Every complex von Neumann algebra $M$ satisfies $H_b^k(M,M)=0$ for every positive integer $k$. In degree one this is the inner-derivation theorem of Kadison and of Sakai. The higher degrees constitute the Kadison-Ringrose cohomology problem, open since 1971. The answer is affirmative, in every positive degree.

For $k \geq 1$ let $C_b^k(M,M)$ denote the Banach space of bounded complex $k$-linear maps $M^k \to M$ with the operator norm
$$
\|f\| = \sup\{ \|f(a_1,\ldots,a_k)\| : a_j \in M,\ \|a_j\| \leq 1\},
$$
and $C_b^0(M,M)=M$. The differential $d : C_b^k(M,M) \to C_b^{k+1}(M,M)$ is
$$
(df)(a_1,\ldots,a_{k+1}) = a_1 f(a_2,\ldots,a_{k+1}) + \sum_{j=1}^k (-1)^j f(a_1,\ldots,a_j a_{j+1},\ldots,a_{k+1}) + (-1)^{k+1} f(a_1,\ldots,a_k) a_{k+1}.
$$
In degree zero this is the inner derivation $(db)(a) = ab - ba$. The algebraic identity $d^2 = 0$ gives $\operatorname{im} d \subset \ker d$, and
$$
H_b^k(M,M) = \ker\bigl(d : C_b^k \to C_b^{k+1}\bigr) \big/ \operatorname{im}\bigl(d : C_b^{k-1} \to C_b^k\bigr).
$$
The denominator is the actual image, not the closure.

Theorem. Let $M$ be a complex von Neumann algebra and $k \geq 2$. Every bounded $k$-cocycle $f$ in $C_b^k(M,M)$ is $dg$ for some bounded $(k-1)$-cochain $g$ in $C_b^{k-1}(M,M)$ with $\|g\| \leq \|f\|$. Consequently $H_b^k(M,M) = 0$.

Combined with $H_b^1(M,M)=0$ from Kadison (1966) and Sakai (1966), the theorem gives $H_b^k(M,M)=0$ for every $k \geq 1$ and every complex von Neumann algebra. Boundedness means the operator norm throughout; complete boundedness is not assumed, and the primitive is a genuine bounded map, not a class modulo closures.

The vanishing in degree one is the derivation equation: Kadison and Sakai proved in 1966 that every derivation of a von Neumann algebra into itself is inner. Kadison and Ringrose proved vanishing in every degree for type I and hyperfinite algebras, and a companion theorem produces a primitive with values in $B(H)$ when a cocycle on a concrete $M \subset B(H)$ is fed with $M$-valued inputs; keeping the primitive $M$-valued is the self-coefficient subtlety. Johnson, Kadison, and Ringrose proved that continuous cohomology equals separately normal cohomology, so any bounded cocycle differs from a separately normal one by a bounded coboundary. Akemann's continuity theorem transfers separate normality into 2-norm continuity on operator-norm balls. These are the classical reductions.

./jkr-1972-reduction-normal-cohomology.pdf
./kadison-ringrose-1971-cohomology-oa-i.pdf
./kadison-ringrose-1971-cohomology-oa-ii.pdf

Until now every vanishing theorem for the finite algebra carried a structural hypothesis. Christensen, Effros, and Sinclair proved vanishing whenever the $II_1$ summand tensorial-absorbs the hyperfinite factor, using completely bounded methods. Sinclair and Smith proved vanishing for separable-predual algebras with a Cartan maximal abelian subalgebra. Cameron removed the separability hypothesis in the $II_1$-factor case. Christensen, Pop, Sinclair, and Smith proved vanishing for every type $II_1$ factor with property Gamma. Qian and Shen extended that to generalised Gamma, and to finite algebras with nontrivial center under mild spectral hypotheses. Pop and Smith proved degree-two vanishing for tensor products of any two type $II_1$ algebras, with no separability assumption. None of these hypotheses is required in the general case, at any degree $k \geq 2$.

./ces-1987-cb-multilinear-maps.pdf
./cpss-2003-hochschild-gamma.pdf
./pop-smith-2010-tensor-products.pdf
./qian-shen-2015-generalized-gamma.pdf
./cameron-2009-cartan-factors.pdf

The freeness input is Popa's. Free-independent sequences supply finite unitary configurations with prescribed small word traces, and the ultraproduct freeness theorem gives exact free Haar unitaries inside a tracial ultrapower. Haagerup's length estimate bounds the operator norm of a word polynomial over free Haar unitaries by the coefficient $\ell^2$ norm times a linear factor in the word length. Voiculescu's and Banica's analyses of the polar decomposition of circular elements control the spectral distribution of a free-Poisson modulus. Furstenberg's martingale construction supplies the $L^2$ convergence used in the Liouville step. On the operator-algebra side, Dixmier and Umegaki give conditional expectations onto subalgebras; Blackadar's monograph fixes the direct-integral decomposition, the central decomposition of a finite algebra into tracial pieces, and the existence of unital matrix systems inside a $II_1$ algebra.

./popa-1995-free-independent-sequences.pdf
./popa-2014-independence-ultraproduct.pdf
./voiculescu-1990-circular-semicircular.pdf
./banica-1996-polar-circular.pdf
./haagerup-1979-non-nuclear-cstar.pdf
./dixxmier-1953-formes-lineaires.pdf
./blackadar-2006-operator-algebras.pdf

A single symmetric probability measure on the unitary group of a finite tracial algebra is built with two properties: the positive-mass atoms are 2-norm dense, and along a subsequence the joint scalar Haar moments of many independent copies of the subsequence convolution power approximate those of a free Haar family. Popa's ultraproduct freeness supplies the finite configurations; a combinatorial lemma about fresh letters in product paths converts a configuration into a bounded-trace unitary tuple; increasingly rare, increasingly large groups of tuples stack into one measure.

A Liouville theorem for that measure says that a bounded linear $T : M \to M$, 2-norm continuous on operator-norm balls and harmonic for the measure, is $T(x) = T(1) x$. The martingale $D(W_n) = T(W_n) W_n^*$ in $L^2(M,\tau)$ has a limit. If the limit is not deterministic, two distinct support points $a,b$ produce, via forward and reverse endpoints of the walk, free Haar unitaries $g_i$ in a tracial ultrapower and a bounded linear $R$ with $R(w) = a w$ or $b w$ on signed words according to their first letter.

