The Kadison-Ringrose cohomology conjecture

Claims Theorem 1.1: for every complex von Neumann algebra $M$ and every $k\ge2$, every bounded $k$-cocycle $f\in C_b^k(M,M)$ with $df=0$ equals $dg$ for some bounded $g\in C_b^{k-1}(M,M)$, so $H_b^k(M,M)=0$. Combined with the Kadison-Sakai theorem in degree one this is the full conjecture. The key new input is a Liouville theorem for harmonic maps under a single unitary random walk built from Popa's free-independent sequences in type $\mathrm{II}_1$ algebras. It concerns ordinary bounded cohomology with coefficients in $M$ itself; it says nothing new about coefficients in $B(H)$ or about completely bounded cohomology beyond what follows.

Claims Theorem 1.1: for every complex von Neumann algebra $M$ and every $k\ge2$, every bounded $k$-cocycle $f\in C_b^k(M,M)$ with $df=0$ equals $dg$ for some bounded $g\in C_b^{k-1}(M,M)$, so $H_b^k(M,M)=0$. Combined with the Kadison-Sakai theorem in degree one this is the full conjecture. The key new input is a Liouville theorem for harmonic maps under a single unitary random walk built from Popa's free-independent sequences in type $\mathrm{II}_1$ algebras. It concerns ordinary bounded cohomology with coefficients in $M$ itself; it says nothing new about coefficients in $B(H)$ or about completely bounded cohomology beyond what follows.