Proof. Apply Theorem 0.1(a) of [17] with every coordinate algebra and every coordinate subalgebra equal to \(P\). The required non-intertwining condition holds: a nonzero corner of the diffuse factor \(P\) cannot embed in the scalar relative commutant, or in a finite matrix amplification of it. Moreover, \(P^\omega\) is a factor, so the relative commutant of the full ultraproduct subalgebra is \(\mathbb C1\). The conclusion for the separable centered subspace \(B\ominus\mathbb C1\) is therefore ordinary scalar-trace freeness. This is the case in which every coordinate subalgebra equals its ambient factor, discussed after that theorem and in [17].

A diffuse abelian tracial algebra contains a Haar unitary. Choose one, adjoin it to \(B\), and repeat. Finite adjunction preserves separable predual: a countable generating set gives a separable tracial Hilbert space by polynomial approximation. Thus all \(d\) choices take place in the same ultrapower. At each step the new algebra is free from the algebra generated at the preceding steps, giving the asserted family. ◻

Proof. The labels \(h_1,\ldots,h_r\) are distinct. Put all nondistinguished unitaries into the coefficient algebra: \[B_0=W^*\bigl(B,u_j:j\notin\{h_1,\ldots,h_r\}\bigr), \qquad C=W^*(u_{h_1},\ldots,u_{h_r}).\] Then \(B_0\) and \(C\) are free. This grouping property follows directly by expanding centered polynomials in a free family into alternating centered products; bounded approximation extends it to the generated von Neumann algebras. Write \[y_i=A_i u_{h_i}^{\epsilon_i}B_i, \qquad A_i,B_i\in\mathcal U(B_0),\quad \epsilon_i\in\{-1,1\}.\] If \(w=x_{i_1}^{\sigma_1}\cdots x_{i_t}^{\sigma_t}\) is nonempty and reduced, its evaluated product has the form \[c_0 u_{h_{i_1}}^{\epsilon_{i_1}\sigma_1}c_1 \cdots u_{h_{i_t}}^{\epsilon_{i_t}\sigma_t}c_t, \qquad c_0,\ldots,c_t\in\mathcal U(B_0).\] The displayed distinguished-letter string is itself reduced. Indeed, an adjacent inverse pair would require \(i_j=i_{j+1}\) and \(\sigma_j=-\sigma_{j+1}\), contrary to the reduction of \(w\).

Expand every \(c_j\), including the two outside coefficients, as its scalar part plus its centered part. In each resulting term, removing scalar coefficients joins distinguished letters into nonempty contiguous subwords of that reduced string. Every such block is a centered element of \(C\). The remaining factors alternate between centered elements of \(B_0\) and centered elements of \(C\), and there is at least one \(C\) factor. Freeness makes the trace zero. If there are no centered coefficient factors, the term is just a scalar multiple of a nonempty reduced Haar word and again has trace zero. This proves the assertion, with no condition on the first and last indices of \(w\). ◻

Proof. First suppose that \(M=P\) is a factor. The constant coefficient algebra \(W^*(F)\subset P^\omega\) has separable predual. Apply Lemma 4 and then Lemma 5. The same free \(d\)-tuple makes every prescribed trace exactly zero in \(P^\omega\), even though different tests can designate different labels.

Every unitary of \(P^\omega\) has unitary representatives. To see this, take a bounded self-adjoint logarithm there, lift it to a bounded sequence of self-adjoint elements of \(P\), and exponentiate coordinatewise. Choose such representatives for the \(d\) unitaries. Each test trace converges to zero along \(\omega\). Since the family of tests is finite, the coordinate sets on which their absolute values are less than \(\varepsilon\) have a common member. Its \(d\) coordinate unitaries solve all the requirements in \(P\).

For general \(M\), the factor case will give successful tuples in almost every fiber. The remaining task is to choose them measurably. Use the central direct integral [3] \[M=\int_\Omega^\oplus M_s\,d\nu(s), \qquad \tau(x)=\int_\Omega\tau_s(x(s))\,d\nu(s),\] where \((\Omega,\nu)\) is a probability space and, almost everywhere, \(M_s\) is a \(\mathrm{II}_1\) factor with its normalized trace \(\tau_s\). Standard separable direct-integral theory provides countably many bounded measurable sections \(a_j\) whose values generate \(M_s\) for almost every \(s\). After removing one null set, all these properties and the unitary identities for the finitely many coefficients hold simultaneously.

There is a countable family \((v_j)\) of measurable unitary sections which is dense in each fiber unitary group for \(\norm{\cdot}_{2,s}\). Here is an explicit construction. Enumerate the sections \[\exp(iq(a_1,a_2,\ldots)),\] where \(q\) ranges over self-adjoint \(*\)-polynomials in finitely many variables with coefficients in \(\mathbb Q+i\mathbb Q\). This includes symmetrizations of arbitrary rational \(*\)-polynomials. In a fixed fiber, any unitary is \(\exp(ih)\) for a bounded self-adjoint \(h\). Kaplansky density and norm approximation by rational polynomials approximate \(h\) in \(2\)-norm by the indicated self-adjoint polynomials. For bounded self-adjoint \(a,b\), the integral identity \[e^{ia}-e^{ib} =i\int_0^1 e^{i(1-t)a}(a-b)e^{itb}\,dt\] implies \(\norm{e^{ia}-e^{ib}}_2\leq\norm{a-b}_2\). Thus these sections have the required fiberwise density. Each section is globally bounded, and exponentiation makes its norm one.

Enumerate all ordered \(d\)-tuples drawn from \((v_j)\) as \(\boldsymbol v^{(1)},\boldsymbol v^{(2)},\ldots\). Let \(E_k\) be the set of fibers where candidate \(k\) satisfies every test with absolute trace less than \(\varepsilon/2\). These sets are measurable, since the traces of products of measurable sections are measurable. They cover almost every fiber. Indeed, the factor case gives a successful tuple in each such fiber, and the requirements are open in the product \(2\)-norm topology. For this last assertion, telescoping products of unitaries gives \[\norm{z_1\cdots z_L-z'_1\cdots z'_L}_2 \leq\sum_{j=1}^L\norm{z_j-z'_j}_2, \qquad |\tau_s(x)|\leq\norm{x}_{2,s},\] and inversion is also a \(2\)-norm isometry on unitaries. The countable dense tuples therefore meet the open set of successful choices.

Take the first successful tuple: on the disjoint measurable sets \[E_k\setminus\bigcup_{j<k}E_j\] use \(\boldsymbol v^{(k)}\), and use the identity tuple on the null complement. The resulting sections \(u_1,\ldots,u_d\) are measurable and unitary, so belong to \(M\). Each test has fiber trace of absolute value less than \(\varepsilon/2\) almost everywhere. Integrating gives an absolute global trace at most \(\varepsilon/2<\varepsilon\), as required. ◻

Proof of Theorem 3. Choose a \(2\)-norm dense sequence \((v_j)_{j\geq1}\) in \(\mathcal U(M)\) and put \[\mu_0=\sum_{j\geq1}2^{-j-1}(\delta_{v_j}+\delta_{v_j^{-1}}).\] For \(\ell\geq1\), fix the numerical parameters \[\begin{align*} p_\ell&=2^{-2^\ell}, &p_0&=1-\sum_{\ell\geq1}p_\ell,\\ n_\ell&=p_\ell^{-3/2}=2^{3\cdot2^{\ell-1}}, &K_\ell&=\ell n_\ell,\\ d_\ell&=K_\ell^4. \end{align*}\] Here \(n_\ell\) and \(d_\ell\) are positive integers, and \(p_0>0\) because \(p_\ell\leq4^{-\ell}\). At time \(n_\ell\) we will sample \(\ell\) paths, choosing level \(j\) with probability \(p_j\) at each increment. These parameters make level \(\ell\) appear in every path with high probability while later levels remain unlikely. The much larger label count \(d_\ell\) makes repeated level-\(\ell\) labels unlikely across the \(K_\ell\) increments. We construct finite symmetric measures \[\mu_\ell=\frac1{2d_\ell} \sum_{j=1}^{d_\ell}(\delta_{u_{\ell,j}} +\delta_{u_{\ell,j}^{-1}})\] successively, and finally set \[\mu=p_0\mu_0+\sum_{\ell\geq1}p_\ell\mu_\ell.\]

At stage \(\ell\), choose a finite symmetric set \(F_\ell\) of base atoms with \(\mu_0(F_\ell)>1-K_\ell^{-2}\). The coefficient alphabet consists of \(F_\ell\) and all \(u_{j,h}^{\pm1}\) from stages \(j<\ell\); it is finite. Introduce \(d_\ell\) unknown unitary labels. Consider every ordered list of \(\ell\) paths of length \(n_\ell\) in this alphabet and the signed unknown labels such that:

1. each path contains at least one unknown label;

2. no unknown label occurs twice anywhere in the entire list, irrespective of sign.

