A counterexample to Naimark's problem
A unital, infinite-dimensional, simple C*-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent, constructed in ZFC: a negative answer to Naimark's problem without additional set-theoretic axioms. Excision of pure states, expectations onto algebras of peaking projections, reduced amalgamated free products and reduced HNN adjunctions give a bridge that joins two chosen classes of pure states while every pure state keeps a unique extension; transfinite iteration with closed subiterations and countable determination treats every pure state. The course is an exposition of OpenAI's construction (2026); it uses the homogeneity theorem of Kishimoto, Ozawa and Sakai without proof.
Seven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Pure states, excision and peaking projections
- Expectations from peaking projections
- Reduced amalgamated free products of tracial algebras
- Reduced HNN adjunctions
- The bridge
- Closed subiterations and countable determination
- The construction in ZFC
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Order, local units and quotients of C*-algebras · Foundations of von Neumann algebras
- Completely positive maps · Foundations of von Neumann algebras
- Hahn–Banach, Baire and the basic theorems on Banach spaces · Foundations of von Neumann algebras
- Kaplansky's density theorem and its consequences · Foundations of von Neumann algebras
- Spectral estimates and Borel selection: two reading routes · Foundations of von Neumann algebras
- Building representations from positive functionals · Foundations of von Neumann algebras
- The universal enveloping von Neumann algebra of a \(C^*\)-algebra, and \(W^*\)-algebras · Foundations of von Neumann algebras
- Adjointable operators · Hilbert C*-modules and Morita equivalence
- Hilbert C*-modules · Hilbert C*-modules and Morita equivalence
- Tensor products and C*-correspondences · Hilbert C*-modules and Morita equivalence
- States, the GNS construction, the enveloping von Neumann algebra, Kadison's transitivity theorem, Tomiyama's theorem and cardinal arithmetic (course Foundations of von Neumann algebras); Hilbert C*-modules, adjointable operators and interior tensor products (course Hilbert C*-modules and Morita equivalence).