By an “-ergodicity question” I mean a question about constructing embeddings of the hyperfinite II factor into another von Neumann factor that’s *ergodic*, in the sense that the action is ergodic, while at the same time verifies various other properties.

I favor the term “ergodic embedding”, rather than previous terminology “irreducible inclusion”, or “inclusion with trivial relative commutant”, because it better emphasizes the dynamical interaction between (sub)algebras. It has indeed become more and more apparent in recent years that analysis in a II factor (and in a von Neumann algebra in general) is best served by considering not only as a collection of elements interacting via “multiplication dynamics”, but also as the host of its subalgebras, generated by subsystems of elements. Especially subsystems of commuting elements, generating abelian von Neumann subalgebras of (notably maximal abelian ones, or *MASA*s) and more generally approximately finite dimensional (*AFD*) subsystems/subalgebras, notably -generating subsystems. But also “canonical subsystems/subalgebras”, that come up from the way is constructed. For instance, if arises as a crossed product construction, where is a tracial von Neumann algebra and is a trace preserving action of a group on it, then and are canonical subalgebras.

The interplay between subalgebras of a II factor is particularly important in deformation-rigidity and intertwining by bimodules techniques, where pairs of subalgebras are “moved around” inside , in the presence of some spectral gap rigidity property intrinsic to , while their interaction is being watched from the perspective of the attract/repel dichotomy (compact versus weak mixing), reflected by ergodicity properties of the embedding .

An initial result about embedding ergodically into another separable factor was obtained in [P81], in case is II, with a complete answer in [P19a], where it is shown that any continuous separable von Neumann factor contains ergodic copies of . This is the same as saying that the action , which is ergodic (due to being a factor), has an “-direction” in which it is ergodic. It is enough for many applications, but the really interesting applications start when additional ergodicity properties can be met.

The typical additional requirement is that be embeddable ergodically into while also being contained in a specific ergodic subfactor of . In other words: given an inclusion of factors with ergodic, does there exist an -direction in which this action is ergodic? It turns out that this is an extremely complex, difficult problem. The answer depends on the nature of the inclusion . It is quite remarkable that three of the most famous and important problems in operator algebras, namely the free group factor problem, Connes Bicentralizer conjecture and Connes Embedding problem, can be approached from this perspective. This adds to the two previous, more `classic'' applications of -ergodicity results, to non-commutative Stone-Weierstrass and vanishing cohomology results. In the rest of this blog-entry I will discuss the specific -ergodicity questions arising from each one of these problems. I will end with a brief comment on`

the possible use of -ergodicity on another classic problem, about whether any AW* II1 factor is a von Neumann algebra.

## (a) The free group factor problem

I have commented at length in [P18b], [P19a], [P19b] and in my blog-entry from November 2019, about the fact that a positive answer to the conjecture below, asserting that any SSG factor can be made “sandwich” between two -ergodic embeddings, would imply that is infinitely generated and that the free group factors , , are non-isomorphic.

TIGHTNESS CONJECTURE: *Any II factor that’s stably single generated, abbreviated SSG (meaning that any amplification of can be generated by two self-adjoint elements), has an -tight decomposition, i.e., there exists a pair of embeddings of the hyperfinite II factor such that is ergodic, or equivalently .*

The “-ergodicity and tightness phenomenology” brought to light two additional refinements of ergodicity for an inclusion of factors : *MASA-ergodicity*, requiring that contains an abelian -subalgebra that’s a MASA (maximal abelian -subalgebra) in ; and *MV-ergodicity*, requiring that has the weak relative Dixmier property, i.e., give any the weak closure of the convex set of averagings intersects the scalars (MV stands here for mean value). It is shown in [P19a] that MASA-ergodicity implies -ergodicity, which in turn obviously implies MV-ergodicity.

But the converse implications “ergodic MV-ergodic” and “MV-ergodic -ergodic” do not hold true in general (see [P19a]). An interesting remaining question is whether -ergodic implies MASA-ergodic. Noticing that if is approximately finite dimensional (AFD) then, once ergodic, an inclusion is automatically MV-ergodic, a first question to answer is *whether any ergodic embedding of the hyperfinite II factor is automatically MASA-ergodic*.

