Proof. Let \(b\in\mathbf Q'\cap\mathbf M\) have a uniformly bounded representative \((b_n)\), and put \[\delta_n=\sup_{w\in\mathcal U(P)} \|wb_nw^*-b_n\|_2.\] Then \(\lim_{n\to\omega}\delta_n=0\). Otherwise there would be an \(\varepsilon>0\) and a set belonging to \(\omega\) on which \(\delta_n>\varepsilon\). Choosing on this set a unitary \(w_n\in P\) with \(\|w_nb_nw_n^*-b_n\|_2>\varepsilon\), and setting \(w_n=1\) elsewhere, would give a unitary in \(\mathbf Q\) that does not commute with \(b\).

For each \(n\), let \(K_n\) be the closed convex hull, in \(L^2(M,\tau)\), of the elements \(wb_nw^*\), \(w\in\mathcal U(P)\). The unique element \(c_n\in K_n\) of smallest \(2\)-norm is fixed by conjugation by every unitary of \(P\): each such conjugation preserves \(K_n\) and its Hilbert space norm. Moreover, \(c_n\) belongs to \(M\) and \(\|c_n\|\leq\|b_n\|\). Indeed, convex combinations from this orbit have this operator norm bound, and an operator-norm-bounded \(2\)-norm limit lies in the same bounded ball of \(M\). To see the latter fact, left multiplication by such a convergent sequence converges strongly on the dense subspace \(M\subset L^2(M)\), since \[\|(x_r-x_s)a\|_2\leq\|x_r-x_s\|_2\|a\| \qquad(a\in M),\] and hence converges strongly on all of \(L^2(M)\) by the uniform operator norm bound. The strong limit belongs to \(M\), and its value at \(1\) is the prescribed \(2\)-norm limit.

Thus \(c_n\in P'\cap M=\mathbb C1\). The trace is constant on \(K_n\) and continuous in \(2\)-norm, so \(c_n=\tau(b_n)1\). Every element of \(K_n\) is at \(2\)-distance at most \(\delta_n\) from \(b_n\). Consequently \[\|b_n-\tau(b_n)1\|_2\leq\delta_n\longrightarrow_{n\to\omega}0.\] The bounded scalar sequence \((\tau(b_n))\) has a scalar ultralimit, which represents \(b\). This proves the assertion. ◻

Proof. The subspace \(X=D\cap\ker\tau\) is norm separable, hence \(2\)-norm separable. For example, applying \(d\mapsto d-\tau(d)1\) to a countable norm-dense subset of \(D\) gives a countable norm-dense subset of \(X\). Apply Theorem 4 to obtain a diffuse abelian algebra \(A\subset\mathbf Q\).

We first choose a Haar unitary in \(A\). Diffuseness gives arbitrarily small nonzero subprojections of every nonzero projection: repeatedly split and retain a piece of at most half the trace. Normality of the trace and maximality under inclusion then give subprojections of every prescribed smaller trace. We may therefore choose recursively partitions of \(1\) by projections \[\{p_{m,k}:0\leq k<2^m\}\subset A, \qquad \tau(p_{m,k})=2^{-m},\] with \(p_{m,k}=p_{m+1,2k}+p_{m+1,2k+1}\). The self-adjoint elements \[h_m=\sum_{k=0}^{2^m-1}\frac{k}{2^m}p_{m,k}\] satisfy \(\|h_{m+1}-h_m\|\leq2^{-(m+1)}\) and converge in norm to an element \(h\in A\). For each continuous function \(f\) on \([0,1]\), \[\tau(f(h)) =\lim_{m\to\infty}2^{-m} \sum_{k=0}^{2^m-1}f(k2^{-m}) =\int_0^1 f(s)\,ds.\] It follows that \(t=\exp(2\pi i h)\) is Haar.

We next remove the centering requirement at the two endpoints, so that (2) can be applied to words with arbitrary outside coefficients. Expand each endpoint into its scalar and centered parts. A term with a centered last endpoint vanishes by (1), since every nonzero power of \(t\) belongs to \(A\) and is centered. A term with a centered first endpoint and scalar last endpoint vanishes after moving the first endpoint to the end by traciality. It remains to consider the word with both endpoints omitted. When \(j=1\) this is the Haar property. When \(j\geq2\), rotate the final power \(t^{n_j}\) next to the first power \(t^{n_1}\). If \(n_j+n_1\ne0\), the resulting word is covered by (1). If \(n_j+n_1=0\), the two powers cancel and leave a word starting and ending with centered elements of \(D\). This word is again covered by (1), except when \(j=2\), in which case it is simply the centered element \(d_1\). This proves (2).

