Bounded primitives

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

This lesson proves Theorem 1.2 of the lesson on the cohomology problem: for a von Neumann algebra $M$ and $k\ge2$, every bounded Hochschild cocycle $f\in C_b^k(M,M)$ has a bounded primitive. First let $f$ be separately normal and $M$ of type $\mathrm{II}_1$ with separable predual and faithful normal tracial state. Averaging $f$ in its last variable along the random walk gives, in the limit, a map that is harmonic in that variable. The Liouville theorem makes it a right module map, and this yields a primitive $g$ with $\|g\|\le\|f\|$ (Section 1). The uniform bound allows a compactness argument that removes the separability assumption (Section 2). An explicit homotopy handles inputs that mix different central pieces (Section 3). Section 4 assembles the pieces, using the classical reductions K1 and K2.

Prerequisites. The cohomology problem: Section 1, Lemmas 2.2 and 2.3, Exercise 4.4, and the reductions K1 and K2 of Section 3. A random walk with asymptotically free products: Theorem 3.1 and Lemma 2.3. The Liouville theorem: Theorem 0.1. From Measurable fields and direct integrals: The central decomposition and the types of the fibres (Facts 1.1, 1.6 and 1.7, Corollary 3.6, Theorems 4.1 and 5.3). From Traces and noncommutative integration: Integration for a trace, the commutation theorem and applications, Theorem 9.1 (trace-preserving conditional expectations). From Foundations of von Neumann algebras: Kaplansky's density theorem and its consequences (Corollary 1.4, Theorem 7.1); The double commutant theorem (Theorem 4.4); Compact and trace-class operators, the predual of B(H) and the operator topologies (Theorems 9.1 and 9.4); Weak topologies (Theorem 2.3, Tychonoff; Theorem 3.1, Banach–Alaoglu); Projections and types of von Neumann algebras (Theorem 7.2, the type decomposition); The universal enveloping von Neumann algebra of a C*-algebra and W*-algebras (Lemma 11.1, supports of normal functionals; Corollary 11.4).

Conventions. As in the lesson on the cohomology problem; throughout $k\ge2$. For $\mathbf a=(a_1,\ldots,a_{k-1})\in M^{k-1}$ and $x\in M$, $(\mathbf a,x)$ is the $k$-tuple with last entry $x$. A net $\Phi_\lambda$ in $C_b^k(M,M)$ converges pointwise ultraweakly to $\Phi$ if $\Phi_\lambda(\mathbf x)\to\Phi(\mathbf x)$ ultraweakly for every $\mathbf x\in M^k$. For a probability measure $\mu$ on $\mathcal U(M)$ with countable support, with atoms $s$ of mass $\mu_s$, put $(P_\mu\Phi)(\mathbf a,x)=\sum_s\mu_s\,\Phi(\mathbf a,xs)\,s^* ,$ a norm-convergent sum.

1. Averaging the last variable

Lemma 1.1 (compactness). Let $c\ge0$ and $\mathcal C=\{\Phi\in C_b^k(M,M):\|\Phi\|\le c\}$.

Every net in $\mathcal C$ has a subnet converging pointwise ultraweakly to an element of $\mathcal C$.
$P_\mu$ maps $\mathcal C$ into itself and is continuous on $\mathcal C$ for pointwise ultraweak convergence.

Proof. (1) For each $\mathbf x$, the values $\Phi(\mathbf x)$, $\Phi\in\mathcal C$, lie in the ball of radius $c\prod_i\|x_i\|$, which is ultraweakly compact (Banach–Alaoglu, the ultraweak topology being $\sigma(M,M_*)$). By Tychonoff's theorem the product of these balls over all $\mathbf x\in M^k$ is compact, so a subnet converges at every $\mathbf x$. The limit is $k$-linear, since addition and multiplication by scalars are ultraweakly continuous, and it lies in $\mathcal C$, since the balls are closed.

