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,...,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,...,a_{k+1}) = a_1 f(a_2,...,a_{k+1}) + sum_{j=1}^k (-1)^j f(a_1,...,a_j a_{j+1},...,a_{k+1}) + (-1)^{k+1} f(a_1,...,a_k) a_{k+1}.
In degree zero this is the inner derivation (db)(a) = ab - ba. The algebraic identity d^2 = 0 gives im d subset ker d, and
H_b^k(M,M) = ker(d : C_b^k to C_b^{k+1}) big/ im(d : C_b^{k-1} to C_b^k).
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.
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pdf/kadison-ringrose-1971-cohomology-oa-i.pdf
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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.
pdf/ces-1987-cb-multilinear-maps.pdf
pdf/cpss-2003-hochschild-gamma.pdf
pdf/pop-smith-2010-tensor-products.pdf
pdf/qian-shen-2015-generalized-gamma.pdf
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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.
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pdf/popa-2014-independence-ultraproduct.pdf
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pdf/banica-1996-polar-circular.pdf
pdf/haagerup-1979-non-nuclear-cstar.pdf
pdf/dixxmier-1953-formes-lineaires.pdf
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, ... is free Haar if tau(q(g_{i_1},...,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 ∞, such that for every r geq 1, every nonempty finite set W subset F_r ∖ {e}, and every eta > 0, if G_1^{(ell)},...,G_r^{(ell)} are independent with law mu^{*n_ell},
Pr( max_{w in W} | tau(w(G_1,...,G_r)) | geq eta ) 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 ℕ. Its tracial ultrapower is P^omega = ell^∞(ℕ,P)/I_omega with 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_{mtoomega} 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,...,u_d in P^omega such that B, W^*(u_1),...,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.
Proof. Apply Popa's Theorem 0.1(a) with every coordinate algebra and every coordinate subalgebra equal to P. The non-intertwining condition holds because a nonzero corner of the diffuse factor P cannot embed in the scalar relative commutant or in a finite matrix amplification of it. Since P^omega is a factor, the relative commutant of the ultraproduct subalgebra is ℂ1, so the conclusion for the separable centered subspace B ominus ℂ1 is ordinary scalar-trace freeness. A diffuse abelian tracial algebra contains a Haar unitary. Adjoining one to B preserves separable predual. Iterating the selection d times in the same ultrapower supplies u_1,...,u_d.
pdf/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),...,W^*(u_d) are free subalgebras with each u_j Haar, and y_1,...,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,...,y_r)) = 0.
Put every non-distinguished u_j into the coefficient algebra and let C = W^*(u_{h_1},...,u_{h_r}); then B_0 and C are free. Writing y_i = A_i u_{h_i}^{ε_i} B_i with A_i, B_i in U(B_0), the string of surviving distinguished letters in w(y_1,...,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.
Proof. Set B_0 = W^*(B, u_j : j notin {h_1,...,h_r}) and C = W^*(u_{h_1},...,u_{h_r}). Then B_0 and C are free. Write y_i = A_i u_{h_i}^{ε_i} B_i with A_i, B_i in U(B_0). If w = x_{i_1}^{sigma_1} ... x_{i_t}^{sigma_t} is a nonempty reduced word, its evaluated product is c_0 u_{h_{i_1}}^{ε_{i_1}sigma_1} c_1 ... u_{h_{i_t}}^{ε_{i_t}sigma_t} c_t with c_j in U(B_0), and the displayed distinguished-letter string is reduced: an adjacent inverse would force i_j = i_{j+1} and sigma_j = -sigma_{j+1}. Expanding every c_j as scalar plus trace-zero gives terms that are alternating centered products from B_0 and C with at least one C-factor; freeness and Haar-ness force the trace to vanish.
Finite path tests. Fix a finite F subset U(M), an integer d geq 1, an ε > 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,...,u_d in U(M) satisfying every test to tolerance ε. 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 ε. For a general M with center, decompose as a direct integral over a probability space (Ω,nu), and choose countably many measurable unitary sections that are fiberwise 2-norm dense (using exponentials of ℚ+iℚ-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.
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For the finite-path-tests step. First M = P a factor. The coefficient algebra W^*(F) subset P^omega has separable predual since F is finite. Popa's freeness in P^omega supplies Haar unitaries g_1,...,g_d free from it. They have coordinatewise-unitary representatives: if u = [(u_m)] in P^omega is a Haar unitary and u_m = e^{i a_m} with a_m in P self-adjoint and |a_m| leq pi, any bounded self-adjoint lift is bounded and u = [(e^{i a_m})]. Let (u_1^{(m)},...,u_d^{(m)}) be coordinatewise-unitary representatives. Each test trace vanishes exactly in P^omega, so the scalar tau(w(y_1^{(m)},...,y_r^{(m)})) tends to zero along omega for every test (y_i, w). Choose m with all test traces in absolute value < ε.
