C*-algebras and their finite-dimensional representations

We investigate C*-algebras whose structure can be recovered from their finite- dimensional representations, i.e. the class of so-called residually finite-dimensional (RFD) C*-algebras. We show that these are exactly the C*-algebras that contain a dense subset of elements that attain their norm under a finite-dimensional representation. Moreover, we prove 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 then use the residual finite-dimensionality of certain universal C*-algebras to formulate and sharpen a von Neumann-type inequality for noncommutative *-polynomials. Finally, we consider a noncommutative *-polynomial analogue to a two-variable von Neumann inequality, showing that it holds for all noncommutative *-polynomials in two variables if and only if Connes' Embedding Problem has a positive solution.