50 Open Problems: Ultraproduct II$_1$ factors

A collection of 50 open problems around the structure theory of ultraproducts of II$_1$ factors is presented, along with some annotations and references.

💡 Research Summary

The paper presents a systematic collection of fifty open problems concerning the structure theory of ultraproducts of II₁ factors. After fixing notation—U denotes a non‑principal ultrafilter on ℕ, N_U the tracial ultrapower of a tracial von Neumann algebra N, M_U the tracial matrix ultraproduct, and G_U the discrete group ultrapower—the authors introduce the notion of elementary equivalence for II₁ factors (N₁ ≡_e N₂ if there exist ultrafilters U₁, U₂ with N₁^{U₁} ≅ N₂^{U₂}).

The first batch of questions (Problems 1–6) asks for a classification of elementary equivalence classes of matrix ultraproducts M_U as U varies, and whether every such class contains a genuine matrix ultraproduct. Problem 2 is the “ergodic Connes‑embedding problem”: given a non‑Γ, Connes‑embeddable II₁ factor N, does there exist an embedding π : N → M_U with trivial relative commutant? Partial results are known for groups with property (T) and for factors with positive 1‑bounded entropy.

Problems 3–5 introduce the concept of pseudo‑compactness: a non‑Γ factor N is pseudo‑compact if it is elementarily equivalent to some M_U. The authors ask whether the free group factors L(F_n) (n ≥ 2) are pseudo‑compact, and whether any group von Neumann algebra L(G) can be pseudo‑compact. Problem 5 asks whether a finite‑index subfactor of a matrix ultraproduct must itself be a matrix ultraproduct.

Problem 6 concerns the symmetric group ultrapower S_U ⊂ M_U and the von Neumann algebra W⁎(S_U) it generates. Since W⁎(S_U) has a Cartan subalgebra while M_U does not, the question is whether W⁎(S_U) contains a copy of a matrix ultraproduct.

The next set (Problems 7–14) deals with existential embeddings. An embedding ι : N → M is existential if it can be extended to an embedding of M into an ultrapower of N that restricts to the diagonal on N. The authors ask for separable II₁ factors N admitting an existential copy of L(F₂) but not decomposing as a non‑trivial free product (Problem 7), for strongly solid N that embed existentially into a factor with a Cartan subalgebra (Problem 8), and for groups G containing an existential copy of F₂ yet not giving rise to a free group factor (Problem 9). Further questions explore primeness of existential extensions of biexact factors (Problem 10) and whether Haagerup property can coexist with Property (T) under existential inclusion (Problem 11).

Problem 12 asks whether a Haagerup factor N with a subalgebra B ⊂ N^U having Property (T) must already arise from a matrix ultraproduct. Problem 13 queries the existence of an existential embedding of L(F₂) into M_U, and whether a random Haar‑sampling embedding is existential. Problem 14 asks for countable groups G such that L(G) is existentially closed, and whether there exist existentially closed II₁ factors not arising from groups.

Problems 15–22 focus on Γ‑type properties and McDuff phenomena. Problem 15 asks whether a sequence of separable non‑Γ factors can have an ultraproduct with Property Γ. Problem 16 asks for many non‑Γ factors that are pairwise non‑elementarily equivalent. Problems 17–19 investigate elementary inequivalence among super‑McDuff factors and whether elementary equivalence preserves the super‑McDuff property. Problem 20 asks for Γ‑factors that are not McDuff yet not elementarily equivalent. Problem 21 asks whether the Thompson group factor L(F) is super‑McDuff. Problem 22 asks whether McDuffness of a tensor product forces one tensor factor to be McDuff.

Problems 23–26 introduce sequential commutation: two Haar unitaries u, v are sequentially commuting if there exists a finite chain of Haar unitaries with successive commutators zero. The commutation diameter O(M) is the supremum of the minimal chain lengths. The authors ask for non‑Γ factors with a unique sequential commutation orbit and prescribed diameter (Problem 23), for Property (T) factors with a unique orbit (Problem 24), for factors without diffuse relative‑(T) subalgebras but with a unique orbit (Problem 25), and for factors with zero 1‑bounded entropy yet continuum many orbits (Problem 26).

Problems 27–34 are about lifting and conjugacy in ultrapowers. Problem 27 asks whether freely independent Haar unitaries in a free product remain free after a suitable ultrapower conjugacy. Problem 28 asks whether a non‑prime factor can have a unique sequential commutation orbit. Problem 29 asks whether N and its tensor powers N^t have the same number of orbits. Problems 30–31 ask whether independent unitaries in N^U can be lifted to independent sequences in N (for all k). Problem 32 asks for a unitary conjugacy between two embeddings of a separable subfactor into N^U when they are asymptotically conjugate via a sequence of u.c.p. maps. Problem 33 asks whether a non‑amenable separable factor has a unique embedding up to unitary conjugacy into its ultrapower. Problem 34 asks whether a Connes‑embeddable factor whose embeddings into R^U are all conjugate must be isomorphic to the hyperfinite factor R.

Problem 35 asks whether the pentagon right‑angle Artin group factor is stable in the sense of tracial stability.

Problems 36–50 are miscellaneous but deep. Problem 36 (the II₁‑Tarski problem) asks whether the free group factors L(F_n) are elementarily equivalent for different n; the C*‑algebraic analogue has been resolved negatively. Problem 37 (Popa) asks whether a Connes‑embeddable factor can embed into R^U with factorial commutant. Problem 38 asks whether an index‑2 subfactor of an ultrapower must arise as an ultraproduct of index‑2 subfactors. Problem 39 asks whether any ultrapower N^U can be a group von Neumann algebra (the answer is expected to be negative). Problem 40 asks for a factor N with infinite 1‑bounded entropy in its ultrapower but zero entropy in N itself. Problem 41 asks whether the double commutant condition (N′∩N^U)′∩N^U = N forces N ≅ R. Problem 42 asks whether M_2(ℂ)⊗M always embeds into M^U (trivial if M is Connes‑embeddable). Problem 43 asks for the existence of an enforceable II₁ factor; a negative answer would imply a negative solution to Connes’ embedding problem (already known). Problem 44 asks whether solidity and U‑solidity coincide. Problem 45 asks for pairs of II₁ factors with the same 2‑quantifier theory but not elementarily equivalent. Problem 46 asks whether elementary equivalence can fail at the C*‑algebra level. Problem 47 asks whether every separable II₁ factor is self‑less as a C*‑algebra (known for ultraproduct factors). Problem 48 asks for a factor whose ultrapower has a non‑full fundamental group. Problem 49 asks whether equality of tensor products of ultrapowers forces elementary equivalence of the factors. Problem 50 asks whether equality of free products of ultrapowers forces equality of index sets and pairwise elementary equivalence.

Each problem is accompanied by brief remarks on known partial results and references to recent work (e.g., Alekseev‑Thom on universal sofic groups, Popa‑Jung on relative commutants, Goldbring‑Hart on small fragment theories, Jekel on 1‑bounded entropy, etc.). The acknowledgments list several experts who contributed comments.

Overall, the paper serves as a roadmap for future research on ultraproducts of II₁ factors, highlighting deep connections between model theory, free probability, group theory, and operator algebraic rigidity. The fifty problems span classification, embedding, rigidity, entropy, and stability phenomena, and solving any of them would represent a significant advance in our understanding of the fine structure of II₁ factors and their ultrapowers.