Locally unital $C^*$-algebras do not admit frames

TL;DR

This paper proves that nonunital $C^*$-algebras with local units do not admit frames, extending previous results from commutative to noncommutative cases and exploring properties and examples of such algebras.

Contribution

It establishes that locally unital $C^*$-algebras cannot have frames and provides new insights into their properties and examples in noncommutative topology.

Findings

Nonunital $C^*$-algebras with local units lack frames

Necessary properties of frames in $C^*$-algebras are identified

Several new examples of $C^*$-algebras are analyzed

Abstract

We study nonunital $C^*$-algebras such that for any element there exists a local unit and prove that in such algebras there are no frames. This fact was previously known only for commutative algebras. Among other results, we establish some necessary properties of frames in $C^*$-algebras (which are of independent interest in the noncommutative topology), and consider several examples of $C^*$-algebras that are new in this context.