The Kadison-Ringrose cohomology conjecture

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VibeMathed record: The Kadison-Ringrose cohomology conjecture evidence for this event

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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. parent of this event

Sinclair and Smith, The Hochschild cohomology problem for von Neumann algebras, PNAS 1998 evidence for this event

The Kadison-Ringrose cohomology conjecture evidence for this event

Lean Comparator challenge KadisonRingrose (not in formalization.yaml at this commit) evidence for this event

The Kadison-Ringrose cohomology conjecture parent of this event

Kadison and Ringrose, Cohomology of operator algebras I, Acta Math. 126 (1971) evidence for this event

Lean solution module OAI/Analysis/BoundedHochschild/MainResult.lean evidence for this event