Open problems in operator algebras
Below is a selected list of open problems in operator algebras and related fields. The list is compiled based solely on my own preference, and so will tend to mostly be about von Neumann algebras and its connection to group theory and ergodic theory. I've chosen to only include problems that can feasibly have a definitive solution, so I will not include many interesting motivational problems like "what properties of a group are remembered by its group von Neumann algebra", or "classify all groups that embed into the unitary group of a free group factor".
I've tried to give the correct attribution to each problem, any errors in attribution are my own. If a problem is not attributed to someone it is because I either do not know the correct attribution, or else it is a folklore problem. For consistency I've phrased everything as a problem. For the problems below that are labeled as a conjecture, the conjecture is always that the problem has an affirmative solution.
To the best of my knowledge all problems remain open as of the last update to this page. If you find any errors, know of any announced/published proofs, or if you know of any open problems I might want to add, please send me an email.
Clicking on some of the problems gives a brief explanation to provide context.
This is very much a work in progress (created October 9, 2020) and I hope to expand this list greatly in the coming months. I will also try to stabilize the presentation and numbering of the problems in this time, but for the moment I will feel free to make adjustments as needed. Last updated February 10, 2023.
General outstanding problems in operator algebras
Problem O.1 (The generator problem; Kadison 1967)
Is every separable von Neumann algebra generated, as a von Neumann algebra, by a single element?
This is one of just a few outstanding problems left from Kadison's famous 1967 list of open problems
[Kadison 1967].
Richard Kadison, Problems on von Neumann algebras, Unpublished, 1967.
By splitting a single operator into its real and imaginary parts it is equivalent to asking if every separable von Neumann algebra is generated, as a von Neumann algebra, by two self-adjoint elements. Also, by using functional calculus and taking exponentials or logarithms the problem is easily seen to be equivalent to asking if every separable von Neumann algebra is generated by two unitaries.
For the abelian case the problem is true, and is a classic result of von Neumann
[von Neumann 1931].
J. von Neumann, Über funktionen von funktionaloperatoren, Ann. of Math. 32 (1931), 191-226.
The problem is also known to hold true for the more general setting of separable type I von Neuamnn algebras
[Pearcy 1968].
C. Pearcy, \(W^*\)-algebras with a single generator, Proc. Amer. Math. Soc. 13 (1962), 831-832.
It is also known to be true for all separable properly infinite von Neumann algebras
[Wogen 1969].
Warren Wogen, On generators for von Neumann algebras, Bull. Amer. Math. Soc., 75 (1969), 95-99.
Using reduction theory, the problem is known to hold true if it can first be proven for all separable factors. Thus, one only needs to consider is the case of II\(_1\) factors.
The problem is known to be true for all II\(_1\) factors containing a Cartan subalgebra
[Popa 1985],
Sorin Popa, Notes on Cartan subalgebras in type II\(_1\) factors, Math. Scand. 57 (1985), 171-188.
and for all II\(_1\) factors that are not prime, or that have property (Gamma)
[Ge+Popa 1998].
Liming Ge and Sorin Popa, On some decomposition properties for factors of type II\(_1\), Duke Math. J. 94, no. 1 (1998), 79-101.
It is also known for the group von Neumann algebras \(L(SL_n(\mathbb Z))\) for \(n \geq 3\)
[Ge+Shen 2002].
Liming Ge and Junhao Shen, Generator problem for certain property T factors, Proc. Natl. Acad. Sci. USA 99(2) (2002), 565-567.
The problem is open for the free group factor \(L\mathbb F_\infty\) (since \(L\mathbb F_\infty\) has full fundamental group
[Radulescu 1992],
Florin Radulescu, The fundamental group fo the von Neumann algebra of a free group with infinitely many generators is \(\mathbb R_+ \setminus \{ 0 \}\), J. Amer. Math. Soc., 5, no. 3 (1992), 517-532.
\(L\mathbb F_\infty\) is singly generated if and only if it is finitely generated
[Wogen 1969].
Warren Wogen, On generators for von Neumann algebras, Bull. Amer. Math. Soc., 75 (1969), 95-99.
). A negative answer here would solve the free group factor problem since \(L\mathbb F_2\) is clearly generated by two unitaries.
