Uniform spatial stability of von Neumann algebras
For von Neumann algebras $M, N \subseteq \mathcal B(H)$ on a complex Hilbert space $H$, let $M_1, N_1$ be their closed operator-norm unit balls, and let $$d(M, N) = \max\left\{ \sup_{x \in M_1} \inf_{y \in N_1} \|x - y\|,\ \sup_{y \in N_1} \inf_{x \in M_1} \|y - x\| \right\}$$ be the Hausdorff distance between them, introduced by Kadison and Kastler. Cameron, Christensen, Sinclair, Smith, White and Wiggins showed, for $n \ge 3$ and a free ergodic probability-measure-preserving action of $\mathrm{SL}_n(\mathbb Z)$ on a standard nonatomic probability space $X$, that whenever $M = (L^\infty(X) \rtimes \mathrm{SL}_n(\mathbb Z)) \mathbin{\overline{\otimes}} R$, with $R$ the hyperfinite II$_1$ factor, is represented on $H$ and $N \subseteq \mathcal B(H)$ is sufficiently close to $M$, there is a unitary $u$ on $H$ close to the identity with $uMu^* = N$: the first nonamenable class of von Neumann algebras with this property.
Problem. Is it true that for every $\varepsilon > 0$ there exists $\delta > 0$, independent of $H$, $M$ and $N$, such that $d(M, N) < \delta$ implies the existence of a unitary $u \in \mathcal B(H)$ satisfying $$uMu^* = N, \qquad \|u - I\| < \varepsilon?$$
Source: Kadison and Kastler (1972); in this form, Peterson's list of open problems in operator algebras, Problem O.4 (the spatial isomorphism conjecture). Stated by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OA — Operator Algebras
- Secondary
- FA
- Keywords
- AIM Open Applied Problems, Kadison–Kastler conjecture, perturbations of von Neumann algebras
References
@article{kadison1972,
author = {Kadison, Richard V. and Kastler, Daniel},
title = {Perturbations of von {N}eumann algebras. {I}. {S}tability of type},
journal = {Amer. J. Math.},
volume = {94},
number = {1},
year = {1972},
pages = {38--54},
doi = {10.2307/2373592},
}
@misc{peterson2026,
author = {Peterson, Jesse},
title = {Open problems in operator algebras},
howpublished = {Web page},
year = {2026},
url = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
note = {Problem O.4; updated 15 May 2026},
}
@article{cameron2014,
author = {Cameron, Jan and Christensen, Erik and Sinclair, Allan M. and Smith, Roger R. and White, Stuart and Wiggins, Alan D.},
title = {{K}adison--{K}astler stable factors},
journal = {Duke Math. J.},
volume = {163},
number = {14},
year = {2014},
pages = {2639--2686},
doi = {10.1215/00127094-2819736},
eprint = {1209.4116},
archivePrefix = {arXiv},
}
@misc{aim-kadison-kastler,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniform spatial stability of von Neumann algebras}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/034-kadison-kastler.md},
note = {Entry 034 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
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.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00691, "Uniform spatial stability of von Neumann algebras", clopen.solutions.