cap M\) is a \(\mathrm{II}_1\) factor: matrix units identify \(M\) with a matrix algebra over one of its corners and \(P_n\) with that corner. For the next requirement \((j,k)\) apply (42) in \(P_n\) to \(E_{P_n}(x_j)\), with error \(1/k\). If \(F\subseteq P_n\) is the resulting matrix algebra, set \(Q_{n+1}=Q_n\vee F\). These commuting matrix algebras generate a full matrix algebra of at least twice the size of \(Q_n\), and \[P_{n+1}=F'\cap P_n,\qquad E_{P_{n+1}}=E_{P_{n+1}}E_{P_n}.\] Hence \(\lVert E_{P_{n+1}}(x_j)-\tau(x_j)I\rVert_2<1/k\), and all earlier bounds persist under subsequent expectations. For \(R=(\bigcup_nQ_n)''\) the expectation onto \(R'\cap M=\bigcap_nP_n\) thus sends every \(x_j\) to \(\tau(x_j)I\). Density gives \(R'\cap M=\mathbb CI\). In particular \(R\) is a factor; it is finite and has unital matrix subalgebras of unbounded size, so it is of type \(\mathrm{II}_1\). Its increasing matrix presentation makes it hyperfinite. ◻

Proof. The commutant of \((M\cup\{e_R\})''\) is \(J_MRJ_M\). Indeed, an element of \(M'\) is right multiplication by an element of \(M\). If it commutes with \(e_R\), applying it to \(\Omega\) shows that its right multiplier lies in \(L^2(R)\cap M=R\), where the equality follows from the trace expectation. Conversely right multiplication by \(R\) reduces \(L^2(R)\). The same argument applies to \(N\). Equality of the basic constructions therefore gives \(J_MRJ_M=J_NRJ_N\). For each \(r\in R\) the two displayed operators take \(\Omega\) to \(r^*\Omega\). The vector \(\Omega\) is separating for the common right algebra, since it is cyclic for \(M\), so the operators are equal. Applying these identities twice proves the last assertion. ◻

Proof. Denote the supremum by \(K\). Start with bounded \(a,b\in M\) with \(\lVert a\rVert_2,\lVert b\rVert_2\le1\), without an operator-norm bound. By (43), finite convex averages of conjugations by unitaries of \(R\) can bring both \(aa^*\) and \(bb^*\) within one of their scalar traces in \(L^2\). One first takes a sufficiently high Haar average and then approximates its integral by a finite sum; rational weights and repetitions give equal weights. Choose such a family \((d_i)\). Independently choose a family \((e_l)\) so that the averages of \(e_l^*a^*ae_l\) and \(e_l^*b^*be_l\) have the same property. Enumerate the Cartesian product of the two families by \(k=1,\ldots,m\) and put \(a_k=d_i a e_l\), \(b_k=d_i b e_l\). Their covariance averages satisfy \[L_a=\frac1m\sum_k a_ka_k^*,\quad Q_a=\frac1m\sum_k a_k^*a_k,\qquad \lVert L_a\rVert_2,\lVert Q_a\rVert_2\le2,\] and the same bounds hold for \(L_b,Q_b\).

Use the same independent fair signs in the two sums \[A=m^{-1/2}\sum_k\epsilon_k a_k,\qquad B=m^{-1/2}\sum_k\epsilon_k b_k.\] Bimodularity and sign orthogonality give \[ \mathbb E\langle VA\Omega,B\Omega\rangle=\langle Va\Omega,b\Omega\rangle, \qquad \mathbb E\lVert A\rVert_2^2,\ \mathbb E\lVert B\rVert_2^2\le1. \tag{45}\] Put \(\lVert x\rVert_4^4=\tau((x^*x)^2)\). The three pairings in the fourth moment give exactly \[\mathbb E\lVert A\rVert_4^4 =\tau(Q_a^2)+\tau(L_a^2) +\frac1{m^2}\sum_{k,l}\tau((a_k^*a_l)^2) -\frac2{m^2}\sum_k\tau((a_k^*a_k)^2).\] Tracial Cauchy–Schwarz gives \(\lvert\tau((a_k^*a_l)^2)\rvert\le\tau(a_ka_k^*a_la_l^*)\). Dropping the last nonpositive term therefore yields \[\mathbb E\lVert A\rVert_4^4\le\lVert Q_a\rVert_2^2+2\lVert L_a\rVert_2^2\le12, \qquad \mathbb E\lVert B\rVert_4^4\le12.\] If \(A=w|A|\), set \(A_s=w|A|1_{[0,s]}(|A|)\) and define \(B_s\) similarly. These operators have norm at most \(s\), their \(L^2\) norms do not increase, and \[\mathbb E\lVert A-A_s\rVert_2^2\le12/s^2, \qquad \mathbb E\lVert B-B_s\rVert_2^2\le12/s^2.\] Substituting them in (45) and applying Cauchy–Schwarz in probability gives \[\lvert\langle Va\Omega,b\Omega\rangle\rvert \le s^2K+\frac{2\sqrt{12}}s\lVert V\rVert.\] Take \(s=4\sqrt{12}\), and then take the supremum over the dense bounded \(L^2\) unit vectors \(a,b\). We obtain \(\lVert V\rVert\le192K+\lVert V\rVert/2\), proving the assertion. ◻

Proof. Set \(T=J_NJ_M-I\). The common right-action identity implies that \(T\) commutes with both actions of \(R\). For \(x\in M_1\) choose \(y\in N_1\) with \(\lVert x-y\rVert<b\). Then \[\lVert Tx\Omega\rVert =\lVert J_Nx^*\Omega-x\Omega\rVert \le\lVert(x^*-y^*)\Omega\rVert+\lVert(x-y)\Omega\rVert<2b.\] Thus the supremum in 25 is at most \(2b\). That lemma gives \(\lVert T\rVert\le768b\), and \(\lVert T\rVert=\lVert J_M-J_N\rVert\). ◻

Proof. Write the map as \(\theta:\mathcal A\to N\), where \(N\) is the target factor. Preservation of the Jordan product gives preservation of \(xyx\), since \(xyx=2x\circ(x\circ y)-x^2\circ y\), where \(x\circ y=(xy+yx)/2\). Polarizing in \(x\) gives preservation of \(xyz+zyx\). Fix \(a,b\), and set \[A=\theta(a),\quad B=\theta(b),\quad D=\theta(ab),\quad E=\theta(ba)=AB+BA-D, \qquad h=D-AB,\quad k=D-BA.\] Compute \(\theta(ab\,c\,ba+ba\,c\,ab)\) in two ways. The polarized triple identity gives \(DCE+ECD\), where \(C=\theta(c)\); applying the unpolarized identity first to the inner and then to the outer products gives \(ABCBA+BACAB\). Expanding their equality yields \[hCk+kCh=0\qquad(C\in N),\] where surjectivity allows every \(C\) in the target. Substitute \(C=C_1kC_2\), and use the same identity with \(C_1\) and \(C_2\) to move \(h\) past the adjacent factors. The two resulting terms are equal, so \[hC_1kC_2k=0\qquad(C_1,C_2\in N).\] A factor is prime: \(xNy=\{0\}\) implies \(x=0\) or \(y=0\), since the central supports of the support projections of nonzero \(x,y\) are both one. If \(k\ne0\), primeness first gives \(hC_1k=0\) for every \(C_1\), and then \(h=0\). Thus, for each pair \((a,b)\), either \(\theta(ab)=\theta(a)\theta(b)\) or \(\theta(ab)=\theta(b)\theta(a)\).

For fixed \(a\), the sets of \(b\) satisfying these two respective identities are additive subgroups whose union is the whole additive group. A group cannot be the union of two proper subgroups: choose an element in each but not the other, and their sum belongs to neither. One identity therefore holds for all \(b\). The same two-subgroup argument, now for those \(a\) for which each identity holds for every \(b\), makes one orientation hold globally. ◻

Proof. Choose a countable strongly dense subgroup \(G\subseteq\mathcal U(M)\). For each \(g\in G\) choose a unitary \(v_g\in N\) with \(\lVert v_g-g\rVert\le3b\): a contraction approximant to \(g\) is invertible for \(b<1\), and its polar part has this bound. Thus \[\sup_{g,h\in G}\lVert v_gv_h-v_{gh}\rVert\le9b.\] Apply 16 to these constant fields on the boundary space \((D,\nu)\) of 15. For sufficiently small absolute \(b\) it gives measurable unitary fields \(W_g:D\to\mathcal U(N)\) satisfying \[ W_g(x)W_h(g^{-1}x)=W_{gh}(x),\qquad \sup_g\lVert W_g-g\rVert_{L^\infty(D)}\le C_1b, \tag{46}\] where \(C_1\) is absolute.

To obtain this correspondence on unitary vectors, we will compare the cocycle with the original left and right actions, and then compare its two boundary values within \(N\). The first comparison uses the separate action on \(D^2\). Form its product cocycle and the constant representation \[\Sigma_{g,h}(x,y)=W_g(x)J_NW_h(y)J_N, \qquad U_{g,h}=gJ_MhJ_M.\] The left and right \(N\) actions commute, so \(\Sigma\) is an exact cocycle for the separate \(G^2\) action. By 26, \[\sup_{g,h}\lVert\Sigma_{g,h}-U_{g,h}\rVert_\infty \le(2C_1+2C_J)b.\] The product version of 17 gives a measurable unitary field \(L:D^2\to\mathcal U(K)\), with \(\lVert L-I\rVert_\infty\le C_2b\), such that \[ L(x,y)U_{g,h}=\Sigma_{g,h}(x,y)L(g^{-1}x,h^{-1}y). \tag{47}\] The constants \(C_1,C_2\), and all subsequent \(C_i\) in this proof, are absolute.