A rigidity theorem forbids that split: for a unitary $V$ in a finite tracial algebra and a bounded linear $S$, power identities $S(V^k) = aV^k$ and $S(V^{-k}) = bV^{-k}$ imply $a = b$. The reduction from first-letter identities to power identities uses a block coefficient estimate over noncrossing partial matchings, Haagerup's length inequality, the polar decomposition of the free-Haar sum $s = r^{-1/2}\sum g_j$ whose squared modulus has free-Poisson moments, and a spectral argument with uniformly bounded sine sums.

Assembly: for a separately normal bounded $k$-cocycle on a separably acting finite tracial algebra, averaging in the last slot and taking a Cesaro limit in the pointwise ultraweak topology produces a bounded multilinear $F$ with $PF = F$. Evaluation at $1$ gives a $(k-1)$-cochain $h$ with $\|h\| \leq \|f\|$ and $F(a,x) = h(a)x$ by Liouville. Averaging the cocycle equation in the same slot yields $dh = \pm f$. Compactness with respect to conditional expectations onto countably generated subalgebras removes separability. A central decomposition into tracial pieces, together with a small homotopy correcting mixed inputs, removes the trace hypothesis.

A tracial von Neumann algebra $(M,\tau)$ with faithful normal trace has the 2-norm $\|x\|_2 = \tau(x^* x)^{1/2}$, finer than the operator norm. A family of unitaries $g_1, g_2, \ldots$ is free Haar if $\tau(q(g_{i_1},\ldots,g_{i_r})) = 0$ for every nonidentity reduced word $q \in F_r$ and every finite subfamily.

Random walk theorem. Let $(M,\tau)$ be a finite tracial algebra of type $II_1$ with separable predual. There exist a symmetric countably supported probability measure $\mu$ on $U(M)$ whose positive-mass atoms are 2-norm dense, and a strictly increasing sequence $n_\ell \to \infty$, such that for every $r \geq 1$, every nonempty finite set $W \subset F_r \setminus \{e\}$, and every $\eta > 0$, if $G_1^{(\ell)},\ldots,G_r^{(\ell)}$ are independent with law $\mu^{*n_\ell}$,
$$
\Pr\bigl( \max_{w \in W} | \tau(w(G_1,\ldots,G_r)) | \geq \eta \bigr) \to 0.
$$

Popa's ultraproduct freeness is the source of true independence. Let $P$ be a $II_1$ factor and $\omega$ a free ultrafilter on $\mathbb{N}$. Its tracial ultrapower is $P^\omega = \ell^\infty(\mathbb{N},P)/\mathcal{I}_\omega$ with $\mathcal{I}_\omega = \{(x_m) : \lim_{m \to \omega} \|x_m\|_2 = 0\}$; it is a finite tracial algebra with trace $\tau^\omega((x_m)^\omega) = \lim_{m\to\omega} \tau(x_m)$. If $B \subset P^\omega$ is a unital von Neumann subalgebra with separable predual, then for every $d \geq 1$ there are Haar unitaries $u_1,\ldots,u_d \in P^\omega$ such that $B, W^*(u_1),\ldots,W^*(u_d)$ are free. The case $d=1$ is Popa's Theorem 0.1(a) with every coordinate algebra and subalgebra equal to $P$; iterating supplies the rest, because a diffuse abelian tracial subalgebra contains a Haar unitary and the constructed abelian subalgebra may be taken free from any separable number of earlier additions.

./popa-2014-independence-ultraproduct.pdf

Distinguished letters. A path expression in a unitary alphabet $\{u_\alpha^{\pm 1}\}$ is a finite product of formal letters with fixed coefficients in $U(M)$. Occurrences are counted before any free reduction. Suppose $B_0, W^*(u_1),\ldots,W^*(u_d)$ are free subalgebras with each $u_j$ Haar, and $y_1,\ldots,y_r$ are path expressions with coefficients in $U(B_0)$ such that each $y_i$ contains a distinguished letter $h_i$ appearing exactly once in the whole list, with the $h_i$ pairwise distinct. Then for every nonidentity $w \in F_r$,
$$
\tau(w(y_1,\ldots,y_r)) = 0.
$$
Put every non-distinguished $u_j$ into the coefficient algebra and let $C = W^*(u_{h_1},\ldots,u_{h_r})$; then $B_0$ and $C$ are free. Writing $y_i = A_i u_{h_i}^{\epsilon_i} B_i$ with $A_i, B_i \in U(B_0)$, the string of surviving distinguished letters in $w(y_1,\ldots,y_r)$ is reduced whenever $w$ is. Expanding each coefficient into its scalar and trace-zero parts, every monomial is a product of alternating centered factors from $B_0$ and $C$ with at least one $C$-factor; freeness and Haar-ness force the trace to vanish. If no centered coefficient factor appears, the monomial is a scalar multiple of a nonempty reduced Haar word and again vanishes.

Finite path tests. Fix a finite $F \subset U(M)$, an integer $d \geq 1$, an $\varepsilon > 0$, and a finite collection of tests, each consisting of a list of path expressions in the alphabet $F \cup \{u_j^{\pm 1} : 1 \leq j \leq d\}$ and a nonidentity word $w \in F_r$, with pairwise-distinct distinguished letters $h_i$, each occurring exactly once in $y_i$ and nowhere else in the whole list. There are $u_1,\ldots,u_d \in U(M)$ satisfying every test to tolerance $\varepsilon$. For a factor $M = P$, apply Popa's freeness in $P^\omega$ to the coefficient algebra generated by $F$; the distinguished-letters lemma gives exactly zero in $P^\omega$; every unitary of $P^\omega$ has a coordinatewise-unitary representative; the finite list of trace conditions therefore gives a common coordinate where all hold to within $\varepsilon$. For a general $M$ with center, decompose as a direct integral over a probability space $(\Omega,\nu)$, and choose countably many measurable unitary sections that are fiberwise 2-norm dense (using exponentials of $\mathbb{Q}+i\mathbb{Q}$-polynomials in countably many generators). Each test holds fiberwise by the factor case, and measurable selection over the good fibers yields unitary sections in $M$ satisfying all tests.

