393 problems

Let $\mathcal{M}$ be a semifinite von Neumann algebra endowed with a faithful normal semifinite trace $\tau$, and let $\mathcal{P}$ be the positive cone of the space of all $\tau$-…

Is it true that every separable unital $AT$ algebra $A$ of real rank zero satisfies $\operatorname{cr}(A)\le 1$? In particular, for every irrational rotation algebra $A_\theta$ wit…

For every second-countable Hausdorff minimal topologically amenable ample groupoid $G$ with Cantor unit space, is $G$ either almost finite or purely infinite?

Given a discrete subgroup $\Gamma\leq \operatorname{PSL}_2(\mathbb{Z})$ in Jones's cusp-form and von Neumann-algebra construction, find an explicit formula for a tracelike vector…

For all locally compact groups $G$ and $H$, prove that $\Lambda_{\mathrm{WH}}(G\times H)=\Lambda_{\mathrm{WH}}(G)\Lambda_{\mathrm{WH}}(H)$.

Let $A$ and $B$ be simple separable $C^*$-algebras, and let $\widehat{A}$ and $\widehat{B}$ denote their spectra, namely the spaces of unitary-equivalence classes of irreducible un…

For all finite graphs $G$ and $H$, the quantum chromatic number of their categorical product satisfies $\chi_q(G\times H)=\min\{\chi_q(G),\chi_q(H)\}$, where $G\times H$ has vertex…

Let $A$ and $B$ be $C^*$-algebras, and let $T:A_+\to B_+$ be surjective and satisfy the Fischer--Muszély identity $\|T(a+b)\|=\|T(a)+T(b)\|$ for all $a,b\in A_+$. Must $T$ be addit…

For every finitely generated, torsion-free, non-elementary discrete subgroup $\Gamma\subset PSL_{2}(\mathbb{R})$, the group von Neumann algebra $L(\Gamma)$ is isomorphic to a free…

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…

For every nonlocal game $G$, if $G$ self-tests a strategy among strategies consisting of a pure state of full Schmidt rank and projective measurements, then $G$ self-tests the same…

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…

$\forall A\,\bigl(A\text{ is an }AW^*\text{-algebra}\Rightarrow A\text{ is normal}\bigr).$

Given two infrared-renormalized van Hove–Miyatake models, determine necessary and sufficient conditions for them to be unitarily equivalent; that is, characterize when there exists…

Every gapped ground state satisfies approximate Haag duality.

Does every type $\mathrm{II}_1$ factor $M$, equipped with its normalized trace $\tau$, admit an orthonormal basis of $L^2(M,\tau)$ consisting of unitaries in $M$? Equivalently, doe…

Let $\mathsf{R}$ be a retrodiction, or quantum Bayesian inversion, assignment for normal quantum processes between von Neumann algebras equipped with faithful states, satisfying th…

Let $A$ be a unital operator algebra generating the $C^*$-algebra $B=C^*(A)$. If every irreducible $*$-representation $\pi:B\to B(H_\pi)$ has the unique completely positive extensi…

For integers $n,k\ge 2$, let $C_{qa}(n,k)$ and $C_{qc}(n,k)$ denote the corresponding approximately quantum and commuting quantum correlation sets, and let…

Let $S=\{v_i\}_{i=1}^g\subseteq\matrices$ be a collection of matrices, and define its Wielength by … where $S^k=\{v_\sigma:\sigma\in\{1,\dots,g\}^k\}$,…

Atiyah conjecture. The von Neumann dimension of the kernel of this map is an integer.

Let $m\in\mathbb{N}$ and $a_i\in M_m(\mathbb{Z})$ be given. Define the completely positive map … Assume that $\eta$ is rank non-decreasing. For $n$ independent GUE random matrices…

Let ${*}\text{-}\mathfrak{Alg}$ be the category of non-commutative, associative, $*$-algebras, and let $\mathfrak{Bkgr}_c$ and $\mathfrak{SpBkgr}_c$ be the categories of spacetime…

Let $\mathcal{H}$ be an $n$-dimensional Hilbert space, and let $\mathcal{P}_k(\mathcal{H})$ denote the set of rank-$k$ orthogonal projectors on $\mathcal{H}$. Assume that $n=2k$ an…

Let $H$ be a quasi-triangular Hopf-$*$ algebra, and consider the corresponding Kitaev model on the plane. Haag duality conjecture. The model satisfies Haag duality for cones. This…