The field \(L\) takes values in \(\mathcal U(K)\); its intertwining identity alone does not supply trace vectors of unitaries in \(N\). For this we use the diagonal comparison. Equation (46) gives \[\sup_g\mathop{\rm ess\,sup}_{x,y\in D} \lVert W_g(x)W_g(y)^*-I\rVert\le2C_1b.\] For sufficiently small \(b\), 18 therefore gives a measurable \(a:D^2\to\mathcal U(N)\) with \(\lVert a-I\rVert_\infty\le C_3b\) and \[ a(x,y)=W_g(x)a(g^{-1}x,g^{-1}y)W_g(y)^*. \tag{48}\] We can now combine the two comparisons: the vector field \(f(x,y)=L(x,y)^*a(x,y)\Omega\) is equivariant for the diagonal representation \(g\mapsto U_{g,g}\). Indeed right multiplication by \(W_g(y)^*\) on trace vectors is \(J_NW_g(y)J_N\), so (48) and (47) give \(f(x,y)=U_{g,g}f(g^{-1}x,g^{-1}y)\). By 19, \(f\) is a constant invariant vector. The invariant vectors for \(g\mapsto gJ_MgJ_M\) are exactly \(\mathbb C\Omega\). To see this, strong density extends invariance from \(G\) to \(\mathcal U(M)\); averaging over \(\mathcal U(Q_j)\) acts on \(x\Omega\) as \(E_j(x)\Omega\) and converges to the projection onto \(\mathbb C\Omega\) by (43). Hence \[ L(x,y)^*a(x,y)\Omega=\lambda\Omega, \qquad |\lambda|=1, \qquad |\lambda-1|\le(C_2+C_3)b. \tag{49}\]

Discard one null set and all its countably many \(G^2\) translates, so that the preceding identities hold at a chosen pair \((x,y)\) and at all its translates. Define the Hilbert-space unitary \[S_0=\lambda a(x,y)^*L(x,y).\] It fixes \(\Omega\). Using (47) for \((g,1)\) and (49) at \((g^{-1}x,y)\) gives \[ S_0g\Omega =a(x,y)^*W_g(x)a(g^{-1}x,y)\Omega =b_g\Omega, \tag{50}\] where \(b_g\in\mathcal U(N)\) and \(\lVert b_g-g\rVert\le C_4b\) uniformly in \(g\).

Every \(u\in\mathcal U(M)\) is the strong limit of a sequence \(g_j\in G\). The vectors \(b_{g_j}\Omega=S_0g_j\Omega\) are Cauchy in trace \(L^2(N)\). For unitaries, traciality gives \(\lVert b_{g_j}^*-b_{g_l}^*\rVert_2=\lVert b_{g_j}-b_{g_l}\rVert_2\). Uniformly bounded \(L^2\) convergence in the standard representation implies strong convergence: test first on the dense right-multiplier vectors \(N'\Omega\) and then use boundedness. Thus \(b_{g_j}\) and their adjoints converge strongly, and their limits are a unitary \(b_u\in N\) and its adjoint. We obtain \[S_0u\Omega=b_u\Omega,\qquad \lVert b_u-u\rVert\le C_4b,\] where the norm inequality follows by weak closedness of the appropriate operator ball.

Every element of \(M\) is a linear combination of unitaries, so the rule \[ \vartheta(z)\Omega=S_0z\Omega\qquad(z\in M) \tag{51}\] defines a linear map into \(N\). It is well-defined because \(\Omega\) is separating for \(N\). Writing a contraction as its real and imaginary selfadjoint parts, each an average of two unitaries, gives \[ \vartheta(I)=I,\qquad \lVert\vartheta(z)-z\rVert\le 2C_4b\lVert z\rVert\quad(z\in M). \tag{52}\] In particular \(\vartheta\) is bounded, and it sends every unitary to a unitary. For \(z=z^*\) expand \(\vartheta(e^{itz})^*\vartheta(e^{itz})=I\) in operator norm. The coefficient of \(t\) says \(\vartheta(z)^*=\vartheta(z)\). Applying this also to \(z^2\) and comparing the coefficient of \(t^2\) gives \(\vartheta(z^2)=\vartheta(z)^2\). Complex linearity and polarization show that \(\vartheta\) preserves adjoints and the Jordan product.

When \(2C_4b<1\), (52) makes its range closed and makes \(\vartheta\) injective. Every contraction of \(N\) has distance at most \((1+2C_4)b\) from this range, by the original gap and (52). For \((1+2C_4)b<1\) the range must be all of \(N\): otherwise the quotient map by this proper closed linear subspace has norm one, contrary to that bound.

27 shows that this surjective Jordan map is either multiplicative or antimultiplicative. The latter possibility is excluded by (52). Indeed, for unitaries \(z,w\in M\) antimultiplicativity would give \[\lVert zw-wz\rVert\le\lVert zw-\vartheta(zw)\rVert +\lVert\vartheta(w)\vartheta(z)-wz\rVert \le6C_4b.\] A unital copy of \(M_2(\mathbb C)\) in \(M\) contains anticommuting selfadjoint unitaries, whose commutator has norm two. Taking \(6C_4b<2\) excludes the anti case. Thus \(\vartheta\) is a unital star-isomorphism, and it is normal because a bijective order isomorphism of von Neumann algebras preserves all bounded increasing suprema. This proves the proposition, with \(C_{\mathrm{st}}=2C_4\) and one absolute sufficiently small \(b_0\). ◻

Proof of 20. Choose \(t> d(M,N)\) below the absolute thresholds specified below. By 21, \(N\) is a finite factor. If either factor is a matrix algebra, 2 for the mutual near inclusions gives the required near-identity conjugacy, with an absolute linear bound. We may therefore suppose both factors are of type \(\mathrm{II}_1\).

Choose \(R\subseteq M\) from 22. By 2, there is a unitary \(w\) with \(\lVert w-I\rVert\le150t\) and \(wRw^*\subseteq N\). Replacing \(N\) by \(N_0=w^*Nw\) makes \(R\) common, and \[d(M,N_0)<301t.\] Take \(g=302t\) and apply 23, provided \(g<1.74\cdot10^{-13}\). Write \(M_s=\pi(M)\), \(N_s=\rho(N_0)\). Their common subfactor remains irreducible in \(M_s\) by faithfulness. By 24 its right actions agree, and \(d(M_s,N_s)\le2\beta\), with \(\beta<50948\sqrt g\). Choose a strict standard-gap bound \(b=3\cdot50948\sqrt g\). If \(b<b_0\), 28 supplies a normal star-isomorphism \(\vartheta_s:M_s\to N_s\) with \(\lVert\vartheta_s-\operatorname{id}\rVert\le C_{\mathrm{st}}b\) on \(M_s\).

Transport it to the original representation: \[\theta=\rho^{-1}\vartheta_s\pi:M\longrightarrow N_0.\] For \(x\in M_1\) choose \(y\in(N_0)_1\) with \(\lVert x-y\rVert<g\). The original-contraction comparison in 23 and faithfulness of \(\rho\) give \[\begin{align*} \lVert\theta(x)-x\rVert &\le\lVert\theta(x)-y\rVert+\lVert y-x\rVert\\ &\le\lVert\vartheta_s\pi(x)-\pi(x)\rVert +\lVert\pi(x)-\rho(y)\rVert+g\\ &\le C_{\mathrm{st}}b+\beta+2g. \end{align*}\] Let \(e(t)=C_{\mathrm{st}}b+50948\sqrt{302t}+604t\). This is an absolute function tending to zero.

On the original \(H\oplus H\), consider the faithful normal paired representation \(x\mapsto x\oplus\theta(x)\) of \(M\), and the selfadjoint switch \(s(\xi,\eta)=(\eta,\xi)\). Its scalar commutator norm on that factor is at most \(e(t)\). 8 therefore gives \[\lVert\operatorname{ad}_s|_{\{x\oplus\theta(x):x\in M\}}\rVert_{\mathrm{cb}} \le C_{\mathrm{amp}}e(t).\] At each matrix level the switch commutator has off-diagonal block \([x_{ij}-\theta(x_{ij})]\), so equivalently \(\lVert\theta-\operatorname{id}\rVert_{\mathrm{cb}}\le C_{\mathrm{amp}}e(t)\). For sufficiently small absolute \(t\), 4 gives a unitary \(v\in\mathcal B(H)\) with \(vMv^*=N_0\) and \(\lVert v-I\rVert\le C_{\mathrm{pair}}C_{\mathrm{amp}}e(t)\), for an absolute constant \(C_{\mathrm{pair}}\). Then \(u=wv\) conjugates \(M\) onto \(N\), and \[\lVert u-I\rVert\le150t+C_{\mathrm{pair}}C_{\mathrm{amp}}e(t).\] All thresholds used above are absolute. Besides \(t<1/10\), they are the bounds in [thm:amenable,lem:paired] and the two bounds in [prop:finite-common-position,prop:finite-standard-isomorphism]. Choose \(t_{\mathrm{fin}}\) so that they hold for every \(0<t<t_{\mathrm{fin}}\), and take \(\eta_{\mathrm{fin}}\) to be the maximum of the displayed bound and the matrix-factor bound. These bounds are nondecreasing functions of \(t\) and tend to zero. This establishes the claimed uniformity on the original Hilbert space. ◻