In the analogous setting of measure equivalence relations, Levitt introduced the notion of cost, which is in a natural sense the "minimal number of generators" for a measure equivalence relation
[Levitt 1995].
Gilbert Levitt, On the cost of generating an equivalence relation, Ergodic Theory and Dynam. Sys. 15(6) (1995), 1173-1181.
The cost is a positive real number and an equivalence relation cannot be generated, as a measure equivalence relation, by fewer elements of the full group than the cost. Gaboriau showed that for a free ergodic measure preserving equivalence relation \(\mathcal R\) generated by \(\mathbb F_n\), the cost is \(n\)
[Gaboriau 2000].
Damien Gaboriau, Coût des relations d'équivalence et des groupes, Invent. Math. 139(1) (2000), 41-98.
Though, the von Neumann algebra \(L\mathcal R\) will be singly generated since it contains a Cartan subalgebra.
The generation problem is discussed in depth in Chapter 16 of the book
[Sinclair+Smith 2008].
Allan M. Sinclair and Roger R. Smith, Finite von Neumann algebras and masas, London Mathematical Society Lecture Note Series, vol. 351, Cambridge University Press, Cambridge, 2008
Problem O.2 (The free group factor problem; Murray+von Neumann 1943)
Do we have \(L \mathbb F_2 \cong L \mathbb F_3\)?
Problem O.3 (Connes's rigidity conjecture 1980)
If \(\Gamma\) and \(\Lambda\) are icc property (T) groups such that \(L\Gamma \cong L\Lambda\) must we have that \(\Gamma \cong \Lambda\)?
Problem O.4 (The spatial isomorphism conjecture; Kadison+Kastler 1972)
For all \(\varepsilon > 0\), does there exist \(\delta > 0\) with the property that if \(M, N \subset \mathcal B(\mathcal H)\) are any two von Neumann algebras with Hausdorff distance between their unit balls less than \(\delta\), then there exists a unitary \(u \in \mathcal B(\mathcal H))\) with \(uMu^* = N\) and \(\| u - 1 \| < \varepsilon\)?
Problem O.5 (The similarity problem; Kadison 1955)
Is every bounded representation of a \(C^*\)-algebra on a Hilbert space similar to a \(*\)-representation?
Problem O.6 (The vanishing cohomology conjecture; Kadison+Ringrose 1967)
Is it true that \(H^k(M, M) = 0\) for all von Neumann algebras \(M\) and all \(k \geq 1\)?
Problem O.7 (Jones's problem on irreducible indices 1983)
Does every number in \( \{ 4 \cos^2 \pi/n \mid n \geq 3 \} \cup [4, \infty]\) appear as the index of an irreducible subfactor of the hyperfinite II\(_1\) factor \(R\)?
Problem O.8 (Von Neumann's problem for II\(_1\) factors)
If \(M\) is a nonamenable II\(_1\) factor, then must \(M\) contain an isomorphic copy of \(L\mathbb F_2\)?
Problem O.9 (The invariant subspace problem for von Neumann algebras)
Let \(M\) be a non-trivial von Neumann algebra and suppose \(x \in M\), is there a non-zero projection \(p \in M\) such that \(p x p = x p\)?
Problem O.10 (Invariance of free entropy dimension; Voiculescu 1994)
Let \(M\) be a II\(_1\) factor and suppose \(\{ x_1, x_2, \ldots, x_n \}, \{ y_1, y_2, \ldots, y_m \} \subset M\) are tuples of self-adjoint elements that separately generate \(M\) as a von Neumann algebra, do we have equality of the free entropy dimensions \(\delta(x_1, \ldots, x_n) = \delta(y_1, \ldots, y_m)\)?
Problems on structural properties of von Neumann algebras
Problem S.1
Is every property (T) von Neumann algebra generated, as a von Neumann algebra, by a single element?
Property (T) factors are always separable and hence this is a subproblem of the generator problem. There is some evidence to suggest that this problem might have a positive solution even if the generator problem has a negative solution. It is known that every property (T) group has an action such that the resulting equivalence relation has cost 1
[Hutchcroft+Pete 2020].
Tom Hutchcroft and Gábor Pete, Kazhdan groups have cost 1, Invent. Math. 221 (2020), 873-891.