For general M with nontrivial center. By Blackadar's direct-integral theory, decompose M over a probability space (Ω, mu) into almost-everywhere type II_1 factors, M = int_Ω^ ⊕ M_s , dmu(s). Countably many measurable unitary sections are 2-norm dense fiberwise, obtained by exponentiating ℚ+iℚ-polynomials in a countable set of generators. Each scalar test trace is measurable in s. In each fiber the factor case supplies a successful tuple, and the requirements are open in the product 2-norm topology, so the successful set E_k for the k-th candidate tuple is a measurable subset of Ω. The E_k cover Ω up to a null set. On E_k ∖ ∪_{j<k} E_j select the k-th tuple, and on the null complement select the identity tuple. Each test trace is now less than ε almost everywhere, and integration gives absolute global value at most ε/2 < ε. Mesh refinement of telescoping products supplies the openness: |z_1... z_L - z'_1 ... z'_L|_2 leq sum_j |z_j - z'_j|_2, and inversion is a 2-norm isometry on unitaries.
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^{3cdot 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 ∪_{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
Pr(bad) leq &; K_ell sum_{j>ell} p_j + K_ell p_0 mu_0(U(M) ∖ F_ell)
&+ binom{K_ell}{2}/d_ell + ell (1 - p_ell)^{n_ell}.
Every term tends to 0. Because p_{ell+1} = p_ell^2, one has sum_{j>ell} p_j leq (16)/(15) p_ell^2 and so K_ell sum_{j>ell} p_j leq (16)/(15)ell √(p_ell); also K_ell p_0 mu_0(U(M)∖ 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_{win 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| < ∞ and |x_j|_2 to 0 jointly imply |R(x_j)|_2 to 0. This is equivalent to the uniform modulus: for every r < ∞ and ε > 0 there is delta > 0 with |x| leq r and |x|_2 < delta implying |R(x)|_2 < ε. 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, 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 ∞ 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) = 𝔼_{U sim mu}[T(x U) U^*] (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, ... be independent with law mu, set W_n = U_1 ... U_n and F_n = sigma(U_1,...,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:
𝔼[Z_{n+1} mid F_n] = 𝔼[T(W_n U_{n+1}) U_{n+1}^* W_n^* mid 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_∞ in L^2(Ω; H) and Z_n = 𝔼[Z_∞ mid F_n]. The error e_k = 𝔼|Z_∞ - Z_k|_2^2 decreases to zero, and orthogonality of increments gives the Cauchy estimate 𝔼|Z_n - Z_k|_2^2 leq e_k for n geq k.
Deterministic case. If Z_∞ = 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_∞ 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 and |D(G^{-1}) - b|_2 < delta) geq eta_delta.
Write G = U_1 ... U_n, h = lfloor n/2 rfloor, and
A_n = D(U_1 ... U_h), B_n = D(U_n^{-1} ... 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
𝔼|D(G) - A_n|_2^2 leq e_h, 𝔼|D(G^{-1}) - B_n|_2^2 leq e_h.
Let p_a = Pr(|Z_∞ - a|_2 < delta/3) > 0 and p_b = Pr(|Z_∞ - b|_2 < delta/3) > 0. Mean-square convergence gives, for large h, Pr(|Z_h - a|_2 < 2delta/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,...,G_m be independent with law mu^{*n}, and v(G) = G_{i_1}^{sigma_1} ... 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
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, |D(G_i^{-1}) - b|_2 < 1/m, 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,...,G_m))| < 1/m, q in Q_m,
for Q_m the nonidentity reduced words of length leq m, and
|D(v(G)) - D(G_{i_1}^{sigma_1})|_2 < 1/m, v in V_m,
for V_m the nonempty signed words on {G_1,...,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 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)},...,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,...,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}... g_{i_t}^{sigma_t} with pairwise distinct unsigned indices,
R(w) = a w if sigma_1 = 1, R(w) = b w 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_∞ 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 ℂ-linear and bounded-ball 2-continuous. Let g_1, g_2, ... be a free Haar family in N. Suppose that for every signed word w = g_{i_1}^{ε_1}... g_{i_q}^{ε_q} with q geq 1, ε_j in {+1,-1}, and i_1,...,i_q pairwise distinct unsigned indices,
R(w) = a w (ε_1 = +1), R(w) = b w (ε_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) ( sum |c_w|^2 )^{1/2}.
