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.

1. Pure states, excision and peaking projections
2. Expectations from peaking projections
3. Reduced amalgamated free products of tracial algebras
4. Reduced HNN adjunctions
5. The bridge
6. Closed subiterations and countable determination
7. 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).