It is well known that while in the first of their “rings-cycle”, [MvN36], Murray and von Neumann have explicitly formulated several problems (all of which having been clarified by now), in their subsequent papers they do not formally state any. There are however several problems that, while not spelled out as such, do come across rather clearly in the last of their papers, [MvN43]. They are easily identifiable:
The problem of whether the II factors arising from non-isomorphic infinite conjugacy class (ICC) groups may or may not give rise to isomorphic group factors . This general “isomorphism (or classification) problem” is very present in the Introduction and in Section 6.3 of [MvN43], but it is not explicitly formulated. It is certainly the “implicit question” with most impact on the subject, a problem that has been the motivation behind many developments in operator algebras (Mc Duff’s work, Connes -invariant and uniqueness of the amenable II factor, Haagerup’s c.b.-approximation properties and -invariants, Voiculescu’s free probability theory, deformation rigidity theory, etc, etc).
Murray and von Neumann have considered the following classes of ICC groups:
locally finite groups, notably ;
free products of groups, such as , ;
a “melange” of free products and usual products, such as , or an infinite alternation of these operations, producing an ICC group satisfying the stability condition (see the Appendix in [MvN43]);
they also consider on page 794 the wreath product groups , which they show are ICC whenever , noticing that the associated II factor coincides with the group measure space factor arising from the Bernoulli action , which they have earlier introduced in [MvN36].
As a consequence of their (Theorem XIV in [MvN43]) on the uniqueness of the approximately finite dimensional (AFD) II factor (later called the hyperfinite II factor), denoted , they show that any locally finite ICC group has associated II factor isomorphic to . But the only result about non-isomorphism of group II factors they are able to prove is that ICC groups of the form give rise to II factors that are not isomorphic to , with , . In particular, are not isomorphic to . They do this by using their property Gamma (introduced in 6.1.1), which accounts for having a non-trivial asymptotic centralizer.
There is much emphasis on factors arising from free products of groups, , as one varies (cf. 6.3 in [MvN43]), in particular on the free groups ‘s. There is also a palpable feeling of frustration that they cannot distinguish between more group factors (see also below).
The free group factors , are thus defined in [MvN43] and the isomorphism problem for factors arising from distinct groups is very much addressed in this paper. So the obvious underlying question concerning the specific groups is clearly implicit in [MvN43]! This problem, which is still open, has circulated ever since people started to read the Murray von Neumann papers and to study algebras of operators. It has been mentioned by Kadison in his unpublished Baton Rouge “Problems in von Neumann algebras” (Problem 2 in [K67]) and by Sakai in his book “C-algebras and W-algebras” (Problem 4.4.44 in [S71]), becoming increasingly popular over time, highlighted for instance by Vaughan Jones in his Millennial Address ([J00]). So I think it can be considered as one of the oldest unsolved problems in this field, going back to the foundations of this subject by Murray and von Neumann and their 1943 paper [MvN43]. It is arguably the most emblematic for this area.
A more modern terminology used in recent years about ICC groups having isomorphic II factors, , is that are W-equivalent, denoted (see e.g., [PS18]). With this terminology, the free group factor problem asks whether the free groups , , are W-inequivalent.
The uniqueness of the AFD II factor in ([MvN43], Theorem XIV) shows that all locally finite ICC groups are W-equivalent. A rather obvious implicit problem in [MvN43] has been to characterize all ICC groups with AFD (thus isomorphic to ), i.e., . This was solved by Connes in [C76], as a consequence of his theorem on the uniqueness of the amenable II factor (which implies that if amenable ICC, then ), combined with a prior result in [Sc63], showing that if then is amenable.
The question of whether any two (separable) II factors can be embedded one into the other, alluded to on (page 717 of [MvN43]): the possibility exists that any factor in the case II is isomorphic to a sub-ring of any other such factor.
It has been shown by the 1960s that this is not the case: by combining results in [Sc63] and [HT67], it follows that if a group factor is embeddable into , then is amenable; thus, cannot be embedded into . A final, culminating point in this direction was Connes’ uniqueness of the amenable II factor in ([C76]), implying that any II factor embeddable into is isomorphic to .
In the non-amenable case, a pioneering “non-embeddability” phenomenon was discovered by Connes and Jones in [CJ84]. It shows that II factors arising from property (T) groups cannot be embedded into . The problem of characterizing all II factors that can be embedded into remains a very hard, beautiful problem. A bold speculation along these lines, formulated for instance in [PS18], is that the only non-hyperfinite such factors are amplifications of free group factors, in other words the interpolated free group factors , , introduced in [D94], [R94]. Another bold conjecture is that the free group factors can be embedded into any non-amenable II factor (the II-factor version of von Neumann’s well known similar question for groups).
