Amenability and unitarizability
OpenAI's answer (2026) to Dixmier's question for discrete groups: a discrete group is amenable exactly when every uniformly bounded representation on a Hilbert space is similar to a unitary one. The course proves the Day–Dixmier theorem and, for every nonamenable group, constructs representations with all norms at most 1 + ε that cannot be made unitary: nonamenability gives a contracting average of translations, Hall's theorem assigns the edges of a Cayley-type graph to their endpoints with small loads, random sign frames turn the assignment into operators with bounded conjugation differences, and a Hilbert–Schmidt energy bound shows that their cocycle has no bounded implementer. Corollaries for Thompson's group F and for group C*-algebras.
Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Uniformly bounded representations
- Contracting averages and sparse assignments
- Random sign frames and masked operators
- Unitarizability implies amenability
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Means, Følner sets, and regular representations (Measured relations and operator algebras)
- Order, local units and quotients of C*-algebras · Foundations of von Neumann algebras
- Hahn–Banach, Baire and the basic theorems on Banach spaces · Foundations of von Neumann algebras
- Hilbert spaces and compact operators · Foundations of von Neumann algebras
- Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian · Foundations of von Neumann algebras
- Kadison's similarity theorem (Kadison's similarity problem)
- Thompson's group F and dyadic partitions (Thompson's group F and amenability)
- Thompson's group F is not amenable (Thompson's group F and amenability)
- Lebesgue measure on Rm: translation invariance and the formula for linear images (core course Measure and Integration, D10: Fremlin, Measure Theory, 134A and 263A).