{"id":"openai-math/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings","pdf":"https://github.com/openai/math/blob/main/preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026/near-inclusions.pdf","url":"https://github.com/openai/math/tree/main/preprints/Near-Inclusions-of-von-Neumann-Algebras-Without-Small-Spatial-Embeddings-October-5-2026","tags":["operator-algebras","von-neumann-algebras","near-inclusions"],"score":0,"title":"Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings","domain":"Operator algebras","family":"289","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.OA","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-05","reactions":{},"formalized":false,"family_title":"Strong Kadison–Kastler stability and its spatial boundaries"}