4 problems

Let $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ be $C^{\ast}$-algebras, and let $T:\mathcal{A}_{1}\to\mathcal{A}_{2}$ be a biseparating map, meaning that both $T$ and $T^{-1}$ preserve…

Let $G$ be a Polish group. Say that $G$ has the countable index property when every subgroup of index at most countable is open, and say that $G$ has the automatic continuity prope…

Automatic continuity conjecture. Every epimorphism from a locally compact Hausdorff group $L$ to $G$ is continuous if and only if $G$ has no non-trivial finite normal subgroups.

A discrete group $G$ is $lcH$-slender if every group homomorphism from a locally compact Hausdorff group into $G$ is continuous. The lcH-slenderness conjecture. All Artin groups ar…