Proof. For each \(n\geq1\) choose isometries \(s_1,\ldots,s_n\) in that common copy such that \(s_i^*s_j=\delta_{ij}1\) and \(\sum_i s_is_i^*=1\). The operator \(V:H^n\to H\) given by \(V(\xi_i)=\sum_i s_i\xi_i\) is unitary. For either \(P=M\) or \(P=N\), \[V M_n(P)V^*=P, \qquad V[x_{ij}]V^*=\sum_{i,j}s_ix_{ij}s_j^*.\] Indeed the inverse coefficient formula is \(x_{ij}=s_i^*xs_j\). The same unitary therefore identifies the two amplified unit balls with the two original unit balls. Taking the supremum over \(n\) proves the assertion. ◻

Proof. Swapping two sites of the same type is implemented by a unitary in a finite site algebra. This unitary preserves the product state, so it belongs to \(D_\rho\). Suppose \(x\in D_\rho'\cap D\). Given a finite site element \(a\), choose bounded finite site approximants \(x_j\) with \(\lVert(x-x_j)\Omega_\rho\rVert\to0\). A product of the indicated swaps moves \(a\) to an element \(a_j\) supported away from \(x_j\). Since \(x\) commutes with the implementing unitary and the state is invariant, \[\rho(ax)=\rho(a_jx),\qquad \rho(a_jx_j)=\rho(a)\rho(x_j).\] State Cauchy–Schwarz bounds the error in the first product by \(\rho(aa^*)^{1/2}\lVert(x-x_j)\Omega_\rho\rVert\); the other error tends to zero as well. Hence \(\rho(ax)=\rho(a)\rho(x)\) for every finite site \(a\). Density gives \((x-\rho(x)1)\Omega_\rho=0\), and faithfulness gives \(x=\rho(x)1\). In particular \(D\) is a factor.

Choose a faithful normal state \(\psi\) on \(A\) and put \(P=A\mathbin{\overline\otimes}D\) with state \(\psi\otimes\rho\). In the regular representation of its modular crossed product on \(L^2(\mathbb R,H_P)\), the generators are \[(\pi(x)\xi)(t)=\sigma^{\psi\otimes\rho}_{-t}(x)\xi(t), \qquad (\lambda_s\xi)(t)=\xi(t-s).\] Here \(H_P\) can be the standard space of \(P\). Let \(Z\) be central in the crossed product. After Fourier transform in \(t\), commutation with every \(\lambda_s\) makes \(Z\) a decomposable operator. The crossed product also commutes with the constant representation of \(P'\). Consequently its decomposable field takes values in \(P\). Commutation with the constant operators \(1\otimes D_\rho\) puts this field in \(A\otimes1\): normal slices and \(D_\rho'\cap D=\mathbb C1\) prove this relative-commutant identity.

An off-diagonal matrix unit at a \(\nu\)-site is a modular eigenoperator with frequency \(\log\nu\) or \(-\log\nu\). Its regular representation, after Fourier transform, is that matrix unit followed by a frequency translation. If \(a(\zeta)\otimes1\) denotes the field of \(Z\), commutation therefore gives \[a(\zeta+\log\lambda)=a(\zeta),\qquad a(\zeta+\log\mu)=a(\zeta) \quad\text{almost everywhere}.\] The generated subgroup is countable and dense in \(\mathbb R\). Test the field on countably many vector coefficients. Translation continuity in local \(L^1\) extends its invariance to every real translation, and convolution with compactly supported approximate identities shows that each coefficient is constant. Thus \(Z=a_0\otimes1\) is a constant field. Commutation with \(\pi(A\otimes1)\), using strong continuity of the modular action and a countable strongly dense set, gives \(a_0\in A'\cap A=\mathbb C1\). The core is a factor, so the continuous-core criterion makes \(P\) a type \(\mathrm{III}_1\) factor.

Finally, adjoining another countable family of \(\lambda\)-sites and reindexing that family gives the asserted absorption. All these tensor products preserve the displayed product states. The usual identification of the two-ratio product with the hyperfinite \(\mathrm{III}_1\) factor is also recorded in (Haagerup 1987, Example 1.6, p. 103); the argument above establishes the additional assertion for every \(A\). ◻

Proof. A countably decomposable infinite factor is properly infinite and contains a unital copy of \(\mathcal B(\ell^2)\): split its identity into countably many equivalent projections and choose corresponding matrix units (Blackadar 2006, III.1.3.5 and III.1.5.1, pp. 243, 247). 2 embeds this amenable subfactor of \(M_0\) into \(N_0\) by a unitary of norm distance at most \(150t\) from 1. Conjugating \(N_0\) in the inverse direction gives \(u_0\) for which the subfactor is common. This also proves that \(N_0\) is infinite. The adjusted ordinary gap is at most \(301t\). 29 makes this also a cb gap. 6 transfers it to the tensor pair because \(D'\) is hyperfinite. By 30, both tensor factors are of type \(\mathrm{III}_1\) and absorb \(R_\lambda\).

For the moment write \(N_1=(u_0N_0u_0^*)\mathbin{\overline\otimes}D\). The bicentralizer and expected-subfactor results just stated give an irreducible hyperfinite \(\mathrm{II}_1\) factor \(R\subseteq M\) and a faithful normal state \(\varphi\) on \(M\) centralizing \(R\). The given representation of \(M\) has a cyclic separating vector realizing \(\varphi\). Here is the representation point needed for this assertion. A normal representation on a separable space splits as a countable sum of cyclic normal representations. Each cyclic summand embeds in the standard representation by the standard-form vector implementing its vector functional (Haagerup 1975, Lemma 2.10(1), p. 278). A countable amplification of the standard representation is unitarily equivalent to it, since its type \(\mathrm{III}\) commutant contains a countable family of isometries with ranges summing to 1. Finally, every nonzero projection in that countably decomposable type \(\mathrm{III}\) commutant is equivalent to 1 (Blackadar 2006, Theorem III.1.7.9). Thus every nonzero normal representation here is equivalent to the standard one. Transport the standard vector for \(\varphi\) to obtain the desired \(\Omega\) on the actual space \(H\otimes K\). No conjugation of either algebra is made in this choice.

Put \(S=J_MRJ_M\subseteq M'\). Centralization gives \[ r\Omega=J_Mr^*J_M\Omega\qquad(r\in R). \tag{53}\] First apply 2 to \(R\subseteq M\) and its near inclusion in \(N_1\). It gives a unitary \(v_1\) with \(v_1Rv_1^*\subseteq N_1\) and \(\lVert v_1-1\rVert=O(t)\). Put \(N_2=v_1^*N_1v_1\), so \(R\subseteq M\cap N_2\). Cb gaps change by at most twice the norm distance of the adjusting unitary from 1. Thus 3 gives \(d(M',N_2')=O(t)\). Apply 2 to \(S\subseteq M'\) and its near inclusion in \(N_2'\). It gives a unitary \(v_2\) with \[v_2Sv_2^*\subseteq N_2',\qquad \lVert v_2-1\rVert=O(t),\qquad v_2\in(S\cup N_2')''\subseteq R'.\] Set \(N=v_2^*N_2v_2\). Since \(v_2\) commutes with \(R\), this algebra still contains \(R\), and now \(S\subseteq N'\). The algebra \(M\), its state \(\varphi\), and the vector \(\Omega\) have remained fixed. The unitary in the statement is \(v=v_2^*v_1^*\), with \(\lVert v-1\rVert=O(t)\), and the resulting cb gap is \(O(t)\).

We next check irreducibility and the vector, rather than assume they survive an arbitrary small perturbation. Choose increasing matrix subfactors \(R_j\) strongly generating \(R\). Haar averaging over \(\mathcal U(R_j)\) converges weakly on \(M\) to \(x\mapsto\varphi(x)1\). Indeed every bounded cluster point commutes with all \(R_j\), hence is scalar, and the averages preserve \(\varphi\). If a nontrivial projection \(p\in R'\cap N\) existed, a close approximant \(x\in M\) would therefore give \[\operatorname{dist}(p,\mathbb C1)\leq\lVert p-x\rVert=O(t).\] The left side is \(1/2\). Taking \(t\) sufficiently small proves \(R'\cap N=\mathbb C1\). The same argument in the commutants, with \(S\) and the vector state on \(M'\), proves \(S'\cap N'=\mathbb C1\).

Let \(p\) be the projection onto \(\overline{N'\Omega}\). It belongs to \(N\). By (53), this subspace reduces \(R\): for \(r\in R\) and \(x'\in N'\), the vector \(rx'\Omega=x'r\Omega\) lies in \(\overline{N'\Omega}\), and the same holds for \(r^*\). Thus \(p\in R'\cap N\), so \(p=1\). Similarly the projection onto \(\overline{N\Omega}\) belongs to \(S'\cap N'\) and is 1. Hence \(\Omega\) is cyclic and separating for \(N\). The common commuting right action in (53) shows that its state on \(N\) is invariant under conjugation by \(\mathcal U(R)\), and hence centralizes \(R\). The two operators \(J_MrJ_M,J_NrJ_N\in N'\) now agree on \(\Omega\); since \(\Omega\) is separating for \(N'\), they agree everywhere. All conditions in 31 are satisfied. ◻

Proof. For \(s\in R\) put \(R_s=J_Ms^*J_M=J_Ns^*J_N\). On either graph, the operation \(g_c(x)\mapsto g_c(rxs)\) for \(r,s\in\mathcal U(R)\) is the restriction of the common unitary \[(rR_s)\oplus\overline{s^*R_{r^*}} \quad\text{on }H\oplus\overline H.\] Both graph subspaces reduce these unitaries. Thus \(T=(1-E_c^N)E_c^M\) commutes with them.