Laurent polynomials in \(t\) are norm dense in \(C^*(t)\). As \(t\) is Haar, the centered Laurent polynomials are precisely the linear combinations of its nonzero powers. Expanding an alternating centered word, and using (2) followed by norm approximation, proves that \(t\) is free from \(D\). ◻

Proof. For any Haar unitary free from \(D\), formula (2) follows immediately by expanding the endpoints into scalar and centered parts and applying freeness to each resulting alternating word. Replacing \(t\) by \(t^*\) merely negates the exponents, so we concentrate on \(v=atb\).

Consider a word \[d_0v^{n_1}d_1\cdots v^{n_j}d_j\] with nonzero integer exponents and centered internal factors \(d_1,\ldots,d_{j-1}\). Expand each power of \(v\) into individual letters \(t\) and \(t^{-1}\), with elements of \(D\) between them. Within one power of \(v\) all these letters have the same sign. A gap between opposite signs can therefore occur only between two successive powers of \(v\). Such a gap equals \[b d_i b^* \quad\text{for a transition from $t$ to $t^{-1}$}, \qquad a^*d_i a \quad\text{for a transition from $t^{-1}$ to $t$}.\] It is centered by traciality. Every other gap joins equal signs. Expand each of these latter gaps into its scalar and centered parts. In a resulting term, removing a scalar gap only joins letters of equal sign. Thus each block joined across scalar gaps is a nonzero power of \(t\), and consecutive blocks are separated by centered elements of \(D\). The endpoint factors remain arbitrary elements of \(D\). Formula (2) shows that every resulting term has trace zero.

We have proved (2) for \(v\). Taking \(j=1\) and \(d_0=d_1=1\) proves that \(v\) is Haar. The general formula and norm approximation by Laurent polynomials then prove its freeness from \(D\). ◻

Proof. For a letter \(a\), let \(B_a\) be the orthogonal projection onto the closed span of reduced words beginning with \(a\). Split \[\lambda(\gamma) =\lambda(\gamma)(1-B_{\gamma^{-1}}) +\lambda(\gamma)B_{\gamma^{-1}}.\] The first summand has range in the range of \(B_\gamma\). These ranges are mutually orthogonal, so for \(\eta\in\ell^2(F)\), \[\begin{align*} \lVert \sum_\gamma b_\gamma\lambda(\gamma) (1-B_{\gamma^{-1}})\eta\rVert^2 &=\sum_\gamma |b_\gamma|^2 \lVert (1-B_{\gamma^{-1}})\eta\rVert^2\\ &\le\left(\sum_\gamma |b_\gamma|^2\right)\lVert \eta\rVert^2. \end{align*}\] For the second summands, the triangle inequality and Cauchy–Schwarz give \[\begin{align*} \lVert \sum_\gamma b_\gamma\lambda(\gamma) B_{\gamma^{-1}}\eta\rVert &\le\sum_\gamma |b_\gamma|\lVert B_{\gamma^{-1}}\eta\rVert\\ &\le\left(\sum_\gamma|b_\gamma|^2\right)^{1/2}\lVert \eta\rVert, \end{align*}\] because the projections \(B_{\gamma^{-1}}\) are mutually orthogonal. Adding the two estimates proves the lemma. ◻

Proof. Fix \(\ell\) and work first in \(\mathbf M=M^\omega\), with \(\mathbf Q=P^\omega\). We use \(\tau\) also for the ultrapower trace.

