# A new proof of the Tikuisis-White-Winter Theorem

## TL;DR

This paper provides a new proof that faithful traces on separable, nuclear C*-algebras in the UCT class are quasidiagonal, utilizing extension theory and a version of the Weyl-von Neumann Theorem.

## Contribution

It introduces an alternative proof of the Tikuisis-White-Winter theorem using extension theory and the Elliott-Kucerovsky Weyl-von Neumann Theorem.

## Key findings

- Traces on nuclear C*-algebras are amenable.
- Faithful traces on certain C*-algebras are quasidiagonal.
- New proof simplifies understanding of trace quasidiagonality.

## Abstract

A trace on a C*-algebra is amenable (resp. quasidiagonal) if it admits a net of completely positive, contractive maps into matrix algebras which approximately preserve the trace and are approximately multiplicative in the 2-norm (resp. operator norm). Using that the double commutant of a nuclear C*-algebras is hyperfinite, it is easy to see that traces on nuclear C*-algebras are amenable. A recent result of Tikuisis, White, and Winter shows that faithful traces on separable, nuclear C*-algebras in the UCT class are quasidiagonal. We give a new proof of this result using the extension theory of C*-algebras and, in particular, using a version of the Weyl-von Neumann Theorem due to Elliott and Kucerovsky.