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

TL;DR

This paper characterizes compact operators on Hilbert $C^*$-modules by a geometric property involving a newly introduced uniform structure, linking algebraic compactness to total boundedness.

Contribution

It introduces a uniform structure on Hilbert $C^*$-modules and characterizes compact operators via total boundedness in this structure.

Findings

Characterization of compact operators via total boundedness

Introduction of a new uniform structure on Hilbert $C^*$-modules

Equivalence between algebraic compactness and geometric total boundedness

Abstract

We introduce a uniform structure on any Hilbert $C^*$-module $\mathcal N$ and prove the following theorem: suppose, $F:{\mathcal M}\to {\mathcal N}$ is a bounded adjointable morphism of Hilbert $C^*$-modules over $\mathcal A$ and $\mathcal N$ is countably generated. Then $F$ belongs to the Banach space generated by operators $\theta_{x,y}$, $\theta_{x,y}(z):=x\langle y,z\rangle$, $x\in {\mathcal N}$, $y,z\in {\mathcal M}$ (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).