** 1. The tightness conjectures **

*stably single generated*(

*SSG*), i.e., if is single generated as a von Neumann algebra for any , then has an –

*tight decomposition*, meaning that it contains hyperfinite subfactors such that . I will refer to this as the

*tightness conjecture*. I have also formulated in that paper a weaker conjecture (5.1(a) of [P18]), stating that if is SSG, then it admits a

*properly infinite*–

*pair*(i.e., hyperfinite subfactors so that is a properly infinite von Neumann algebra in ) hereafter called

*weak tightness conjecture*.

These two conjectures were in fact stated again in (7.2 of [P19a]) and then discussed in details in my recent paper [P19b]. The purpose of this Blog-entry is to go over parts of those comments, in a more informal manner. A novelty with respect to ([P19a], [P19b]) is that my comments will now integrate some recent progress in this direction, due to Sayan Das and Jesse Peterson. I am grateful to them, for allowing me to include here their result, and to Stefaan Vaes, for his active part in this Blog-discussion.

** 2. Relation to the free group factor problem **

*infinitely generated*).

To see this, note first that if a II factor has non-trivial fundamental group, , then is SSG if and only if it is finitely generated. Thus, since has non-trivial fundamental group by [V88] (one even has by [R91]), if is finitely generated and (5.1(a) in [P18]) holds true, and are hyperfinite subfactors such that the von Neumann algebra is properly infinite, then it admits a cyclic vector, i.e., there exists such that , contradicting (Theorem 4.2 in [GP98], based on Voiculescu’s free entropy theory [V96]).

Also, recall from (Corollary 4.7 in [R92]), that once one can establish that is infinitely generated, it also follows that the Murray-von Neumann free group factors ([MvN43]) are all non-isomorphic.

** 3. Why should tightness occur? **

*coarse decomposition*, in the sense that there exist embeddings of the hyperfinite II factor, , such that is finite (thus isomorphic to ). More precisely, they were motivated by the method I have used to prove this result: the coarse pair of hyperfinite II subfactors in is constructed recursively, as inductive limits of dyadic finite dimensional factors , , so that at each step more and more of the vectors in a countable dense subset implement asymptotically a specific type of state on , namely the trace .

It struck me right away that it should be possible to carry out such an “iterative construction with constraints” of the increasing sequences so that the vectors in implement asymptotically states that “stay away” from , a fact that’s equivalent to being properly infinite.

But at the same time, I knew this was not possible for all , that this first reaction has to be reconciled with a result from a joint paper of Liming Ge and myself in ([GP98]), obtained by using Voiculescu’s free entropy theory [V96]. Thus, one can easily deduce from (Theorem 4.2 in [GP98]) that if is a free group factor then any choice of an increasing sequence of dyadic factors ends up producing a pair of hyperfinite factors with having a coarse part (see Theorem 2.9 in [P19b]). In other words, no matter what one does at the “local levels”, some of the vectors in will necessarily implement on on the inductive limit , making it impossible for to be properly infinite.

I find this phenomenon, which I like to call the “coarseness trap”, fascinating: the free group factors “lure into coarseness” any attempt of building pairs of AFD subalgebras as an inductive limit of finite dimensional local data!

Nevertheless, that initial “reactive intuition” does prove right for many classes of factors. Indeed, I have been able to confirm tightness in just about any known example of a II factor that has “good decomposability” (or “regularity”) properties, like non-prime, existence of Cartan subalgebras, property Gamma, crossed product constructions, etc. It is worth noticing that these classes of factors are consistent with the ones satisfying Kenley Jung’s “strongly 1-bounded” property ([J05]) and Ben Hayes condition , on the 1-bounded free entropy of ([H15]). In fact, Hayes calculations of zero 1-bounded entropy for classes of II satisfying conditions of “regularity over an AFD core” use systematically the fact that one can “build” from AFD subalgebras through operations that avoid coarseness.

** 4. An approach to weak-tightness **

To have these conditions satisfied it would be sufficient that at each step the support of is majorized by a projection of the form where , are projections which on the right give a partition of , , while satisfy (or at least approximately, in an appropriate sense) for all and so that the set of all ‘s for which one has satisfies .

This implies that the left support of should have expectation on that’s supported “in large part” by a projection of trace , for all and all . Since the partition can be made with arbitrarily small projections independently of the set , the trace of left supports can indeed be made small. But it seems quite difficult to use the SSG assumption to prove the existence of the the finite dimensional factors so that this left supports “avoid coarseness”, i.e., so that avoid being orthogonal to some finite dimensional factor that contains .

All these difficulties resemble the kind of problems that have been encountered when trying to define “densely defined” cohomology theories, which then depend on proving non-independence to “change of generators”. It also resembles the difficulty of proving that Voiculescu’s free entropy does not depend on the set of generators. There is also a similarity with difficulties in proving that certain II factors are strongly 1-bounded. In other words, if SSG property is to be used successfully to prove weak tightness in this manner, then it should be possible to prove that SSG implies is strongly 1-bounded, or . While it is tempting to try this, there has already been much effort put in such directions, to no avail.

** 5. A dynamical approach to -tightness **

The “dynamical” viewpoint is due to the fact that the -tightness of amounts to proving that has a pair of “hyperfinite directions” along which the ergodicity of the action is being realized. This type of results have been proved before, albeit in an “-environment”, rather than , as we have here.

Indeed, -tightness requires constructing two increasing sequences of finite dimensional factors , so that by averaging over the unitaries in at each step , one obtains that “larger and larger” finite subsets of a countable dense subset of the space get “more and more annihilated”. Indeed, by (Proposition 2.5 in [P19]) this condition is equivalent to the fact that , meaning that , give an -tight decomposition of .