For each such list and every nonempty reduced word of length at most \(\ell\) on \(\ell\) generators, require its evaluated trace to have absolute value less than \(1/\ell\). There are finitely many lists and finitely many words: all alphabet sizes, path lengths, and word lengths have been fixed before making this choice. Designating any one unknown occurrence in each path puts every test under Lemma 6. Apply that lemma to choose \(u_{\ell,1},\ldots,u_{\ell,d_\ell}\) satisfying all these requirements simultaneously. This completes the induction.

We estimate probabilities on a labeled sampling space. An increment first chooses a level \(j\) with probability \(p_j\). At level zero it chooses an atom with law \(\mu_0\); at level \(j\geq1\) it chooses one of \(d_j\) labels uniformly and an independent uniform sign. Its value is then the corresponding unitary. This description gives law \(\mu\) even when distinct labels happen to represent the same element of \(M\).

Sample \(\ell\) independent paths of length \(n_\ell\), involving \(K_\ell\) increments in all. Let \(\mathcal B_\ell\) be the union of the following four events: a level greater than \(\ell\) appears; a level-zero increment lies outside \(F_\ell\); a level-\(\ell\) label is repeated; or some path has no level-\(\ell\) increment. Union bounds give \[\begin{align*} \Pr(\mathcal B_\ell) &\leq K_\ell\sum_{j>\ell}p_j +K_\ell p_0\mu_0(\mathcal U(M)\setminus F_\ell)\\ &\quad+\frac{\binom{K_\ell}{2}}{d_\ell} +\ell(1-p_\ell)^{n_\ell}. \tag{4}\end{align*}\] For the collision term, any specified pair of increments has probability \(p_\ell^2/d_\ell\leq1/d_\ell\) of using the same level-\(\ell\) label; the signs do not matter. For the last term, a specified path misses that level with probability \((1-p_\ell)^{n_\ell}\).

Every term on the right of (4) tends to zero. Indeed, \(p_{\ell+1}=p_\ell^2\) and \(p_\ell\leq1/4\), so \[\sum_{j>\ell}p_j \leq\frac{p_\ell^2}{1-p_\ell^2} \leq\frac{16}{15}p_\ell^2.\] Consequently, the four terms admit the respective bounds \[\begin{align*} \frac{16}{15}\ell\sqrt{p_\ell},\qquad K_\ell^{-1},\qquad \frac{1}{2K_\ell^2},\qquad \ell\exp(-p_\ell^{-1/2}), \end{align*}\] all converging to zero.

Outside \(\mathcal B_\ell\), the sampled paths form exactly one of the lists tested at stage \(\ell\). All their earlier-level factors belong to the coefficient alphabet, each path has a level-\(\ell\) label, and all occurrences of those labels are distinct. Thus, simultaneously for every \(w\in\mathbb F_\ell\) with \(1\leq|w|\leq\ell\), the trace of the evaluated word has absolute value less than \(1/\ell\).

Now fix \(r\), a nonempty finite \(\mathcal W\), and \(\eta>0\). For \(\ell\geq\max\{r,\max_{w\in\mathcal W}|w|\}\), append independent paths to the given \(r\) paths to obtain \(\ell\) paths. On the preceding good event every word in \(\mathcal W\) has absolute trace below \(1/\ell\). Once \(1/\ell<\eta\), the probability in the theorem is at most \(\Pr(\mathcal B_\ell)\) and therefore tends to zero.

The measure \(\mu\) is symmetric and countably supported by construction. Its base component has positive weight, so each \(v_j\) is a positive-mass atom. Since bounded \(2\)-norm convergence implies strong convergence in the tracial representation, the von Neumann algebra generated by these atoms contains every unitary and hence is \(M\). Finally, the identity group word always has trace one, and every \(*\)-monomial in unitaries reduces to a group word. The asserted convergence therefore describes all joint \(*\)-moments. ◻

Proof. If (6) fails for some \(r,\varepsilon\), choose \(x_j\) with \(\norm{x_j}\le r\), \(\norm{x_j}_2<1/j\), and \(\norm{R(x_j)}_2\ge\varepsilon\). This contradicts (5); the converse is immediate. Applying the uniform assertion to differences gives uniform \(2\)-norm continuity on each fixed operator norm ball.

Use the tracial ultrapower and its ideal \(\mathcal I_\omega\) from (3), now for \((N,\rho)\). Its tracial norm is \[\norm{[(x_j)]_\omega}_2=\lim_{j\to\omega}\norm{x_j}_2.\] By (6), the coordinate map induced by \(R\) preserves \(\mathcal I_\omega\). Applying this observation to the difference of two bounded representatives proves that (7) is well-defined. The quotient norm gives \[\norm{R^\omega(X)} \le \norm{R}\inf_{[(x_j)]_\omega=X}\sup_j\norm{x_j} =\norm{R}\norm{X}.\]

To transfer the modulus, fix \(r<\infty\) and \(\varepsilon>0\), and choose \(\delta\) in (6) for the radius \(r+1\) and the output tolerance \(\varepsilon/2\). If \(\norm{X}\le r\), the quotient norm allows a representative with \(\sup_j\norm{x_j}\le r+1\). When \(\norm{X}_2<\delta/2\), the set \[\{j:\norm{x_j}_2<\delta\}\] belongs to \(\omega\). Consequently \[\norm{R^\omega(X)}_2 =\lim_{j\to\omega}\norm{R(x_j)}_2 \le\varepsilon/2<\varepsilon.\] Thus \(\delta/2\) is a valid input tolerance for \(R^\omega\) on the radius-\(r\) ball. The proof used no separability of \(N\) and applies again with \(N^\omega\) in its place, proving the last assertion. ◻

Proof. Write \(C=\norm{T}\) and \(H=L^2(M,\tau)\). Expectations in (8) are norm-convergent sums, since \(\mu\) is countably supported and \(\norm{T(xU)U^*}\le C\norm{x}\). For \(g\in\mathcal U(M)\) put \[D(g)=T(g)g^*.\] Let \(U_1,U_2,\ldots\) be independent with law \(\mu\), and set \(W_0=1\), \(W_n=U_1\cdots U_n\), and \(\mathcal F_n=\sigma(U_1,\ldots,U_n)\). The \(H\)-valued random variables \(Z_n=D(W_n)\) have countable range and satisfy \(\norm{Z_n}_2\le\norm{Z_n}\le C\). Moreover, \[\begin{align*} \mathbb E[Z_{n+1}\mid\mathcal F_n] &=\mathbb E\bigl[T(W_nU_{n+1})U_{n+1}^*W_n^* \mid\mathcal F_n\bigr] \\ &=T(W_n)W_n^*=Z_n. \end{align*}\] Thus \((Z_n)\) is a martingale in \(L^2(\Omega;H)\). Its increments are orthogonal, and \[\norm{Z_0}_2^2+ \sum_{j=0}^{n-1}\mathbb E\norm{Z_{j+1}-Z_j}_2^2 =\mathbb E\norm{Z_n}_2^2\le C^2.\] It follows that \(Z_n\) converges in \(L^2(\Omega;H)\) to a random variable \(Z_\infty\), and continuity of conditional expectation gives \(Z_n=\mathbb E[Z_\infty\mid\mathcal F_n]\). In particular, with \[ e_k=\mathbb E\norm{Z_\infty-Z_k}_2^2\longrightarrow0, \tag{10}\] orthogonality gives the uniform Cauchy estimate \[ \mathbb E\norm{Z_n-Z_k}_2^2=e_k-e_n\le e_k \qquad(n\ge k). \tag{11}\]

