144 problems

Let $A$ be a simple separable unital nuclear $C^*$-algebra. The Toms–Winter conjecture. Finite nuclear dimension of $A$, tensorial absorption of the Jiang–Su algebra $\mathcal{Z}$,…

Elliott's classification conjecture. Unital separable simple nuclear $C^*$-algebras are classified up to isomorphism by $K$-theoretic invariants.

Blackadar–Handelman conjecture. The set

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

Let $R$ be a ring, and let $\operatorname{Nil}^*(R)$ denote its upper nilradical, the largest nil ideal of $R$. A one-sided ideal $J$ of $R$ is called nil if every element of $J$ i…

Let $A$ be a real C-algebra, let $K$ denote nonconnective algebraic $K$-theory, and let $\mathscr{C}([0,1];A)$ be the ring of continuous functions from $[0,1]$ to $A$. Rosenberg's…

The three regularity properties of $\mathcal{Z}$-stability, finite nuclear dimension, and the relevant comparison property of positive elements are considered for separable simple…

Let $\mathbb{F}_2$ be the free group on two generators, and let $\mathrm{C}^*_r(\mathbb{F}_2)$ denote its reduced C-algebra. Kadison's conjecture. The algebra…

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 countable torsion-free amenable group, let $D$ be a Kirchberg algebra, and let $BG$ denote the classifying space of $G$. Consider outer actions of $G$ on $D\otimes K$,…

Let $A$ be a simple, unital, separable, nuclear, infinite-dimensional C$^*$-algebra. Winter–Tikuisis regularity conjecture. The following are equivalent: 1. $A$ has finite nuclear…

Let $A$ be a $C^*$-algebra, let $p\in A$ be a projection, and let $G_A$ denote the group of invertible elements of $A$. Write ${\cal E}_p$ for the projections equivalent to $p$, an…

Let … be a reduced free product of unital $C^*$-algebras with faithful states $\phi_1$ and $\phi_2$. Pure infiniteness conjecture. Whenever $A$ is simple and at least one of…

Let $A$ be a Banach algebra, and let $C(\mathbf{S}^2,A)$ denote the algebra of continuous $A$-valued functions on the $2$-sphere. Write $Bsr(A)$ for the Bass stable rank of $A$. St…

A limit algebra is an inductive-limit nonselfadjoint operator algebra of the type considered in the paper; for such an algebra ${\mathcal A}$, write $C^*({\mathcal A})$ for its…

Infinite-degree conjecture. There exists a $C^*$-algebra $A$ such that

Let $A$ be a simple, separable, nuclear approximately subhomogeneous $C^*$-algebra with strict slow dimension growth. The algebra $A$ is $\mathcal{Z}$-stable when…

Let $A$ and $B$ be separable nuclear $C^*$-algebras in the class ${\mathcal{N}}$; Proposition 3.1 and Theorem 3.2 of the cited work concern the associated extension and $K$-theoret…

Let $A$ be a unital and stably finite C$^*$-algebra with no nonzero finite-dimensional representations. Let $iota$ denote the natural map from the positive part of the Cuntz semigr…

Let $A$ be a simple, unital, finite, exact C$^*$-algebra that absorbs the JiangSu algebra $Z$ tensorially. Suppose its Cuntz semigroup admits the complete order embedding into an o…

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…

Jorgensen–Schmitt–Werner conjecture. If $\lVert T\rVert<1$, then the resulting $C^*$-algebra is isomorphic to the Cuntz–Toeplitz $C^*$-algebra $\mathcal{O}_d$. A sufficient conditi…

Jorgensen–Schmitt–Werner conjecture. For any $|q_{ij}|<1$, the algebra $\mathcal{O}_n^{q_{ij}}$ is the Cuntz–Toeplitz algebra. This is the limiting case obtained by setting the dia…

Let $d$ be a positive integer, and let $a_i,a_i^*$, $i=1,\dots,d$, generate a $C^*$-algebra subject to … where $q_i\in(-1,1)$ and $|q_{ij}|\leq 1$. Jorgensen–Schmitt–Werner conject…

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…