For instance, if is an ergodic inclusion of hyperfinite II factors, then the basic construction inclusion follows ergodic, and the question is whether it is in fact MASA-ergodic. First cases to check would be when , for some infinite amenable group acting freely on , and , with a trace preserving action of an ICC amenable group on a tracial von Neumann algebra , that acts ergodically on the center of , and .

A question that came out naturally from the tightness conjecture was whether the left-right action of the unitary group of a II factor on , which is obviously ergodic, is in fact MV-ergodic. This has been clarified by Das-Peterson in [DP19], through their *double ergodicity* theorem: indeed, given any countably generated (equivalently separable) II factor , if one takes to be the Markov (or Laplacian) u.c.p. map given by a (at most) countable set of generators and positive weights summing up to , the left-right averaging by and pushes any to the scalars.

With this at hand, a way to approach the tightness conjecture would be to use the SSG property combined with the fact the Laplacian can be taken to be supported by just two generating unitaries, , in any amplification of , to construct recursively a tight pair (see [P19a], [P19b] for more comments on this).

Returning to the general framework of arbitrary factors, I’d like to point out that deciding whether a specific ergodic inclusion of factors is -ergodic, MASA-ergodic, or MV-ergodic, or that it does not satisfy one of these conditions, is usually quite subtle. Related to this, note that MASA-ergodicity and -ergodicity are both “stable properties”, in that if satisfies any of these properties then any “amplification” of the inclusion satisfies it as well. But it is not evident at all whether the MV-ergodicity is always stable.

## (b) On Connes’s bicentralizer (CB) conjecture

In its original form, as formulated by Connes in 1976-1978, this conjecture states that any separable III factor has a normal faithful state whose bicentralizer is trivial. This means that if one denote by the asymptotic centralizer of , i.e., , then the bicentralizer is reduced to the scalars. In [H87] Haagerup showed that the above conjecture is equivalent to the existence of an ergodic embedding of the hyperfinite II factor with normal conditional expectation. It has been noticed in [P19a] that, due to [H87], the conjecture is also equivalent to the fact that the II core , in the continuous decomposition of , which is known to be ergodic by Connes-Takesaski theorem, is in fact -ergodic (and also equivalent to it being MASA-ergodic, and respectively MV-ergodic).

III factors are amazing mathematical objects and this conjecture is certainly the most important open problem in this area. Two key background results to understand the full depth of this question are the Connes-Takeaski “Flow of weight” paper [CT76] (notably their relative commutant theorem) and Haagerup’s solution to the conjecture in the injective/amenable case (which finished off the proof of the uniqueness of the amenable/AFD factor of type III). There has been steady progress on this problem in recent years, due to work by Cyril Houdayer, Yusuke Isono, Amine Marrakchi, Dima Shlyakhtenko, Stefaan Vaes, with many classes of III factors being shown to check the CB property.

I would like to propose here a related problem: While Connes had since 1976 a proof that if an injective/amenable III factor satisfies CB property then it is isomorphic to the unique Araki-Woods III factor (cf. [C85]) and Haagerup produced an alternative proof of this fact, I think there is room for a “better proof”.

More precisely, I think that one can use in a more direct way the amenability of together with the fact that has a MASA with normal conditional expectation on it (which by [H87] is what CB property implies) to show that must then have a Cartan subalgebra, i.e., that it is a Krieger-factor. A possible proof of this could be in the spirit of the proof to Connes’ uniqueness of the amenable II factor in [P85] and of the proof of Connes-Feldman-Weiss theorem in [CFW81], about classifying amenable OE-relations of all types. Ideally, such a proof may not even need the assumption that is of type III (a case which however allows using Connes-Stormer transitivity theorem). In other words, it is worth trying to prove that given any amenable factor having a MASA with normal expectation, , one can “reconstruct” around a new MASA with expectation , obtained as a limit diagonals in some dyadic AFD approximation of .

The fact that the CB problem can be formulated as whether or not can be embedded ergodically, with expectation, into any (separable) III factor, makes it fit into the general theme of this Blog-entry, which is about “-ergodicity questions”. But there is an additional reason for which I wanted to include it here, a reason that also explains why I put it in line with the free group factor problem and Connes Embedding.