The problem of whether two given non-amenable II factors can (or cannot) be embedded one into the other remains an extremely interesting and difficult question. It is of course a natural companion to the isomorphism/classification problem for II factors. Both problems have seen much progress over the last 20 years due to deformation-rigidity theory (see e.g., [P06] and Sections 3-5 in [P18a])
It is interesting to note that, by using their property Gamma alone, Murray and von Neumann were able to prove that there exist ICC groups such that , can be embedded one into the other, but . They do this in Appendix, where they construct a group with the property that . Thus, is so that doesn’t have property Gamma (see 1 above) and it embeds into , while this latter factor does have property Gamma and obviously embeds into .
The notation has been proposed in [PS18], when two ICC groups are in the subordination relation given by the embedding of their associated II factors, . With this notation, the above conjecture states that if is non-amenable ICC then , and a conjecture in [PS18] states that for ICC groups one has iff is either amenable or treeable.
The problem of whether any II factor is isomorphic to its opposite (see page 742 in [MvN43]). This question was particularly compelling to Murray and von Neumann because is isomorphic to the commutant of in its standard representation, , and the relation between and has been a central point of interest throughout their work. Related to this, they noticed that any II factor in the classes of examples they were able to construct (the group measure space factors in [MvN36], the group factors in [MvN43]) had a natural anti-automorphism, thus satisfying . This question was then formulated as (Problem 4 in [K67]) and (Problems 4.4.30, 4.4.31 in [S71]).
The problem was answered by Connes in [C75], who produced examples of II factors that are not isomorphic to their opposite, and are thus not group factors, nor group measure space factors. More such examples were constructed in [P01].
Problems concerning the fundamental group of a II factor , the invariant Murray-von Neumann have defined as the multiplicative group of positive numbers for which (the amplification of by ) is isomorphic to .
Thus, after proving that for the hyperfinite II factor one has (as a consequence of their theorem showing that any approximately finite dimensional II factor is isomorphic to ), they mention: there is no reason to believe that for all II factors ; the general behavior of this invariant remains an open question (cf page 742 in [MvN43]).
So a “first implicit question” along these lines is whether is always equal to , which they clearly believed was not the case. The “second and third implicit questions” are more broad and less specific, addressing the problem of what are all possible values that may take, and respectively the actual calculation of this invariant in concrete examples. These questions about are also alluded to on page 714 (see citation below).
There have been many results by now that go a long way towards clarifying the first and second of these questions: existence of factors with countable fundamental group, but without specific calculation, in [C80]; examples of II factors with in [P01]; examples of factors that have an arbitrary countable subgroup as fundamental group, , in [P03]; examples of factors with uncountable but in [PV08]. If one allows the II factors to be non-separable, then by taking suitable products of uncountably many Connes-Stormer Bernoulli actions of the group (in [P03]), or free products of uncountably many amplifications of (in [IPP05]), it was shown that any group can appear as fundamental group of a (possibly non-separable) II factor.
However, the problem of completely characterizing the set of subgroups that can appear as fundamental groups of separable II factors is still open: it has been pointed out in [PV08] that such a must be a Borel set and Polishable (so there are continuously many subgroups that can occur as fundamental groups of separable II factors, while the set of all subgroups of has strictly larger cardinality!); it remains a valid conjecture that coincides with the set of Borel, Polishable subgroups of (cf. [PV08]). Independently of this, a conjecture in [PV08] predicts that any can be realized as the fundamental group of a II factor arising from a free ergodic measure preserving action of on a probability space.
The problem of calculating for a concrete II factor , constructed out of some specific data (e.g., the group factors constructed from ICC groups , as in [MvN43], or the group measure space factors constructed from free ergodic p.m.p. group-actions , as in [MvN36]), remains a very challenging problem. Nevertheless, a large number of concrete classes of factors were shown to have trivial fundamental group, notably in [P01] and the factors arising from Bernoulli actions of certain non-amenable groups, such as groups with property (T) in [P03], non-amenable product groups in [P06], free products of property (T) groups in [IPP05], as well as , for arbitrary free ergodic p.m.p. actions of , in ([PV11]; relying also on [G01]).
It is manifest that Murray-von Neumann were thrilled about the fact that II factors can be “amplified by arbitrary ”, as well as about the ensuing invariant , two concepts that were going hand in hand with their discovery of “continuous dimension”. They comment on page 718 of [MvN43]: is probably of greater general significance, but it is the property Gamma (which they introduce in 6.1.1 of [MvN43]) that has so far been put to greater practical use. One can certainly feel their frustration for not being able to use the fundamental group as an effective invariant, and that they could not calculate it in any of the concrete examples they consider, except for .
