An explicit counterexample to Connes' embedding conjecture: a concretely presented type II$_1$ factor not embedding into an ultrapower of the hyperfinite factor

The theorem MIP* = RE of Ji, Natarajan, Vidick, Wright, and Yuen (2020) shows that approximating the entangled value of a two-prover nonlocal game is as hard as the Halting Problem, and thereby refutes Connes' 1976 embedding conjecture (CEC) from the theory of von Neumann algebras. The refutation, however, is purely nonconstructive: the separating object is produced by a diagonal fixed-point construction tied to undecidability, and no counterexample factor is written down. The first open question posed in the paper is to construct an explicit type II$_1$ factor that fails the Connes embedding property, i.e. that does not embed into any tracial ultrapower $R^\omega$ of the hyperfinite II$_1$ factor $R$; a stronger form asks for an explicit non-hyperlinear group (which, since every sofic group is hyperlinear, would necessarily also be non-sofic), whose existence is itself open. As of 2026 no explicit counterexample of any kind is known: later work (Goldbring–Hart, Manzoor) has strengthened only the nonconstructive existence side. The complexity-theoretic companion questions from the same section of the source paper (MIP$^{co}$ =? coRE and the zero-gap classification) have been resolved or claimed resolved in the 2022–2025 literature, so this task concerns only the explicit construction.

Background

A von Neumann algebra is a self-adjoint algebra of bounded operators on a Hilbert space closed under weak limits; a factor is one with trivial center, and a type II$_1$ factor is an infinite-dimensional factor carrying a faithful normal tracial state $\tau$ (a positive linear functional with $\tau(1)=1$ and $\tau(ab)=\tau(ba)$ for all elements $a,b$). The hyperfinite II$_1$ factor $R$ is the smallest infinite-dimensional factor, obtained as the weak closure of an increasing chain of matrix algebras, and it is the building block of approximately finite-dimensional operator algebras. A tracial ultrapower $R^\omega$ is a nonstandard enlargement of $R$ built from bounded sequences of its elements modulo a free ultrafilter $\omega$ on the natural numbers. Connes' 1976 embedding conjecture predicted that every separable II$_1$ factor embeds into some $R^\omega$; equivalently, that every finite part of every II$_1$ factor can be approximated in trace norm by finite-dimensional matrices. The conjecture organized a large web of equivalences, notably with Kirchberg's QWEP conjecture (the assertion that every C*-algebra is a quotient of one with a suitable weak-expectation approximation property, shown by Kirchberg to be equivalent to CEC) and with Tsirelson's problem in quantum information (the question of whether the tensor-product and commuting-operator correlation sets below coincide).

A nonlocal game is a one-round test in which a classical randomized verifier sends questions to two cooperating provers who cannot communicate and must produce answers that the verifier accepts or rejects. The entangled value $\operatorname{val}^*(G)$ of a game $G$ is the maximal winning probability when the provers may share an arbitrary finite-dimensional entangled state and apply local measurements (tensor-product model); the commuting-operator value $\operatorname{val}^{co}(G)$ allows all measurement operators of both provers to act on one Hilbert space, with operators belonging to different provers required to commute. MIP* is the class of languages decidable by a polynomial-time classical verifier interacting with several entangled provers. Ji–Natarajan–Vidick–Wright–Yuen proved MIP* = RE: distinguishing $\operatorname{val}^*(G)=1$ from $\operatorname{val}^*(G)\le 1/2$ is complete for the recursively enumerable languages. Through the equivalence chain of Fritz, Junge–Navascués–Palazuelos–Pérez-García–Scholz–Werner, and Ozawa, this separates the commuting-operator from the (closure of the) tensor-product correlation sets and refutes CEC. The refutation is nonconstructive: the witnessing game arises from iterating a compression procedure (a transformation that replaces a nonlocal game by a smaller one while approximately preserving its value) and taking a fixed point indexed by a Turing machine, and the authors explicitly leave the explicit description of a non-embeddable factor as their first open question.

Two constructions make the question concrete. First, in a synchronous nonlocal game (both provers receive questions from the same set and must return identical answers when given identical questions), Kim–Paulsen–Schafhauser associate to the game a -algebra generated by projections satisfying the game's constraint relations; perfect commuting-operator strategies correspond to nontrivial representations of this algebra, while approximability of the game value by finite-dimensional strategies is governed by the tensor-product value. The separating game of the MIP = RE paper is synchronous with $\operatorname{val}^*(G)<\operatorname{val}^{co}(G)=1$, so this correspondence is the route the authors point to for converting their game into a factor failing the Connes embedding property. Second, a countable group is sofic if its left-multiplication action can be approximated by permutations of finite sets, and hyperlinear if the group can be approximated by unitary matrices in normalized Hilbert–Schmidt distance; every sofic group is hyperlinear, and by a theorem of Rădulescu a group is hyperlinear exactly when its group von Neumann algebra $L(G)$ — the von Neumann algebra generated by the unitary operators of left multiplication acting on the group — is Connes-embeddable. Hence an explicit non-hyperlinear group that is i.c.c. (infinite conjugacy classes, so that $L(G)$ is a factor; and, since every sofic group is hyperlinear, necessarily also non-sofic) would yield an explicit non-embeddable II$_1$ factor; a merely non-sofic group would not do so by itself, since a non-sofic group could still be hyperlinear and hence Connes-embeddable. Whether any non-sofic or non-hyperlinear group exists at all is a long-standing open problem, with no example of either kind known.

Problem Statement

Construct an explicit counterexample to Connes' embedding conjecture: give an explicit description of a separable type II$_1$ factor $M$ that does not admit a trace-preserving embedding into any tracial ultrapower $R^\omega$ of the hyperfinite II$_1$ factor $R$. The description must be concretely presentable — for example, a finitely presented group together with the construction of its group von Neumann algebra, a finitely generated *-algebra given by explicit generators and relations, or a finite synchronous nonlocal game (explicit question set, answer set, and acceptance predicate) together with an explicit construction such as the Kim–Paulsen–Schafhauser correspondence from which $M$ is obtained — and the submission must include a complete proof of non-embeddability. The listed formats are illustrative rather than mandatory: any other finite, explicit presentation from which $M$ is concretely obtained qualifies. A pure existence argument, or a construction whose parameters are defined only through non-halting computations of Turing machines, does not answer the question: the presentation itself must be written down and the non-embeddability must be proven. A stronger form of the problem, also posed in the source, is an explicit finitely presented i.c.c. group that is not hyperlinear — and therefore, since every sofic group is hyperlinear, not sofic either; this subsumes the factor form because the group von Neumann algebra of an i.c.c. group is a factor and that of a non-hyperlinear group fails Connes embeddability, and note that even the mere existence of such a group is currently unknown. The factor $M$ is the required deliverable; the group form is an optional stronger alternative that also resolves the task. (The existence of a merely non-sofic group is a weaker open question and would not by itself produce a non-embeddable factor, since a non-sofic group could still be hyperlinear.)

References

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