Kadison's similarity problem

Kadison's similarity problem, solved: every bounded unital homomorphism of a unital C*-algebra into the bounded operators on a Hilbert space is similar to a *-homomorphism (OpenAI, 2026). Completely bounded homomorphisms and Paulsen's similarity theorem; implementation of derivations with half the completely bounded norm and Arveson's distance formula; the noncommutative little Grothendieck inequality, row and column estimates, finitely generated homomorphisms and cyclic domains; Kirchberg's passage from inner derivations to similarity with a polynomial bound; words in free subalgebras; Schur-Weyl invariants, propagation from small corners and a Hankel test, which give a free-product corner estimate independent of the corner; transfer to II_1 factors through Popa's free independence in ultrapowers; the uniform commutator estimate for all von Neumann algebras; inner derivations, the similarity theorem with the bound ||pi||_cb <= ||pi||^(2C), and universal hyperreflexivity of von Neumann algebras.

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

1. Completely bounded homomorphisms and similarity
2. Row, column and cyclic estimates
3. From inner derivations to similarity
4. Words in free subalgebras
5. Invariant tensors and the word structure
6. Propagation from small corners
7. The Hankel test
8. Matrix commutators in finite factors
9. The uniform commutator estimate
10. Kadison's similarity theorem

Theorems and assumptions · Reader and editable sources · Course record

Used in the course Strong Kadison–Kastler stability: Corollary 4.5 of the first lesson gives the distance from an operator to a commutant, and the uniform commutator estimate of the ninth lesson turns a norm estimate into a completely bounded one in the proof for finite factors.

Prerequisites

• Completely bounded extension and factorization (Positive maps and finite-dimensional approximation)
• The sharp noncommutative bilinear inequality (Positive maps and finite-dimensional approximation)
• Schur–Weyl duality · Open Mathematics Courses (Representations of finite groups)
• Order, local units and quotients of C*-algebras · 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 spectral theorem for bounded self-adjoint operators · Foundations of von Neumann algebras
• Free independence in ultraproducts (Generators of II₁ factors)
• Haar unitaries, freeness and tracial ultraproducts (Generators of II₁ factors)
• Haar measure on locally compact groups (Harmonic analysis on locally compact groups)
• 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
• Completely bounded maps and the noncommutative Grothendieck inequality (OA-APPROX); C*-algebras, Kaplansky's density theorem, projections and types, the spectral theorem (Foundations of von Neumann algebras); Schur-Weyl duality (RT-FIN); traces, the commutation theorem and conditional expectations (Traces and noncommutative integration); Haar unitaries, tracial ultraproducts and Popa's free independence (Generators of II₁ factors); the central decomposition (Measurable fields and direct integrals); Haar measure (Harmonic analysis on locally compact groups).