The operator norm ball \(K_C=\{x\in M:\norm{x}\le C\}\) is closed in \(H\). Indeed, from a \(2\)-norm-convergent sequence in \(K_C\), ultraweak compactness provides an ultraweakly convergent subnet with limit in \(K_C\). Testing against \(\tau(y^*\,\cdot)\), for \(y\in M\), identifies this limit with the \(H\)-limit, since \(M\) is dense in \(H\). A subsequence of \(Z_n\) converges almost surely in \(H\), so \(Z_\infty\in K_C\) almost surely.

If \(Z_\infty=c\) almost surely, conditional expectation gives \(Z_0=Z_1=c\) almost surely. Hence \(c=T(1)\) and \(D(u)=c\) at every atom \(u\) of \(\mu\). These atoms contain a \(2\)-norm-dense subset of \(\mathcal U(M)\) by Theorem 3. For any \(u\in\mathcal U(M)\), choose atoms \(u_j\) with \(\norm{u_j-u}_2\to0\). Bounded-ball continuity gives \(T(u_j)\to T(u)\) in \(2\)-norm, while \(\norm{cu_j-cu}_2\le\norm{c}\norm{u_j-u}_2\to0\). Since \(T(u_j)=cu_j\), we obtain \(T(u)=cu\). Every element of a unital complex \(C^*\)-algebra is a finite linear combination of unitaries, proving (9) in this case.

Suppose, for a contradiction, that \(Z_\infty\) is not deterministic. Its law is a Borel probability measure on the separable Hilbert space \(H\), so its support contains two distinct points \(a,b\). Because \(K_C\) is closed and has full measure, \(a,b\in K_C\subset M\).

Forward and reverse endpoints. We claim that, for every \(\delta>0\), there are numbers \(\eta_\delta>0\) and \(N_\delta\) such that, whenever \(n\ge N_\delta\) and \(G\) has law \(\mu^{*n}\), \[ \Pr\bigl(\norm{D(G)-a}_2<\delta, \ \norm{D(G^{-1})-b}_2<\delta\bigr) \ge\eta_\delta. \tag{12}\] Realize \(G=U_1\cdots U_n\), let \(h=\lfloor n/2\rfloor\), and put \[A_n=D(U_1\cdots U_h),\qquad B_n=D(U_n^{-1}\cdots U_{n-h+1}^{-1}).\] The two variables use disjoint sets of increments and are independent. Symmetry of \(\mu\) implies that both have the law of \(Z_h\). Apply (11) to the forward string and separately to the reversed inverse string, which is also a string of independent increments with law \(\mu\). It gives \[ \mathbb E\norm{D(G)-A_n}_2^2\le e_h, \qquad \mathbb E\norm{D(G^{-1})-B_n}_2^2\le e_h. \tag{13}\] For fixed \(\delta\), the support property gives \[p_a=\Pr(\norm{Z_\infty-a}_2<\delta/3)>0, \qquad p_b=\Pr(\norm{Z_\infty-b}_2<\delta/3)>0.\] Mean-square convergence implies, for all sufficiently large \(h\), \[\Pr(\norm{Z_h-a}_2<2\delta/3)\ge p_a/2, \qquad \Pr(\norm{Z_h-b}_2<2\delta/3)\ge p_b/2.\] Independence of \(A_n,B_n\), the triangle inequality, and Markov’s inequality therefore bound the probability in (12) from below by \[\frac{p_ap_b}{4}-\frac{18e_h}{\delta^2}.\] For large \(n\) this is at least \(p_ap_b/8\), proving the claim. The independence used here is that of the disjoint prefixes \(A_n,B_n\); the variables \(G\) and \(G^{-1}\) remain dependent.

Concatenation of independent blocks. Let \(G_1,\ldots,G_m\) be independent with law \(\mu^{*n}\), and consider \[v(G)=G_{i_1}^{\sigma_1}\cdots G_{i_t}^{\sigma_t}, \qquad \sigma_j\in\{1,-1\},\] where \(i_1,\ldots,i_t\) are pairwise distinct. Each inverse block can be represented by reversing and inverting its own increments. By symmetry, concatenating these distinct signed blocks is a length-\(tn\) walk with independent increments of law \(\mu\). In particular, \[\bigl(D(v(G)),D(G_{i_1}^{\sigma_1})\bigr) \quad\hbox{has the law of}\quad (Z_{tn},Z_n).\] Consequently \[ \mathbb E\norm{D(v(G))-D(G_{i_1}^{\sigma_1})}_2^2\le e_n. \tag{14}\] The restriction is on unsigned indices: using both \(G_i\) and \(G_i^{-1}\) would reuse a block and would not justify this estimate.

Finite requirements and diagonal selection. Fix an integer \(m\ge1\) first, and use tolerance \(1/m\) in (12). For \(m\) independent length-\(n\) products, let \(E_{m,n}\) be the event \[\norm{D(G_i)-a}_2<1/m,\qquad \norm{D(G_i^{-1})-b}_2<1/m \quad(1\le i\le m).\] Independence between the paths gives numbers \(c_m>0\) and \(N_m\) such that \[ \Pr(E_{m,n})\ge c_m\qquad(n\ge N_m). \tag{15}\] For example, one may take \(c_m=(\eta_{1/m})^m\).

Let \(\mathcal Q_m\) be the finite set of nonidentity reduced words of length at most \(m\) on the symbols \(x_1,\ldots,x_m\). Let \(\mathcal V_m\) be the finite set of all nonempty signed words on these symbols with pairwise distinct unsigned indices. Along the times \(n_\ell\) in Theorem 3, the probability that any of the requirements \[\begin{align*} |\tau(q(G_1,\ldots,G_m))|&<1/m &&(q\in\mathcal Q_m),\tag{16}\\ \norm{D(v(G))-D(G_{i_1}^{\sigma_1})}_2&<1/m &&\bigl(v=x_{i_1}^{\sigma_1}\cdots x_{i_t}^{\sigma_t} \in\mathcal V_m\bigr) \tag{17}\end{align*}\] fails tends to zero. For the first line, this is a finite union of the moment convergence statements in Theorem 3. For the second line, the union bound and (14) give the upper bound \(|\mathcal V_m|m^2e_{n_\ell}\).

Choose \(\ell_m>\ell_{m-1}\), after fixing \(m\), so large that \(n_{\ell_m}\ge N_m\) and this combined failure probability is less than \(c_m/2\). Then the intersection of \(E_{m,n_{\ell_m}}\) with all the requirements (16)–(17) has probability at least \(c_m/2>0\). Choose a deterministic tuple \[(u_1^{(m)},\ldots,u_m^{(m)})\in\mathcal U(M)^m\] from that intersection. The constants \(c_m\) may tend to zero as fast as they please: the choice of \(\ell_m\) is made separately for each fixed \(m\), after the positive lower bound has been obtained.

