A new proof of the Tikuisis–White–Winter theorem

Abstract

Abstract A trace on a C * {\mathrm{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 * {\mathrm{C}^{*}} -algebra is hyperfinite, it is easy to see that traces on nuclear C * {\mathrm{C}^{*}} -algebras are amenable. A recent result of Tikuisis, White, and Winter shows that faithful traces on separable, nuclear C * {\mathrm{C}^{*}} -algebras in the UCT class are quasidiagonal. We give a new proof of this result using the extension theory of C * {\mathrm{C}^{*}} -algebras and, in particular, using a version of the Weyl–von Neumann Theorem due to Elliott and Kucerovsky.