Tensor products of maximal injective subalgebras
Ge and Kadison's work shows that $B_1 \mathbin{\overline\otimes} B_2$ is maximal injective in $M_1 \mathbin{\overline\otimes} M_2$ when $M_1$ is injective, and Fang proved the same when $M_1$ has separable predual and $B_1$ is a factor (more generally, when the centre of $B_1$ is atomic).
Problem. Let $M_1$ and $M_2$ be II$_1$ factors and $B_1 \subseteq M_1$, $B_2 \subseteq M_2$ maximal injective von Neumann subalgebras. Is $B_1 \mathbin{\overline\otimes} B_2$ a maximal injective von Neumann subalgebra of $M_1 \mathbin{\overline\otimes} M_2$?
Source: S. Popa, Maximal injective subalgebras in factors associated with free groups (1983), Problem 4.5(1); listed as S.9 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).
Imported from math.uwaterloo.ca/~j37peter/problems.html.
Classification
- Primary
- OA — Operator Algebras
- Keywords
- II₁ factors, maximal injective subalgebras, Peterson's operator algebra problems, tensor products
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{popa1983,
author = {Popa, Sorin},
title = {Maximal injective subalgebras in factors associated with free groups},
journal = {Adv. Math.},
volume = {50},
number = {1},
pages = {27--48},
year = {1983},
doi = {10.1016/0001-8708(83)90033-6},
}
@misc{peterson-s9,
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 S.9; version of 15 May 2026},
}
@article{ge1996,
author = {Ge, Liming},
title = {On maximal injective subalgebras of factors},
journal = {Adv. Math.},
volume = {118},
number = {1},
pages = {34--70},
year = {1996},
doi = {10.1006/aima.1996.0017},
}
@article{gekadison1996,
author = {Ge, Liming and Kadison, Richard},
title = {On tensor products of von {N}eumann algebras},
journal = {Invent. Math.},
volume = {123},
number = {3},
pages = {453--466},
year = {1996},
doi = {10.1007/s002220050036},
}
@article{fang2007,
author = {Fang, Junsheng},
title = {On maximal injective subalgebras of tensor products of von {N}eumann algebras},
journal = {J. Funct. Anal.},
volume = {244},
number = {1},
pages = {277--288},
year = {2007},
doi = {10.1016/j.jfa.2006.12.006},
}
@article{shen2006,
author = {Shen, Junhao},
title = {Maximal injective subalgebras of tensor products of free group factors},
journal = {J. Funct. Anal.},
volume = {240},
number = {2},
pages = {334--348},
year = {2006},
doi = {10.1016/j.jfa.2006.03.017},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00967, "Tensor products of maximal injective subalgebras", clopen.solutions.