(2) The bound $\|P_\mu\Phi\|\le\|\Phi\|$ is clear. Let $\Phi_\lambda\to\Phi$ in $\mathcal C$, fix $(\mathbf a,x)$ and $\omega\in M_*$, and enumerate the atoms as $s_1,s_2,\ldots$. For each $j$, $y\mapsto\omega(ys_j^*)$ is normal, so the first $L$ terms of $\omega((P_\mu\Phi_\lambda)(\mathbf a,x))$ converge to those of $\omega((P_\mu\Phi)(\mathbf a,x))$. The remaining terms are bounded in absolute value by $c\|\omega\|\,\|x\|\prod_i\|a_i\|\sum_{j>L}\mu_{s_j}$, for $\Phi_\lambda$ and for $\Phi$. Letting $L\to\infty$ proves the convergence. $\square$

Proposition 1.2 (the separable case). Let $M$ be of type $\mathrm{II}_1$ with separable predual and faithful normal tracial state, and let $f\in C_b^k(M,M)$ be a separately normal cocycle. There is $g\in C_b^{k-1}(M,M)$ with $dg=f$ and $\|g\|\le\|f\|$.

Proof. Let $\mu$ be the measure of Theorem 3.1 of the lesson on the random walk, and $P=P_\mu$. Let $U_1,U_2,\ldots$ be independent with law $\mu$, $W_0=1$ and $W_j=U_1\cdots U_j$; expectations are norm-convergent countable sums. By induction on $j$, $(P^jf)(\mathbf a,x)=\mathbb E\bigl[f(\mathbf a,xW_j)\,W_j^*\bigr]:$ applying $P$ to the right side multiplies $W_j$ on the left by an independent increment of law $\mu$, and the product has the law of $W_{j+1}$. Put $F_n=\frac1n\sum_{j=0}^{n-1}P^jf,\qquad\|F_n\|\le\|f\|.$ By Lemma 1.1(1), a subnet $(F_{n_\lambda})$ converges pointwise ultraweakly to some $F\in C_b^k(M,M)$ with $\|F\|\le\|f\|$. Since $\begin{gathered} PF_n-F_n=\frac{P^nf-f}n ,\\ \|PF_n-F_n\|\le\frac{2\|f\|}n , \end{gathered}$ and $n_\lambda\to\infty$, Lemma 1.1(2) gives $PF=F$.

Each $F(\mathbf a,\cdot)$ is a left multiplication. Fix $\mathbf a$, and put $T(x)=F(\mathbf a,x)$. The identity $PF=F$ says that $T$ is $\mu$-harmonic. The map $B(x)=f(\mathbf a,x)$ is normal, so by Lemmas 2.3 and 2.2(1) of the lesson on the cohomology problem, for $R<\infty$ and $\varepsilon>0$ there is $\delta>0$ with $\|B(y)\|_2<\varepsilon$ whenever $\|y\|\le R$ and $\|y\|_2<\delta$. Let $\|x\|\le R$ and $\|x\|_2<\delta$. For every unitary $v$, $xv$ satisfies the same bounds, so $\|B(xv)v^*\|_2=\|B(xv)\|_2<\varepsilon$. Averaging gives $\|(P^jf)(\mathbf a,x)\|_2\le\varepsilon$ and $\|F_n(\mathbf a,x)\|_2\le\varepsilon$. The $2$-norm is ultraweakly lower semicontinuous (proof of Lemma 2.3 there), so $\|T(x)\|_2\le\varepsilon$. Thus $T$ is continuous on balls for the $2$-norm, and the Liouville theorem gives $F(\mathbf a,x)=h(\mathbf a)\,x,\qquad h(\mathbf a)=F(\mathbf a,1).$ So $h\in C_b^{k-1}(M,M)$ and $\|h\|\le\|f\|$.

The coboundary. Put $h_n(\mathbf a)=F_n(\mathbf a,1)$. Evaluate $df=0$ at $(a_1,\ldots,a_k,v)$ for a unitary $v$, multiply on the right by $v^*$, and average over $v=W_j$ and over $0\le j<n$. As $f(a_1,\ldots,a_k)vv^*=f(a_1,\ldots,a_k)$, this gives $\begin{aligned} 0={}&a_1h_n(a_2,\ldots,a_k)\\ &+\sum_{j=1}^{k-1}(-1)^jh_n(a_1,\ldots,a_ja_{j+1},\\ &\qquad\ldots,a_k)\\ &+(-1)^kF_n(a_1,\ldots,a_k)\\ &+(-1)^{k+1}f(a_1,\ldots,a_k). \end{aligned}$ Along the subnet each term converges ultraweakly; left multiplication by $a_1$ is ultraweakly continuous. In the limit $F(a_1,\ldots,a_k)=h(a_1,\ldots,a_{k-1})a_k$, so $\begin{aligned} 0={}&(dh)(a_1,\ldots,a_k)\\ &+(-1)^{k+1}f(a_1,\ldots,a_k). \end{aligned}$ Hence $g=(-1)^kh$ satisfies $dg=f$ and $\|g\|\le\|f\|$. $\square$

