1,045 problems

Baum–Connes conjecture for discrete groups. The analytic assembly map

Coarse Baum–Connes conjecture. The assembly map is an isomorphism for every $i$. This conjecture relates the large-scale geometry of $X$ to the $K$-theory of its Roe algebra. The s…

A separable operator system $S$ is a unital, $*$-closed subspace of a $C^*$-algebra, and a unital completely positive map has the unique extension property (u.e.p.) if every unital…

Connes' rigidity conjecture. If

Let $G$) be an exponential solvable Lie group, and let ${\mathcal J}_1\subseteq{\mathcal J}_2$ be closed two-sided ideals of $C^*(G)$ such that ${\mathcal J}_2/{\mathcal J}_1$ is a…

Let $\Gamma$ be a discrete group and consider the assembly map from the $K$-homology of an appropriate classifying space of $\Gamma$ to the $K$-theory of the reduced $C^*$-algebra…

HK conjecture. For $* = 0,1$, there is an isomorphism

Kadison–Kastler conjecture. For a $\mathrm{C}^*$-algebra $A$, there exists $\varepsilon>0$ such that whenever $d_{KK}(A,B)<\varepsilon$, the algebras $A$ and $B$ are isomorphic. A…

Let $\mathbb{F}$ be the free group on two generators, and let $C^*(\mathbb{F})\otimes_{\mathrm{max}}C^*(\mathbb{F})$ and $C^*(\mathbb{F})\otimes_{\mathrm{min}}C^*(\mathbb{F})$ deno…

Let a $C^*$-algebra have the weak expectation property (WEP) when it satisfies Kirchberg's WEP condition, and call a $C^*$-algebra QWEP if it is a quotient of a $C^*$-algebra with…

Let $\mathbb{F}_\infty$ be the free group with countably infinitely many generators, and let $C^*(\mathbb{F}_\infty)$ be its full group $C^*$-algebra. A $C^*$-algebra has the weak…

Let $\Gamma$ be a torsion-free discrete group, and let $C^*_r(\Gamma)$ denote its reduced group $C^*$-algebra. An idempotent is nontrivial if it is neither the zero element nor the…

Let $M$ be a type ${\rm III}_1$ factor and let $\varphi\in M_*$ be a faithful normal state. The bicentralizer conjecture. The bicentralizer satisfies … This is Connes' bicentralize…

Let $X$ be a locally compact space and let $Gamma$ act on $X$, with $C_0(X)\rtimes\Gamma$ the associated crossed-product $C^*$-algebra. Assume the action is amenable and second cou…

Let $\Gamma$ be a countable discrete group and let $f\in\mathbf{Z}[\Gamma]$. Let $\boldsymbol{X}_f$ be the algebraic action obtained from the Pontryagin dual of…

Coarse Novikov conjecture. The induced map

Real-rank-zero conjecture. Then $G$ is locally finite.

Let $\mathcal{A}$ be an exact C$^\ast$-algebra, and let $I(\mathcal{A})$ denote its injective envelope. Ozawa's conjecture. There is a nuclear C$^\ast$-algebra $N(\mathcal{A})$ suc…

Let $(M,\tau)$ be a finite von Neumann algebra with separable predual, and let $R$ be the hyperfinite $\mathrm{II}_1$-factor over ${\mathbbm k}\in\{{\mathbb C},{\mathbb R}\}$. Writ…

Let $E$ and $F$ be graphs and let $\mathsf{k}$ be a field. Consider the Leavitt path algebras $L_\mathsf{k}(E)$ and $L_\mathsf{k}(F)$, and the graph $C^*$-algebras $C^*(E)$ and…

For $r\in(1,\infty]$, let $L(\mathbb F_r)$ be the interpolated free group factor, and let the first-order fundamental group of $L(\mathbb F_r)$ be the group of positive scalars $t$…

For a set $S\subseteq\mathbb{N}$, let $P_S$ be the projection onto the coordinates indexed by $S$, and let $E(T)$ denote the diagonal operator of an operator…

Let $\mathcal{A}$ be a separable, nonunital C-algebra, and let its corona be the quotient $\mathcal{Q}(\mathcal{A})=\mathcal{M}(\mathcal{A})/\mathcal{A}$ of its multiplier algebra…

Let $A$ be a UHF algebra and let $(\sigma_t)_{t\in\mathbb R}$ be a flow on $A$, that is, a point-norm continuous one-parameter group of automorphisms of $A$. Powers–Sakai conjectur…

Let $H$ be a Hilbert space, let $B(H)$ denote the algebra of bounded operators on $H$, and let $M$ be an atomic masa in $B(H)$ with conditional expectation $E_M:B(H)\to M$. A pure…