18 problems

Let $(\mathcal{M},\rho)$ be a quantum probability space, and let $A_1,\ldots,A_k$ be self-adjoint elements with mean $0$ and symmetric covariance matrix. For each $t$, write … and…

Second-eigenvalue conjecture. If $\phi$ is unital, then

Equality conjecture. For each fixed $g\in{\cal G}$,

Let $A_1,\ldots,A_n\in\mathbb R^{n\times n}$ be symmetric matrices satisfying $\|A_i\|_{\mathrm{op}}\le 1$. A coloring is a vector $x\in\{\pm1\}^n$. Variance-sensitive Matrix Spenc…

Let $(\mathcal{A}_1,E_1,E_1'),\ldots,(\mathcal{A}_k,E_k,E_k')$ be OVI $C^*$-probability spaces over $\mathcal{B}$, and let $(\mathcal{A},E)$ be their operator-valued $C^*$-free, Bo…

Christoffel-Darboux approximation conjecture. Under these hypotheses, the measures $\mu_{\tau,k}$ should provide good random-matrix approximants to the noncommutative distribution…

A noncommutative Bernoulli shift is a noncommutative Bernoulli shift in the sense of the source, and partial spreadability is the property defined there for such shifts. Existence…

Let $ES^+_r$ be the extended semi-cosimplicial monoid generated by $h_n$ and $d_n$ for $n\in\mathbb{N}_0$, with the defining relations given in the source, and let…

Let $b$ be a centred Lévy $L^p$-process, let $M_p(t)=\tau(|b_t|^p)$, and let $\tilde{\beta}_p$, $\tilde{\beta}_p$ and the quantities $\tilde{\beta}_p$ be as in the preceding growth…

Let $\mathcal{T}$ be a tree whose edges are assigned weights, and modify the operators $\lambda_{\mathcal{T},j}(x)$ by multiplying their restrictions to…

In algebraic probability spaces, a notion of stochastic independence is an algebraically defined rule describing how random variables or subalgebras combine. Schürmann's conjecture…

Conditional-expectation conjecture. The Betti numbers can be obtained as cumulants with respect to a suitable conditional expectation.

Non-commutative central limit theorem. The limit exists for all $omega$ in a full-measure set $tilde{\Omega}\subset\Omega$, is independent of $omega$ in this set, and uniquely defi…

Let $X=\sum_{k=1}^s g_kA_k$ be the Gaussian matrix model from the noncommutative Khintchine theorem, where $g_k$ are independent standard Gaussian variables and $A_k$ are the assoc…

Let $\Phi$ be an Orlicz function with Boyd indexes $p_{\Phi}$ and $q_{\Phi}$. Consider the noncommutative Khintchine inequality in the range $q_{\Phi}<2$ for finite sequences in th…

Let $n$ and $m$ be positive integers, and let $\{\mathbf{A}_i\}_{i=1}^n$ be positive definite matrices. Variant of the conjecture. The corresponding inequality with the operator no…

Let $\bm{A}_1,\dots,\bm{A}_n$ be positive-semidefinite matrices, and let $m\leq n$. Ducci's conjecture. For each $m\leq n$, … This variant places the matrix operator norm inside th…

Let $\bm{A}_1,\ldots,\bm{A}_n$ be $d\times d$ positive semidefinite matrices. Define the arithmetic mean and symmetrized geometric mean by … where $S_n$ is the group of permutation…