The proof averages the cocycle identity at fixed inputs; it uses neither a commutation relation between $P$ and $d$ nor normality of $F$.

2. Removing separability

Lemma 2.1. Let $(M,\tau)$ be a tracial von Neumann algebra and $N\subseteq M$ a von Neumann subalgebra with $1\in N$ generated by a countable set. Then $N$ has separable predual.

Proof. We may take the generating set $S$ closed under adjoints. Let $A_0$ be the countable set of polynomials in $S$ with coefficients in $\mathbb Q+i\mathbb Q$, and $A$ its norm closure, the C*-algebra generated by $S$. By the double commutant theorem the strong closure of $A$ is $N$. By Kaplansky's density theorem every element of the unit ball of $N$ is the strong limit of a net in the unit ball of $A$, hence its $2$-norm limit (Exercise 4.4 of the lesson on the cohomology problem). So $A_0$ is $2$-norm dense in $N$, and $L^2(N)$ is separable.

The map $\lambda:N\to B(L^2(N))$ of Lemma 2.3(2) of the lesson on the random walk is an injective unital $*$-homomorphism onto a von Neumann algebra, with $\lambda$ and $\lambda^{-1}$ normal; that part of its proof does not use a separable predual. The predual of $\lambda(N)$ is a quotient of $B(L^2(N))_*$ (Theorem 9.4(a) of the lesson on operator topologies). The latter is separable: by Theorem 9.1(ii) there every normal functional is $x\mapsto\sum_n\langle x\xi_n,\eta_n\rangle$ with square-summable sequences, so it is a norm limit of finite sums, and $\|\langle\cdot\,\xi,\eta\rangle-\langle\cdot\,\xi',\eta'\rangle\|$$\le\|\xi-\xi'\|\,\|\eta\|+\|\xi'\|\,\|\eta-\eta'\|$ lets one take $\xi,\eta$ from a countable dense set. Finally $\omega\mapsto\omega\circ\lambda$ is an isometry of $\lambda(N)_*$ onto $N_*$. $\square$

Lemma 2.2. Let $N$ be a von Neumann algebra with separable predual and a faithful normal tracial state. Suppose that for every $m\ge2$, $N$ contains matrix units $e^{(m)}_{ij}$, $1\le i,j\le m$, with $\sum_ie^{(m)}_{ii}=1$. Then $N$ is of type $\mathrm{II}_1$.

Proof. Let $N$ act on $L^2(N)$ (Lemma 2.3 of the lesson on the random walk) and write $N=\int^\oplus N(\gamma)\,d\mu(\gamma)$ with factors $N(\gamma)$ (Theorem 4.1 of the lesson on the central decomposition). By Corollary 3.6 there, there are normal positive tracial functionals $\varphi_\gamma$ on $N(\gamma)$, faithful for almost every $\gamma$. The fibre map is almost everywhere a unital $*$-homomorphism on the countably many matrix units.