Write \(a_1=a\) and \(a_{-1}=b\). The endpoint conditions and (17) imply, for every \(v=x_{i_1}^{\sigma_1}\cdots x_{i_t}^{\sigma_t}\in\mathcal V_m\), \[ \norm{T(v(u^{(m)}))-a_{\sigma_1}v(u^{(m)})}_2 =\norm{D(v(u^{(m)}))-a_{\sigma_1}}_2<2/m. \tag{18}\] The equality uses \(T(v)=D(v)v\) and the \(2\)-norm isometry of right multiplication by the unitary \(v\).

The exact family in the ultrapower. Fix a free ultrafilter \(\omega_1\) on the tuple index \(m\). Set \(u_i^{(m)}=1\) when \(m<i\), and define \[N=M^{\omega_1},\qquad g_i=[(u_i^{(m)})_{m\ge1}]_{\omega_1}\quad(i\ge1), \qquad R=T^{\omega_1}.\] Lemma 9 makes \(R\) a bounded linear map on \(N\), with \(\norm{R}\le C\), and supplies its bounded-ball \(2\)-continuity. Each \(g_i\) is unitary. Every fixed nonidentity reduced word \(q\) lies in \(\mathcal Q_m\) for all sufficiently large \(m\), so (16) gives \[ \tau^{\omega_1}(q(g_1,g_2,\ldots))=0. \tag{19}\] These are precisely the joint moments of a countable free Haar family: powers of one generator are centered, and expansion of alternating centered Laurent polynomials in different generators reduces to nonidentity words. The corresponding freeness of their generated von Neumann algebras follows by bounded \(2\)-norm approximation.

Identify \(a,b\in M\) with their constant images in \(N\). For every nonempty word \(w=g_{i_1}^{\sigma_1}\cdots g_{i_t}^{\sigma_t}\) with pairwise distinct unsigned indices, (18) holds for all sufficiently large tuple indices. Hence it becomes the exact identity \[ R(w)= \begin{cases} aw,&\sigma_1=1,\\ bw,&\sigma_1=-1. \end{cases} \tag{20}\] These are all the hypotheses of Theorem 11, applied to \((N,\tau^{\omega_1})\) and \(R\). That theorem yields \(a=b\) in \(N\). The constant embedding preserves the faithful trace, so this also means \(a=b\) in \(M\), contrary to their choice. Thus \(Z_\infty\) is deterministic, and the earlier argument proves (9). ◻

Proof. Write \(L=\sum_{|w|=q}c_w\lambda(w)\). On the standard basis of \(\ell^2(\mathbb F_r)\), separate each product \(wx\) according to its exact number \(t\in\{0,\ldots,q\}\) of cancellations at the joining boundary. This gives \(L=\sum_{t=0}^q L_t\) on finitely supported vectors. Fix \(t\). Such a product has the form \[w=uv,\qquad x=v^{-1}h,\qquad wx=uh, \qquad |u|=q-t,\quad |v|=t,\] where all three displayed concatenations \(uv\), \(v^{-1}h\), and \(uh\) are reduced. Conversely, these conditions describe exactly the products with \(t\) cancellations, allowing any of \(u,v,h\) to be empty when its required length permits this. In particular, \(h\) may be empty when the input is exhausted.

For this fixed \(t\), an input basis word of length at least \(t\) has a unique prefix of length \(t\) and hence a unique suffix \(h\). An output basis word of length at least \(q-t\) likewise determines \(h\) uniquely by removing its prefix of length \(q-t\). Thus the input spaces indexed by different \(h\) are orthogonal, as are the corresponding output spaces. Relative to these two decompositions, \(L_t\) is a direct sum of matrices \(B_h\), with columns indexed by \(v\) and rows by \(u\). An entry is \(c_{uv}\) when the three reduction conditions hold and is zero otherwise. The fixed cutting length makes \((u,v)\mapsto uv\) injective within each matrix. Therefore \[\norm{B_h}\leq\norm{B_h}_{\mathrm{HS}} \leq\left(\sum_{|w|=q}|c_w|^2\right)^{1/2}.\] Taking the direct-sum norm bounds \(L_t\) by the same quantity. Summing over the \(q+1\) possible values of \(t\) proves (23).

For the norm transfer, recall that if \(A\geq0\) belongs to a von Neumann algebra with faithful tracial state \(\rho\), then \[ \norm{A}=\lim_{n\to\infty}\rho(A^n)^{1/n}. \tag{24}\] The upper bound is immediate. If \(0<c<\norm{A}\), the spectral projection \(e=1_{(c,\infty)}(A)\) is nonzero and has positive trace, so \(\rho(A^n)\geq c^n\rho(e)\). Taking roots and then letting \(c\uparrow\norm{A}\) proves the formula. For a word polynomial \(p\), the free Haar moments give, for every \(n\geq1\), \[\tau\bigl((p(g)^*p(g))^n\bigr) =\tau_{\mathbb F_r}\bigl((p(\lambda)^*p(\lambda))^n\bigr).\] Both traces are faithful; here \(\tau_{\mathbb F_r}\) is the canonical trace on the free group von Neumann algebra. Applying (24) gives \(\norm{p(g)}=\norm{p(\lambda)}\). The asserted \(2\)-norm formula follows from orthonormality of reduced words. ◻

Proof. Fix this product, and let \(L=\sum_{j=1}^k(2d_j+1)\) be its degree. Every term in the expansion of one block \(B_d(s_r)\) has the odd alternating sign pattern \[+,-,+,-,\ldots,+.\] Deleting adjacent inverse letters deletes two consecutive opposite signs. The remaining pattern is still alternating, begins and ends positively, and is nonempty: the excess of positive over negative signs remains one. Hence the free reduction of each block is a nonempty word with positive endpoint signs. After reducing the blocks internally, their boundaries join two positive letters and admit no further cancellation. Every reduced word in \(P(s_r)\) is therefore nonempty and begins and ends positively. Taking adjoints reverses the block order and all signs, so the corresponding assertion for \(P(s_r)^*\) has negative endpoint signs.

We next bound the coefficient of every possible reduced word, including words much shorter than \(L\). Write \[P(s_r)=\sum_{q=1}^L\sum_{|w|=q}c_w(r)w(g).\] For a fixed reduced word \(w\) of length \(q\), record a free reduction of an original term by pairing its original positions whenever their letters cancel. For definiteness, the reduction may be performed by reading from left to right and canceling against the last surviving letter whenever possible. The recorded pairs form a noncrossing partial matching of the \(L\) positions: there are \((L-q)/2\) pairs, and the unmatched positions spell \(w\) in order. All nested and newly adjacent cancellations are included in this record.

There are at most \(3^L\) noncrossing partial matchings of \(L\) positions. Indeed, mark each position as an opening endpoint, a closing endpoint, or unmatched; noncrossing forces a closing endpoint to pair with the last unpaired opening endpoint, so this encoding is injective. We may count every such partial matching, including ones not realized by a reduction. For a fixed pattern and fixed \(w\), the labels at the unmatched positions are prescribed. Each canceled pair has equal unsigned labels and opposite signs and hence has at most \(r\) label choices. Coincidences between different pairs or with unmatched positions can only reduce the number of choices. Each original term has coefficient \(r^{-L/2}\). Consequently, for every \(q\) and \(w\), \[ |c_w(r)|\leq C_L r^{(L-q)/2}r^{-L/2} =C_Lr^{-q/2},\qquad C_L=3^L. \tag{28}\] An impossible parity or reduction pattern contributes zero. The same bound applies to the adjoint polynomial.

