Geometric conditions for compactness of operators between uncountably generated Hilbert $C^*$-modules

E. V. Troitsky proved the following criterion. Let $F\colon M\to N$ be a bounded adjointable morphism of Hilbert $C^*$-modules over a $C^*$-algebra $A$. Suppose that $N$ is countably generated. Then $F$ is $A$-compact (i.e., it is a norm limit of finite rank operators) iff the image of the unit ball of $M$ under $F$ is totally bounded with respect to a certain uniform structure on $N$. In this talk, we discuss possible generalizations of this criterion to uncountably generated modules. Both positive and negative results will be presented.