Title: Elements of $C^*$-algebras Attaining Their Norm in a Finite-Dimensional Representation,
Abstract: 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.