Let \(E_{r,q}\) be the sum of those length-\(q\) terms whose surviving reduced word has a repeated unsigned index. Original repetitions which disappear in the reduction remain included in the coefficients of the retained words. For \(q=1\) this sum is zero. For \(q\geq2\) the number of possible words in the sum is at most \[2^q\binom q2 r^{q-1}:\] choose a pair of equal-index positions, choose their common label and the other \(q-2\) labels, and choose all signs. Counting nonreduced strings here only enlarges the bound. By orthonormality and (28), \[ \norm{E_{r,q}}_2^2 \leq C_L^2 2^q\binom q2 r^{-1}\qquad(2\leq q\leq L). \tag{29}\] Lemma 12 now yields the aggregate operator norm estimate \[ E_r=\sum_{q=1}^L E_{r,q},\qquad \norm{E_r}\leq\sum_{q=2}^L(q+1)\norm{E_{r,q}}_2 =O_L(r^{-1/2})\longrightarrow0. \tag{30}\] There is no scalar term, by the block reduction argument.

Put \(Q_r=P(s_r)-E_r\). Each word in this finite sum has pairwise distinct unsigned indices and begins positively. Thus (22) and linearity give \(R(Q_r)=aQ_r\). Moreover, \(\norm{P(s_r)}\leq2^L\), so (30) also proves that the sequence \((Q_r)\) is bounded. In \(N\) we have \[\norm{R(P(s_r))-aP(s_r)} \leq(\norm{R}+\norm{a})\norm{E_r}\longrightarrow0.\] Passing to \(\widehat N\) proves the first identity in (27). For the adjoint, the retained sum is \(Q_r^*\): its reduced words have distinct unsigned indices and begin negatively. Therefore \(R(Q_r^*)=bQ_r^*\), and the identical estimate with \(b\) proves the second identity. ◻

Proof. We first show that \(s\) has zero kernel, which makes its polar part unitary in the finite algebra. We then transfer the block identities to powers of that unitary. Set \(A=s^*s\), so \(0\leq A\leq4\). To determine its kernel, we compute its moments. For \(d\geq1\), the quantity \(\tau((s_r^*s_r)^d)\) is \(r^{-d}\) times the number of assignments of labels in \(\{1,\ldots,r\}\) to the alternating \(2d\) positions \[-,+,-,+,\ldots,-,+\] whose word reduces to the identity. An identity reduction pairs all positions noncrossingly, so it uses at most \(d\) unsigned labels. Assignments using at most \(d-1\) labels number \(O_d(r^{d-1})\): for each number of labels, choose an ordered list of those labels and then assign a member of that list to each of the \(2d\) positions.

If exactly \(d\) labels occur in an identity word, every label occurs twice. Its two occurrences must be paired in any cancellation matching, so the matching is uniquely determined by the labeling. Conversely, in a noncrossing perfect matching, every pair has an even number of positions inside it, since those positions must be matched among themselves. The endpoints therefore have opposite parities, and hence opposite signs in our alternating string. Assigning distinct labels to its \(d\) pairs gives an identity word: erase innermost pairs successively. It follows that the number of leading assignments is exactly \[C_d(r)_d,\qquad (r)_d=r(r-1)\cdots(r-d+1),\] where \(C_d\) is the number of noncrossing perfect matchings on \(2d\) positions. Choosing the partner of the first position gives \(C_0=1\) and \(C_d=\sum_{j=0}^{d-1}C_jC_{d-1-j}\). The generating function thus satisfies \(C(z)=1+zC(z)^2\); its solution with constant term one gives \[C_d=\frac1{d+1}\binom{2d}{d}.\] Consequently, \[ \widehat\tau(A^d) =\lim_{r\to\omega_2}\tau((s_r^*s_r)^d) =\lim_{r\to\infty}r^{-d} \bigl(C_d(r)_d+O_d(r^{d-1})\bigr) =C_d. \tag{32}\] The zeroth moment is also \(C_0=1\).

Consider the measure \[ d\nu(x)=\frac1{2\pi}\sqrt{\frac{4-x}{x}}\, 1_{(0,4)}(x)\,dx. \tag{33}\] For every integer \(d\geq0\), substitution \(x=4t\) and the beta integral give \[\begin{split} \int_0^4 x^d\,d\nu(x) &=\frac{4^{d+1}}{2\pi} \int_0^1 t^{d-1/2}(1-t)^{1/2}\,dt\\ &=\frac{4^{d+1}}{2\pi} \frac{\Gamma(d+\tfrac12)\Gamma(\tfrac32)}{\Gamma(d+2)} =\frac{(2d)!}{d!(d+1)!}=C_d. \end{split}\] In particular, \(\nu\) is a probability measure. Both \(\nu\) and the spectral distribution of \(A\) are supported in \([0,4]\). Equality of all their moments implies equality on polynomials and then on all continuous functions by uniform polynomial approximation. They are therefore the same measure. Since (33) has no atom at zero, \[\widehat\tau(1_{\{0\}}(A))=0, \qquad 1_{\{0\}}(A)=0\] by faithfulness. In the polar decomposition \(s=VA^{1/2}\) this gives \(V^*V=1\). Traciality and faithfulness then give \[\widehat\tau(1-VV^*)=1-\widehat\tau(V^*V)=0, \qquad VV^*=1.\] Thus \(V\) is a unitary.

It remains to transfer the identities. For a fixed regularization parameter we will use polynomial approximation in operator norm; we will then remove the regularization using bounded-ball \(2\)-continuity. For \(\varepsilon>0\) put \[x_\varepsilon=s(A+\varepsilon1)^{-1/2} =V\left(\frac{A}{A+\varepsilon1}\right)^{1/2}.\] Fix \(k\geq1\). A polynomial \(p(t)=\sum_{d=0}^D c_dt^d\) has the literal ordered expansion \[ [sp(A)]^k =\sum_{d_1,\ldots,d_k=0}^D c_{d_1}\cdots c_{d_k} [sA^{d_1}]\cdots[sA^{d_k}]. \tag{34}\] Every summand is a product covered by Lemma 13. That lemma and complex linearity yield \[S([sp(A)]^k)=a[sp(A)]^k, \qquad S(([sp(A)]^k)^*)=b([sp(A)]^k)^*.\] For the second identity one takes the adjoint of the expansion (34), conjugating its scalar coefficients and reversing the block order, and applies the separate adjoint identity in Lemma 13. Choose polynomials converging uniformly on \([0,4]\) to \((t+\varepsilon)^{-1/2}\). Functional calculus and boundedness of \(S\) pass the last two identities to the norm limit, giving \[ S(x_\varepsilon^k)=ax_\varepsilon^k, \qquad S((x_\varepsilon^k)^*)=b(x_\varepsilon^k)^*. \tag{35}\]

The \(x_\varepsilon\) are contractions, and their distance to \(V\) satisfies \[\norm{x_\varepsilon-V}_2^2 =\int_0^4 \left(\sqrt{\frac{x}{x+\varepsilon}}-1\right)^2\,d\nu(x) \longrightarrow0\] by dominated convergence and \(\nu(\{0\})=0\). Multiplication satisfies \(\norm{xyz}_2\leq\norm{x}\norm{y}_2\norm{z}\) in a tracial algebra. The noncommutative telescoping identity \[x_\varepsilon^k-V^k =\sum_{j=0}^{k-1}x_\varepsilon^j (x_\varepsilon-V)V^{k-1-j}\] therefore shows that \(\norm{x_\varepsilon^k-V^k}_2\leq k\norm{x_\varepsilon-V}_2\to0\); the same holds for their adjoints. All these powers are contractions. Using the bounded-ball \(2\)-norm continuity of \(S\) in (35), along for example \(\varepsilon=1/n\), proves (31). ◻

