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$C^{\ast }$-algebras Attaining their Norm in a Finite-dimensional Representation
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.
