The Ubiquitous Hyperfinite II_1 factor

Q2: CE and Vanishing 2-Cohomology for \text{Out}(R)\curvearrowright R \vee (R'\cap R^\omega)

I resume here my comments on the eight questions pertaining to the hyperfinite II{_1} factor listed in an early September 2020 blog-entry. I have labelled the questions Q1-Q8 and have already discussed Q1. I will now discuss Q2, which is about an equivalent formulation of the Connes Embedding (CE) problem. It is a re-iteration of comments I made in a June 2019 Blog-entry, but with more details. I am grateful to Stefaan Vaes for lots of useful feed-back.

The starting point of this remark is the well known fact that any automorphism {\theta} of the hyperfinite II{_1} factor {R} is approximately inner, thus being implemented by a unitary {U_\theta=(u_n)_n} in the normalizer {\mathcal{N}} of {R} in {R^\omega} (i.e., Ad{(U_\theta)=\theta} on {R}). Since {U_\theta RU_\theta^*=R}, it follows that {U_\theta} normalizes the centralizer factor {R_\omega=R'\cap R^\omega} ([C75]) as well.

The bicommutant relation {(R'\cap R^\omega)'\cap R^\omega=R} (implicit in [C75], [C76]; for a complete proof see for instance Section 2 in [P13]) implies that Ad{(U_\theta)} implements an outer automorphism on the centralizer factor {R'\cap R^\omega} iff {\theta} on {R} is outer, and if this is the case then the automorphism {\theta_\omega} implemented by Ad{(U_\theta)} on {R\vee (R'\cap R^\omega)} is outer. The choice of {U_\theta} is of course not unique, but any other {U'_\theta\in \mathcal{N}} that implements {\theta} on {R} is a perturbation of {U_\theta} by a unitary in {R'\cap R^\omega}. So the class of {\theta_\omega} in Out{(R\vee R_\omega)} is unique. Since {\text{\rm Out}(R) \ni \theta \mapsto \theta_\omega} is clearly a group morphism when viewed with values in Out{(R\vee R_\omega)}, this gives a free cocycle action of {\text{\rm Out}(R)} on {R \vee R_\omega}, with the corresponding crossed product II{_1} factor {(R\vee R_\omega) \rtimes \text{\rm Out} R} equal to the von Neumann algebra generated by {\mathcal{N}} in {R^\omega}.

In case {\Gamma \curvearrowright^\sigma R} is a free action of a countable group (e.g., a Bernoulli action), this map provides a free cocycle-action {\{\sigma_\omega(g)\}_g} on {R\vee R_\omega}, with the corresponding 2-cocycle {v^\sigma_\omega(g,h):=U_{\sigma(g)}U_{\sigma(h)}U_{\sigma(gh)}^*} lying in {\mathcal{U}(R_\omega)}. If the 2-cocycle {v^\sigma_\omega} is co-boundary, i.e. if there exist {w_g\in \mathcal{U}(R_\omega)} so that {U'_g=w_gU_{\sigma(g)}} gives a representation of {\Gamma} (N.B. This is equivalent to {v^\sigma_\omega(g,h)=(w_g\tilde{\sigma}_\omega(g)(w_h))^*w_{gh}}, {\forall g,h}; it is also equivalent to being able to choose the unitaries {U_{\sigma(g)}, g\in \Gamma}, in a “group-manner”), then {R\vee \{U'_g\}_g} is clearly isomorphic to {R\rtimes_\sigma \Gamma}, with {U'_g} playing the role of the “canonical unitaries” in this crossed-product, implying that {M=R\rtimes_\sigma \Gamma} embeds into {R^\omega}, so {M} satisfies the CE property.

Conversely, since any {R\simeq R_1\simeq^\gamma R_2\simeq R} in {R^\omega} are unitary conjugate, with a unitary implementing the specific isomorphism {\gamma}, if {M\hookrightarrow R^\omega} then we may also assume the resulting copy of {R} in {M=R\rtimes_\sigma \Gamma} goes on the constant sequences {R\subset R^\omega}, thus the canonical unitaries {\{u_g\}_g\subset M\subset R^\omega} are necessarily {R_\omega}-perturbations of {\{U_{\sigma(g)}\}_g}. Since they give a representation of {\Gamma}, this means the 2-cocycle {v^\sigma_\omega} is co-boundary.