./blackadar-2006-operator-algebras.pdf

Level construction of $\mu$. Fix a 2-norm dense sequence $(v_j)$ in $U(M)$, and $\mu_0 = \sum_{j \geq 1} 2^{-j-1}(\delta_{v_j} + \delta_{v_j^{-1}})$. For $\ell \geq 1$ put $p_\ell = 2^{-2^\ell}$, $n_\ell = p_\ell^{-3/2} = 2^{3\cdot 2^{\ell-1}}$, $K_\ell = \ell n_\ell$, $d_\ell = K_\ell^4$, and $p_0 = 1 - \sum_{\ell \geq 1} p_\ell > 0$ (since $\sum p_\ell \leq \sum 4^{-\ell} < 1$). Inductively at stage $\ell$: choose a finite symmetric $F_\ell \subset U(M)$ of base atoms with $\mu_0(F_\ell) > 1 - K_\ell^{-2}$; let the coefficient alphabet be $F_\ell \cup \bigcup_{j < \ell} \{u_{j,h}^{\pm 1}\}$; enumerate every ordered list of $\ell$ length-$n_\ell$ paths with coefficients in this alphabet, each path containing an unknown label from a fresh $d_\ell$-element set with no repeats anywhere in the list; impose $|\tau| < 1/\ell$ for every nonidentity reduced word of length $\leq \ell$; and apply the finite-path-tests corollary to choose $\{u_{\ell,h} : 1 \leq h \leq d_\ell\}$. Put $\mu_\ell = (2 d_\ell)^{-1} \sum_{h=1}^{d_\ell}(\delta_{u_{\ell,h}} + \delta_{u_{\ell,h}^{-1}})$ and $\mu = p_0 \mu_0 + \sum_{\ell \geq 1} p_\ell \mu_\ell$. The measure is symmetric and countably supported by construction. Every $v_j$ has positive mass in $\mu_0$, so $(v_j)$ is a positive-mass atom set and is 2-norm dense in $U(M)$.

Bad events. Sample $\ell$ independent paths of length $n_\ell$, for $K_\ell$ increments in total. Each increment is a level $j$ drawn with probability $p_j$ and then either a base atom drawn with law $\mu_0$ (level $0$) or a level-$j$ label drawn uniformly from $d_j$. Four failure modes matter:
(a) some increment has level $> \ell$;
(b) some level-0 increment lies outside $F_\ell$;
(c) some level-$\ell$ label is repeated;
(d) some path has no level-$\ell$ increment.
Union bounds give
$$
\begin{aligned}
\Pr(\text{bad}) \leq &\; K_\ell \sum_{j>\ell} p_j + K_\ell p_0 \mu_0(U(M) \setminus F_\ell) \\
&+ \binom{K_\ell}{2}/d_\ell + \ell (1 - p_\ell)^{n_\ell}.
\end{aligned}
$$
Every term tends to $0$. Because $p_{\ell+1} = p_\ell^2$, one has $\sum_{j>\ell} p_j \leq \frac{16}{15} p_\ell^2$ and so $K_\ell \sum_{j>\ell} p_j \leq \frac{16}{15}\ell \sqrt{p_\ell}$; also $K_\ell p_0 \mu_0(U(M)\setminus F_\ell) \leq 1/K_\ell$; further $\binom{K_\ell}{2}/d_\ell \leq 1/(2 K_\ell^2)$; and $\ell (1-p_\ell)^{n_\ell} \leq \ell \exp(-p_\ell^{-1/2})$.

Conclusion. Let $\ell$ satisfy $\ell \geq r$, $\ell \geq \max_{w\in W}|w|$, and $1/\ell < \eta$. Sampling $r$ independent paths of length $n_\ell$, the bad event has probability $\to 0$, while on the good event the sampled paths are exactly one of the length-$\ell$ lists tested at stage $\ell$, so their scalar traces lie within $1/\ell < \eta$. This gives the theorem.

Continuity and ultrapowers. Let $(N,\rho)$ be a finite tracial algebra. A linear map $R : N \to N$ is bounded-ball 2-continuous if $\sup_j \|x_j\| < \infty$ and $\|x_j\|_2 \to 0$ jointly imply $\|R(x_j)\|_2 \to 0$. This is equivalent to the uniform modulus: for every $r < \infty$ and $\varepsilon > 0$ there is $\delta > 0$ with $\|x\| \leq r$ and $\|x\|_2 < \delta$ implying $\|R(x)\|_2 < \varepsilon$. Equivalence: fail one and diagonalize, to contradict the other. If bounded-ball 2-continuity holds, the induced map
$$
R^\omega : N^\omega \to N^\omega, \qquad R^\omega([(x_j)]_\omega) = [(R(x_j))]_\omega,
$$
is well defined on the tracial ultrapower, bounded by $\|R\|$, and again bounded-ball 2-continuous; the same holds after a second ultrapower, on $(N^\omega)^\nu$. Well-definedness: a bounded representative with $\|x_j\|_2 \to 0$ along $\omega$ has $(R(x_j))$ also converging to zero by the modulus. Modulus transfer to $N^\omega$: pick a representative with $\sup_j \|x_j\| \leq r + 1$ from the quotient-norm hypothesis $\|X\| \leq r$; then the pointwise modulus applies.

Liouville theorem. Let $(M,\tau)$ be a finite type $II_1$ algebra with separable predual, $\mu$ a symmetric countably supported probability measure on $U(M)$ with 2-norm dense positive-mass atoms and with a sequence $n_\ell \to \infty$ satisfying the joint Haar moment condition, and $T : M \to M$ a bounded linear bounded-ball 2-continuous map harmonic for $\mu$, in the sense that
$$
T(x) = \mathbb{E}_{U \sim \mu}[T(x U) U^*] \qquad (x \in M).
$$
Then $T(x) = T(1) x$ for every $x \in M$.

