Strong Kadison–Kastler stability

Close von Neumann algebras are conjugate by a unitary near the identity, with a tolerance that depends only on the required closeness, after OpenAI's 2026 papers. Finite factors through boundary means of countable groups, measurable cocycles, double ergodicity and common tracial position; infinite factors through a fixed III_1 tensor factor, rescaled Tomita graphs, infinitesimal normalizers, analytic multipliers, a common modular action and common crossed products; all algebras through Dixmier's theorem and the central decomposition. Two counterexamples: close separable C*-algebras without unitary conjugacy, and one-sided near inclusions without small spatial embeddings.

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

1. Close operator algebras
2. Averaging and the completely positive criterion
3. Boundary means and measurable cocycles
4. Double ergodicity and diagonal comparison
5. Irreducible hyperfinite subfactors
6. Common tracial position and Jordan maps
7. Stability of finite factors
8. Preparing infinite factors
9. Rescaled Tomita graphs
10. Infinitesimal normalizers of type III factors
11. Frequency nests and analytic multipliers
12. Near normalizers at every time
13. A common modular action
14. Common crossed products
15. Stability of separably acting factors
16. Centers and measurable assembly
17. Stability on arbitrary Hilbert spaces
18. Convergent sequences and a relative tensor norm
19. A free-group deformation and close separable algebras
20. A circle family and a near inclusion
21. Almost commuting matrices and small embeddings

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• The core trace and its changes of weight · Open Mathematics Courses (Operator algebras: crossed products, spectra and flows)
• Approximate unitary homogeneity of normal states · Open Mathematics Courses (Operator algebras: crossed products, spectra and flows)
• The fixed center and the factor criterion · Open Mathematics Courses (Operator algebras: crossed products, spectra and flows)
• The full weight spectral identity and the modular intersection comparison · Open Mathematics Courses (Operator algebras: crossed products, spectra and flows)
• Zero in every modular spectrum and the type III spectral trichotomy · Open Mathematics Courses (Operator algebras: crossed products, spectra and flows)
• Traces on von Neumann algebras
• Fixed observations, density weights and modular time · Modular theory and weights (Open Course — Modular Theory & Weights)
• Modular time from a faithful normal state · Modular theory and weights (Open Course — Modular Theory & Weights)
• The KMS boundary condition determines the modular group · Modular theory and weights (Open Course — Modular Theory & Weights)
• Conditional expectations from modular invariance · Modular theory and weights (Open Course — Modular Theory & Weights)
• The positive cone of a standard representation · Modular theory and weights (Open Course — Modular Theory & Weights)
• Holomorphy in Banach spaces, Stone's theorem and resolvent convergence · Analytic elements, strips and KMS
• 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
• Completely positive maps · Foundations of von Neumann algebras
• Kaplansky's density theorem and its consequences · Foundations of von Neumann algebras
• Borel models, measurable inversion and Polish group quotients · Foundations of von Neumann algebras
• Projections and types of von Neumann algebras · Foundations of von Neumann algebras
• Derivations of von Neumann algebras: an alternative proof (Foundations of von Neumann algebras)
• The double commutant theorem · 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
• Haar measure on locally compact groups (Harmonic analysis on locally compact groups)
• Injective von Neumann algebras · Injective factors
• Uniqueness of the injective II₁ factor · Injective factors
• Completely bounded homomorphisms and similarity (Kadison's similarity problem)
• The uniform commutator estimate (Kadison's similarity problem)
• The central decomposition and the types of the fibres · Measurable fields and direct integrals (Measurable fields and direct integrals — lifting and differentiation)
• An injective subalgebra with the same core commutant (Relative bicentralizers and core commutants)
• Spatial tensor products of von Neumann algebras · Tensor products of operator algebras
• Integration for a trace, the commutation theorem, and applications · Traces and noncommutative integration
• Measurable operators for a trace: examples, convergence and the commutant · Traces and noncommutative integration
• Ultraproducts and the asymptotic centralizer · Type III factors: central sequences, full factors and almost periodic states
• Operator algebras: C*-algebras and von Neumann algebras, completely positive and completely bounded maps, factors and their types, Tomita–Takesaki theory and the continuous core; free groups and their regular representations; standard Borel spaces and measurable selection.