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Peer-reviewedCC BYCompositio Mathematica

There is an equivalence relation whose von Neumann algebra is not Connes embeddable

  • Aareyan Manzoor1
  1. 1University of Waterloo
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Abstract

The landmark quantum complexity result MIP Superscript asterisk ∗ * equals = = RE has been used to prove the existence of a non-Connes-embeddable tracial von Neumann algebra. Recently, similar ideas have been used to give a negative solution to the Aldous–Lyons conjecture: there is a non-co-sofic invariant random subgroup (IRS) on any non-abelian free group. We define a notion of hyperlinearity for an IRS and show that there is a non-co-hyperlinear IRS on any non-abelian free group. As a corollary, we prove that there is a relation whose von Neumann algebra is not Connes embeddable. We do this by significantly simplifying the reduction of Aldous–Lyons to non-local games, removing the need for subgroup tests entirely.

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Cite

BibTeXRIS
BibTeX
@article{manzoor2026there,
  title = {{There is an equivalence relation whose von Neumann algebra is not Connes embeddable}},
  author = {Aareyan Manzoor},
  journal = {Compositio Mathematica},
  year = {2026},
  month = aug,
  doi = {10.1017/s0010437x26103431},
  url = {https://doi.org/10.1017/s0010437x26103431},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - There is an equivalence relation whose von Neumann algebra is not Connes embeddable
AU  - Aareyan Manzoor
PY  - 2026
DA  - 2026/08/01
JO  - Compositio Mathematica
DO  - 10.1017/s0010437x26103431
UR  - https://doi.org/10.1017/s0010437x26103431
AB  - The landmark quantum complexity result MIP Superscript asterisk ∗ ^* equals = = RE has been used to prove the existence of a non-Connes-embeddable tracial von Neumann algebra. Recently, similar ideas have been used to give a negative solution to the Aldous–Lyons conjecture: there is a non-co-sofic invariant random subgroup (IRS) on any non-abelian free group. We define a notion of hyperlinearity for an IRS and show that there is a non-co-hyperlinear IRS on any non-abelian free group. As a corollary, we prove that there is a relation whose von Neumann algebra is not Connes embeddable. We do this by significantly simplifying the reduction of Aldous–Lyons to non-local games, removing the need for subgroup tests entirely.
SN  - 0010-437X
SN  - 1570-5846
ER  -