Free variables and prefixes. Set \(D_0=C^*(1,u,z,\xi)\). By Lemma 5, choose successively \(t_j\in\mathcal U(\mathbf Q)\) Haar and free from \(D_j\), and put \(D_{j+1}=C^*(D_j,t_j)\), for \(0\le j\le\ell\). Each of these \(C^*\)-algebras is norm separable. Define \[s_1=t_0,\qquad s_{j+1}=u t_j\quad(1\le j\le\ell),\qquad g_k=s_1\cdots s_k\quad(1\le k\le\ell),\] and \[W=g_\ell s_{\ell+1} =t_0\prod_{k=1}^{\ell}(u t_k).\] Thus \(g_k\) is the prefix immediately preceding the \(k\)th occurrence of \(u\) in \(W\). Since \(D_j=C^*(D_0,s_1,\ldots,s_j)\), Lemma 6 shows that \(s_{j+1}\) is Haar and free from \(D_j\). In particular, \(s_1,\ldots,s_\ell\) are free Haar unitaries. Every nonidentity reduced word in them has trace zero, so the group they generate can be identified with the free group \(G\) on these generators. Since \(W=(g_\ell u)t_\ell\) and \(g_\ell u\in\mathcal U(D_\ell)\), the same lemma shows that \(W\) is Haar and free from \(D_\ell\). Applying it to \(W^*\) and \(z\in\mathcal U(D_\ell)\) also shows that \[y=zW^*\] is Haar and free from \(D_\ell\).

For \(p\in G\) set \(Y_p=pyp^{-1}\). These form a free family of Haar unitaries. Indeed, if \(a_1,\ldots,a_r\) are nonzero integers and successive labels \(p_1,\ldots,p_r\) are distinct, then \[Y_{p_1}^{a_1}\cdots Y_{p_r}^{a_r} =p_1y^{a_1}(p_1^{-1}p_2)y^{a_2}\cdots (p_{r-1}^{-1}p_r)y^{a_r}p_r^{-1}.\] Each internal ratio is a nonidentity element of \(G\) and hence is centered in \(D_\ell\). Equation (2) gives zero trace. It follows, by applying this to the reduction of \(v^*w\), that distinct reduced words in the \(Y_p\) are orthonormal in \(L^2(\mathbf M,\tau)\), including the empty word.

A left perturbation of \(u\) at its \(k\)th occurrence in \(W\) is conjugated by \(g_k\) when moved to the front of the word. We therefore use the conjugates \(g_j^{-1}yg_j\): conjugating the \(j\)th one by \(g_j\) returns \(y\). Taking imaginary parts will give a self-adjoint perturbation, while freeness makes their average small in \(2\)-norm. Put \(\operatorname{Im}(b)=(b-b^*)/(2i)\) and define \[h=\frac\alpha\ell\sum_{j=1}^{\ell} \operatorname{Im}(g_j^{-1}yg_j),\qquad u'=e^{ih}u,\qquad W'=t_0\prod_{k=1}^{\ell}(u't_k).\] The \(g_j^{-1}\) are distinct in \(G\), so the summands defining \(h\) are orthogonal centered imaginary parts of free Haar unitaries. Each has squared \(2\)-norm \(1/2\). Consequently, \[h=h^*,\qquad \lVert h\rVert\le\alpha,\qquad \lVert h\rVert_2=\frac\alpha{\sqrt{2\ell}}.\] Write \(H_k=g_khg_k^{-1}\), \(p_{kj}=g_kg_j^{-1}\), and \(c=\alpha/(2\ell)\). Cancelling consecutive prefixes gives \[ W'=\left(\prod_{k=1}^{\ell}e^{iH_k}\right)W, \qquad iH_k=c\sum_{j=1}^{\ell} (Y_{p_{kj}}-Y_{p_{kj}}^{-1}). \tag{4}\] For the first identity, use \(e^{iH_k}=g_ke^{ih}g_k^{-1}\) and \(g_k^{-1}g_{k+1}=ut_k\) for \(1\le k<\ell\); the product order is therefore essential.

Let \(S=\{p_{kj}:1\le k,j\le\ell\}\) and let \(n_p\) count the ordered pairs \((k,j)\) with \(p_{kj}=p\). For the identity \(e\) of \(G\), one has \(n_e=\ell\). If \(k\ne j\), the word \[p_{kj}=s_1\cdots s_k s_j^{-1}\cdots s_1^{-1}\] is reduced, since its joining letters have different indices. Its initial positive segment and its final negative segment determine \(k\) and \(j\). All off-diagonal pairs therefore give distinct nonidentity words, and \[ \sum_{p\in S}n_p^2=\ell^2+\ell(\ell-1)=2\ell^2-\ell. \tag{5}\]

