The Ubiquitous Hyperfinite II_1 Factor

Ever since it has been defined by Murray and von Neumann in their famous “Rings of operators” papers some eight decades ago ([MvN36]-[MvN43]), the hyperfinite II{_1} factor {R} played a central role in operator algebras (both C{{}^*} and W{^*}). It is certainly a mathematical object of fundamental importance.

Its construction as a limit of dyadic matrices, {R:=\overline{\otimes}_n (\mathbb{M}_{2\times 2}(\mathbb{C}), tr)_n}, makes it the natural non-commutative version of the probability measure space {([0,1], \mu)}, and it perfectly illustrates the phenomenon of continuous dimension.

The hyperfinite factor is the only II{_1} factor that has so far been characterized by an abstract property: already identified in [MvN43] as the unique approximately finite dimensional (AFD) II{_1} factor, {R} was then shown by Connes to even be the unique amenable II{_1} factor ([C76]). Since any II{_1} factor arising from “amenable data” is amenable as a II{_1} factor, this provides many ways of representing {R}, most notably as group factors {L\Gamma}, arising from ICC (infinite conjugacy class) amenable groups {\Gamma}, or as group measure space factors {L(\Gamma \curvearrowright X)=L^\infty X\rtimes \Gamma}, arising from free ergodic probability measure preserving actions of amenable groups {\Gamma \curvearrowright X}.

This provides the “most extreme”, and indeed the prototype, of the many-to-one paradigm for von Neumann algebras, where a large class {\mathcal{G}} of distinct geometric objects are shown to give rise to the same II{_1} factor, i.e., are all W{^*}equivalent.

Connes theorem also implies that any II{_1} subfactor of {R} is isomorphic to {R}. When combined with the Murray-von Neumann result that {R} embeds into any other factor (see [MvN43]), this shows that {R} is the “smallest” II{_1} factor, with respect to the embedding order relation. As it turns out, a number of important problems in operator algebras can be reduced to constructing “special” embeddings of {R} into factors, especially embeddings that satisfy various degrees of ergodicity properties (see [P19]). Some of these problems have been solved, but most of these “{R}-ergodicity” questions are wide open.

In fact, despite being the “most understood II{_1} factor”, it is surprising how many questions about the hyperfinite II{_1} factor are still un-answered. I will comment on several of them, which I have grouped into eight parts, as follows:

  • {Q1} {R}-ergodicity questions.
  • {Q2} CE and vanishing 2-cohomology for {\text{\rm Out}(R) \curvearrowright R \vee (R'\cap R^\omega)}
  • {Q3} Bicommutant characterization of {R}
  • {Q4} Can one embed \mathbb{F}_2 discretely into {\mathcal{U}(R)}?
  • {Q5} Is {R} quasidiagonal ?
  • {Q6} Malleability characterization of {R}
  • {Q7} Classifying smooth and discrete dynamics on {R}
  • {Q8} Symmetries of {R} and the commuting square problem

Some of these questions are notoriously hard and long standing, and answering them would be a major achievement (e.g. {Q2}, {Q7}, {Q8}). The first question {Q1} is somewhat vague as stated. I meant it as a “theme”, that in fact encompasses many sub-questions, which in turn are very specific and have deep consequences. While not of central importance, questions {Q3}, {Q4}, {Q6} are intriguing. They illustrate well how mysterious the structure of {R} still remains. Most certainly question {Q5} has a negative answer, and it is quite surprising that this fact has not been established to this day.

My intention is to discuss these questions one by one, over several blog entries, including motivation, historical background, possible strategies to solve, etc. To a certain extent, the first three questions were also discussed in my recent papers ([P18], [P19], [P13]). But on this blog the style will be more informal and the comments more detailed.

For a “fast track” account of the basic results on the hyperfinite II{_1} factor, see:
(this is a series of lectures I have given at RIMS in Kyoto in April 2019, under the title “The ubiquitous hyperfinite II{_1} factor”).


  • [C76] A. Connes: Classification of injective factors, Ann. of Math. 104 (1976), 73-115.
  • [MvN36] F. Murray, J. von Neumann: On rings of operators, Ann. Math. 37 (1936), 116-229.
  • [MvN43] F. Murray, J. von Neumann: On rings of operators IV, Ann. Math. 44 (1943), 716-808.
  • [P13] S. Popa: Independence properties in subalgebras of ultraproduct II{_1} factors, Journal of Functional Analysis 266 (2014), 5818-5846 (math.OA/1308.3982)
  • [P18] S. Popa: On the vanishing cohomology problem for cocycle actions of groups on II{_1} factors, to appear in Ann. Ec. Norm. Super., math.OA/1802.09964
  • [P19] S. Popa: On ergodic embeddings of factors, to appear in Communications Math. Phys., arXiv:1910.06923

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s