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

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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.