A Hilbert C*-module with extremal properties

TL;DR

This paper constructs a specific Hilbert C*-module that demonstrates limitations of existing theorems on A-compact operators and shows that such modules can lack frames, highlighting extremal properties.

Contribution

It provides a counterexample of a non-countably generated Hilbert C*-module with extremal properties, challenging previous generalizations of Troitsky's theorem.

Findings

Counterexample of a non-countably generated Hilbert C*-module

Shows the module admits no frames

Demonstrates limitations of Troitsky's theorem

Abstract

We construct an example of a Hilbert C*-module which shows that Troitsky's theorem on the geometrical essence of A-compact operators between Hilbert C*-modules is not extendable to a not countably generated module case (even in the case of a stronger uniform structure, which is also introduced). In addition, the constructed module admits no frames.