Pith Number

pith:HZDLCCG3

pith:2026:HZDLCCG3UGPOAD672TM4QCVZVG

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On ultraproduct approximations and property (T) factors

Jesse Peterson

A framework for deformation and rigidity in continuous logic shows that L(SL3(Z)) and L(F2) are not elementarily equivalent.

arxiv:2605.16669 v1 · 2026-05-15 · math.OA · math.GR

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Claims

C1strongest claim

We show that the group von Neumann algebras L(SL3(Z)) and L F2 are not elementarily equivalent, and we show that the group von Neumann algebra L F2 is not pseudomatricial. We also show a Bass-Serre type strong rigidity result in the setting of ultraproducts to provide an infinite family of pairwise non-elementarily equivalent full factors, each of which embeds into an ultraproduct of the hyperfinite II1 factor.

C2weakest assumption

The introduced framework correctly transfers key aspects of deformation/rigidity theory into the continuous model theory setting for II1 factors without essential loss of applicability or introduction of artifacts that would invalidate the equivalence and rigidity conclusions. (Stated in the opening of the abstract as the basis for solving the listed open problems.)

C3one line summary

A framework is introduced to transfer deformation/rigidity methods into the continuous model theory of II1 factors, proving non-elementary equivalence of L(SL3(Z)) and LF2, non-pseudomatriciality of LF2, and existence of infinite families of pairwise non-equivalent full factors.

References

16 extracted · 16 resolved · 0 Pith anchors

[1] Anantharaman-Delaroche,Amenable correspondences and approximation properties for von Neu- mann algebras, Pacific J 1995

[2] [AKE21] Scott Atkinson and Srivatsav Kunnawalkam Elayavalli,On ultraproduct embeddings and amenability for tracial von Neumann algebras, Int. Math. Res. Not. IMRN (2021), no. 4, 2882–2918. [BCI17] R´ 2021

[3] 30 JESSE PETERSON [Bek07] Bachir Bekka,Operator-algebraic superridigity forSL n(Z),n≥3, Invent. Math.169(2007), no. 2, 401–425. [BO08] Nathanial P. Brown and Narutaka Ozawa,C ∗-algebras and finite-dim 2007

[4] Brown,Topological dynamical systems associated toII 1-factors, Adv 2011

[5] [GH17] Isaac Goldbring and Bradd Hart,On the theories of McDuff’sII 1 factors, Int. Math. Res. Not. IMRN (2017), no. 18, 5609–5628. [GH23] ,A survey on the model theory of tracial von Neumann algebras 2017

Formal links

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Receipt and verification

First computed	2026-05-20T00:02:35.455721Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

3e46b108dba19ee00fdfd4d9c80ab9a9bf7debd47b5c28e30cd2426142c8f527

Aliases

arxiv: 2605.16669 · arxiv_version: 2605.16669v1 · doi: 10.48550/arxiv.2605.16669 · pith_short_12: HZDLCCG3UGPO · pith_short_16: HZDLCCG3UGPOAD67 · pith_short_8: HZDLCCG3

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/HZDLCCG3UGPOAD672TM4QCVZVG \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 3e46b108dba19ee00fdfd4d9c80ab9a9bf7debd47b5c28e30cd2426142c8f527

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b0aa3af01b117c50a49083ec7b014c0926842fc871a31d8e738d8423f050f65e",
    "cross_cats_sorted": [
      "math.GR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.OA",
    "submitted_at": "2026-05-15T22:11:21Z",
    "title_canon_sha256": "1ce14febbfb5ec412b5a1a4165cbcaaedc7c30c1df52a5789e8a2698fd548fd1"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16669",
    "kind": "arxiv",
    "version": 1
  }
}