OPEN PROBLEMS
NCGOA 2010, VANDERBILT UNIVERSITY
Below is a list of open problems compiled for the NCGOA Spring Institute on von Neumann algebras
held at Vanderbilt University, May 10-19, 2010. Along with some of the problems is the name of the person
who suggested the problem for the conference which may or may not be the same as the person(s) who
originally formulated the problem. This list includes some problems which have appeared on other lists such
as [Kadison67], [Jones00], [Ge03], and [Popa09].
1. General problems on von Neumann algebras
Problem 1.1 (Suggested by Popa). Assume a II1 non-amenable factor M has a “free s-malleable deformation”, i.e. there exists a continuous group morphism α : R → Aut(M ∗ M ) such that α1 (M ∗ 1) = 1 ∗ M and
such tat there exists β ∈ Aut(M ∗ M ) with β 2 = id, βαt = α−t β and βM ∗1 = id. Is then M necessarily a
(interpolated) free group factor? Related to this: If M factor and classic flip on M ⊗ M is path connected
to id, then M ' R?
Problem 1.2 (Suggested by Vaes). Give an intrinsic description of the subgroups of R∗+ that arise as the
fundamental group F(M ) of a II1 factor M with separable predual.
Background reading: [PV08a, PV08b].
Problem 1.3. (M 0 ∩ M ω )0 ∩ M ω = M =⇒ M ' R?
Problem 1.4. Is every separable von Neumann algebra generated as a von Neumann algebra by a single
element?
Problem 1.5 (Connes’ embedding problem). Is every separable II1 factor a subfactor of Rω ?
2. Problems on group actions/group-measure space constructions
Conjecture 2.1 (Suggested by Popa). If Γ non-amenable & Γ y X Bernoulli, then L∞ (X) o Γ has unique
Cartan?
(2
Conjecture 2.2 (Suggested by Popa). If β1 (Γ) 6= 0, then L∞ (X) o Γ has unique Cartan ∀ Γ y X. (Maybe
(2
even for βn (Γ) 6= 0, for some n ≥ 1.) Related conjecture: If Γ = Fn , and Fn y B is an arbitrary action on
a finite vN algebra (not necessarily free nor ergodic), and A is a Cartan subalgebra of M = B o Fn , then
A ≺M B.
Problem 2.3 (Suggested by Popa). Γ y X rigid ⇒ strongly ergodic?
Problem 2.4 (Suggested by Vaes). For a pretty large family G of groups Γ, including all free products
Γ = Γ1 ∗ Γ2 of an infinite property (T) group Γ1 and a non-trivial group Γ2 , it is shown in [PV09] that for all
free ergodic probability measure preserving actions Γ y (X, µ), the II1 factor L∞ (X) o Γ has, up to unitary
conjugacy, a unique group measure space Cartan subalgebra.
Does it actually hold that L∞ (X) o Γ has, up to unitary conjugacy, a unique Cartan subalgebra?
In order to solve this problem, it suffices to generalize the transfer of rigidity lemmas in [PV09, Section
3] to arbitrary Cartan subalgebras.
Background reading: [PV09].
Date: May 10, 2010.
1
2
NCGOA 2010, VANDERBILT UNIVERSITY
Problem 2.5 (Suggested by Vaes). In [OP07] it is shown that for all free ergodic profinite probability
measure actions Fn y (X, µ) of the free group Fn , the II1 factor L∞ (X) o Fn has, up to unitary conjugacy,
a unique Cartan subalgebra.
Does the same result hold for arbitrary free ergodic probability measure preserving actions of the free
group?
Background reading: [OP07].
Problem 2.6 (Suggested by Chifan). Suppose N1 and N2 are finite von Neumann algebras and B ⊂ N =
N1 ∗ N2 is a nonamenable subfactor which contains a Cartan subalgebra, is it true that either B ≺N N1 or
B ≺ N N2 .
Problem 2.7. In [Popa07] Popa showed that any cocycle for the Bernoulli action of a property (T) group
with values in Ufin (the class of closed subgroups of the unitary group of a separable II1 factor) is cohomologous
to a homomorphism. Find larger classes of groups U for which this is still true.
Problem 2.8. Find the class CS (resp. OES) of groups such that any cocycle for the Bernoulli action with
values in Ufin (resp. Udis , the class of all countable discrete groups) is cohomologous to a homomorphism.
Problem 2.9. Calculate H 2 (RΓyX ) for some action Γ y X, e.g., for the Bernoulli action.
Problem 2.10. Extend Bowen’s entropy invariant [Bowen10] to Bernoulli actions of arbitrary nonamenable
groups.
Problem 2.11. Suppose Fn y X is a free ergodic action for 2 ≤ n < ∞ is it true that we always have
(2)
F(L∞ (X) o Fn ) = {1}? What if instead of Fn we consider Γ with β1 (Γ) 6= 0, ∞?
(2)
(2)
Problem 2.12. Show that if L(R) = L(S) for some equivalence relations R and S, then βn (R) = βn (S),
for all n ∈ N.
