# Q3: Bicommutant characterization of the hyperfinte II1 factor

In the introduction of his 1976 “Classification of injective factors” paper (see bottom of page 73 in [C76]), Alain Connes makes the following remark: Another remarkable property of the factor ${R}$ is that if ${S\subset R}$ is any self-adjoint subset, the von Neumann subalgebra ${M}$ of ${R}$ generated by ${S}$ can be characterized by a bicommutation property, analogue to the bicommutation in a type I factor…. In the case ${S=R}$, the condition he then formulates amounts to the bicommutant relation ${(R'\cap R^\omega)'\cap R^\omega=R}$. This fact is stated so casually (as an evident fact), that it may have been broadly known at the time, being probably of “folklore nature”. Indeed, the proof is an easy exercise.

In informal discussions during that period, both Connes and Takesaki have apparently asked whether this bicommutant condition characterizes the hyperfinite factor, i.e., whether if ${M}$ is a (separable) II ${_1}$ factor with the property ${(M'\cap M^\omega)'\cap M^\omega=M}$, then necessarily ${M \simeq R}$. However, the problem didn’t really “take-off” at the time.

The question resurfaced again some 15 years later, raised by Masamichi Takesaki during a seminar at UCLA in the early 1990s. Masamichi actually formulated it as: characterize all (separable) II ${_1}$ factors ${M}$ satisfying the bicommutant condition ${(M'\cap M^\omega)'\cap M^\omega=M}$.

It is at that time that I “connected” with the problem. I made right away the easy observation that if the (necessarily McDuff) II ${_1}$ factor ${M}$ satisfies the above bicommutant condition, then it cannot be of the form ${N_0\otimes N_1}$ with ${N_0}$ a non-Gamma II ${_1}$ factor, i.e., ${M}$ cannot “split-off” a non-Gamma II ${_1}$ factor. That’s because if this would be the case, then the commutant ${M'\cap M^\omega}$ would follow equal to ${N_1'\cap N_1^\omega}$ (as a consequence of Connes’ spectral gap property for non-Gamma factors), whose commutant in ${M^\omega}$ contains the non-separable factor ${N_0^\omega}$, implying that the bicommutant ${(M'\cap M^\omega)'\cap M^\omega}$ is non-separable, thus not equal to ${M}$.

While I cannot argue that this is an important, consequential problem, it certainly is one of my preferred open problems in II ${_1}$ factors. At a first glance it looks surprising that the question has not been answered to this day. But anybody who would take some time to reflect on it would soon recognize that it is a very hard, beautiful problem. I have discussed this problem repeatedly with colleagues over the years, notably with Uffe Haagerup, Kenley Jung, Dima Shlyakhtenko, who seemed to share this opinion.

I have formulated the problem, and various generalizations of it, in my paper [P13], where a version for orbit equivalence relations is stated as well.

I could venture into describing ideas towards a possible proof, but I think none of them is “encouraging” enough to be worth mentioning here.

• [C76] A. Connes: Classification of injective factors, Ann. of Math. 104 (1976), 73-115.
• [P13] S. Popa: Independence properties in subalgebras of ultraproduct II ${_1}$ factors, Journal of Functional Analysis 266 (2014), 5818-5846 (math.OA/1308.3982)