Free Haar unitaries h_1,...,h_r determine a trace-preserving normal *-isomorphism of L(F_r) = W^*(lambda(F_r)) onto W^*(h_1,...,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,...,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|_{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 ℕ, set Nhat = N^{omega_2}, tauhat = tau^{omega_2}, s = (s_r)_{omega_2} in Nhat, 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,...,d_k geq 0, set B_d(z) = z(z^* z)^d and P(z) = B_{d_1}(z)... B_{d_k}(z). Then
S(P(s)) = a P(s), 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}, 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 Nhat 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 -+-...-+ 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)...(r-d+1). Hence
tauhat(A^d) = lim_{r to omega_2} tau((s_r^* s_r)^d) = C_d, d geq 0.
The measure
dnu(x) = (1)/(2pi) √((4-x)/(x)), 1_{(0,4)}(x), dx
has moments int_0^4 x^d dnu = 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 tauhat 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 ε > 0, and put x_ε = s(A+ε 1)^{-1/2} = V(A/(A+ε 1))^{1/2}. For a scalar polynomial p(t) = sum_{d=0}^D c_d t^d,
[sp(A)]^k = sum_{d_1,...,d_k = 0}^D c_{d_1}... c_{d_k}, [s A^{d_1}]...[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+ε)^{-1/2} give, via boundedness of S, S(x_ε^k) = a x_ε^k and S((x_ε^k)^*) = b(x_ε^k)^*. As ε to 0, x_ε^k to V^k and (x_ε^k)^* to V^{-k} in operator norm; bounded-ball 2-continuity of S, equivalently operator-norm continuity on the bounded sequence x_ε^k together with boundedness of S, transfers again:
S(V^k) = a V^k, S(V^{-k}) = b V^{-k}, 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],
| sum_{k=p}^q sin(kt) | leq | sum_{k=p}^q e^{ikt} | leq 1/sin(t/2) leq pi/t.
Abel summation from p = K+1 gives | sum_{k=p}^m sin(kt)/k | leq pi uniformly in m, so |Im H_m(e^{itheta})| leq B = 1+pi.
Fix m and lambda in 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 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^{itheta}) 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 (C)/(cos(1) h_m) =: delta_m to 0.
Partition T into m half-open arcs of length 1/m with projections e_1,...,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_{sep} by a bounded coboundary db, with b a bounded (k-1)-cochain. A bounded primitive for f_{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,...,a_{k-1},x) bounded-ball 2-continuous on M for every fixed (k-1)-tuple a.
pdf/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
(PPhi)(a_1,...,a_{k-1},x) = sum_s p_s, Phi(a_1,...,a_{k-1}, x u_s), u_s^*.
The sum converges in operator norm, and |PPhi| leq |Phi|. Iteration gives
(P^j Phi)(a,x) = 𝔼[Phi(a, x W_j) W_j^*], W_j = U_1 ... 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 2Cprod_i|a_i||x|sum_{s>L} p_s, so P Phi_lambda to PPhi; 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) = 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,...,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,...,a_k) + sum_{j=1}^{k-1}(-1)^j h_n(a_1,...,a_j a_{j+1},...,a_k) + (-1)^k F_n(a_1,...,a_k) + (-1)^{k+1} f(a_1,...,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(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 Υ_E : M to N_E be the normal trace-preserving conditional expectation; then f_E = Υ_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|.
pdf/dixxmier-1953-formes-lineaires.pdf
Extend each g_E to a bounded (k-1)-cochain on M by G_E(x_1,...,x_{k-1}) = g_E(Υ_E x_1, ..., Υ_E x_{k-1}). The bound |G_E| leq |f| is inherited. For a tuple (a_1,...,a_k) in E^k the inputs a_j, the adjacent products a_j a_{j+1}, and f(a_1,...,a_k) all lie in N_E; the differential dG_E at that tuple agrees with f(a_1,...,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,...,a_k) the finite set E_0 = {a_1,...,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| < ∞. 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,...,x_k) = z_i f(z_i x_1, ..., 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: btilde_i(x_1,...,x_{k-1}) = b_i(z_i x_1, ..., z_i x_{k-1}). Then
dbtilde_i(a_1,...,a_k) = z_i f(z_i a_1,...,z_i a_k).
The right side is the central cut 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,...,a_{n-1}) = sum_{i=1}^{n-1}(-1)^i psi(z a_1, ..., z a_{i-1}, q, q a_i, a_{i+1}, ..., 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 = id - cut_z, |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 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,...,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 = btilde_i + J_{z_i}(z_i f), and when z_0 neq 0 set H_0 = btilde_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^∞}_{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,...,x_{k-1}) the family {H_i(x_1,...,x_{k-1})}_{i in I cup {0}} is bounded by C prod_j |x_j| and defines an element G(x_1,...,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_{orig} = f_{sep} + db with f_{sep} separately normal. The separable-predual tracial argument applied to f_{sep} yields G with dG = f_{sep}, and d(G+b) = f_{sep} + db = f_{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.