Proof. Write $C = \|T\|$ and $H = L^2(M,\tau)$. For $g \in U(M)$ set $D(g) = T(g) g^*$. Let $U_1, U_2, \ldots$ be independent with law $\mu$, set $W_n = U_1 \cdots U_n$ and $\mathcal{F}_n = \sigma(U_1,\ldots,U_n)$, and $Z_n = D(W_n) \in H$. Each $Z_n$ lies in the operator-norm ball $K_C = \{x \in M : \|x\| \leq C\}$, and $K_C$ is closed in $H$: from a 2-norm convergent sequence in $K_C$, ultraweak compactness gives a subnet converging ultraweakly to some $x \in K_C$, and testing against $\tau(y^*\,\cdot)$ for $y \in M$ identifies $x$ with the $H$-limit. Conditional expectation:
$$
\mathbb{E}[Z_{n+1} \mid \mathcal{F}_n] = \mathbb{E}[T(W_n U_{n+1}) U_{n+1}^* W_n^* \mid \mathcal{F}_n] = T(W_n) W_n^* = Z_n,
$$
using the harmonicity identity with $x = W_n$. So $(Z_n)$ is an $H$-valued martingale with $\|Z_n\| \leq C$; by Hilbert-space martingale convergence, $Z_n \to Z_\infty$ in $L^2(\Omega; H)$ and $Z_n = \mathbb{E}[Z_\infty \mid \mathcal{F}_n]$. The error $e_k = \mathbb{E}\|Z_\infty - Z_k\|_2^2$ decreases to zero, and orthogonality of increments gives the Cauchy estimate $\mathbb{E}\|Z_n - Z_k\|_2^2 \leq e_k$ for $n \geq k$.

Deterministic case. If $Z_\infty = c$ almost surely for some $c \in H$, then $Z_0 = c$, so $c = T(1)$. For every positive-mass atom $u$ of $\mu$, $D(u) = c$ in $L^2$, hence $T(u) = c u$ in the operator norm (both sides lie in $M$ and faithfulness kills any zero-2-norm difference). Since positive-mass atoms are 2-norm dense in $U(M)$, bounded-ball 2-continuity extends $T(u) = cu$ to every unitary: for $u \in U(M)$ and atoms $u_j \to u$ in 2-norm, $\|T(u) - T(u_j)\|_2 \to 0$ and $\|c u - c u_j\|_2 \leq \|c\|\,\|u - u_j\|_2 \to 0$. Finite linear combinations of unitaries exhaust $M$, so $T(x) = c x$.

Non-deterministic case. Suppose $Z_\infty$ is not deterministic. Its law is a Borel probability on the separable $H$, so its support contains two distinct points $a, b \in K_C$, hence $a, b \in M$ with $\|a\|, \|b\| \leq C$. For every $\delta > 0$ there are $\eta_\delta > 0$ and $N_\delta$, such that whenever $n \geq N_\delta$ and $G$ has law $\mu^{*n}$,
$$
\Pr(\|D(G) - a\|_2 < \delta \text{ and } \|D(G^{-1}) - b\|_2 < \delta) \geq \eta_\delta.
$$
Write $G = U_1 \cdots U_n$, $h = \lfloor n/2 \rfloor$, and
$$
A_n = D(U_1 \cdots U_h), \qquad B_n = D(U_n^{-1} \cdots U_{n-h+1}^{-1}).
$$
These involve disjoint increments and are independent; symmetry of $\mu$ gives $A_n \sim B_n \sim Z_h$. The martingale Cauchy estimate applied to the forward increment string and, separately, to its reverse-inverse conjugate (also a string of independent law-$\mu$ increments) gives
$$
\mathbb{E}\|D(G) - A_n\|_2^2 \leq e_h, \qquad \mathbb{E}\|D(G^{-1}) - B_n\|_2^2 \leq e_h.
$$
Let $p_a = \Pr(\|Z_\infty - a\|_2 < \delta/3) > 0$ and $p_b = \Pr(\|Z_\infty - b\|_2 < \delta/3) > 0$. Mean-square convergence gives, for large $h$, $\Pr(\|Z_h - a\|_2 < 2\delta/3) \geq p_a/2$ and likewise with $b$. Independence plus Markov bounds the two-point probability from below by $p_a p_b / 4 - 18 e_h / \delta^2$, positive uniformly in large $n$.

Independent blocks. Let $G_1,\ldots,G_m$ be independent with law $\mu^{*n}$, and $v(G) = G_{i_1}^{\sigma_1} \cdots G_{i_t}^{\sigma_t}$ with pairwise distinct unsigned indices. Concatenating the inverted and reversed blocks makes a length-$tn$ random walk with independent law-$\mu$ increments; the pair $(D(v(G)), D(G_{i_1}^{\sigma_1}))$ thus has the law of $(Z_{tn}, Z_n)$, and
$$
\mathbb{E}\|D(v(G)) - D(G_{i_1}^{\sigma_1})\|_2^2 \leq e_n.
$$

Finite selection. For $m \geq 1$ and tolerance $1/m$, let $E_{m,n}$ be the event
$$
\|D(G_i) - a\|_2 < 1/m, \qquad \|D(G_i^{-1}) - b\|_2 < 1/m, \qquad 1 \leq i \leq m.
$$
Independence gives $\Pr(E_{m,n}) \geq c_m > 0$ for large enough $n$ (take $c_m = (\eta_{1/m})^m$). Impose moreover
$$
|\tau(q(G_1,\ldots,G_m))| < 1/m, \qquad q \in \mathcal{Q}_m,
$$
for $\mathcal{Q}_m$ the nonidentity reduced words of length $\leq m$, and
$$
\|D(v(G)) - D(G_{i_1}^{\sigma_1})\|_2 < 1/m, \qquad v \in \mathcal{V}_m,
$$
for $\mathcal{V}_m$ the nonempty signed words on $\{G_1,\ldots,G_m\}$ with pairwise distinct unsigned indices. The random-walk theorem uniformly controls the trace conditions along $n_\ell$; Markov plus a union bound over $\mathcal{V}_m$ controls the block conditions. The intersection of $E_{m,n}$ and the constraint sets has probability at least $c_m / 2$ for suitable $n = n_{\ell_m}$. Choose a deterministic tuple $(u_1^{(m)},\ldots,u_m^{(m)})$ in the intersection.