Proof. We use analytic polynomials whose imaginary parts remain uniformly bounded on the unit circle while their real parts grow near \(1\). Rotating these polynomials will bound \(a-b\) on each short spectral arc; the trace will then combine the compression estimates without a loss depending on the number of arcs. Write \(\mathbb T=\{z\in\mathbb C:|z|=1\}\) and \[H_m(z)=\sum_{k=1}^m\frac{z^k}{k},\qquad h_m=\sum_{k=1}^m\frac1k.\] We first verify the uniform sine-sum bound \[ \sup_{m\geq1}\sup_{\theta\in\mathbb R} \left|\sum_{k=1}^m\frac{\sin(k\theta)}k\right| \leq B,\qquad B=1+\pi. \tag{36}\] By periodicity and oddness it suffices to use the angular distance \(t\in[0,\pi]\) from \(2\pi\mathbb Z\). At \(t=0\) the sum is zero. For \(t>0\) put \(K=\lfloor1/t\rfloor\). The part with \(k\leq\min(K,m)\) has absolute sum at most \(Kt\leq1\), since \(|\sin(kt)|\leq kt\). Geometric summation gives, for every interval of integers \(p\leq q\), \[\left|\sum_{k=p}^q\sin(kt)\right| \leq\left|\sum_{k=p}^q e^{ikt}\right| \leq\frac1{\sin(t/2)}\leq\frac\pi t.\] If \(m\geq K+1\), put \(p=K+1\) and \(A_j=\sum_{k=p}^j\sin(kt)\). Abel summation gives \[\sum_{k=p}^m\frac{\sin(kt)}k =\frac{A_m}{m} +\sum_{j=p}^{m-1}A_j\left(\frac1j-\frac1{j+1}\right).\] Its absolute value is at most \(\pi/(tp)\leq\pi\), including when \(K=0\). This proves (36).

For \(\lambda\in\mathbb T\), write \[H_m(\lambda V)=X_{m,\lambda}+iY_{m,\lambda}, \qquad X_{m,\lambda}=X_{m,\lambda}^*,\quad Y_{m,\lambda}=Y_{m,\lambda}^*.\] Continuous functional calculus and (36) give \(\norm{Y_{m,\lambda}}\leq B\). The identities for positive and negative powers imply \[\begin{split} S\bigl(H_m(\lambda V)-H_m(\lambda V)^*\bigr) &=aH_m(\lambda V)-bH_m(\lambda V)^*\\ &=(a-b)X_{m,\lambda}+i(a+b)Y_{m,\lambda}. \end{split}\] Since the argument of \(S\) has norm at most \(2B\), we obtain \[ \norm{(a-b)X_{m,\lambda}}\leq C, \qquad C=B(2\norm{S}+\norm{a}+\norm{b}), \tag{37}\] uniformly in \(m\) and \(\lambda\).

Fix \(m\) and any half-open arc \(I\subset\mathbb T\) of angular length at most \(1/m\), and let \(e=1_I(V)\). Choose \(\lambda\) to rotate the midpoint of \(I\) to \(1\). On the rotated arc the argument \(\theta\) satisfies \(|\theta|\leq1/m\), so \[\operatorname{Re}H_m(e^{i\theta}) =\sum_{k=1}^m\frac{\cos(k\theta)}k \geq\cos(1)h_m.\] Thus, for \(X=X_{m,\lambda}\), Borel functional calculus gives an element \(F\in W^*(V)\) equal to \(1/X\) on \(I\) and to zero off \(I\), with \[XF=e,\qquad \norm{F}\leq\frac1{\cos(1)h_m}.\] Writing \(c=a-b\) and multiplying (37) on the right by \(F\) gives \[ \norm{ce}=\norm{cXF} \leq\frac{C}{\cos(1)h_m}=:\delta_m. \tag{38}\] In the corner \(e\mathcal A e\), this says \(ec^*ce\leq\delta_m^2 e\).

Partition \(\mathbb T\) into finitely many disjoint half-open arcs of angular length at most \(1/m\), and denote their spectral projections by \(e_1,\ldots,e_{J_m}\). They are orthogonal and sum to \(1\). Traciality and (38) imply \[\begin{split} \norm{c}_2^2 &=\sum_{j=1}^{J_m}\rho(c^*ce_j) =\sum_{j=1}^{J_m}\rho(e_jc^*ce_j)\\ &\leq\delta_m^2\sum_{j=1}^{J_m}\rho(e_j) =\delta_m^2\longrightarrow0, \end{split}\] because \(h_m\geq\log(m+1)\to\infty\). Each arc estimate is weighted by the trace of its own projection; the number of arcs introduces no factor. Faithfulness gives \(c=0\), as required. The right multiplication by \(F\) and the tracial compression identity apply to arbitrary \(a,b\); the argument imposes no commutation condition and uses no particular spectral distribution of \(V\). ◻

Proof of Theorem 11. Construct \(s\in\widehat N\) and \(S=R^{\omega_2}\) as in (26). Lemma 13 propagates the first-letter identities to every ordered block product. Lemma 14 then provides a unitary \(V\) satisfying the positive and negative power identities. Lemma 15 gives equality of the constant images of \(a\) and \(b\) in \(\widehat N\). The constant embedding is trace preserving and faithful, so \(a=b\) in \(N\). ◻

Proof. We argue by contradiction. A failure of continuity will give elements with mutually orthogonal two-sided supports whose images stay uniformly away from zero in \(2\)-norm. Random signs will then contradict the boundedness of \(B\).

If the first assertion fails, there are \(R>0\), \(\eta>0\), and a sequence \((x_j)\) such that \[\norm{x_j}\le R,\qquad \norm{x_j}_2\longrightarrow0, \qquad \norm{B(x_j)}_2\ge\eta.\] Choose \(\alpha>0\) with \(\norm{B}\alpha<\eta/4\). In the polar decomposition \(x_j=v_j|x_j|\), set \[r_j=1_{(\alpha,\infty)}(|x_j|),\qquad y_j=v_j|x_j|r_j,\qquad \ell_j=v_jr_jv_j^*,\qquad e_j=r_j\vee\ell_j.\] Here \(r_j\) and \(\ell_j\) are the right and left support projections of \(y_j\). In particular, \[y_j=e_jy_je_j,\quad \norm{y_j}\le R,\quad \norm{x_j-y_j}\le\alpha,\quad \norm{B(y_j)}_2\ge 3\eta/4.\] The trace and the spectral truncation give \[\tau(e_j)\le\tau(r_j)+\tau(\ell_j) =2\tau(r_j)\le2\alpha^{-2}\norm{x_j}_2^2.\] Pass to a subsequence for which \(\sum_j\tau(e_j)<\infty\), and put \(p_j=\bigvee_{h\ge j}e_h\). Then \(p_j\) decreases to zero: indeed, \(\tau(p_j)\le\sum_{h\ge j}\tau(e_h)\to0\), and \(\tau\) is faithful and normal.

For fixed \(j\) and \(h>j\), the compressions \[(p_j-p_h)y_j(p_j-p_h)\] converge ultraweakly to \(y_j\) as \(h\to\infty\). Applying \(B\) preserves this convergence. The \(2\)-norm is ultraweakly lower semicontinuous, as is seen from \[\norm{y}_2= \sup\{ |\tau(b^*y)|:b\in M,\ \norm{b}_2\le1\}.\] Starting with \(j_1=1\), we may therefore choose successively \(j_{s+1}>j_s\) so that \[t_s=(p_{j_s}-p_{j_{s+1}})y_{j_s}(p_{j_s}-p_{j_{s+1}}), \qquad \norm{B(t_s)}_2\ge\eta/2.\] The \(t_s\) have pairwise orthogonal two-sided supports and \(\norm{t_s}\le R\). Hence \(\norm{\sum_{s=1}^N\epsilon_st_s}\le R\) for every choice of signs \(\epsilon_s\in\{-1,1\}\). Taking independent uniform signs, Hilbert space orthogonality of these scalar random variables yields \[\frac{N\eta^2}{4} \le\sum_{s=1}^N\norm{B(t_s)}_2^2 =\mathbb E\norm{B\Bigl(\sum_{s=1}^N\epsilon_st_s\Bigr)}_2^2 \le\norm{B}^2R^2,\] which is impossible for large \(N\). This proves the asserted modulus. Applying it to differences, which lie in the ball of radius \(2R\), gives continuity on the ball of radius \(R\). ◻

Proof. We first construct an averaged cochain that is a right module map in its last input. Evaluation at \(1\) will give a \((k-1)\)-cochain, and averaging the original equation \(df=0\) will identify its differential.