Fix $\gamma$ outside a null set, with $N(\gamma)\ne0$. Then $\varphi_\gamma$ is faithful, so $\varphi_\gamma(1)>0$, and $N(\gamma)$ contains the matrix units $e^{(m)}_{ij}(\gamma)$. The projections $e^{(m)}_{ii}(\gamma)$ are equivalent through the $e^{(m)}_{i1}(\gamma)$, so each has $\varphi_\gamma$-value $\varphi_\gamma(1)/m$. Suppose $p\in N(\gamma)$ is a nonzero abelian projection. In the factor $N(\gamma)$ its only subprojections are $0$ and $p$ (Fact 1.6(a) there), and comparison (Fact 1.6(b)) gives $p\precsim e^{(m)}_{11}(\gamma)$ or $e^{(m)}_{11}(\gamma)\precsim p$; in the second case $e^{(m)}_{11}(\gamma)$ is equivalent to a nonzero subprojection of $p$, that is to $p$. Either way $\varphi_\gamma(p)\le\varphi_\gamma(1)/m$, for every $m$. Hence $\varphi_\gamma(p)=0$, contradicting faithfulness. So $N(\gamma)$ is not of type I; it is finite because it has a faithful tracial functional. By Theorem 5.3 there, almost every $\gamma$ with $N(\gamma)\ne0$ lies in the set $E_{\mathrm{II}_1}$ of the type decomposition. Hence the central projections $z_{\mathrm I},z_{\mathrm{II}_\infty},z_{\mathrm{III}}$ of $N$ vanish (Fact 1.1(f)), and $N$ is of type $\mathrm{II}_1$. $\square$

Proposition 2.3 (the tracial case). Let $M$ be of type $\mathrm{II}_1$ with a faithful normal tracial state $\tau$, and let $f\in C_b^k(M,M)$ be a separately normal cocycle. There is $g\in C_b^{k-1}(M,M)$ with $dg=f$ and $\|g\|\le\|f\|$.

Proof. For each $m\ge2$, Fact 1.7(b) of the lesson on the central decomposition gives $1=e_1+\cdots+e_m$ with mutually orthogonal, mutually equivalent projections. With partial isometries $v_i$, $v_i^*v_i=e_1$, $v_iv_i^*=e_i$, the elements $e^{(m)}_{ij}=v_iv_j^*$ are matrix units with $\sum_ie^{(m)}_{ii}=1$.

Order the finite subsets $E\subseteq M$ by inclusion. For each $E$, let $N_E$ be the von Neumann subalgebra generated by $E$, the finite set $f(E^k)$ and all the $e^{(m)}_{ij}$. It is generated by a countable set, so it has separable predual (Lemma 2.1); the restriction of $\tau$ is a faithful normal tracial state; and $N_E$ is of type $\mathrm{II}_1$ by Lemma 2.2. Let $\mathcal E_E:M\to N_E$ be the trace-preserving conditional expectation of Theorem 9.1 of the integration lesson: a normal projection of norm one onto $N_E$ with $\mathcal E_E(axb)=a\,\mathcal E_E(x)\,b$ for $a,b\in N_E$.

The map $f_E=\mathcal E_E\circ f|_{N_E^k}$ is separately normal, $\|f_E\|\le\|f\|$, and $df_E=\mathcal E_E\circ(df)=0$ on $N_E$, by bimodularity. By Proposition 1.2 there is $g_E\in C_b^{k-1}(N_E,N_E)$ with $dg_E=f_E$ and $\|g_E\|\le\|f\|$. Put $\begin{aligned} &G_E(x_1,\ldots,x_{k-1})\\ &\quad=g_E(\mathcal E_Ex_1,\ldots,\mathcal E_Ex_{k-1}), \end{aligned}$ an element of $C_b^{k-1}(M,M)$ with $\|G_E\|\le\|f\|$. If $a_1,\ldots,a_k\in E$, all inputs and products of neighbours in $(dG_E)(a_1,\ldots,a_k)$ lie in $N_E$, where $\mathcal E_E$ is the identity, so $\begin{aligned} (dG_E)(a_1,\ldots,a_k)&=(dg_E)(a_1,\ldots,a_k)\\ &=\mathcal E_Ef(a_1,\ldots,a_k)\\ &=f(a_1,\ldots,a_k), \end{aligned}$ the last because $f(E^k)\subseteq N_E$.