Ultrapower. Fix a free ultrafilter $\omega_1$, set $u_i^{(m)} = 1$ for $m < i$, define $N = M^{\omega_1}$, $g_i = [(u_i^{(m)})]_{\omega_1}$, and $R = T^{\omega_1}$. Each $g_i$ is unitary. The trace conditions become exact,
$$
\tau^{\omega_1}(q(g_1,\ldots,g_r)) = 0
$$
for every nonidentity reduced $q$, so $\{g_i\}$ is a free Haar family in $N$. The block conditions and endpoint conditions become, for every signed word $w = g_{i_1}^{\sigma_1}\cdots g_{i_t}^{\sigma_t}$ with pairwise distinct unsigned indices,
$$
R(w) = a w \text{ if } \sigma_1 = 1, \qquad R(w) = b w \text{ if } \sigma_1 = -1,
$$
with $a, b$ the constant images in $N$ of the two support points. The rigidity theorem applied to $(N,\tau^{\omega_1})$, $R$, $\{g_i\}$, $a$, $b$ forces $a = b$ in $N$; the trace-preserving constant embedding then forces $a = b$ in $M$, contradicting distinctness. So $Z_\infty$ is deterministic and the deterministic case concludes the Liouville theorem.

Rigidity theorem. Let $(N,\tau)$ be a finite tracial algebra with faithful normal trace. Let $R : N \to N$ be bounded $\mathbb{C}$-linear and bounded-ball 2-continuous. Let $g_1, g_2, \ldots$ be a free Haar family in $N$. Suppose that for every signed word $w = g_{i_1}^{\varepsilon_1}\cdots g_{i_q}^{\varepsilon_q}$ with $q \geq 1$, $\varepsilon_j \in \{+1,-1\}$, and $i_1,\ldots,i_q$ pairwise distinct unsigned indices,
$$
R(w) = a w \ \ (\varepsilon_1 = +1), \qquad R(w) = b w \ \ (\varepsilon_1 = -1)
$$
for some fixed $a, b \in N$. Then $a = b$.

Length estimate. Let $\lambda$ be the left regular representation of $F_r$ on $\ell^2(F_r)$. For finitely supported $c = (c_w)_{|w| = q}$,
$$
\Bigl\| \sum_{|w| = q} c_w \lambda(w) \Bigr\| \leq (q+1) \bigl( \sum |c_w|^2 \bigr)^{1/2}.
$$
Free Haar unitaries $h_1,\ldots,h_r$ determine a trace-preserving normal $*$-isomorphism of $L(F_r) = W^*(\lambda(F_r))$ onto $W^*(h_1,\ldots,h_r)$ sending $\lambda(w)$ to $w(h)$. Scalar-coefficient word polynomials therefore have the same operator norm and the same 2-norm in either algebra. The bound itself: act by $L = \sum_{|w|=q} c_w \lambda(w)$ on the standard basis $e_x$, and for each product $wx$ record the number $t \in \{0,\ldots,q\}$ of cancellations at the joining boundary. This decomposes $L = \sum_t L_t$, with each $L_t$ acting as a block-diagonal sum over the surviving right-tail $h$; within a block $B_h$, the map $(u,v) \mapsto uv$ into length-$(q-t)$ reduced words is injective, so $\|B_h\| \leq \|B_h\|_{\mathrm{HS}}$ and $\|L_t\| \leq \|c\|_{\ell^2}$; the triangle inequality over $q+1$ values of $t$ gives the bound.

Second ultrapower and free summations. Put $s_r = r^{-1/2}\sum_{j=1}^r g_j$, so $\|s_r\|_2 = 1$ and, by the length estimate applied to coefficients $(c_w)_{|w|=1}$ given by the $r$ generator labels, $\|s_r\| \leq 2$. Fix a free ultrafilter $\omega_2$ on $\mathbb{N}$, set $\widehat{N} = N^{\omega_2}$, $\widehat{\tau} = \tau^{\omega_2}$, $s = (s_r)_{\omega_2} \in \widehat{N}$, and $S = R^{\omega_2}$. The map $S$ is bounded and bounded-ball 2-continuous; $a$ and $b$ embed as constant sequences.

Block identities. For $k \geq 1$ and $d_1,\ldots,d_k \geq 0$, set $B_d(z) = z(z^* z)^d$ and $P(z) = B_{d_1}(z)\cdots B_{d_k}(z)$. Then
$$
S(P(s)) = a P(s), \qquad S(P(s)^*) = b P(s)^*.
$$
Write $P(s_r)$ as an ordered sum of length-$L$ products of factors of $s_r$ and $s_r^*$ with $L = \sum_j (2d_j + 1)$. Every product begins and ends positively, and no cancellation occurs at the boundary between adjacent blocks, so every reduced word in the expansion begins and ends positively. Group the terms by the reduced word $w$ that survives. For a reduced word $w$ of length $q$ with $1 \leq q \leq L$, the number of noncrossing partial matchings of the $L$ positions that produce $w$ as the survivor is at most $3^L$, encoded injectively by labeling each position as opening endpoint, closing endpoint, or unmatched. Each canceled pair contributes at most $r$ choices for its shared label. Each surviving letter carries coefficient $r^{-1/2}$ from $s_r$. Hence
$$
|c_w(r)| \leq 3^L \cdot r^{(L-q)/2} \cdot r^{-L/2} = 3^L r^{-q/2}.
$$
Let $E_r$ be the sum of those terms with survivor of length $\leq L$ and at least one repeated unsigned index. For fixed $q \geq 2$, the number of such survivor words is at most $2^q \binom{q}{2} r^{q-1}$: choose the repeated pair, its common label, the labels of the $q-2$ remaining survivors, and all signs. Orthonormality of Haar words and the coefficient bound gives
$$
\|E_{r,q}\|_2^2 \leq C_L 2^q \binom{q}{2} r^{-1}, \qquad 2 \leq q \leq L,
$$
with $C_L$ depending only on $L$. The length estimate then gives
$$
\|E_r\| \leq \sum_{q=2}^L (q+1)\|E_{r,q}\|_2 = O_L(r^{-1/2}) \to 0.
$$
No scalar term can occur, because at least one positive letter survives in each product. Set $Q_r = P(s_r) - E_r$. Every surviving word in $Q_r$ has distinct unsigned indices and starts positively, so $R(Q_r) = a Q_r$. Boundedness $\|P(s_r)\| \leq 2^L$ keeps $(Q_r)$ bounded. In $N$,
$$
\|R(P(s_r)) - aP(s_r)\| \leq (\|R\| + \|a\|)\|E_r\| \to 0,
$$
and passage to $\widehat{N}$ proves $S(P(s)) = a P(s)$. The adjoint version with $b$ in place of $a$: $P(s_r)^*$ expands with reversed block order and every reduced word starts and ends negatively, and the retained sum $Q_r^*$ has all surviving words starting negatively with distinct unsigned indices.

