Existence of the enforceable II$_1$ factor

After the negative resolution of the Connes embedding problem (CEP) via the complexity-theoretic identity MIP*=RE, the model-theoretic analysis of tracial von Neumann algebras leaves a central question open: does there exist an enforceable II$_1$ factor $E$, a separable II$_1$ factor whose isomorphism type is forced by a winning strategy in the model-building game for tracial von Neumann algebras? Goldbring's dichotomy theorem states that exactly one of the following holds: either the necessarily unique enforceable factor $E$ exists, or every enforceable property of tracial von Neumann algebras is shared by continuum many pairwise nonisomorphic separable algebras. A positive answer would produce a canonical II$_1$ factor rivaling the hyperfinite factor $R$ and embedding into every existentially closed factor; a negative answer is a strong, generic failure of Connes embedding. The problem remains open as of August 2026.

Background

A von Neumann algebra is a unital self-adjoint algebra of bounded operators on a Hilbert space that is closed in the weak operator topology. A II$_1$ factor is an infinite-dimensional von Neumann algebra with trivial center that carries a faithful normal tracial state $\tau$ (a positive linear functional with $\tau(1)=1$ and $\tau(xy)=\tau(yx)$ for all elements $x$ and $y$). The hyperfinite II$_1$ factor $R$ is the unique separable II$_1$ factor containing an increasing sequence of finite-dimensional matrix algebras with dense union; it embeds into every II$_1$ factor and plays the role of the smallest II$_1$ factor. Given a free ultrafilter $\mathcal U$ on the natural numbers, the tracial ultrapower $R^{\mathcal U}$ is the quotient of the algebra of bounded sequences in $R$ by the ideal of sequences whose 2-norms $\|x\|_\tau=\tau(x^*x)^{1/2}$ tend to zero along $\mathcal U$; it is a nonseparable II$_1$ factor that serves as an infinite-dimensional approximation scheme for $R$.

The Connes embedding problem (CEP), raised in Connes' 1976 classification of the injective factors (those that are retracts of any von Neumann algebra containing them), asked whether every separable II$_1$ factor embeds into some $R^{\mathcal U}$, that is, whether every II$_1$ factor is approximable by matrices in trace norm. In 2020 the problem was resolved negatively: Ji, Natarajan, Vidick, Wright, and Yuen proved the complexity-theoretic identity MIP*=RE (equality of the class of languages verifiable by multi-prover interactive proofs with entangled quantum provers with the recursively enumerable languages), which, combined with the equivalence of the CEP with Tsirelson's problem on whether quantum correlations of commuting and of tensor-product observables coincide, yields separable II$_1$ factors with no embedding into any $R^{\mathcal U}$.

The question comes from the model-theoretic aftermath of that refutation. In continuous logic, a first-order logic for metric structures in which truth values live in $[0,1]$ and quantifiers are suprema and infima, separable tracial von Neumann algebras form an elementary class. Goldbring adapted to this setting a classical model-construction technique known as model-theoretic forcing or building models by games. Fix a countably infinite set $C$ of constant symbols. Two players, $\forall$ and $\exists$, take turns playing finite sets $\Sigma$ of expressions of the form $|\,\|p(c)\|_\tau-r\,|<\epsilon$, where $p$ is a *-polynomial (a noncommutative polynomial in variables and their adjoints), $c$ is a tuple from $C$, and $r$ and $\epsilon$ are rationals with $\epsilon>0$; each move must extend the previous one and be satisfiable in some tracial von Neumann algebra. A definitive play, one that pins down the 2-norm of every *-polynomial at every tuple of constants, compiles to a unique separable tracial von Neumann algebra, called the compiled structure. A property $P$ of tracial von Neumann algebras is enforceable if player $\exists$ has a strategy guaranteeing that the compiled structure has $P$ regardless of the opponent's moves, and a tracial von Neumann algebra $M$ is enforceable if the property of being isomorphic to $M$ is enforceable; at most one algebra can be enforceable up to isomorphism.

Enforceable properties turn out to be abundant: being a II$_1$ factor, being McDuff (isomorphic to one's tensor product with $R$), and being existentially closed are all enforceable, and a countable conjunction of enforceable properties is enforceable. An existentially closed (e.c.) tracial von Neumann algebra is one in which any behavior that could approximately occur in an extension already approximately occurs inside the algebra, the analog for this setting of an algebraically closed field. Every e.c. algebra is a McDuff II$_1$ factor all of whose automorphisms are approximately inner; every tracial von Neumann algebra embeds into an e.c. one; and by a theorem of Farah, Goldbring, Hart, and Sherman, $R$ itself is e.c. exactly if the CEP holds, so after the refutation no e.c. algebra embeds into $R^{\mathcal U}$.

Goldbring proved that the CEP is equivalent to $R$ being the enforceable tracial von Neumann algebra, so the refutation of the CEP shows that $R$ is not enforceable, but it leaves open whether some other factor is. In many neighboring classes the enforceable object exists and is the canonical member of the class: the enforceable graph is the Rado random graph, the enforceable field of a fixed characteristic is the algebraic closure of the prime field, the enforceable metric space is the Urysohn space, and the enforceable Hilbert space is the separable infinite-dimensional one; for discrete groups, by contrast, no enforceable object exists. The dichotomy theorem of 'Enforceable operator algebras' states that exactly one of the following holds: (1) for every enforceable property $P$ of tracial von Neumann algebras there exist continuum many pairwise nonisomorphic separable tracial von Neumann algebras with property $P$; or (2) the enforceable II$_1$ factor $E$ exists. Should $E$ exist, it would be e.c. and McDuff with only approximately inner automorphisms, would embed into every e.c. factor just as $R$ embeds into every II$_1$ factor, would have the Jung property (every embedding of $E$ into its own ultrapower $E^{\mathcal U}$ is unitarily conjugate to the diagonal embedding; by a theorem of Kenley Jung, $R$ has this property, and Atkinson and Kunnawalkam Elayavalli showed that $R$ is the only $R^{\mathcal U}$-embeddable factor that does), and could not be isomorphic to $M\otimes M$ for any tracial von Neumann algebra $M$; indeed it could not even satisfy the same continuous first-order sentences as $E\otimes E$.

Problem Statement

Does the enforceable II$_1$ factor exist? That is, does there exist a separable II$_1$ factor $E$ such that the property of being isomorphic to $E$ is enforceable in the model-building game for tracial von Neumann algebras, equivalently such that player $\exists$ has a strategy guaranteeing that the compiled structure of any definitive play is isomorphic to $E$?

By the dichotomy theorem, this is equivalent to deciding which of two alternatives holds for the class of tracial von Neumann algebras:

(1) for every enforceable property $P$, there exist continuum many pairwise nonisomorphic separable tracial von Neumann algebras with property $P$; or

(2) the enforceable II$_1$ factor $E$ exists, necessarily unique up to isomorphism.

A positive answer should identify the factor $E$, or otherwise prove its existence, together with a proof of its enforceability. A negative answer should establish alternative (1).

References

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