Thus, while exploring ways of proving that cannot be finitely generated during the Summer 2019, I thought it was plausible for the following *general principle* to be true: if an ergodic inclusion of factors (i.e., with trivial relative commutant) is so that is stably single generated, abbreviated SSG (I recalled in Part what this means for a II factor, while for properly infinite factors this is automatic), then is necessarily MASA ergodic.

The intuition behind this speculation is the following: the ergodicity of the inclusion means that any that wo-approximately commutes with the two unitary generators , is close to scalars on arbitrarily large “finite dimensional window” (see Lemma 5.1 in [P19b]); so one may hope that this should imply that if almost commutes with either one of , on a large window, then this may already have an effect on how looks in a much smaller window; by approximating that unitary spectrally, this could be the starting point for constructing recursively partition of that give that’s a MASA in .

As it turns out, this intuition is too simplistic, and in this generality the “MASA-ergodicity principle” is false, something it is worth discussing. But before explaining this, let me mention that such a general “principle” would imply that cannot be single generated, and that both CB and Connes Embedding conjecture would hold true!

Indeed, if assumed to be finitely generated, would follow SSG, because it has non-trivial fundamental group (cf. [V88]; see above and [P18b], [P19a], [P19b] for this discussion), so if one takes an irreducible , and then apply the “principle” to , one gets a MASA of that sits in . This implies that the span of is dense in , for some vector , contradicting [GP96].

For the CB conjecture this is trivial, because if is of type III and is its II core (continuous decomposition) then is SSG (because properly infinite), so the principle would imply contains a MASA of , and CB would follow by Haagerup’s results in [H87].

As for the CE conjecture, this is because MASA-ergodicity implies -ergodicity (cf. [P19a]) and because I knew for some time that if an ergodic inclusion that’s an -amplification of certain basic construction inclusions, such as , i.e., , can be shown to be -ergodic, then CE would hold true (I will explain this in a forthcoming paper). Thus, since is properly infinite, the “principle” would lead to CE being true.

But, as I mentioned before, the “MASA-ergodicity principle” is totally false in this generality. In fact, the various “higher levels” of ergodicity of an ergodic inclusion of factors (MV-ergodicity, -ergodicity, MASA-ergodicity) depend heavily on the nature of the inclusion, not just on the properties of . This is much explained in [P19a]. I will show in a forthcoming paper that there is a “sharp obstruction” for the -ergodicity of such an inclusion, and more generally for the existence of “tight” Hilbert -bimodules (i.e., ) that satisfy “almost orthogonality” properties , for finitely many given (in the style of [P18b]), a property that would imply CE as well.

However, all these -ergodicity statements that imply the CE property are not “iff” statements, they are just sufficient conditions. So showing they do not hold true leaves CE problem untouched. I recall below the CE problem in its original form ([C76]), for completeness, sending to [O04], [O13] for surveys on this problem and on its many equivalent formulations, notably in C-algebras (Kirchberg QWEP conjecture) and quantum information theory (Tsirelson’s conjecture).

## (c) Connes Embedding (CE) conjecture

**Connes Embedding (CE) conjecture/problem** asks whether any II factor (and more generally any tracial von Neumann algebra) can be simulated in moments by matrices. This is equivalent to whether any separable II factor can be embedded in an ultrapower , of the hyperfinite II factor, where is a free ultrafilter on . This problem is particularly interesting for group von Neumann algebras, .

There has been a lot of excitement in recent months about the CE problem being solved in the negative by Z. Ji, A. Natarajan, T. Vidick, J. Wright, H. Yuen (see arXiv:2001.04383), as a consequence of a more general result in complexity theory, whose statement gives the title of their paper: *MIP=RE*. The authors have developed an impressive array of deep, new techniques to prove this result, with several preparatory papers starting in 2017. There is an ongoing effort towards checking this work, by several groups of people, which due to its sheer length and novelty of methods may take some time.

Note that even if this work is confirmed to be correct, it leaves open the question of whether there exist groups whose von Neumann algebra is not CE. I have explained in my June 2019 Blog-entry of how the CE property for a group (in fact, for the wreath product ) is equivalent to a vanishing 2-cohomology problem for the cocycle action arising from the Bernoulli -action with base , and a strategy for proving that certain groups do not have this property (see also Sec. 6 in [P18a]). I will get back to this in Q2.

