A universal Banach–Mazur stability radius for von Neumann algebras

We prove cohomological criteria for ordinary complex Banach–Mazur stability of von Neumann algebras, without separability, type, or density restrictions. A fixed algebra has a positive stability radius if its second bounded Hochschild cohomology vanishes and its third coboundary space is norm closed. For any class closed under countable bounded direct products, vanishing in degrees two and three gives one radius for the entire class. Assuming the degree-two and degree-three vanishing theorem in the September 2026 preprint of OpenAI, this yields one $\varepsilon>0$ such that, for all complex von Neumann algebras $M,N$,

$$\min\{d_{\mathrm{BM}}(M,N),d_{\mathrm{BM}}(M_*,N_*)\}<1+\varepsilon\quad\Longrightarrow\quad M\cong N\text{ as Jordan }*\text{-algebras}.$$

Here $M_*,N_*$ are the preduals, and the resulting Jordan isomorphism is normal. Relative to that external cohomology input, this answers the ordinary-distance stability question studied by Jean Roydor. The proof applies a known Jordan splitting criterion inside a norm ultraproduct, using uniformly bounded invertible commutators that also cover odd finite matrix sizes. A bounded product argument makes the cohomological correction bounds uniform. The proof gives no numerical value for the resulting radius.

GitHub repository: https://github.com/eulogica/universal-banach-mazur-stability