Like the “free group factor problem”, the problem of calculating the fundamental group of the free group factors , which are key examples of II factors considered in [MvN43], is evidently another “implicit” problem in [MvN43], falling within the “third” type of questions about fundamental groups mentioned above, of studying the behavior of the invariant for special/remarkable examples of II factors . Aspects of this problem have been re-iterated in (Problem 3 in [K67], Problem 4.4.46 in [S71]).
In the case when is finite, this problem is still open. But for , Voiculescu showed in [V89] that contains all positive rationals and then his techniques were further exploited in [R92] to show that in fact .
As it happens, the two problems (the free group factor problem and the calculation of ) are in fact related. Thus, it was shown in [Dy94], [R94] that the free group factors are either all isomorphic or all non-isomorphic, with this latter possibility being equivalent to the fundamental group being trivial for some (equivalently all) finite . More on this in our previous Blog entry “Tight decomposition of factors and the single generation problem”.
The group factors in [MvN43] were actually the second class of II factors considered by Murray and von Neumann. Their first examples of II factors, constructed in their first paper, [MvN36], were the so-called group measure space II factors , arising from free ergodic pmp actions of infinite groups on the diffuse probability space, .
In [MvN43], Murray and von Neumann seem to be more excited by this new, somewhat simpler construction, of group factors. Nevertheless, they do address the isomorphism/classification problem for group measure space factors as well. Thus, they prove in Lemma 5.2.2 that if is a countable locally finite group, then is AFD and thus isomorphic to . Right after that, Lemma 5.2.3 states that the same holds true for an arbitrary abelian group, accompanied by the comment: the proof of this lemma is somewhat complicated. It requires some rather deep results on the decompositions of mappings of measurable sets, which will be published elsewhere. We shall not pursue this matter further on this occasion.
Like for group factors, an isomorphism between two group measure space factors can be viewed as a (very weak!) equivalence between the two underlying free ergodic pmp actions , which in recent years has been called W-equivalence of group-actions. In an important follow up to Murray-von Neumann’s work on group measure space algebras and their classification, Henry Dye developed in [D59], [D63] the study of group actions up to isomorphisms of probability spaces that take full groups one onto the other. This is obviously
same as an isomorphism between the II factors of “special type”, that carries their Cartan subalgebras one onto the other (see e.g., [Si55] or Section 6 in [D63]). For countable, it is also “same as” an orbit equivalence (OE) of the actions (see page 130 in [Si55]).
So the classification problem for group measure space factors has as an intermediate step the classification of their associated OE relations (or full groups). This latter problem has developed into a parallel subject, which has the special feature that it can be approached by either using a more operator algebra angle (the “full group” point of view, favored by Dye), or a purely measure theory approach, which uses points in the a ground space and the “orbit equivalence” point of view (cf. [FM77]).
In his papers, Dye pursues the same paradigms as in [MvN43]: On the one hand, he identifies the AFD property for OE relations, proves the uniqueness of such OE relations, then establishes that any free ergodic pmp action of a group that’s either locally finite or abelian (see Corollary 4.1 in [D63]; note that Theorem 1 in [D63] is in fact “close to” showing that any group with subexponential growth has AFD OE relation!), and mentions in (Section 6 of [D63]) that all corresponding II factors are isomorphic to the Murray-von Neumann unique AFD factor . On the other hand, he uses a “paradoxical decomposition” trick to prove that any full group (OE relation) arising from a free ergodic pmp action , of a group that contains , is not AFD. He misses however to prove that for these group actions (something that will follow from [Sc63], [HT67]), and altogether does not produce any group measure space factor .
Like for group factors, a first implicit question about group measure space factors emanating from [MvN43] (and also from [D63]) was to identify all group actions that give rise to the hyperfinite II factor . It already followed from [Sc63], [HT67] that is amenable iff is amenable, so Connes’ uniqueness of the amenable II factor answered this problem as well: iff amenable. The analogue question for OE relations, which was an “implicit problem” in [D59], [D63], has in turn been solved by Ornstein-Weiss in [OW80], where it was shown that if is amenable then is OE to the unique AFD orbit relation.
The problem of identifying specific groups that give rise to non-isomorphic group measure space factors and OE relations is quite present in Dye’s papers (a disappointment for not producing such actions can be felt…). Group actions involving free groups do appear in [D59], [D63] but not in [MvN43]. All things considered, I view the problem of whether free ergodic pmp actions are non-OE, for distinct ‘s, solved by Gaboriau in [G00], as originating in Dye’s papers. So does the problem of the non-isomorphism of group measure space factors arising from actions of free groups of different rank $n$, the analogue for group measure space factors of the free group factor problem, solved by Popa-Vaes in [PV11].
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