Geometric essence of "compact" operators on Hilbert $C^*$-modules

Abstract

We introduce a uniform structure on any Hilbert C * -module N and prove the following theorem: suppose, F : M → N is a bounded adjointable morphism of Hilbert C * -modules over A and N is countably generated. Then F belongs to the Banach space generated by operators θ x,y , θ x,y (z) := x y, z , x ∈ N , y, z ∈ M (i.e. F is A-compact, or "compact") if and only if F maps the unit ball of M to a totally bounded set with respect to this uniform structure (i.e. F is a compact operator).