# 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 October 12, 2020.

## 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?

#### 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?

#### 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?)

#### Problem S.5 (Popa+Vaes)

Let $$\Gamma$$ and $$\Lambda$$ be torsion free groups such that $$C^*(\Gamma) \cong C^*(\Lambda)$$. 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 $$\Gamma \cong \Lambda$$?

#### 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 $$\omega$$?

#### 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 $$M \subset \prod_\omega N_n \subset N^\omega$$?

## Problems with groups

#### Problem G.1

Is there a nonsofic group?

#### Problem G.2

Is Thompson's group $$F$$ amenable?

## 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?

#### 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 follows 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?

## 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?