connes_embedding_problem

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Connes' embedding problem (1976): Can every finite von Neumann algebra with a faithful normal tracial state be embedded into an ultrapower of the hyperfinite II₁ factor? Refuted in 2020 (Ji–Natarajan–Vidick–Wright–Yuen via MIP*=RE), showing the answer is NO. This formalization captures a weaker matrix approximation version.

Why this node was retired

The posted statement is

import Mathlib

theorem connes_embedding_problem :
    ∀ (A : Type*) [Ring A] [StarRing A] [Algebra ℂ A]
      (tau : A →+ ℂ) (htrace : ∀ a b : A, tau (a * b) = tau (b * a))
      (hpos : ∀ a : A, 0 ≤ (tau (star a * a)).re)
      (hnorm : tau 1 = 1),
    ∃ (n : ℕ) (embed : A →*  Matrix (Fin n) (Fin n) ℂ),
      Function.Injective embed := by
  sorry

A single finite matrix algebra replaces an ultrapower/approximation statement, and trace faithfulness and von Neumann structure are absent.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Historical formulation: every separable type II₁ factor with faithful normal tracial state admits a trace-preserving embedding into an ultrapower R^ω of the hyperfinite II₁ factor. This conjecture was refuted; restoring it does not produce a true theorem. Specify an equivalent finite-set moment-approximation formulation if desired, rather than an exact injection into one finite matrix algebra.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.