Proposition 12. Let M be of type II₁ with separable predual and faithful normal tracial state τ . If k≥2 and f∈C_b^k(M,M) is a separately normal cocycle, then there is g∈C_b^k-1 (M,M) such that dg=f, g.

We first construct an averaged cochain that is a right module map in its last input. Evaluation at 1 will give a (k-1) -cochain, and averaging the original equation df=0 will identify its differential. Choose the symmetric countably supported measure μ for which Theorem 3.1 applies, and write μ=∑_ s≥1 p_sδ_ u_s . For a bounded k -cochain Φ , define (PΦ)( a,x) =∑_ s≥1 p_sΦ( a,xu_s)u_s^*, a=(a_1,…,a_ k-1 ). The sum converges in operator norm, and P is a contraction for the cochain norm. If W_j=U_1⋯U_j , where the U_i are independent with law μ , and W_0=1 , induction gives (P^j f)( a,x) =[f( a,xW_j)W_j^*]. Set F_n=1n∑_ j=0 ^ n-1 P^j f. We use compactness for pointwise ultraweak convergence. For each fixed tuple x=(x_1,…,x_k) , all values F_n( x) lie in the ultraweakly compact ball of radius f∏_ix_i . The product of these balls, indexed by all tuples in M^k , is compact. Multilinearity is closed for this topology, since addition and scalar multiplication are ultraweakly continuous. Thus some subnet of (F_n) converges at every tuple to a bounded multilinear map F with F . On any common cochain norm ball, P is continuous for this topology. To see this, suppose Φ_λ→Φ pointwise ultraweakly and Φ_λ,Φ≤C . For a fixed tuple ( a,x) and a normal functional ρ∈M_* , truncate after L terms. The finite sum passes to the limit, and the remaining difference has absolute value at most 2C ∏_ i=1 ^ k-1 a_i∑_ s>L p_s. Letting L→∞ proves the claim. Since PF_n-F_n=(P^nf-f)/(n), PF_n-F_n≤2fn, the subnet limit satisfies PF=F . Fix a , and put B(x)=f( a,x) . This is a bounded normal linear map. For any R<∞ and ε>0 , Lemma 11 provides δ>0 such that x≤R, x_2<δ B(xv)v^*_2<ε every v∈(M). Indeed, right multiplication by a unitary preserves both the operator norm and the 2 -norm. The same bound holds after averaging over any W_j and after taking the Ces\`aro average. Ultraweak lower semicontinuity of the 2 -norm then gives F( a,x)_2≤ε under these input bounds. Using ε/2 in place of ε proves the required continuity modulus for T(x)=F( a,x) . The equality PF=F is exactly the harmonicity identity for this map. Consequently Theorem 3.1 gives F( a,x)=h( a)x, h( a)=F( a,1). In particular h is a bounded (k-1) -cochain and h . It remains to obtain its coboundary identity. Let ν_n=n^ -1 ∑_ j=0 ^ n-1 μ^ *j and h_n( a)=F_n( a,1) . Evaluate the original equation df=0 at (a_1,…,a_k,v) , multiply the output on the right by v^* , and average with respect to v∼ν_n . This gives the exact equality 0= , a_1h_n(a_2,…,a_k) +∑_ j=1 ^ k-1 (-1)^j h_n(a_1,…,a_ja_ j+1 ,…,a_k); , +(-1)^k F_n(a_1,…,a_ k-1 ,a_k) +(-1)^ k+1 f(a_1,…,a_k). The final term is unchanged because vv^*=1 . Along the chosen subnet, each displayed evaluation converges ultraweakly; multiplication by a fixed a_i preserves that convergence. Passing to the limit and using therefore yields 0=(dh)(a_1,…,a_k)+(-1)^ k+1 f(a_1,…,a_k). Thus g=(-1)^kh satisfies. This calculation averages df at the displayed tuples; it requires neither a commutation relation between P and d nor separate normality of the limiting cochain F .