Proof. The upper-left block of the projection onto a closed operator graph is \((1+A^*A)^{-1}\). Applied to (54), this gives \((1+c^2\Delta_P)^{-1}\). With \(g(s)=(1+e^s)^{-1}\), 33 consequently yields \[\lVert g(L_M-a)-g(L_N-a)\rVert\leq C\gamma\qquad(a\in\mathbb R).\] Let \(p=1_{(-\infty,a-1/2]}(L_N)\) and \(q=1_{[a+1/2,\infty)}(L_M)\), and regard \(pq\) as an operator \(qH\to pH\). Let \(A\) and \(B\) be the restrictions of \(g(L_N-a)\) and \(g(L_M-a)\) to those respective subspaces. Then \[A\geq g(-1/2)1,\qquad B\leq g(1/2)1, \qquad\lVert A(pq)-(pq)B\rVert\leq C\gamma.\] For \(D=A(pq)-(pq)B\), integration of the derivative of \(e^{-tA}(pq)e^{tB}\) gives \[pq=\int_0^\infty e^{-tA}D e^{tB}\,dt, \qquad \lVert pq\rVert\leq\frac{\lVert D\rVert}{g(-1/2)-g(1/2)}.\] The denominator is a fixed positive number. Reversing the two algebras proves the other estimate. ◻

Proof. Put \(d=\lVert\operatorname{ad}_h|_P\rVert\). Choose a unital copy \(S\) of \(\mathcal B(\ell^2)\) in \(P\), with matrix units \((e_{ij})\) whose diagonal sum is \(I\). The finite-dimensional unital algebras \[S_n=\operatorname{span}\{e_{ij}:1\le i,j\le n\} +\mathbb C\Bigl(I-\sum_{i=1}^ne_{ii}\Bigr)\] increase and generate \(S\). Average \(h\) over \(\mathcal U(S_n)\) and take a weak-operator cluster point \(h_0\). Each average is self-adjoint, within \(d\) of \(h\), and has scalar commutator bound at most \(d\) on \(P\). These properties pass to the cluster point. Haar invariance at every earlier stage gives \(h_0\in S'\).

For any \(k\), choose isometries \(s_1,\ldots,s_k\in S\) with \(s_i^*s_j=\delta_{ij}I\) and \(\sum_i s_is_i^*=I\). The unitary \(V:H^k\to H\), \(V(\xi_i)=\sum_i s_i\xi_i\), identifies \(M_k(P)\) with \(P\) and \(h_0^{(k)}\) with \(h_0\). Consequently \[\lVert\operatorname{ad}_{h_0}|_P\rVert_{\mathrm{cb}}\le d.\] By 3, \(\operatorname{dist}(h_0,P')\le d\), so in fact \(\operatorname{dist}(h,P')\le2d\). The stated larger constant will be convenient. ◻

Proof. Suppose there is no common \(K\). By scaling a counterexample and subtracting a self-adjoint sum from \(P+P'\), we can choose factors \(P_j\subset\mathcal B(H_j)\) and self-adjoint \(h_j\) with \[\operatorname{dist}(h_j,P_j+P_j')=1, \qquad \lVert h_j\rVert\le2, \qquad \varepsilon_j:=\sup_{x\in(P_j)_1}\operatorname{dist}([h_j,x],P_j)\longrightarrow0.\] Taking real parts justifies using a self-adjoint approximating sum. Its subtraction does not change either distance in the assertion, because its commutator with \(P_j\) belongs to \(P_j\).

Fix a free ultrafilter \(\omega\) on \(\mathbb N\) and form the norm ultraproducts \[A=\prod_\omega P_j\subset \prod_\omega\mathcal B(H_j)=E.\] Here a norm ultraproduct is the algebra of bounded sequences modulo the sequences whose operator norms tend to zero along \(\omega\). The algebra \(A\) is simple. Indeed, a nonzero positive element \(a\) has uniformly bounded positive representatives \(a_j\) and a number \(t>0\) such that \(e_j=1_{[t,\infty)}(a_j)\) is nonzero on a set in \(\omega\). In a separably acting type \(\mathrm{III}\) factor each nonzero projection is equivalent to \(I\). Choose isometries \(v_j\) with \(v_jv_j^*=e_j\) on that set. Then \(v_j^*a_jv_j\ge tI\) there. The class \(v^*av\) is invertible in \(A\) and belongs to the ideal generated by \(a\). Thus every nonzero closed ideal of \(A\) contains \(I\).

Commutation with \([h_j]\in E\) defines a bounded derivation \(D:A\to A\). To see that its range is in \(A\), choose, for every bounded sequence \(x_j\in P_j\), elements \(y_j\in P_j\) with \[\lVert[h_j,x_j]-y_j\rVert \le (\varepsilon_j+j^{-1})\sup_i\lVert x_i\rVert.\] The \(y_j\) are uniformly bounded, and their class is the commutator in \(E\). Its definition is therefore independent of the choices; linearity and the derivation identity follow in \(E\).

Sakai’s theorem for simple unital \(C^*\)-algebras gives \(a\in A\) with \(D(x)=[a,x]\) (Sakai 1971, sec. 1, p. 259; Section 2, p. 260). No separability of the norm ultraproduct is required. Since the \(h_j\) are self-adjoint, replacing \(a\) by \((a+a^*)/2\) leaves this identity unchanged. Choose uniformly bounded self-adjoint representatives \(a_j\in P_j\). The equality of the two derivations implies \[\lim_\omega\lVert\operatorname{ad}_{h_j-a_j}|_{P_j}\rVert=0.\] Otherwise contractions witnessing a fixed positive commutator norm could be selected at each coordinate of an ultrafilter-large set, contradicting equality on their class in \(A\). 36 now gives \[1=\operatorname{dist}(h_j,P_j+P_j') \le\operatorname{dist}(h_j-a_j,P_j') \le4\lVert\operatorname{ad}_{h_j-a_j}|_{P_j}\rVert\longrightarrow_\omega0,\] a contradiction. ◻

Proof. Write \(b(t)=\operatorname{dist}(v(t),\mathcal G_P)\). This is a continuous function, since distance to any nonempty subset is \(1\)-Lipschitz. Suppose \(b=b(t)\) is smaller than a fixed absolute radius, and choose \(p\in\mathcal G_P\) with \(\lVert p^*v(t)-I\rVert<b+\eta\). For sufficiently small \(b+\eta\), principal functional calculus writes \(p^*v(t)=e^{ih}\) with \(h=h^*\) and \(\lVert h\rVert\le2(b+\eta)\). As \(p\) normalizes \(P\), this conjugation has the same one-sided approximation bound \(e\). The twice-integrated conjugation formula gives \[\lVert e^{ih}xe^{-ih}-x-i[h,x]\rVert\le2\lVert h\rVert^2\lVert x\rVert.\] It follows that \[\sup_{x\in P_1}\operatorname{dist}([h,x],P)\le e+2\lVert h\rVert^2.\] Apply 37 and approximate \(h\) by a self-adjoint sum \(a+b'\) with \(a\in P\), \(b'\in P'\). Since \(a\) and \(b'\) commute, \(e^{i(a+b')}=e^{ia}e^{ib'}\) belongs to \(\mathcal G_P\). The inequality \(\lVert e^{ih}-e^{ik}\rVert\le\lVert h-k\rVert\) for self-adjoint \(h,k\) therefore shows, on letting both approximation slacks vanish, that \[ b(t)\le K e+8K b(t)^2 \quad\text{whenever }b(t)\text{ is in that radius}. \tag{61}\] Choose \(r>0\) within the radius and with \(8Kr\le1/4\). If \(Ke<r/2\), the continuous function \(b\), starting at zero, cannot first reach \(r\): (61) would give \(r<3r/4\). Thus \(b(t)<r\) throughout the interval between \(0\) and any chosen \(t\in I\). Absorbing the quadratic term in (61) gives \(b(t)\le2Ke\) uniformly. This proves the assertion with suitably reduced \(e_0\). ◻

Proof. Write \(w=(t-i)/(t+i)\) and give the unit circle normalized measure \(d\theta/(2\pi)\). The transformation \[(Jf)(t)=\frac{1}{\sqrt\pi(t+i)} f\left(\frac{t-i}{t+i}\right)\] is unitary from \(L^2(\mathbb T,H)\) onto \(\mathcal K\): the change of variables has \(d\theta/(2\pi)=dt/(\pi(1+t^2))\). For \(n\geq0\), \(J(w^n)\) is a linear combination of \((t+i)^{-k}\), \(1\leq k\leq n+1\), whose Fourier transforms are supported in \([0,\infty)\). For \(n<0\) its expression is a linear combination of \((t-i)^{-k}\), whose Fourier transforms are supported in \((-\infty,0]\). These support statements follow directly from \[(t+i)^{-k}=\frac{(-i)^k}{(k-1)!} \int_0^\infty u^{k-1}e^{-u}e^{itu}\,du\] and its complex conjugate, interpreted under the \(L^2\) Fourier transform. Since the images of all circle modes form an orthonormal basis, \(J\) identifies the nonnegative-mode subspace with \(P_0\mathcal K\).

