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.