4 problems

Operator amenability characterization conjecture. The Fourier–Stieltjes algebra $\mathrm{B}(G)$ is operator amenable if and only if

Lau–Loy conjecture. To capture the amenability of $G$ through an amenability condition on $B(G)$ or $B_r(G)$, the condition must take into account both the operator-space structure…

Compactness conjecture. Either of these algebras is operator amenable if and only if $G$ is compact.

Let $\mathbb{G}$ be a locally compact quantum group and let $\hat{\mathbb{G}}$ be its dual. The associated convolution algebras are $L^1(\mathbb{G})$ and $L^1(\hat{\mathbb{G}})$. D…