Conjugating the multiplier by \(J\) gives the circle multiplier \(\widetilde B(w)=B(t)\). Apply its invariant-subspace property to every constant vector. All negative Fourier coefficients of \(\widetilde B\) vanish. If \(b_n\) denotes its \(n\)th coefficient, defined by matrix coefficients, then \[F(w)=\sum_{n\geq0} b_nw^n,\qquad |w|<1,\] converges in operator norm locally uniformly and is operator-norm holomorphic. Its value at \(w=\rho e^{i\theta}\) is the circle Poisson integral of \(\widetilde B\), since all its negative coefficients vanish. The Poisson kernel is positive and has integral one, so \(\lVert F(w)\rVert\leq\lVert\widetilde B\rVert_\infty\). Changing variables in the circle Poisson integral gives exactly (62); explicitly, harmonic measure at \(t+ir\) pushes forward to \(r\,ds/(\pi((t-s)^2+r^2))\).

For completeness, multiplication of these extensions can be checked without presupposing pointwise boundary limits. The radial Poisson integrals converge in \(L^2(\mathbb T,H)\) on each fixed vector, for both \(\widetilde B\) and \(\widetilde B^*\), and likewise for \(\widetilde S\) and \(\widetilde S^*\). Choose a sequence of radii tending to one along which these convergences hold pointwise for a countable dense family of vectors. Uniform boundedness extends the pointwise convergence to every vector. Along this sequence the products converge strongly-* almost everywhere to \(\widetilde B(w)x\widetilde S(w)\). Each Fourier coefficient of the bounded analytic product \(F(w)xG(w)\) therefore converges, by dominated convergence on matrix coefficients, to the corresponding coefficient of that boundary product. In particular its negative coefficients vanish, and its nonnegative coefficients are precisely the Taylor coefficients of \(F xG\). This proves the asserted product identity. The same argument applies to products with a scalar bounded analytic function. ◻

Proof. The letters \(C\) below denote constants depending only on the universal constants in the stated inputs, and can increase between occurrences. All required smallness conditions are met by decreasing one universal \(\gamma_0\) at the end.

Proof. Write \(v=e^{ih}\) by the principal logarithm, with \(h=h^*\) and \(\lVert h\rVert\le2\lVert v-I\rVert\). For \(r_0\) small, the norm-convergent logarithm series on \(\mathcal B(\mathcal B(H))\) gives \[\log(\operatorname{Ad}_v)=i\operatorname{ad}_h.\] Indeed \(\operatorname{Ad}_v=\exp(i\operatorname{ad}_h)\) and \(\lVert\operatorname{ad}_h\rVert\) is within the radius where logarithm and exponential are inverse analytic functions. Each term of the logarithm series preserves \(P\), since \(vPv^*=P\). Hence \(i\operatorname{ad}_h\) restricts to a bounded \(*\)-derivation of \(P\).

The factor \(P\) is a simple unital \(C^*\)-algebra, by the same projection argument as in 37. Sakai’s theorem therefore gives a self-adjoint \(a_0\in P\) such that \(h-a_0\in P'\). A scalar may be subtracted from \(a_0\) and added to this commutant element. Choose the scalar at the midpoint of the spectrum of \(a_0\). We claim that the resulting \(a_0\) has \(\lVert a_0\rVert\le\lVert h\rVert\).

For any self-adjoint \(a\in P\), its spectral diameter is at most \(\lVert\operatorname{ad}_a|_P\rVert\). In fact, choose nonzero spectral projections near the two spectral endpoints and a norm-one partial isometry between nonzero subprojections of them. Compressing \([a,w]\) by \(w^*\) shows that its norm is at least the endpoint separation minus the two spectral tolerances. Let those tolerances tend to zero. Apply this to \(a_0\), using \(\lVert\operatorname{ad}_{a_0}|_P\rVert=\lVert\operatorname{ad}_h|_P\rVert\le2\lVert h\rVert\). After the midpoint shift, \(\lVert a_0\rVert\le\lVert h\rVert\) and \(b_0'=h-a_0\) has norm at most \(2\lVert h\rVert\). They commute, so \(v=e^{ia_0}e^{ib_0'}\); these are the required factors, with total displacement at most \(3\lVert h\rVert\). ◻

Proof. Let \(m_s\) denote the invariant operator mean just described. Put \[F_t(s)=v_{t+s}v_s^*,\qquad A_t=m_sF_t(s).\] Every \(F_t(s)\) is a unitary normalizer and \(\lVert F_t(s)-v_t\rVert\le e\). Hence \(\lVert A_t-v_t\rVert\le e\). Translation invariance and the exact factorization \[F_{t+r}(s)=F_t(r+s)F_r(s)\] give \[\begin{align*} A_{t+r}-A_tA_r &=m_s\bigl[(F_t(r+s)-A_t)(F_r(s)-A_r)\bigr],\\ \lVert A_{t+r}-A_tA_r\rVert&\le4e^2. \end{align*}\] The centered terms have norm at most \(2e\), and their means vanish; expanding verifies the first equality with the written operator order. Likewise \[I-A_t^*A_t =m_s(F_t(s)-A_t)^*(F_t(s)-A_t),\] so its norm is at most \(4e^2\). Taking polar parts in \(\mathcal B(H)\) alone would not ensure membership in \(\mathcal N_P\). We instead use the local product structure of exact normalizers.

By 41, choose, for every \(s,t\), \[F_t(s)v_t^*=a_{t,s}b_{t,s}', \quad a_{t,s}\in\mathcal U(P),\quad b_{t,s}'\in\mathcal U(P'), \quad \lVert a_{t,s}-I\rVert+\lVert b_{t,s}'-I\rVert\le C_0e.\] All these choices can be arbitrary. Their averages \(a_t=m_sa_{t,s}\in P\) and \(b_t'=m_sb_{t,s}'\in P'\) satisfy \[\lVert m_s(a_{t,s}b_{t,s}')-a_tb_t'\rVert\le4C_0^2e^2.\] This follows by expanding the centered product; its two factors have norm at most \(2C_0e\). Unitarity of the individual factors also gives \[\lVert I-a_t^*a_t\rVert\le4C_0^2e^2, \qquad \lVert I-(b_t')^*b_t'\rVert\le4C_0^2e^2.\] For small \(e\) the averages are invertible, since they are within \(C_0e\) of \(I\). Their polar unitaries \(\widehat a_t\in P\) and \(\widehat b_t'\in P'\) differ from them by at most \(Ce^2\), with an absolute \(C\): use \(\lVert|a_t|-I\rVert\le \lVert a_t^*a_t-I\rVert/(1+\sqrt{1-4C_0^2e^2})\), and similarly on the other side. Therefore \[w_t=\widehat a_t\widehat b_t'v_t\in\mathcal N_P, \qquad \lVert w_t-A_t\rVert\le C_2e^2.\] It follows, uniformly in \(t,r\), that \[\lVert w_t-v_t\rVert\le e+C_2e^2, \qquad \lVert w_tw_r-w_{t+r}\rVert\le(4+3C_2)e^2.\] For the second inequality use \(\lVert A_t\rVert\le1\) and \(\lVert w_t\rVert=1\) to compare products, then the preceding bound on the defect of \(A\).

Iterate this operation, always with the same absolute constants. Choose \(e_1\) so small that every new defect is at most half the preceding one and every displacement is at most twice the preceding defect. The total displacement is then at most \(4e\). The families converge uniformly in \(t\) in operator norm to unitaries \(z_t\), satisfying \(z_tz_r=z_{t+r}\). The group \(\mathcal N_P\) is norm closed: if unitaries \(u_j\to u\) normalize \(P\), norm closure of \(P\) gives \(uPu^*\subset P\) and, by applying the same argument to \(u_j^*\), the reverse inclusion. Thus \(z_t\in\mathcal N_P\) for all \(t\). The exact multiplication law implies \(z_0=I\) and \(z_{-t}=z_t^*\), as required. ◻

Proof of 35. Put \(U_t=\Delta_N^{it}\) and \(W(t)=\Delta_N^{it}\Delta_M^{-it}\). 40 supplies \(n_t\in\mathcal N_M\) with \(\sup_t\lVert n_t-W(t)\rVert\le C\gamma\). Set \(v_t=n_t\Delta_M^{it}\), so \(v_t\in\mathcal N_M\) and \(\sup_t\lVert v_t-U_t\rVert\le C\gamma\). Since \(U\) is an exact representation, \(\sup_{t,s}\lVert v_tv_s-v_{t+s}\rVert\le3C\gamma\). For sufficiently small universal \(\gamma\), apply 42 to obtain an exact representation \(z_t\in\mathcal N_M\) with \[\sup_t\lVert z_t-U_t\rVert\le C'\gamma.\]

Take the invariant mean \(T=m_tz_tU_t^*\). It is within \(C'\gamma\) of \(I\) and hence invertible for small \(\gamma\). For every \(s\), translation invariance and the two exact representation laws imply \[z_sTU_s^*=m_tz_{s+t}U_{s+t}^*=T.\] Its polar unitary \(w=T|T|^{-1}\) therefore also satisfies \(z_sw=wU_s\). Moreover \(\lVert w-I\rVert\le2\lVert T-I\rVert\le2C'\gamma\). In particular \(z_s=wU_sw^*\); this proves strong continuity of \(z\) without any regularity assumption on the selected normalizers.

