The invariant subspace problem relative to a von Neumann algebra
A projection $p \in M$ satisfies $pxp = xp$ exactly when its range is an invariant subspace of $x$. For $M = \mathcal B(H)$, $H$ separable and infinite-dimensional, the question is the invariant subspace problem.
Problem. Let $M \ne \mathbb C 1$ be a von Neumann algebra and $x \in M$. Is there a projection $p \in M$ with $p \ne 0, 1$ and $pxp = xp$?
Source: Listed without attribution as O.9 in J. Peterson, Open problems in operator algebras (version of 15 May 2026); the invariant subspace problem relative to a von Neumann algebra, in the form of U. Haagerup's ICM 2002 lecture.
Imported from math.uwaterloo.ca/~j37peter/problems.html.
Classification
- Primary
- OA — Operator Algebras
- Secondary
- FA
- Keywords
- invariant projections, invariant subspace problem, Peterson's operator algebra problems, von Neumann algebras
References
@misc{peterson-o9,
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.9; version of 15 May 2026},
}
@misc{haagerup2002,
author = {Haagerup, Uffe},
title = {Random matrices, free probability and the invariant subspace problem relative to a von {N}eumann algebra},
year = {2002},
eprint = {math/0212396},
archivePrefix = {arXiv},
}
@article{haagerup2009,
author = {Haagerup, Uffe and Schultz, Hanne},
title = {Invariant subspaces for operators in a general {II}$_1$-factor},
journal = {Publ. Math. Inst. Hautes {\'E}tudes Sci.},
volume = {109},
pages = {19--111},
year = {2009},
doi = {10.1007/s10240-009-0018-7},
}
@article{dykema2004,
author = {Dykema, Ken and Haagerup, Uffe},
title = {Invariant subspaces of the quasinilpotent {DT}-operator},
journal = {J. Funct. Anal.},
volume = {209},
number = {2},
pages = {332--366},
year = {2004},
doi = {10.1016/S0022-1236(03)00167-8},
}
Resolution claims
For every irrational angle, we construct a continuous nonnegative circle weight with exactly one zero and logarithmic integral $-\infty$ whose weighted rotation has no nontrivial invariant projection in the associated hyperfinite type $\mathrm{II}_1$ factor. This answers negatively the question of Zhu, Fang, and Shi, with the angle prescribed in advance. The resulting operator is nonzero and norm-quasinilpotent.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-29 | Needs verification → Solution claimed | — | Recorded here: math @ 7ef5c0c, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01061, "The invariant subspace problem relative to a von Neumann algebra", clopen.solutions.