12 problems

A separable nuclear $C^*$-algebra is a $C^*$-algebra whose underlying $C^*$-algebra is separable and nuclear. The bootstrap category is the class of $C^*$-algebras to which the Uni…

Let $G$ be a topological group with a universal proper space $\mathcal E G$. Let $A$ be a $C^*$-algebra and let $B$ be a $G$-algebra. Define the naive topological $KK$-groups by ……

Functoriality conjecture. The class $\Omega_!(g\circ f)$ should be identified, up to a canonical bordism, with

KK-rigidity conjecture. If

UCT conjecture. Every Kirchberg algebra is KK-equivalent to a commutative C-algebra; equivalently, every Kirchberg algebra satisfies the UCT.

A $C^*$-algebra is separable if it has a countable dense subset, and nuclear if its $C^*$-tensor product with any $C^*$-algebra is uniquely determined by the algebraic tensor produ…

Let $G$ be a simply-connected, connected, compact Lie group. Multiplication by each element $z\in Z(G)$ of the centre induces an invertible element $J_z\in{}^\kappa KK_G(G,G)$, whi…

Let $X$ be a second-countable compact Hausdorff space, and consider the flat-bundle index theorem proved in the paper for flat bundles of finitely generated projective modules over…

Let $G$ be a discrete, torsion-free, amenable group, and let $I(G)$ be the augmentation ideal, namely the kernel of the trivial representation $C^*(G)\to\mathbb{C}$. For a separabl…

Let $A$ be a $C^*$-algebra and let $E$ be a correspondence over $A$, with associated Cuntz–Pimsner algebra $O_E$ and mapping cone $M(A,O_E)$. Let $\hat{\mathcal{D}}$ denote the rel…

Let $A$ and $B$ be separable, unital $C^*$-algebras, and let $\alpha\colon \mathbb{T}\to\operatorname{Aut}(A)$ and $\beta\colon \mathbb{T}\to\operatorname{Aut}(B)$ be circle action…

Spectrum conjecture. For every finite group $G$, the natural map