By Lemma 1.1(1) a subnet of $(G_E)$ converges pointwise ultraweakly to some $g\in C_b^{k-1}(M,M)$ with $\|g\|\le\|f\|$. Fix $a_1,\ldots,a_k\in M$. Along the subnet the index $E$ eventually contains $\{a_1,\ldots,a_k\}$, so $(dG_E)(a_1,\ldots,a_k)=f(a_1,\ldots,a_k)$ eventually. The left side converges ultraweakly to $(dg)(a_1,\ldots,a_k)$, being a finite sum in which fixed elements multiply on the left or right. So $dg=f$. $\square$

3. Central cuts

Let $z\in M$ be a central projection and $q=1-z$. The ideal $zM$ is an $M$-bimodule under multiplication, and $q$ acts on it as $0$ from both sides. Let $C_b^n(M,zM)$ be the bounded $n$-linear maps $M^n\to zM$, with $C_b^0(M,zM)=zM$ and the differential $d$ of the lesson on the cohomology problem; $d\circ d=0$ by the same proof. Define $\operatorname{cut}_z$ by $(\operatorname{cut}_z\psi)(a_1,\ldots,a_n)=\psi(za_1,\ldots,za_n),$ with $\operatorname{cut}_z=\mathrm{id}$ on $C_b^0(M,zM)$. For $n\ge2$ define $J_z:C_b^n(M,zM)\to C_b^{n-1}(M,zM)$ by $\begin{aligned} &(J_z\psi)(a_1,\ldots,a_{n-1})\\ &=\sum_{i=1}^{n-1}(-1)^i\psi(za_1,\ldots,za_{i-1},q,\\ &\qquad qa_i,a_{i+1},\ldots,a_{n-1}), \end{aligned}$ and $J_z=0$ on $C_b^0$ and $C_b^1$.

Lemma 3.1 (central homotopy). On $C_b^n(M,zM)$, for every $n\ge0$, $dJ_z+J_zd=\mathrm{id}-\operatorname{cut}_z ,$ and $\|J_z\psi\|\le(n-1)\|\psi\|$ for $n\ge2$. In particular, for a cocycle $\psi\in C_b^n(M,zM)$ with $n\ge2$, $d(J_z\psi)=\psi-\operatorname{cut}_z\psi$.

Proof. The bound holds term by term, as $\|za\|\le\|a\|$ and $\|q\|\le1$. In degree $0$ both sides vanish. In degree $1$, $J_z\psi=0$ and $\begin{aligned} (J_zd\psi)(a)&=-(d\psi)(q,qa)\\ &=-q\psi(qa)+\psi(qa)-\psi(q)qa\\ &=\psi(qa), \end{aligned}$ because $q$ annihilates $zM$; and $\psi(qa)=\psi(a)-\psi(za)$.

Let $n\ge2$. Both sides are $n$-linear in $(a_1,\ldots,a_n)$, so it suffices to take each $a_j$ nonzero and in $zM$ or in $qM$; call such tuples pure. Products of elements of $zM$ and $qM$ are in $zM$, in $qM$ or $0$. Let $b$ be a pure tuple and $\varphi$ a cochain of degree at least $2$. If all entries of $b$ lie in $zM$, then $(J_z\varphi)(b)=0$, since $qb_i=0$. Otherwise, if $b_r$ is the first entry in $qM$, only the term $i=r$ survives — $qb_i=0$ for $i<r$ and $zb_r=0$ for $i>r$ — and $\begin{aligned} &(J_z\varphi)(b)=(-1)^r\varphi(b_1,\ldots,b_{r-1},\\ &\qquad q,b_r,\ldots).\qquad(3.1) \end{aligned}$

Let $a$ be a pure $n$-tuple. If all entries lie in $zM$, so do all tuples obtained by multiplying neighbours, and both sides vanish. Otherwise let $a_r$ be the first entry in $qM$. Then $(\operatorname{cut}_z\psi)(a)=0$, as $za_r=0$, and by (3.1) $(J_zd\psi)(a)=(-1)^r(d\psi)(c)$ with $c=(a_1,\ldots,a_{r-1},q,a_r,\ldots,a_n).$ Compare the terms of $(-1)^r(d\psi)(c)$ with those of $(dJ_z\psi)(a)$, evaluating the latter by (3.1).

