Bimodule decomposition of a II-1 factor and the SSG property

1. The tightness conjectures

I wanted to popularize here a conjecture that I have formulated in (5.1(b) of [P18]), asserting that if a II{_1} factor {M} is stably single generated (SSG), i.e., if {M^t} is single generated as a von Neumann algebra for any {t>0}, then {M} has an {R}tight decomposition, meaning that it contains hyperfinite subfactors {R_0, R_1 \subset M} such that {R_0 \vee R_1^{op}=\mathcal{B}(L^2M)}. 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 {M} is SSG, then it admits a properly infinite {R}pair (i.e., hyperfinite subfactors {R_0, R_1\subset M} so that {R_0\vee R_1^{op}} is a properly infinite von Neumann algebra in {\mathcal{B}(L^2M)}) 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

The interest of these conjectures comes from the fact that, if true, then they would imply that the II{_1} factor {L(\mathbb{F}_\infty)}, of the free group with infinitely many generators {\mathbb{F}_\infty}, cannot be generated as a von Neumann algebra by finitely many elements (i.e., {L(\mathbb{F}_\infty)} is infinitely generated).

To see this, note first that if a II{_1} factor {M} has non-trivial fundamental group, {\mathcal{F}(M)\neq 1}, then {M} is SSG if and only if it is finitely generated. Thus, since {M=L(\mathbb{F}_\infty)} has non-trivial fundamental group by [V88] (one even has {\mathcal{F}(L(\mathbb{F}_\infty))=\Bbb R_+} by [R91]), if {L(\mathbb{F}_\infty)} is finitely generated and (5.1(a) in [P18]) holds true, and {R_0, R_1\subset M} are hyperfinite subfactors such that the von Neumann algebra {R_0\vee R_1^{op}\subset \mathcal{B}(L^2M)} is properly infinite, then it admits a cyclic vector, i.e., there exists {\xi \in L^2M} such that {[R_0\xi R_1]=L^2M}, 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 {L(\mathbb{F}_\infty)} is infinitely generated, it also follows that the Murray-von Neumann free group factors {L(\mathbb{F}_n), 2 \leq n \leq \infty} ([MvN43]) are all non-isomorphic.

3. Why should tightness occur?

The above conjectures have been triggered by a result I have obtained in [P18], showing that any separable II{_1} factor {M} has a coarse decomposition, in the sense that there exist embeddings of the hyperfinite II{_1} factor, {R_0, R_1\hookrightarrow M}, such that {R_0\vee R_1^{op}} is finite (thus isomorphic to {R_0\overline{\otimes} R_1^{op}}). More precisely, they were motivated by the method I have used to prove this result: the coarse pair of hyperfinite II{_1} subfactors {R_0, R_1} in {M} is constructed recursively, as inductive limits of dyadic finite dimensional factors {R_{0,n}\nearrow R_0}, {R_{1,n}\nearrow R_1}, so that at each step {n} more and more of the vectors in a countable dense subset {\mathcal{L} \subset L^2M} implement asymptotically a specific type of state on {R_{0,n}\vee R_{1,n}^{op} \simeq R_{0,n}\otimes R_{1,n}^{op}}, namely the trace {\tau\otimes \tau}.

It struck me right away that it should be possible to carry out such an “iterative construction with constraints” of the increasing sequences {R_{0,n}, R_{1,n}} so that the vectors in {\mathcal{L} \subset L^2M} implement asymptotically states that “stay away” from {\tau\otimes \tau}, a fact that’s equivalent to {R_0\vee R_1^{op}} being properly infinite.

But at the same time, I knew this was not possible for all {M}, 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 {M} is a free group factor then any choice of an increasing sequence of dyadic factors {R_{0,n}, R_{1,n}} ends up producing a pair of hyperfinite factors {R_0, R_1\subset M} with {R_0\vee R_1^{op}} 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 {L^2M} will necessarily implement {\tau\otimes \tau} on on the inductive limit {R_0\vee R_1^{op}}, making it impossible for {R_0\vee R_1^{op}} 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{_1} 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 {h(M)=0}, on the 1-bounded free entropy of {M} ([H15]). In fact, Hayes calculations of zero 1-bounded entropy {h(M)=0} for classes of II{_1} satisfying conditions of “regularity over an AFD core” use systematically the fact that one can “build” {M} from AFD subalgebras through operations that avoid coarseness.