Set \(q=w^*\). Since \(z_s\) normalizes \(M\), the intertwining identity gives \(U_s(qMq^*)U_s^*=qMq^*\) for every \(s\). At each matrix level conjugation by \(q^{(k)}\) moves contractions by at most \(2\lVert q-I\rVert\). Consequently \[d_{\mathrm{cb}}(qMq^*,N)\le\gamma+2\lVert q-I\rVert\le C_{\rm mod}\gamma.\] Enlarging \(C_{\rm mod}\) and reducing \(c_{\rm mod}\) to meet the finitely many universal smallness conditions proves all assertions. ◻

Proof. The generators \(\pi(x)\) and \(\lambda_r\) lie in \(P\mathbin{\overline\otimes}\mathcal B(L^2(\mathbb R))\) and are fixed by every \(\alpha_s\). For the reverse inclusion, take a fixed operator \(X\) in that tensor product. If \(f,g\in C_c(\mathbb R)\), put \[\lambda(f)=\int_\mathbb Rf(s)\lambda_s\,ds, \qquad Y=\lambda(f)X\lambda(g)^*.\] These integrals are understood on vectors. The operator \(Y\) remains fixed because translations commute with \(U_s\otimes\rho_s\). Define bounded operators from \(\mathcal K\) to \(H\) by \[R_{f,t}\xi=\int_\mathbb Rf(t-s)\xi(s)\,ds.\] They have norm at most \(\lVert f\rVert_2\) and depend continuously on \(t\) in operator norm. On compactly supported vector functions, Fubini’s theorem gives the integral kernel \[ k(t,r)=R_{f,t}XR_{g,r}^*,\qquad (Y\xi)(t)=\int_\mathbb Rk(t,r)\xi(r)\,dr. \tag{81}\] The kernel is bounded and jointly norm continuous. Each value belongs to \(P\): the evaluation operators intertwine the constant action of \(P'\), and \(X\) commutes with that action. Fixedness of \(Y\) gives \[k(t+s,r+s)=U_s^*k(t,r)U_s.\] For fixed \(s\) the identity first holds almost everywhere by testing the two kernels on compactly supported vectors. Norm continuity then makes it hold at every \((t,r)\). Consequently \[ k(t,r)=U_t^*k(0,r-t)U_t. \tag{82}\]

To reconstruct a bounded operator from this kernel inside the generated algebra, use the masks \[\chi_j(v)=(1-|v|/j)_+\qquad(j\ge1).\] The function \(\chi_j\) is the Fourier transform of the probability measure with density \[\frac{j}{2\pi} \left(\frac{\sin(ja/2)}{ja/2}\right)^2\,da,\] with the continuous value at \(a=0\). Thus multiplication of an operator kernel by \(\chi_j(t-r)\) is a contraction: it is the weak integral of \(\operatorname{Ad}(Q_a)\) against this measure. Let \(Y_j\) be the masked operator. Using (82) and \(\rho_v\xi(t)=\xi(t+v)\), \[ Y_j=\int_\mathbb R \chi_j(v)\pi(k(0,v))\rho_v\,dv\ \in C(P,U). \tag{83}\] Indeed the integrand has compact support and is strongly continuous, since \(v\mapsto k(0,v)\) is norm continuous. Its vector integral is a strong limit of finite sums of elements of \(C(P,U)\). Formula (81) verifies the equality on compactly supported vector functions. On these tests the kernels are bounded on a set of finite measure and \(\chi_j(t-r)\to1\), so dominated convergence gives \(Y_j\to Y\) weakly. The uniform norm bound extends this convergence to all vector tests. Hence \(Y\in C(P,U)\).

Choose nonnegative \(f_n\in C_c(\mathbb R)\) of integral one with supports shrinking to zero. Strong continuity of translations gives \(\lambda(f_n)\to I\) and \(\lambda(f_n)^*\to I\) strongly. Applying the preceding conclusion with \(f=g=f_n\) gives \(\lambda(f_n)X\lambda(f_n)^*\to X\) strongly. This proves (79).

For the second assertion, \[\beta_a(\pi(x))=\pi(x),\qquad \beta_a(\lambda_s)=e^{ias}\lambda_s.\] Thus \(\beta\) preserves \(C(P,U)\). An operator fixed by all \(\beta_a\) commutes with the scalar multiplication algebra on \(L^2(\mathbb R)\). If it also belongs to \(P\mathbin{\overline\otimes}\mathcal B(L^2(\mathbb R))\), it is multiplication by a bounded measurable \(P\)-valued field \(F(t)\). Here and below field measurability is weak operator measurability; on bounded sets, separability of \(H\) permits countably many coefficient tests. The remaining \(\alpha\)-fixedness says, for every \(s\), \[F(t+s)=U_s^*F(t)U_s\quad\hbox{for almost every }t.\] It follows that \(G(t)=U_tF(t)U_t^*\) is translation invariant in \(L^\infty(\mathbb R;\mathcal B(H))\). Such a field is almost everywhere constant. To see this without choosing uncountably many exceptional sets, test against a countable dense family of vector pairs. Each scalar coefficient is translation invariant as an \(L^\infty\) class; convolving with a smooth compactly supported approximate identity gives continuous translation-invariant functions, hence constants. Their local \(L^1\) convergence proves the assertion for the coefficients and then for the field. Since \(U_t\) normalizes \(P\), the constant value \(x\) belongs to \(P\). Therefore \(F(t)=U_t^*xU_t\) almost everywhere, which proves (80). ◻

Proof. By 6, tensoring the pair with \(\mathcal B(L^2(\mathbb R))\) preserves the upper bound. Average on the common ambient operator algebra using \(V_s=U_s\otimes\rho_s\) in 7. For a contraction \(X\) in any matrix level of \(C(P,U)\), choose a contraction \(Y\) in the same matrix level of \(Q\mathbin{\overline\otimes}\mathcal B(L^2(\mathbb R))\) within the chosen gap bound. The matrix amplification of \(E_V\) fixes \(X\), takes \(Y\) into \(C(Q,U)\) by 43, and preserves both the norm bound and the approximation error. Reverse \(P,Q\). This proves the assertion at every matrix level with the same bound. ◻

Proof. Write \(\mathcal A=C(P,U)\) and \(\mathcal B=C(Q,U)\). Use a slightly smaller gap bound if necessary so that 44 gives \(d_{\mathrm{cb}}(\mathcal A,\mathcal B)<\gamma\). The semifinite factor \(\mathcal B\) contains matrix units \((e_{ij})_{i,j\ge1}\) with \(\sum_i e_{ii}=I\) strongly and \(e=e_{11}\) finite and nonzero. Here is the usual projection construction. A faithful normal semifinite trace, together with separability of the representation, gives an increasing sequence of finite-trace projections with supremum \(I\). Split their successive differences by trace continuity, and group the resulting pieces into countably many projections of one fixed positive finite trace. The infinite total trace ensures that the grouping exhausts \(I\). Equal-trace comparison gives the required matrix units. Their weak closed span \(S\) is a unital copy of \(\mathcal B(\ell^2)\), hence amenable.

Apply 2 to \(S\subset_\gamma\mathcal A\). For \(\gamma<1/100\) it gives a unitary \(w\) with \[wSw^*\subseteq\mathcal A,\qquad \lVert w-I\rVert\le150\gamma.\] Put \(\mathcal A_1=w^*\mathcal A w\), so \(S\subseteq\mathcal A_1\cap\mathcal B\) and \[ d_{\mathrm{cb}}(\mathcal A_1,\mathcal B)<301\gamma=:\kappa. \tag{84}\] Compression by the common projection \(e\) gives \[A_0=e\mathcal A_1e|_{e\mathcal K},\qquad B_0=e\mathcal Be|_{e\mathcal K},\qquad d(A_0,B_0)<\kappa.\] The algebra \(B_0\) is a type \(\mathrm{II}_1\) factor. When \(\kappa<1/10\), 21 first makes \(A_0\) finite and then makes it a factor. Thus 20 applies to this pair on the separable Hilbert space \(e\mathcal K\). Choosing its orientation appropriately gives a unitary \(v_0\) on \(e\mathcal K\) with \[v_0A_0v_0^*=B_0, \qquad \lVert v_0-e\rVert\le\eta_{\rm fin}(\kappa).\]

Inflate this unitary using the common matrix units: \[ V=\sum_{i=1}^{\infty}e_{i1}v_0e_{1i}. \tag{85}\] The sum converges strongly and is a unitary, because it is the direct sum of copies of \(v_0\) on the orthogonal spaces \(e_{ii}\mathcal K\). Moreover \[\lVert V-I\rVert=\lVert v_0-e\rVert,\qquad Ve_{ij}=e_{ij}V.\] For \(x\in\mathcal A_1\), its transported matrix entries \(e_{1i}xe_{j1}\) belong to \(A_0\). The entries of \(VxV^*\) therefore belong to \(B_0\). Its finite matrix compressions lie in \(\mathcal B\) and converge strongly to \(VxV^*\); hence \(V\mathcal A_1V^*\subseteq \mathcal B\). The reverse argument gives equality. Thus \[ Z=Vw^*,\qquad Z\mathcal A Z^*=\mathcal B, \qquad \lVert Z-I\rVert\le150\gamma+\eta_{\rm fin}(301\gamma). \tag{86}\]

It remains to descend to \(H\). Average the character action \(\beta\) on the entire \(\mathcal B(\mathcal K)\), obtaining a unital completely positive map \(E_\beta\) from 7, with \(V_a=Q_a\). By 43, it maps \(\mathcal B\) into \(\pi(Q)\) and fixes \(\pi(x)\) for every \(x\in\mathcal B(H)\). Therefore \[ F(x)=\pi^{-1}\bigl(E_\beta(Z\pi(x)Z^*)\bigr),\qquad x\in P, \tag{87}\] defines a unital completely positive map \(P\to Q\). Since the same map \(\pi\) is completely isometric on all of \(\mathcal B(H)\), \[\lVert F-\operatorname{id}_P\rVert_{\mathrm{cb}} \le2\lVert Z-I\rVert \le300\gamma+2\eta_{\rm fin}(301\gamma).\] Here \(\operatorname{id}_P\) denotes the original inclusion into \(\mathcal B(H)\). The ordinary gap \(d(P,Q)\) is less than \(\gamma\). Apply 5. Its absolute thresholds hold for all sufficiently small \(\gamma\), because \(\eta_{\rm fin}(t)\to0\). Its displacement bound supplies \(\zeta(\gamma)\). A constant multiple of \(\gamma+\eta_{\rm fin}(301\gamma)\), enlarged to a nondecreasing function if needed, is sufficient. This proves the proposition. ◻

Proof. If either factor is finite, 20 applies, with the roles reversed if necessary. Its finite-neighbor conclusion also shows that for a sufficiently small gap both factors are finite in that case. We may therefore suppose that both are infinite.

