Locally unital $C^*$-algebras and limitations of seminorm criteria for compact operators

We study the problem of geometric characterization of Banach-compact and $\mathcal A$-compact operators on Hilbert $C^*$-modules in terms of uniform structures generated by seminorms. Previously, such a criterion for $\mathcal A$-compactness of adjointable operators was obtained by Troitsky for countably generated range modules with a suitable system of seminorms. Using locally unital $C^*$-algebras as modules over themselves, we establish fundamental limitations of such characterizations. Namely, we show that for nonunital locally unital $C^*$-algebras not only does Troitsky's criterion fail to extend, but, more generally, no system of seminorms can characterize $\mathcal A$-precompactness by total boundedness, nor Banach-compactness of bounded operators or $\mathcal A$-compactness of adjointable operators by total boundedness of the images of the unit ball. Moreover, the same statement holds for Hilbert $C^*$-modules $\mathcal N$ whenever the algebra $\mathbf K(\mathcal N)$ of $\mathcal A$-compact operators is locally unital and nonunital. We also study Troitsky-type systems of seminorms and their relation to $\mathcal A$-precompactness. We prove that every $\mathcal A$-precompact subset is totally bounded with respect to the strongest of such structures. Consequently, for Hilbert $C^*$-modules admitting a standard frame, all the corresponding notions of total boundedness coincide.