Problem 2.13. Is it true that for any nonamenable group Γ there exists a free ergodic action Γ y X such
that F(L∞ (X) o Γ) = {1}?
Problem 2.14. Can the fundamental group of any separable II1 factor can be realized as the fundamental
group of a factor coming from a free ergodic action of F∞ ?
Problem 2.15. If Γ has property (T) and Γ y X is a free ergodic action is F(L∞ (X) o Γ) ⊂ Q+ ?
Problem 2.16. What are the groups Γ for which there exists a free ergodic action Γ y X such that
(L∞ (X) ⊂ L∞ (X) o Γ) is rigid?
Problem 2.17. Find ICC groups Γ, Λ such that LΓ ' LΛ but such that Γ and Λ are not measure
equivalent, i.e. there does not exist stably orbit equivalent actions of Γ and Λ. Counter examples to the
converse implication have been constructed by Chifan and Ioana [CI09].
3. Problems on groups/group von Neumann algebras
Problem 3.1 (Suggested by Valette). Recall that a 2nd countable, locally compact group G is Howe-Moore
if every unitary representation of G without fixed vectors, has coefficients vanishing at infinity. Equivalently,
every probability measure-preserving, ergodic action of G, is mixing. The only known examples are, on
the one hand simple algebraic groups over local fields [HM79], on the other hand automorphism groups of
regular or bi-regular trees [LM91]. It is known that, among analytic real or p-adic groups, there are no other
examples as the ones provided by the Howe-Moore Theorem [CCLTV09]. Here are two questions:
1) Find new examples of totally disconnected, Howe-Moore groups.
2) Does there exist a discrete Howe-Moore group?
OPEN PROBLEMS
3
Problem 3.2 (Suggested by Ozawa). Although I do not have any applications in mind, I am curious
about the following problem about Radulescu’s correspondence. Let SL(2, R) act on the upper half plane
H and πm be its discrete series representation on H 2 (H, µm ). Let Γ be a lattice in SL(2, R). The Toeplitz
operator Tf associated with f ∈ L∞ (H) is the multiplication operator by f truncated to H 2 (H, µm ), i.e.,
Tf = PH 2 Mf |H 2 . If f is Γ-invariant, then Tf belongs to the II1 factor πm (Γ)0 (which is stably isomorphic to a
direct summand of LΓ). Radulescu [Rădulescu98] proved that the completely positive map T : L∞ (Γ\H) →
πm (Γ)0 is continuous as an L2 space operator and the extension is injective and with dense range. (It is
odd that there is such a completely positive map from a commutative von Neumann algebra into a nonamenable factor.) Thus the polar decomposition yields a unitary isomorphism U : L2 (Γ\H) → L2 (πm (Γ)0 )
which intertwines several information, including the Hecke operators (see Rădulescu, op. cit.). I wonder if
the invariant Laplacian ∆ on L2 (Γ\H) is a quantum Dirichlet form on the noncommutative side L2 (πm (Γ)0 ).
Problem 3.3 (Suggested by Popa). Find a new proof (i.e., without using free probability) of the VoiculescuGe result [Voiculescu96, Ge97] that M = L(Fn ) has no MASAs A ⊂ M such that A∨JAJ is maximal abelian
in B(L2 M ) (e.g., by using deformation/rigidity).
Problem 3.4 (Suggested by Peterson). If B1 , B2 ⊂ LFn are two amenable von Neumann subalgebras such
that B1 ∩ B2 is diffuse, is the von Neumann subalgebra generated by B1 and B2 also amenable?
Conjecture 3.5 (Connes’ rigidity conjecture). If Γ and Λ are ICC property (T) groups, does LΓ ' LΛ
imply Γ ' Λ? More generally, if θ : LΓ → LΛt is an isomorphism, must we have that t = 1, and there exist
u ∈ U(LΛ), δ : Γ → Λ, and χ ∈ Hom(Γ, T) such that Ad(u) ◦ θ(Σγ∈Γ cγ uγ ) = Σγ∈Γ χ(γ)cγ uδ(γ) ?
Problem 3.6. LΓn ' LΓm =⇒ n = m, for Γn = P SL(n, Z)? What about for Γn = Zn o SL(n, Z)?
Problem 3.7. Is L(SL(3, Z)) solid in the sense of Ozawa [Ozawa04]?
Problem 3.8. LFn ' LFm =⇒ n = m?
Problem 3.9. Can LF∞ be finitely generated as a von Neumann algebra?
4. Problems on subfactors/standard invariants
Problem 4.1 (Suggested by Bisch). Find methods to construct “exotic” infinite depth subfactors (e.g. with
non-integer index).
Problem 4.2 (Suggested by Bisch). Determine planar algebras generated by two or several bi-projections.
This will depend on their relative angles.
Problem 4.3 (Suggested by Bisch). We say that a subfactor N ⊂ M is n-supertransitive, if N 0 ∩ Mk is
just Temperley-Lieb for 0 ≤ k ≤ n. Are there subfactors which are n-supertransitive, but not (n + 1)supertransitive, for each n ≥ 1? The answer is known for some small n.
