71 problems

For every type $\mathrm{III}_1$ factor $M$ and every faithful normal state $\varphi$ on $M$, the bicentralizer algebra $B(M,\varphi)$ is trivial: $B(M,\varphi)=\mathbb{C}1$, where…

Let $A\subset M$ be a MASA in a von Neumann algebra, and suppose first that there is a normal conditional expectation from $M$ onto $A$. The app-paving conjecture. Every MASA in a…

Let $G$ be an ICC property (T) group, let $H$ be an arbitrary countable ICC group, and let $d:G\times G\to\mathbb{T}$ and $c:H\times H\to\mathbb{T}$ be $2$-cocycles. For $t>0$, sup…

Let $N \subset M$ be an inclusion of von Neumann algebras with expectation, and let $\varphi \in N_*$ be a faithful normal state. Write $c(N)=N \rtimes_{\sigma^\varphi} \mathbb{R}$…

Let $\mathcal H$ be a Hilbert space, let $\mathcal B(\mathcal H)$ denote the bounded operators on $\mathcal H$, and let $d(M,N)$ be the Hausdorff distance between the unit balls of…

Let $N$ be a tracial von Neumann algebra, let $\mathcal U$ be a free ultrafilter, and let $u,v\in N^{\mathcal U}$ be freely independent unitaries. Is it true that, for every…

Let $\Gamma$ be a higher-rank lattice and let a character mean the character object considered in the source. Connes' character rigidity conjecture. Every character of $\Gamma$ is…

Type II$_1$ factor approximation conjecture. Suppose that $T$ is an operator in $\mathcal{B}(\mathcal{H})$ such that $W^*(T)$ is a type $\mathrm{II}_1$ factor. Then for every…

Characteristic-invariant conjecture. The action $\alpha$ is classified up to cocycle conjugacy by the characteristic invariant $(H,\chi)$.

Let $L(\mathbb{F}_n)$ be a free group factor and let $B\subset L(\mathbb{F}_n)$ be a diffuse amenable subalgebra. A maximal amenable extension of $B$ is an amenable subalgebra of…

Let an AW-algebra be a unital C-algebra whose left annihilator of every nonempty subset is a principal right ideal generated by a projection. An AW-factor is an AW-algebra that is…

Rationality and integrality conjecture. For every degree $p$,

Let $M$ be a separable II$_1$ factor. Let $G$ be a countably infinite group, let $(B,\tau)$ be a tracial von Neumann algebra, and let $(\sigma,c)$ be a trace-preserving cocycle act…

Let $M$ be a separable II$_1$ factor, and let $P\subset M$ be a regular von Neumann subalgebra, meaning that the normalizer of $P$ in $M$ generates $M$. Margulis-type conjecture. T…

Peterson–Thom conjecture for hyperbolic groups. If $\mathcal{A}\capmathcal{B}$ is diffuse, then the von Neumann algebra they generate is amenable:

Peterson–Thom conjecture. There exists a unique maximal amenable von Neumann subalgebra $\mathcal{P}$ of $\mathcal{L}(\mathbb{F}_r)$ such that

Kawahigashi's uniqueness conjecture. The following assertions are equivalent: $\alpha$ is cocycle conjugate to $\sigma^{\infty}$; $\alpha$ satisfies the Rokhlin property; $\alpha$…

A countable discrete group $G$ is inner amenable if the conjugation action of $G$ on $G\setminus\{1\}$ admits an invariant mean, and it is strongly 1-bounded if its group von Neuma…

Let $G$ be an icc property (T) group. For a $\mathrm{II}_1$ factor $\mathcal M$, let $\mathcal F_s(\mathcal M)$ be the set of all $t\in\mathbb R_+$ for which there exists a $*$-emb…

Let $G$ be a group that is hyperbolic relative to a finite family of subgroups $\{H_1,\ldots,H_n\}$. Denote by $\mathcal M=\mathcal L(G)$ the von Neumann algebra corresponding to…

Let $\mathcal{M}$ be a local von Neumann factor, let $\sigma_s$ be its modular action, and let $\mathcal{R}(\mathcal{M},\sigma_s)$ be the algebra generated by $\mathcal{M}$ togethe…

Let $(M,g)$ be the spacetime under consideration and let $\mathcal{O}\in\mathcal{K}(M,g)$. Write $\mathrm{QuantLie}$ for the set … Let…

Quantum-gravity tensor-structure conjecture. The tensor-product structure above describes the first steps away from the semiclassical approximation toward a full theory of quantum…

Let $X=(X_1,\ldots,X_n)$ be a noncommutative random vector with joint law $\mu_X$, let $V_0$ and $V_1$ be potentials, and let $A=A^{\dagger}$ be a free Stein kernel for $X$ with re…

Let $\mathsf{G}$ be a semisimple algebraic group over $\mathbb{C}$, let $\Gamma$ be a finitely generated subgroup of $\mathsf{G}(\mathbb{C})$, fix a central character $\chi$, and l…