IRA Rigidity and boomerang subalgebras of group von Neumann algebras
Abstract
We introduce the notion of Boomerang subalgebras of group von Neumann algebras and study their general properties. We show that every Boomerang subalgebra of L(Γ) is the group von Neumann algebra of a Boomerang subgroup when Γ is a countable C*-simple acylindrically hyperbolic group or a finite product of such groups. Consequently, every such group is invariant random algebra (IRA) rigid: every IRA arises as the pushforward of an invariant random subgroup (IRS) under the group von Neumann algebra construction. We also establish IRA rigidity for adjoint K-split higher rank S-arithmetic groups, as well as for all countable mixed identity-free (MIF) groups and their finite products. Consequently, IRA rigidity holds for countable groups admitting a topologically free minimal extremely proximal action on a compact Hausdorff space with more than two points. The same conclusion holds for countable groups admitting a faithful minimal action of general type on a tree whose induced boundary action is topologically free, and for finite products of groups admitting either of these actions.
Related
- Intermediate subalgebras of cross-product von Neumann algebras arising from weakly compact actions
- Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras
- Distances Between von Neumann Subalgebras: Spin Models, Commuting Squares, and Free Group Factors
- Coarse rigidity in von Neumann algebras via derivations
- Associativity of operator convolutions for group actions on von Neumann algebras
BibTeX
@misc{amrutam2026rigidity,
title = {{IRA Rigidity and boomerang subalgebras of group von Neumann algebras}},
author = {Tattwamasi Amrutam and Yair Glasner and Yair Hartman and Yongle Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41417},
eprint = {2610.41417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41417},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - IRA Rigidity and boomerang subalgebras of group von Neumann algebras
AU - Tattwamasi Amrutam
AU - Yair Glasner
AU - Yair Hartman
AU - Yongle Jiang
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41417
UR - https://arxiv.org/abs/2610.41417
AB - We introduce the notion of Boomerang subalgebras of group von Neumann algebras and study their general properties. We show that every Boomerang subalgebra of L(Γ) is the group von Neumann algebra of a Boomerang subgroup when Γ is a countable C^*-simple acylindrically hyperbolic group or a finite product of such groups. Consequently, every such group is invariant random algebra (IRA) rigid: every IRA arises as the pushforward of an invariant random subgroup (IRS) under the group von Neumann algebra construction. We also establish IRA rigidity for adjoint K-split higher rank S-arithmetic groups, as well as for all countable mixed identity-free (MIF) groups and their finite products. Consequently, IRA rigidity holds for countable groups admitting a topologically free minimal extremely proximal action on a compact Hausdorff space with more than two points. The same conclusion holds for countable groups admitting a faithful minimal action of general type on a tree whose induced boundary action is topologically free, and for finite products of groups admitting either of these actions.
ER -