16 problems

For a type $\mathrm{II}_1$ factor $\mathcal{M}$, define … and … where the minimum is defined to be $\infty$ if $\mathcal{M}$ cannot be generated as stated. Here…

Let $r\in\mathbb{N}$. For each $N$, let $U_1^{(N)},\dots,U_r^{(N)}$ be independent Haar random unitaries, and let $u_1,\dots,u_r$ be the canonical generators of $L(\mathbb{F}_r)$.…

Let $V$ be a self-adjoint convex potential, taken to be polynomial for simplicity, with associated free Gibbs state $\tau_V$, and let $F$ be a vector of self-adjoint multiple Wigne…

A II$_1$ factor is nonamenable if it is not amenable, and $L(\mathbb F_2)$ denotes the group II$_1$ factor of the free group on two generators. von Neumann algebraic von Neumann pr…

A nonamenable II₁ factor is a nonamenable finite factor with separable predual. For II₁ factors $N$ and $M$, write $N\hookrightarrow M$ when there is a unital embedding of $N$ into…

Fix an integer $r\geq 2$. For each $k$, let $\mu^{(k)}$ be the $r$-fold product of the GUE distribution on self-adjoint $k\times k$ matrices, set $\mu=\prod_k\mu^{(k)}$, and choose…

Let $r>1$ and let $Q$ be a von Neumann subalgebra of the free group factor $L({\mathbb F}_r)$. Assume that $Q$ is diffuse and nonamenable. Positivity conjecture for 1-bounded entro…

Maximal-amenable noncoarse pair implication. If $Q$ and $P$ are maximal amenable in $M$ and not coarse, then $M$ has a properly infinite $R$-bimodule decomposition.

Let $M=L(\mathbb F_t)$ be an interpolated free group factor with $1<t\leq\infty$. A hyperfinite subfactor $L\subset M$ is called ergodic when the inclusion has the ergodicity prope…

Let $1<t\leq\infty$, let $M=L(\Bbb F_t)$, and let $B\subset M$ be a maximal amenable von Neumann subalgebra. Multiple mixing conjecture. The subalgebra $B$ is multiple mixing in…

Let $1<t\leq\infty$, let $B$ be a maximal amenable von Neumann subalgebra of the interpolated free group factor $L(\Bbb F_t)$, and let $\mathcal U(L(\Bbb F_t))$ denote its unitary…

Let $1<t\leq\infty$, and let $B$ be a maximal amenable, equivalently maximal AFD, von Neumann subalgebra of the interpolated free group factor $L(\Bbb F_t)$. Coarseness conjecture.…

Let $M$ be a $\rm II_1$ factor with the stable single generation (SSG) property, meaning that some sequence $t_n\searrow0$ has each amplification $M^{t_n}$ singly generated. Let…

Let $M$ be a free group factor and let $A\subset M$ be a maximal amenable subalgebra. A subalgebra $B\subset A$ is diffuse if it has no minimal projections. Peterson's conjecture.…

Let $t\in(1,\infty]$ and let $N\subseteq L(\mathbb{F}_{t})$ be a maximal amenable von Neumann subalgebra. Regard the following as $N$-$N$ bimodules, where…

Let $N\subseteq L(\mathbb{F}_{2})$ be a diffuse hyperfinite von Neumann subalgebra. Define its wq-normalizing subalgebra to be the smallest von Neumann subalgebra $Q$ of…