Problem S.2 (Kadison 1967)
Does every II\(_1\) factor have an orthonormal (with respect to the trace) basis consisting of unitaries?
Problem S.3
For \(\mathbb F_2 = \langle a, b \rangle\), is the radial masa \(\{ u_a + u_a^* + u_b + u_b^* \}''\) conjugate to the generator masa \(\{ u_a \}''\) by an autormorphism of \(L\mathbb F_2\)?
Problem S.4 (Popa)
If \(\Gamma\) is an icc property (T) group, does there exist \(n \in \mathbb N\) such that \(L\Gamma\) is not isomoprhic to a tensor product of more than \(n\) II\(_1\) factors? (This is also an open problem on the group level?)
There are examples of non-trivial finitely generated groups that are isomorphic to their own square (e.g.,
[Jones 1974],
Tyrer Jones, J.M., Direct products and the Hopf property, J. Aust. Math. Soc. 17 (1974), 174-196.
[Meier 1982],
David Meier, Non-Hopfian groups, J. Lond. Math. Soc., II. Ser. 26 (1982), 265-270.
[Baumslag+Miller 1988]
Gilbert Baumslag and Charles F. Miller III, Some odd finitely presented groups, Bull. Lond. Math. Soc., no. 3 (1988), 239-244.
). ICC groups in this class clearly give rise to II\(_1\) factors that have no bound on the number of factors that appear in a tensor product decomposition. However, it is unknown if any of these groups have property (T).
It is shown in Proposition 9.8 of
[Isono+Marrakchi 2022]
Yusuke Isono and Amine Marrakchi, Tensor product decompositions and rigidity of full factors, Ann. Sci. Éc. Norm. Supér. (4) 55 (2022), no.1, 109-139.
that Meiers group \(G\) from above is ICC and not inner amenable, hence \(LG\) is an example of a full type II\(_1\) factor that is isomorphic to its square.
Problem S.5 (Popa+Vaes)
Let \(\Gamma\) and \(\Lambda\) be torsion free groups such that \(C_r^*(\Gamma) \cong C_r^*(\Lambda)\). Does it follow that \(\Gamma \cong \Lambda\)?
Problem S.6
Is \(C_r^* (\mathbb F_\infty)\) finitely generated?
Problem S.7
Suppose \(M \cong M_1 \, \overline \otimes \, M_2\) is McDuff. Does it follow that either \(M_1\) or \(M_2\) is McDuff?
Problems on rigidity in von Neumann algebras
Problem R.1 (Connes's rigidity conjecture for higher-rank lattices)
Let \(G\) be a higer-rank simple Lie group with trivial center, and suppose \(\Gamma < G\) is a lattice. If \(\Lambda\) is another group such that \(L\Gamma \cong L\Lambda\), then must we have \(\Gamma \cong \Lambda\)?
Problem R.2 (Popa's strengthening of Connes's rigidity conjecture)
Let \(G\) be a higer-rank simple Lie group with trivial center, and suppose \(\Gamma < G\) is a lattice. Suppose \(\Lambda\) is any group and we have a von Neumann algebra isomorphism \(\theta: L\Gamma \to (L\Lambda)^t\) for some \(t \in (0, \infty)\), then must we have \(t = 1\), and must there exist a group isomorphism \(\alpha: \Gamma \to \Lambda\), a character \(\chi: \Gamma \to \mathbb T\), and a unitary \(u \in \mathcal U(L\Gamma)\) such that for all \(t \in \Gamma\), we have \(\theta(u u_t u^*) = \chi(t) u_{\alpha(t)}\)?
Problem R.3
Do we have \(\mathcal F( L(PSL_n(\mathbb Z))) = \{ 1 \}\) for all \(n \geq 3\)?
Problem R.4
For \(n, m \geq 3\), do we have \(L(PSL_m (\mathbb Z)) \cong L(PSL_n (\mathbb Z))\) only if \(n = m\)?
Problem R.5
Is there a non-trivial icc property (T) group such that if \(\Lambda\) is any group satisfying \(L\Gamma \cong L\Lambda\), then we must have \(\Gamma \cong \Lambda\)?