Multiplying $q$ and $a_r$ gives $(-1)^r(-1)^r\psi(a)=\psi(a)$.
Multiplying $a_{r-1}$ and $q$ gives $0$; so does multiplying $a_{r-1}$ and $a_r$ in $dJ_z\psi$, as $zMqM=0$.
Left outer terms: for $r=1$ both vanish, as $q$ and $a_1\in qM$ annihilate $zM$. For $r\ge2$ they are $(-1)^ra_1\psi(a_2,\ldots,a_{r-1},$$q,a_r,\ldots,a_n)$ and $a_1(J_z\psi)(a_2,\ldots,a_n)$, where the first $qM$-entry sits at position $r-1$; they cancel.
Products $a_ja_{j+1}$ with $j\le r-2$: both have index $j$, and the first $qM$-entry of the shortened tuple sits at position $r-1$ in $dJ_z\psi$. The signs $(-1)^{r+j}$ and $(-1)^{j+r-1}$ cancel.
Products $a_ja_{j+1}$ with $j\ge r$: index $j+1$ in $d\psi(c)$ and $j$ in $dJ_z\psi$. If the product is $0$, both vanish. Otherwise the first $qM$-entry stays at position $r$, and the signs $(-1)^{r+j+1}$ and $(-1)^{j+r}$ cancel.
Right outer terms: for $r=n$ both vanish, since $a_n\in qM$ annihilates $zM$ and $(a_1,\ldots,a_{n-1})$ lies in $zM$. For $r<n$ they are $(-1)^{r+n+1}\psi(a_1,\ldots,a_{r-1},$$q,a_r,\ldots,a_{n-1})a_n$ and $(-1)^{n+r}$ times the same; they cancel.

This accounts for all $n+2$ terms of $(d\psi)(c)$ and all $n+1$ terms of $(dJ_z\psi)(a)$. The sum is $\psi(a)=(\psi-\operatorname{cut}_z\psi)(a)$. $\square$

4. The proof of the main theorem

Lemma 4.1 (tracial pieces). Let $M\ne0$ be of type $\mathrm{II}_1$. There are mutually orthogonal nonzero central projections $z_i$, $i\in I$, with $\sum_iz_i=1$, such that every $z_iM$ is of type $\mathrm{II}_1$ and has a faithful normal tracial state.

Proof. Let $Z$ be the centre. By Zorn's lemma there is a maximal family $(z_i)$ of mutually orthogonal nonzero projections of $Z$, each the support of a normal state of $Z$ (Lemma 11.1 of the lesson on W*-algebras). If $w=1-\sum_iz_i\ne0$, a vector state $\varphi$ of $Z$ with $\varphi(w)>0$ gives the normal state $\varphi(w\,\cdot\,)/\varphi(w)$, whose support is a nonzero projection below $w$, contradicting maximality. So $\sum_iz_i=1$. The centre $z_iZ$ of $z_iM$ carries the faithful normal state given by $z_i$, and $z_iM$ is finite. By Fact 1.7(a) of the lesson on the central decomposition, $z_iM$ has a faithful normal tracial state; it is of type $\mathrm{II}_1$ as a central summand of $M$. $\square$

Lemma 4.2 (central partitions). Let $z_i$, $i\in I$, be mutually orthogonal central projections of $M$ with $\sum_iz_i=1$. For every family $(y_i)$ with $y_i\in z_iM$ and $\sup_i\|y_i\|<\infty$, there is exactly one $y\in M$ with $z_iy=y_i$ for all $i$; moreover $\|y\|=\sup_i\|y_i\|$.

Proof. If $z_iy=0$ for all $i$, then $y=\sum_iz_iy=0$, the sum converging strongly (Corollary 1.4 of the Kaplansky lesson). For existence, the finite partial sums $y_F=\sum_{i\in F}y_i$ form a net with $\begin{aligned} &\|(y_F-y_{F'})\xi\|^2\\ &=\sum_{i\in F\triangle F'}\|y_iz_i\xi\|^2\\ &\le\sup_i\|y_i\|^2\sum_{i\in F\triangle F'}\|z_i\xi\|^2 , \end{aligned}$ which is small for large $F,F'$, since $\sum_i\|z_i\xi\|^2=\|\xi\|^2$. The strong limit $y$ lies in $M$, satisfies $z_iy=y_i$, and $\|y\xi\|^2=\sum_i\|y_iz_i\xi\|^2\le\sup_i\|y_i\|^2\|\xi\|^2$; also $\|y\|\ge\|z_iy\|=\|y_i\|$. $\square$

Proof of Theorem 1.2 of the lesson on the cohomology problem. Let $M\ne0$, $k\ge2$, and let $\phi\in C_b^k(M,M)$ be a cocycle. By K1 there are a separately normal cocycle $f$ and $b\in C_b^{k-1}(M,M)$ with $\phi=f+db$.

