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

We study the problem of geometric characterization of Banach-compact and A-compact operators on Hilbert C^*-modules in terms of uniform structures generated by seminorms. Previously, such a criterion for 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 A-precompactness by total boundedness, nor Banach-compactness of bounded operators or A-compactness of adjointable operators by total boundedness of the images of the unit ball. Moreover, the same statement holds for Hilbert C^*-modules N whenever the algebra K( N) of A-compact operators is locally unital and nonunital. We also study Troitsky-type systems of seminorms and their relation to A-precompactness. We prove that every 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.