4. An approach to weak-tightness

Proving the weak-tightness conjecture amounts to constructing recursively two increasing sequences of finite dimensional factors {R_{0,m}}, {R_{1,m}}, {m\geq 1}, inside {M}, so that if we take {\{\xi_k\}_k\subset L^2M} to be a dense sequence in the set of vectors of {\| \ \|_2}-norm equal to {1}, then at each “next step” {n}, the finite dimensional factors {R_{0,n}\supset R_{0,n-1}, R_{1,n}\supset R_{1,n-1}} of {M} have to be chosen so that the restrictions of the vector states {\omega_{\xi_k}} to {R_{0,n}\vee R_{1,n}^{op}=R_{0,n}\otimes R_{1,n}^{op}} have Radon-Nykodim derivative with respect to {\tilde{\tau}=\tau \otimes \tau}, denoted {a_k}, to satisfy that “most” of its {\tilde{\tau}}-trace is concentrated on a projection of small {\tilde{\tau}}-trace, for all {1\leq k \leq n}. This amounts to requiring that {1-\tilde{\tau}(a_ke_{[1, \infty)}(a_k) \leq \varepsilon_{n}}, {\forall 1\leq k \leq n}, for some constants {\varepsilon_n \searrow 0} (e.g., {\varepsilon_n=2^{-n}}).

To have these conditions satisfied it would be sufficient that at each step {n} the support of {s(\varphi_k) \in R_{0,n}\otimes R_{1,n}^{op}} is majorized by a projection of the form {\sum_i f_i \otimes g_i} where {f_i \in R_{0,n}}, {g_i \in R_{1,n}} are projections which on the right give a partition of {1}, {\sum_i g_i=1}, while {f_i} satisfy {\xi_i g_i=f_i \xi_k g_i} (or at least approximately, in an appropriate sense) for all {1\leq k\leq n} and so that the set {J} of all {i}‘s for which one has {\tau(f_i)\leq \varepsilon_n} satisfies {\sum_{i\in J} \tau(g_i)\geq 1- \varepsilon_n}.

This implies that the left support {l(\xi_kg_i)} of {\xi_kg_i} should have expectation on {R_{0,n}} that’s supported “in large part” by a projection of trace {\leq \varepsilon_n}, for all {k} and all {i\in J}. Since the partition {\{g_i\}_i} can be made with arbitrarily small projections independently of the set {\xi_1, ..., \xi_n}, the trace of left supports {\vee_k l(\xi_kg_i)} 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 {R_{0,n}, R_{1,n}} so that this left supports “avoid coarseness”, i.e., so that {l(\xi_kg_i)} avoid being orthogonal to some finite dimensional factor {R_{1,n}} that contains {R_{0,n-1}}.

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{_1} factors {M} 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 {M} SSG implies {M} is strongly 1-bounded, or {h(M)=0}. 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 {R}-tightness

By contrast, the tightness conjecture is more prompt to a “dynamical” approach, a new angle that has not been much explored. Paradoxically, although this is a stronger statement, to me it looks more feasible and more intuitive.

The “dynamical” viewpoint is due to the fact that the {R}-tightness of {M} amounts to proving that {\mathcal{U}(M)} has a pair of “hyperfinite directions” along which the ergodicity of the action {\mathcal{U}(M) \times \mathcal{U}(M^{op})} { \curvearrowright^{\text{\rm Ad}} \mathcal{B}(L^2M)} is being realized. This type of results have been proved before, albeit in an “{L^2}-environment”, rather than {L^1}, as we have here.

Indeed, {R}-tightness requires constructing two increasing sequences of finite dimensional factors {R_{0,n}}, {R_{1,n}\subset M} so that by averaging over the unitaries in {R_{0,n}, R_{1,n}^{op}} at each step {n}, one obtains that “larger and larger” finite subsets of a countable dense subset {\{x_k\}_k} of the space {\{x\in \mathcal{B}(L^2M)\mid \|x\|_{1,Tr}:=Tr(|x|)\leq 1, Tr(x)=0\}} get “more and more annihilated”. Indeed, by (Proposition 2.5 in [P19]) this condition is equivalent to the fact that {(\cup_n R_{0,n}\cup \cup_n R_{1,n}^{op})'\cap \mathcal{B}(L^2M)=\Bbb C}, meaning that {R_0=\overline{\cup_n R_{0,n}}^w}, {R_1=\overline{\cup_n R_{1,n}}^w} give an {R}-tight decomposition of {M}.