The first-order correlation. Put \(C_1=\sum_{k=1}^{\ell}iH_k\). Each diagonal term \(cY_{p_{kk}}W\) equals \(cz\), because \(p_{kk}=e\) and \(Y_eW=z\). We show that all other linear terms vanish when paired with \(\xi\). Equation (2) for \(W\) and \(D_\ell\) gives \(\tau(\xi^*W)=0\). For \(p\ne e\), the element \(p^{-1}\) is centered in \(D_\ell\), so \[\tau(\xi^*Y_pW) =\tau(\xi^*p z W^*p^{-1}W)=0.\] For every \(p\in G\), one also has \[\tau(\xi^*Y_p^{-1}W) =\tau(\xi^*p W z^*p^{-1}W)=0.\] Indeed, split the internal factor \(z^*p^{-1}\in D_\ell\) into its centered and scalar parts. The first contribution vanishes by (2); the second is a scalar multiple of \(\tau(\xi^*pW^2)\) and vanishes by the same formula. It follows that \[\tau\bigl(\xi^*(1+C_1)W\bigr) =c n_e\tau(\xi^*z)=\frac\alpha2\tau(\xi^*z).\]

The uniform remainder estimate. It remains to bound the contribution of \(\prod_k e^{iH_k}-1-C_1\) uniformly in \(\ell\). Group the ordered product of exponentials by its original degree: \[C_d=\sum_{\substack{m_1+\cdots+m_\ell=d\\m_k\ge0}} \frac{(iH_1)^{m_1}\cdots(iH_\ell)^{m_\ell}} {m_1!\cdots m_\ell!}.\] The term \(C_0\) is \(1\), and the degree-one term is the \(C_1\) above. For fixed \(\ell\), the expansion is absolutely convergent in operator norm, because \[\sum_{d\ge0}\lVert C_d\rVert \le\prod_{k=1}^{\ell} \sum_{m\ge0}\frac{\lVert H_k\rVert^m}{m!} =\exp\left(\sum_{k=1}^{\ell}\lVert H_k\rVert\right) \le e^{\ell\alpha}.\] The Cauchy product may thus be grouped in this way, and its sum is \(\prod_k e^{iH_k}\). We will prove the bound \[ \lVert \prod_{k=1}^{\ell}e^{iH_k}-1-C_1\rVert_2 \le\frac{4\alpha^2}{1-2\alpha}, \tag{6}\] which, unlike the preceding norm-convergence bound, is uniform in \(\ell\).

Let \(F_S\) be the formal free group with generators \(T_p\), \(p\in S\). In particular \(T_e\) is a generator, not the identity of \(F_S\). Let \(\Phi:\mathbb C[F_S]\to\mathbf M\) be the homomorphism \(\Phi(T_p)=Y_p\). By the orthonormality proved above, for every finite group-ring sum \(f=\sum_{w\in F_S}f(w)w\) one has \[ \lVert \Phi(f)\rVert_2^2=\sum_{w\in F_S}|f(w)|^2. \tag{7}\] Define \[B_k=c\sum_{j=1}^{\ell} (T_{p_{kj}}-T_{p_{kj}}^{-1}),\qquad A_k=c\sum_{j=1}^{\ell} (T_{p_{kj}}+T_{p_{kj}}^{-1}),\] and \[A=\sum_{k=1}^{\ell}A_k =c\sum_{p\in S}n_p(T_p+T_p^{-1}).\] For a group-ring element \(f\), write \(|f|\) for its coefficientwise absolute value, and compare nonnegative coefficient functions coefficientwise. Group multiplication gives \[|fg|(w)=\left|\sum_{ab=w}f(a)g(b)\right| \le\sum_{ab=w}|f(a)||g(b)|=(|f|\,|g|)(w).\] This comparison already includes all free reductions and all contributions collected at the same resulting word. Thus the formal element \[\mathcal C_d= \sum_{\substack{m_1+\cdots+m_\ell=d\\m_k\ge0}} \frac{B_1^{m_1}\cdots B_\ell^{m_\ell}} {m_1!\cdots m_\ell!}, \qquad \Phi(\mathcal C_d)=C_d,\] satisfies \[ |\mathcal C_d| \le\sum_{m_1+\cdots+m_\ell=d} \frac{A_1^{m_1}\cdots A_\ell^{m_\ell}} {m_1!\cdots m_\ell!} \le A^d. \tag{8}\] For the last inequality, each tuple \((m_1,\ldots,m_\ell)\) corresponds to one nondecreasing sequence of row indices in the distributive expansion of \((A_1+\cdots+A_\ell)^d\). Its factorial weight is at most one, and every other summand has nonnegative coefficients. No commutation of the \(A_k\) is used. The index \(d\) records the degree before reduction; resulting words may have shorter length, including zero.