Choose the symmetric countably supported measure \(\mu\) for which Theorem 10 applies, and write \(\mu=\sum_{s\ge1}p_s\delta_{u_s}\). For a bounded \(k\)-cochain \(\Phi\), define \[ (P\Phi)(\mathbf a,x) =\sum_{s\ge1}p_s\Phi(\mathbf a,xu_s)u_s^*, \qquad \mathbf a=(a_1,\ldots,a_{k-1}). \tag{40}\] The sum converges in operator norm, and \(P\) is a contraction for the cochain norm. If \(W_j=U_1\cdots U_j\), where the \(U_i\) are independent with law \(\mu\), and \(W_0=1\), induction gives \[(P^j f)(\mathbf a,x) =\mathbb E\bigl[f(\mathbf a,xW_j)W_j^*\bigr].\] Set \[F_n=\frac1n\sum_{j=0}^{n-1}P^j f.\]

We use compactness for pointwise ultraweak convergence. For each fixed tuple \(\mathbf x=(x_1,\ldots,x_k)\), all values \(F_n(\mathbf x)\) lie in the ultraweakly compact ball of radius \(\norm{f}\prod_i\norm{x_i}\). The product of these balls, indexed by all tuples in \(M^k\), is compact. Multilinearity is closed for this topology, since addition and scalar multiplication are ultraweakly continuous. Thus some subnet of \((F_n)\) converges at every tuple to a bounded multilinear map \(F\) with \(\norm{F}\le\norm{f}\).

On any common cochain norm ball, \(P\) is continuous for this topology. To see this, suppose \(\Phi_\lambda\to\Phi\) pointwise ultraweakly and \(\norm{\Phi_\lambda},\norm{\Phi}\le C\). For a fixed tuple \((\mathbf a,x)\) and a normal functional \(\rho\in M_*\), truncate (40) after \(L\) terms. The finite sum passes to the limit, and the remaining difference has absolute value at most \[2C\norm{\rho}\norm{x} \prod_{i=1}^{k-1}\norm{a_i}\sum_{s>L}p_s.\] Letting \(L\to\infty\) proves the claim. Since \[PF_n-F_n=\frac{P^nf-f}{n},\qquad \norm{PF_n-F_n}\le\frac{2\norm{f}}n,\] the subnet limit satisfies \(PF=F\).

Fix \(\mathbf a\), and put \(B(x)=f(\mathbf a,x)\). This is a bounded normal linear map. For any \(R<\infty\) and \(\varepsilon>0\), Lemma 17 provides \(\delta>0\) such that \[\norm{x}\le R,\quad \norm{x}_2<\delta \quad\Longrightarrow\quad \norm{B(xv)v^*}_2<\varepsilon \quad\hbox{for every }v\in\mathcal U(M).\] Indeed, right multiplication by a unitary preserves both the operator norm and the \(2\)-norm. The same bound holds after averaging over any \(W_j\) and after taking the Cesàro average. Ultraweak lower semicontinuity of the \(2\)-norm then gives \(\norm{F(\mathbf a,x)}_2\le\varepsilon\) under these input bounds. Using \(\varepsilon/2\) in place of \(\varepsilon\) proves the required continuity modulus for \(T(x)=F(\mathbf a,x)\). The equality \(PF=F\) is exactly the harmonicity identity for this map. Consequently Theorem 10 gives \[ F(\mathbf a,x)=h(\mathbf a)x, \qquad h(\mathbf a)=F(\mathbf a,1). \tag{41}\] In particular \(h\) is a bounded \((k-1)\)-cochain and \(\norm{h}\le\norm{f}\).

It remains to obtain its coboundary identity. Let \(\nu_n=n^{-1}\sum_{j=0}^{n-1}\mu^{*j}\) and \(h_n(\mathbf a)=F_n(\mathbf a,1)\). Evaluate the original equation \(df=0\) at \((a_1,\ldots,a_k,v)\), multiply the output on the right by \(v^*\), and average with respect to \(v\sim\nu_n\). This gives the exact equality \[\begin{align*} 0={}&a_1h_n(a_2,\ldots,a_k) +\sum_{j=1}^{k-1}(-1)^j h_n(a_1,\ldots,a_ja_{j+1},\ldots,a_k)\\ &+(-1)^k F_n(a_1,\ldots,a_{k-1},a_k) +(-1)^{k+1}f(a_1,\ldots,a_k). \tag{42}\end{align*}\] The final term is unchanged because \(vv^*=1\). Along the chosen subnet, each displayed evaluation converges ultraweakly; multiplication by a fixed \(a_i\) preserves that convergence. Passing to the limit and using (41) therefore yields \[0=(dh)(a_1,\ldots,a_k)+(-1)^{k+1}f(a_1,\ldots,a_k).\] Thus \(g=(-1)^kh\) satisfies (39). This calculation averages \(df\) at the displayed tuples; it requires neither a commutation relation between \(P\) and \(d\) nor separate normality of the limiting cochain \(F\). ◻

Proof. Order the finite subsets \(E\subset M\) by inclusion. For each \(E\), choose a countably generated unital von Neumann subalgebra \(N_E\) containing \(E\), the finite set \(f(E^k)\), and unital matrix systems of every size \(m\ge2\). Such systems exist in a type \(\mathrm{II}_1\) algebra: the unit can be divided into \(m\) equivalent orthogonal projections [3], and partial isometries between them supply the matrix units. Adjoining one system for each \(m\) adds only countably many generators.

The restriction of the trace to \(N_E\) is faithful and normal. The tracial Hilbert space \(L^2(N_E)\) is separable: \(*\)-polynomials with coefficients in \(\mathbb Q+i\mathbb Q\) in a countable set of generators are dense in the generated \(C^*\)-algebra in norm, and bounded strong approximation makes that algebra \(2\)-norm dense in \(N_E\). The faithful tracial representation thus gives \(N_E\) separable predual. Moreover \(N_E\) has no type I central part. A nonzero homogeneous type \(\mathrm{I}_n\) central summand would inherit a unital copy of \(M_m(\mathbb C)\) for every \(m\), whereas its \(n\)-dimensional fibers cannot admit a unital copy when \(m>n\). Every finite type I algebra decomposes into its homogeneous type \(\mathrm{I}_n\) summands. Since \(N_E\) is finite, it is therefore of type \(\mathrm{II}_1\).

Let \(\mathcal E_E:M\to N_E\) be the normal trace-preserving conditional expectation. Its existence and bimodularity are classical; see [8] and [23]. Define \[f_E=\mathcal E_E\circ f\big|_{N_E^k}.\] This map is separately normal and has norm at most \(\norm{f}\). For inputs in \(N_E\), bimodularity of \(\mathcal E_E\) gives \(df_E=\mathcal E_E(df)=0\). By Proposition 18, choose \(g_E\in C_b^{k-1}(N_E,N_E)\) such that \(dg_E=f_E\) and \(\norm{g_E}\le\norm{f}\). Extend it to a cochain on \(M\) by \[G_E(x_1,\ldots,x_{k-1}) =g_E(\mathcal E_Ex_1,\ldots,\mathcal E_Ex_{k-1}).\] This extension has the same norm bound. If \((a_1,\ldots,a_k)\in E^k\), all its inputs and adjacent products belong to \(N_E\). Expanding the differential consequently gives \[ (dG_E)(a_1,\ldots,a_k) =(dg_E)(a_1,\ldots,a_k) =\mathcal E_Ef(a_1,\ldots,a_k) =f(a_1,\ldots,a_k), \tag{43}\] where the final equality uses \(f(E^k)\subset N_E\).

