Strong Kadison–Kastler stability and its spatial boundaries

Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C^∗-algebras need not be ambiently unitarily conjugate. (Lean)

Manuscripts (3)

Universal strong Kadison–Kastler stability

September 23, 2026 · source · cite as \cite{OAI:Universal-strong-Kadison-Kastler-stability-September-23-2026}

We prove that sufficiently close unital von Neumann algebras are conjugate by a unitary arbitrarily close to the identity. The tolerance depends only on the prescribed distance of that unitary from the identity, uniformly over all algebras, representations, and Hilbert spaces. This resolves the strong Kadison–Kastler conjecture.

Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings

October 5, 2026 · source · cite as \cite{OAI:Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026}

We give a negative answer to the unrestricted small-spatial-embedding problem for one-sided near inclusions of von Neumann algebras. On separable complex Hilbert spaces, we construct pairs of unital von Neumann algebras with common identity whose one-sided gaps tend to zero. Spatial embeddings of the source into the target exist, but every implementing unitary stays a fixed positive distance from the identity. The obstruction therefore concerns small implementing unitaries, rather than the existence of spatial embeddings.

Close Separable C*-Algebras Without Spatial Conjugacy

October 5, 2026 · source · cite as \cite{OAI:Close-Separable-Cstar-Algebras-Without-Spatial-Conjugacy-October-5-2026}

We disprove the separable C^∗-algebraic spatial form of the Kadison–Kastler conjecture. For every ε > 0, we construct unital, norm-separable C^∗-algebras on a common separable complex Hilbert space, with the same identity and Kadison–Kastler distance less than ε, that are not conjugate by any unitary. The two algebras have the same von Neumann closure.

Result 289 · 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.