53 problems

Let $\mathcal{A}$ be a unital C-algebra with the invariant basis number property, and let $\mathcal{E}$ be an $n$-rank Banach module over $\mathcal{A}$. For $k\in\mathbb{N}$, write…

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…

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…

Real-rank inequality conjecture. For every $C^*$-algebra $A$, one has

Let $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ be $C^{\ast}$-algebras, and let $T:\mathcal{A}_{1}\to\mathcal{A}_{2}$ be a biseparating map, meaning that both $T$ and $T^{-1}$ preserve…

Let $g$ and $h$ be minimal diffeomorphisms of $S^m$ and $S^n$, respectively, with $m,n\geq 3$ and odd. Let the associated transformation group C-algebras be $C^*(\mathbf{Z},S^m,g)$…

Let $(A,\varphi)$ and $(B,\psi)$ be nontrivial $C^*$-probability spaces inducing faithful GNS representations. Form their reduced free product $(A,\varphi)*(B,\psi)$. Choda–Dykema…

The Toms–Winter conjecture. The following conditions are equivalent:

Graph-algebra characterization conjecture. The following conditions are equivalent:

Generalized Toms--Winter Conjecture. The following conditions are equivalent:

Toms--Winter Conjecture. The following conditions are equivalent: $$

Let $G$ be a compact group. The equivariant Kasparov theory $KK^G$ is said to be very special when it has the special properties defined in the surrounding discussion. Compact-grou…

Let $(X,\varphi)$ be a mixing Smale space, and let $S$ and $U$ denote its stable and unstable algebras, respectively. Stable–unstable rank conjecture. One has … This equality predi…

Rosenberg's conjecture. 1. For $n>0$, the map

Equivariance-implies-admissibility conjecture. If $\mu$ is $G$-equivariant, then $\mu$ is $G$-admissible.

Let $A$ be a unital, separable, nuclear ${\rm C}^*$-algebra of real rank zero and stable rank one, and let $\underline{\mathrm{Cu}}_u(A)$ denote its total Cuntz semigroup invariant…

Let $A$ be a separable, nuclear, K-pure ${\rm C}^*$-algebra of real rank zero and stable rank one. Let $({\rm\underline{K}}(A),{\rm\underline{K}}(A)_+,\Sigma A)_{\rm \Lambda}$ cons…

Let $\mathcal{A}$ be a commutative C-algebra, let $n\bru001ein\breed Mathbb{N}$, and let … be a polynomial over $\mathcal{A}$ with $a_1,\dots,a_n\bru001ein\breed \mathcal{A}$. For…

Let $\mathcal{A}$ be a C-algebra, let $d\in\mathbb{N}\setminus\{1\}$, and let … be a polynomial over $\mathcal{A}$ with $a_1,\ldots,a_d\in\mathcal{A}$. Suppose that … on…

Let $\mathcal{A}$ be a C-algebra, let $d\in\mathbb{N}\setminus\{1\}$, and let … be a polynomial over $\mathcal{A}$ with $a_1,\ldots,a_d\in\mathcal{A}$. Suppose that its formal deri…

Let $\mathcal{A}$ be a unital C-algebra, let $n\in\mathbb{N}\setminus\{1\}$, and let … with $a_1,\dots,a_n\in\overline{\mathbb{D}^*(0,1)}$. Define … where the hatted factor is omit…

Let $\mathcal{A}$ be a unital commutative C-algebra whose group of invertible elements $G(\mathcal{A})$ is dense in $\mathcal{A}$. Let $n\in\mathbb{N}\setminus\{1\}$ and … where…

For a family of coefficients $T_{ij}^{kl}$, let $W(T)$ be the Wick $*$-algebra generated by $a_j,a_j^*$, and let $T\colon\mathcal H^{\otimes 2}\to\mathcal H^{\otimes 2}$ be its ope…

Let $A$ be a $C^*$-algebra. Blackadar–Kirchberg conjecture. If $A$ is separable, stably finite and nuclear, then $A$ is quasidiagonal. This conjecture concerns the converse to the…

Let $G$ be a second countable locally compact group and let $\partial_F G$ be its Furstenberg boundary. For $z\in\partial_F G$, assume that $z$ is a continuity point of the stabili…