A tracial state on a C*-algebra A is called amenable (resp. quasidiagonal) if there is a sequence of completely positive contractive maps from A into matrix algebras which approximately preserve the trace and approximately preserve the multiplication in the 2-norm (resp. operator norm). It is clear that every quasidiagoal trace is amenable, but the converse remains open. In nuclear setting, it is a classical result that all traces on nuclear C*-algebras are amenable, but even here, the quasidiagonality question is unsolved.

A major step towards this problem was taken recently by Tikuisis, White, and Winter where they showed every faithful trace on a nuclear C*-algebra in the UCT class is quasidiagonal. We will discuss this problem and a more direct proof of this result which avoids much of the regularity and classification theory used in the original proof. They key tool is a Hilbert module version of Voiculescu’s Weyl-von Neumann Theorem due to Elliott and Kucerovsky.