Bounded Hochschild cohomology of von Neumann algebras

Every bounded Hochschild cocycle of degree at least two on a von Neumann algebra is the coboundary of a bounded cochain, after OpenAI's 2026 proof; with the Kadison-Sakai theorem this settles the Kadison-Ringrose cohomology problem. A rigidity theorem for first-letter identities on free Haar families; a random walk on the unitary group whose long products are asymptotically free; a Liouville theorem for its harmonic maps; bounded primitives by averaging, removal of separability and an explicit central homotopy.

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. The cohomology problem
2. Rigidity of first-letter identities
3. A random walk with asymptotically free products
4. The Liouville theorem
5. Bounded primitives

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Order, local units and quotients of C*-algebras · Foundations of von Neumann algebras
• Compact and trace-class operators, the predual of B(H), and the operator topologies · Foundations of von Neumann algebras
• Kaplansky's density theorem and its consequences · Foundations of von Neumann algebras
• Projections and types of von Neumann algebras · Foundations of von Neumann algebras
• The double commutant theorem · Foundations of von Neumann algebras
• The spectral theorem for bounded self-adjoint operators · Foundations of von Neumann algebras
• The universal enveloping von Neumann algebra of a \(C^*\)-algebra, and \(W^*\)-algebras · Foundations of von Neumann algebras
• Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian · Foundations of von Neumann algebras
• Free independence in ultraproducts (Generators of II₁ factors)
• Haar unitaries, freeness and tracial ultraproducts (Generators of II₁ factors)
• The central decomposition and the types of the fibres · Measurable fields and direct integrals (Measurable fields and direct integrals — lifting and differentiation)
• Integration for a trace, the commutation theorem, and applications · Traces and noncommutative integration
• Ultraproducts and the asymptotic centralizer · Type III factors: central sequences, full factors and almost periodic states
• Von Neumann algebras (course Foundations of von Neumann algebras), Haar unitaries and Popa's free independence theorem (course Generators of II1 factors), the central decomposition (course Measurable fields and direct integrals), conditional expectations (course Traces and noncommutative integration), and products of probability spaces (core course Measure and Integration).