16 problems

AF-embeddability characterization conjecture. A C-algebra is AF embeddable if, and only if, it is separable, exact, and quasidiagonal.

Let $H$ be a separable, infinite-dimensional Hilbert space, let $\mathcal K=\mathcal K(H)$ be the algebra of compact operators, and let $\mathcal Q(H)=\mathcal B(H)/\mathcal K$ be…

Let $A$ be a simple, separable, unital ASH algebra with strict slow dimension growth. Let $\mathrm{W}(A)$ be its Cuntz semigroup and let $\widetilde{\mathrm{W}}(A)$ denote the orde…

Let $A$ be a unital C-algebra. An algebraically irreducible representation of ${\rm Max}\,A$ is a representation that is atomic as a sup-lattice and whose every atom is a generator…

Let $0\to I\to E\to B\to 0$ be an asymptotically split extension of NF algebras, meaning that the quotient map admits an asymptotic completely positive contractive splitting. Brown…

Nine dual spaces conjecture. There are precisely nine isomorphism types of dual Banach spaces arising from Type I AF C-algebras.

Let $\mathcal{A}$ be a unital C$^*$-algebra. A MASA is a maximal abelian C$^*$-subalgebra of $\mathcal{A}$, and a W$^*$-algebra is a C$^*$-algebra admitting a Banach-space predual.…

The 2-quastrace conjecture. Every $2$-quasitrace on a unital C$^*$-algebra is a trace.

Let $A$ and $D$ be separable and unital $\mathrm{C}^*$-algebras, with $D$ strongly self-absorbing. For a $\mathrm{C}^*$-algebra $B$, write $\mathcal Q(B)=\mathcal M(B)/B$ for its c…

Let $A$ be a separable $A\mathbb{T}$-algebra of real rank zero that is not AF. Its complexity-rank-one conjecture. complexity rank is always one. In particular, the complexity rank…

Let $D$ be a stably finite strongly self-absorbing C-algebra, meaning that $D$ is infinite dimensional and admits an isomorphism $D\to D\otimes_{\mathrm{min}}D$ approximately unita…

Let $A$ be a semiprojective $C^*$-algebra and let $p$ be a full projection in $A$, so that the corner $pAp$ is a full corner of $A$. Blackadar's conjecture. Passage of semiprojecti…

Let $S\subset\mathbb{N}^{\times}\setminus\{1\}$ be a family of relatively prime numbers with $\lvert S\rvert\geq 2$. Let $\mathcal{A}_S$ be the torsion subalgebra associated to $S$…

Maximal abelian subalgebra conjecture. The subalgebra $M$ is a maximal abelian subalgebra of $C^*(\Lambda)$.

Consider simple, separable, nuclear $C^*$-algebras and regularity properties expressing that they are sufficiently well-behaved, such as Jiang–Su stability, unperforation of the Cu…

An ASH algebra is an approximately subhomogeneous $C^{*}$-algebra, and its topological dimension refers to the dimension bound in an ASH decomposition. Let $A$ be a simple ASH alge…