Strong Kadison–Kastler stability and its spatial boundaries

2 · Going further

The main problem concerns unital von Neumann algebras $M$ and $N$ acting on the same Hilbert space. Their Kadison–Kastler distance measures the operator-norm distance between their unit balls, requiring approximation in both directions. Universal strong stability says that, for every prescribed $\epsilon >0$, there is a tolerance $\delta >0$ such that $d_{\mathrm{KK}}(M,N)<\delta$ yields a unitary $u$ with $uM{u}^{\ast }=N$ and $\Vert u-I\Vert <\epsilon$. Crucially, $\delta$ depends only on $\epsilon$, uniformly over the algebras, representations, and Hilbert spaces.

The main abstract describes this as resolving the strong Kadison–Kastler conjecture. Its central assertion is the passage from two-sided approximation of unit balls to exact conjugacy by a unitary close to the identity. The supplied abstracts do not describe the argument that establishes this passage, so no proof mechanism can be extracted from them. The main result has a Lean formalization, although the supplied material gives no more detailed account of its scope.

The second manuscript examines one-sided near inclusions instead. It constructs pairs of unital von Neumann algebras with common identity on separable complex Hilbert spaces whose one-sided gaps tend to zero. Spatial embeddings of the source into the target exist, but every implementing unitary remains a fixed positive distance from the identity. Thus the obstruction shown in the manuscript is specifically to small implementing unitaries, not to spatial embeddings themselves.

The third manuscript changes the class of algebras. For every $\epsilon >0$, it constructs unital, norm-separable C*-algebras with the same identity on a common separable complex Hilbert space, at Kadison–Kastler distance less than $\epsilon$, that are not conjugate by any unitary. The two algebras even have the same von Neumann closure. This is presented as a negative answer to the separable C*-algebraic spatial form of the conjecture, rather than a contradiction of the von Neumann algebra result.

Useful notions

• Operators on a Hilbert space
• Operator norm and unit balls
• Unital operator algebras
• Unitary conjugacy
• One-sided versus two-sided approximation

What was known before / what is new

Limits and assumptions

The positive result concerns sufficiently close unital von Neumann algebras on the same Hilbert space, with closeness measured in both directions between unit balls. The one-sided counterexamples concern small implementing unitaries despite the existence of spatial embeddings; the C*-algebra counterexamples rule out any ambient unitary conjugacy. The main result has a Lean formalization, but the supplied material does not specify its detailed coverage or state that either companion result is formalized. No argument details, explicit tolerance, or further remaining open questions are supplied.

Mini glossary

Kadison–Kastler distance

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A measure of how closely the unit balls of two operator algebras approximate one another in operator norm, in both directions.

Unitary conjugacy

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A relation in which a unitary change of coordinates takes one algebra exactly onto the other, expressed as $uM{u}^{\ast }=N$.

One-sided near inclusion

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An approximation condition from a source algebra toward a target algebra, without requiring approximation in the reverse direction.

Spatial embedding

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An embedding of one algebra into another implemented by a unitary change of coordinates.

Universal tolerance

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A closeness threshold depending only on the desired distance of the implementing unitary from the identity, not on the algebras, representations, or Hilbert spaces.

Keywords : Kadison–Kastler stability · Unitary conjugacy · Near inclusions

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family