The common norm bound and the same product compactness used above give a pointwise ultraweakly convergent subnet of \((G_E)\), with limit \(g\in C_b^{k-1}(M,M)\) and \(\norm{g}\le\norm{f}\). For a fixed \(k\)-tuple \((a_1,\ldots,a_k)\), the indices of this subnet eventually contain \(E_0=\{a_1,\ldots,a_k\}\). Equation (43) therefore holds eventually along the subnet. Passing to the limit in its finitely many coboundary terms gives \(dg=f\) at this tuple. This holds at every tuple, proving the proposition. The choices of \(N_E\) and \(\mathcal E_E\) can be made independently for different \(E\); eventual exactness is supplied by the directed finite subsets. ◻

Proof. The norm estimate follows directly from (44). To verify the identity, expand each input as \(a_i=za_i+qa_i\). By multilinearity it suffices to use tuples whose entries each belong to \(zM\) or to \(qM\). On such a tuple with first \(qM\) entry at position \(r\), formula (44) consists of the single term inserting \(q\) immediately before that entry, with sign \((-1)^r\); on an all-\(zM\) tuple it is zero.

Consider such an \(n\)-tuple with first \(qM\) entry \(a_r\). In \((J_zd)\psi\), merging the inserted \(q\) with \(a_r\) has total sign \((-1)^r(-1)^r=1\) and contributes \(\psi(a_1,\ldots,a_n)\). A merger strictly before the insertion has the same arguments as its counterpart in \((dJ_z)\psi\), but the latter inserts at position \(r-1\), so their signs are opposite. A merger strictly after the pair \((q,a_r)\) also has the same arguments as its counterpart in \((dJ_z)\psi\), and its merger index is larger by one, again reversing the sign. The merger of \(a_{r-1}\) with the inserted \(q\), when \(r>1\), vanishes because \(zMqM=0\); the corresponding merger across \(a_{r-1},a_r\) in \((dJ_z)\psi\) vanishes for the same reason.

The left outer terms cancel when \(r>1\), with insertion indices \(r\) and \(r-1\). When \(r=1\), both left outer terms vanish because \(qM\) annihilates the output module. For the right outer terms, if \(r<n\) the same insertion index and the two consecutive differential degrees give opposite signs. If \(r=n\), the last input is in \(qM\), so both right outer terms vanish. Thus the only surviving term is \(\psi(a_1,\ldots,a_n)\), as required because the cut of this tuple is zero. On an all-\(zM\) tuple, both sides of (45) vanish.

For completeness, in degree one the formula reads \[(J_zd\psi)(a)=-(d\psi)(q,qa)=\psi(qa) =\psi(a)-\psi(za),\] since both outer module actions in \((d\psi)(q,qa)\) vanish. In degree zero both sides are zero by the definitions. This proves the identity in all degrees. ◻

Proof of Theorem 1. The zero algebra is immediate, so assume \(M\ne0\). Fix \(k\ge2\) and an arbitrary bounded cocycle \(f_{\mathrm{orig}}\in C_b^k(M,M)\). By the normal reduction in Proposition 2, there are a separately normal cocycle \(f\) and a bounded \((k-1)\)-cochain \(b\) such that \[f_{\mathrm{orig}}=f+db.\] Let \(z_{\mathrm{II}_1}\) be the central projection of the type \(\mathrm{II}_1\) summand, and let \(z_0=1-z_{\mathrm{II}_1}\). We first partition \(z_{\mathrm{II}_1}\) into central projections whose algebras admit faithful normal tracial states.

In the abelian algebra \(Z(z_{\mathrm{II}_1}M)\), choose a maximal orthogonal family \((z_i)_{i\in I}\) of nonzero supports of normal states. Its join is \(z_{\mathrm{II}_1}\): a nonzero complementary projection would support a further normal state, contradicting maximality. A state is faithful on its support, so each \(z_iZ(M)\) has a faithful normal state \(\varphi_i\). Composing \(\varphi_i\) with the faithful normal center-valued trace of \(z_iM\) gives a faithful normal tracial state \(\tau_i\) on \(z_iM\); see [3]. The family \(I\) may be uncountable. Omit \(z_0\) if it is zero, and otherwise keep the entire complementary summand as one piece.

For each piece define the restricted cocycle \[f_i(x_1,\ldots,x_k)=z_i f(x_1,\ldots,x_k), \qquad x_j\in z_iM.\] Multiplication by a central projection is a bimodule map, so \(f_i\) is a separately normal cocycle and \(\norm{f_i}\le\norm{f}\). For \(i\in I\), Proposition 19 supplies a primitive \(b_i\in C_b^{k-1}(z_iM,z_iM)\) with \[db_i=f_i,\qquad \norm{b_i}\le\norm{f}.\] If \(z_0\ne0\), the vanishing result for algebras without a type \(\mathrm{II}_1\) central summand in Proposition 2 supplies a bounded primitive \(b_0\) of \(f_0\). Its norm is some finite number.

Pull each primitive back to \(M\) through the homomorphism \(x\mapsto z_ix\): \[\widetilde b_i(x_1,\ldots,x_{k-1}) =b_i(z_ix_1,\ldots,z_ix_{k-1}).\] The outer module actions also pull back, since the outputs lie in \(z_iM\). Thus \[d\widetilde b_i(x_1,\ldots,x_k) =z_i f(z_ix_1,\ldots,z_ix_k) =\operatorname{cut}_{z_i}(z_if)(x_1,\ldots,x_k).\] Define cochains with values in \(z_iM\) on all of \(M\) by \[H_i=\widetilde b_i+J_{z_i}(z_if).\] Equation (47) gives \[dH_i=z_if, \qquad \norm{H_i}\le k\norm{f}\quad(i\in I),\] and, when present, \[\norm{H_0}\le\norm{b_0}+(k-1)\norm{f}.\] There is consequently a single finite constant \(C\) bounding the norms of all these cochains; one can take the maximum of the two displayed bounds, omitting the second when \(z_0=0\).

For an arbitrary central partition of unity, the map \[M\longrightarrow\prod_i^{\ell^\infty}z_iM, \qquad x\longmapsto(z_ix)_i,\] is an isometric \(*\)-isomorphism onto the bounded direct product. For surjectivity, the finite partial sums of a bounded family \((x_i)_i\), \(x_i\in z_iM\), form a net directed by inclusion of the finite index sets. This net converges strongly, since the central supports are orthogonal; its limit belongs to \(M\). For each tuple \((x_1,\ldots,x_{k-1})\), the family \((H_i(x_1,\ldots,x_{k-1}))_i\) belongs to this product and has norm at most \(C\prod_j\norm{x_j}\). Let \(G(x_1,\ldots,x_{k-1})\in M\) be the corresponding element. Coordinatewise multilinearity shows that \(G\) is a bounded multilinear cochain with \(\norm{G}\le C\). Central projection commutes with every term of its coboundary, so \[z_i(dG)=dH_i=z_if \qquad\hbox{for every central piece }i.\] Since the projections have join \(1\), it follows that \(dG=f\). Finally, \[d(G+b)=f+db=f_{\mathrm{orig}}.\] We have obtained an actual bounded \((k-1)\)-cochain whose coboundary is the given cocycle. Hence \(H_b^k(M,M)=0\) for every complex von Neumann algebra \(M\) and every \(k\ge2\), with the actual image of \(d\) in the definition of cohomology and with no complete-boundedness hypothesis. ◻