This may be solved. An example is presented in the preprint
[Chifan+Ioana+Osin+Sun 2021].
Ionut Chifan, Adrian Ioana, Denis Osin, and Bin Sun Wreath-like product groups and rigidity of their von Neumann algebras, arXiv:2111.04708
Problem R.6
If \(\Gamma\) is an icc property (T) group are there at most finitely many isomorphism classes of groups giving rise to group von Neumann algebras that are isomorphic to \(L\Gamma\)?
Problem R.7
Let \(\Gamma\) be an icc property (T) hyperbolic group, is \({\rm Out}(L\Gamma)\) finite?
Problem R.8 (Popa)
If \(\Gamma\) is a non-amenable group such that \(\beta_1^{(2)}(\Gamma) = 0\), then is the Bernoulli shift action \(\mathcal U_{\rm fin}\) cocycle superrigid?
Problem R.9 (Popa+Vaes)
If \(\Gamma\) is a non-amenable group with \(\beta_1^{(2)}(\Gamma) < \infty\), then does it follow that the Bernoulli shift crossed product has trivial fundamental group?
Algebraic problems
Problem A.1 (Kadison 1967)
Let \(\mathcal H\) be a Hilbert space and suppose \(A \subset \mathcal B(\mathcal H))\) is a finitely generated self-adjoint algebra having the property that every self-adjoint operator in \(A\) has finite spectrum. Must \(A\) be finite dimensional?
Problems with ultrafilters and approximations
Problem U.1
Is there a II\(_1\) factor \(M\) so that \(\mathcal F(N^\omega) \not= \mathbb R_+^*\)?
Problem U.2
Are there two full II\(_1\) factors \(M_1\) and \(M_2\) such that \(M_1^\omega \not\cong M_2^\omega\) for every free ultrafilter \(\omega\)?
This may be solved. Examples are presented in the preprint
[Chifan+Ioana+Kunnawalkam Elayavalli 2022].
Ionut Chifan, Adrian Ioana, and Srivatsav Kunnawalkam Elayavalli Two non-elementarily equivalent II\(_1\) factors without property Gamma, arXiv:2209.10645
It is still unknown if there are three non-elementarily equivalent II\(_1\) factors without property Gamma.
Problem U.3
Is there a II\(_1\) factor \(M\) such that \(M^\omega \not\cong ({ M^{\rm op} })^\omega\) for any free ultrafilter \(\omega\)?
Problem U.4
Is there a free ultrafilter \(\omega\) so that \((L \mathbb F_2)^\omega \cong (L \mathbb F_3)^\omega\)?
Problem U.5 (Conjecture of Peterson 2018)
Let \(N\) be a II\(_1\) factor with the Haagerup property, let \(\omega\) be a free ultrafilter and suppose \(M \subset N^\omega\) is a subfactor with propety (T), does there exist finite dimensional von Neumann subalgebras \(N_n \subset N\) so that \(M \subset \prod_\omega N_n \subset N^\omega\)?
Problem U.6 (Popa)
If \(M\) is a separable II\(_1\) factor that embeds into a tracial ultraproduct of matrix algebras, then does there exist such an embedding with trivial relative commutant?
Problems with groups
Problem G.1
Is there a nonsofic group?
Problem G.2
Is Thompson's group \(F\) amenable?
Problem G.3
Do we have \(MA(G) \not= M_{cb}A(G)\) for any non-amenable locally compact group \(G\)?
Problems with free group factors
Problem F.1
Is every nonamenable subfactor of \(L\mathbb F_2\) an interpolated free group factor?
Problem F.2 (Peterson+Thom conjecture 2011)
Is every diffuse amenable von Neumann subalgebra of \(L \mathbb F_2\) contained in a unique maximal amenable von Neumann subalgebra?
This may be solved. The result is claimed in the preprint
[Belinschi+Capitaine 2022].
Serban Belinschi and Mireille Capitaine Strong convergence of tensor products of independent G.U.E. matrices, arXiv:2205.07695
A different solution is claimed in the preprint
[Bordenave+Collins 2023].