Note that a necessary condition for {R}-tightness is that the (ergodic) left-right action of entire unitary group {\mathcal{U}(M)\times \mathcal{U}(M^{op})} on {\mathcal{B}(L^2M)} satisfies the following mean-value (or MV) property (cf. 7.3 in [P19a]): the weak closure of the convex hull of {uv^{op}T{v^{op}}^*u^*}, over unitaries {u, v \in M}, intersects the scalars, for any {T \in \mathcal{B}(L^2M)}. By (2.5 in [P19a]), this is equivalent to the following {L^1}-type mean value property: the {\| \ \|_{1,Tr}}-norm closure of the convex hull of {uv^{op}x{v^{op}}^*u^*}, over unitaries {u, v \in M}, contains {0}, for any {x\in L^1_0(\mathcal{B}, Tr)} (the space of trace class operators in {\mathcal{B}=\mathcal{B}(L^2M)} with {Tr(x)=0}).

So in order to construct the tight pair {R_0, R_1\subset M} recursively, one possible strategy (which has been used in {L^2}-framework in [P81]) is to first establish that {M} 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 {\text{\rm II}_1} 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 {M} 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{_1} factors {M}, including the free group factors {L(\mathbb{F}_n)}, {2\leq n \leq \infty} !

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” {B_0, B_1 \subset M}, that refine a previously constructed pair of finite dimensional factors, so that the averaging over {B_0 \vee B_1^{op}} “diminishes” the {\| \ \|_{1,Tr}}-norm of a given finite set {F\subset L^1_0(\mathcal{B}, Tr)} by a fixed universal constant {c<1}.

Usually, this is being done by first finding an “abelian direction” (a finite partition of {1} by projections) on a “corner” of {M} (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 {\| \frac{1}{|J|} \sum_{(i,j)\in J} u_iv_j^{op} x u^*_i{v_j^{op}}^*\|_{1,Tr} < \varepsilon}, one should deduce that there two points on the circle, i.e., {u_{i_0}, u_{i_1}}, {v_{j_0}, v_{j_1}}, such that {\|\frac{1}{2} (u_{i_0}v^{op}_{j_0}xu_{i_0}^*{v_{j_0}^{op}}^* + u_{i_1}v^{op}_{j_1}xu_{i_1}^*{v_{j_1}^{op}}^*)\|_{1,Tr} < c}, {x\in F}. If this were to be the case, then {u_{i_0}u_{i_1}^*} and {v_{j_0}v_{j_1}^*} 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 {R} in {M}. But unlike [P81], where the ambient norm is the {L^2}-norm {\| \ \|_2} on {M}, in this attempted construction of {R_0, R_1\subset M} with {R_0\vee R_1^{op}} ergodic in {\mathcal{B}(L^2M)} we are dealing with the {L^1} norm {\| \ \|_{1,Tr}}. So the intuition about existence of two points on the circle with its mid-point diminishing the {L^1}-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 {R_{0,n}, R_{1,n}\subset M} 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 {n} one has to apply again all of the above to a “small corner” of {M}.

Such an approach to proving “SSG {\Rightarrow} {R}-tight” may benefit from having some sort of “uniform spectral gap” , that would be the same on each corner of {M}. The additional assumption that the fundamental group of {M} is {\Bbb R_+}, or just {1/3 \in \mathcal{F}(M)} ({M^{1/3} \simeq M}), may be useful in this respect. Indeed, this would allow taking an isomorphism {\theta: M \simeq pMp} for {\tau(p)=1/3}, choosing two triadic subfactors with same diagonal {B_0, B_1\subset M} that generate {M} (as in Section 5 of [P19b]), and then taking {\theta^n(B_0)}, {\theta^n(B_1)} as generators on the corners at steps {n=1, 2, ....}. Note that at each of these steps one would still have the possibility of conjugating {B_0}, {B_1} 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 {1/3 \in \mathcal{F}(M)} would still imply that {L(\mathbb{F}_\infty)} is infinitely generated and that all {L(\mathbb{F}_n), 2\leq n \leq \infty}, follow non-isomorphic.

References

  • [AP17] C. Anantharaman, S. Popa: “An introduction to II{_1} factors”, \newline http://www.math.ucla.edu/{\sim}popa/Books/IIun-v13.pdf

  • [DP19] S. Das, J. Peterson: Poisson boundaries of II{_1} factors, 2019, in preparation.

  • [GP98] L. Ge, S. Popa: On some decomposition properties for factors of type II{_1}, 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 {1}-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{_1} 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 {\Bbb R_+}, 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.

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