Let \(\lambda\) now denote the left regular representation of \(F_S\), and let \(\delta_{\mathrm{id}}\) be the unit vector at its identity. By (7) and (8), \[\lVert C_d\rVert_2 \le\lVert A^d\rVert_{\ell^2(F_S)} =\lVert \lambda(A)^d\delta_{\mathrm{id}}\rVert_{\ell^2(F_S)} \le\lVert \lambda(A)\rVert^d.\] This uses only the coefficient \(2\)-norm identity (7), with no comparison of operator norms between the two representations. Lemma 7 and (5) give \[\lVert \lambda(A)\rVert \le 2\left(2c^2\sum_{p\in S}n_p^2\right)^{1/2} =2\alpha\left(1-\frac1{2\ell}\right)^{1/2} \le2\alpha<1.\] Consequently \(\lVert C_d\rVert_2\le(2\alpha)^d\). The operator-norm convergence established above implies convergence of the same partial sums in \(2\)-norm. Taking their tails and using the triangle inequality therefore yields \[\lVert \prod_{k=1}^{\ell}e^{iH_k}-1-C_1\rVert_2 \le\sum_{d\ge2}(2\alpha)^d =\frac{4\alpha^2}{1-2\alpha}.\] In particular, orthogonality between different degrees is not required. This proves (6).

Combining the first-order correlation with (4), (6), and Cauchy–Schwarz gives \[ \left|\tau(\xi^*W')-\frac\alpha2\tau(\xi^*z)\right| \le\lVert \xi\rVert_2\frac{4\alpha^2}{1-2\alpha}. \tag{9}\]

Lifting for the fixed word length. Each \(t_j\) has a representative consisting of unitaries in \(P\). To see this, start with a bounded representative \((b_n)\) of any unitary in \(P^\omega\). The polar part of \(b_n\) extends to a unitary \(w_n\in P\), since its initial and final complementary projections have equal trace in the finite factor \(P\). Then \(b_n=w_n|b_n|\) and functional calculus gives \[\lVert b_n-w_n\rVert_2=\lVert |b_n|-1\rVert_2 \le\lVert b_n^*b_n-1\rVert_2\longrightarrow_\omega0.\] Here the inequality follows from \(|\sqrt t-1|\le|t-1|\) for \(t\ge0\).

Choose unitary representatives \(t_{j,n}\in P\) for the finitely many \(t_j\). Using the constant representatives of \(u,z\), form \(g_{k,n},W_n,y_n,h_n,u_n',W_n'\) by exactly the formulas above. For this fixed \(\ell\) these are bounded sequences, with \(h_n=h_n^*\) and \(\lVert h_n\rVert\le\alpha\). Bounded algebraic operations and continuous functional calculus in the quotient show that they represent the corresponding ultrapower elements. In particular, \[\lim_{n\to\omega}\lVert h_n\rVert_2 =\frac\alpha{\sqrt{2\ell}},\qquad \lim_{n\to\omega}\tau(\xi^*W_n')=\tau(\xi^*W').\] Moreover \(\lVert u_n'-u\rVert_2\le\lVert h_n\rVert_2\), by the scalar inequality \(|e^{it}-1|\le|t|\) and functional calculus. Choose one index \(n\) for which \[\lVert h_n\rVert_2<\frac\alpha{\sqrt{2\ell}}+\frac1\ell, \qquad |\tau(\xi^*W_n')-\tau(\xi^*W')|<\frac1\ell.\] Both conditions hold on sets belonging to \(\omega\), so their intersection is nonempty. Set \(\widetilde u=u_n'\) and \(\widetilde t_j=t_{j,n}\). Equation (9) proves (3). This choice is made separately for each \(\ell\), as the proposition permits. ◻

Proof. The unit ball of \(M\) is compact and metrizable in the weak-star topology, so it has a countable weak-star dense subset \(S\). For each \(\xi\in L^2(M,\tau)\), the functional \(a\mapsto\langle a,\xi\rangle\) on \(M\) is normal. Indeed, normal functionals form a norm-closed subspace of \(M^*\), and approximation of \(\xi\) by elements of \(M\) in \(L^2\) gives norm approximation of this functional by normal functionals, since \[\sup_{\lVert a\rVert\leq1} |\langle a,\xi-\eta\rangle| \leq\lVert \xi-\eta\rVert_2.\] If \(\xi\) is orthogonal to \(S\), normality implies that it is orthogonal to the unit ball, and hence to \(M\). Thus \(S\) has dense linear span in \(L^2(M,\tau)\), proving separability.

