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

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.