Use 32, retaining its changes of representation explicitly. On \(H\otimes K\) it gives the fixed hyperfinite tensor factor \(D\subseteq\mathcal B(K)\) and unitaries \(u_0\in\mathcal B(H)\) and \(v\in\mathcal B(H\otimes K)\) such that \[M=M_0\mathbin{\overline\otimes}D,\qquad N=v\bigl((u_0N_0u_0^*)\mathbin{\overline\otimes}D\bigr)v^*,\] are a prepared type \(\mathrm{III}_1\) pair, and, for absolute constants, \[\lVert u_0-I\rVert+\lVert v-I\rVert\le C_1t, \qquad d_{\mathrm{cb}}(M,N)<C_2t.\] An enlargement of \(C_2\) absorbs strict upper-bound slack. Let \(U_s=\Delta_N^{is}\) be the modular group belonging to the common vector in that preparation. By 35, a unitary \(q\) satisfies \[\lVert q-I\rVert\le C_3t,\qquad P=qMq^*\text{ is normalized by every }U_s, \qquad d_{\mathrm{cb}}(P,N)<C_4t.\] The algebra \(N\) and its modular group have not changed. Consequently \(C(N,U)\) is its type \(\mathrm{II}_\infty\) modular core. No modular claim about the action on \(P\) is necessary. Apply 45 with \(\gamma=C_4t\) to obtain \[aPa^*=N,\qquad \lVert a-I\rVert\le\zeta(C_4t).\]

All these operations have taken place on the actual tensor product of the original Hilbert space. Their inverse adjustments give \[ b=(u_0^*\otimes I_K)v^*aq, \qquad b(M_0\mathbin{\overline\otimes}D)b^*=N_0\mathbin{\overline\otimes}D, \tag{88}\] and \[ \lVert b-I\rVert\le(C_1+C_3)t+\zeta(C_4t)=:r(t). \tag{89}\] The right side tends to zero universally.

Choose a unit vector \(\xi\in K\), and let \(\omega_\xi\) be its normal vector state on all of \(\mathcal B(K)\). Slicing the conjugation in (88) defines \[F_0(x)=(\operatorname{id}\mathbin{\overline\otimes}\omega_\xi) \bigl(b(x\otimes I_K)b^*\bigr),\qquad x\in M_0.\] This is a unital completely positive map into \(N_0\). The slice is defined on the entire ambient tensor product, so comparison with \(x\otimes I_K\) gives at every matrix level \[\lVert F_0-\operatorname{id}_{M_0}\rVert_{\mathrm{cb}}\le2\lVert b-I\rVert\le2r(t).\] The reverse ordinary gap is less than \(t\). A second application of 5 therefore gives the desired conjugating unitary on the original \(H\), with displacement at most an absolute constant times \(r(t)\), for all sufficiently small \(t\).

Choose \(t_{\rm fac}\) below the finite, preparation, modular-alignment, core-descent, and two UCP thresholds just used. Every threshold is absolute, and the only non-linear modulus used here is \(\eta_{\rm fin}\), which is uniform and tends to zero. The maximum of \(\eta_{\rm fin}(t)\) and a sufficiently large constant multiple of \(r(t)\), enlarged if needed to be positive and nondecreasing, defines \(\eta_{\rm fac}\). This establishes the stated uniform quantifiers. ◻

Proof. Fix \(z\in Z(M)_1\) and choose \(y\in N_1\) with \(\lVert z-y\rVert<g\). For \(v\in\mathcal U(N)\), choose \(a\in M_1\) with \(\lVert v-a\rVert<g\). Since \(z\) commutes with \(a\), \[\lVert[z,v]\rVert<2g, \qquad \lVert vyv^*-z\rVert \leq\lVert y-z\rVert+\lVert v z v^*-z\rVert<3g.\] Dixmier’s averaging theorem for von Neumann algebras (Blackadar 2006, Definition III.2.5.16 and Theorem III.2.5.18) says that the norm-closed convex hull of \(\{vyv^*:v\in\mathcal U(N)\}\) meets \(Z(N)\). An element of this intersection is a contraction and is within \(3g\) of \(z\). Interchanging \(M,N\) and then letting \(g\downarrow d(M,N)\) proves the gap estimate.

The centers are amenable. Apply the mutual conclusion of 2, with near-inclusion bound \(3g\), to obtain \(w\). Finally, conjugation by \(w\) changes an ordinary gap by at most \(2\lVert w-I\rVert\), giving the asserted bound for \(N_0\). ◻

Proof. Choose separable unital \(C^*\)-subalgebras \(A\subseteq M\) and \(B\subseteq N\) whose strong closures are \(M,N\). Such subalgebras exist because the strong topology on each bounded operator ball is metrizable and separable when \(H\) is separable. Choose countable norm-dense subsets \(\{a_j\}\subseteq A_1\) and \(\{b_j\}\subseteq B_1\). Off one null set, their fiber representations satisfy \[M_t=A(t)'',\qquad N_t=B(t)''.\]

We first check the unit-ball density needed below. The map \(A\to A(t)\) is a quotient \(*\)-homomorphism, possibly with a nonzero kernel. A contraction in \(A(t)\) has lifts of norm at most \(1+s\) for every \(s>0\). Rescaling these lifts to contractions shows that the image of \(A_1\) is norm dense in \(A(t)_1\). It follows that \(\{a_j(t)\}\) is norm dense in \(A(t)_1\) and, by Kaplansky density, strongly dense in \((M_t)_1\). The same argument applies to \(B\).

Put \(d_0=d(M,N)\). For every \(j,n\geq1\), choose \(y_{j,n}\in N_1\) such that \[\lVert a_j-y_{j,n}\rVert<d_0+1/n.\] The essential-supremum norm formula and the countability of these choices give a common conull set on which \[y_{j,n}(t)\in(N_t)_1, \qquad \lVert a_j(t)-y_{j,n}(t)\rVert\leq d_0+1/n \quad(j,n\geq1).\] Fix such a \(t\), an \(x\in(M_t)_1\), and an \(n\). Choose a net of \(a_j(t)\) converging strongly to \(x\). Along a subnet, the corresponding \(y_{j,n}(t)\) converge weakly to a contraction \(y\in N_t\). Operator-norm balls are weak-operator closed, so \(\lVert x-y\rVert\leq d_0+1/n\). Letting \(n\) increase gives \(\operatorname{dist}(x,(N_t)_1)\leq d_0\). Make the same countable choices in the opposite direction and discard their null exceptions. This proves both near inclusions on one conull set. ◻

Proof. Work on the underlying standard Borel model of \(X\), retaining the countable contraction fields \(a_j(t),b_j(t)\) from the preceding proof. Choose Borel versions of the dimension function, the trivializations on its countably many level sets, and the countably many contraction fields. Discard a Borel null set containing all exceptions to these identifications, the asserted density and algebraic properties, and the implementer hypothesis. On each remaining dimension stratum, the fibers are identified with one fixed separable Hilbert space \(K\). Zero-dimensional fibers contribute nothing to the direct integral and may be omitted. It suffices to select implementers on each of these standard Borel strata.

The unitary group \(\mathcal U(\mathcal B(K))\), with the strong-\(*\) topology, is Polish. Fix a dense sequence \((\xi_l)\) in the unit sphere of \(K\). Consider the relation on pairs \((t,u)\) defined by \(\lVert u-I\rVert\leq a\) and the following two families of conditions: \[\begin{align*} &\text{for every }j,m,k\text{ there is }q\text{ such that } \max_{l\leq m} \lVert\bigl(u a_j(t)u^*-b_q(t)\bigr)\xi_l\rVert<1/k,\\ &\text{for every }j,m,k\text{ there is }q\text{ such that } \max_{l\leq m} \lVert\bigl(u^* b_j(t)u-a_q(t)\bigr)\xi_l\rVert<1/k. \end{align*}\] All these predicates are Borel. Indeed, multiplication of bounded operator fields and strong-\(*\) unitary fields is Borel, and the norm condition on \(u-I\) is a countable collection of vector-norm tests. The strong density of the contraction fields shows that the first family is equivalent to \(uM_tu^*\subseteq N_t\), and the second to the reverse inclusion. Thus the relation describes exactly the implementers with the imposed bound. Its sections are nonempty by hypothesis.

