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

We study nonunital $C^*$-algebras such that for any element there exists a local unit and prove that in such algebras there is no frames -- a fact previously known only for commutative algebras. Among other, we establish some necessary properties of frames in $C^*$-algebras, and also consider several examples of $C^*$-algebras that are new in this context.