Note that a necessary condition for -tightness is that the (ergodic) left-right action of entire unitary group on satisfies the following *mean-value* (or *MV*) property (cf. 7.3 in [P19a]): the weak closure of the convex hull of , over unitaries , intersects the scalars, for any . By (2.5 in [P19a]), this is equivalent to the following -type mean value property: the -norm closure of the convex hull of , over unitaries , contains , for any (the space of trace class operators in with ).

So in order to construct the tight pair recursively, one possible strategy (which has been used in -framework in [P81]) is to first establish that has the above MV-property. It was recently pointed out to me by Jesse Peterson that he and Sayan Das have in fact already solved this problem in [DP19], where they use non-commutative boundary methods to prove the following:

Theorem 5.1. (Das-Peterson)Any separable factor has the MV-property.

I should point out that when I have formulated the “MV-property” question in ([P19a], [P19b]), it was not clear whether one may need the SSG property for at this stage already. The Das-Peterson result shows that one doesn’t: the property holds true in full generality, for arbitrary countably generated II factors , including the free group factors , !

This is very important, because now one knows that if this approach to solving the tightness conjecture is to work, then it is at the next step that SSG needs to intervene.

This “next step”, or *Step* 2, as I labelled it in [P19b], should consist in using SSG to extract from the MV-property that there exist two “finite dimensional directions” , that refine a previously constructed pair of finite dimensional factors, so that the averaging over “diminishes” the -norm of a given finite set by a fixed universal constant .

Usually, this is being done by first finding an “abelian direction” (a finite partition of by projections) on a “corner” of (and of all inclusions involved), like for instance in [P81]. Viewing the terms of the average as sitting on a circle, this means that from the average inequality , one should deduce that there two points on the circle, i.e., , , such that , . If this were to be the case, then and would give the two “abelian directions” that would allow moving on with the iterative construction of the finite dimensional algebras.

This is how the iterative construction in ([P81], or [P16]) is done, to obtain an ergodic hyperfinite copy of in . But unlike [P81], where the ambient norm is the -norm on , in this attempted construction of with ergodic in we are dealing with the norm . So the intuition about existence of two points on the circle with its mid-point diminishing the -norm no longer works. This fact, and the mysterious way in which SSG may play a role in all this, are the major hurdles for accomplishing this *Step* 2.

Finally, the “last step” (labelled *Step* 3 in [P19b]) of such an approach would consist in doing all the above recursively, to get the pair of increasing sequences of finite dimensional factors that satisfy the desired properties, somewhat like in the proof of (Theorem 4.1 in [P81]) or of (Theorem 5.4 in [P19b]). The fact that SSG is a stable property is crucial for this iterative procedure, because at each step one has to apply again all of the above to a “small corner” of .

Such an approach to proving “SSG -tight” may benefit from having some sort of “uniform spectral gap” , that would be the same on each corner of . The additional assumption that the fundamental group of is , or just (), may be useful in this respect. Indeed, this would allow taking an isomorphism for , choosing two triadic subfactors with same diagonal that generate (as in Section 5 of [P19b]), and then taking , as generators on the corners at steps . Note that at each of these steps one would still have the possibility of conjugating , by random unitaries on both left and right, an operation that may be of additional help to obtain a suitable “uniform spectral gap” property.

Of course, by the same arguments explained in Section 1 above, solving the tightness conjecture under the assumption would still imply that is infinitely generated and that all , follow non-isomorphic.

** References **

- [AP17] C. Anantharaman, S. Popa: “An introduction to II factors”, \newline http://www.math.ucla.edu/popa/Books/IIun-v13.pdf
- [DP19] S. Das, J. Peterson:
*Poisson boundaries of*II*factors*, 2019, in preparation. - [GP98] L. Ge, S. Popa:
*On some decomposition properties for factors of type*II, Duke Math. J.,**94**(1998), 79-101. - [H15] B. Hayes,
*1-Bounded entropy and regularity problems in von Neumann algebras*, International Mathematics Research Notices,**1**(3), 57-137, 2018. - [J05] K. Jung:
*Strongly -bounded von Neumann algebras*, Geom. Funct. Anal.,**17**(2007), 1180-1200. - [MvN43] F. Murray, J. von Neumann:
*On rings of operators*IV, Ann. Math.**44**(1943), 716-808. - [P81] S. Popa:
*On a problem of R.V. Kadison on maximal abelian *-subalgebras in factors*, Invent. Math.,**65**(1981), 269-281. - [P16] S. Popa:
*Constructing MASAs with prescribed properties*, to appear in Kyoto J. of Math, math.OA/1610.08945 - [P18] S. Popa:
*Coarse decomposition of*II*factors*, math.OA/1811.11016 - [P19a] S. Popa:
*On ergodic embeddings of factors*, math.OA/1910.06923 - [P19b] S. Popa:
*Tight decomposition of factors and the single generation problem*, math.OA/1910.14653 - [R91] F. Radulescu:
*The fundamental group of the von Neumann algebra of a free group with infinitely many generators is*, J. Amer. Math. Soc.**5**(1992), 517-532. - [R92] F. Radulescu:
*Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index*, Invent. Math.**115**(1994), 347-389. - [V88] D. Voiculescu:
*Circular and semicircular systems and free product factors*, Prog. in Math.**92**, Birkhauser, Boston, 1990, pp. 45-60. - [V96] D. Voiculescu:
*The analogues of entropy and Fisher’s information measure in free probability theory*III:*absence of Cartan subalgebras*, GAFA**6**(1996), 172-199.