We characterize the class of RFD
$C^{\ast }$
-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
$C^{\ast }$
-algebra is finite-dimensional, which is equivalent to the
$C^{\ast }$
-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
$C^{\ast }$
-algebras whose norms in finite-dimensional representations fit certain prescribed properties.
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