7 problems

Let $\mathcal{A}$ be a commutative Banach algebra, and let $D:\mathcal{A}\to\mathcal{A}$ be a derivation, meaning that $D(ab)=aD(b)+bD(a)$ for all $a,b\in\mathcal{A}$. Singer–Werme…

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.

Let $A$ and $B$ be $C^*$-algebras. A linear map $T:A\to B$ is biseparating when it is bijective and both $T$ and $T^{-1}:B\to A$ preserve zero products: $ab=0$ implies $T(a)T(b)=0$…

Power's algebraic-isometric isomorphism conjecture. Algebraic isomorphism may be equivalent to isometric isomorphism for subalgebras of AF $C^*$-algebras containing a regular canon…

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…

Let a group $H$ be cm-slender if the kernel of every abstract homomorphism from a completely metrizable topological group to $H$ is open. Conner's conjecture. A countable group is…

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…