Geometric essence of "compact" operators on Hilbert $C^{\ast }$-modules

Abstract

We introduce a uniform structure on any Hilbert $C^{\ast }$-module $\mathcal{N}$ and prove the following theorem: suppose, $F:\mathcal{M}\to \mathcal{N}$ is a bounded adjointable morphism of Hilbert $C^{\ast }$-modules over $\mathcal{A}$ and $\mathcal{N}$ is countably generated. Then $F$ belongs to the Banach space generated by operators ${\theta }_{x,y}$, ${\theta }_{x,y}(z):=x⟨y,z⟩$, $x\in \mathcal{N}$, $y,z\in \mathcal{M}$ (i.e. $F$ is $\mathcal{A}$-compact, or "compact") if and only if $F$ maps the unit ball of $\mathcal{M}$ to a totally bounded set with respect to this uniform structure (i.e. $F$ is a compact operator).

15 pages