Locally unital $C^*$-algebras and Frame Theory

In non-unital algebras, one can consider the situation where for every element there exists another element that acts as a unit only for it. This property of local unitality turns out to be quite exotic for $C^*$-algebras. We will consider examples of such algebras, their properties, characterization in topological terms and discuss the result that in these algebras there are no frames in the sense of Hilbert $C^*$-modules.