Let \((u_n)\) be a Cauchy sequence in \(\mathcal U(M)\) for \(d\). The adjoints are also Cauchy, because \(\lVert u_n^*-u_m^*\rVert_2=\lVert u_n-u_m\rVert_2\). For \(a\in M\), \[\lVert (u_n-u_m)a\rVert_2\leq\lVert a\rVert\,\lVert u_n-u_m\rVert_2.\] The left multiplication operators associated with \(u_n\) therefore converge strongly on the dense subspace \(M\subset L^2(M,\tau)\). Their uniform operator norm bound extends this convergence to the whole Hilbert space. The same holds for the left multipliers associated with \(u_n^*\). Strong closedness of the left regular representation of \(M\) gives limits \(u,v\in M\), and passage to inner products gives \(v=u^*\). Multiplication is jointly strongly continuous on bounded sets, so \(uv=vu=1\). Finally, convergence on the vector \(1\) gives \(\lVert u_n-u\rVert_2\to0\). ◻

Proof. Let \(\mathcal P_P\) denote the family of finite \(*\)-polynomials in a unitary variable with coefficients in \(P\). For each fixed \(u\), the values \(b(u)\), \(b\in\mathcal P_P\), are dense in \(L^2(N_u,\tau)\). To see this, take the operator norm closure of the polynomial algebra. Its von Neumann closure is \(N_u\), and Kaplansky density approximates each element of \(N_u\) strongly by bounded elements of this \(C^*\)-algebra. Evaluation on \(1\in L^2(M,\tau)\) gives \(2\)-norm approximation, which can then be followed by operator norm approximation by polynomials. The density of \(N_u\) in \(L^2(N_u,\tau)\) finishes the assertion.

For fixed \(b\in\mathcal P_P\), the map \(u\mapsto b(u)\) is continuous in \(2\)-norm. This follows by telescoping products, using the fixed operator norm bounds of the coefficients, the fact that unitaries have norm one, and the identity \(\lVert u^*-v^*\rVert_2=\lVert u-v\rVert_2\). Consequently \[d_\xi(u):=\operatorname{dist}\bigl(\xi,L^2(N_u,\tau)\bigr) =\inf_{b\in\mathcal P_P}\lVert \xi-b(u)\rVert_2\] is upper semicontinuous: an infimum of any family of continuous real functions is upper semicontinuous. Orthogonality gives \[f_\xi(u)^2=\lVert \xi\rVert_2^2-d_\xi(u)^2,\] so \(f_\xi\) is lower semicontinuous and \(0\leq f_\xi(u)\leq\lVert \xi\rVert_2\).

For completeness, a bounded lower semicontinuous real function \(f\) on a complete metric space is continuous on a dense \(G_\delta\) set. For \(\varepsilon>0\), let \(G_\varepsilon\) be the union of all open sets on which the oscillation of \(f\) is less than \(\varepsilon\). This union is dense. Indeed, in any nonempty open set \(O\), write \(s=\sup_O f\) and choose \(x\in O\) with \(f(x)>s-\varepsilon/3\). Lower semicontinuity gives an open neighborhood \(V\subset O\) of \(x\) on which \(f>f(x)-\varepsilon/3\). The oscillation on \(V\) is less than \(\varepsilon\). Baire’s theorem now shows that \(\bigcap_{n\geq1}G_{1/n}\) is dense, and \(f\) is continuous at each of its points.