Polar unitary. Put $A = s^* s$, so $0 \leq A \leq 4$. For $d \geq 1$ the quantity $\tau((s_r^* s_r)^d) = r^{-d}$ times the number of labelings of the alternating $2d$-position word $-+-\cdots-+$ that reduce to the identity. Identity reduction pairs all positions noncrossingly, using at most $d$ unsigned labels; assignments with at most $d-1$ distinct labels are $O_d(r^{d-1})$ in number. For each $d$, the number of labelings using exactly $d$ labels equals $C_d (r)_d$ where $C_d = \binom{2d}{d}/(d+1)$ and $(r)_d = r(r-1)\cdots(r-d+1)$. Hence
$$
\widehat{\tau}(A^d) = \lim_{r \to \omega_2} \tau((s_r^* s_r)^d) = C_d, \qquad d \geq 0.
$$
The measure
$$
d\nu(x) = \frac{1}{2\pi} \sqrt{\frac{4-x}{x}}\, 1_{(0,4)}(x)\, dx
$$
has moments $\int_0^4 x^d d\nu = C_d$ for every $d \geq 0$: substitution $x = 4t$ reduces the moment to a beta integral $B(d+1/2, 3/2)$ up to the normalizing constant $2 \cdot 4^d/\pi$, and the value is the Catalan number. Equality of all moments on $[0,4]$ gives equality of $\nu$ and the spectral distribution of $A$. Because $\nu$ has no atom at $0$ and $\widehat{\tau}$ is faithful, $1_{\{0\}}(A) = 0$. The polar decomposition $s = V A^{1/2}$ then gives $V^* V = 1$; traciality plus faithfulness gives $V V^* = 1$, and $V$ is unitary.

Transfer to $V^k$. Fix $k \geq 1$ and $\varepsilon > 0$, and put $x_\varepsilon = s(A+\varepsilon 1)^{-1/2} = V(A/(A+\varepsilon 1))^{1/2}$. For a scalar polynomial $p(t) = \sum_{d=0}^D c_d t^d$,
$$
[sp(A)]^k = \sum_{d_1,\ldots,d_k = 0}^D c_{d_1}\cdots c_{d_k}\, [s A^{d_1}]\cdots[s A^{d_k}],
$$
and each summand is a product of blocks covered by the block identities. Complex linearity of $S$ yields $S([sp(A)]^k) = a[sp(A)]^k$; the same on adjoints yields $S(([sp(A)]^k)^*) = b([sp(A)]^k)^*$. Polynomials converging uniformly on $[0,4]$ to $(t+\varepsilon)^{-1/2}$ give, via boundedness of $S$, $S(x_\varepsilon^k) = a x_\varepsilon^k$ and $S((x_\varepsilon^k)^*) = b(x_\varepsilon^k)^*$. As $\varepsilon \to 0$, $x_\varepsilon^k \to V^k$ and $(x_\varepsilon^k)^* \to V^{-k}$ in operator norm; bounded-ball 2-continuity of $S$, equivalently operator-norm continuity on the bounded sequence $x_\varepsilon^k$ together with boundedness of $S$, transfers again:
$$
S(V^k) = a V^k, \qquad S(V^{-k}) = b V^{-k}, \qquad k \geq 1.
$$

Sine sums and identity of $a, b$. Write $H_m(z) = \sum_{k=1}^m z^k/k$ and $h_m = \sum_{k=1}^m 1/k$. Periodicity and oddness of the imaginary part reduce to $t = |\theta| \in [0,\pi]$. For $t > 0$ put $K = \lfloor 1/t \rfloor$; then $|\sum_{k \leq \min(K,m)} \sin(kt)| \leq K t \leq 1$, and for any integer interval $[p,q]$,
$$
\bigl| \sum_{k=p}^q \sin(kt) \bigr| \leq \bigl| \sum_{k=p}^q e^{ikt} \bigr| \leq 1/\sin(t/2) \leq \pi/t.
$$
Abel summation from $p = K+1$ gives $\bigl| \sum_{k=p}^m \sin(kt)/k \bigr| \leq \pi$ uniformly in $m$, so $|\Im H_m(e^{i\theta})| \leq B = 1+\pi$.

Fix $m$ and $\lambda \in \mathbb{T}$. Set $X_{m,\lambda} = \Re H_m(\lambda V)$ and $Y_{m,\lambda} = \Im H_m(\lambda V)$, so $H_m(\lambda V) = X_{m,\lambda} + i Y_{m,\lambda}$ with $X_{m,\lambda} = X_{m,\lambda}^*$ and $Y_{m,\lambda} = Y_{m,\lambda}^*$. Functional calculus of $V$ gives $\|Y_{m,\lambda}\| \leq B$. From $S(V^k) = aV^k$ and $S(V^{-k}) = bV^{-k}$,
$$
S(H_m(\lambda V) - H_m(\lambda V)^*) = a H_m(\lambda V) - b H_m(\lambda V)^* = (a-b)X_{m,\lambda} + i(a+b)Y_{m,\lambda}.
$$
The argument of $S$ has norm at most $2B$, so
$$
\|(a-b)X_{m,\lambda}\| \leq C = B(2\|S\| + \|a\| + \|b\|),
$$
uniformly in $m$ and $\lambda$. Fix $m$, let $I \subset \mathbb{T}$ be a half-open arc of angular length at most $1/m$, choose $\lambda$ to rotate the midpoint of $I$ to $1$, and set $e = 1_I(V)$. On $I$ the rotated argument $\theta$ satisfies $|\theta| \leq 1/m$, so $\Re H_m(e^{i\theta}) \geq \cos(1) h_m$. Borel functional calculus produces $F$ supported on $I$ with $X_{m,\lambda} F = e$ and $\|F\| \leq 1/(\cos(1) h_m)$. Therefore, writing $c = a-b$,
$$
\|c e\| = \|c X_{m,\lambda} F\| \leq \frac{C}{\cos(1) h_m} =: \delta_m \to 0.
$$
Partition $\mathbb{T}$ into $m$ half-open arcs of length $1/m$ with projections $e_1,\ldots,e_m$; then $1 = \sum_{j=1}^m e_j$ and $\|c\| \leq \sum_{j=1}^m \|ce_j\| \leq m \delta_m \to 0$. Therefore $c = 0$ and $a = b$.