**(d) Vanishing cohomology problems**

The next set of problems related to “-ergodicity” that I want to discuss consists of vanishing cohomology questions. The reason -ergodicity is relevant to such problems is due to the fact that if an embedding is ergodic in some appropriate augmentation of , the amenability of can be used to “push” any into by averaging over unitaries in , via the Ad-action. When applied to suitable , this amounts to “untwisting” a cocycle.

This philosophy applies to two types of vanishing cohomology problems: the Hochschild-type cohomology of II factors (of Kadison and Ringrose); and the 2-cohomology for cocycle actions of groups on II factors.

The first of these problems is about calculating for the Hochschild -cohomology space , obtained as the quotient between the space of -cocycles of the II factor into an -bimodule , and the space of inner such cocycles. For instance, amounts to the space of *derivations* (i.e., linear maps satisfying ) moded-out by the space of *inner derivations* , where , .

In theory, such spaces provide a multitude of isomorphism invariants for II factors. But unfortunately, all attempts to find some class of bimodules for which this invariant is non-trivial and calculable have failed so far. In the absence of that, the focus has been on proving that for many classes of II factors and bimodules , one has (any derivation of into is inner), and more generally . As I mentioned before, if can be shown to contain a “large” ergodic copy or , or pairs of such copies, then “integrating” over such -directions is often sufficient to show innerness of the derivation (more generally of the -cocycle).

The second of these problems is about showing that given some free cocycle action of a group on a II factor, , it “untwists”, i.e., each , can be perturbed by an inner automorphism Ad of so that , becomes a “genuine” action. I am refering to [P18a] for a detailed discussion of this problem and the definitions involved.

Within these two types of “vanishing cohomology” questions, let me mention a few specific ones where -ergodicity may play a role.

While all derivations of a II factor into itself have been shown to be inner by Kadison and Sakai in the 1960s, the problem of whether (more generally , ) for all II factors remained open. Christensen, Pop, Sinclair and Smith proved that this indeed the case for II factors that admit various “thin” bimodule decomposition over a pair of hyperfinite subfactors, , and more generally over a pair of AFD subalgebras (see [CPSS03] and the references therein). Whether for all factors probably depends on whether this is true for the free group factors . More precisely, I believe that if one can prove it for , then almost certainly for all factors, and some appropriate -ergodicity should play a key role in the proof. A related famous problem, known to be equivalent to the *similarity problem*, is whether for all II factors, notably for , see Pisier’s book [Pi01] for a comprehensive account on this. While this may well not be the case, this cohomology does probably vanish whenever has good bimodule-decomposition over pairs of AFD subalgebras, i.e., for all “thin” factors, a class that includes factors with Cartan subalgebras, and property Gamma factors (see [CPSS03] for this latter case, and for other supporting evidence).

What is the class of all countable groups for which any free cocycle action untwists to a genuine action? Do there exist groups in this class for which there exists a free cocycle action on some II factor (e.g., on the free group factor ) that doesn’t untwist ? See [P18a] for more on this.

## (e) The Stone-Weirestrass (S-W) problems

In operator algebras a statement qualifies as “S-W result/problem” if it is of the form: “*If are C-algebras usually taken separable such that separates some special class of states of , then *”. The “special class” of states has to be so that when is a commutative C-algebra, becomes the space of pure states of . This type of problems have been quite fashionable in the early years of operator algebras, and mathematicians such as Kaplansky, Kadison, Glimm, Sakai, Pedersen, Akemann, Anderson have put much effort into it. The most general problem along these lines, called the *pure state S-W problem*, is the following: given two C-algebras , is it true that if separates the pure states of , then ? This is still open, though there have been many nice partial results along these lines in 1950-1970 (see e.g. [AB81] for an account).

One of the best S-W results to this date remains Glimm’s theorem [G60], showing that if separates the states obtained as the -closure of the pure states of , then . Note that if is commutative, then is closed, so one indeed has .

