Compact Operators and Uniform Structures in Hilbert $C^*$-Modules

Abstract

Quite recently a criterion for the $\mathcal{A}$-compactness of an ajointable operator $F\colon {\mathcal M} \to\mathcal{N}$ between Hilbert $C^*$-modules, where $\mathcal{N}$ is countably generated, was obtained. Namely, a uniform structure (a system of pseudometrics) in $\mathcal{N}$ was discovered such that $F$ is $\mathcal{A}$-compact if and only if $F(B)$ is totally bounded, where $B\subset {\mathcal M}$ is the unit ball.

We prove that (1) for a general $\mathcal{N}$, $\mathcal{A}$-compactness implies total boundedness, (2) for $\mathcal{N}$ with $\mathcal{N}\oplus K\cong L$, where $L$ is an uncountably generated $\ell_2$-type module, total boundedness implies compactness, and (3) for $\mathcal{N}$ close to be countably generated, it suffices to use only pseudometrics of “frame-like origin” to obtain a criterion for $\mathcal{A}$-compactness.