Let $z_{\mathrm{II}_1}$ be the central projection of the type $\mathrm{II}_1$ summand (Theorem 7.2 of the lesson on projections and types) and $z_0=1-z_{\mathrm{II}_1}$. If $z_{\mathrm{II}_1}\ne0$, Lemma 4.1 applied to $z_{\mathrm{II}_1}M$ gives central projections $z_i$, $i\in I$, of $z_{\mathrm{II}_1}M$, hence of $M$, with sum $z_{\mathrm{II}_1}$. Let $I'$ consist of $I$ and, if $z_0\ne0$, the index $0$. For $i\in I'$, the map $\psi_i=z_if$ belongs to $C_b^k(M,z_iM)$ and is a cocycle there, as multiplication by the central $z_i$ commutes with $d$. Its restriction $f_i$ to $(z_iM)^k$ is a separately normal cocycle on the von Neumann algebra $z_iM$, with $\|f_i\|\le\|f\|$.

For $i\in I$, Proposition 2.3 gives $b_i\in C_b^{k-1}(z_iM,z_iM)$ with $db_i=f_i$ and $\|b_i\|\le\|f\|$. If $z_0\ne0$, the algebra $z_0M$ has no type $\mathrm{II}_1$ central summand, and K2 gives $b_0\in C_b^{k-1}(z_0M,z_0M)$ with $db_0=f_0$. Put $\tilde b_i(x_1,\ldots,x_{k-1})=b_i(z_ix_1,\ldots,z_ix_{k-1}).$ Since $b_i$ takes values in $z_iM$, $x\,b_i(\ldots)=(z_ix)\,b_i(\ldots)$ and $b_i(\ldots)\,x=b_i(\ldots)\,(z_ix)$; also $z_i(x_jx_{j+1})=(z_ix_j)(z_ix_{j+1})$. Hence $d\tilde b_i=\operatorname{cut}_{z_i}\psi_i$. Let $H_i=\tilde b_i+J_{z_i}\psi_i\in C_b^{k-1}(M,z_iM).$ By Lemma 3.1, $dH_i=\operatorname{cut}_{z_i}\psi_i+\psi_i-\operatorname{cut}_{z_i}\psi_i=z_if$. Moreover $\|H_i\|\le k\|f\|$ for $i\in I$, and $\|H_0\|\le\|b_0\|+(k-1)\|f\|$. Let $C$ be the larger of these two bounds.

For $\mathbf x\in M^{k-1}$, let $G(\mathbf x)\in M$ be the element with $z_iG(\mathbf x)=H_i(\mathbf x)$ for all $i\in I'$ (Lemma 4.2). By uniqueness, $G$ is $(k-1)$-linear, and $\|G\|\le C$. As each $z_i$ is central, $z_i(dG)(\mathbf y)=(dH_i)(\mathbf y)=z_if(\mathbf y)$ for every $\mathbf y\in M^k$, so $dG=f$ by uniqueness in Lemma 4.2. Finally $d(G+b)=f+db=\phi .$ So every bounded cocycle of degree $k\ge2$ is the coboundary of a bounded cochain, and $H_b^k(M,M)=0$. $\square$

The proof uses the classical theorems K1 and K2 without proof; all other steps are proved in this course or in the lessons it cites.

5. Exercises

5.1 (easy). Show directly from the definition that $f_E=\mathcal E_E\circ f|_{N_E^k}$ is a cocycle on $N_E$.

5.2 (medium). Show that the proof of Proposition 1.2 also works for $k=1$, and deduce: if $M$ is of type $\mathrm{II}_1$ with separable predual and faithful normal tracial state, every normal derivation $\delta$ of $M$ is inner, $\delta(a)=ha-ah$ with $\|h\|\le\|\delta\|$.

5.3 (easy). Check Lemma 3.1 in degree $2$ on the pure tuple $(a_1,a_2)$ with $a_1\in qM$ and $a_2\in zM$.

5.4 (easy). Show that $M_n(\mathbb C)$ does not contain matrix units $e_{ij}$, $1\le i,j\le m$, with $\sum_ie_{ii}=1$ when $m>n$. Which step of the proof of Lemma 2.2 does this reflect?

