Small-unitary implementations of one-sided von Neumann near-inclusions
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Does arbitrarily small one-sided near-inclusion of unital von Neumann algebras on separable Hilbert space guarantee an embedding by a unitary arbitrarily close to the identity? The counterexample target permits spatial embeddings but requires all embedding unitaries to stay a fixed distance away.
Why it matters and what counts as progress
Why it matters: It tests the stability of one-sided spatial embeddings. Two-sided Kadison–Kastler stability is a separate problem.
What counts as progress: Near-included algebras requiring a fixed-size spatial embedding perturbation, or a small-unitary theorem under additional hypotheses; two-sided distance is separate.
How the problem developed
- 1972 · Conjecture [1]
Richard V. Kadison and Daniel Kastler Introduced the perturbation problem asking whether sufficiently close von Neumann algebras are related by a unitary close to the identity.
- 1980 · Partial result [2]
Erik Christensen Proved a spatial embedding theorem for sufficiently small near-inclusions when the source algebra is amenable; this does not cover arbitrary source algebras.
- 5 Oct 2026 · Counterexample claim [3]
OpenAI Reported separable von Neumann algebras with one-sided near-inclusion gap tending to zero although every unitary embedding stays a fixed distance from the identity.
AI contribution
Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings
The OpenAI-authored manuscript constructs unital von Neumann algebra pairs on separable Hilbert spaces with one-sided near-inclusion gaps tending to zero, yet every unitary embedding the source into the target stays a fixed distance from the identity. Spatial embeddings do exist; this does not refute two-sided Kadison-Kastler stability.
Sources
- Kadison and Kastler, Perturbations of von Neumann algebras. I. Stability of type I factors
- Erik Christensen, Near inclusions of C*-algebras
- OpenAI: near inclusions without small spatial embeddings
- Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings
- Near-inclusion theorem and context
- Matrix obstruction and construction
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