Elements of $C^*$-algebras Attaining Their Norm in a Finite-Dimensional Representation

We characterize the class of RFD $C^*$-algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the $C^*$-algebra is finite-dimensional, which is equivalent to the $C^*$-algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of $C^*$-algebras whose norms in finite-dimensional representations fit certain prescribed properties.