Problem 4.4 (Suggested by Bisch). Which subfactor planar algebras arise as the standard invariant of
hyperfinite subfactors?
Problem 4.5 (Suggested by Bisch). It is known that there are only finitely many finite depth subfactors
for a given pair of principal graphs (Ocneanu rigidity). Is it possible to have infinitely many infinite depth
subfactors with the same index and the same principal graphs? This seems plausible.
Problem 4.6 (Suggested by Morrison). The dimension of any object in a fusion category (in particular, the
N -N and M -M bimodules of a subfactor N ⊂ M ) is a real cyclotomic integer which is maximal amongst
its Galois conjugates. All such numbers between 2 and 76/33 are known, and all are realised by objects in
some fusion category. (See [CMS10]) Are there such numbers which are not realised? What’s the smallest
one?
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NCGOA 2010, VANDERBILT UNIVERSITY
Problem 4.7 (Suggested by Morrison). Are there any subfactors (finite or infinite depth) with principal
graph beginning like the ”bad seed”:
Problem 4.8 (Suggested by Morrison). We know of 8 principal graphs for subfactors below index 5, and
have a subfactor realising each. Is this subfactor unique?
Problem 4.9 (Suggested by Morrison). Can you find generators and relations for the Asaeda-Haagerup
planar algebra? Can it be defined over a cyclotomic field? (The Haagerup planar algebra can not be, see
[MS10])
Conjecture 4.10 (Suggested by Xu). Let Ni ⊂ Mi , i = 1, 2 be
Ntwo irreducible
N subfactors with finite index.
Then
the
number
of
minimal
intermediate
subfactors
in
N
N
⊂
M
M2 which is not of the form
1
2
1
N
N
N1 P, P
N2 is less or equal to (n1 − 1)(n2 − 1) where ni is the dimension of second higher relative
commutant of Ni ⊂ Mi , i = 1, 2.
References
[Bowen10] L. Bowen, Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), no.
1, 217–245.
[CMS10] F. Calegari, S. Morrison, and N. Snyder, Cyclotomic integers, fusion categories, and subfactors, Preprint.
arXiv:1004.0665
[CI09]
I. Chifan and A. Ioana, On a Question of D. Shlyakhtenko, Preprint. arXiv:0906.5345
[CCLTV09] R. Cluckers, Y. Cornulier, N. Louvet, R. Tessera, and A. Valette, The Howe-Moore property for real and p-adic
groups, Preprint. arXiv:1003.1484
[Ge97]
L. Ge, Applications of free entropy to finite von Neumann algebras, Amer. J. Math. 119 (1997), no. 2, 467–485.
[Ge03]
L. Ge, On “Problems on von Neumann algebras by R. Kadison, 1967”. Acta Math. Sin. (Engl. Ser.) 19 (2003), no.
3, 619–624.
[HM79]
R. E. Howe and C. C. Moore, Asymptotic properties of unitary representations. J. Funct. Anal. 32 (1979), no. 1,
72–96.
[Jones00] V. F. R. Jones, Ten Problems. Mathematics: frontiers and perspectives, 79–91, Amer. Math. Soc., Providence, RI,
2000.
[Kadison67] R. Kadison, Problems on von Neumann algebras, Unpublished, 1967.
[LM91]
A. Lubotzky and S. Mozes, Asymptotic properties of unitary representations of tree automorphisms. Harmonic
analysis and discrete potential theory (Frascati, 1991), 289–298, Plenum, New York, 1992.
[MS10]
S. Morrison and N. Snyder, Non-cyclotomic fusion categories, Preprint. arXiv:1002.0168
[Ozawa04] N. Ozawa, Solid von Neumann algebras. Acta Math. 192 (2004), no. 1, 111–117.
[OP07]
N. Ozawa and S. Popa, On a class of II1 factors with at most one Cartan subalgebra. Ann. Math., to appear.
arXiv:0706.3623
[Popa07] S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170
(2007), no. 2, 243–295.
[Popa09] S.
Popa,
Some
results
and
problems
in
W ∗ -rigidity,
Unpublished,
2009.
Available
at
http://www.math.ucla.edu/p̃opa/
[PV08a] S. Popa and S. Vaes, Actions of F∞ whose II1 factors and orbit equivalence relations have prescribed fundamental
group. J. Amer. Math. Soc. 23 (2010), 383-403.
[PV08b] S. Popa and S. Vaes, On the fundamental group of II1 factors and equivalence relations arising from group actions.
In Noncommutative geometry, Proceedings of the Conference in honor of A.Connes’ 60th birthday, to appear.
arXiv:0810.0706
[PV09]
S. Popa and S. Vaes, Group measure space decomposition of II1 factors and W∗ -superrigidity. Preprint.
arXiv:0906.2765
OPEN PROBLEMS
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[Rădulescu98] F. Rădulescu, The Γ-equivariant form of the Berezin quantization of the upper half plane. Mem. Amer. Math.
Soc. 133 (1998), no. 630, viii+70 pp.
[Voiculescu96] D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. III. The
absence of Cartan subalgebras. Geom. Funct. Anal. 6 (1996), no. 1, 172–199.