Charles Bordenave and Benoit Collins Norm of matrix-valued polynomials in random unitaries and permutations, arXiv:2304.05714
Problem F.3 (Popa)
Suppose \(M\) is a II\(_1\) factor such that the "free flip" \(x * y \mapsto y * x\) is path connected to the identity in \({\rm Aut}(M * M)\). Does it follow that \(M\) is an interpolated free group factor? (Popa)
Problems with the hyperfinite II\(_1\) factor
Problem H.1 (Ozawa)
Is there an embedding of \(\mathbb F_2\) into \(\mathcal U(R)\) whose range is discrete in the strong operator topology?
Problem H.2
Let \(M\) be a II\(_1\) factor and suppose \(\pi: \mathbb F_2 \to \mathcal U(M)\) is an embedding whose range is discrete in the strong operator topology. Does \(\pi\) extend to a \(*\)-isomorphism embedding \(L\mathbb F_2\) into \(M\), i.e., must we have \(\tau \circ \pi = \delta_{\{ e \}}\) where \(\tau\) is the trace on \(M\)?
Problem H.3 (Popa)
Suppose \(M\) is a II\(_1\) factor such that the "classic flip" \(x \otimes y \mapsto y \otimes x\) is path connected to the identity in \({\rm Aut}(M \, \overline \otimes \, M)\) does it follows that \(M \cong R\)?
Problems with invariants
Problem I.1 (Connes+Shlyakhtenko 2005)
Are \(\ell^2\)-Betti numbers of groups an invariant of their von Neumann algebras?
Problem I.2 (Anantharaman-Delaroche 1995)
Do the Haagerup property and the compact approximation property describe the same class of II\(_1\) factors?
This may be solved. The result is claimed in the preprint
[Ding+Kunnawalkam Elayavalli+Peterson 2022].
Changying Ding, Srivatsav Kunnawalkam Elayavalli, and Jesse PetersonProperly proximal von Neumann algebras, arXiv:2204.00517
Problems with Cartan subalgebras
Problem C.1 (Conjecture of Popa)
Let \(\Gamma\) be a non-amenable group, and let \((X_0, \mu_0)\) be a non-trivial standard probability space. Does the Bernoulli shift crossed product \(L^\infty( \prod_{t \in \Gamma} (X_0, \mu_0) ) \rtimes \Gamma\) have a unique, up to unitary conjugacy, Cartan subalgebra?
Problem C.2 (Popa)
Is there a free mixing p.m.p. action of a nonamenable group \(\Gamma\) such that \(L^\infty(X) \rtimes \Gamma\) does not have a unique Cartan subalgebra, up to unitary conjugation?
Problem C.3 (Popa)
Does \(L^\infty(\mathbb T^n) \rtimes SL_n(\mathbb Z)\) have a unique Cartan subalgebra, up to unitary conjugacy, for all \(n \geq 3\)?
Problem C.4
Suppose \(\beta_1^{(2)}(\Gamma) \not= 0\). Does every free ergodic measure-preserving action of \(\Gamma\) give rise to a II\(_1\) factor with unique, up to unitary conjugacy, Cartan subalgebra.
Problem C.5 (Conjecture of Popa and Vaes)
Suppose \(\beta_n^{(2)}(\Gamma) \not= 0\) for some \(n \geq 1\). Does every free ergodic measure-preserving action of \(\Gamma\) give rise to a II\(_1\) factor with unique, up to unitary conjugacy, Cartan subalgebra.
Problem C.4
Is there a II\(_1\) factor that has exactly two Cartan subalgebras, up to unitary conjugation?
Problems with bounded geometry
Problem B.1 (Anantharaman-Delaroche 1995)
Let \((X, \mathcal E)\) and \((Y, \mathcal E')\) be uniformly locally finite coarse spaces such that \(C^*_u(X, \mathcal E)\) and \(C^*_u(Y, \mathcal E')\) are isomorphic. Does it follow that \((X, \mathcal E)\) and \((Y, \mathcal E')\) are coarsely equivalent?
This may be solved in the case of metric spaces. See the preprint
[Baudier+Braga+Farah+Khukhro+Vignati+Willett 2021].
Florent P. Baudier, Bruno de Mendonça Braga, Ilijas Farah, Ana Khukhro, Alessandro Vignati, and Rufus Willett Uniform Roe algebras of uniformly locally finite metric spaces are rigid, arXiv:2106.11391