For the cohomology, fix $k \geq 2$ and $f \in C_b^k(M,M)$ with $df = 0$. By Johnson-Kadison-Ringrose normal reduction, every bounded $k$-cocycle on $M$ with values in $M$ differs from a separately normal bounded $k$-cocycle $f_{\mathrm{sep}}$ by a bounded coboundary $db$, with $b$ a bounded $(k-1)$-cochain. A bounded primitive for $f_{\mathrm{sep}}$ yields a bounded primitive for $f$ by adding $b$. Assume that $f$ is separately normal. Akemann's theorem makes $B_a(x) = f(a_1,\ldots,a_{k-1},x)$ bounded-ball 2-continuous on $M$ for every fixed $(k-1)$-tuple $a$.

./jkr-1972-reduction-normal-cohomology.pdf

Separable-predual, tracial case. Let $(M,\tau)$ be type $II_1$ with separable predual and faithful normal trace. Fix the walk measure $\mu = \sum_s p_s \delta_{u_s}$ from the random-walk theorem. For a bounded $k$-cochain $\Phi$ put
$$
(P\Phi)(a_1,\ldots,a_{k-1},x) = \sum_s p_s\, \Phi(a_1,\ldots,a_{k-1}, x u_s)\, u_s^*.
$$
The sum converges in operator norm, and $\|P\Phi\| \leq \|\Phi\|$. Iteration gives
$$
(P^j \Phi)(a,x) = \mathbb{E}[\Phi(a, x W_j) W_j^*], \qquad W_j = U_1 \cdots U_j,
$$
with the $U_i \sim \mu$ independent. Set $F_n = n^{-1}\sum_{j=0}^{n-1} P^j f$. Then $\|F_n\| \leq \|f\|$ and $\|P F_n - F_n\| \leq 2\|f\|/n$.

Compactness and the fixed point. For fixed $(a, x)$ the values $F_n(a,x)$ lie in the operator-norm ball of radius $\|f\|\prod_i\|a_i\|\|x\|$, ultraweakly compact. The product over tuples is compact by Tychonoff. Multilinearity is closed for the pointwise ultraweak topology, so some subnet of $(F_n)$ converges pointwise ultraweakly to a bounded multilinear $F$ with $\|F\| \leq \|f\|$. For $\Phi_\lambda \to \Phi$ pointwise ultraweakly with uniform bound $C$, truncating the sum after $L$ terms leaves a remainder of size at most $2C\prod_i\|a_i\|\|x\|\sum_{s>L} p_s$, so $P \Phi_\lambda \to P\Phi$; hence $PF = F$.

Liouville. For each $(k-1)$-tuple $a$ put $T_a(x) = F(a, x)$. Akemann's theorem applied to $B_a(x) = f(a, x)$, together with the identity that $x \mapsto B_a(x u)u^*$ carries the same modulus for every $u \in U(M)$, gives $T_a$ bounded-ball 2-continuous; the 2-norm is ultraweakly lower semicontinuous, so passing to the pointwise ultraweak limit of $F_n(a, \cdot)$ preserves the uniform modulus. Harmonicity reads $T_a(x) = \mathbb{E}[T_a(x U)U^*]$, which is $PT_a = T_a$. The Liouville theorem gives $T_a(x) = T_a(1) x = h(a) x$ for $h(a) = F(a,1)$, and $h$ is a bounded $(k-1)$-cochain with $\|h\| \leq \|f\|$.

Averaging. Evaluate $df = 0$ at $(a_1,\ldots,a_k,v)$, multiply on the right by $v^*$, and average in $v \sim \nu_n = n^{-1}\sum_{j=0}^{n-1}\mu^{*j}$:
$$
0 = a_1 h_n(a_2,\ldots,a_k) + \sum_{j=1}^{k-1}(-1)^j h_n(a_1,\ldots,a_j a_{j+1},\ldots,a_k) + (-1)^k F_n(a_1,\ldots,a_k) + (-1)^{k+1} f(a_1,\ldots,a_k),
$$
with $h_n(a) = F_n(a,1)$. The last term is unchanged because $vv^* = 1$. Passing to the limit gives $dh = (-1)^k f$; set $g = (-1)^k h$ to obtain $dg = f$ and $\|g\| \leq \|f\|$.

Separability removal. Let $M$ be type $II_1$ with a faithful normal trace, separability of predual not assumed. Order finite subsets $E \subset M$ by inclusion. For each $E$ choose a unital von Neumann subalgebra $N_E \subset M$ containing $E$ and $f(E^k)$, countably generated, and containing unital copies of $M_m(\mathbb{C})$ for every $m \geq 2$ (available by splitting $1_M$ into $m$ orthogonal equivalent projections with matrix units, for every $m \geq 2$). The algebra $N_E$ is then a countably generated type $II_1$ algebra with separable predual. Let $\Upsilon_E : M \to N_E$ be the normal trace-preserving conditional expectation; then $f_E = \Upsilon_E \circ (f|_{N_E^k})$ is a separately normal bounded $k$-cocycle on $N_E$, and the separable case supplies $g_E \in C_b^{k-1}(N_E, N_E)$ with $dg_E = f_E$ and $\|g_E\| \leq \|f\|$.