Apply this construction to \(f_\eta\) for \(\eta\) in a countable dense subset of \(L^2(M,\tau)\) and intersect the resulting dense \(G_\delta\) sets. By Baire’s theorem, their intersection \(\mathcal C\) is again a dense \(G_\delta\) set. For arbitrary \(\xi,\eta\in L^2(M,\tau)\), contractivity of \(e_u\) gives \[ |f_\xi(u)-f_\eta(u)|\leq\lVert \xi-\eta\rVert_2 \qquad(u\in\mathcal U(M)). \tag{10}\] Thus, for \(u\in\mathcal C\) and \(v\in\mathcal U(M)\), \[|f_\xi(v)-f_\xi(u)| \leq2\lVert \xi-\eta\rVert_2+|f_\eta(v)-f_\eta(u)|.\] Approximating \(\xi\) by an \(\eta\) from the countable dense subset, and then letting \(v\to u\), proves continuity of every \(f_\xi\) at \(u\). ◻

Proof of Theorem 1. Take \(u\in\mathcal C\) from Lemma 10, and suppose that \(N_u\ne M\). There is a unitary \(z\in M\setminus N_u\), since the unitaries linearly span \(M\). Let \(E_{N_u}:M\to N_u\) be the trace-preserving conditional expectation, and put \[\xi=z-E_{N_u}(z)\in M,\qquad s=\lVert \xi\rVert_2>0.\] The conditional expectation is the restriction to \(M\) of \(e_u\). Therefore \[e_u\xi=0,\qquad f_\xi(u)=0,\qquad \tau(\xi^*z)=s^2.\] Fix \[ 0<\alpha<\min\{1/4,s/32\}. \tag{11}\] For every integer \(\ell\geq1\), Proposition 8 provides a unitary \(u_\ell\in M\) and unitaries \(t_{0,\ell},\ldots,t_{\ell,\ell}\in P\) such that, with \[W_\ell=t_{0,\ell}\prod_{k=1}^\ell(u_\ell t_{k,\ell})\in\mathcal U(N_{u_\ell}),\] one has \[\begin{align*} \lVert u_\ell-u\rVert_2 &\leq\frac{\alpha}{\sqrt{2\ell}}+\frac1\ell, \tag{12}\\ \left|\tau(\xi^*W_\ell)-\frac{\alpha s^2}{2}\right| &\leq s\frac{4\alpha^2}{1-2\alpha}+\frac1\ell. \tag{13}\end{align*}\] The fixed choice (11) gives \[s\frac{4\alpha^2}{1-2\alpha} <8s\alpha^2<\frac{\alpha s^2}{4}.\] It follows from (13) that \[|\tau(\xi^*W_\ell)|\geq\frac{\alpha s^2}{4}-\frac1\ell \geq\frac{\alpha s^2}{8}\] for all sufficiently large \(\ell\). On the other hand, \(\lVert W_\ell\rVert_2=1\) and \(W_\ell\in L^2(N_{u_\ell},\tau)\), so \[|\tau(\xi^*W_\ell)| =|\langle W_\ell,e_{u_\ell}\xi\rangle| \leq f_\xi(u_\ell).\] By (12), \(u_\ell\to u\) in \(2\)-norm. Continuity of \(f_\xi\) at \(u\) gives \(f_\xi(u_\ell)\to f_\xi(u)=0\), a contradiction. Thus \(M=W^*(P,u)\) for every \(u\in\mathcal C\).

To see that the full set of relative generators is \(G_\delta\), choose \((\eta_j)_{j\geq1}\) dense in \(L^2(M,\tau)\). With \(d_{\eta_j}\) as in Lemma 10, this set is \[\bigcap_{j,n\geq1}\{v\in\mathcal U(M):d_{\eta_j}(v)<1/n\}.\] Each set in the intersection is open by upper semicontinuity. Vanishing of every distance is equivalent to \(L^2(N_v,\tau)=L^2(M,\tau)\), hence to \(N_v=M\) by the trace-preserving conditional expectation. The set is dense because it contains \(\mathcal C\). ◻

Proof. By uniqueness of the hyperfinite \(\mathrm{II}_1\) factor with separable predual (Popa 2014, sec. 1.3), we may use its standard tensor model \[R=\overline{\bigotimes}_{m\geq1} \bigl(M_2(\mathbb C),\operatorname{tr}_2\bigr).\] On the \(m\)th tensor site, let \[d_m=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad q_m=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\] with the identity on the other sites. On that site, \(2d_mq_m(1-d_m)\) is the matrix unit \(e_{12}\), so \(d_m,q_m\) generate the full matrix algebra. Each of the two families \((d_m)\) and \((q_m)\) consists of commuting projections.

