379 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 $\tau$-…

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…

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 $A$ 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…

Refined embedding conjecture. The only von Neumann algebras that can embed into

Folk conjecture. The only von Neumann algebras that can embed into a uniform Roe algebra or a quasi-local algebra associated to a uniformly locally finite metric space are those of…

Let $A$ be a unital $C^*$-algebra. Define the classical Cuntz semigroup $\mathrm{W}(A)$ from positive elements in $M_\infty(A)$, and let $\mathrm{DF}(A)$ be the set of normalized d…

Anti-deformation conjecture. Any Woronowicz algebra is a dimension-preserving $R^+$-deformation of a Woronowicz-Kac algebra.

Finiteness conjecture. There are only finitely many Woronowicz algebras having a given $(R^+,list)$ invariant.

Let $(X,\f8\sigma)$ and $(Y,\f8\tau)$ be multivariable dynamical systems, and let ${\mathcal{A}}(X,\sigma)$ and ${\mathcal{A}}(Y,\tau)$ denote their tensor algebras. Piecewise conjug…

Analytical zero-divisor conjecture. If $\Gamma$ is torsion-free, then $\lambda_{\Gamma}(f)$ is injective for every non-zero $f\in\mathbf{C}\Gamma$. This is a classical conjecture c…

Let $A$ be a unital Kirchberg algebra, and let $\alpha \colon G \to \operatorname{Aut}(A)$ be an action of a finite group $G$ on $A$ such that $\alpha_g$ is outer for all…

Let $\Gamma$ be a torsion-free group such that $(\Gamma,\sigma)$ satisfies the twisted Bost conjecture, and suppose that the classifying space $B\Gamma$ is a smooth, compact, orien…

Twisted Bost conjecture. The map $\mu_\sigma^{\mathcal A}$ is an isomorphism for $j=0,1$. The conjecture is presented as an unconditional analogue of the twisted Baum–Connes conjec…

Let $\Gamma_k$ denote the graphs in the paper's Figure 2. Haagerup graph principal-graph conjecture. For every $k>1$, the graph $\Gamma_k$ is not a principal graph of a subfactor.…

Let $(A,G,\alpha)$ be a separable dynamical system, where $A$ is a $C^*$-algebra, $G$ is a locally compact group, and $\alpha$ is an action of $G$ on $A$. The system satisfies the…

Let $A$ be a simple, unital, separable, and nuclear $C^*$-algebra of stable rank one. Write $V(A)$ for its Murray–von Neumann semigroup of projections and $[1_A]$ for the class of…

The coarse geometric Novikov conjecture. The map $Ind$ is injective.

Let $\mathcal{M}$ be a finite-rank, quasi-free, $p$-reductive Hilbert module over $A(\Omega)$, and let $\mathcal{S}$ be a submodule such that $\mathcal{S}^{\bot}$ is spanned by gen…

Let $S({\bf C}^{N})^{\infty}_{\mathrm{AP}}$ denote the asymptotically periodic sequences and let $S({\bf C}^{N})^{\infty}_{\mathrm{EP}}$ denote the eventually periodic sequences. W…

Let $(N,M,\alpha,\beta,\Gamma,\mu,T_L,T_R)$ be an adapted measured quantum groupoid, meaning a measured quantum groupoid with the stated adaptedness property, and suppose that $M$…

Let $(N,M,\alpha,\beta,\Gamma,\mu,T_L,T_R)$ be a measured quantum groupoid, and suppose that $M$ is commutative. Let $G$ denote a locally compact groupoid, with unit space…

Quasitrace conjecture. Quasitraces on arbitrary C-algebras are traces.

Let an AW-algebra be a unital C-algebra whose left annihilator of every nonempty subset is a principal right ideal generated by a projection. An AW-factor is an AW-algebra that is…

Let $M$ be a connected compact smooth manifold with $\dim(M)>0$, and let $h\colon M\to M$ be a uniquely ergodic minimal diffeomorphism. Let…