./dixxmier-1953-formes-lineaires.pdf

Extend each $g_E$ to a bounded $(k-1)$-cochain on $M$ by $G_E(x_1,\ldots,x_{k-1}) = g_E(\Upsilon_E x_1, \ldots, \Upsilon_E x_{k-1})$. The bound $\|G_E\| \leq \|f\|$ is inherited. For a tuple $(a_1,\ldots,a_k)$ in $E^k$ the inputs $a_j$, the adjacent products $a_j a_{j+1}$, and $f(a_1,\ldots,a_k)$ all lie in $N_E$; the differential $dG_E$ at that tuple agrees with $f(a_1,\ldots,a_k)$. Ultraweak compactness of operator-norm balls in the space of bounded multilinear maps passes to a pointwise ultraweakly convergent subnet of $(G_E)$, with limit $g$ satisfying $\|g\| \leq \|f\|$. For a fixed tuple $(a_1,\ldots,a_k)$ the finite set $E_0 = \{a_1,\ldots,a_k\}$ lies in every $N_E$ with $E \supset E_0$, so $dG_E$ agrees with $f$ at that tuple for every large $E$; passage to the limit gives $dg = f$.

Central decomposition. Let $M$ be any complex von Neumann algebra, $z_{II_1}$ the central projection onto its type $II_1$ summand, and $z_0 = 1 - z_{II_1}$. On $z_0 M$ the classical cases (type $I$ and hyperfinite, plus the completely bounded methods of Christensen-Effros-Sinclair) supply a bounded primitive $b_0$ of $f_0 = z_0 f$, with $\|b_0\| < \infty$. On $z_{II_1} M$ choose a maximal family $(z_i)_{i \in I}$ of central projections, mutually orthogonal with join $z_{II_1}$, such that every $z_i M$ is a type $II_1$ factor. Each $z_i M$ carries a unique faithful normal trace, and the separability-removed case applied to $f_i(x_1,\ldots,x_k) = z_i f(z_i x_1, \ldots, z_i x_k)$, $x_j \in z_i M$, supplies a bounded $(k-1)$-cochain $b_i$ with $db_i = f_i$ and $\|b_i\| \leq \|f\|$.

Pull back to $M$: $\tilde{b}_i(x_1,\ldots,x_{k-1}) = b_i(z_i x_1, \ldots, z_i x_{k-1})$. Then
$$
d\tilde{b}_i(a_1,\ldots,a_k) = z_i f(z_i a_1,\ldots,z_i a_k).
$$
The right side is the central cut $\mathrm{cut}_{z_i}(z_i f)$, not generally $z_i f$: the $q_i = 1 - z_i$ parts of the inputs are still missing. A homotopy fixes the difference. For a central projection $z$ and $q = 1-z$, define on $n$-cochains with values in $zM$, for $n \geq 2$,
$$
(J_z \psi)(a_1,\ldots,a_{n-1}) = \sum_{i=1}^{n-1}(-1)^i \psi(z a_1, \ldots, z a_{i-1}, q, q a_i, a_{i+1}, \ldots, a_{n-1}),
$$
and $J_z = 0$ in degrees $0$ and $1$. Multilinearity and $z^2 = z$, $q^2 = q$, $zq = 0$ give
$$
dJ_z + J_zd = \mathrm{id} - \mathrm{cut}_z, \qquad \|J_z \psi\| \leq (n-1)\|\psi\|.
$$
Case check on tuples split into $zM$-entries and $qM$-entries. On an all-$zM$ tuple $J_z \psi = 0$ and $\mathrm{cut}_z \psi = \psi$, so both sides vanish. On a tuple whose first $qM$-entry is at position $r$, one term of $J_z \psi$ on the shortened tuple survives; in $(d J_z \psi)$ that term merges the inserted $q$ with $a_r$ with total sign $(-1)^r(-1)^r = +1$ and yields $\psi(a_1,\ldots,a_n)$, matching the target. Merges of a $zM$-entry with a $qM$-entry before the insertion point and after the pair cancel against corresponding terms in $(J_z d)\psi$, with opposite signs from the index shift. Merges of a $zM$-entry with a $qM$-entry elsewhere vanish because $zM \cdot qM = 0$.

Assembly. For $i \in I$ set $H_i = \tilde{b}_i + J_{z_i}(z_i f)$, and when $z_0 \neq 0$ set $H_0 = \tilde{b}_0 + J_{z_0}(z_0 f)$. Then $dH_i = z_i f$ for every $i \in I \cup \{0\}$. The bounds $\|H_i\| \leq k\|f\|$ hold uniformly for $i \in I$, and $\|H_0\| \leq \|b_0\| + (k-1)\|f\|$. Put $C = \max(k\|f\|, \|b_0\| + (k-1)\|f\|)$ when $z_0 \neq 0$ and $C = k\|f\|$ otherwise.

The map $M \to \prod^{\ell^\infty}_{i \in I \cup \{0\}} z_i M$, $x \mapsto (z_i x)_{i \in I \cup \{0\}}$, is an isometric $*$-isomorphism onto bounded families. For any $(k-1)$-tuple $(x_1,\ldots,x_{k-1})$ the family $\{H_i(x_1,\ldots,x_{k-1})\}_{i \in I \cup \{0\}}$ is bounded by $C \prod_j \|x_j\|$ and defines an element $G(x_1,\ldots,x_{k-1}) \in M$. Coordinatewise multilinearity and bound give a bounded $(k-1)$-cochain $G$ with $\|G\| \leq C$. Central projection commutes with every term of the Hochschild differential, so $z_i(dG) = dH_i = z_i f$ for every $i \in I \cup \{0\}$. The central projections have join $1$, hence $dG = f$.

Restoring the original cocycle. Write $f_{\mathrm{orig}} = f_{\mathrm{sep}} + db$ with $f_{\mathrm{sep}}$ separately normal. The separable-predual tracial argument applied to $f_{\mathrm{sep}}$ yields $G$ with $dG = f_{\mathrm{sep}}$, and $d(G+b) = f_{\mathrm{sep}} + db = f_{\mathrm{orig}}$. Therefore $H_b^k(M,M) = 0$ for every complex von Neumann algebra $M$ and every $k \geq 2$, in the actual-image sense with no complete-boundedness condition.