Notice also that in the commutative case a pure state is the same as a factor state. So another legitimate S-W problem is whether if separates the factor states of then . Work of Anderson and Bunce in [AB81] showed that if one could prove that any (separable) factor contains a semi-regular MASA, then this *factorial state S-W problem* would have a positive solution. One way to prove existence of such MASAs in arbitrary factors is by showing that any factor contains an ergodic AFD subfactor with a regular MASA that’s still a MASA in . In [P81], [P83], I have shown that this is indeed the case whenever is of type II or III, . The remaining type III case has been settled by Longo and me, independently, in ([P84], [L84]), thus solving the factorial S-W. My recent result in [P19a] makes this more precise: in fact any separable continuous factor contains an ergodic -embedding with a diagonal of that’s a MASA in .

It is also shown in [AB81] that the pure state S-W would hold true if one could prove that any (separable) factor contains a regular MASA. But it is by now known that many group II factors arising from countable ICC groups do not have regular MASAs, see [V94], [OP07], [PV11]. (N.B.: in the II case, a regular MASA has normal expectation onto it, so it is what one calls a *Cartan subalgebra*). Nevertheless, I think that ideas in [AB81], and prior to that in [G60], [S70], should be re-examined, in the hope that the pure state S-W problem could be reduced to proving an appropriate -ergodicity statement.

I should add that, while the commutative S-W theorem is a fundamental result in classic analysis, I am not aware of any application of the S-W statements for non-commutative C-algebras. Perhaps because of that, the interest on this problem has faded over the last few decades… Nevertheless, the pure state S-W problem remains a “classic problem” in this subject, an intrinsically beautiful problem, over 60 years old by now, and certainly worth solving!

## (f) **The quasi-trace problem for AW* II_1 factors**

I end this Blog-entry with another “classic” unsolved problem in C and W-algebras, for which -ergodicity results may be useful. To state it, let me recall that after Murray-von Neumann’s papers [MvN36]-[MvN43], a key problem faced by people interested in this subject was to characterize abstractly the “rings of operators” (later called von Neumann algebras, or W-algebras). This effort culminated with Sakai’s 1956 paper showing that von Neumann algebras are C-algebras that are dual Banach spaces. But prior to that, Kaplansky speculated in the early 1950s that any C-algebra with the property that any of its maximal abelian -subalgebras is a von Neumann algebra (C-algebras with this property have been called AW-algebras), is a von Neumann (or W) algebra , i.e., it can be represented on a Hilbert space as a wo-closed -algebra of operators.

There has been much work on this problem over the years, for instance showing that the geometry of projections basically works the same way for AW-algebras, leading to a similar classification into type I, II and III, finite/infinite AW-algebras, and factoriality. I refer to https://encyclopediaofmath.org/wiki/AW*-algebra for a brief historical account.

For us here, we’ll only need to know that an *AW* II *factor* is a unital C-algebra with the following properties: all MASAs of are diffuse W-algebras; satisfies the finiteness axiom “ with implies ” (any isometry in must be a unitary); is a factor (center reduced to scalars). The remaining and most interesting case of Kaplansky’s problem is whether any AW II factor is a W-algebra (or von Neumann algebra).

It is not hard to see that for an AW II factor the Murray-von Neumann construction of a completely additive dimension function goes exactly as in [MvN36], leading to a unique faithful completely additive *quasi-trace* , i.e., a map that satisfies the positivity/traciality property , is faithful, linear and completely additive on any MASA, and satisfies for self-adjoint.

Taking all this into account, Kaplansky’s problem reduces to:

**The quasi-trace problem for AW II factors**, which asks whether the canonical quasi-trace on such an algebra is necessarily linear.

Haagerup proved in 1991 that this is indeed the case whenever contains a nuclear (more generally exact) C-subalgebra that’s dense in with respect to a metric defined out of the quasi-trace of (see [H14]). This is the best result so far in this direction, and for anybody who wants to get acquainted with this problem, [H14] is a “must read” paper.

I tend to believe that the “quasi-trace problem” for AW II factors has a positive answer in full generality. One can adapt the iterative method of constructing -embeddings in [P81], [P18b] from W II (tracial) factors to AW II factors , to get ergodic embeddings into such with “room to spare” for additional properties to be satisfied, and this may be of use for solving the problem.

**References**

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