For any countable family of commuting projections \((e_m)\), the operator norm convergent series \[b=\sum_{m\geq1}2\,3^{-m}e_m\] recovers every \(e_m\) by spectral calculus. Inductively, after recovering \(e_1,\ldots,e_{m-1}\), form \[r_m=b-\sum_{j<m}2\,3^{-j}e_j.\] The restriction of \(r_m\) to \(1-e_m\) lies between \(0\) and \(3^{-m}\), whereas its restriction to \(e_m\) lies between \(2\,3^{-m}\) and \(3\,3^{-m}\). Since \(e_m\) commutes with \(r_m\), \[e_m=1_{((3/2)3^{-m},\,\infty)}(r_m)\in W^*(b).\] Applying this encoding separately to \((d_m)\) and \((q_m)\) gives two self-adjoint elements whose generated von Neumann algebra contains every tensor site, and hence is all of \(R\). ◻

Proof. Let \(M\) be such a factor. By Theorem 11, applied with \(N=M\), it contains a hyperfinite \(\mathrm{II}_1\) subfactor \(P\) with \(P'\cap M=\mathbb C1\). The trace-preserving conditional expectation onto \(P\) embeds \(P_*\) isometrically into \(M_*\) by precomposition, so \(P\) has separable predual. Lemma 12 gives self-adjoint \(a,b\in P\) with \(P=W^*(a,b)\), and Theorem 1 gives \(u\in\mathcal U(M)\) with \(M=W^*(P,u)\). Hence \[M=W^*\bigl(a,b,\operatorname{Re}u,\operatorname{Im}u\bigr).\qedhere\] ◻

Proof. Write \(N=W^*(a_1,a_2,a_3,a_4)\) with \(a_j=a_j^*\), and choose \(C>\lVert a_1\rVert+\lVert a_2\rVert\). Put \[A=\begin{pmatrix}a_1&0\\0&a_2+C1_N\end{pmatrix},\qquad B=\begin{pmatrix}a_3&1_N\\1_N&a_4\end{pmatrix},\qquad Q=W^*(A,B).\] Let \(E_{ij}\) be the standard matrix units of \(M_2(N)\). The first diagonal block of \(A\) has spectrum in \([-\lVert a_1\rVert,\lVert a_1\rVert]\), and the second has spectrum in \([C-\lVert a_2\rVert,C+\lVert a_2\rVert]\). For \(\lVert a_1\rVert<\lambda<C-\lVert a_2\rVert\), spectral calculus therefore gives \[E_{11}=1_{(-\infty,\lambda)}(A)\in Q, \qquad E_{22}=1-E_{11}\in Q.\] The off-diagonal cuts of \(B\) give \(E_{12}=E_{11}BE_{22}\in Q\) and \(E_{21}=E_{12}^*\in Q\). The first corner then contains \[\begin{align*} a_1E_{11}&=E_{11}AE_{11},& a_2E_{11}&=E_{12}AE_{21}-CE_{11},\\ a_3E_{11}&=E_{11}BE_{11},& a_4E_{11}&=E_{12}BE_{21}. \end{align*}\] Thus \(NE_{11}\subset Q\). Multiplying this corner by the matrix units yields \(NE_{ij}\subset Q\) for every \(i,j\), and hence \(Q=M_2(N)\). ◻

Proof of Theorem 2. Let \(M\) be a \(\mathrm{II}_1\) factor with separable predual and normalized trace \(\tau\). Choose a projection \(p\in M\) with \(\tau(p)=1/2\). Set \(N=pMp\), with normalized trace \(\tau_N(a)=2\tau(a)\). This corner is a \(\mathrm{II}_1\) factor with separable predual: precomposition with the normal compression \(a\mapsto pap\) embeds \(N_*\) isometrically into \(M_*\). Since \(p\) and \(1-p\) have the same trace, there is a partial isometry \(v\in M\) with \(v^*v=p\) and \(vv^*=1-p\). The corresponding matrix units give a normal unital \(*\)-isomorphism \[\Phi:M_2(N)\longrightarrow M,\qquad \Phi\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix} =b_{11}+b_{12}v^*+vb_{21}+vb_{22}v^*.\] Proposition 13, applied to \(N\), and Lemma 14 provide self-adjoint \(A,B\in M_2(N)\) with \(W^*(A,B)=M_2(N)\). Define \(x=\Phi(A+iB)\in M\). Since the real and imaginary parts of \(x\) are \(\Phi(A)\) and \(\Phi(B)\), respectively, \[W^*(x)=W^*(\Phi(A),\Phi(B))=\Phi(M_2(N))=M.\qedhere\] ◻