26 problems

Let $F$ be an infinite field, let $n\ge 2$, and let $p\in F\langle \mathcal{X} \rangle$ be a multilinear polynomial … where $m$ is a positive integer and each $\lambda_\sigma\in F$…

Let $F$ be a field, let $n\ge 2$, and let $p\in F\langle x_1,\ldots,x_m\rangle$ be a multilinear polynomial. The image $p(\operatorname{M}_n(F))$ is the set of values of $p$ on the…

Let $d4aa$ be either $d53d$ or $d54a$, let $d4aa\langle\bar X\rangle$ be the free polynomial algebra in self-adjoint variables, and let $f\ind4aa\langle\bar X\rangle$. If…

The paper considers conditions for the existence of multilinear operations on skew primitive elements, expressed through linear dependence of particular noncommutative polynomials…

Let $f_1,\ldots,f_l,g\in\mathbb{C}\langle\underline{x}\rangle$. For a complex Hilbert space $\mathcal{H}$, let $\mathcal{B}(\mathcal{H})$ denote the bounded operators on…

Let $\mathbb{K}$ be a field, let $f_1,\ldots,f_l,g\in\mathbb{K}\langle\underline{x}\rangle$ be noncommutative polynomials, and for each $n\in\mathbb{N}$ and…

Kernel equality conjecture. For every such tuple,

Rank Nullstellensatz conjecture. The following are equivalent: (i) there is $K\in\mathbb{N}$ such that for every $n\in\mathbb{N}$ and every…

Makar-Limanov's conjecture.

Invariant-subspace certificate conjecture. The following are equivalent: (i) for every $n\in\mathbb{N}$ and every $\underline X\in\operatorname{M}_n(\mathbbm k)^d$, every joint inv…

Let $K$ be an algebraically closed field, let $n\geq 2$ and $m\geq 1$ be integers, and let $T_n(K)$ denote the algebra of $n\times n$ upper triangular matrices over $K$. For…

Let $A$ be the universal unital $C^*$-algebra generated by two projections, let $M$ be the concrete magic unitary from the paper, and let $u$ be the matrix containing the generator…

Let $K$ be a field, let $n\geq 2$ be an integer, and let $f(x_1,\dots,x_m)$ be a multilinear polynomial. Say that $f$ satisfies $(S1)$ when it is a central polynomial, so its image…

Let $\mathbb K\langle\underline{x}\rangle$ denote the free noncommutative polynomial algebra, let $\mathfrak r=(p_{ij})\in\mathbb K\langle\underline{x}\rangle^{L\times L}$ have non…

Eigenvector generation conjecture. For each $n\geq 1$, a generating set of $S_{\mathbb Q}\langle a,b\rangle_n$, consisting only of eigenvectors, may be obtained by applying all pos…

Kernel conjecture. For each $n\geq 1$,

Finite-type representing measure conjecture. The following are equivalent:

Let $\mathcal{R}$ be a finite von Neumann algebra, with its trace, and let $p$ be a non-commutative polynomial in $n$ variables. Let the Murray-von Neumann algebra affiliated with…

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…

Let $P$ be a homogeneous polynomial in noncommuting variables with the property that, whenever $X_1,\dots,X_n$ are free copies of a fixed random variable $X$, the distribution of…

Let $f$ be a multilinear polynomial with complex coefficients, and let $d\in\mathbb{N}$. Consider the set of values obtained by evaluating $f$ on $M_d(\mathbb{C})$. Kaplansky–Lvov…

Let $p$ be a prime, let $s_1\geq s_2\geq\cdots\geq s_r$, and let $M_{p^{s_i}}^d$ denote the ring of $d\times d$ matrices over $\mathbb{Z}/p^{s_i}\mathbb{Z}$. For an $r$-tuple…

Let $Q_{\mathrm{nc}}$ be the union, over all $d\geq 1$, of tuples $(X_1,\ldots,X_n)\in(\mathcal S^d)^n$ satisfying $I-X_i^2\succeq0$ for every $i\in[n]$. Let…

Helton–McCullough conjecture. If $p$ is irreducible and $\mathcal D_p$ is convex, then $\mathcal D_p$ is a ball.

Let $q$ be the partial Hessian of a symmetric polynomial, let $Z(0,A)$ be the associated middle matrix evaluated at $A\in\mathbb S_n(\mathbb R^g)$, let $\mu_\pm$ denote the number…