5.5 (easy). Let $p\ne0,1$ be a projection of $M$. Show that the von Neumann algebra generated by $p$ is $\mathbb Cp+\mathbb C(1-p)$, which is of type I. Why does the proof of Proposition 2.3 put matrix units of every size into $N_E$?

6. Solutions

5.1. For $a_1,\ldots,a_{k+1}\in N_E$, every term of $(df_E)(a_1,\ldots,a_{k+1})$ is $\mathcal E_E$ applied to the corresponding term of $(df)(a_1,\ldots,a_{k+1})$, because $\mathcal E_E(a_1y)=a_1\mathcal E_E(y)$ and $\mathcal E_E(ya_{k+1})=\mathcal E_E(y)a_{k+1}$. So $(df_E)(a)=\mathcal E_E((df)(a))=0$.

5.2. For $k=1$ there is no $\mathbf a$: $F$ is a limit of the averages $F_n(x)=\frac1n\sum_{j<n}\mathbb E[\delta(xW_j)W_j^*]$, it is $\mu$-harmonic and continuous on balls, so $F(x)=hx$ with $h=F(1)$ and $\|h\|\le\|\delta\|$. The identity $(d\delta)(a,v)=a\delta(v)-\delta(av)+\delta(a)v=0$, multiplied by $v^*$ and averaged, gives $ah_n-F_n(a)+\delta(a)=0$ with $h_n=F_n(1)$. In the limit $ah-ha+\delta(a)=0$, that is $\delta(a)=ha-ah$.

5.3. Here $r=1$ and $c=(q,a_1,a_2)$. By (3.1), $\begin{aligned} &(J_zd\psi)(a_1,a_2)=-(d\psi)(q,a_1,a_2)\\ &=-q\psi(a_1,a_2)+\psi(qa_1,a_2)\\ &\quad-\psi(q,a_1a_2)+\psi(q,a_1)a_2. \end{aligned}$ The first term vanishes, $qa_1=a_1$, and $a_1a_2=0$, so $(J_zd\psi)(a_1,a_2)=\psi(a_1,a_2)+\psi(q,a_1)a_2$. In $dJ_z\psi$, $(J_z\psi)(a_2)=0$ as $a_2\in zM$, $(J_z\psi)(a_1a_2)=0$, and $(J_z\psi)(a_1)=-\psi(q,a_1)$, so $\begin{aligned} &(dJ_z\psi)(a_1,a_2)\\ &=a_1(J_z\psi)(a_2)-(J_z\psi)(a_1a_2)\\ &\quad+(J_z\psi)(a_1)a_2=-\psi(q,a_1)a_2. \end{aligned}$ The sum is $\psi(a_1,a_2)$, and $(\operatorname{cut}_z\psi)(a_1,a_2)=\psi(za_1,za_2)=0$.

5.4. The $e_{ii}$ would be $m$ mutually orthogonal nonzero projections in $M_n(\mathbb C)$, whose ranks add up to at most $n$. In the proof of Lemma 2.2, a factor of type I with a faithful trace is excluded in the same way: its minimal projection would have trace at most $1/m$ for every $m$.

5.5. The set $\mathbb Cp+\mathbb C(1-p)$ is a unital $*$-algebra, two-dimensional and hence closed, and every von Neumann algebra containing $p$ contains it. It is abelian, so every projection in it is abelian, and it is of type I. A subalgebra generated by $E$ and $f(E^k)$ alone may therefore be of type I, where Proposition 1.2 does not apply; the matrix units force $N_E$ to be of type $\mathrm{II}_1$ (Lemma 2.2).

Sources

[OpenAI-295] OpenAI, Vanishing of higher bounded Hochschild cohomology, OpenAI Math Release preprint, 23 September 2026, Section 5 (bounded primitives, removal of separability, central cuts, completion of the proof). https://github.com/openai/math/tree/main/preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026
E. Christensen, F. Pop, A. M. Sinclair and R. R. Smith, Hochschild cohomology of factors with property $\Gamma$, Ann. of Math. 158 (2003), 635–659, Section 7: uniform primitive bounds and pointwise ultraweak compactness, used there to remove separability.