10 problems

Let $\chi_\mathrm{orb}$ denote orbital free entropy and let $i^*$ denote free mutual information for tuples of subalgebras in a tracial $W^*$-probability space. Orbital free entrop…

Let $X$ be a tuple of non-commutative random variables, let $S$ be a free semicircular family free from $X$, and let $\chi$ and $\chi^*$ denote Voiculescu's microstates and non-mic…

Orbital free entropy mutual information conjecture. It would be further conjectured that

Projection mutual free information conjecture. The quantity $-\chi_\mathrm{proj}(p,q)$ should coincide with

Let $(A,\tau)$ be a tracial $C^*$-probability space, meaning that $A$ is a unital $C^*$-algebra equipped with a faithful tracial state $\tau$. It is inner amenable if there is a se…

Let $f:\mathbb{R}\rightarrow\textbackslash \mathbb{R}$ be continuous and satisfy the assumptions denoted by $(\mathrm{gibbs})$. For $\lambda\textbackslash in\mathbb{R}$, let the fr…

Let $(\mathcal{M},\tau)$ be a tracial $W^\ast$-probability space, let $X=(X_1,\dots,X_n)$ be a tuple of selfadjoint noncommutative random variables in $\mathcal{M}$, and let…

Projection Unification Conjecture. One has

Unification Conjecture. The mutual free information satisfies

Let $(\mathcal{M},\tau)$ be a tracial non-commutative $W^*$-probability space, and let $\mathbf{X}_1,\dots,\mathbf{X}_n$ be finite tuples of self-adjoint elements of $\mathcal{M}$.…