In other words, a crossed product II{_1} factor {M=R\rtimes_\sigma \Gamma}, corresponding to a free action {\sigma} of a group {\Gamma} on {R}, is CE if and only if the 2-cocycle {v^\sigma_\omega}, corresponding to the cocycle action {\{\sigma_\omega(g)\}_g} of {\Gamma} on {R\vee R_\omega} (or just considered on {R_\omega}), is co-boundary. Or alternatively, iff the unitaries {U_{\sigma(g)}, g\in \Gamma}, can be chosen in a group-manner.

One should note that if one takes {\sigma} to be the Bernoulli {\Gamma} action with base {R}, the CE property for {R\rtimes_\sigma \Gamma} amounts to the wreath product group {H\wr \Gamma} having the CE property (i.e., {R^\omega} contains a copy of the left regular representation of {H\wr \Gamma}), whenever {H} is an amenable ICC group, like {H=S_\infty}.

I have made this remark some twenty years ago. For me it was an important moment, because after being convinced for many years that CE holds true, it made me change my mind to being convinced it was not true… I think that anybody who is familiar with Alain Connes’ invariant {\chi(M)} and his paper [C75], or with Vaughan Jones hand written Notes on Connes’ Invariant {\chi(M) } ( and his paper [J80], would have the reaction that such “universal vanishing 2-cohomology” for any action of a group {\Gamma} on {R} (equivalently, a group-manner choice of {U_{\sigma(g)}, g\in \Gamma}), is very unlikely.

I have shared this remark with several colleagues over the years, but only made it public in [P18], where however I did not propose any concrete way of approaching CE from this angle. There are in fact several ways to go, all based on the idea of creating a framework similar to [P01], and using deformation-rigidity arguments.

I will explain here one such venue.

I’ll begin by only assuming {\Gamma \curvearrowright^\sigma R} is a free {\Gamma}-action with the property that {M=R\rtimes_\sigma \Gamma} is embeddable into {R^\omega} (so {M} is CE), in other words that one can choose the {U_{\sigma(g)}\in \mathcal{N}} implementing {\sigma} in a group-manner. By “doubling” them as {U_{\sigma(g)}\otimes U_{\sigma(g)}} one gets unitaries in the normalizer of {R\otimes R} in {(R\otimes R)^\omega} that implement the double action {\sigma\otimes \sigma} of {\Gamma} on {R\otimes R} (in all that follows, tensor product notation means “von Neumann tensor product”). Assume in addition that {R = R\otimes 1 \subset R \otimes R} has an s-malleable deformation {(\alpha, \beta)} (see [P01]), like for instance the case when {\sigma} is a Bernoulli {\Gamma}-action with base {\mathbb{M}_2(\mathbb{C})}. Denote {(\alpha^\omega, \beta^\omega)} the corresponding ultrapower actions they implement on {(R\otimes R)^\omega}. Note that while {\alpha} is a continuous {\mathbb{R}}-action on {R}, its ultrapower {\alpha^\omega} is not necessarily continuous on (all) {(R\otimes R)^\omega}.

So far, all of this comes from the assumption that {R\rtimes_\sigma \Gamma} has CE property. But now I would also like that the unitaries {U_{\sigma(g)}\otimes U_{\sigma(g)}} are fixed by {\alpha^\omega}. This is because I am trying to create a framework where the deformation-rigidity arguments “a la [P01]” can be carried out. This means to choose the {U_{\sigma(g)}} so that in addition to being group-like, they are so that the doubling {U_{\sigma(g)}\otimes U_{\sigma(g)}} lies in the fixed point algebra of {\alpha^\omega}.

At this point I should mention that if {\sigma} is the Bernoulli {\Gamma}-action of base {\mathbb{M}_2(\mathbb{C})} (or of base {R}) one can indeed choose each individual {U_{\sigma(g)}} in this manner, by using permutation tensors and doubling them. But making this choice group-like is much harder. This is however easy to do for residually finite groups {\Gamma} and more generally for “quotient-limits” of residually finite groups, i.e. for groups {\Gamma} which are limits of a sequence {\Gamma_n} of consecutive quotients of a group (like in Gromov’s well known construction of “many” simple groups with the property (T)), if one assumes each {\Gamma_n} is residually finite.

Moreover, I would like the action {\tilde{\sigma}_\omega} implemented via Ad by {\{U_{\sigma(g)}\otimes U_{\sigma(g)}\}_g} on {(R\otimes R) \vee (R\otimes R)_\omega} to have a large “interesting part” that’s {\alpha^\omega}-continuous and {\alpha^\omega}-invariant.

