A Hilbert C*-module admitting no frames

TL;DR

This paper demonstrates that certain infinite-dimensional commutative unital C*-algebras have Hilbert C*-modules without frames, highlighting limitations of Kasparov's stabilization theorem for general modules.

Contribution

It constructs examples of Hilbert C*-modules with no frames over infinite-dimensional commutative unital C*-algebras, showing the theorem's limitations.

Findings

Existence of Hilbert C*-modules without frames

Limitations of Kasparov's stabilization theorem

Counterexamples in infinite-dimensional settings

Abstract

We show that every infinite-dimensional commutative unital C*-algebra has a Hilbert C*-module admitting no frames. In particular, this shows that Kasparov's stabilization theorem for countably generated Hilbert C*-modules can not be extended to arbitrary Hilbert C*-modules.