Kadison's similarity conjecture

Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C^∗-algebra to operators on a Hilbert space becomes a *-homomorphism after conjugation by a bounded invertible operator. (Lean)

Manuscript

Kadison's similarity theorem through uniform derivation estimates

September 23, 2026 · source · cite as \cite{OAI:Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026}

We prove that every bounded complex-linear unital algebra homomorphism from a unital complex C^∗-algebra into the bounded operators on an arbitrary Hilbert space is similar to a *-homomorphism. This resolves Kadison's similarity conjecture positively. We also obtain one universal hyperreflexivity constant for all unital von Neumann algebras on arbitrary complex Hilbert spaces.

Result 288 · Operator algebras. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.