The continuous part of {\alpha^\omega} on {(R\otimes R)^\omega} is easily seen to be a von Neumann algebra. Identifying it is perhaps challenging. But one should not need all of this algebra. Note for instance that if {\alpha} is the s-malleable deformation of a Bernoulli {\Gamma}-action with base {\mathbb{M}_2(\mathbb{C})}, as in [P01], then if {x=(x_n)_n \in R^\omega = (R\otimes 1)^\omega} is so that {x_n} have uniformly bounded “tensor-length” (i.e., supported by a tensor product of at most {K} of the {(\mathbb{M}_2(\mathbb{C}))_g}, for all {n}, where {K<\infty}), then {x} is {\alpha^\omega}-continuous. The von Neumann algebra {\mathcal{R}} of all such elements in {(R\otimes 1)^\omega} is an AFD factor (non-separable!). Its doubling {\mathcal{R}\otimes \mathcal{R}} is normalized by {\tilde{\sigma}_\omega} when the unitaries {\{U_{\sigma(g)}\otimes U_{\sigma(g)}\}_g} that implement this action are chosen tensor permutations. (In fact, the algebra of all “bounded tensors” in {(R\otimes R)^\omega} is invariant to all “tensor-permutation” unitaries of the double form {U\otimes U}).

Thus, it is possible to “create” out of a CE-type assumption (in the above, we have in fact used the stronger assumption that {\Gamma} is a quotient limit of residually finite groups) a large AFD II{_1} factor {\mathcal{R}} with a {\Gamma}-action on it {\sigma_\omega}, and with a (continuous) s-malleable deformation of {\mathcal{R}\otimes \mathcal{R}}, so that the double action {\tilde{\sigma}_\omega} commutes with this malleable deformation, in the spirit of [P01]. Note that this {\Gamma} action is far from being ergodic on {\mathcal{R}\otimes \mathcal{R}}.  The hope for this approach is that one could construct some 1-cocycle for the {\Gamma}-action {\tilde{\sigma}_\omega} that “lives” on {\mathcal{R}\otimes 1}. If so, and if {\Gamma} has property (T) and is simple, then one could use the argument in [P01], plus the fact that for such {\Gamma} the double action is automatically weak mixing relative to its fixed points. This should imply that any such 1-cocycle is equivalent to a group morphism from {\Gamma} into the fixed point algebra of {\sigma_\omega}. For {\Gamma} simple with (T), this would be a contradiction, because the fixed point algebra is AFD. 

Note that this already implies that not all hyperbolic groups are residually finite, thus settling a famous open problem in geometric group theory! But in order to show the existence of a group {\Gamma} that’s not CE by using this line of thought, one needs to be able to “lift” {U_{\sigma(g)}\in \mathcal{N}} as above, with its doubling lying in the fixed point algebra of {\alpha^\omega}. This is a first challenge for this approach. The second and much bigger challenge is to produce (one sided) 1-cocycles for {\tilde{\sigma}_\omega}.

Let me also recall here Remark 6.2.3{^\circ} in [P18], where I have asked whether the whole free cocycle action of the group Out{(R)} on {R\vee R_\omega} has non-vanishing 2-cohomology. In other words, whether one can choose the {U_\theta \in \mathcal{N}, \theta \in \text{\rm Out}(R)}, in a group-manner. This is of course much stronger than having CE property for all {H\wr \Gamma}, with {H} ICC amenable and {\Gamma} countable group, and should definitely have a negative answer, but even this is still open for now.


  • [C75] A. Connes: Sur la classification des facteurs de type II, C. R. Acad. Sci. Paris Sr. A–B 281 (1975), A13–A15.
  • [C76] A. Connes: Classification of injective factors, Ann. of Math. 104 (1976), 73–115.
  • [J80] V. Jones:  A II{_1} factor anti-isomorphic to itself without involutory antiautomorphisms, Math. Scand., 46 (1980), 103-117
  • [P01] S. Popa: Some rigidity results for non-commutative Bernoulli shifts, J. Fnal. Analysis 230 (2006), 273–328 (MSRI preprint No. 2001–005).
  • [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, Ann. Ec. Norm Sup, 54 (2021), 409–445 (math.OA/1802.09964)

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