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.

Is every separable von Neumann algebra generated, as a von Neumann algebra, by a single element?

Do we have \(L \mathbb F_2 \cong L \mathbb F_3\)?

If \(\Gamma\) and \(\Lambda\) are icc property (T) groups such that \(L\Gamma \cong L\Lambda\) must we have that \(\Gamma \cong \Lambda\)?

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 \(\| u - 1 \| < \varepsilon\)?

Is every bounded representation of a \(C^*\)-algebra on a Hilbert space similar to a \(*\)-representation?

Is it true that \(H^k(M, M) = 0\) for all von Neumann algebras \(M\) and all \(k \geq 1\)?

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 \(R\)?

If \(M\) is a nonamenable II\(_1\) factor, then must \(M\) contain an isomorphic copy of \(L\mathbb F_2\)?

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 \(p x p = x p\)?

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 \(\delta(x_1, \ldots, x_n) = \delta(y_1, \ldots, y_m)\)?

Is every property (T) von Neumann algebra generated, as a von Neumann algebra, by a single element?

Does every II\(_1\) factor have an orthonormal (with respect to the trace) basis consisting of unitaries?

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 \(L\mathbb F_2\)?

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?)

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

Is \(C_r^* (\mathbb F_\infty)\) finitely generated?

Suppose \(M \cong M_1 \, \overline \otimes \, M_2\) is McDuff. Does it follow that either \(M_1\) or \(M_2\) is McDuff?

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 \(\Gamma \cong \Lambda\)?

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 \(\theta(u u_t u^*) = \chi(t) u_{\alpha(t)}\)?

Do we have \(\mathcal F( L(PSL_n(\mathbb Z))) = \{ 1 \}\) for all \(n \geq 3\)?

For \(n, m \geq 3\), do we have \(L(PSL_m (\mathbb Z)) \cong L(PSL_n (\mathbb Z))\) only if \(n = m\)?

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 \(\Gamma \cong \Lambda\)?

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 \(L\Gamma\)?

Let \(\Gamma\) be an icc property (T) hyperbolic group, is \({\rm Out}(L\Gamma)\) finite?

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?

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?

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?

Is there a II\(_1\) factor \(M\) so that \(\mathcal F(N^\omega) \not= \mathbb R_+^*\)?

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 \(\omega\)?

Is there a II\(_1\) factor \(M\) such that \(M^\omega \not\cong ({ M^{\rm op} })^\omega\) for any free ultrafilter \(\omega\)?

Is there a free ultrafilter \(\omega\) so that \((L \mathbb F_2)^\omega \cong (L \mathbb F_3)^\omega\)?

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 \(M \subset \prod_\omega N_n \subset N^\omega\)?

Is there a nonsofic group?

Is Thompson's group \(F\) amenable?

Is every nonamenable subfactor of \(L\mathbb F_2\) an interpolated free group factor?

Is every diffuse amenable von Neumann subalgebra of \(L \mathbb F_2\) contained in a unique maximal amenable von Neumann subalgebra?

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)

Is there an embedding of \(\mathbb F_2\) into \(\mathcal U(R)\) whose range is discrete in the strong operator topology?

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 \(M\)?

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 \(M \cong R\)?

Are \(\ell^2\)-Betti numbers of groups an invariant of their von Neumann algebras?

Do the Haagerup property and the compact approximation property describe the same class of II\(_1\) factors?

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?

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?

Does \(L^\infty(\mathbb T^n) \rtimes SL_n(\mathbb Z)\) have a unique Cartan subalgebra, up to unitary conjugacy, for all \(n \geq 3\)?

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.

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.

Is there a II\(_1\) factor that has exactly two Cartan subalgebras, up to unitary conjugation?

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