The Jankov–von Neumann uniformization theorem (Kechris 1995, Theorem 18.1) gives a selection measurable for the sigma algebra generated by analytic sets, for an analytic relation between standard Borel spaces; our relation is Borel and hence analytic. Analytic sets are universally measurable (Kechris 1995, Theorem 21.10), so the selector is measurable for the completed measure \(\mu\). Its direct integral is a unitary \(v\) with \(\lVert v-I\rVert\leq a\). For \(x\in M\), the field \(v_t x_t v_t^*\) is a bounded measurable field in \(N_t\), so the field-reconstruction part of central decomposition puts \(vxv^*\) in \(N\). The reverse inclusion follows in the same way. ◻

Proof. The zero Hilbert space is immediate. By 46, choose \(\tau>0\) such that any pair of factors on a separable Hilbert space at distance less than \(\tau\) has an implementer within \(a/2\) of the identity. Choose \(\delta_{\mathrm{sep}}(a)>0\) so small that 47 applies with \(g=\delta_{\mathrm{sep}}(a)\) and \[450\delta_{\mathrm{sep}}(a)<a/2, \qquad 901\delta_{\mathrm{sep}}(a)<\tau.\] For a pair satisfying the hypothesis, align the centers by the unitary \(w\) in that lemma, and set \(N_0=w^*Nw\). The common-center pair \(M,N_0\) has distance less than \(\tau\). By 48, almost every fiber pair has distance less than \(\tau\) as well. Apply the factor theorem on each fiber and then 49, with bound \(a/2\). The resulting unitary \(v\) satisfies \[vMv^*=N_0,\qquad\lVert v-I\rVert\leq a/2.\] Now \(u=wv\) implements the original conjugacy and \(\lVert u-I\rVert\leq\lVert w-I\rVert+\lVert v-I\rVert<a\). Every tolerance used above is independent of the fibers and of \(H\). ◻

Proof. Choose \(r\) with \(d(M,N)<r<g\), and start with \(A_0=C^*(E,I)\), \(B_0=C^*(F,I)\). Suppose that separable unital \(A_j\subseteq M\), \(B_j\subseteq N\) have been constructed. For each element of a countable norm-dense subset of \((A_j)_1\), choose an \(N_1\)-approximant within \(r\), and adjoin all these approximants to \(B_j\). In the same way, adjoin to \(A_j\) approximants in \(M_1\) for a countable dense subset of \((B_j)_1\). Let \(A_{j+1},B_{j+1}\) be the unital \(C^*\)-algebras generated by these additions. Put \[A=\overline{\bigcup_j A_j}^{\,\lVert\cdot\rVert}, \qquad B=\overline{\bigcup_j B_j}^{\,\lVert\cdot\rVert}.\] They are separable and contain the required tests. The stage unit balls are norm dense in the unit balls of these closures: approximate in norm and rescale if necessary. The choices of approximants therefore give \(d(A,B)\leq r<g\).

Let \(K=\overline{\operatorname{span}(C^*(A,B)V)}\). Because the algebra is separable and \(V\) finite, \(K\) is separable. It contains \(V\) and is reducing for both algebras. To transfer the gap through restriction, take \(x\in(A|_K)_1\). For every \(s>0\) there is a quotient lift \(a\in A\) with \(a|_K=x\) and \(\lVert a\rVert\leq1+s\). The contraction \(a/(1+s)\) has a \(B_1\)-approximant within \(r\) plus arbitrary slack. Restricting this approximant gives \[\operatorname{dist}\bigl(x,(B|_K)_1\bigr) \leq \frac{s}{1+s}+r.\] Let \(s\downarrow0\) and reverse the roles. Thus \(d(A|_K,B|_K)\leq r\).

Finally, for a contraction in \((A|_K)''\), Kaplansky density gives a strongly convergent net of contractions from \(A|_K\). Approximate these in \((B|_K)_1\) with errors at most \(r\) plus arbitrary slack, and take a bounded weak-operator cluster subnet. Its limit lies in \(((B|_K)'')_1\) and has the same error bound. Interchanging the algebras proves the final assertion. ◻

Proof of 1. Let \(\varepsilon>0\). The conclusion of 5 is uniform in the algebras and the Hilbert space, and its implementing-unitary bound tends to zero with the cb error. Choose \(a>0\) and \(g_0>0\) so small that this proposition gives an implementer within \(\varepsilon\) of the identity whenever the reverse ordinary gap is at most \(g_0\) and the cb error of the UCP map is at most \(2a\). All its smallness requirements are included in this choice. Choose \[0<\delta<\min\{g_0,\delta_{\mathrm{sep}}(a)\}.\] We prove the theorem for any \(M,N\subseteq\mathcal B(H)\) with \(d(M,N)<\delta\).

If \(H=\{0\}\), take its identity unitary. Otherwise fix a unit vector \(\xi_0\in H\) and a number \(g\) satisfying \[d(M,N)<g<\delta_{\mathrm{sep}}(a).\] Consider the directed set of quadruples \(i=(E_i,F_i,V_i,k_i)\), where \(E_i\subseteq M\), \(F_i\subseteq N\), and \(V_i\subseteq H\) are finite, \(\xi_0\in V_i\), and \(k_i\in\mathbb N\) is positive. The order is inclusion in the three finite sets and the usual order in \(k_i\).

For each \(i\), apply 51 to obtain \(A_i,B_i,K_i\). Set \[P_i=(A_i|_{K_i})'',\qquad Q_i=(B_i|_{K_i})''.\] These are unital von Neumann algebras on the nonzero separable space \(K_i\), and \(d(P_i,Q_i)<g\). By 50, there is a unitary \(u_i\in\mathcal B(K_i)\) with \[u_iP_iu_i^*=Q_i,\qquad\lVert u_i-I\rVert<a.\] Extend it by the identity on \(K_i^\perp\) to a unitary \(U_i\in\mathcal B(H)\). For all matrix sizes, conjugation satisfies \[ \lVert\operatorname{Ad}(U_i)|_M-\operatorname{id}_M\rVert_{\mathrm{cb}}\leq2\lVert U_i-I\rVert<2a. \tag{90}\]

We next record what the local conjugacy says about the finite tests on the original space. If \(x\in E_i\), then \(K_i\) reduces \(x\), and the restriction of \(U_ixU_i^*\) belongs to \(Q_i\) with norm at most \(\lVert x\rVert\). Kaplansky density approximates this restriction on the finitely many coefficients from \(V_i\) by an element of \(B_i|_{K_i}\) of norm at most \(\lVert x\rVert\) (with arbitrarily small slack if needed). Lift that element through the restriction quotient. Choosing the slack small enough gives \(b_{i,x}\in B_i\subseteq N\) such that \[ \lVert b_{i,x}\rVert\leq\lVert x\rVert+1, \qquad \lvert\langle (U_ixU_i^*-b_{i,x})\xi,\eta\rangle\rvert<1/k_i \quad(\xi,\eta\in V_i). \tag{91}\] The displayed coefficients are coefficients on \(H\), since all the vectors belong to \(K_i\). At this stage the whole conjugated algebra need not lie in \(N\); (91) is the property that will survive as the tests increase.

For each \(x\in M\), the operators \(U_ixU_i^*\) lie in the weak-operator compact ball of radius \(\lVert x\rVert\) in \(\mathcal B(H)\). Compactness of the product of these balls gives a subnet for which \[F(x)=\operatorname*{WOT\!\!-\!lim}_i U_ixU_i^* \quad\text{exists for every }x\in M.\] Linearity and unitality pass to this limit. At each matrix size, positive operators form a weak-operator closed cone, so \(F\) is completely positive. Also, the norm ball of radius \(2a\lVert X\rVert\) is weak-operator closed for every fixed matrix \(X\) over \(M\). Taking limits in (90) therefore gives \[ \lVert F-\operatorname{id}_M\rVert_{\mathrm{cb}}\leq2a. \tag{92}\]

It remains to verify the range of \(F\). Fix \(x\in M\). Cofinality of the subnet implies that eventually \(x\in E_i\), so choose \(b_{i,x}\) as in (91) on this tail. Every fixed pair \(\xi,\eta\in H\) eventually belongs to \(V_i\), while \(k_i\to\infty\). Hence (91) and the definition of \(F\) imply \(b_{i,x}\to F(x)\) in the weak-operator topology. As \(N\) is weak-operator closed, \(F(x)\in N\). This holds for every \(x\in M\), and proves that \(F:M\to N\) is UCP. Normality of this cluster map is neither asserted nor needed.

The original inclusion \(M\hookrightarrow\mathcal B(H)\) is faithful, normal, and unital. Its range has reverse ordinary gap less than \(\delta<g_0\) from \(N\), and (92) supplies the cb bound. Apply 5. It yields a unitary \(u\in\mathcal B(H)\) with \(uMu^*=N\) and \(\lVert u-I\rVert<\varepsilon\).

The choices of \(a,g_0\), and \(\delta\) depend only on \(\varepsilon\). The intervening choices of finite tests, separable subspaces, and quotient lifts change none of their bounds. Thus the same tolerance works for all algebras, all representations, and all Hilbert spaces, as claimed. ◻

Proof. Choose \(\delta\) from 1 with unitary displacement \(\varepsilon/2\). The resulting unitary satisfies \(uMu^*=N\) and \(\lVert u-I\rVert<\varepsilon/2\). Conjugation by its diagonal amplification identifies the unit balls of \(M_n(M)\) and \(M_n(N)\) and moves each contraction by at most \(2\lVert u-I\rVert\), independently of \(n\). The same estimate applies to the commutants because \(uM'u^*=N'\). ◻